Toric 2-group anomalies via cobordism
Joe Davighi, Nakarin Lohitsiri
TL;DR
This paper classifies anomalies in toric 2-group symmetries, combining cobordism theory with physics, and applies it to Maxwell theory and 2d U(1) symmetries, revealing new insights into their anomaly structures.
Contribution
It introduces a cobordism-based classification of anomalies in toric 2-group symmetries, extending the understanding of anomalies in higher-form symmetries in physics.
Findings
Computed the 5th spin bordism group for 2-groups with U(1) parts.
Reproduced known anomaly results in 4d QED.
Showed 2d U(1) anomalies reduce to order 2 with spin structures.
Abstract
2-group symmetries arise in physics when a 0-form symmetry and a 1-form symmetry intertwine, forming a generalised group-like structure. Specialising to the case where both and are compact, connected, abelian groups (i.e. tori), we analyse anomalies in such `toric 2-group symmetries' using the cobordism classification. As a warm up example, we use cobordism to study various 't Hooft anomalies (and the phases to which they are dual) in Maxwell theory defined on non-spin manifolds. For our main example, we compute the 5th spin bordism group of where is any 2-group whose 0-form and 1-form symmetry parts are both , and is the geometric realisation of the nerve of the 2-group . By leveraging a variety of algebraic methods, we show that $\Omega^{\mathrm{Spin}}_5(B|\mathbb{G}|) \congâŠ
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Taxonomy
TopicsOphthalmology and Eye Disorders · Homotopy and Cohomology in Algebraic Topology · Intracranial Aneurysms: Treatment and Complications
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aainstitutetext: Physics Institute, University of Zurich, Switzerland
bbinstitutetext: Department of Mathematical Sciences, Durham University, United Kingdom
ccinstitutetext: Department of Mathematics, Purdue University, USA
Toric 2-group anomalies via cobordism
Joe Davighi b ââ
and Nakarin Lohitsiri c ââ
with a mathematical appendix by Arun Debray
Abstract
2-group symmetries arise in physics when a 0-form symmetry and a 1-form symmetry intertwine, forming a generalised group-like structure. Specialising to the case where both and are compact, connected, abelian groups (i.e. tori), we analyse anomalies in such âtoric 2-group symmetriesâ using the cobordism classification. As a warm up example, we use cobordism to study various ât Hooft anomalies (and the phases to which they are dual) in Maxwell theory defined on non-spin manifolds. For our main example, we compute the 5th spin bordism group of where is any 2-group whose 0-form and 1-form symmetry parts are both , and is the geometric realisation of the nerve of the 2-group . By leveraging a variety of algebraic methods, we show that where is the modulus of the Postnikov class for , and we reproduce the expected physics result for anomalies in 2-group symmetries that appear in 4d QED. Moving down two dimensions, we recap that any (anomalous) global symmetry in 2d can be enhanced to a toric 2-group symmetry, before showing that its associated local anomaly reduces to at most an order 2 anomaly, when the theory is defined with a spin structure.
1 Introduction
In quantum field theory, anomalies are loosely defined to be quantum obstructions to symmetries. More precisely, anomalies can themselves be identified with (special classes of) quantum field theories in one dimension higher than the original theory, via the idea of âanomaly inflowâ.111Mathematically, any anomalous theory in dimensions can be described by a relative quantum field theory Freed:2012bs between an extended field theory in one dimension higher (the âanomaly theoryâ), and the trivial extended field theory (see e.g. Freed:2014iua ). This modern viewpoint led to an algebraic classification of anomalies via cobordism, which was made rigorous by Freed and Hopkins Freed:2016rqq following many important works (including Witten:1985xe ; Dai:1994kq ; Kapustin:2014dxa ; Witten:2015aba ; Witten:2019bou ). The cobordism classification includes all known anomalies afflicting chiral symmetries of massless fermions in any number of dimensions, as well as other more subtle anomalies that involve discrete spacetime symmetries (see e.g. Tachikawa:2018njr ; Witten:2016cio ; Wang:2018qoy ).
