A classification of 3+1D bosonic topological orders (I): the case when point-like excitations are all bosons
Tian Lan, Liang Kong, and Xiao-Gang Wen
TL;DR
This paper proposes a classification scheme for 3+1D bosonic topological orders with only bosonic point-like excitations, linking them to finite groups and 4-cocycles, and shows they can be realized by Dijkgraaf-Witten theories.
Contribution
It introduces a partial classification of 3+1D bosonic topological orders based on unitary pointed fusion 2-categories and connects them to Dijkgraaf-Witten gauge theories.
Findings
Classifies 3+1D bosonic topological orders with bosonic point-like excitations.
Establishes a one-to-one correspondence with finite groups and 4-cocycles.
Demonstrates realization via Dijkgraaf-Witten gauge theories.
Abstract
Topological orders are new phases of matter beyond Landau symmetry breaking. They correspond to patterns of long-range entanglement. In recent years, it was shown that in 1+1D bosonic systems there is no nontrivial topological order, while in 2+1D bosonic systems the topological orders are classified by a pair: a modular tensor category and a chiral central charge. In this paper, we propose a partial classification of topological orders for 3+1D bosonic systems: If all the point-like excitations are bosons, then such topological orders are classified by unitary pointed fusion 2-categories, which are one-to-one labeled by a finite group and its group 4-cocycle up to group automorphisms. Furthermore, all such 3+1D topological orders can be realized by Dijkgraaf-Witten gauge theories.
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A classification of 3+1D bosonic topological orders (I):
the case when point-like excitations are all bosons
Tian Lan
Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
Liang Kong
Yau Mathematical Science Center, Tsinghua University, Beijing, 100084, China
Xiao-Gang Wen
Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
Abstract
Topological orders are new phases of matter beyond Landau symmetry breaking. They correspond to patterns of long-range entanglement. In recent years, it was shown that in 1+1D bosonic systems there is no nontrivial topological order, while in 2+1D bosonic systems the topological orders are classified by a pair: a modular tensor category and a chiral central charge. In this paper, we propose a partial classification of topological orders for 3+1D bosonic systems: If all the point-like excitations are bosons, then such topological orders are classified by unitary pointed fusion 2-categories, which are one-to-one labeled by a finite group and its group 4-cocycle up to group automorphisms. Furthermore, all such 3+1D topological orders can be realized by Dijkgraaf-Witten gauge theories.
Contents
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III Some general properties of 3+1D topological orders of boson systems
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III.3 Untwisted sector of dimension reduction is the Drinfeld center of
-
IV Condensing all the point-like excitations to obtain a trivial topological order
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IV.1 There is no 3+1D topological order with only nontrivial string-like excitations
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IV.3 Condensing all the point-like excitations gives rise to a trivial 3+1D topological order
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VI The classification of unitary pointed fusion 2-categories
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B A general discussion of string and membrane operators in 3+1D topological orders
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C More general properties of string-like excitations in 3+1D topological orders
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C.1 Pure string-like excitations and sectors in dimension reduction
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C.2 Pure string-like excitations are labeled by the conjugacy classes of
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C.3 String-like excitations are -flux, even in generic 3+1D AB topological orders
I Introduction
In history, we have completely classified some large class of matter states only for a few times. The first time is the classification of all spontaneous symmetry breaking orders Landau (1937); Landau and Lifschitz (1958). We find that all symmetry breaking orders can be described by a pair:
[TABLE]
where is the symmetry group of the system and , a subgroup of , is the symmetry group of the ground state.
The second time is the classification of all 1-dimensional gapped quantum phases. We find that 1-dimensional gapped quantum phases with on-site symmetry can be classified by a triple (even for strongly interacting bosons/fermions) Chen et al. (2011); Schuch et al. (2011):
[TABLE]
where pRep is a projective representation of Pollmann et al. (2012). We see that all the 1-dimensional gapped quantum phases are described by symmetry breaking plus an addition structure described by . The additional structure is the symmetry-protect topological (SPT) orderGu and Wen (2009).
The third time is the classification of 2-dimensional gapped quantum phases. In the absence of any symmetry, a gapped phase may have a nontrivial topological order Wen (1990); Wen and Niu (1990); Keski-Vakkuri and Wen (1993). We find that all 2+1D bosonic topological orders are classified by a pair:Keski-Vakkuri and Wen (1993); Rowell et al. (2009); Wen (2016)
[TABLE]
where MTC is a unitary modular tensor category and is the chiral central charge of the edge states. Physically, the tensor category theory MTC is just a theory that describes the fusion and the braiding of quasiparticles, which correspond to fractional/non-abelian statistics. Modular means that every nontrivial quasiparticle has a nontrivial mutual statistics with some quasiparticles.
For fermion systems, the classification is different: 2+1D fermionic topological orders are classified by a triple:Lan et al. (2016)
[TABLE]
where is the symmetric fusion category (SFC). In general, a SFC describes particles with trivial mutual statistics with each others. In the SFC , the particles carry the representations of the fermion-number-parity symmetry where the nontrivial representation is assigned Fermi statistics. So just describes the underlying fermions that form the fermionic systems. Also, BFC is a unitary braided fusion category that describes all the quasiparticles (the anyons and the underlying fermions). Those quasiparticles all have trivial mutual statistics with the underlying fermions in . We also require BFC to have minimal modular extensions.
In the presence of unitary finite on-site symmetry , all the 2+1D gapped bosonic phases are classified by Barkeshli et al. (2014); Lan et al. (2017a)
[TABLE]
where is a SFC that describes bosonic quasiparticles with trivial mutual statistics that carry the representations of , BFC describes all the quasiparticles that have trivial mutual statistics with all the quasiparticles in , and MTC is a minimal modular extension of the BFC. Physically speaking, MTC include both quasiparticles (described by BFC) and symmetry-twist defects (added by hand), such that every nontrivial quasiparticle/defect has a nontrivial mutual statistics with some quasiparticles/defects, and every defect has a nontrivial mutual statistics with some quasiparticles in . Such a classification includes symmetry breaking orders (described by ), as well as the SPT orders Chen et al. (2013, 2012) [described by ] and topological orders [described by ].
For fermion systems with unitary finite on-site symmetry , we have a very similar classification: all such 2+1D gapped fermionic phases are classified by Lan et al. (2017a)
[TABLE]
where describes quasiparticles with trivial mutual statistics that carry the representations of where some representations are assigned Fermi statistics.
After those fairly complete classification results in 1+1D and 2+1D, in this paper, we are going to study the classification of 3+1D topological orders. We will only deal with the simpler case, the 3+1D topological orders for bosonic systems. 3+1D bosonic topological orders are gapped quantum liquids Zeng and Wen (2015); Swingle and McGreevy (2016) without any symmetry. Note that there are gapped non-liquid states in 3+1D, such as stacked fractional-quantum Hall layers Zeng and Wen (2015), and fractal/fracton topological states Chamon (2005); Bravyi et al. (2011); Vijay et al. (2016); Ma et al. (2017); Vijay (2017); Hsieh and Halász (2017); Slagle and Kim (2017) which include Haah’s model Haah (2011). Unlike the non-liquid states, the gapped quantum liquids have point-like excitations and string-like excitations, that can move in all directions and have nontrivial braidings among them. But the statistics for point-like excitations alone is simpler than in 2+1D; they are bosons or fermions with trivial mutual statistics. In other words, the point-like excitations in a 3+1D topological order are described fully by a SFC.
If the point-like excitations are all bosons, the corresponding SFC will always have the form for some finite group . In other words, the point-like excitations of a 3+1D topological order can always be viewed as carrying irreducible representations of the group.111In this paper, we will also call those pointed-like excited states carrying reducible group representations as a (composite) point-like excitation; they in general have accidental degeneracy. This is unconventional in physics literature, but beneficial when we discuss the universal properties, such as fusion, of topological orders. The composite (accidentally degenerate) excitations also play an important role. They exactly behave like the quasiparticle excitations above a product state with symmetry. This is a quite amazing result: a 3+1D topological order whose quasiparticles are all bosonic is always related to a finite group .222This is true even for the general cases when some quasiparticles are fermions. We will refer to such (point-like-excitations-are-)all-boson topological order as AB topological order.
