Higher lattices, discrete two-dimensional holonomy and topological phases in (3+1) D with higher gauge symmetry
Alex Bullivant, Marcos Calcada, Zolt\'an K\'ad\'ar, Jo\~ao Faria, Martins, Paul Martin
TL;DR
This paper develops a Hamiltonian model for higher lattice gauge theory in 3+1 dimensions, extending Kitaev's model with 2-group connections, and proves topological invariance of ground-state degeneracy.
Contribution
It introduces a well-defined higher lattice gauge theory Hamiltonian using 2-group connections on CW-decompositions, generalizing Kitaev's model to higher dimensions.
Findings
Ground-state degeneracy is a topological invariant.
The model is well-defined on arbitrary CW-decompositions.
Discrete 2-dimensional holonomy is rigorously constructed.
Abstract
Higher gauge theory is a higher order version of gauge theory that makes possible the definition of 2-dimensional holonomy along surfaces embedded in a manifold where a gauge 2-connection is present. In this paper, we will continue the study of Hamiltonian models for discrete higher gauge theory on a lattice decomposition of a manifold. In particular, we show that a previously proposed construction for higher lattice gauge theory is well-defined, including in particular a Hamiltonian for topological phases of matter in 3+1 dimensions. Our construction builds upon the Kitaev quantum double model, replacing the finite gauge connection with a finite gauge 2-group 2-connection. Our Hamiltonian higher lattice gauge theory model is defined on spatial manifolds of arbitrary dimension presented by slightly {\it combinatorialised} CW-decompositions (2-lattice decompositions), whose 1-cells and…
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Higher lattices, discrete two-dimensional holonomy and topological phases in (3+1)D with higher gauge symmetry
Alex Bullivant111E-mail address: [email protected]
School of Physics and Astronomy, University of Leeds, Leeds, LS2 9JT, United Kingdom
Marcos Calçada222E-mail address: [email protected]
Zoltán Kádár333E-mail address: [email protected]
João Faria Martins444E-mail address: [email protected]
Paul Martin555E-mail address: [email protected]
Abstract
Higher gauge theory is a higher order version of gauge theory that makes possible the definition of 2-dimensional holonomy along surfaces embedded in a manifold where a gauge 2-connection is present. In this paper, we will continue the study of Hamiltonian models for discrete higher gauge theory on a lattice decomposition of a manifold. In particular, we show that a previously proposed construction for higher lattice gauge theory is well-defined, including in particular a Hamiltonian for topological phases of matter in 3+1 dimensions. Our construction builds upon the Kitaev quantum double model, replacing the finite gauge connection with a finite gauge 2-group 2-connection. Our Hamiltonian higher lattice gauge theory model is defined on spatial manifolds of arbitrary dimension presented by slightly combinatorialised CW-decompositions (2-lattice decompositions), whose 1-cells and 2-cells carry discrete 1-dimensional and 2-dimensional holonomy data. We prove that the ground-state degeneracy of Hamiltonian higher lattice gauge theory is a topological invariant of manifolds, coinciding with the number of homotopy classes of maps from the manifold to the classifying space of the underlying gauge 2-group.
The operators of our Hamiltonian model are closely related to discrete 2-dimensional holonomy operators for discretised 2-connections on manifolds with a 2-lattice decomposition. We therefore address the definition of discrete 2-dimensional holonomy for surfaces embedded in 2-lattices. Several results concerning the well-definedness of discrete 2-dimensional holonomy, and its construction in a combinatorial and algebraic topological setting are presented.
Keywords: Kitaev Model; topological phases in 3+1D; topological quantum computing; topological quantum field theory; higher gauge theory; surface holonomy; crossed module; lattice gauge theory.
1 Introduction
In the absence of external symmetries, a topological phase of matter is characterised by a local, gapped, quantum many-body Hamiltonian whose effective (infra-red) field theory is described by a topological quantum field theory (TQFT) [34, 54, 71, 68]. A topological phase is therefore diffeomorphism invariant, and thus insensitive to local perturbations in the sense that the amplitudes of physical processes are global topological invariants. It is this latter property which makes topological phases of matter candidates for the implementation of fault tolerant quantum computing [68, 56, 45, 54].
Due to a lack of local observables, experimentally distinguishing different topological phases can be a difficult task from a microscopic point of view. Instead the characterising properties of topological phases are most efficiently described by their emergent behaviours. Signatures for the presence of topological order include ground state degeneracies which depend on the spatial topology of the material in question [69, 43], universal negative corrections to the entanglement entropy [44, 37, 48] and the presence of stable topological excitations which provide non-trivial representations of their respective motion groups [47, 41]. This means the braid group for point particles (anyons) in 2+1D [54] and the loop braid group for loop excitations in 3+1D [66].
In 2+1D there exist several constructions for TQFTs (see for instance [64]). Path-integral models arise from Chern-Simons-Witten theory [73] and from BF theory [1], while the discrete realisation of BF-theory coincides with the Turaev-Viro [64]/Barrett-Westbury [6] state-sum (see [2]). In contrast, in 3+1D a framework general enough to capture all features of 4D topology is still lacking. Nevertheless, we have the Crane-Yetter TQFT [23, 64] and its generalisations [71, 25]; and the Yetter homotopy 2-type TQFT [74, 59, 33], derived from a strict finite 2-group [4]. All of these 3+1D TQFTs give rise to topological invariants which at most depend on the homotopy 2-type, signature and spin-structure of space-time [64, 33], or are conjectured to do so.
One successful approach to understanding candidate models for D topological phases has been to define Hamiltonian realisations of D TQFTs [60, 71]. This means that a finite dimensional Hilbert space , and an exactly solvable (the sum of mutually commuting projectors) Hamiltonian is assigned to each -manifold , with a given lattice decomposition (e.g. can be a triangulation or a CW-decomposition of ). The constructions of both and should be local on . To say that such a Hamitonian schema [60] is a realisation of the TQFT roughly means that given a -manifold each ground state vector space of is canonically isomorphic to . (In particular this implies that the ground state degeneracy does not depend on and it is a topological invariant of .) In 2+1D, this Hamiltonian realisation approach has been successfully achieved in the case of Dijkgraaf-Witten topological gauge theories [26, 45, 40, 67] and the Turaev-Viro TQFT, giving rise to the so-called string-net models [49]. The Kitaev quantum-double model [45] can be seen as a Hamiltonian realisation for the Dijkgraaf-Witten TQFT [26] with trivial cocycle and thus also for finite-group BF-theory. Similar ideas were applied to the 3+1D Crane-Yetter TQFT [66, 65], giving rise to the Walker-Wang model.
A Hamiltonian realisation of Yetter’s homotopy 2-type TQFT was constructed in [71, 21]. This is a higher gauge theory version of Kitaev quantum-double model [45]. It is this that we continue to develop in this paper. We note that topological phases protected by higher gauge symmetry are also proposed in [42].
Higher gauge theory [3, 5] is a generalisation of ordinary gauge theory with further levels of structure and symmetry. A key feature of higher gauge theory is parallel transport along surfaces embedded in a manifold where a gauge 2-connection is present [3, 5, 32, 63]. In higher gauge theory, instead of local gauge symmetry groups we have local gauge symmetry 2-groups. These, recall, are equivalent to crossed modules of groups [18, 7, 4]. (Recall is a map of groups and is a left action of on by automorphisms, satisfying some compatibility relations: the 1st and 2nd Peiffer relations.).
In this paper, completing the programme initiated in [21], we define an exactly solvable Hamiltonian model for higher lattice gauge theory on manifolds of any dimension, here called the “higher Kitaev model”. This lifts Kitaev’s quantum-double model [45] for topological phases from finite group topological gauge theory to finite 2-group topological higher gauge theory [36, 31]. Typically will be a 3-dimensional manifold, and the higher Kitaev model is proposed to be a model for (3+1)-dimensional topological phases [66, 71, 67, 21, 65, 47, 22, 42, 50]. We prove that the ground state degeneracy of the higher Kitaev model is a topological (in fact homotopy) invariant of manifolds. Specifically, we show that the ground state degeneracy is given by the number of homotopy classes of maps from to the classifying space of the underlying gauge symmetry 2-group [18, 33]; hence the ground state degeneracy is closely related [33] to Yetter homotopy 2-type TQFT. (The precise relation appears in [21], where a proof that the higher Kitaev model is a Hamiltonian realisation of Yetter homotopy 2-type TQFT is given.)
1.1 Higher lattices and higher lattice gauge theory
Our model utilises ideas from higher lattice gauge theory [57, 42]. Similarly to [45], we take lattice gauge theory as the starting point (with its good connection to physical observation [72, 46]) and lift the structure through the process of categorification. We thereby replace the gauge group with a gauge 2-group and a gauge connection discretised on a lattice with a discretised higher gauge connection, here called a fake-flat 2-gauge configuration. Therefore we enrich the local variables of lattice gauge theory (holonomies along edges) to include non-abelian 2-dimensional holonomies along the faces of the lattice; recall again that 2-dimensional holonomies feature prominently in higher gauge theory; see [5, 32, 3, 63].
The model constructed in this paper extends and formalises the proposal of [21] for a Hamiltonian model for the Yetter homotopy 2-type TQFT [74], from triangulated manifolds to manifolds with a slightly combinatorialized version of CW-decompositions (here called 2-lattice decompositions — see Definition 21). Hence a 2-lattice decomposition represents a manifold as a disjoint union of -cells, here is an arbitrary non-negative integer, where each -cell homeomorphic to the interior of the -disk . As costumary, [math]-cells are called vertices, 1-cells are called edges, 2-cells are called faces (or plaquettes), and 3-cells are called blobs. These 2-lattice decomposition are considerably less rigid than triangulations. Therefore using 2-lattice decompositions of manifolds, as opposed to triangulations, to decompose a manifold into smaller pieces has the advantage that fewer cells are needed to decompose a manifold, leading to microscopic Hilbert spaces of much smaller rank. We illustrate this fact by describing two small models for discrete higher gauge theory in the 3-sphere.
Many constructions in this paper would still work if we use CW-complex decompositions of manifolds rather than 2-lattice decomposition; however a lot of the combinatorial flavour presented in the final construction of the Hamiltonian model would be lost. By using 2-lattice decompositions instead of triangulations some combinatorics is taken away; therefore, despite the fact that our model is fully combinatorial, some algebraic topology will be required in proving that it is well-defined.
By passing from triangulations to 2-lattices, we hence demonstrate the internal consistency of the model in [21], which tacitly assumed that discrete 2-dimensional holonomy of a discrete higher gauge field is well-defined, for instance when proving in loc cit that the ground state degeneracy is a topological invariant derived from Yetter TQFT, and as such that our model is a Hamiltonian realisation of Yetter TQFT.
Prominent in this paper is the concept of a fake-flat 2-gauge configuration in a 2-lattice, to be a discretised model of a higher gauge field; as well as the construction of discrete 2-dimensional holonomy operators for surfaces cellularly embedded in a 2-lattice. (Fake-flat 2-gauge configurations are in line with the framework for higher lattice gauge theory of [57, 33] and also appear in formal homotopy quantum field theory constructions; see [58]). We carefully construct these discrete 2-dimensional holonomy operators, in an algebraic topological (§3.4) and in a combinatorial manner (§3.5), and, using algebraic topology, prove that this discrete 2-dimensional holonomy is gauge invariant and independent of the way we combine the faces of a particular CW-decomposition of the 2-sphere and of the 2-disk. These are results, of intrinsic interest. They provide a combinatorial construction of the 2-dimensional holonomy of a higher order bundle, completing its differential geometrical construction discussed for example in [5, 61, 32, 3].
In sections 2, 3 and 4 we lift the construction of ordinary lattice gauge theory to a higher setting, as outlined in [21, 57]. Let us summarise the general procedure.
A gauge configuration of ordinary lattice gauge theory with gauge group is given by a map from the set of (by definition oriented) edges of the lattice into . The well-definedness of lattice gauge theory can be expressed by saying that there is a lattice groupoid supporting well-defined groupoid maps (here called discrete parallel transport functors §3.1) to a gauge group anytime a gauge configuration is given. The ‘lattice groupoid’ is a groupoid version of the free category over a graph (see for example [52, 39]) for a suitable graph derived from the lattice. It is the freeness that makes discrete parallel transport functors well defined. A ‘suitable graph’ is (it is claimed) the 1-skeleton of a suitable CW-complex decomposition of physical space. If we aim for topological field theory then in principle any sufficiently regular CW-complex will do. Normally there is a notion of local structure — chunks of space that are independent of each other, which collectively encode extended structure. In this sense, the ‘big story’ of lattice gauge theory is that the free groupoid over a suitable lattice is an adequate model of physical space.
Our first task here is to construct a well-defined lift of these notions to the higher setting. The main tool is the concept of a lattice 2-groupoid (to be a model of space in lattice higher gauge theory), which in this paper is constructed in an algebraic topological language as the fundamental crossed module (see [11] and [18, Chapter 6]) of a certain filtered space associated to a 2-lattice decomposition of the manifold ; see §3.3. We will make a very strong use of a freeness result for the lattice 2-groupoid, which essentially is a classical freeness theorem of Whitehead [70, 10], transported to the groupoid setting by Brown and Higgins; see [18, 6.8] and [13, 14, 15]. Whitehead’s theorem provides also an equivalent combinatorial definition of the lattice 2-groupoid of , i.e. of a pair consisting of a manifold with a 2-lattice decomposition.
Let be a manifold with a 2-lattice decomposition . Given a crossed module , representing the underlying gauge symmetry 2-group, a 2-gauge configuration is defined as a map assigning an element of the group to each (pointed and oriented) face of and an element of to each (oriented) edge of . Physically relevant configurations furthermore satisfy a certain compatibility condition — called fake-flatness. This is a discretised version of the well-established fake-flatness condition for differential geometrical 2-connections; see [3, 5, 61, 32]. The term fake-flatness was first used in [9].
In analogy to lattice gauge theory, we prove that any fake-flat 2-gauge configuration extends uniquely to a crossed module map (called a discrete parallel transport 2-functor) from the lattice 2-groupoid of into the underlying gauge 2-group ; see §3.3. These discrete parallel transport 2-functors are a discrete version of the differential-geometrical parallel transport 2-functors of [61, 32]. Given an oriented 2-disk or 2-sphere embedded in , as a subcomplex, and a vertex of , we can then combine the 1-dimensional and 2-dimensional holonomies of the constituting pieces of , and obtain an -valued 2-dimensional holonomy of the fake-flat 2-gauge configuration along . These are the 2-dimensional holonomy operators previously referred to. By using some basic algebraic topology, and the fact that the oriented mapping class groups of the 2-sphere and of the 2-disk both are trivial, we can then provide algebraic-topological and combinatorial descriptions of , and also show that the discrete 2-dimensional holonomy of along depends only on the base-point (in a way controlled by the action of on ) and on the surface orientation, and not on any other data such as the order of multiplication of constituent 2-cells. This latter result does not apply (in this form) to other surfaces since the mapping class group is then more complicated: in general an isotopy class of embeddings is needed to define the 2-dimensional holonomy of a 2-gauge connection along an embedded surface. For discussion see [32, 63].