The cobordism group that classifies anomalies, for spacetime dimensions and symmetry type ,222A given symmetry type , as defined precisely in Freed:2016rqq , includes both the spacetime symmetry and an internal symmetry group, as we as maps between them, extended to all dimensions. is the Anderson dual of bordism Freed:2016rqq ,
[TABLE]
where denotes the MadsenâTillman spectrum associated to symmetry type , denotes the Anderson dual of the sphere spectrum (with denoting the -fold suspension), and denotes the homotopy classes of maps between a pair of spectra and . In fact, this cobordism classification goes beyond what we would normally think of as âanomaliesâ to cover all reflection positive invertible field theories (with or without fermions) with symmetry in -dimensions.
To unpack the meaning and significance of the cobordism group , it is helpful to recall that it fits inside a defining short exact sequence Freed:2016rqq
[TABLE]
where is the stable homotopy group of a spectrum . For the special case of chiral fermion anomalies in dimensions, the invertible field theory in question is precisely the exponentiated -invariant Witten:2019bou of Atiyah, Patodi and Singer Atiyah:1975jf ; Atiyah:1976jg ; Atiyah:1976qjr in dimension . In that case, integrating the gauge-invariant anomaly polynomial provides an element of the right factor . Theories with , which are in the kernel of the right map and thus in the image of the left map, correspond to residual âglobal anomaliesâ which are thus captured by the left-factor. For theories with symmetry type , where is the internal symmetry group, we have that classifies global anomalies, where denotes spin bordism. A straightforward corollary is that if the group vanishes there can be no global anomalies, which has been applied to various particle physics applications in recent years Freed:2006mx ; Garcia-Etxebarria:2017crf ; Garcia-Etxebarria:2018ajm ; Wan:2018bns ; Seiberg:2018ntt ; Hsieh:2019iba ; Davighi:2019rcd ; Wan:2019gqr ; Davighi:2020uab ; Lee:2020ewl ; Davighi:2020kok ; Debray:2021vob ; Davighi:2022fer ; Davighi:2022bqf ; Wang:2022eag ; Lee:2022spd ; Davighi:2022icj ; Debray:2023yrs .
Concurrent with this development of the cobordism classification of anomalies, in the past decade the notion of âsymmetryâ has been generalised beyond the action of groups on local operators, in various exciting directions (for recent reviews, see Ref. Cordova:2022ruw ; Bah:2022wot ). This includes the notion of -form symmetries Gaiotto:2014kfa , which act not on local operators (the case ) but more generally on extended operators of dimension . These symmetries couple to background fields that are -form gauge fields. Such higher-form symmetries are found in a plethora of quantum field theories. The most well-known examples include a pair of 1-form symmetries in 4d Maxwell theory, and a 1-form symmetry in 4d YangâMills theory; the latter exhibits a mixed anomaly with charge-parity symmetry when the -angle equals , which led to new insights regarding the vacua of QCD Gaiotto:2017yup . Other important classes of generalised symmetry include non-invertible and categorical symmetries, as well as subsystem symmetries. These shall play no role in the present paper.
Soon after the notion of -form symmetries was formalised in Gaiotto:2014kfa , it was realised that higher-form symmetries of different degrees can âmixâ, leading to the notion of -group symmetry. The most down-to-earth example of this symmetry structure, which in general is described using higher category theory (see e.g. Gripaios:2022yjy ), is the notion of a 2-group symmetry whereby a 1-form symmetry and a 0-form symmetry combine non-trivially. This occurs, for example Hsin:2020nts ; Lee:2021crt , when a bunch of Wilson lines (which are charged under a discrete 1-form centre symmetry) can be screened by a dynamical fermion (which is charged under a 0-form flavour symmetry). 2-group symmetries were first observed, in their gauged form, in string theory in the context of the GreenâSchwarz mechanism, in which a 1-form symmetry combines non-trivially with a diffeomorphism Baez:2005sn ; Sati:2009ic ; Fiorenza:2010mh ; Fiorenza:2012tb . Their appearance as global symmetries in field theory was appreciated first by Sharpe Sharpe:2015mja , before many new examples were identified in Refs. Cordova:2018cvg ; Benini:2018reh . Further instances of 2-group symmetry have since been discovered and analysed in 4d Hsin:2020nts ; Lee:2021crt ; Bhardwaj:2021wif ; Cvetic:2022imb ; Carta:2022fxc , 5d Cvetic:2022imb ; Benini:2018reh ; Apruzzi:2021vcu ; DelZotto:2022fnw ; DelZotto:2022joo , and 6d Cordova:2020tij ; DelZotto:2020sop ; Apruzzi:2021mlh quantum field theories. A 2-group symmetry structure has recently been studied in the Standard Model Cordova:2022qtz , in the limit of zero Yukawa couplings.