One may naturally wonder if a 3+1D AB topological order is always described by a -gauge theory, since the point-like excitations in a -gauge theory are indeed described by . In fact, the above statement is not true. There are 3+1D topological orders arising from the Dijkgraaf-Witten gauge theory Dijkgraaf and Witten (1990), whose point-like excitations are also described by . So we cannot say that all 3+1D topological order with point-like excitations are described by the usual -gauge theory. But, do we have something even more general than Dijkgraaf-Witten gauge theory that also produce point-like excitations? In this paper, we like to show that there is nothing more general:
[TABLE]
In this paper, “classified” always means a correspondence in a one-to-one fashion. Furthermore,
[TABLE]
The above result is obtained by condensing all the point-like excitations in a 3+1D topological order to form a new topological order (which is possible when all the point-like excitations are bosons), and argue that
The new phase is a trivial phase. Therefore, has a 2+1D gapped boundary induced by such condensation, which carries only string-like excitations. 2. 2.
The above string-like excitations on the boundary are labeled by the elements of a finite group , and their fusion rule is given by the group multiplication. It is the same group whose representations are carried by the point-like excitations in the bulk. 3. 3.
The string-only boundary form a unitary pointed fusion 2-category whose only nontrivial level are the objects. The different pointed fusion 2-categories are classified by a finite group and its 4-cocycle in , up to group automorphisms. 4. 4.
The bulk topological order is the center of the fusion 2-category : Kong and Wen (2014); Kong et al. (2017), which is a Dijkgraaf-Witten gauge theory with . Furthermore, each bulk topological order corresponds to a unique unitary pointed fusion 2-category .
In the following, we will discuss some general properties of 3+1D topological orders. Then we will show the main result of the paper following the above four steps.
II Excitations in topologically ordered state
II.1 Point-like excitations
II.1.1 Use trap Hamiltonian to define excitations
Consider a bosonic system defined by a local gapped Hamiltonian in dimensional space without boundary. A collection of quasiparticle excitations labeled by and located at can be produced as gapped ground states of where is non-zero only near . By choosing different ’s we can create (or trap) all kinds of point-like excitations. The gapped ground states of may have a degeneracy which depends on the quasiparticle types and the topology of the space . The degeneracy is not exact, but becomes exact in the large space and large particle separation limit. We will use to denote the space of the degenerate ground states, which will also be called fusion space. If the Hamiltonian is not gapped, we will say (i.e., has zero dimension). If is gapped, but if also creates quasiparticles away from ’s (indicated by the bump in the energy density away from ’s), we will also say . (In this case quasiparticles at ’s do not fuse to trivial quasiparticles.) So, if , only creates/traps quasiparticles at ’s.
For topologically ordered state with no spontaneous symmetry breaking, the fusion space on -dimensional sphere with no particles is always one dimensional. Thus in the presence of point-like excitations, dimension of the fusion space, represents the total number of internal degrees of freedom for the quasiparticles . To obtain the number of internal degrees of freedom for type- quasiparticle, we consider the dimension of the fusion space on type- particles on . In large limit has a form
[TABLE]
Here is called the quantum dimension of the type- particle, which describe the internal degrees of freedom the particle. For example, a spin-0 particle has a quantum dimension , while a spin-1 particle has a quantum dimension .
II.1.2 Simple type and composite type
Two excitations (trapped by ) and (trapped by ) are said to have the same type if the corresponding fusion spaces and can smoothly deform into each other as we change the trap Hamiltonian from to . Two excitations and are of the same type iff they only differ by some local operators. If an exaction can be created by local operators from the ground state, the excitation will be said to have a trivial type, and denoted as .
Even after quotient out the local excitations of trivial type, topological quasiparticle type still have two kinds: simple type and composite type: If the ground-state degenerate subspace cannot not be splitted by any small local perturbations near , then the particle is said to be simple. Otherwise, the particle type is said to be composite.
When is composite, the fusion space has a direct sum decomposition (after splitting by a generic perturbation of ):
[TABLE]
where , , , etc. are simple types. The above decomposition allows us to denote the composite type as
[TABLE]
II.1.3 Fusion of point-like excitations
When we fuse two simple types of topological particles and together, it may become a topological particle of a composite type:
[TABLE]
Here, we will use an integer tensor to describe the quasiparticle fusion, where label simple types:
[TABLE]
Such an integer tensor is referred as the fusion coefficients of the topological order, which is a universal property of the topologically ordered state.
The internal degrees of freedom (*i.e. *the quantum dimension ) for the type- simple particle can be calculated directly from . In fact is the largest eigenvalue of the matrix , whose elements are .
II.2 String-like excitations
Similarly, we can also use gapped trap Hamiltonians to define string-like excitations, where is no zero only near a loop. The ground state subspace of is called the fusion space of strings . If the fusion spaces and can smoothly deform into each other, we say the strings and are of the same type.
If the ground-state degenerate subspace cannot not be split by any small non-local perturbations along the string , then the string is said to be simple. Otherwise, the string is said to be composite. We stress that here we allow non-local perturbations along the string . In other word, any degrees of freedom near the string can interact no matter how far are them. But the interactions do not involve degrees of freedom far away from the string. The non-local perturbations is necessary. If we used local perturbations to define string types, we would have too many string types that are not related to the topological orders in the ground state.
The fusion of the string loops is also described by integer tensors
[TABLE]
The string loops can also shrink and become point-like excitations
[TABLE]
We like to conjecture that
[TABLE]
If a simple string satisfy , we say is a pure simple string.
II.3 The on-string excitations are always gappable
The strings, as 1D extended objects, may carry excitations that travel along them. Those excitations can some times be gapless. In the following, we like to argue that, if the types of point-like and string-like excitations are finite, those on-string excitations can always be gapped by adding proper interactions.
We do this by contradiction. Assuming that on-string excitations cannot be gapped by interactions, based on what we know about 1+1D system, there are only two situations.
When the on-string excitations are chiral with a non-zero chiral central charge . But in this case, when fusing string together, the new string will have on-string excitations with chiral central charge . This means that the fusion will produce infinite types of strings. The finite string-type assumption excludes the possibility of on-string excitations with a non-zero chiral central charge. 2. 2.
When the on-string excitations are described by the edge of certain fractional-quantum Hall states (which have chiral central charge ) Levin (2013). In this case, those on-string excitations have a gravitational anomaly described by a non-invertible 2+1D topological order Wen (2013); Kong and Wen (2014) In this case, the open membrane operator that creates such a string on its boundary must creates a non-invertible 2+1D topological order on the membrane. Multiplying membrane operators corresponds to stacking 2+1D topological orders together, and stacking non-invertible topological order can never produce a trivial topological order Kong and Wen (2014). Thus fusing string together will always produce a new non-trivial string. Again, finite string-type assumption exclude this possibility.
We like to remark that ungappable strings can appear in gapped non-liquid states Zeng and Wen (2015); Swingle and McGreevy (2016), such as the 3+1D gapped states obtained by stack fractional quantum Hall layers. But in that case, the strings are not mobile in all the directions. It appears that the liquid assumption of topological order Zeng and Wen (2015); Swingle and McGreevy (2016) makes all strings gappable in 3+1D. In the rest of this paper, we will always assume the on-string excitations to be gapped.
III Some general properties of
3+1D topological orders of boson systems
III.1 The group structure in 3+1D topological order
We note that the point-like excitations in 3+1D topological orders are described by a symmetric fusion category (SFC). Physically, a SFC is just a collection of particles which are all bosons or fermions with trivial mutual statistics.
Mathematically, it has been shown that a SFC must be either (a braided fusion category (BFC) formed by the representations of with all the irreducible representations being assigned Bose statistics) or (a braided fusion category (BFC) formed by the representations of with some of the irreducible representations being assigned Bose statistics while other irreducible representations being assigned Fermi statistics) for some group .
The above implies that
[TABLE]
In this paper, we will use this fact heavily to gain a systematic understanding of 3+1D topological orders. (In fact, each higher dimensional topological order is also related to a group in the same fashion.) In some sense, 3+1D topological orders can all be viewed as gauge theories with some old or new twists.
However, those point-like excitations have trivial mutual statistics among them. One cannot use the point-like excitations to detect other point-like excitations by remote operations. In general, we believe Levin (2013); Kong and Wen (2014)
[TABLE]
Here “anomaly-free” means realizable by a local bosonic model on lattice Wen (2013). Thus the remote detectability condition is also the anomaly-free condition.
The above implies that an anomaly-free (*i.e. *realizable) 3+1D topological order must contain string-like topological excitations, so that every point-like topological excitation can be detected by some string-like topological excitations via remote braiding, and every string-like topological excitation can be detected by some point-like and/or string-like topological excitations via remote braiding. We see that the properties of string-like topological excitations are determined by the point-like topological excitations (i.e. or ) to a certain degree.