Playing a prominent role in the construction of our model, we introduce gauge transformations between fake-flat 2-gauge configurations. Gauge transformations initially come in two different types: vertex and edge types. These correspond to the thin and fat gauge transformation of [31]. Vertex and edge gauge transformations obey a semi-direct product structure, and can be assembled into a group of gauge operators, which acts on the set of fake-flat 2-gauge configurations. This action is explicitly constructed using a double category derived from the crossed module [15, 32], and ultimately originates from a groupoid of fake-flat 2-gauge configurations and ‘full gauge transformations between them’, which we will carefully construct in §4.3. We prove in §4.3.2 that gauge transformations preserve the 2-dimensional holonomy of fake-flat 2-gauge configurations along cellularly embedded 2-spheres in .
As mentioned in the previous paragraph, a major underpinning construction is that of a groupoid of fake-flat 2-gauge configurations and full gauge transformation between them §4.3. The latter groupoid can be seen as a combinatorial description of a certain groupoid of crossed complex (a generalisation of crossed modules) maps and their homotopies, which appeared in the work of Brown and Higgins on tensor products and homotopies of crossed complexes; see [16, 11]. This point of view will be essential when we discuss the ground state degeneracy of the higher Kitaev model in §5.2.
Let be a crossed module, representing the underlying gauge symmetry 2-group. Let be a 2-lattice decomposition of . A fake-flat 2-gauge configuration in is said to be 2-flat along a cellularly embedded 2-sphere if the 2-dimensional holonomy of along is the identity element of . This 2-flatness of along a 2-sphere is preserved by gauge transformations. A fake-flat 2-gauge configuration in is said to be 2-flat if it is 2-flat along the boundaries of all 3-cells of .
A crucial fact that we will use in this paper is the following one, a consequence of the work of Brown and Higgins [14, 15, 16, 17]; a more modern reference is [18]. A 2-flat 2-gauge configuration naturally yields a map , defined up to homotopy, from into the classifying space of the crossed module ; classifying spaces of crossed modules are defined in [18, 17, 11] and also [33, 28]. Moreover, by [17, THEOREM A] and [18, §11.4]), it follows that given two 2-flat 2-gauge configurations and , then are homotopic if, and only if, the 2-flat 2-gauge configurations and are connected by a full gauge transformation. These facts will play a primary role in the proof that the ground state degeneracy of our model is a topological invariant of manifolds , counting the number of homotopy classes of maps from into ; see §5.2.
Overview of the paper
In Section 2, we recap and fix conventions for: crossed modules, fundamental crossed modules, CW-complexes and 2-lattices, defined in §2.4. In section 3, we firstly define and discuss fake-flat 2-gauge configurations (called “cellular formal -maps” in [58]); see §3.2. In §3.3, we define the lattice 2-groupoid for a pair , consisting of a manifold with a 2-lattice decomposition , and show how fake-flat 2-gauge configurations give rise to 2-dimensional parallel transport 2-functors, from the lattice 2-groupoid of into the gauge crossed module . In §3.4 we give an algebraic topological definition of the 2-dimensional holonomy of a fake-flat 2-gauge configuration along a 2-sphere and along a 2-disk. In §3.5, we give a combinatorial definition of 2-dimensional holonomy along 2-disks and 2-spheres, and prove that the two definitions of 2-dimensional holonomy coincide.
In Section 4 we discuss gauge transformations between fake-flat 2-gauge configurations defined on a 2-lattice. In particular we define a group of gauge operators and prove that it acts on the set of fake-flat 2-gauge configurations in a way such that the 2-dimensional holonomy along cellularly embedded 2-spheres is preserved. The underpinning groupoid of fake-flat 2-gauge configurations and full gauge transformations between them is constructed in 4.3.
In Section 5, we address a Hamiltonian model for higher lattice gauge theory on a pair , consisting of a manifold with 2-lattice decomposition . This will be our proposal for a higher gauge theory version of Kitaev quantum-double model for topological phases: the higher Kitaev model. The underlying Hilbert space of our model is the free vector space on the set of all fake-flat 2-gauge configurations, and hence coincides with the Hilbert space in [21] for triangulated manifolds. In §5.1 we explicitly construct the higher Kitaev model, and give detailed description of all operators involved. In 5.1.3 we define the local operator algebra. The Hamiltonian 5.1.4 for the higher Kitaev model is a sum of three mutually commuting terms. We have two sums over 1-cells and 2-cells, respectively, constructed by using the action of the group of gauge operators, which impose higher gauge invariance along gauge transformations of vertex and edge types; and one sum over 3-cells, imposing 2-flatness along their boundary 2-sphere. A comparison with the Kitaev model is done in §5.1.6.
In §5.2 we show that the dimension of the ground state of the higher Kitaev model is given by the number of homotopy classes of maps from the space manifold to the classifying space of the gauge crossed module and therefore ground state degeneracy is a topological (in fact homotopy) invariant of . (At this point we needed again to appeal to some basic algebraic topology for crossed modules and crossed complexes as given in [18, 28, 33].) This ground state degeneracy can be proven to coincide with Yetter’s invariant on (the level invariant of the TQFT); see [21].
Contents
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2 Preliminaries on crossed modules, CW-complexes and 2-lattices
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3 Higher order gauge configurations and discrete 2D holonomy for surfaces embedded in 2-lattices
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3.1 Gauge configurations, discrete 1D parallel transport and holonomy along circles
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3.3 On Whitehead theorem, 2-gauge configurations and the lattice 2-groupoid
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3.3.1 The discrete 2-dimensional (2D) parallel transport of a fake-flat 2-gauge configuration
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3.4 Algebraic topological definition of 2D holonomy along 2-disks and 2-spheres
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3.5 Combinatorial definition of 2D holonomy along 2-disks and 2-spheres
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3.5.2 A combinatorial description of the 2D holonomy along embebbed 2-disks
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3.5.3 Algebraic topology preliminaries for the 2-sphere case
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3.5.4 A combinatorial description of the 2D holonomy along embedded 2-spheres
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4.3 Full gauge transformations between fake-flat 2-gauge configurations
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4.3.1 Groupoid of fake-flat 2-gauge configurations and full gauge transformations
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4.3.2 Full gauge transformations preserve 2D holonomy along embedded 2-spheres
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4.3.3 Groupoid of 2-flat 2-gauge configurations and full gauge transformations
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5.1.3 The local operator algebra of higher lattice gauge theory
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5.1.4 The higher Kitaev model for (3+1)-dimensional topological phases
2 Preliminaries on crossed modules, CW-complexes and 2-lattices
In §3.2 we give the definition of a fake-flat 2-gauge configuration on a 2-lattice decomposition of a manifold. This makes extensive use of crossed modules; we assemble the key definitions in §2.1. (Crossed modules of groups are well-known to be equivalent to strict 2-groups [18, §2.5] and [4], thus from now on only crossed modules will be mentioned.) Then in §2.3 we recall some facts about CW-complexes which we will need in §2.4 for defining 2-lattice decompositions of manifolds.
Remark 1**.**
In this paper we will use to denote the boundary of a manifold . (We avoid the common notation , in order not to overuse the symbol , which appears in several other contexts.)
2.1 Crossed modules (of groups and of groupoids)
Crossed modules of groups are discussed in [4, 11, 28]. Crossed modules of groupoids, discussed extensively in this paper, appear in [18, §6.2] and [11, 33, 19]. Crossed modules of groups and groupoids can be used for formalising 2-dimensional (2D) notions of holonomy (surface holonomy), in the same way that groups appear in the formulation of the holonomy of a connection in a principal bundle.
Definition 2** (Crossed modules of groups; Peiffer relations).**
Let and be groups. A crossed module of groups is given by a group map , together with a left action of on by automorphisms, such that the relations below, called Peiffer relations, hold for each and :
[TABLE]
Example 3**.**
The crossed module , where is the additive group of integers modulo and acts on as . The boundary map sends everything to .
Example 4** (From groups to crossed modules I).**
Given a group , let be the group of automorphisms of . Clearly acts in by automorphisms as , for each and each . Let be the morphism that sends to the inner automorphism , obtained by conjugating by . Then is a crossed module.
Example 5** (From groups to crossed modules II).**
If is a group, then and , where is the adjoint action, are crossed modules. If is abelian then is also a crossed module.
Let us now discuss crossed modules of groupoids. Let denote a groupoid [39, 11] with set of objects ; set of morphisms ; and source and target maps . We represent the morphisms as . Thus and . Given , the set of morphisms is . The composition map in yields for each triple of objects a map , which we represent as (notice composition order):
[TABLE]
A totally intransitive groupoid is a groupoid of the form . (Thus source and target maps coincide.) Given , we let , which is a group. And then is isomorphic to the totally intransitive groupoid given by , with the obvious composition and map . Hence a totally intransitive groupoid can been seen as being given by a disjoint union of groups.
A left groupoid action [11, 19], by automorphisms, of the groupoid on , a totally intransitive groupoid with the same set of objects as , is given by a set map:
[TABLE]
such that whenever compositions and actions are well-defined:
[TABLE]
Definition 6** (Crossed module of groupoids).**
Let and be groupoids with the same object set, with totally intransitive. A crossed module of groupoids is given by a groupoid map , which is the identity on objects, together with a left action of on , by automorphisms, such that the Peiffer relations (1,2) are satisfied, whenever actions and compositions make sense. (Full equations are in [19].)
Given , we call the top groupoid of and the underlying groupoid of .
Definition 7** (Crossed module map).**
A map of crossed modules of groupoids is given by two groupoid maps and , which are compatible with actions and boundary maps in the obvious way. (Full equations are in [33, §1.1.1].)
Since groups can be considered to be groupoids with a single object, we will see group crossed modules as particular cases of crossed modules of groupoids.
2.2 Example: the fundamental crossed module
The main example of a crossed module of groupoids is a topological one crucial to our construction. Our main references are [18, §2.1, §2.2 and §6] and [11]. We will need to review some algebraic topology definitions.
Definition 8**.**
(See e.g. [24, p.17] and [18, 11].) Let be a locally path-connected space, and any subset (in this paper will always be finite). The fundamental groupoid of , with object set , denoted , is as follows. The set of objects of is . Given , the set of morphism is the set of equivalence classes of paths , such that and , where two paths are equivalent if they are homotopic in , relative to the end-points (i.e. end-points remain stable during the homotopy). The composition in is given by concatenation (and rescaling) of representative paths.
If is a path in , the equivalence class to which it belongs in is denoted by . A morphism in from to is denoted as or simply by if no ambiguity arises.
Remark 9**.**
Let . The group of morphisms in the groupoid is exactly the fundamental group . Let , with a base point at ; recall Rem. 1. Morphisms hence can equivalently be seen as pointed homotopy classes of maps .
Relative homotopy groups, including , of pointed pairs of spaces ( being the base-point) are classical in homotopy theory and are defined e.g. in [38, p.343]. In this paper, we will use relative homotopy groupoids , with a set of base-points; see [18, §1.6, §6.2 and §6.3]. These are totally intransitive groupoids built as . Let us give a quick review.
Definition 10** (The totally intransitive groupoid ).**
Let be a locally path-connected space. Let be a locally path-connected subspace of . Choose a subset of . In this paper, will always intersect non-trivially each path-component of and of . For each , consider the relative homotopy group . This group is made out of homotopy classes of maps such that:
\Gamma\big{(}([0,1]\times\{0\})\cup(\{0,1\}\times[0,1])\big{)}={\{c\}}, 2. 2.
.
Specifically, two such maps are said to be homotopic if there exists a homotopy , connecting and , such that for all the slice of at , namely , satisfies the properties 1 and 2. The multiplication in is through horizontal juxtaposition of maps , followed by rescaling in the horizontal direction.
We can thus define a totally intransitive groupoid with set of objects .
Let be as in Def. 10. The elements , or simply , if no confusion arises, are visualised as:
[TABLE]
Let . As indicated by the diagram above, if we restrict a to the top of the square , this gives rise to an element . This yields a group map Putting all of these group maps together, yields a groupoid map , which is the identity on objects. We also have an action of the groupoid on the totally intransitive groupoid , as indicated in figure 1. Details are in [18, §2.2 and §6.1] and (in the pointed case) [38, pp 355].
Theorem 11**.**
(JHC Whitehead, [18, §2.3, §6] and [11, 13]) Let be a triple of spaces, as in Def. 10. Considering the natural action of the groupoid on the totally intransitive groupoid , and the boundary map , we have a crossed module of groupoids, called the fundamental crossed module of . The fundamental crossed module of is denoted as:
[TABLE]
Remark 12**.**
Let be a pair of spaces and . Recall that the underlying set of the group can also be defined as the set of all maps , such that , where , and , up to a homotopy , such that, for each , and . The boundary map is obtained by restricting to ; c.f. Rem. 9.
Analogously [38, Chapter IV], the underlying set of the relative homotopy group can be defined as the set of all maps such that , where , and , up to a homotopy such that, for each , and . We also have a boundary map obtained by restricting to .
Example 13** (The fundamental crossed module of the disk).**
Let and . Let be any point. Then , where , for each . To see this, look at the end of the homotopy long exact sequence of ; see e.g. [38, Chapter IV]. This yields Details are in e.g. [28].
2.3 CW-complexes
Let denote the closed -disk in the form . The open -disk is . Also put:
[TABLE]
— the boundary of the -disk. Let .
Let us briefly review the definition of CW-complexes [38, Appendix], [35] and [51]. We will use the definition given in [38, Prop A2] and [51, Chapter II].
Definition 14** (CW-complex).**
A CW-complex is a Hausdorff topological space , a collection of sets , , , …, and, for each , a family of continuous maps (the ‘characteristic maps of the closed -cells’) satisfying conditions 1,2,3 and 4, below.
Let the set . It is called an open cell of dimension , and is given the induced topology. Put . It is called a closed cell of dimension , and is given the induced topology. Put . It is called the boundary of . (Note that need not be a -manifold, hence might not be a manifold boundary, though this will be imposed when we define 2-lattices.) Then:
Each characteristic map restricts to a homeomorphism . 2. 2.
The open cells where and , form a partition of . (I.e. they are pairwise disjoint and their union is .) 3. 3.
Each is contained in the union of a finite number of open cells of dimension . 4. 4.
A set is closed if, and only if, is closed in , for each and each .
A CW-complex is called finite if is finite for each and for all but a finite subset of .
We write for . The data is called a CW-decomposition of .
Definition 15**.**
A subcomplex of a CW-complex is a subspace which is the union of open cells of , such that the closure in of each of these open cells is contained in .
A subcomplex can be made into a CW-complex , where for each , we put . (For a proof see e.g. [38, pg 16].)
Definition 16**.**
The -skeleton of a CW-complex is the subspace given by the union of all the open cells of dimensions , with the induced topology. Note that is a subcomplex of , hence a CW-complex.
Remark 17** (CW-complexes: properties and nomenclature).**
For proofs see e.g. [38, Appendix] and [51, 35].