In this paper we study how the cobordism classification of anomalies can be applied to 2-group symmetries. As a first step, we here specialise to 2-groups for which both and are compact, connected, abelian groups ergo tori ; we refer to such a structure, at times, as a âtoric 2-groupâ. In this case, there are 1-form and 2-form Noether currents and associated with the 0-form and 1-form symmetries respectively, and the non-trivial 2-group âmixingâ between these symmetries is reflected in a fusion rule for these currents, schematically
[TABLE]
for a known (singular) function . Toric 2-group symmetries appear, for example, in chiral gauge theories in 4d when there are mixed âoperator-valuedâ anomalies between global chiral symmetry currents and the gauge current Cordova:2018cvg , as well as in 3d Damia:2022rxw and in hydrodynamics Iqbal:2020lrt . Toric 2-group symmetries thus naturally appear in fermionic systems, for which anomalies can be more fruitfully analysed using cobordism (compared to purely bosonic anomalies).
For such a toric 2-group , we study the bordism theory of manifolds equipped with âtangential -structureâ. This can be defined analogously to a tangential -structure for ordinary symmetry type , given now the particular 2-group plus an appropriate form of spin structure. When the 2-group symmetry does not further mix with the spacetime symmetry, this reduces to studying the spin bordism groups of the topological space that classifies -2-bundles.
Thankfully much is known about the classifying space of -2-bundles that will be of use to us, given the physics applications we have in mind. Quite generally, the classifying space of a 2-group can be computed as , where is the geometric realization of the nerve of the 2-group viewed as a category bartels2006higher ; Baez2008 , which is itself a topological 1-group â albeit often an infinite-dimensional one. This means that Freed and Hopkinsâ classification of invertible field theories with fixed symmetry type can be applied directly, taking the tangential structure to be .
In the special case of a pure 1-form symmetry valued in an abelian group , this formula for the classifying space of the corresponding 2-group reduces to , recovering the well-known result for classifying abelian gerbes brylinski2007loop . This special case, which is of significant physical interest, has already been used in the physics literature to study 1-form anomalies, for many examples with variously Spin, SO, and O structures in Ref. Wan:2018bns . An extension of this to a pure 2-form symmetry and its anomalies in 6d gauge theories was studied via cobordism in Ref. Lee:2020ewl .
More generally for any 2-group , the classifying space always sits in a âdefining fibrationâ
[TABLE]
which allows one to leverage the usual spectral sequence methods to compute e.g. the cohomology and/or the (co)bordism groups of . For ânon-trivialâ 2-groups, i.e. those for which the fibration (1.4) is non-trivial, there is to our knowledge only a smattering of calculations in the literature; in particular, the case was studied by Wan and Wang Wan:2018bns (Section 4), and was studied by Lee and Tachikawa Lee:2020ewl (Appendix B.6). In this paper we extend these works with a dedicated cobordism analysis of theories with 2-group symmetry and their anomalies.