III.2 Dimension reduction of topological orders
To understand better the relation between the point-like and the string-like excitations, we will introduce the dimension reduction in the next section, which turns out to be a very useful tool in our approach. We can reduce a D topological order on space-time to D topological orders on space-time by making the circle small (see Fig. 1). In this limit, the D topological order can be viewed as several D topological orders , which happen to have degenerate ground state energy. We denote such a dimensional reduction process by
[TABLE]
where is the number of sectors produced by the dimensional reduction.
For example, let us use to denote the D topological order described by the gauge theory with the finite gauge group . We find that, for (see Table 1) Moradi and Wen (2015),
[TABLE]
where sums over all different conjugacy classes of , and is a subgroup of formed by all the elements that commute with an element in . In fact, each dimension reduced D topological order, , is produced by threading a -gauge flux described by the conjugacy classes through the in the space-time . The -flux breaks the gauge symmetry down to . Thus the corresponding D topological order is a -gauge theory.
For Dijkgraaf-Witten theories (gauge theories twisted by group-cocycles of the gauge group), the dimension reduction have a form Wang and Wen (2015)
[TABLE]
where is a -group-cocycle , is a -group-cocycle , and is the topological order described by Dijkgraaf-Witten theory with gauge group and cocycle twist .
To understand the number of sectors in the dimension reduction, we note that the different sectors come from the different holonomy of moving point-like excitations around the (see Fig. 1). For gauge theory, this so called holonomy comes from the gauge flux going through the compactified . For more general topological orders, this holonomy comes from threading co-dimension 2 topological excitations through the .
From this picture, we see that the number of sectors is bounded by the number of types of the co-dimension 2 pure topological excitations. Also, if two co-dimension 2 topological excitations cannot be distinguished by their braiding with point-like excitations, then threading them through the will not produce different sectors. Thus, the number of sectors is the number of the classes of co-dimension 2 topological excitations that can be distinguished by the braiding with the point-like excitations. In particular, in 3+1D, the number of the sectors is the number of the classes of string-like topological excitations that can be distinguished by the braiding with the point-like excitations.
From the above discussion, we also see that the dimension reduction always contain a sector where we do not thread any nontrivial string through and the holonomy of moving any point-like excitations around the is trivial. Such a sector will be called the untwisted sector. Also, if a topological order has no nontrivial point-like excitation, then its dimension reduction contains only one sector – the untwisted sector.
In the untwisted sector, there are three kinds of anyons. The first kind of anyons correspond to the 3+1D point-like excitations. The second kind of anyons correspond to the 3+1D pure string-like excitations wrapping around the compactified . The third kind of anyons are bound states of the first two kinds (see Fig. 1).
We like to point out that the untwisted sector in the dimension reduction can even be realized directly in 3D space without compactification. Consider a 2D sub-manifold in the 3D space (see Fig. 2), and put the 3D point-like excitations on the 2D sub-manifold. We can have a loop of string across the 2D sub-manifold which can be viewed as an effective point-like excitation on the 2D sub-manifold. We can also have a bound state of the above two types of effective point-like excitations on the 2D sub-manifold. Those effective point-like excitations on the 2D sub-manifold can fuse and braid just like the anyons in 2+1D. The principle of remote detectability requires those effective point-like excitations to form a MTC. When we perform dimension reduction, the above MTC becomes the untwisted sector of the dimension reduced 2+1D topological order. We like to mention that the dimension reduction introduce new types of the perturbations that may not be local from 3+1D point of view. But those new perturbations are local in the dimension reduced 2+1D theory. MTC is very rigid which cannot be changed by any 2+1D local perturbations. This is why the untwisted sector is still described by the same MTC that describes the effective point-like excitations on the 2D sub-manifold.
This way, we show that
[TABLE]
Table 1 describes the dimension reduction of a 3+1D topological order described by -gauge theory:
[TABLE]
The 2+1D topological order is the untwisted sector. The three types of particles in 3+1D , , that form a SFC becomes three types of particles in the untwisted sector , , that also form a . We also see that in other sectors, the distinct point-like excitations in 3+1D may not be reduced to distinct simple objects in the dimension reduced 2+1D topological orders.
Since the dimension reduced 2+1D topological orders must be anomaly-free, they must be described by modular tensor category. Since the untwisted sector always contains , we conclude that
[TABLE]
In next section, we will show that such a modular extension must be a minimal one.
III.3 Untwisted sector of dimension reduction is the Drinfeld center
of
In the following we will show a stronger result, for the untwisted sector. Given a 3+1D bosonic topological order, let the symmetric fusion category formed by the point-like excitations be , or .
[TABLE]
Note that Drinfeld center is the minimal modular extension of .
First, let us recall the definition of Drinfeld center. The Drinfeld center of a fusion category , is a braided fusion category, whose objects are pairs , where is an object in , is a set of isomorphisms , satisfying natural conditions. is called a half braiding. Morphisms between the pairs is a subset of morphisms between , such that they commute with the half braidings . The fusion and braiding of pairs is given by
[TABLE]
In other words, to half-braid , one just half-braids and successively, and the braiding between pairs is nothing but the half braiding.
is the consequence that the strings in the untwisted sectors are in fact shrinkable. From the effective theory point of view, we can shrink a string (including bound states of particles with strings, in particular, point-like excitations viewed as bound states with the trivial string) to a point-like excitation in
[TABLE]
So if we only consider fusion, the particles in the dimension reduced untwisted sector can all be viewed as the particles in , regardless if they come from the 3D particles or 3D strings. In particular, the particles from the 3+1D strings can be viewed as composite particles in (see eqn. (19)). To obtain the Drinfeld center we need to introduce braiding among those particles ’s and ’s.
In the untwisted sector, the braiding between strings , denoted by , requires string moving through string , which prohibits shrinking string . However, there is no harm to consider the shrinking if we focus on only the initial and end states of the braiding process.
In particular, the braiding between a string and a particle , induces an isomorphism between the initial and end states where the string is shrunk (see Fig. 3)
[TABLE]
which is automatically a half-braiding on the particle . Thus, , by definition, is an object in the Drinfeld center .
Shrinking induces a functor
[TABLE]
which is obviously monoidal and braided, *i.e. *, preserves fusion and braiding. It is also fully faithful, namely bijective on the morphisms. Physically this means that the local operators on both sides are the same. On the left side, morphisms on a string are operators acting on near (local to) the string ; on the right side, morphisms in the Drinfeld center are morphisms on the particle which commute with the half braiding . From the shrinking picture, morphisms on can be viewed as the operators acting on both near the string and the interior of the string (namely on a disk ). But in order to commute with for all , which can be represented by string operators for all going through the interior of the string (this includes all possible string operators, because string operators for all particles form a basis), we can take only the operators that act trivially on the interior of the string. Therefore, morphisms on the right side are also operators acting on only near the string. This establishes that the functor is fully faithful, thus a braided monoidal embedding functor; in other words, can be viewed as a full sub-MTC of . However, is already a minimal modular extension of , which implies that
[TABLE]
As is known well, many properties can be easily extracted. For example, objects in have the form , where is a conjugacy class, is a representation of the subgroup that centralizes . One then concludes
[TABLE]
We also see that a 3+1D bosonic topological order is similar to a gauge theory of a finite group . The following properties are the same:
- the quantum dimensions of point-like and string-like
excitations.
-
the fusion rule of those excitatios.
-
particle-loop and two-loop braidings.
IV Condensing all the point-like excitations to obtain a trivial topological order
Starting from this section, we are going to show the main result of the paper via the four steps outlined at the end of introduction. First, we like to show that condensing all the point-like excitations in 3+1D always gives us a trivial topological order. To do so, we first like to show the following:
IV.1 There is no 3+1D topological order with only
nontrivial string-like excitations
Such a result can be shown using the principle of remote detectability in Section III.1. When there is no nontrivial point-like excitations, the remote detectability condition requires that a single loop of string can be remotely detected by braiding other string-like excitations around the loop. Such a braiding is the two-string braiding described by Fig. 4a where a string is braided around a loop .
We can also use the dimension reduction picture to show that the anomaly-free condition requires that the two-string braiding must be nontrivial. Since there is no nontrivial point-like excitations, the dimension reduction contains only the untwisted sector. In the 2+1D dimension reduced topological order, all the nontrivial anyons come from the pure strings in the 3+1D topological order, and correspond to the pure strings wrapping around the compactified . The 2+1D topological order is anomaly-free and the anyons form a modular tensor category. Physically, it means that any nontrivial anyon must have nontrivial mutual statistics with some anyons. This implies that any nontrivial pure strings in 3+1D must have nontrivial two-string braiding with some strings.