- •
Condition 4. of the definition of a CW-complex is redundant if has only a finite number of cells; see [38, pp 521]. (Essentially this follows since a finite union of closed sets is always closed). In this paper we will only deal with finite CW-complexes, so condition 4. of Def. 14 will not be mentioned again.
- •
Cf. [51, pg 6], as the notation suggests, the closed cell is the closure in of the open cell .
- •
The attaching map of each closed -cell is the restriction of to , namely:
[TABLE]
The underlying topological space of the -skeleton of is homeomorphic to the space obtained from by attaching to it, along the attaching maps of the closed -cells.
Definition 18**.**
Given CW-complexes and , a map is called cellular if , for all .
Definition 19** (Abstract cells).**
If is a CW-complex, we call the set of abstract -cells.
Abstract -cells are in one-to-one correspondence with open -cells and with closed -cells. If is an abstract -cell, the closed and open -cells it corresponds to are (respectively) and .
Remark 20** ((Geometric) vertices, edges, plaquettes (or faces), and blobs).**
Abstract 0, 1, 2 and 3-cells of a CW-complex will sometimes be called called vertices, edges, plaquettes (or faces), and blobs, respectively. Closed 0, 1, 2 and 3-cells will sometimes be called geometric vertices, geometric edges, geometric plaquettes (or faces), and geometric blobs.
2.4 2-lattices
Simplicial complexes give rise to CW-complexes; but simplicial complexes are very rigid, therefore a large number of simplices is usually required to triangulate a manifold. CW-complexes allow for the decomposition of a manifold into fewer cells; however they are too general for our purposes, since the attaching maps of the closed cells can be highly singular, making it harder to use CW-complexes in combinatorial frameworks. In order to simplify our discussion later, we will consider CW-complexes which are 2-lattices, defined below.
If is the -sphere, the base-point of it is defined to be .
Definition 21** (2-lattices. Base point of a cell).**
Let be a topological manifold, with CW-complex . This is called a 2-lattice for if, for each and each :
(1) a CW-decomposition of is given for which the base-point is a 0-cell, and such that the attaching map of the corresponding closed -cell is cellular (as in Def.18).
(Note that in particular (1) implies that is a closed 0-cell of , for each and each . The image is called the base-point of the closed -cell .)
(2) one of the following two conditions holds:
- •
The attaching map of the corresponding -cell is constant.
- •
The attaching map of the corresponding -cell is an embedding (i.e. it is a homeomorphism onto its image). Moreover, for each closed -cell of , it holds that is a closed -cell of , and the restriction of to is a homeomorphism .
(3) If , we impose that the attaching map of the closed 3-cell is an embedding and furthermore that the boundary of the 3-cell is a subcomplex of .
The space is then said to have a 2-lattice decomposition.
A 2-lattice will usually be denoted as , or \big{(}M,L=(L^{0},L^{1},\dots)\big{)}.
Remark 22**.**
In practice, when defining a particular 2-lattice decomposition of , normally only the closed -cells will be made explicit, as it will always be clear that, for each -cell , an attaching map can be found which is cellular by using a suitable CW-decomposition of the -sphere. This does not fully determine a CW-decomposition, as some ambiguity rests on the actual characteristic maps of the -cells. However the topological space , all closed cells, all -skeletons , and hence the crossed modules and will be defined with no ambiguity. This is all we need for this paper.
Remark 23** (Lax 2-lattice).**
A CW-complex satisfying only of the definition of 2-lattices is called a lax-2-lattice. All combinatorial constructions in this paper are still true for lax-2-lattices, with the obvious modifications. In particular, the combinatorial construction of the 2-dimensional holonomy operators in §3.5 remains almost unaltered. The only issue is that the description of edge and vertex gauge spikes in §5.1.1 then requires a lot more cases, especially when it comes to edge operators.
Remark 24**.**
Let be a subcomplex of a CW-complex . If is a 2-lattice then clearly so is .
Example 25**.**
Evidently, 1-dimensional CW-complexes are always 2-lattices. The circle can be given a 2-lattice decomposition with two vertices (i.e. 0-cells) at and closed 1-cells at and . Here denotes the imaginary part of .
Example 26**.**
An example of a CW-complex which cannot be made into a 2-lattice is given by attaching to along , prolonged by continuity to . This is the type of singular attaching maps we want to avoid by restricting to 2-lattices.
Example 27**.**
Consider the 2-sphere , with the CW-decomposition arising from the polyhedral structure of . Let be the space obtained from by attaching along defined as:
[TABLE]
This CW-decomposition of is not a 2-lattice since the attaching map of its unique -cell is not an embedding.
Example 28** (Two 2-lattice decompositions of the 3-sphere ).**
Let us in this example model the 2- and 3-spheres as being and . The following are two 2-lattice decompositions of the 3-sphere.
: We consider the 3-sphere with the globe decomposition as follows. We firstly consider a CW-decomposition of with a unique closed 0-cell at the point of zero latitude and longitude, and a unique closed 1-cell making the equator, oriented eastwards. We have two closed 2-cells, , one for each hemisphere, attaching along the equator (oriented eastwards). See Fig. 2.
To get from to we now need to add two additional 3-cells attaching on each side of the 2-sphere.
: We can choose an even simpler 2-lattice decomposition of , having unique [math]- and -cells (resulting in ), and two 3-cells and , as above, attaching along each side of the 2-sphere.
Definition 29** (Notation: and ).**
Let be a 2-lattice. Let be a geometric 2-cell (i.e. plaquette; Def. 20). Let be the attaching map of the corresponding closed 2-cell (i.e. geometric plaquette). By definition, and is a closed 0-cell of . Hence is a pointed map . Passing to the pointed homotopy class of yields an element ; cf. Rem. 9. Here means inclusion of a groupoid into another.
Analogously, if is a blob (i.e. an abstract 3-cell), then the attaching map of the corresponding closed 3-cell (geometric blob) sends the base-point of to a 0-cell (the base-point of ; see Def. 21). Hence the attaching map is a pointed map . Passing to the pointed homotopy class of gives rise to an element .
Definition 30** (Notation: and ).**
Cf. Rem. 9 and 12. Continuing Def. 29, let be a plaquette. Then the characteristic map of the corresponding closed 2-cell induces a pointed map . Passing to the relative homotopy class of yields an element . Analogously if is an abstract 3-cell, then the characteristic map yields an element .
Note that given and it holds that:
[TABLE]
A CW-pair [38, pp16] is a pair of spaces where has a CW-decomposition and is a subcomplex of ; cf. Rem. 17.
Definition 31** (Relative 2-lattice decomposition of pairs and triples of spaces).**
Given a pair of topological manifolds (i.e. is a submanifold of ), we say that a 2-lattice decomposition of is a 2-lattice decomposition of if is a subcomplex of . (Note that the CW-decomposition of rendered from the fact that is a subcomplex of is always a 2-lattice decomposition; see Rem. 24.) If is a 2-lattice decomposition of that yields a relative 2-lattice decomposition of , we let be the induced 2-lattice decomposition of . CW-decompositions of a triple of manifolds are defined analogously.
If is a 2-lattice decomposition of , we say that is cellularly embedded in .
2.5 Paths on the lattice: the lattice groupoid of
Free groupoids on graphs are discussed in [12, 39, 18]. Ref. [11] in addition addresses groupoid presentations.
Definition 32** (Directed graph; totally intransitive graph).**
A directed graph is a pair of sets and , the sets of vertices and of edges of , together with a pair of maps and , called the source and target maps.
The maps identify, given an edge , its source and target (also called initial and end-points). Edges of may be represented as , where and .
A graph map is given by a pair of set maps and compatible with initial and end-point of edges.
A totally intransitive graph is a graph for which source and target maps coincide.
Definition 33**.**
The functor sends a groupoid to its underlying graph — simply forget the composition in . Its left adjoint takes a graph to the free groupoid on the graph; see [39].
A directed graph gives rise to another graph obtained by adding formal reverses to the edges of . Here is the set of symbols , and we put and .
Definition 34** (Free groupoid on a graph. Quantised path on a graph.).**
Let be a directed graph. A quantised path from vertex to on is a path on , i.e. a sequence where and , such that the initial point of coincides with the final point of , and also and . Quantised paths include the empty path at .
We define an equivalence relation on quantised paths as follows. Firstly quantised paths are related if we can modify into by deleting a subpath of the form . (Initial and end points of quantised paths remain stable under this relation.) Now take the symmetric-reflexive-transitive closure of this relation. If is a quantised path the equivalent class to which it belongs is denoted .
The (free) groupoid is the groupoid with object set ; arrows given by the set of equivalence classes of quantised paths; and arrows and are composed by \big{(}v\xrightarrow{[\gamma][\gamma^{\prime}]}u\big{)}=\big{(}v\xrightarrow{[\gamma\gamma^{\prime}]}u\big{)}. (Note that this composition is well-defined.)
The notion of freeness will be important in our construction. We will use freeness in the context of crossed module of groupoids. The latter is a non-trivial construction, so to prepare for this we now take the opportunity to recall what it means to say that the groupoid is ‘free’ in Def. 34 above.
Definition 35**.**
[39] A groupoid is free over a graph if there is a graph map satisfying the following property. For every groupoid and each graph map there is a unique groupoid map so that the diagram below commutes:
[TABLE]
Straightfoward computations, analogous to the free-group construction prove that:
Lemma 36**.**
[12, 8.2.1],[11],[39]* The groupoid is free over . ∎*
Let be a 2-lattice (more generally a CW-complex). Recall that is the -skeleton of . Note that the characteristic maps of the 1-cells give the structure of a directed graph. Given put and , where we identified and .
Definition 37** (Quantised path in a 2-lattice).**
A quantised path on a 2-lattice is a quantised path on the graph ; Def. 34. Hence quantised paths in are obtained by formally chaining together closed 1-cells of and their reverses: , where are closed 1-cells, such that the initial point of is the end-point of
The fundamental groupoid is defined in Def. 8. Its set of objects is . Let be an edge in . Let be the closed 0-cells corresponding to the abstract 0-cells . The characteristic map of is such that and . Passing to the homotopy class of , relative to the boundary of , yields a morphism in the homotopy groupoid ; cf. Rem. 30. Since is free, extends to a groupoid map
[TABLE]
The following can be seen as a generalisation of the van Kampen theorem [38], for spaces with a set of base points. Proofs are in [12, 9.1.5],[18],[11]. This holds more generally for CW-complexes.
Theorem 38**.**
Let be a 2-lattice. The groupoid map is an isomorphism. Hence is isomorphic to the free groupoid , with set of objects being and with a free generator for each edge . (Here and are the source and target of .) ∎
In particular, for any group and for any map there exists a unique groupoid map whose value on each , an edge, is . (The same holds if is a groupoid, except that we must pay attention to sources and targets.) As we will see in §3.2, this observation lies at the heart of the realisation of gauge theory that we will lift to the higher case.
We will hence see as being the lattice groupoid of .
3 Higher order gauge configurations and discrete 2D holonomy for surfaces embedded in 2-lattices
In order to establish a template for the ‘higher’ construction, we start with a suitable characterisation of ordinary gauge configurations, and of their holonomy along cellularly embedded circles .
3.1 Gauge configurations, discrete 1D parallel transport and holonomy along circles
Definition 39**.**
Let be a group. A gauge configuration on a 2-lattice is a map
[TABLE]
We write for each edge .
By the freeness of (Lem. 36 and Thm. 38) a gauge configurations extends to a uniquely defined groupoid morphism
[TABLE]
Here is regarded as a groupoid with one object. All groupoid maps arise this way. Hence:
Theorem 40** (The discrete parallel transport of a gauge configuration).**
Let be a 2-lattice. Let be a group. The correspondence yields a one-to-one correspondence between gauge configurations and groupoid maps . ∎
Those groupoid maps associated to a gauge configuration will sometimes be called discrete parallel transport functors, in analogy with the differential-geometrical construction in [62, 32].
Definition 41**.**
Let be a quantised path in . Let be a gauge configuration. Put:
[TABLE]
Let be the equivalence class of . Given Thm. 38, it is clear that:
Definition 42** (Holonomy along a circle: combinatorial definition).**
Let be a gauge configuration on a 2-lattice decomposition . Let be an oriented circle embedded in . Suppose that is a 2-lattice decomposition of ; Def. 31. Let . Starting at the vertex , the path around the circle in the positive direction therefore traces a quantised path , connecting to . The holonomy of , along , with initial point , is defined as:
[TABLE]
Note that the holonomy of along depends on the chosen starting point only by conjugation by an element of .
Remark 43** (Holonomy along a circle: algebraic topological definition).**
Recall . Choose a homeomorphism , preserving the orientation, sending the base-point of to . By elementary algebraic topology (as ), any two such homeomorphisms are homotopic, relative to . Let , where is the induced map on homotopy groups. Clearly , as in Def. 42. Let be the restriction of to the induced lattice decomposition of ; see Def. 31. It hence clearly holds that:
[TABLE]
Here is the discrete parallel transport of .
Remark 44**.**
Although gauge configurations can formally be defined separately from Hamiltonians, as above, they have no physical meaning without an associated Hamiltonian. In particular parts of the structure of space-time are encoded in a model not in the gauge configuration but in the Hamiltonian. We are not ready to give the ‘higher’ Hamiltonian §5.1.4 (the higher Kitaev model) that will be the central focus of this paper, but we can already give an illustrative ‘standard’ example, which also serves as a template for the Kitaev quantum double model [45]; see §5.1.6. Given a lattice and a group , and hence the set of functors between and , we may define for each character a Wilson action by (cf. Def. 29):
[TABLE]
where is the real part (see e.g. Wilson [72], [46, §8], [53, §10.2] or [55, §1.2]). Note that this depends strongly on the cell-decomposition of , as well as . The main thing to note at this point is that the sum is over plaquettes, thus the Hamiltonian is sensitive to the 2-dimensional structure in the lattice (whereas the gauge configuration ‘sees’ only the underlying graph ). We will return to this point later.
3.2 Higher order gauge configurations
In this paper, we consider fake-flat 2-gauge configurations on a 2-lattice , as discretised models for higher gauge fields [5, 3, 32]. Instead of a gauge group we have a crossed module of groups; Def. 2. The main aim is to extend Thm. 40, Def. 42 and Rem. 43 to the case of fake-flat 2-gauge configurations. This yields 2-dimensional (2D) notions of parallel transport which restrict to notions of 2D holonomy along surfaces, cellularly embedded in . We will address the 2-sphere and 2-disk case, which play an important role in higher Kitaev models.
3.2.1 Fake-flat 2-gauge configurations
Continuing the work of Yetter and Porter [74, 59], fake-flat 2-gauge configurations on CW-complexes were defined in [28, 33, 27]. Their algebraic topology interpretation was developed therein, following the work of Brown and Higgins on fundamental crossed modules of pairs of spaces and 2-dimensional van Kampen theorem [13, 14, 15, 18]. Homotopy quantum field theory applications of fake-flat 2-gauge configurations appear in [58] (and were there called “formal maps”). The inherent (and independently addressed) differential-geometric higher gauge theory for 2-bundles with a 2-connection appears in [5, 62, 61, 32, 63]. The term “fake-flatness” appeared originally in the context of gerbes with connection; see [9].