The content of this paper is as follows. After reviewing 2-groups and their appearances in quantum field theory in §2, we then describe the topological spaces that classify 2-group background fields, and show how these spaces can be computed, in §3. By considering appropriate tangential structures for 2-group symmetries, this then leads to a definition of the cobordism groups relevant for describing 2-group symmetries and their anomalies (§3.2). We then apply this cobordism theory to study a small selection of examples in detail, focussing on toric 2-group symmetries as an especially tractable case. The main examples that we study in this paper are as follows. In §4, which is a warm up example, we study anomalies involving the pair of 1-form symmetries in 4d Maxwell theory, defined on non-spin manifolds. As well as the -valued âlocalâ mixed anomaly between the two 1-form symmetries Gaiotto:2014kfa ; Wang-Senthil:2016a ; Hsin:2019fhf ; Brennan:2022tyl , there is a trio of -valued global anomalies (involving gravity) that only appear on non-spin manifolds; the associated phases have been studied in Jian:2020qab . In §5 we turn to our main example of toric 2-group symmetries, with non-trivial Postnikov class, occurring in 4d abelian chiral theories. We see, using cobordism, how turning on the Postnikov class transmutes a -valued local anomaly into a torsion-valued global anomaly Ă la the GreenâSchwarz mechanism â we precisely recover the physics results derived by CĂłrdova, Dumitrescu and Intriligator in Ref Cordova:2018cvg . Finally, in §6 we drop down to two dimensions, where anomalous 0-form symmetries can be recast as 2-group symmetries by using the trivially conserved 1-form symmetry whose 2-form current is simply the volume form Sharpe:2015mja . Our cobordism analysis highlights the prominent role played by spin structures; when the anomaly coefficient is odd, the GreenâSchwarz mechanism cannot in fact absorb the anomaly completely but leaves a -valued anomaly, which we identify as an anomaly in the spin structure; we also show how this -anomaly is trivialised upon passing to bordism.333These examples emphasize that bordism computations are useful tools for keeping track of such subtle mod 2 effects, by automatically encoding various characteristic classesâ normalisation that can differ in spin vs. non-spin theories â for example, the class is even on a spin manifold but neednât be on a non-spin one.
2 Review: 2-groups and their classifying spaces
In this Section we recall the notion of a 2-group (§2.1), and briefly reprise their appearance as generalised symmetry structures in quantum field theory (§2.2).
2.1 What is a 2-group?
We begin by sketching some equivalent definitions of a 2-group, for which our main reference is Ref. Baez2008 by Baez and Stevenson. First and most concise is probably the definition of a 2-group as a higher category. Loosely, a 2-group is in this language a 2-category that contains a single object, with all 1-morphisms and 2-morphisms being invertible.
(Slightly) less abstractly, a 2-group can also be described using âordinaryâ category theory, without recourse to higher categories, and this can be done in two ways; either as a groupoid in the category of groups, or as a group in the category of groupoids.444It will be implicitly assumed that all groups and groupoids discussed carry a topology. Expanding on the former viewpoint, a 2-group is itself a category whose objects form a group , and whose morphisms also form a group, where the latter can be written as a semi-direct product between and another group , viz.
[TABLE]
such that all the various maps involved in the definition of a groupoid are continuous group homomorphisms (see §3 of Baez2008 ). We will usually be interested in situations where both and are Lie groups.
This definition involving a pair of groups is closely related to yet another, somewhat more practical, definition of a 2-group as a topological crossed module. The data specifying a topological crossed module consists of a quadruplet
[TABLE]
where and can be identified with the groups that appear in (2.1), and , are continuous homomorphisms such that the pair of conditions
[TABLE]
hold , which can be thought of as equivariance conditions on the maps. To define a 2-group from this data, one takes the group of objects and the group of morphisms , where the semi-direct product is defined using the map as , and the various source, target, and composition maps from , as well as identity maps from , are defined using simple formulae (for which we refer the interested reader to §3 of Baez2008 .)
We can tie together this circle of definitions by linking this final crossed module definition of to the first definition of as a 2-category; in that picture, is the group of 1-morphisms, and is the group of 2-morphisms from the identity 1-morphism to all other 1-morphisms Kapustin:2013uxa . When and are both Lie, and when the maps and are both smooth, is a Lie 2-group.
The final description of a 2-group that we just gave, as a topological crossed module, is still of somewhat limited use in physics because the groups and are not preserved under equivalence of 2-groups as 2-categories. Moreover, a physical system with a 2-group symmetry (possibly with associated background fields) is only sensitive to the equivalence class of the 2-group at long distance, not the 2-group itself, as described by Kapustin and Thorngren Kapustin:2013uxa . It is therefore more convenient to work with equivalence classes of 2-groups from the beginning, which we describe next.