Next we like to show that the two-string braiding is always trivial when there is no nontrivial point-like excitations. This is because the braiding path of -string around a -string in Fig. 4a is a torus wrapped around the loop . Such a torus can be deformed into a sphere around with a thin tube going through its center (see Fig. 4b). If the total space is a 3-sphere , we can deform the sphere into a small sphere on the other side of . This deforms the braiding path of into a thin torus, that describes a small string braiding around the loop (see Fig. 4c). This is like shrinking the string into a point and let the point braid around the loop , Since there is no nontrivial point-like excitations, the point that represents the small must have the trivial braiding phase around the loop . This way, we show that the two strings must have trivial braiding around each other when there is no nontrivial point-like excitations. Therefore,
[TABLE]
IV.2 There is no nontrivial string-like excitations that have trivial
braiding with all point-like excitations
Let us assume that there is a nontrivial string-like excitation , that has trivial braiding with all point-like excitations. If all the point-like excitations are bosons, then we can condense all the point-like excitations to obtain a new 3+1D topological order, which will have no nontrivial point-like excitations. But since the string has trivial braiding with all point-like excitations, it can survive the condensation and become a nontrivial string-like excitation in the new 3+1D topological order.
However, in the last section, we have shown that 3+1D topological orders with only string-like excitations cannot exist. This contradiction implies that
[TABLE]
This result also implies that
[TABLE]
This is because, in the untwisted sector, other anyons beside all come from strings in 3+1D, which all have nontrivial braiding with the particles in . This implies the modular extension to be minimal. However, the above result is weaker than that obtained in Section. III.3.
IV.3 Condensing all the point-like excitations gives rise to a trivial
3+1D topological order
When all the point-like excitations are bosons, we can obtain a new topological order by condensing all the point-like excitations. The new topological order has no nontrivial point-like excitations (since they are all condensed) and has no nontrivial string-like excitations (they are confined due to the nontrivial braiding with the point-like excitations). Thus the new topological order must be an invertible topological order . But in 3+1D all the invertible topological orders are the trivial one Kapustin (2014); Kong and Wen (2014); Freed (2014). Hence condensing all the point-like excitations gives us a trivial 3+1D topological order.
In a gauge theory, condensing all the point-like excitations corresponds to condensing all the charged excitations, which breaks all the “gauge symmetry”. This will give us an Anderson-Higgs phase, which is a trivial phase with no topological order.
V String-only boundary of 3+1D topological order
In this section, we are going to study a particular boundary of 3+1D topological orders. We note that there is no gravitational Chern-Simons term in 3+1D. Thus all 3+1D bosonic topological orders can have a gapped boundary Kong and Wen (2014). We call such a gapped boundary an anomalous 2+1D topological order.
From the last section, we see that all 3+1D AB topological orders (where all point-like excitations are bosons) can have a gapped boundary obtained by condensing all the point-like excitations. For such a boundary, the anomalous 2+1D topological order on the boundary has no point-like excitations, and has only string-like excitations. We will call such a boundary string-only boundary. Thus
[TABLE]
V.1 Unitary pointed fusion 2-category
We will show that the string-only gapped boundary is described by a so called unitary pointed fusion 2-category. But what is a fusion 2-categories? In general, a fusion category describes the fusion of codimension-1 excitations, *i.e. *domain-wall excitations. In 1-dimensional space, the domain-wall excitations are point-like. The fusion of those point-like excitations in 1D space is described by a fusion 1-category (which is also called fusion category). In 2-dimensional space, the domain-wall excitations are string-like. The fusion of those string-like excitations in 2D space is described by a fusion 2-category.
We like to point out that the fusion 2-categories that describes the string-only boundary of 3+1D AB topological order are very special: (1) the string-like excitations on the boundary are labeled by the group elements of : . (2) The fusion of the boundary string-like excitations (see Fig. 5) is very simple and is given by the group multiplication
[TABLE]
The fusion 2-categories with the above type of fusion rule are called pointed fusion 2-categories. Such an amazing result is a consequence of condensing all the point-like excitations described by on the boundary.
One way to show the above result is to consider the untwisted sector of the dimension reduction, which is a 2+1D topological order. We have shown that the untwisted sector is a minimal modular extension of in Sections III.3 and IV.2. The 2+1D boundary with only strings corresponds to a 1+1D boundary with only particles of the untwisted sector in the dimension reduced 2+1D topological order. Such a 1+1D boundary is obtained by condensing all the anyons in . The corresponding mathematical problem has already been solved, see for example LABEL:DGNO09; we reorganized the related mathematical results and provided physical interpretations in LABEL:LKW1602.05946 (Section VI D). We find that the particles on such 1+1D boundary of the untwisted sector are labeled by group elements in with a fusion given by group multiplication. Those 1+1D boundary particles correspond to the strings on the 2+1D boundary (see Fig. 1), this allows us to show eqn. (23). In the next a few sections, we will give a different argument without using dimension reduction.
V.2 Tannaka Duality in more explicit language
Our argument relies heavily on the Tannaka duality, or Tannaka reconstruction theorem for group representations. It is exactly how we extract the group from an abstract symmetric fusion category (SFC). A naive example is that for an abelian group, the tensor product of its irreducible representations, has exactly the same group structure, which can be viewed as a Fourier transformation.
In more general cases, one can reconstruct a group from its representation category , by the automorphisms of a fiber functor, namely a functor from to the category of vector spaces , that preserves the fusion and braiding. We know that the category of vector spaces describes particles in a trivial phase (*i.e. *in a product state with no symmetry). So one way to physically realize a fiber functor is by condensing (or other ways such as symmetry breaking) a nontrivial phase to a trivial phase. With a fiber functor , we have
[TABLE]
To understand the physical significance of the above amazing result, let us consider a physical problem: given a system with a symmetry whose ground state is a product state with the symmetry, if we only measure the system via probes that do not break the symmetry, can we determine the symmetry group of the system? Here symmetric probes correspond to operators that commute with all group actions, . Generic group actions are not symmetric probes, unless they are in the center of the group. On the other hand, the fusion and braiding of the point-like excitations above the ground state correspond to symmetric operation. The representation category contains only those symmetric probes. Tannaka duality tells us that we can indeed determine the symmetry group via only symmetric probes. Although the fiber functor seems to break the symmetry if we realize it physically, mathematically it is proven that such fiber functor always exists and is unique up to isomorphisms. Therefore, from the data of symmetric probes (fusion and braiding) in , we can obtain (formally calculate) the group up to isomorphisms, without really breaking the symmetry of the system.
Now let us try to break the abstract theorem into more explicit terms. Firstly, the fiber functor means nothing but realizing the abstract fusion and braiding in category with the tensor product and (trivial) braiding of concrete Hilbert spaces in a quantum system. It is helpful to consider how we build in : a group representation is a vector space equipped with a group action . Moreover, there is a monoidal structure for the representations, which is taking the tensor product of the vector spaces and the new group action is (which is called the fusion of group representations).
The Tannaka duality goes exactly the other direction. Assuming that we know a representation category , which contains only information on symmetric operations such as how the representations fuse with each other, can we obtain the group actions and also the group? The answer of the theorem then goes:
If we have a collection of invertible linear maps for each irreducible representation , acting on the vector space assigned to by a fiber functor , such that 2. 2.
They are compatible with the fusion, in the sense that ,
[TABLE]
It is possible that is a reducible representation. We extend the linear maps to being reducible representations by direct sums, *i.e. *if is the direct sum of irreducible representations , , is given by the corresponding direct sum .
This collection of invertible linear maps , must correspond to the action of some group element , .
Moreover, take all collections of such invertible linear maps, they form a group under composition, namely the automorphism group of the fiber functor, . It is isomorphic to . In other words, if at the beginning we are given an abstract bosonic SFC , with a fiber functor , we can use the above reconstruction to extract the group underlying , via .
V.3 Fusion of boundary strings recover the group
Let us focus on the loop excitations on the string-only boundary. A loop excitation shrunk to a point may become a direct sum of point-like excitations (see eqn. (A.4))
[TABLE]
where is the trivial point-like excitation and represent other possible nontrivial point-like excitations. When , the string is not pure. Another possibility is that . In this case the string is unstable; it has accidental degeneracy which can be lifted by perturbations. So the pure simple strings have .
Since there is only trivial particle on the boundary, when we shrink a loop on the boundary, it must become a multiple of the trivial particle, . Thus, it suffices to consider only the simple loops () on the boundary, which shrink to the trivial particle . In other words, simple loops on the boundary shrinks to nothing; this is an essential property in the following discussions. We note that such simple loops have a quantum dimension , and their fusion is group-like. For the moment, we denote the group formed by the simple loops on the boundary under fusion (see Fig. 5), by .