Definition 45** (2-gauge configuration).**
Let be a crossed module of groups. Given , a 2-gauge configuration , based on a 2-lattice (M,L)=\big{(}M,L=(L^{0},L^{1},L^{2},\dots)\big{)}, is given by:
- •
A map , denoted: , or .
- •
A map , denoted: , or .
A 2-gauge configuration gives rise to a groupoid map ; see Thm. 40.
We mainly consider fake-flat 2-gauge configurations. Let us explain what this means.
Definition 46** (Fake-flat 2-gauge configuration).**
A 2-gauge configuration , based on a 2-lattice , is said to be fake-flat if for each plaquette it holds that (recall the notation of Rem. 29):
[TABLE]
Given a crossed module , we denote the set of fake-flat 2-gauge configurations in as .
Let us give more explanation on the definition of fake-flatness. This is one of the points where the fact that we are restricting to 2-lattices Def. 21 makes our discussion a lot simpler. One more definition is needed.
Definition 47** (Quantised boundary of a plaquette).**
Let be a plaquette of a 2-lattice . Let be the attaching map of the correspondent closed 2-cell We are given a CW-decomposition of , which contains as a [math]-cell, such that is cellular and satisfies the conditions of Def. 21. We will in addition suppose that all characteristic maps of the closed 1-cells of are oriented counterclockwise; see Fig. 3. We also assume that the 1-cells of appear in that order, as we “travel” counterclockwise from to around .
We let be the base-point of the closed 2-cell corresponding to . Suppose that is not constant. Then for each , is a closed 1-cell of and restricts to a homeomorphism . The closed 1-cell is oriented by its characteristic map. We put if the restriction of preserves orientation and , otherwise. The quantised boundary of is defined to be the following quantised path in (Def. 37), connecting to : .
Otherwise, if satisfies , we define the quantised boundary of as .
An example appears in Ex. 49.
Let . By passing to the equivalence class of the quantised path (cf. the construction of in Def. 34 and Prop. 36) yields in Def. 29. Hence:
Proposition 48**.**
Let be a 2-lattice. Let be a crossed module of groups. A 2-gauge configuration is fake-flat if, and only if:
- •
For each plaquette for which is not constant, putting , it holds that:
[TABLE]
- •
If is a plaquette for which it should hold that .
Example 49**.**
Consider the square , with the 2-lattice decomposition indicated in the middle of Fig. 4, namely . (Abstract cells and the corresponding closed cells are denoted in the same way.) The geometric 2-cell attaches along the identity map , hence the attaching map is positively oriented. The CW-decomposition of (Def. 21) has a vertex for each corner and a positively oriented edge for each side of .
For this example, the quantised boundary of the plaquette is Hence a 2-gauge configuration of is given by four elements of , the colours of the edges, , , , , and an element , colouring . The fake flateness conditions says:
Let denote the set of fake-flat 2-gauge configurations. Note that is non-empty. In particular the ‘naive vacuum’ given by for all plaquettes of and for all edges is fake-flat. Here and denote the identities of and .
3.3 On Whitehead theorem, 2-gauge configurations and the lattice 2-groupoid
Let be a 2-lattice. Passing to the 0, 1 and 2-skeletons of the corresponding CW-decomposition of , yields a triple of locally path-connected spaces, where intersects non-trivially any path-connected component of and . Utilising Def. 10, we can form the fundamental crossed module ; Thm. 11. This crossed module plays the role of lattice 2-groupoid of .
Observe that to make use of our fake-flat 2-gauge configurations we need corresponding lifts of Lem. 36 and Thm. 38. Analogously to Thm. 38, the crossed module is free on the attaching maps of the geometric plaquettes (i.e. closed 2-cells) of . This result (which holds in the general case of CW-complexes) is an old result due to JHC Whitehead [70]. Modern treatments can be found in [18, 11, 10, 13, 8].
Consider groupoids and . Throughout this subsection, we use the following notation. If is a groupoid map, put to be the restriction of to morphisms and to be the restriction of to objects. If is a a crossed module of groupoids, thus and have the same set of objects, it holds that is the identity map.
In order not to excessively load our formulae, we use the same notation for the groupoid and for its set of morphisms, and the same for . Which one is meant is clear from the context. The coinciding source and target maps in the groupoid are given by the obvious map:
[TABLE]
First we specify what “crossed module freeness” is. Let be a groupoid. Let also be a set mapping to , through a map . Suppose also that we have a map that makes the diagram below commute (therefore and :
[TABLE]
(This in particular means that is a map from into the set of arrows (morphisms) in that have the same source and target.) Let be a totally intransitive groupoid with the same set of objects as . We say that a crossed module of groupoids is free on (or more precisely on and as in (6)) if there exists an inclusion (set) map making the diagram below commute:
[TABLE]
such that the following universal property is satisfied: Given any crossed module of groupoids, where and , and any groupoid map , and any set map , such that , there exists a unique groupoid map , with , making the diagram below commutative:
[TABLE]
*and so that the pair of groupoid maps is a crossed module map . *
Lemma 50**.**
Given as in (6), the totally intransitive groupoid , i.e. the top groupoid of the free crossed module on , is uniquely specified up to isomorphism by the universal property above. A model for is the following. First of all note that we have a totally intransitive graph having as set of vertices and the set of edges being the set of all pairs , where and are such that . The coinciding source and target maps of are given by . Edges of therefore take the form:
[TABLE]
We then form the free groupoid , which is a totally intransitive groupoid having as set of objects. We have a groupoid map which is the identity on objects and on generating morphisms is:
[TABLE]
The groupoid has a natural left action by automorphisms of the groupoid . On generators of , the action takes the following form: If are such that , and is such that , put . Then together with the map , nearly all conditions that crossed modules of groupoids must satisfy (Def. 2 and 6) hold, except for the 2nd Peiffer relation. The groupoid is obtained from by dividing out the 2nd Peiffer relations, in the obvious way.
Proof.
Routine. Details are in [18, §7.3(ii)] and [11]. ∎
Let be a 2-lattice. Recall Rem. 29 and 30, and Def. 21. Going back to , to each plaquette we can associate elements and , where is the base-point of the closed 2-cell corresponding to . Also . In particular we have a commutative diagram as in (6,7,8):
[TABLE]
Hence we can form the free crossed module on , where . And we have a unique groupoid morphism , which is the identity on objects, that makes the diagram below commutative, and so that is a crossed module map:
[TABLE]
Theorem 51** (Whitehead Theorem).**
Let be a 2-lattice, or indeed any CW-complex. Then the crossed module is free on . Specifically, the map defined from (9) is an isomorphism of groupoids, and, moreover, the pair is an isomorphism of crossed modules.
Proof.
See [18, §6] and [13, 11, 14, 15], where Whitehead’s theorem is deduced from the more general 2-dimensional van Kampen theorem, and also [10, 8]. (We note that Whitehead’s original proof was done for crossed modules of groups rather than of groupoids, and only considered spaces with a single base-point.)∎
Remark 52**.**
Note that Whitehead’s theorem together with the construction of free crossed modules (Lem. 50) implies that the totally intransitive groupoid is generated by , where and is such that: ; recall that and are the initial and end-points of and is the base-point of . This will have a primary role in the construction of the 2-dimensional holonomy of a fake-flat 2-gauge configuration along a cellularly embebbed surface in §3.4.
We will only use the universal property (8) in the case when is a crossed module of groups. In this case there is not much to worry about maps on object sets of groupoids, as and are groupoids with a single object. We hence will not display the related part of the commutative diagrams. Given groupoid maps and we put and . In , we put . Recall that we use the same notation for the groupoid and for its set of morphisms, and the same for .
Whitehead’s theorem implies the following. Consider the inclusion map , with , cf. Rem. 29 and 30. If is a group, is a crossed module of groups, and is a groupoid map, then given any set map such that , there exists a unique groupoid map making the diagram below commutative and also making the pair a crossed module map (thus is compatible with boundaries and groupoid actions):
[TABLE]
3.3.1 The discrete 2-dimensional (2D) parallel transport of a fake-flat 2-gauge configuration
As promised in 3.1, we now state and prove the analogue of Thm. 40 for fake-flat 2-gauge configurations.
Let be a 2-lattice and be a group crossed module.
Theorem 53** (The discrete 2D parallel transport of a fake-flat 2-gauge configuration).**
There exists a one-to-one correspondence between fake-flat 2-gauge configurations in and crossed module maps (Note and therefore are groupoid maps, compatible with boundary maps and groupoid actions in the obvious way.)
In analogy with the differential-geometric construction of 2-dimensional parallel transport 2-functors attached to 2-connections on 2-bundles [61, 32], the crossed module map associated to a fake-flat 2-gauge configurations will be called the discrete 2D parallel transport 2-functor of .
Proof.
Recall the notation introduced after Thm 51. A fake-flat 2-gauge configuration is an assignment of an element of to each 1-cell of , and an assigment of an element of to each 2-cell , satisfying the fake-flatness condition of Def. 46. Whitehead’s theorem (Thm 51) states that the fundamental crossed module , with set of base points – a crossed module of groupoids, is isomorphic to the free crossed module on the map .
Since the groupoid is free on the 1-cells, the assignment uniquely extends to a groupoid map . The fake-flatness condition means that the outer part of the diagram below commutes:
[TABLE]
And by applying the universal property defining free crossed modules of groupoids, in the particular form of (10), we can see that a gauge configuration can be extended (uniquely) to a crossed module map , and all crossed module maps arise this way. ∎
Cf. 3.1. We have now explained the crossed module analogue of discrete parallel transport functors (Thm 40), in terms of discrete 2D parallel transport 2-functors. In the next two subsections §3.4 and §3.5, we address how the 2D parallel transport of a fake-flat 2-gauge configuration can be used to define notions of discrete 2D holonomy along surfaces cellularly embedded in a 2-lattice . We will only deal with the case when is the 2-disk or the 2-sphere . In these cases, which are the ones needed to define higher Kitaev models, a 2D holonomy can be associated to cellularly embedded surfaces . 2D holonomy along 2-disks and 2-spheres is particularly simple to formulate, given that the corresponding oriented mapping class groups are trivial.
For surfaces not homeomorphic to or to , additional information is needed to define a meaningful 2D holonomy for a cellular embedding of into . Namely (assuming orientability) we must choose an isotopy class of homeomorphisms , where is the boundary of an unknotted handlebody in ; see [32, 63]. We will address this more general 2D holonomy in a forthcoming publication.
3.4 Algebraic topological definition of 2D holonomy along 2-disks and 2-spheres
Let us fix a crossed module of groups . In this subsection we use elementary algebraic topology to define precisely and concisely the 2-dimensional (2D) holonomy of a fake-flat 2-gauge configuration along cellularly embedded 2-disks and 2-spheres; as such we present the 2D analogue of Rem. 43. A combinatorial definition of this 2D holonomy (therefore the analogue of Def. 42) will be dealt with in §3.5.
3.4.1 The 2-disk case
Let be a 2-lattice. Let be a surface embedded in . Suppose that is homeomorphic to the 2-disk . Suppose in addition that is oriented. Furthermore (cf. Def. 31) suppose that is a 2-lattice decomposition of the triple , where is a pair homeomorphic to . We have an induced relative CW-decomposition of . Note (the 2-skeleton of ) and
Choose a base point . Since is trivial, the homotopy exact sequences of the triple and of the pair imply that the inclusion yields injections and . We can thus see the crossed module (Ex. 13) as canonically included in .
Let be the common base point of and . By elementary algebraic topology – since – any two pointed homeomorphisms preserving orientation are homotopic as maps of pointed pairs . This is used in the definition below.
Definition 54** (Notation: and ).**
Cf Rem. 29 and 30. Let be an oriented surface homeomorphic to . Let , the boundary of . We will be mainly interested in the case when . Consider a pointed orientation preserving homeomorphism . We let be given by the induced map on homotopy groups. Let be where is the positive (counterclockwise) generator of . Hence is a positively oriented loop along the boundary of the 2-disk , starting and ending at . Analogously put , where is now the positive generator of .
Note that by construction (cf. Ex 13):
[TABLE]
Lemma 55** (Dependence of and on ).**
Suppose that . Choose another . Consider a path in , connecting to . (There are two different possible homotopy classes for .) Then passing to the corresponding element , it holds that:
[TABLE]
Proof.
Follows from geometric considerations and the fact that acts trivially on . ∎
Let now be a crossed module of groups.
Definition 56** (2D holonomy of a fake-flat 2-gauge configuration along , with initial point ).**
Let be a topological manifold. Let be an oriented disk embedded in . Let be a 2-lattice decomposition of ; see Def. 31. Let . Let be a fake-flat 2-gauge configuration in , and be its restriction to the induced 2-lattice decomposition of . Let be the 2D parallel transport 2-functor of ; Thm. 53. We define the 2D holonomy of along , with initial point , as:
[TABLE]
In the conditions of Def. 56, note that:
[TABLE]
This is because, by (11) and the fact that is a crossed module map:
[TABLE]
Remark 57** (Dependence of 2D holonomy on base points).**
The 2D holonomy of a fake-flat 2-gauge configuration along a cellularly embedded 2-disk depends on the choice of a base point . However, the dependence is mild. Cf. Rem. 55. Choose any quantised path , in the boundary of the disk , from the new base point to the initial base point . Let be the corresponding element of . Then, since is a crossed module map:
[TABLE]
Hence:
[TABLE]
3.4.2 The 2-sphere case
We resume the notation and ideas of 3.4.1. Let be the base-point of . Let be a topological manifold. Cf. Def. 31, let be a 2-lattice decomposition of , where is oriented and homeomorphic to the 2-sphere . Choose a base point . By elementary algebraic topology, any two orientation-preserving homeomorphisms preserving base-points are pointed homotopic.
Since , the final bits of the homotopy exact sequence of the pointed pair , namely , yield a monomorphism , thus an isomorphism . Hence can be seen as included in the set of morphisms of the groupoid .
Definition 58** (Notation: ).**
Let carry the orientation arising from its embedding into . Let be an oriented manifold homeomorphic to . Choose a base point . Choose an orientation preserving homeomorphism . Let be the positive generator of . We define , to be , where is the induced map on homotopy. Note that (cf. Def. 34), it hence follows that \partial\big{(}\overline{\iota_{v}}(\Sigma)\big{)}=\emptyset_{v}, where is the boundary map in the crossed module of groupoids .
In what follows, we will frequently not distinguish from .
Remark 59**.**
Let and be different points in the 0-skeleton of . Let be any path in connecting to , considered up to homotopy relative to the end-points. Then .
Definition 60** (2D holonomy of a fake-flat 2-gauge configuration along , with initial point ).**
Let be an oriented manifold homeomorphic to the 2-sphere. Let be a 2-lattice decomposition of . Let . Let be a fake-flat 2-gauge configuration in . Let be the restriction of to the induced 2-lattice decomposition of . Let be the 2D parallel transport 2-functor of ; see Thm. 53. We define the 2D holonomy of along , with initial point , as:
[TABLE]
Remark 61**.**
Continuing Def. 60, note that since is a crossed module map:
[TABLE]
So it always holds that if This is not the case for the 2-disk; cf. (13).
Lemma 62** (Dependence of 2D holonomy along 2-spheres on base points and orientations).**
We resume the notation of Def. 60. Let be two base points. Let be a quantised path in , from to ; see Def. 37. Recall ; see Def. 41. We then have:
[TABLE]
Furthermore, if is with the opposite orientation, then:
[TABLE]
Proof.
Let be (Thm. 53) the crossed module map (i.e. the discrete parallel transport 2-functor) yielded by the restriction of to . Then:
[TABLE]
Let be with the opposite orientation. Then i\big{(}\overline{\iota_{v}}(\Sigma^{*})\big{)}=\big{(}i\big{(}\overline{\iota_{v}}(\Sigma)\big{)})^{-1}. Hence:
[TABLE]
∎
3.5 Combinatorial definition of 2D holonomy along 2-disks and 2-spheres
We now prepare a combinatorial description of the 2D holonomy of a fake-flat 2-gauge configuration along cellularly embedded 2-disks and 2-spheres. Some algebraic topology preliminaries are yet still needed.
3.5.1 Algebraic topology preliminaries for the 2-disk case
Let be an oriented manifold homeomorphic to the 2-disk . Hence we have an orientation of as well. Let be a 2-lattice decomposition (Def. 31) of . Choose , to be a 0-cell of . It will look more or less like the pattern in Fig. 5. (Here and in other diagrams later, we put oriented circles inside the plaquettes in order to indicate the orientation of their attaching maps; this is redundant as orientations can be inferred from the form of their quantised boundary.)
Recall that the definition of and , which are given in Def. 54.
Remark 63**.**
The homotopy exact sequence of the pointed pair gives an exact sequence:
[TABLE]
(see [38, pp 344]). Therefore is an isomorphism. Hence, if :
[TABLE]
Definition 64** (Quantised boundary of a 2-disk ).**
Choose a base-point , to be a 0-cell. Now go around , following its orientation, starting at until you go back to . Along the way we pass by the geometric 1-cells of , in that order. Put if the characteristic map of is oriented positively, and otherwise. Cf. Fig 5, the quantised boundary of is the following quantised path (cf. Defs. 34, 37) in , from to :
[TABLE]
By passing to morphisms in , we hence have ; Def. 34, 54.
If we allow the cancellation of consecutive pairs of a 1-cell and its inverse, thus considering quantised paths up to equivalence (Def. 34,) we can express – however not uniquely – as a product of quantised boundaries (cf. Def. 47 and Thm. 38) of plaquettes (or their inverses), each of which is in addition conjugated by a (possibly trivial) quantised path in , connecting the base-point of with the base-point of each plaquette. More precisely:
Lemma 65**.**
Let be a relative 2-lattice decomposition of . Choose a base point , to be a 0-cell. There exists a positive integer , plaquettes in (plaquettes can be repeated), integers (where ), as well as quantised paths in , connecting to the base point of each plaquette , such that the following equivalence between quantised paths holds:
[TABLE]
Here is the equivalence relation on quantised paths in Def. 34. Hence we can pass from the left-hand-side of (17) to the right-hand-side by sucessfully inserting or removing pairs where is a 1-cell of .
Remark 66**.**
Note that, by Rem. 63, equation (17) holds if, and only if, in (cf. Def. 54, 30):
[TABLE]
This is because by (16), equation above holds if, and only if, in we have:
[TABLE]
Example 67**.**
In Fig. 5, we can put , , and . Also put , and . Then, as indicated in Fig. 5:
[TABLE]
This follows from a simple calculation, which we recommend the reader to do. From (18) it follows that:
[TABLE]
Proof.
(Lemma 65) Consider the fundamental crossed modules , and the elements and of ; see Rem. 54. Recall \partial\big{(}\iota_{v}(\Sigma,L)\big{)}=\partial_{v}(\Sigma,L) in .
Cf. Rem. 29 and Thm. 51. We know that is a free-crossed module on the map , where is the free groupoid on the 1-cells; see Thm. 38. Let us apply Rem. 52. Hence there exists a positive integer , such that we can choose plaquettes , quantised paths , from the base-point of to the base point of and integers , such that, in :
[TABLE]
By using the first Peiffer Law in Def. 2, and Rem. 29 and 30, it follows that in , and where means equivalence class of quantised paths (Def. 34):
[TABLE]
Hence we can go from to the quantised path in a finite number of steps by inserting, or removing pairs , where is any 1-cell of . ∎
Remark 68**.**
The choice of a positive integer and of an assignment:
[TABLE]
where such that we have an equivalence of quantised paths:
[TABLE]
– equivalently (cf. Rem. 63) such that, in :
[TABLE]
or equivalently such that, in :
[TABLE]
– is far from being unique.
3.5.2 A combinatorial description of the 2D holonomy along embebbed 2-disks
Cf. [58]. Let be a 2-lattice. Suppose that is homeomorphic to the 2-disk and that is a decomposition of the triple ; Def. 31. Fix an orientation on . Choose a base-point . Consider a fake-flat 2-gauge configuration in . Let be the induced 2-lattice decomposition of . Let be the quantised boundary of ; Def. 64.
Cf. 3.5.1, choose a positive integer and plaquettes (plaquettes might be repeated), integers (where ), and quantised paths in , from to the base point of , such that we have an equivalence of quantised paths (cf. Def. 34, 37, 47):
[TABLE]
(*)
Recall that by Rem. 63 and 68, equation (23) is the same as saying that in (cf. Rem. 54):
[TABLE]
Fix a crossed module . Recall the construction of the 2D holonomy of a fake-flat 2-gauge configuration (cf. Def. 46) along , with initial point ; Def. 56.
Theorem 69**.**
Suppose that is a 2-lattice decomposition of , where is a surface cellularly embebbed in . Let . Let be as in . If
[TABLE]
is a fake-flat 2-gauge configuration on , then can be calculated as:
[TABLE]
Cf. Def. 41, here is the product of the elements of assigned to the 1-cells of the quantised path (or their inverses), and the same for . In other words and .
As an immediate consequence, we have the promised independence theorem of the 2D holonomy of a fake-flat 2-gauge configuration along a pointed 2-disk on the way we combine the group elements associated to the edges and plaquetes, as long as the rules of the assigment (*) are followed.
Theorem 70**.**
Fix . The evaluation in (24) does not depend on the assignment as in () chosen; see Rem. 68. Moreover (13) holds, i.e. *
Example 71**.**
Let us consider a fake-flat 2-gauge configuration on the 2-lattice decomposition of in Fig. 5. In the figure below, we put and . For to hold, we can put (see Ex. 67): , and ; and , and . Hence:
[TABLE]
Proof.
(Theorem 69) We use Rem. 66, 63. Given an assignment as in , then
[TABLE]
where this holds in , and, now in :
[TABLE]
The restriction of to gives a crossed module map . Thus:
[TABLE]
where, N.B., firstly is in and then it is in . Analogously:
[TABLE]
∎
3.5.3 Algebraic topology preliminaries for the 2-sphere case
Let be a 2-lattice. Suppose that is an oriented surface homeomorphic to the 2-sphere. Choose a base-point , to be a 0-cell. Recall that the homotopy exact sequence of yields:
[TABLE]
(which is exact), and hence we have an isomorphism .
Cf. Def. 58, 60, Rem. 30 and the construction in 3.5.2. In order to find a combinatorial definition of , we must express (see Def. 58) as a product of terms like , where is a quantised path from to the base point of the plaquette , and is the element in it yields; Def. 37 and 34. A crucial point in 3.5.1 is that the kernel of the boundary map is trivial if , whereas if we have ; see (25). In order to identify , we will need to use the Hurewicz map between homotopy and homology long exact sequences of pairs; see [38, pp 374]. Our main tool is the fact that the Hurewicz map is an isomorphism, if .
Before continuing, let us define, given a plaquette , with characteristic map :
[TABLE]
Lemma 72**.**
There exist a positive integer , and an assignment , where , is a quantised path from to the base point of the plaquette , and , such that:
we have an equivalence of quantised paths (Def. 37 and 34):
[TABLE] 2. 2.
given any , .
Moreover, given a positive integer and an assignment , where , is a quantised path from to the base point of the plaquette , and , then:
[TABLE]
(cf. Def. 58 and Rem. 30) happens if, and only if, conditions 1. and 2. of the lemma are satisfied.
Proof.
Cf. Rem. 29, Thm. 51 and Rem. 52. Since is the free crossed module on , there exist a positive integer , and an assignment , where , is a quantised path from to the base point of the plaquette , and , such that:
[TABLE]
We claim that satisfies items 1 and 2, of the statement of the lemma.
Item 1. Since \partial(i\big{(}\overline{\iota_{v}}(\Sigma,L)\big{)})=\emptyset_{v}, combining with:
[TABLE]
yields that satisfies item 1.
Item 2. Consider the map of exact sequences obtained from the Hurewicz map between homotopy and homology long exact sequences, [38, pp 374]:
[TABLE]
The group is the free abelian group on the relative homology classes determined by the plaquettes ; [38, pp 137]. Moreover , for each plaquette and each path connecting to the base-point of . We let . Then is the positive generator of . We now need the following claim:
Claim .
Proof of the claim (sketch) This is seemingly well know, however we could not find a proof anywhere. Since is the free abelian group on the , we know that there exist unique , where , such that . We need to prove that , for each .
Let . Let be an interior point of open cell corresponding to . We have a commutative diagram (28), where all morphisms are induced by inclusion. The vertical line corresponds to the identity map , in the sense that it sends the positive generator to the positive generator of . (Note is oriented, so it makes sense to speak about those positive generators.)
[TABLE]
Then , by definition of . Whereas if is another plaquette, then since the corresponding closed 2-cell is contained in it holds . Hence . QED.
Having proven the claim, item 2 of the statement of the lemma now follows from the fact that:
[TABLE]
We note that is the free abelian group on the , where
To finalise, let us be given an assignment , where , is a quantised path from to , and . Let . From (27) it follows:
[TABLE]
∎
3.5.4 A combinatorial description of the 2D holonomy along embedded 2-spheres
Let be an oriented embedded in a manifold . Let be a 2-lattice decomposition of . Let . Let be a fake-flat 2-gauge configuration in . Recall the definition of the 2D holonomy of along as {\rm Hol}_{v}^{2}({\cal F},\Sigma,L)=\Psi_{{\cal F}_{\Sigma}}\big{(}i(\overline{\iota_{v}}(\Sigma,L)\big{)}\in\ker(\partial)\subset E; Def. 60.
Theorem 73**.**
Let be a 2-lattice decomposition of . Let be the induced 2-lattice decomposition of . Let be a fake-flat gauge 2-configuration in . Let be its restriction to . Recall Lem. 72.
Find a positive integer , and, for each , a plaquette (plaquettes might be repeated), an integer , and a quantised path , connecting to the base point of , such that:
we have an equivalence of quantised paths (Def. 37 and 34):
[TABLE] 2. 2.
given any , ; cf. (26).
(**)
Then we have the following combinatorial formula for
[TABLE]
Here is the product of the elements of assigned to the 1-cells of the quantised path (or their inverses). And in particular, fixing , to be a 0-cell of , the expression (29) for does not depend on the assignment as in chosen. Moreover, Lem. 62 holds.
Proof.
By Lem. 72, condition (**) is equivalent to: [\gamma_{1}]\triangleright\iota_{L}(P_{1})^{\theta_{1}}\,\,[\gamma_{2}]\triangleright\iota_{L}(P_{2})^{\theta_{2}}\dots[\gamma_{N}]\triangleright\iota_{L}(P_{N})^{\theta_{N}}=i\big{(}\overline{\iota_{v}}(\Sigma)\big{)}. Hence:
[TABLE]
since is a crossed module map. ∎
Example 74**.**
Consider the 2-lattice decomposition of the 2-sphere with a single 0-cell and a single 2-cell , whose characteristic map is positively oriented; cf. Ex. 28. The base point of is . A 2-gauge configuration is simply an element , colouring its unique plaquette . Fake-flateness imposes Let , positively oriented. An assignment as in is such that , , and Hence , as it should.
Example 75**.**
To facilitate drawing diagrams, let us now see the 2-sphere has being the square , where we squash the upper edge and the lower edge to be single points (the north and south poles and ), and we identify the left and right boundary edges. We give the reverse orientation to the one induced by . Consider the 2-lattice decomposition of the 2-sphere, with two zero cells, at and , and four one cells , all connecting to . We have 2-cells , indicated in figure below. All plaquettes are based at the south pole. The characteristic map of each plaquette preserves orientation, so . The quantised boundary of each plaquette is indicated in figure below.
Let , with the same orientation. Let . An assignment (for ) satisfying can be , , and . Any cyclic permutation will also work.
A 2-gauge configuration is given by elements , colouring the edges , and elements colouring , as indicated in the figure below. Conditions for fake-flatness to hold are also made explicit in figure below. Hence .
By Thm. 73, cyclic permutations of must yield the same value for , as (**) is still satisfied. The former can be directly proven: note , thus is central, by the second Peiffer law of the definition of crossed modules (Def. 2). Hence .
Example 76**.**
Consider the standard tetrahedron displayed below. Hence the boundary , of , with the induced orientation, is given by the two triangles below identified along their boundaries. We give a 2-lattice decomposition derived from the obvious triangulation of . The quantised boundary of each plaquette is indicated in the figure below. Note, and are based in , whereas is based in .
[TABLE]
Let consider 2D holonomy along based at . An assignment satisfying (**) is such that , and:
[TABLE]
The general form of a fake-flat 2-gauge configuration of is presented below:
[TABLE]
Hence, by (29) it follows that: .
3.6 2-flat 2-gauge configurations
Let be a 2-lattice. Let . The corresponding closed 3-cell (called a blob) is also denoted by . From the definition of 2-lattices (Def. 21), the attaching map of is an embedding and is a subcomplex of , called the boundary of the blob . Orient by using . Cf. Rem. 29, 30 and Def. 58, we have \overline{\iota_{v}}\big{(}\mathrm{bd}(b)\big{)}=\partial_{L}(b), in , where is the base-point of .
Definition 77** (2-flat 2-gauge configuration).**
Let be a crossed module. Consider a fake-flat 2-gauge configuration based on . The boundary of each blob inherits a 2-lattice decomposition. The fake-flat 2-gauge configuration is said to be 2-flat if for every blob , we have:
[TABLE]
Recalling (Def. 46) that denotes the set of fake-flat 2-gauge configurations in , the set of 2-flat 2-gauge configurations is denoted .
More generally, a fake-flat 2-gauge configuration is said to be 2-flat along a cellularly embedded 2-sphere if, for some , hence – by Lem. 62 – for all , it holds that .
Example 78**.**
The fake-flat 2-gauge configuration from the end of §3.2.1 (the naive vacuum) is 2-flat.