To apply the Tannaka duality, we need a physical realization of the fiber functor. Consider a simple topology for a string-only boundary: put the 3+1D topological order in a 3-disk , the boundary on , and outside is the trivial phase . When there is only a particle in the 3-disk, with no string and no other particles, we associate the corresponding fusion space (the physical states with such a configuration) to the particle , and denote this fusion space by (see Fig. 6). Viewed from very far away, a 3-disk containing a particle is like a “local excitation” in the trivial phase, thus mimics a local Hilbert space. When there are two 3-disks, each containing only one particle, and respectively, the fusion space is . Moreover, as adiabatically deforming the system will not change the fusion space, we can “merge” the two 3-disks to obtain one 3-disk containing one particle . Therefore . Similarly, also preserves the braiding of particles. In other words, the assignment gives rise to a fiber functor. By Tannaka duality, we can, at least formally, reconstruct a group , such that the particles in the bulk are identified with . Our goal is to show that the fusion group of the simple loops on the boundary, is the same as .
To do this we consider the process of adiabatically moving a particle around a simple loop on the boundary, as shown in Fig. 7. As the simple loop shrinks to nothing, inserting simple loops will not change the fusion space. But an initial state , after such an adiabatically moving process, can evolve into a different end state . Thus, braiding around induces an invertible (since we can always move backwards) linear map on the fusion space , .
Next, consider that we have two particles in the bulk. If we braid them together (fusing them to one particle ) around the simple loop , we obtain the linear map . If the fusion of the bulk particles is given by , we can split to the irreducible representations , and braid with ; in other words,
But it is also equivalent if we move one after the other. More precisely, we can first separate into another 3-disk, braid with , and then merge back to the original 3-disk. Thus moving alone corresponds to the linear map . Similarly, moving alone corresponds to and in total we have the linear map . Therefore, , or using only irreducible representations,
[TABLE]
These linear maps are compatible with the fusion of bulk particles.
Moreover, the simple loop provides such an invertible linear map for each particle in , thus by Tannaka duality, these linear maps must correspond to the action of certain group element , . In other words, we obtain a map from the simple loops to , . It is compatible with the fusion of simple loops, because the path of braiding around two concentric simple loops, (as in Fig. 5), separately, can be continuously deform to the braiding path around the two loops together, or around their fusion . This implies that , namely, is a group homomorphism.
What we really want is that is an isomorphism and . This is a consequence of the remote detectability condition. Before proving it, we explain in detail the principle of remote detectability near the string-only boundary. The general idea is the same, that everything must be detectable remotely. Near a string-only boundary, the only way to perform remote detection is the half-braiding between bulk particles and boundary strings Levin (2013); Kong and Wen (2014). Therefore,
[TABLE]
One may have doubts in (2): even if bulk particles can not be detected by boundary strings, we may still have bulk strings to detect them. The reason for (2) is that we believe generalized boundary-bulk duality, that the bulk strings can always be viewed as certain “lift” of boundary strings to the bulk.Kong and Wen (2014); Kong et al. (2017) If a bulk particle has trivial half-braiding with all boundary strings, it also has trivial braiding with all the “lift” of boundary strings, *i.e. *all the bulk strings, which conflicts with the remote detectability condition in the bulk.
A typical half-braiding path is shown in Fig. 7. It is important to note that the (non-Abelian) geometric phase depends on the half-braiding path; however, we can extract a universal path-independent half-braiding invariant, by complementing the half-braiding path into a full loop with another half loop of path in the trivial phase outside the boundary. Different half loop of path in the trivial phase with the same starting and end points on the boundary always contribute the same geometric phase (because closed paths in the trivial phase has no geometric phase). This way we obtain the half-braiding invariant as the expectation value of such whole loop adiabatically moving process (half in the bulk, half in the trivial phase).333It is similar in 2+1D topological orders. The topological -matrix is the invariant of braiding, but it is the expectation value of double braidings, *i.e. *moving one particle a whole loop around another. Trivial half-braiding means that such half-braiding invariant is trivial. Immediately we see that the linear maps are directly related to the half-braidings, in the sense that gives the above half-braiding invariant. If is the identity map, it implies trivial half-braiding between and .
Now, we are ready to show that is an isomorphism:
is injective. Consider , namely the simple loops that induce just identity linear maps on all bulk particles. In other words, consists of simple loops that have trivial half-braiding with all bulk particles. By the remote detectability condition (1), must be trivial, which means is injective. 2. 2.
is surjective. We already showed that is injective, so we can view as a subgroup of .
Now consider a special particle in the bulk, which carries the representation , linear functions on the right cosets . More precisely, consists of all linear functions on , , such that (takes the same value on a coset). The group action is the usual one on functions, .
The linear maps induced by the simple loops are all actions of group elements in , and they are all identity maps on the special particle . In other words, the bulk particle has trivial half-braiding with all the boundary strings. By the remote detectability condition (2), it must be the trivial particle carrying the trivial representation. In other words, we have .
To conclude, the simple loop excitations on the string-only boundary, forms a group under fusion. It is exactly the same group whose representations are carried by the point-like excitations in the bulk.
If we insert a bulk loop excitation in the 3-disk and perform a similar braiding process, it also induces a linear map on the underlining fusion space. One may wonder if this also associates group elements to the bulk strings. This is in general not true. Unlike the boundary simple loops, inserting a bulk string, even if it is pure and simple, will enlarge the fusion space of only particles, as long as the quantum dimension of such string is greater than 1. As a result, only those bulk strings with quantum dimension can be associated with group elements. In section III.3 we have shown that all the bulk strings can be associated with conjugacy classes of the group (even if topological order is not a -gauge theory). Some further discussions can be found in Appendix C.
VI The classification of unitary pointed fusion 2-categories
VI.1 A mathematical formulation
First, let us consider the so-called unitary pointed fusion (1-)categories. A pointed fusion category consists of a finite number of simple objects. A simple object is an object such that Hom. For each simple object , there is a simple object such that , where is the tensor unit, i.e. . In other words, the set of simple objects form a finite group . We will also denote the simple object by , , , etc.
In this case, the only not-yet-fixed structure is the associator isomorphism:
[TABLE]
Note that both domain and target are the same simple object. But the associator isomorphisms can be nontrivial. By the simpleness, the isomorphism is just a non-zero complex number . If the theory is unitary, one needs to further require that this number is a phase (in ). Then the pentagon condition implies that is a 3-cocycle. Different cocycles give equivalent fusion categories if they differ by a 3-coboundary. Moreover, we may permute the simple objects by group automorphisms, thus two different cocycles also give equivalent fusion categories if they got mapped to each other by group automorphisms. In conclusion, unitary pointed fusion categories, are classified by where up to group automorphisms.
Now let us consider a pointed unitary fusion 2-category Gurski (2007). We will not define it in full detail here, but only describe some physically relevant ingredients of it. It has only finite number of simple objects. A generic object is a direct sum of simple objects. For two simple objects , we have and for , where [math] is the category consisting of only the [math]-vector space. The tensor unit is simple. For each simple object , there is a simple object such that , where is the tensor unit. So, again the set of simple objects is a finite group . We will denote simple objects by group elements , etc.
For a simple object , the identity 1-morphism is (the only invertible object in ). The 2-morphisms form Hom, and there are unit 1-isomorphisms and associator 1-isomorphisms: (1) The unit 1-isomorphisms: is just the identity morphism . (2) The associator 1-isomorphism:
[TABLE]
is also still the identity 1-morphism . There are two ways to go from to via identity 1-morphisms. Therefore, two paths both give the identity map . So the commutative of the pentagon diagram is clear. But we can introduce for each pentagon a 2-isomorphism: , which is a phase, denoted by . These 2-isomorphisms need satisfy a higher coherence relation. Then this coherence relation implies that is a 4-cocycle. Again, 4-cocycles differing only by a coboundary give equivalent pointed fusion 2-categories. One can do the same for the triangle relation. Namely, one can introduce a 2-isomorphism for each triangle. We believe that these 2-isomorphisms should give the same unitary fusion 2-category.
The same structure is discussed in LABEL:EN170202148, under a different name, -graded 2-vector spaces , where they also believe that is enough to determine a unitary pointed fusion 2-category.
Note that the equivalence between unitary pointed fusion 2-categories must preserve the tensor product of simple objects, thus must correspond to some group automorphism . Such automorphism also acts on the cocycles (necessarily change the cocycle if it is an outer automorphism, namely, not of the form for some ). Under such automorphism , and , where is a -cocycle, correspond to the same pointed unitary fusion -category. Therefore, we believe that pointed unitary fusion 2-categories one-to-one correspond to the pairs where , up to group automorphisms.