Example 79**.**
The fake-flat 2-gauge configurations in Ex. 76 is 2-flat if, and only if,
Let us provide an algebraic-topological interpretation of 2-flat 2-gauge configurations. Let be a fake-flat 2-gauge configuration in . Cf. the construction of in 3.4.2. Consider the discrete 2D parallel transport 2-functor of ; see Thm. 53. By the construction in 3.4.2 and 3.5.4, it holds that . Recall that gives a one-to-one correspondence between fake-flat 2-gauge configurations and crossed module maps .
The map of crossed modules induced by the inclusion is denoted by . In components , where is a surjection and is the identity .
Cf. Rem. 29. Given any 3-cell , note that , the identity of , where is the base-point of . These are the only relations we need to impose in order to pass from to ; what is meant by this is in Lem. 80. This follows from the long homotopy exact sequence of the triple , applied to each choice of base point , namely:
[TABLE]
together with the fact that the group is isomorphic to the free \mathbb{Z}\big{(}\pi_{1}(M^{1},x)\big{)}-module on ; [38, Lemma 4.38].
Cf. the diagram below, a crossed module map is said to descend to if there exists a (necessarily unique) crossed module map such that .
[TABLE]
Lemma 80**.**
A crossed module map descends to if, and only if, for each blob we have .
Note that , by the cellular approximation theorem.
Proof.
As mentioned above, this follows from the long homotopy exact sequence of the triple , applied to each choice of base point ; details can be found in [28], for the case of CW-complexes with a single base-point. Alternatively we can also use the higher-dimensional van Kampen theorem of Brown and Higgins; see [13, 14, 15, 11] and [18, §6], stating that (under mild conditions) the fundamental crossed module functor preserves colimits. Note that the conditions of 2-lattices (Def. 21) imply that for each , the corresponding closed 3-cell is a subcomplex of , homeomorphic to . Moreover , and . From [11, §6.3], it follows that the diagram (30) below is a pushout diagram in the category of crossed modules of groupoids:
[TABLE]
In the diagram (30) above, all arrows are induced by inclusions. Also, . Given , we pass from to by quotienting by the normal closure of ; we are using the notation of Def 58. By inspecting (30), and applying the universal property of pushouts, it hence follows that a crossed module map descends to if, and only if, for each , and for each , it holds that . by Lem. 62, in order for the latter to happen, it suffices to check that , if is the base-point of , which is the same as saying that . ∎
Combining Lem. 80 with Thm. 53, yields the following interpretation of fake-flat 2-gauge configurations.
Theorem 81** (2-flat 2D parallel transport 2-functors).**
The bijection of Thm. 53 yields a bijection between 2-flat configurations and crossed module maps , from now on called 2-flat 2D parallel transport 2-functors. The correspondence sends to the crossed module map that descends to.
4 Gauge transformations
Throughout this section, we fix a crossed module of groups §2.1 and a 2-lattice §2.4. Recall that has the structure of a directed graph , see §2.5. We denote the edges (1-cells) of as , where and are the source and target of . (It may be that .)
4.1 The group of gauge operators
If is a group and is a set we put to denote the group , with pointwise multiplication.
Definition 82** (The group of gauge operators).**
There is a left-action of on by automorphisms. Given and , the action has the form:
[TABLE]
for each in . (Note that denotes the underlying action of on , which exists since is a crossed module; Def. 2.) We define the group of gauge operators to be:
[TABLE]
Here denotes semidirect product. In particular we take:
[TABLE]
Example 83**.**
Recall the 2-lattice decomposition of from Fig. 2. We have a unique edge and a unique vertex. Hence If we extend to be a 2-lattice decomposition of , by adding 3-cells (Ex. 28), it also holds that . This is because groups of gauge operators on 2-lattices depend only on 1-skeletons.
The group of gauge operators acts on the set of fake-flat 2-gauge configuration on in a way such that the 2D holonomy is preserved; see §4.4, below. Moreover, this action restricts to an action of on the set of 2-flat configurations; Def. 77. In order to present the action, we now define a double groupoid out of the crossed module ; see [18, §6.6], [15] and [32].
4.2 The double groupoid
The definition of a double groupoid appears e.g. in [18, §6.1] and [32, 63]. In this paper double groupoids are edge-symmetric and have a unique object. We explain the definition of such double groupoids as we elaborate how a crossed module of groups gives rise to one, denoted ; [18, §6.6].
We have a unique object , and sets and of horizontal and vertical 1-squares in ; [32]. These sets of horizontal and vertical 1-squares in consist of diagrams of the form:
[TABLE]
Horizontal and vertical 1-squares in are composed in the obvious way, here shown for horizontal 1-squares:
[TABLE]
We therefore have horizontal and vertical groupoids, also denoted and , with a single object. The sets of morphisms are in one-to-one correspondence with . The groupoids and are isomorphic. An obvious isomorphism is obtained by clockwise rotation.
We have a set of squares in . This set consists of diagrams of the form below:
[TABLE]
Remark 84** (Squares in and fake-flat 2-gauge configurations of ).**
Elements hence can be seen as fake-flat 2-gauge configurations on a certain 2-lattice decomposition of ; see Ex. 49.
Several maps exist connecting , and ; see below ( is as in (34)):
[TABLE]
[TABLE]
Horizontal and vertical compositions of 2-squares can be done when squares match on the relevant sides:
\hskip-11.38092pt{\begin{CD}&*@>{W}>{}>&*\\ &@A{Z}A{}A\hskip-36.98866pt\scriptstyle{e}&@A{}A{Y}A\\ &*@>{}>{X}>&*\end{CD}\quad\quad{\begin{CD}&*@>{W^{\prime}}>{}>&*\\ &@A{Y}A{}A\hskip-36.98866pt\scriptstyle{e^{\prime}}&@A{}A{Y^{\prime}}A\\ &*@>{}>{X^{\prime}}>&*\end{CD}}\quad=\quad\begin{CD}&*@>{WW^{\prime}}>{}>&*\\ &@A{Z}A{}A\hskip-36.98866pt\scriptstyle{(X\triangleright e^{\prime})e}&@A{}A{Y^{\prime}}A\\ &*@>{}>{XX^{\prime}}>&*\end{CD}}
and
{\begin{CD}\begin{CD}&*@>{W^{\prime}}>{}>&*\\ &@A{Z^{\prime}}A{}A\hskip-36.98866pt\scriptstyle{e^{\prime}}&@A{}A{Y^{\prime}}A\\ &*@>{}>{W}>&*\end{CD}\\ \begin{CD}&*@>{W}>{}>&*\\ &@A{Z}A{}A\hskip-36.98866pt\scriptstyle{e}&@A{}A{Y}A\\ &*@>{}>{X}>&*\end{CD}\end{CD}\quad\quad=\quad\quad\begin{CD}&*@>{W^{\prime}}>{}>&*\\ &@A{ZZ^{\prime}}A{}A\hskip-36.98866pt\scriptstyle{eZ\triangleright e^{\prime}}&@A{}A{YY^{\prime}}A\\ &*@>{}>{X}>&*\end{CD}}
(37)
These compositions are associative. Hence the set of squares in is the set of morphisms of two categories, called horizontal and vertical categories. The correspondent sets of objects are the sets of vertical and horizontal squares in , respectively. Source and target maps are in (35). Unit maps are in (36).
Remark 85** (Interchange law in ).**
Horizontal and vertical compositions in satisfy the interchange law, which says that the composition indicated below does not depend on the order whereby it is done.
[TABLE]
This means that we can either first perform horizontal compositions, and then vertical compositions, or vice-versa, yielding the same result. To prove the interchange law we must make explicit use of the 2nd Peiffer condition in Def. 2, for crossed modules of groups. (All other mentioned properties follow from the 1st Peiffer relation in Def. 2, and the fact that acts on by automorphisms.)
The horizontal and vertical categories are both groupoids. Given , the inverses and of , with respect to the vertical and horizontal compositions, called vertical and horizontal reverses of , are given in (38), below. This finishes the construction of the double groupoid .
[TABLE]
4.3 Full gauge transformations between fake-flat 2-gauge configurations
The action of the group §4.1 of gauge operators on the set §3.2.1 of fake flat 2-gauge configurations is given in §4.4. We still need some technicalities.
4.3.1 Groupoid of fake-flat 2-gauge configurations and full gauge transformations
Definition 86** (Full gauge transformation).**
A full gauge transformation , starting in the fake-flat 2-gauge configuration , is given by a pair of maps: and , such that:
- •
Let . Put: U_{1}(v)=\left(\,\,\,\begin{CD}&*\\ &@A{\,\,g_{v}}A{}A\\ &*\end{CD}\right). Let be the set of edges of incident to . If then:
[TABLE]
Hence if two edges and share a vertex, the corresponding vertical sides of and match.
- •
For each edge , it holds that d_{u}\big{(}U_{2}(t))={{\cal F}^{1}}(t). (For notation see (35).)
Definition 87**.**
An example of a full gauge transformation starting in is . It is such that , for each , hence is an identity vertical 1-square. assigns in (36) to each .
Remark 88** (Full gauge transformations and crossed module homotopies).**
In the language of [19, §2.1] and [20], full gauge transformations, starting at , boil down to crossed module homotopies starting on the associated crossed module map of Thm. 53. (An explanation of crossed module homotopies in a language close to this paper’s is in [33, 27, 30, 29].) In order to prove this fact, we must use the fact that the groupoid is free on the set of 1-cells of . Therefore, a crossed module homotopy starting on the crossed module map can be arbitrarily (and uniquely) specified by its value on the set of 0 and 1-cells of , yielding our full gauge transformations.
For an explanation of crossed module homotopy in the general framework of crossed complexes, see [18, §9] and [16].
A full gauge transformation , starting in the fake-flat 2-gauge configuration , transforms into another fake-flat 2-gauge configuration, denoted
[TABLE]
The definition of is given below. 666 This is a consequence of the general construction in [19, §2.1], [20] and [18, §9] of crossed module and crossed complex homotopy. Some explicit calculations are in [33, 29, 30].
We give a combinatorial explanation of , based on the framework of this paper.
At the level of edge colourings, if then is the bottom colour of the square , i.e. ; see (35). Let us describe , where is a plaquette. We consider two cases, depending on the attaching map of the closed 2-cell (also denoted ); cf. Def. 21.
If is constant, then , where is a 0-cell of . And we then put:
[TABLE]
where is defined in Def. 86. Since (cf. Prop. 48), is fake-flat at , because:
[TABLE] 2. 2.
Otherwise, we now make explicit use of the fact that must then be an embedding; as such the characteristic map of is a homeomorphism. Consider , with the obvious product lattice decomposition, where has unique 0-cells at [math] and . The 2-dimensional lattice made out of the top and lateral sides of can be given a fake-flat 2-gauge configuration, obtained by putting together and . We refer to Fig. 7. It depicts a fake-flat 2-gauge configuration in the boundary of the cylinder . The base-point of plaquette , whose attaching map is oriented counterclockwise, is . The top of the cylinder is coloured by the restriction of to . The sides of the cylinder are coloured by the 1 and 2-squares and , where is a vertex of and is an edge of . (Each can be seen as a fake-flat 2-gauge configuration of the 2-disk ; see Rem. 84. The direction of the edges of gives an unambiguous way to transport the fake-flat 2-gauge configuration of onto the correspondent lateral square in Fig 7.) Finally, the bottom of the cylinder is coloured with
By definition, the ‘gauge-transformed’ colour of the plaquette (which is based at ) is the 2D holonomy, based at , along the 2-disk consisting of the top and lateral sides of the cylinder in Fig. 7, with the fake-flat 2-gauge configuration obtained by putting together and . By Thm. 70, the 2D holonomy along this 2-disk is well defined. Also by (13) is fake-flat at .
Remark 89**.**
A more concrete expression for can be derived by using the double groupoid . In the example in Fig. 7, we evaluate the following composition in . (We note that the elements of assigned to the three squares in the bottom right arise from the horizontal reverses of the squares , and , above; cf. (38).) And then is the element of assigned to the resulting square in .
[TABLE]
Remark 90** (Notation).**
If is a full gauge transformation starting at , and transforming into , we use the notation: By construction it clearly follows that ; see Def. 87.
Lemma 91**.**
Consider a sequence of full gauge transformations . A full gauge transformation , starting in , can be defined. Its underlying 1 and 2-squares in are obtained by vertically composing the 1- and 2-squares in of and , in the obvious way. Therefore the squares in making will be put under the squares in making in (37). Then connects to ; i.e. .
Proof.
(Sketch) We must prove that \big{(}({\cal U}^{\prime}*{\cal U})\triangleright({\cal F}^{1})\big{)}(t)={{\cal F}^{\prime\prime}}^{1}(t) for each , and that \big{(}({\cal U}^{\prime}*{\cal U})\triangleright({\cal F}^{2})\big{)}(P)={{\cal F}^{\prime\prime}}^{2}(P), for each . This is trivial to verify for edges, and for plaquettes attaching along constant maps. Otherwise, cf. Fig. 7. Put the squares of the full gauge transformation on the bottom of the ones of . This yields a fake-flat 2-gauge configuration , defined on the 2-disk made out of the top and lateral faces of , where has 0-cells at and 2. There are two different ways to explicitly compute the 2D holonomy of along . They must yield the same element of ; see Thm. 70. a) Either we firstly multiply the squares standing on top of each other, and then compose with the top 2-disk, and in this case the result will be \big{(}({\cal U}^{\prime}*{\cal U})\triangleright({\cal F}^{2})\big{)}(P). Or b) compose with and only after that compose with ; and then the result will be {\big{(}{\cal U}^{\prime}\triangleright({\cal U}\triangleright{\cal F}^{2})\big{)}(P)=\big{(}{\cal U}^{\prime}\triangleright{{\cal F}^{\prime}}^{2}\big{)}(P)={{\cal F}^{\prime\prime}}^{2}(P).}
Cf. Rem. 89, if we put the squares of under those of in (40), then the statement of the lemma also follows from the interchange law for the vertical and horizontal compositions in ; see Rem. 85. ∎
Cf. (38). By using the vertical reverse of 2-squares in , we conclude that full gauge transformations can be reversed. Namely if we have , then , obtained by applying vertical reverses to the 1-squares , , and the 2-squares , , is such that . Also and ; see Def. 87. Therefore we have the following result.
Theorem 92** **(The groupoid of fake-flat 2-gauge
configurations and full gauge transformations).
Let be a 2-lattice and a crossed module. We have a groupoid , whose objects are the fake-flat 2-gauge configurations . The morphisms are the full gauge transformations .
Remark 93** (Algebraic topological definition of – following Brown and Higgins).**
Recall (Thm. 53) that we have a bijection , between fake-flat 2-gauge configurations and crossed module maps As mentioned in Rem. 88, there exists a relation of homotopy between crossed module maps , where and are crossed modules of groupoids, discussed in [19, 20, 33]. This relation is a particular case of homotopy of crossed complex maps [18, §7.1.vii, §9.3], [16]. By using the notation in [18, §9.3.i], given crossed modules and , we have a groupoid of crossed module maps and homotopies between them. When , where a CW-complex, and is a crossed module of groups, a homotopy connecting and boils down to a full gauge transformation . For discussion see [33, 27, 29]. Hence in [18, §9.3.i].
Remark 94** (A 2-groupoid of fake-flat 2-gauge configurations, full gauge transformations and 2-fold gauge transformations).**
The groupoid is part of a more general construction. Let be crossed module of groupoids. By considering 2-fold homotopies between crossed module homotopies (see [19, 20, 33] and [18, §9.3.i]), we can furthermore define a 2-groupoid , whose objects are crossed module maps , 1-morphisms are homotopies between 2-crossed module maps, and 2-morphisms are 2-fold homotopies between homotopies. Explicit formulae are in [33, 27, 30, 29]. This leads to a notion of 2-fold gauge transformation between full gauge transformations, prominent in higher gauge theory [32, 5, 61]. These 2-fold gauge transformation between full gauge transformations do not appear to have large importance for this paper. However they have prime importance for addressing algebraic topology descriptions of higher gauge theory invariants of manifolds (namely Yetter invariant [74, 59]), as explained in [33, 27].
4.3.2 Full gauge transformations preserve 2D holonomy along embedded 2-spheres
Full gauge transformations preserve the 2D holonomy of fake-flat 2-gauge configurations along embedded surfaces. Let us address how to prove this using some basic algebraic topology, closely following the work of Brown and Higgins [16, 15, 13, 17]. Our proof is done in very identical lines to the one of [32], which was done for the case of differential-geometric 2-connections.
We temporarily denote the fundamental crossed module of a CW-complex by . If is a CW-complex, let be the product CW-complex, where is given the obvious CW-decomposition with 0-cells at [math] and . If is a fake-flat 2-gauge configuration in a 2-lattice , we hence denote the (Thm. 53) 2D parallel transport 2-functor of by .
The main result underpinning our discussion is the following interpretation of full gauge transformations between fake-flat 2-gauge configurations. It essentially appears in [18, §9.3.i, §9.7 and §9.8] and [16, 17], in the more general case of crossed complexes.
Lemma 95**.**
Let be a 2-lattice. Let be a group crossed module. Let be a full gauge transformation connecting and . We then have a crossed module map , making the diagram below (in the category of crossed modules) commute:
[TABLE]
Here and are induced by and .
Proof.
Let us for simplicity assume that the product CW-decomposition of is a 2-lattice . The general case in analogous. Recall the construction of the usual CW-decomposition of , where and embed as subcomplexes, and we have an additional -cell of for each -cell of . (See [38, Page 523].)
Cf. the discussion in §4.3.1, particularly the construction of , and Fig. 7. The fake-flat 2-gauge configurations , , and the full gauge transformation , can together be assembled to yield a fake-flat 2-gauge configuration of . The restriction of to is obtained from , and the restriction of to is obtained from . Finally the restriction of to the 1- and 2-cells (where ) and (where ) is obtained from and ; for conventions on how to do this see the discussion in §4.3.1.
We have a 3-cell of for each 2-cell . And is 2-flat along , given the explicit construction of the value of at . Since there are no more 3-cells in , we hence conclude that is a 2-flat 2-gauge configuration in . We now just need to apply Thm 81 to . Clearly H_{\cal U}=(\Psi_{\cal M},\Phi_{\cal M})\colon\Pi_{2}\big{(}M^{2}\times[0,1])\to{\cal G} makes the diagram in (41) commute. ∎
Remark 96**.**
A stronger result can be proved, and is implicit in [16] and [18, Chapter 9]. Namely, there is a one-to-one correspondence between full-gauge transformations and maps , making (41) commute. This can be inferred by combining the beginning of [18, §9.3.i] with Thm [18, 9.8.1].
We state the result concerning invariance of 2D holonomy under full gauge transformations for a 2-sphere cellularly embedded in a 2-lattice, only; see §3.4.2, §3.5.4. This is the case whose behaviour under full gauge transformations is the neatest and it is the generality needed to formulate higher Kitaev models in §5.1.
Theorem 97**.**
Let be a crossed module of groups. Let be a 2-lattice. Let be a 2-sphere cellularly embedded in . Let be a fake-flat 2-gauge configuration on . Let be a full gauge transformation starting in . Let , to be a 0-cell of . Let be the element of associated to ; see Def. 86. Then:
[TABLE]
Proof.
We strongly use the previous lemma, and resume the notation there introduced.
Let . Cf. (41), put . Let be as in Def. 58. By Def. 60, we have that:
[TABLE]
Let be the following path in , connecting to :
[TABLE]
Then, passing to the correspondent element in the underlying groupoid \pi_{1}\big{(}(M^{2}\times[0,1])^{1},(M^{2}\times[0,1])^{0}\big{)} of the crossed module , it holds that:
[TABLE]
By construction we have that . Cf. (41), it hence follows that:
[TABLE]
∎
4.3.3 Groupoid of 2-flat 2-gauge configurations and full gauge transformations
Let be a crossed module of groups. Let be a 2-lattice. Recall the definition of a 2-flat 2-gauge configuration in §3.6 and details therein. Let be a fake-flat 2-gauge configuration. The 2D holonomy of along the boundary of a 3-cell is invariant under full gauge transformations, in the sense of Thm. 97. Suppose that is 2-flat, hence that , for each . Since acts on by automorphisms, if is any full gauge transformation, starting in , it follows that , for each . Hence full gauge transformations transform 2-flat 2-gauge configurations into 2-flat 2-gauge configurations.
In particular, the groupoid of fake-flat 2-gauge configurations and full gauge transformations of Thm 92 has a full subgroupoid , whose objects are the 2-flat 2-gauge transformations.
Remark 98** (Algebraic topological definition of – following Brown and Higgins).**
Cf. Rem. 93. Recall (Thm. 81) that we have a bijection , between 2-flat 2-gauge configurations and crossed module maps Cf. [18, §9.3.i], given crossed modules and we have a groupoid of crossed module maps and homotopies between them. When , where a CW-complex, is a group crossed module, and are 2-flat 2-gauge configurations, a homotopy , connecting to , boils down to a full gauge transformation . Hence in [18, §9.3.i].
4.4 Gauge operators on fake-flat 2-gauge configurations
Let be a crossed module of groups. We now finally define a left-action of the group of gauge operators of §4.1 on the set of fake-flat 2-gauge configurations . Our main tool is the groupoid of fake-flat 2-gauge configurations and full gauge transformations between them; see Thm 92. We will define a left-action whose action-groupoid is isomorphic to .
The main observation is that given a fake-flat 2-gauge configuration , the set of full gauge transformations starting in can be put in one-to-one correspondence with elements . Given and , we have a full gauge transformation , starting at , defined as:
[TABLE]
Lemma 99**.**
Let . Let . We have:
[TABLE]
N.B.: See §4.1 for conventions on the product of The composition of of full gauge transformation is made explicit in Lem. 91.
Proof.
This follows by construction, by looking at the conventions (42) for . Just compare the explicit definition of the group operation in in (33) with the explicit form of the vertical composition of squares in in (37). The latter yields the composition of full gauge transformations in Lem. 91. ∎
An operation of the group on can then be defined by:
[TABLE]
By Lem. 99, is indeed a left-action of on and is its action groupoid.
By construction of and Thm. 97, it follows that:
Theorem 100**.**
Let be a 2-lattice. Let be a 2-sphere cellularly embedded in . Let . Let be a fake-flat 2-gauge configuration in . Let . Then:
[TABLE]
Cf. 4.3.3, in particular restricts to an action of on the set of 2-flat 2-gauge configurations.
5 The Hamiltonian models
5.1 A Hamiltonian model for higher gauge theory
In this subsection, we fix a manifold , a 2-lattice decomposition of (Def. 21), and a crossed module of groups ; Def. 2. We suppose that is compact (thus that is finite) and that is finite, meaning that both and are finite groups. Hence the set of fake-flat 2-gauge configurations is finite. Note that is non-empty, as the naive vacuum is in ; see §3.2.1.
Recall from §4.1 the group of gauge operators. We have §4.4 a left-action of the group of gauge operators on , preserving 2D holonomy, as in Thm. 100.
Definition 101** (Hilbert space ).**
The Hilbert space for Hamiltonian higher gauge theory is the free vector space on the set of fake-flat 2-gauge configurations, with the unique inner product that renders different fake-flat 2-gauge configurations orthonormal.
The group algebra of has a representation on , obtained by linearising the action of on . Given , we denote the corresponding linear operator by Thus , for each fake-flat 2-gauge configuration .
By construction, if , then the operator is unitary, i.e. , where denotes Hermitian adjoint.
5.1.1 Vertex and edge gauge spikes and
Definition 102** (Edge gauge spikes and vertex gauge spikes).**
A vertex gauge spike, supported in , is a gauge operator such that for each , and such that for each it holds that , unless . Analogously, given an edge , a gauge spike supported in is a gauge operator such that for each , and such that for each , it holds that , unless .
For a vertex and , we let be the unique vertex gauge spike supported in such that . For and , we let be the unique edge gauge spike, supported in , such that .
The linear operators are also called vertex and edge gauge spikes.
Lemma 103**.**
(I) Vertex gauge spikes supported in different vertices commute, and edge gauge spikes supported in different edges commute. I.e., given , with , and , with (including the case when and share a vertex), it holds that, given and :
[TABLE]
(II) For each vertex and each edge , then given any and any , we have:
[TABLE]
*(III) If t=\big{(}\sigma(t)\xrightarrow{t}\tau(t)\big{)}\in L^{1}, including the case when , and is such that , then given and , it follows that and commute: .
(IV) Given an edge t=\big{(}\sigma(t)\xrightarrow{t}\tau(t)\big{)}\in L^{1}, including the case , and , we have:*
[TABLE]
(V) Moreover, any gauge operator can be obtained as a product of vertex and edge gauge spikes.
Proof.
This all follows immediately from the definition of the group of gauge operators as a semidirect product in §4.1. Equations (31) and (33) translate exacty to the formulae in the lemma. ∎
Remark 104**.**
Note that (I–IV) also hold for the associated operators , since they are constructed from a representation of the group on the Hilbert space .
Let us now unpack the construction in §4.3 and §4.4, and give an explicit description of how vertex and edge gauge spikes act on . Let be a fake-flat 2-gauge configuration; Def. 46. If , recall is the operator, . Put \widehat{m}({\cal F})=\big{(}\widehat{m}({\cal F}^{2})\colon L^{2}\to E\,,\,\widehat{m}({\cal F}^{1})\colon L^{1}\to G\big{)}. Recall (42):
Action of vertex gauge spikes: Let and . Let \big{(}\sigma(t)\xrightarrow{t}\tau(t)\big{)}\in L^{1}. Then:
[TABLE]
Let . Let be the base-point of . Then:
[TABLE]
Action of edge gauge spikes: Let and . Let , then, cf. (42):
[TABLE]
Let . Let be the base-point of . Some bits of notation are necessary to describe \big{(}\widehat{U_{\gamma}^{e}}({\cal F}^{1})\big{)}(P). Let \partial_{L}^{Q}(P)=\big{(}x_{P}\xrightarrow{t_{1}^{\theta_{1}}t_{2}^{\theta_{2}}...t_{n}^{\theta_{n}}}x_{P}\big{)} be the quantised boundary of ; Def. 47. Put , . Given that the attaching map of the 2-cell is either constant or an embedding, an arbitrary edge can appear in the list where , at most once.
Definition 105**.**
Let \partial_{L}^{Q}(P)=\big{(}x_{P}\xrightarrow{t_{1}^{\theta_{1}}t_{2}^{\theta_{2}}...t_{n}^{\theta_{n}}}x_{P}\big{)}. We say that the edge :
[TABLE]
Suppose is an embedding. Given a vertex , with , there exists a unique quantised path made from the and contouring in the positive direction (where orientation is provided by the attaching map of ). There is another quantised path contouring in the negative direction. We also put and .
We let and be the 1D holonomy of along and ; see Def. 41. And then:
[TABLE]
Example 106**.**
Consider a 2-lattice decomposition that close to a plaquette looks like figure 8, below:
The only edge and vertex gauge spikes which have a non-trivial action on the restriction of a fake-flat 2-gauge configuration to are the vertex gauge spikes , and the edge gauge spikes , where and are arbitary. The actions of some of these on a fake-flat 2-gauge configuration at are below. (We can also see that fake-flatness at is preserved in these examples.)
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
Remark 107**.**
We note that if is a triangulation of , then the vertex and edge gauge spikes here defined coincide with the vertex and edge gauge transformations appearing in [21, III-A & III-B].
5.1.2 Vertex operators, edge operators and blob operators
Given a set , we put to denote the cardinality of .
Definition 108** (Vertex operators and edge operators ).**
Let be a vertex and be an edge, of . The elements below of the group algebra are called vertex and edge operators:
[TABLE]
The corresponding operators are also called vertex and edge operators.
Note that the operators are all self-adjoint. This is because, if :
[TABLE]
and analogously for .
From Lem. 103 we have the following.
Lemma 109**.**
Given arbitrary vertices and edges we have:
[TABLE]
These relations also hold for and , since acts on . ∎
Let be a 3-cell (i.e a blob; Rem. 20). Cf. §3.6, by definition of 2-lattices, is a subcomplex of homeomorphic to , with a base-point , which is a 0-cell, and an orientation. We can thus consider the 2D holonomy , of along .
Definition 110** (Blob operator).**
Let . Let also . The diagonal idempotent blob operator is given by the formula below, for each basis element :
[TABLE]
Here if we put , if and , if
See [21, III-B] and Ex. 76, for the explicit form of blob operators when our lattice is a triangulation of .
Since fake-flat 2-gauge configurations form an orthonormal basis of , it is easy to see that each is a self-adjoint operator.
As an immediate application of Thm. 100, follows:
Lemma 111**.**
Let , and (it may be that ). Let and . Let . We have:
[TABLE]
Hence edge gauge-spikes always commute with blob operators , regardeless of being an edge in , or not. A vertex gauge-spike commutes with a blob operator , unless is the base point of .
5.1.3 The local operator algebra of higher lattice gauge theory
The algebra , which underpins the construction of the higher Kitaev model in 5.1.4, is our proposal for the local operator algebra of higher lattice gauge theory.
Definition 112** (Local operator algebra for higher lattice gauge theory).**
Let be a 2-lattice. Let be a crossed module of finite groups. We define the -algebra as formally generated by the
[TABLE]
imposing the relations appearing in Lem. 103 and 111.
Note that is a *-algebra, where:
[TABLE]
Given the discussion in 5.1.1 and 5.1.2, we hence have a unitary representation of on the Hilbert space of higher lattice gauge theory.
5.1.4 The higher Kitaev model for (3+1)-dimensional topological phases
We now propose a higher gauge theory version (the “higher Kitaev model”) of Kitaev quantum-double model for (2+1)-dimensional topological phases of matter [45, 40]. This higher Kitaev model for (3+1)-dimensional topological phases is formulated for manifolds , of any dimension, with a 2-lattice decomposition ; see Def. 21. For a description of higher Kitaev model in the particular case of triangulated manifolds we refer the reader to [21], and to [71], in a more general context. Topological phases protected by higher gauge symmetry are also proposed in [42].