VI.2 A physical argument
In the following, we will try to understand the above mathematical result from a physical point of view. Let the 3-dimensional space to be a 3-disk . Consider the boundary strings and on the surface of the 3-disk (see Fig. 8a). The process for the boundary strings to fuse to a non-string state can be represented by a membrane-net in (see also LABEL:EN170202148). The same boundary strings can fuse to a non-string state through a different process which is represented by another membrane-net in . To compare the two processes, we can glue the boundary of the above two membrane together along the , to form a membrane-net in . Such a membrane-net in describe the process of creating boundary strings from a no-string state, and then fuse those boundary strings to no-string state. Such a process induce a geometric phase , since the fusion space of the boundary strings is always 1-dimensional. So we assign such a geometric phase to the membrane-net on .
But such a geometric phase may not have a local expression. Let us assume that the membrane-net on is formed by the 2-simplices of a triangulation of . The vertices of the triangulation are labeled by . “Having no local expression” means that we cannot assign a phase factor to each 3-simplex of the triangulation to express the total geometric phase as a product of those local phases:
[TABLE]
We see that the process of creating some boundary strings from nothing and then fusing them to nothing can be represented by a membrane-net on space . Such a process correspond to a phase factor . Two different processes of creating some boundary strings from nothing and then fusing them to nothing give rise to two phase factors and . The two processes can be compared by a “time”-evolution from the membrane-net on that correspond to the first process, to the membrane-net on that correspond to the second process. In other words, the comparison of the two processes is represented by a 3-brane-net on , where is the space and the segment represents the “time” direction (see Fig. 9). The first process corresponds to the membrane-net on one boundary of which is one boundary of the 3-brane-net on . The second process corresponds to the membrane-net on the other boundary of which is the other boundary of the 3-brane-net on .
In 4-dimensions, a 3-brane-net is dual to a string-net where each 3-brane in the 3-brane-net intersects with a string in the string-net (see Fig. 9 and 8). So the strings in the string-net is also labeled by . In the 3-brane-net, only the 3-branes that satisfy the fusion rule eqn. (23) can intersect along a line (see Fig. 9 and 8b). This means that the labels of the strings in the string-net satisfies
[TABLE]
[Note the same string with opposite orientations is labeled by and respectively. The orientation of strings in the string-net is chosen to from a branching structure (see Appendix E) of the string-net .] The above happen to be the flat connection condition if we view on a string as the gauge connect between the two vertices connected by the string. So the evolution from one process to the other can be represented by a string-net on .
The two different processes may differ by a phase factor . So we can assign the string-net on by such a phase factor, to represent the difference of the two processes. Let us assume the string-net on is formed by the edges of a triangulation of . Then starting from one boundary of , we can build the whole triangulation of by adding one pentachoron (*i.e. *one 4-simplex) at a time. We note that adding a pentachoron corresponds to change one process to its neighboring process. The difference of the two neighboring processes is described by the added pentachoron with edges labeled by , (where label the five vertices of the pentachoron). We may assign the phase difference of the two neighboring processes to the added pentachoron. So each pentachoron is assigned to a phase factor . Due to the flat connection condition eqn. (35), ’s for the pentachoron are not independent. So can be rewritten as . Such a 4-variable function on can be viewed as a group 4-cochain.
So the total phase difference of the two processes can be written as
[TABLE]
where multiply over all the pentachorons in the triangulation of , and describes the two different orientations of the pentachorons which arises from the branching structure (see Appendix E).
If we choose the two processes described by the boundary of to be the do nothing process that the leave the no-string state unchanged, then the string-net on can be viewed as a string-net on . The total phase difference of the two do-nothing processes (which is actually the same process) should be zero:
[TABLE]
and the above should hold for any triangulation of and any assignment of the label on the edges (as long as the flat connection condition eqn. (35) is satisfied). It implies that is a group 4-cocycle in . This is a physical way to explain why
[TABLE]
VII From boundary to bulk
We have shown that all 3+1D AB bosonic topological orders can have a boundary described by pointed unitary fusion 2-category whose fusion is given by the group . It is believed the boundary anomalous topological order completely determine the bulk topological order Kong and Wen (2014); Kong et al. (2017). More precisely, the bulk topological order should be given by the center , which can be explicitly defined by the 2-category of --bimodule 2-functors.
But their relation can be many-to-one: several different anomalous boundary topological orders may correspond to the same bulk topological order; in other words, the same bulk topological order can have several different gapped boundaries. Since 3+1D topological orders always have gapped boundaries, all 3+1D topological orders are determined by some anomalous 2+1D boundary topological orders. Mathematically, we say that there is surjective map from the set of anomalous 2+1D boundary topological orders to the set of 3+1D topological orders
[TABLE]
Furthermore, since all 3+1D AB topological orders have a string-only boundary described by unitary pointed fusion 2-categories, we further have
[TABLE]
In this paper the string-only boundary is obtained by condensing all the point-like particles that form . A natural question is whether such condensation process, and also the resulting boundary, are unique or not.
Firstly, we believe that the condensation of particles in 3+1D follows the same rule as that for condensation of anyons in 2+1D (at least if we restrict the 3+1D condensation to a 2+1D sub-manifold, see Fig. 2). Anyon condensation in 2+1D has been thoroughly studied. It is fully controlled by the so-called condensable algebraKong (2014) in the category of anyons. In other words, the condensable algebra completely determines the condensed phase and the domain wall/boundary between the old phase and the condensed phase.
Thus, we should focus on the condensable algebras in . They are already classified in Theorem 2.2 in LABEL:KO01 (see also Theorem 3.7 in LABEL:LKW1602.05936). There is a unique condensable algebra that condenses all the particles in . It is given by , the algebra of all functions on . Therefore, there is only one way to condense all particles . We obtain a unique condensed phase, which is trivial. As result, there is a unique, also canonical, string-only boundary.
This way, we got an even stronger result. Each 3+1D AB topological order only have a unique boundary that corresponds to the condensation of all point-like excitations. In other words, each 3+1D topological order corresponds to a unique unitary pointed fusion 2-category.
[TABLE]
Such a result is similar to a result in one lower dimension:
[TABLE]
Let us briefly explain why the approach used in this paper for 3+1D topological orders does not apply in 2+1D, which gives a flavour why we can obtain a stronger result in 3+1D. In 3+1D, all point-like excitations have trivial statistics; if they are all bosons, it is a natural and canonical choice to condense all of them and we obtain a unique string-only 2+1D gapped boundary. In 2+1D, there are only point-like excitations with non-trivial statistics between them. One can similar choose a subset of quasiparticles to condense; if the subset is big enough one can also obtain a gapped 1+1D boundary. However, in general there are several such subsets to condense, among which none is special. As a result, there is no canonical gapped 1+1D boundary. This essential difference makes the classification of topological orders in 3+1D simpler than those in 2+1D.
VIII Realization by Dijkgraaf-Witten models
Combining the results from the last a few sections, we obtain that
[TABLE]
A finite group and its group 4-cocycle happen to be the data needed to construct the Dijkgraaf-Witten model. In fact all the 3+1D AB topological orders can be realized by Dijkgraaf-Witten models.
We note that 3+1D Dijkgraaf-Witten models Dijkgraaf and Witten (1990) are defined on a 4-dimensional simplicial complex with branching structure (see Appendix E). Let us use to label the vertices of the complex. The degrees of freedoms live on the links of the complex, which are labeled by where is a finite group. ’s satisfies a flat-connection condition
[TABLE]
for any triangles . The Dijkgraaf-Witten models are defined via a path integral
[TABLE]
where multiply over all the 4-cells whose vertices are ordered as . Also, describes the orientation of the 4-cell (see Appendix E), and is a group 4-cocycle .
When the space-time has a boundary, we can obtain an exactly soluble boundary by setting on all the links on the boundary. Such an exactly soluble boundary is actually the string-only boundary discussed in this paper. The world-lines of topological point-like excitations are described by Wilson lines in the bulk
[TABLE]
where is a representation of . But on the boundary, and is an identity matrix. All the different topological point-like excitations becomes the same trivial excitation on the boundary. However, there are non-trivial string-like excitations on the boundary. The world-sheet of those boundary string-like excitations is given by the following: Draw a membrane on the 3-dimension boundary of space-time. Change on the links that intersect the membrane from to . Such a change still satisfy the flat-connection condition. We see that different boundary strings are labeled by the group elements and their fusion is given by the group multiplication. Therefore, Dijkgraaf-Witten models can realize all unitary pointed fusion 2-category on the boundary. Using the boundary-bulk relation Kong and Wen (2014); Kong et al. (2017), we can show that Dijkgraaf-Witten models can realize all 3+1D AB topological orders.