Definition 113** (Higher Kitaev model).**
(Cf. the notation in 5.1.2). Let be a finite crossed module of groups. Let be a compact topological manifold, of any dimension, with a 2-lattice decomposition . Our proposal for a totally solvable (the sum of mutually commuting projection operators) higher lattice gauge theory Hamiltonian, which we call the “higher Kitaev model”:
[TABLE]
(where is as in Def. 101), with respect to the 2-lattice is (where is the identity of ):
[TABLE]
The commutation relations of Lem. 103, 109 and 111, ensure that, if , and :
[TABLE]
(Observe that these relations also hold for the , and .) And moreover we have that:
[TABLE]
Note that by construction each term is Hermitian, hence so is .
Typically will be a 3-dimensional manifold, and the higher Kitaev model should be considered to be a model for (3+1)-dimensional topological phases [66, 71, 67, 21, 65, 47, 22, 42, 50]. The higher Kitaev model also makes sense if is a surface, but in this case blob operators will not appear in the model.
Note that vertex and edge operators, which implement gauge invariance at vertices and edges of a 2-lattice, and the blob operators, which enforce 2-flatness at a blob, are very different in nature.
Lemma 114**.**
Let be a 2-lattice. Let be a finite crossed module of groups. Let . Let
[TABLE]
Then is invariant under full gauge transformations and in particular it is invariant under the action of vertex and edge gauge operators. Moreover .
Proof.
The first bits follow from Thm. 100: given , then {\rm Hol}_{\beta_{0}(b)}^{2}\big{(}{\cal F},\mathrm{bd}(b),L\big{)} is invariant under the action of gauge operators, up to acting by an element of , which acts on by automorphisms. On the other hand, the fact that follows from the definition of .∎
5.1.5 Example: higher gauge theory in the 3-sphere
Let us give an explicit description of the higher Kitaev model, if the underlying manifold is . We consider two different 2-lattice decompositions of Let be a finite crossed module of groups.
**First case:
**Cf. Ex. 28 and 74. Consider the 3-sphere with the lattice decomposition , with a unique [math]-cell, no 1-cells, one 2-cell (thus the 2-skeleton is ) and two blobs, attaching on each side of the 2-sphere. By Ex. 74, the Hilbert space is thus isomorphic to , the vector space generated by the orthonormal basis . The 2D holonomy along the 2-sphere of a fake-flat 2-gauge configuration associated with is itself: .
For this 2-lattice we have no edge operators. The higher Kitaev Hamiltonian therefore has the form: , where:
[TABLE]
**Second case:
**For a more substantial example, let us give the 2-lattice decomposition: of Ex. 28. A 2-gauge configuration is given by a and a pair . The fake flateness condition enforces that . Therefore, we have:
[TABLE]
The 2D holonomy of a configuration , along the 2-sphere , based at , is:
[TABLE]
The vertex and edge gauge spikes on and , and the blob operators along and , have the form:
[TABLE]
(Cf. §5.1.1.) Here , and . Note that , for each .
The commutation relations of Lem. 103 and 111 here boil down to:
[TABLE]
This gives the local operator algebra ; see Def. 112.
The higher Kitaev hamiltonian has the form , where:
[TABLE]
5.1.6 Comparison with the Kitaev model
Though constructed in a similar way, the higher Kitaev model and the Kitaev model [45] (also known as Kitaev quantum double model) are subtly different constructions. In the following, we will demonstrate that a subspace of the Kitaev model is equivalent to a class of higher Kitaev models, while differing in the whole Hilbert space.
In the language of this paper, the Kitaev model takes as input: a 2-lattice and a finite group , realising a lattice model with local operator algebra, which is at the base point of each plaquette isomorphic to , the quantum double of [45]. The Hilbert space {\cal H}_{K}\big{(}M,L,G\big{)} of the Kitaev model is the free vector space on the set of gauge configurations ; see Def. 39. Considering the group as the crossed module, (see Ex. 5), it follows:
[TABLE]
Here is the 1-skeleton of . Note the use of as opposed to , so that the fake-flatness condition becomes void. The Kitaev model is defined by the Hamiltonian
[TABLE]
Here the operator is as in 5.1.2, defined for the crossed module , with action on \mathcal{H}_{K}\big{(}M,L,G\big{)} given by the action on {\cal H}\big{(}M^{1},L,(1\to G)\big{)}. Whereas, is defined from a new type of self-adjoint operators , which act on gauge configurations as follows (for notation see Def. 42):
[TABLE]
Lemma 115**.**
Given and the following relations hold:
[TABLE]
Hence is a sum of mutually commuting projection operators.
We now compare the Kitaev model with group to the higher Kitaev model with crossed module , with fixed 2-lattice . We begin by defining the flat sub-Hilbert space of the Kitaev model :
[TABLE]
It is straightforward to show
[TABLE]
This is due to the requirement of fake-flat 2-gauge configurations of on being an equivalent condition to requiring .
For the crossed module , both the blob and edge operators act as the identity. In this way the higher Kitaev Hamiltonian reduces to:
[TABLE]
This model is equivalent to the Kitaev model defined on the flat sub-Hilbert space {\cal H}^{\text{flat}}_{K}\big{(}M,L,G\big{)}:
[TABLE]
This is because, by definition, the operator has trivial action on {\cal H}^{\text{flat}}_{K}\big{(}M,L,G\big{)}={\cal H}\big{(}M,L,(1\to G)\big{)}, while the operator has the same action on both Hilbert spaces. In this way we can identify the higher Kitaev model for with the Kitaev model defined on the sub-Hilbert space {\cal H}^{\text{flat}}_{K}\big{(}M,L,G\big{)}. However, the Kitaev model diverges from the higher Kitaev model for outside of the sub-space {\cal H}^{\text{flat}}_{K}\big{(}M,L,G\big{)} due to the presence of non-fake-flat configurations in H_{K}\big{(}M,L,G\big{)}.
5.2 Ground state degeneracy
Let be a compact manifold and be a finite group crossed module. Let be a 2-lattice decomposition of . Hence is a finite dimensional Hilbert space, which explicitly depends on the 2-lattice decomposition of . In this subsection, we prove that the dimension of the ground state space of the higher Kitaev model in 5.1.4 is a topological invariant of , meaning that depends only on alone. Specifically, we will show that has a basis in canonical one-to-one correspondence with the set of homotopy classes of maps from to the classifying space of the crossed module . (Classifying spaces of crossed modules are defined in [18, §2.4] and [17, 11, 33, 27].) It therefore follows that the dimension of the ground state space is a homotopy invariant of manifolds, as expected given the relation [21] of our model to Yetter’s invariant of manifolds [74, 59]. Yetter’s invariant was proven in [33, 27] to be a homotopy invariant of manifolds.
This subsection is less self-contained than the remainder of the paper. Cf. Rems. 93 and 94. Let be a CW-complex. We use deep results of Brown and Higgins on the description of the weak homotopy type of the function space , where is a crossed complex (a generalisation of crossed modules) and is its classifying space; the bits we need can be found in [17, Thm. A] and [18, Thm. 11.4.19]. Let denote the fundamental crossed complex of . The main tool we use is the fact that the weak homotopy type of , with the -iffication of the compact-open topology on the space of continuous maps , is represented by the crossed complex ; an explanation of this is in [33, §2.6.1]. The crossed complex is made of crossed complex maps and -fold homotopies, .
For a crossed module , the underlying groupoid of the crossed complex is the groupoid of crossed module maps and their homotopies, referred to in Rem 88, 93 and 98; see [18, §7.1.vii and §9.3.i]. Combining with Thm 81, it hence follows that if is a 2-lattice, then the underlying groupoid of is isomorphic to the groupoid of 2-flat 2-gauge configurations and full gauge transformations between them: see 4.3.3.
Given a 2-lattice , the results of [17] and [18, §11.4] hence tell us that path-connected components of the function space – i.e. homotopy classes of maps – are in canonical one-to-one correspondence with connected components of the groupoid of 2-flat 2-gauge configurations and full gauge transformations between them; see §4.3.3. This will be the main tool used in this subsection.
Let us now connect the discussion in the previous paragraphs with the ground state degeneracy of the higher Kitaev model. We start by looking at the expression (46) for :
[TABLE]
Each of the operators , where is a vertex; , where is an edge; and , where is a blob, is a Hermitian projector. All of those projectors commute. We can choose an eigenspace decomposition with respect to which all those projectors are diagonal.
If we apply or to , we get a non-zero linear combination of fake-flat 2-gauge configurations with non-negative coefficients. Let us write for the subset of non-zero -linear combinations in . Recall the naive vacuum is given by for all cells ; see §3.2.1 and Ex. 78. Since and both take an to an element of , and indeed take an element of to an element of , we have that is non-zero. Since , for each , it follows that there is an -eigenspace of (containing ) with eigenvalue 0. (Note that is in the kernel of all of the , and , where , and , given the commutation relations in (47).)
Now projectors have eigenvalues [math] or , thus the ground state has energy zero, meaning:
[TABLE]
And, furthermore, a vector belongs to the ground state if, and only if, it is in the kernel of all of the projectors , and , where , and .
Lemma 116**.**
A state , where , is in if and only if:
- (i)
unless is 2-flat then ; see Def. 77 for the definition of a 2-flat configuration; 2. (ii)
given any , any vertex and any , it holds ; 3. (iii)
given any , any edge and any , it holds .
Proof.
First the ‘only if’ part. Cf. the discussion just before the Lemma. In order that it must be that (i) ; that (ii) ; and that (iii) .
- (i)
For each , C_{b}^{1_{E}}(\Psi)=\displaystyle\sum_{{\cal F}\in\Theta(M,L,{\cal G})}\lambda_{\cal F}\delta\Big{(}{\rm Hol}_{v}^{2}\big{(}{\cal F},\mathrm{bd}(b)\big{)},1_{E}\Big{)}{\cal F}. In order that , it must be that whenever : \delta\Big{(}{\rm Hol}_{v}^{2}\big{(}{\cal F},\mathrm{bd}(b)\big{)},1_{E}\Big{)}=1,\forall b\in L^{3}. Hence is 2-flat. 2. (ii)
Suppose that , for all . Then:
[TABLE]
Now apply . And since \widehat{U_{v}^{gh}}({\cal F})=\widehat{U_{v}^{g}}\big{(}\widehat{U_{v}^{h}}({\cal F})\big{)}, we have for each and any :
[TABLE] 3. (iii)
Analogously, it order that be in the kernel of all , then .
Conversely, if satisfies (i),(ii) and (iii), then will be in the kernel of all operators , and , where and . As such will be in the ground state . ∎
Let be the set of 2-flat 2-gauge configurations in ; Def. 77. We say that and in are equivalent () if we can go from to through acting by a sequence of vertex and edge gauge spikes; see 5.1.1. By Lem. 116: we hence have:
[TABLE]
The composite of gauge spikes is a full gauge transformation and, Lem. 103 (V), any full gauge transformation is the composition of gauge spikes; as such to say that and in are equivalent is to say that they are connected by a full gauge transformation. In other words and are equivalent if, and only if, they can be connected by a morphism in the groupoid , of 2-flat 2-gauge configurations and full gauge transformations between them; see §4.3.3.
The set of connected components of the groupoid is denoted by \pi_{0}\big{(}\Theta_{\rm 2flat}^{\sharp}(M,L,{\cal G})\big{)}. I.e. \pi_{0}\big{(}\Theta_{\rm 2flat}^{\sharp}(M,L,{\cal G})\big{)} is the set of equivalence classes of objects of , where two 2-flat 2-gauge configurations are equivalent if a full gauge transformation connects the two. We have a basis for the ground space , in one-to-one correspondence with \pi_{0}\big{(}\Theta_{\rm 2flat}^{\sharp}(M,L,{\cal G})\big{)}, namely:
[TABLE]
As explained in [17, Thm. A] or [18, Thm. 11.4.19] (in the general case of crossed complexes), there is a natural bijection between elements of \pi_{0}\big{(}\Theta_{\rm 2flat}^{\sharp}(M,L,{\cal G})\big{)}=\pi_{0}({\rm CRS}_{1}\big{(}\Pi_{2}(M,M^{1},M^{0}),{\cal G})\big{)} and homotopy classes of maps ; cf. Rems. 93 and 98. (For an explanation of this in the crossed module case see [33, 27].) In particular the cardinality of the set \pi_{0}\big{(}\Theta_{\rm 2flat}^{\sharp}(M,L,{\cal G})\big{)} does not depend on .
Theorem 117**.**
Let be a compact manifold. Let be a 2-lattice decomposition . Let be a finite crossed module. Consider the higher Kitaev Hamiltonian of Def. 113. Then the ground state of has a basis in canonical one-to-one correspondence with the set of homotopy classes of maps , where is the classifying space of the crossed module . Hence depends only on the topology of and not on the chosen 2-lattice decomposition of .
Proof.
Compare the preceding observation with (55). ∎
Remark 118**.**
It was proven in [21] that the dimension of the ground state coincides with Yetter invariant [74, 59] of . We note that the Yetter invariant of is not quite the same as since uses finer information on the space of functions than its number of connected components; see [33].
Remark 119**.**
We can write as , since it does not depend on , and only on . Indeed depends only on the homotopy type of and the weak homotopy type of the crossed module [33], since the homotopy type of depends only on the weak homotopy type of .
Acknowledgement
We would like to thank a referee for a previous version of this paper for useful comments. Alex Bullivant, Zoltan Kádár, and Paul Martin thank EPSRC for funding under Grant EP/I038683/1.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] J. C. Baez. Four-dimensional bf theory as a topological quantum field theory. Letters in Mathematical Physics , 38(2):129–143, 1996.
- 2[2] John C. Baez. An introduction to spin foam models of B F 𝐵 𝐹 BF theory and quantum gravity. In Geometry and quantum physics (Schladming, 1999) , volume 543 of Lecture Notes in Phys. , pages 25–93. Springer, Berlin, 2000.
- 3[3] John C Baez and John Huerta. An invitation to higher gauge theory. General Relativity and Gravitation , 43(9):2335–2392, 2011.
- 4[4] John C. Baez and Aaron D. Lauda. Higher-dimensional algebra. V: 2-Groups. Theory Appl. Categ. , 12:423–491, 2004.
- 5[5] John C. Baez and Urs Schreiber. Higher gauge theory. In Categories in algebra, geometry and mathematical physics. Conference and workshop in honor of Ross Street’s 60th birthday, Sydney and Canberra, Australia, July 11–16/July 18–21, 2005 , pages 7–30. Providence, RI: American Mathematical Society (AMS), 2007.
- 6[6] John Barrett and Bruce Westbury. Invariants of piecewise-linear 3-manifolds. Transactions of the American Mathematical Society , 348(10):3997–4022, 1996.
- 7[7] John W. Barrett and Marco Mackaay. Categorical representations of categorical groups. Theory Appl. Categ. , 16:529–557, 2006.
- 8[8] Hans Joachim Baues. Combinatorial homotopy and 4 4 4 -dimensional complexes , volume 2 of De Gruyter Expositions in Mathematics . Walter de Gruyter & Co., Berlin, 1991. With a preface by Ronald Brown.