IX Relation to 3+1D bosonic SPT orders
There are two kinds of SPT orders when the symmetry group is unitary and finite: the ones whose boundary have a pure gauge-anomaly will be called pure SPT orders Chen et al. (2013, 2012), and the ones whose boundary have a mixed gauge-gravity-anomaly will be called mixed SPT orders Wen (2015). In 3+1D space-time, the pure SPT orders are classified by group cohomology , while all the mixed SPT orders are described by some elements in Wen (2015)
[TABLE]
For many groups, . But a non zero does not implies the existence of mixed SPT, since not all the elements in correspond to existing SPT orders.
Since 3+1D AB topological orders can be obtained by gauging Levin and Gu (2012) the symmetry of 3+1D bosonic SPT states, and since Dijkgraaf-Witten models only correspond to gauging the pure SPT states, we see that the classification results in this paper implies that
[TABLE]
In fact, using SPT invariant, we can directly show that for unitary finite symmetry group , there is no mixed SPT orders in 3+1D. (However, if contains time reversal, there are mixed SPT orders in 3+1D Vishwanath and Senthil (2013); Kapustin (2014).) To obtain SPT invariant, we gauge the symmetry and put a flat-connection on a closed orientable space-time . The partition function of the system on with a fixed flat-connection is the so called SPT invariant Wen (2014); Hung and Wen (2014); Wang et al. (2015). If there is a mixed 3+1D SPT order described by , its SPT invariant will have a form
[TABLE]
where are topological -cocycles in , and is also the Stiefel-Whitney class. There are many relations between Stiefel-Whitney classes and cocycles . For example, by calculating in two different ways, we find that on orientable , Wen (2015, 2017). (Here is the Steenrod operation.) Thus
[TABLE]
Similarly, . Thus
[TABLE]
We see that the SPT order described by the above SPT invariant is actually a pure SPT order described by Kapustin (2014); Wen (2015) and hence, there is no mixed SPT order in 3+1D for unitary finite symmetry group. This result supports our classification of 3+1D AB topological orders in terms of Dijkgraaf-Witten models. In 4+1D, there is a mixed bosonic SPT state Wen (2015). Gauging such a mixed SPT state will produce a 4+1D AB topological order that is beyond Dijkgraaf-Witten theory.
X Walker-Wang models and particle-only boundaries
We like to remark that Walker-Wang models Walker and Wang (2011); von Keyserlingk et al. (2013); Williamson and Wang (2016); Chen et al. (2014) is another quite systematic way to construct 3+1D bosonic topological orders. In fact, Walker-Wang models realize all 3+1D bosonic topological orders who have a particle-only boundary, which is described by a premodular tensor category. Such particle-only boundary can exist for a 3+1D topological order, if condensing the maximum set of strings that have trivial mutual braiding will change the 3+1D topological order to a trivial phase.
It is known that Walker-Wang models (and the related 3+1D string-net models Levin and Wen (2005)) can realize 3+1D bosonic topological orders with emergent fermionic point-like excitations. von Keyserlingk et al. (2013) It appears that Walker-Wang models cannot realize all 3+1D Dijkgraaf-Witten models (*i.e. *not all 3+1D bosonic topological orders whose point-like excitations are all bosons).
XI Summary
3+1D topological orders contain both point-like and string-like excitations. At first, it appears that 3+1D topological orders, with all the fusion and braiding of those point-like and string-like excitations, have a very complicated structure, which may be hard to classify. However, in this paper, we obtain a very simple classification of 3+1D topological orders for bosonic systems, when all the point-like excitations are bosons: they are classified by unitary pointed fusion 2-categories, which in turn are classified by pairs up to group automorphisms. This gives us hope that the 3+1D topological orders may not be that complicated. We may get a simple classification even for the general case when some point-like excitations are emergent fermions. We hope that the arguments developed in this paper are helpful for such a task, which we plan to carry out in a forthcoming work.
We like to thank Meng Cheng, Zhenghan Wang, and Edward Witten for helpful discussions. XGW is supported by NSF Grant No. DMR-1506475 and NSFC 11274192. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research.
Appendix A An example: 3+1D -gauge theory
To gain an intuitive understanding of 3+1D topological orders and to introduce the related concepts, let us study an exactly soluble local bosonic model whose ground state has a topological order described by a 3+1D gauge theory of a finite group . Our lattice bosonic model is defined on a 3D spatial lattice whose sites are labeled by . The degrees of freedom live on the links labeled by . On an oriented link , such degrees of freedom are labeled by . ’s on links with opposite orientations satisfy
[TABLE]
The Hamiltonian of the exactly soluble model is expressed in terms of string operators and membrane operators.
A.1 The string operators
The string operators are labeled by ’s, the irreducible representations of the gauge group (where is the matrix of the irreducible representation):
[TABLE]
We note that
[TABLE]
We use to denote the usual tensor product of matrices or vector spaces over the complex numbers , while to denote the fusion of excitations. Using
[TABLE]
we see that
[TABLE]
The ends of the strings are point-like topological excitations and the above are the fusion coefficients of those topological excitations. Let be the quantum dimension of those topological excitations which satisfy
[TABLE]
and let
[TABLE]
We have
[TABLE]
Thus, is a projection operator. In fact, it is a projection operator into the subspace with .
A.2 The membrane operators
A membrane is formed by the faces of the dual lattice, which is also a cubic lattice. The faces of the dual lattice correspond to the links in the original lattice and are also labeled by .
A membrane operator is given by
[TABLE]
where the operator is defined as
[TABLE]
and is the conjugacy class of . Also ’s are on one side of the membrane and ’s are on the other side of the membrane,
We note that
[TABLE]
The above expression allows us to see that are non-negative integers. Clearly and , which imply that
[TABLE]
Let , and we can rewrite the second equation in the above as
[TABLE]
For example, the permutation group of three elements has three conjugacy classes: , , and . We find that
[TABLE]
Let be a common eigenvector of whose components are all non-negative. (Such common eigenvector exists since the matrix elements of are all non-negative.) The eigenvalue of such a eigenvector is for . We choose the scaling factor of to satisfy
[TABLE]
In this case
[TABLE]
A.3 A commuting-projector Hamiltonian
Let be the smallest membrane operator that creates a small membrane corresponding to the surface of a cube in the dual lattice. Such a membrane wraps a site in the original cubic lattice. We note that is a sum of gauge transformation operators . Since the string operators are gauge invariant, we have
[TABLE]
Therefore, we can construct the following commuting projector Hamiltonian Kitaev (2003); Mesaros and Ran (2013)
[TABLE]
where
[TABLE]
and labels the loops around the squares of the original cubic lattice.
The ground state of the above exactly soluble Hamiltonian has a nontrivial topological order. The low energy effective theory is the -gauge theory.
A.4 The point-like and string-like excitations
What are the excitations for the above Hamiltonian? There are local point-like excitations created by local operators. There are also topological point-like excitations that cannot be created by local operators. Two topological point-like excitations are said to be equivalent if they differ by local point-like excitations. The equivalent topological point-like excitations are said to have the same type.
The different types of topological point-like excitations are created at the ends of the open string operators that we discussed before. Thus we see that types of topological point-like excitations one-to-one correspond to the irreducible representations of . In other words, topological point-like excitations are described by in a -gauge theory.
Similarly, there are also topological string-like excitations. They are created at the boundary of the open membrane operators. However, the types of membrane operators are not one-to-one correspond to the types of string-like excitations. There are pure string-like excitations which one-to-one correspond to the conjugacy classes of . There are also mixed string-like excitations which are bound state of pure string-like excitations and point-like excitations Kong and Wen (2014); Wang and Levin (2014); Else and Nayak (2017). In general, the types (pure and mixed) of string-like excitations in a -gauge theory are labeled by a pair , where is a conjugacy class of , is a representation of , and is a subgroup of whose elements all commute with a fixed element in (a centralizer subgroup).
For example, in -gauge theory, the -flux-loop breaks the gauge “symmetry” down to gauge symmetry. So there are two types of -flux-loop excitations, one carries no charge (which is the pure one denoted by ) and the other carries charge 1 (denoted by ). Similarly, the -flux-loop breaks the gauge “symmetry” down to gauge symmetry. So there are three types of -flux-loop excitations, charring charge (denoted by ). Thus, the string excitations in -gauge theory are given by . (Note that and .) Those string like excitations also have a shrinking rule: if we shrink a string to a point, it will behave like a point-like excitation:
[TABLE]
In general, the string-excitations whose shrinking rule contain the trivial excitation are the pure string excitations, which are and in the -gauge example. We see that the types of pure string excitations are labeled by , the conjugacy classes of .
The -flux-loop (*i.e. *trivial flux-loop) does not break the gauge. So there are three types of -flux-loop excitations, carrying a trivial representation (denoted by or ), a nontrivial 1D irreducible representation (denoted by or ), or a 2D irreducible representation (denoted by or ) of . In fact, those -flux-loop excitations (or trivial-string excitations) correspond to the point-like excitations. The fusions of all those string-like excitations are given by Table 2.
We may regard loop-like excitations with the same conjugacy class but different representations as equivalent and introduce a notion of pure-type: the loop-like excitation is said to have an pure-type . So the fusion of the membrane operators correspond to the fusion of pure-types, which is closely related to the fusion of string-like excitations, after we quotient out the -representations , by identifying
[TABLE]
in the fusion rule table 2. The general identification formula is
[TABLE]
where is the dimension of the -representations .
Appendix B A general discussion of string and membrane operators in 3+1D topological orders
B.1 String operators in 3+1D topological orders
For a generic 3+1D topological order, the type- particle-like excitations are still described by string operators
[TABLE]
We say that two closed string operators are equivalent if they differ by a local unitary transformation. More precisely, two closed string operators are equivalent iff they can deform into each other while keep all the local operators having short ranged correlations. We also normalize the string operators such that is independent of string length when the string is closed. Such normalized string operators satisfy the following fusion algebra
[TABLE]
We can show that are non-negative integers, by viewing the string operators as the world-lines in time direction.
B.2 Membrane operators in 3+1D topological orders
Similarly, the type- string-like excitations are described by membrane operators
[TABLE]
which is tensor network operator. Also two closed membrane operators are equivalent if they differ by a local unitary transformation. More precisely, two closed membrane operators are equivalent iff they can deform into each other while keep all the local operators having short ranged correlations.
The equivalent classes of the membrane operators can be different, for membrane operators with different topology. The equivalent classes of the spherical closed membrane operators, , correspond to the pure membrane types. The pure membrane type corresponds to the type for pure string-like excitations. For toric closed membrane operators, , the number of the equivalent classes will in general be different from the number of the equivalent classes of spherical closed membrane operators. This is because toric closed membrane operators may contain closed string operators wrapping around the non-contractible loops, which generate different equivalent classes. Clearly, if we do not have nontrivial point-like excitations, then there will be a one-to-one correspondence between the spherical membrane operators and toric membrane operators.
Since the loop of string operator on is always contractible, the spherical membrane operators does not contain loops of string operator. Thus the spherical membrane operators are labeled by the conjugacy classes only. The spherical membrane operators also satisfy the following fusion algebra
[TABLE]
In particular
[TABLE]
which can be calculated from the fusion of the conjugacy classes and (see eqn. (57)).
Appendix C More general properties of
string-like excitations in 3+1D topological orders
C.1 Pure string-like excitations and sectors in dimension reduction
In Section III.2, we have shown that in 3+1D, the number of the sectors in the dimension reduction is the number of the classes of string-like topological excitations that can be distinguished by the braiding with the point-like excitations. But in Section IV.2, we have shown that all string-like topological excitations can be distinguished from each other via their braiding properties with the point-like excitations. Therefore,
[TABLE]
Let GSD be the ground state degeneracy of a D topological order on a closed -dimensional space manifold . We note that
[TABLE]
Now, let us consider a more general dimension reduction where we reduce -dimensional space to -dimensional space by shrinking the . be the number of sectors of the dimension reduced topological orders. We find that
[TABLE]
We see that
[TABLE]
C.2 Pure string-like excitations are labeled by the conjugacy classes
of
From last section, we see that for 3+1D AB topological order
[TABLE]
In the following, we like to show that
[TABLE]
We consider the GSD on . We note that the path integral on space-time gives us a particular ground state on . To obtain other ground states on we insert a type- string operator along the in . The string operator is inserted at a particular point on . The insertion of different string operators generate linear independent states. This is because the point-like excitations represented by the string operators have non-degenerate braiding with the pure string-like excitations.
The braiding between the inserted point-like excitations and the string-like excitations is described by creating a small loop of strings on . Then we enlarge the loop and let the loop wrap around the . Such a braiding process is equivalent to applying the sphere membrane operator on . The eigenvalues of the sphere membrane operators should distinguish all the state created by inserting the string operators . Therefore,
[TABLE]
This allows us to show that
[TABLE]
Moreover, from the fact that the untwisted sector of dimension reduction is the Drinfeld center , we know that
[TABLE]
Now, the dimension reduction of a generic bosonic D topological order can be written as
[TABLE]
where is the conjugacy class of the group whose representations form the SFC of , and sums over all the conjugacy classes of .
From the dimension reduction eqn. (79), we can also compute the ground state degeneracy on 3-torus
[TABLE]
Those degenerate ground states form a representation of the mapping class group of , which is .
The dimensional reduction leads to reduction of the representation of to the representations of that characterize the 2+1D dimension reduced topological orders . We consider subgroup and the reduction of the representation to the representations :
[TABLE]
The representation describes the 3+1D topological order and the representations describe the 2+1D topological orders . The decomposition eqn. (81) gives us the dimensional reduction eqn. (79).
C.3 String-like excitations are -flux, even in generic 3+1D
AB topological orders
From the fact that the untwisted sector of dimension reduction is the Drinfeld center , we even know that,
[TABLE]
Alternatively, we can argue the above claim using the results for the string-only boundary. This is because the bulk string-like excitations can be obtained by lifting the boundary string-like excitations. Since a bulk string-like excitation can braid around a boundary string-like excitation , their fusion satisfy
[TABLE]
This allows us to show that
[TABLE]
where is a conjugacy class of . Therefore, even in a generic 3+1D topological order, we may still view string-like excitations as the -gauge flux which is described the conjugacy classes of the group . In particular, the bulk pure string-like excitations fuse like the conjugacy classes (see eqn. (57)). As a result, the quantum dimension of a pure string-like excitation is given by the size of the conjugacy class: . This is one of the key result of this paper.
There is a simple physical way to understand the relation between the bulk and boundary string-like excitations. Since the boundary is induced by the condensation of all the point-like excitations, it corresponds to breaks all the “gauge symmetry”. So there is no gauge equivalence of , and there is no degeneracy between the flux-loop that induce monodromy and monodromy. So, if we bring a bulk string near the boundary, it will split (see eqn. (83)).
Appendix D More about dimension reduction
We argued that the string-like excitations are -flux, even in generic 3+1D AB bosonic topological orders. Now we can say more about the 2+1D topological orders that appear in the dimension reduction eqn. (79). We first note that the point-like excitations in 3+1D topological order are described by for a group . In the dimension reduction, those 3+1D point-like excitations becomes the 2+1D point-like excitations with trivial mutual statistics between them; they form symmetric fusion subcategories of the 2+1D dimension reduced topological orders .
For the conjugacy class , *i.e. *the untwisted sector, and is a minimal modular extension of . But what about the other conjugacy classes?
Because is induced by threading a -flux described by through , such a -flux will break the “gauge symmetry” from to , where is a subgroup of that commutes with a fixed element in conjugacy class . Therefore, the SFC in is given by . The 3+1D point-like excitations described by will split into 2+1D point-like excitations described by in each sector (see Table 1).
Similarly for the other sectors we have
[TABLE]
The minimal modular extension means that the anyons in that are not in all have nontrivial mutual statistics with the bosons in . This condition comes from the result in Section IV.2. Note that unlike the untwisted sector, is in general not the Drinfeld center of .
Appendix E The branching structure of space-time lattice
In order to define a generic lattice theory on the space-time complex using local tensors, it is important to give the vertices of each simplex a local order. A nice local scheme to order the vertices is given by a branching structure.Costantino (2005); Chen et al. (2013, 2012) A branching structure is a choice of orientation of each link in the -dimensional complex so that there is no oriented loop on any triangle (see Fig. 10).
The branching structure induces a local order of the vertices on each simplex. The first vertex of a simplex is the vertex with no incoming links, and the second vertex is the vertex with only one incoming link, *etc *. So the simplex in Fig. 10a has the following vertex ordering: .
The branching structure also gives the simplex (and its sub-simplices) a canonical orientation. Fig. 10 illustrates two -simplices with opposite canonical orientations compared with the 3-dimension space in which they are embedded. The blue arrows indicate the canonical orientations of the -simplices. The black arrows indicate the canonical orientations of the -simplices.
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