An equivalence class of 2-groups, with a particular 2-group above as a representative, can also be described by a quadruplet where Kapustin:2013uxa
[TABLE]
The homomorphism is so named because it descends from the homomorphism . The new component is the so-called Postnikov class
[TABLE]
More precisely, two crossed modules and are said to be equivalent when , and there are homomorphisms (not necessarily isomorphisms) and that makes the diagram
[TABLE]
of two exact sequences commutative and compatible with the actions of on and on . These equivalence classes are classified by the group cohomology  Brown-K-S:1982 , which is isomorphic to the ordinary cohomology .
For completeness, we also define the notion of 2-group homomorphisms, in terms of ordinary group homomorphisms between different elements of the associated crossed modules Baez2008 . Represent two 2-groups and by crossed modules and . A 2-group homomorphism , which is a functor such that and are continuous homomorphisms of topological groups, can be represented by a pair of maps and such that the diagram
[TABLE]
is commutative and that are compatible with :
[TABLE]
for all .
2.2 2-group symmetries in quantum field theory
Physically, an equivalence class of 2-groups describes a symmetry structure appearing in quantum field theories which have both a 0-form symmetry group and a 1-form symmetry group . When the two symmetries are not independent, the Postnikov class of the corresponding 2-group symmetry is non-trivial, and the 0-form and 1-form parts cannot be analysed in isolation.
While we will eventually specialise to the case of toric 2-groups in the bulk of this paper, we first recap how 2-groups appear in field theory more broadly. It is convenient to distinguish two broad categories of 2-groups which in general have different physical origin:
A continuous 2-group is one with a continuous 1-form symmetry group . This class of 2-groups arises in field theory when, for example, the gauge transformation law for the 1-form symmetry background gauge field is modified so that there is no operator-valued ât Hooft anomaly involving the background gauge field for the 0-form symmetry, in an analogue of the GreenâSchwarz mechanism for global symmetries Cordova:2018cvg . The 0-form symmetry can here be abelian or non-abelian, connected or disconnected, and compact or non-compact. In this paper we focus our attention on the special case where both the 0-form and 1-form symmetry are connected, compact, and abelian groups, which we refer to as a âtoric 2-groupâ. 2. 2.
A discrete 2-group is one with a discrete 1-form symmetry group . This class of 2-group symmetries arises in gauge theories when the gauge Wilson lines, which are charged objects under the 1-form symmetry , are not completely screened in the presence of the background gauge field for the 0-form symmetry group . The local operator that screens the gauge Wilson lines when the 0-form background gauge field is turned off is also charged under the 0-form symmetry. Hence, when the background gauge field of is turned on, transmutes a gauge Wilson line into a flavour Wilson line Benini:2018reh ; Bhardwaj:2021wif ; Lee:2021crt .
3 Cobordism with 2-group structure
In this Section we describe the background gauge fields on associated principal -2-bundles which play an important role in field theories with 2-group symmetries. We focus on describing the topological spaces that classify these background fields, and show how to calculate such classifying spaces in elementary examples, before introducing the cobordism groups that are central to this paper in §3.2.
3.1 Background fields and their classifying spaces
A theory with a 2-group symmetry can be coupled to a background gauge field (which a physicist might wish to decompose into components of an ordinary 1-form gauge field and a 2-form gauge field),555A more explicit description of the background fields will be sketched in §5 where we discuss the 2-group symmetries appearing in 4d abelian chiral gauge theory. which is a connection on a principal -2-bundle in the sense defined in bartels2006higher .
Mathematically, Bartels moreover shows that such 2-bundles over a manifold are classified by the Äech cohomology group with coefficients valued in the 2-group . Baez and Stevenson prove Baez2008 that there is a bijection
[TABLE]
where denotes the set of homotopy classes of maps from to , and is the geometric realization of the nerve of the 2-group when viewed as a groupoid as in (2.1). The nerve of the category with and is a set of simplices that we can construct out of these objects and morphisms â we will see some explicit examples shortly.
Quantum field theories with 2-group symmetry are thus defined on spacetime manifolds equipped with maps to , just as a theory with an ordinary symmetry is defined on spacetimes equipped with maps to .
3.1.1 Elementary examples of
Since the classifying space will play a central role in what follows, we pause to better acquaint the reader with and how it can be computed by looking at some simple examples. The calculations in this subsection are purely pedagogical â readers who are familiar with (or not interested in) such constructions might wish to skip ahead to §3.1.2.
Pure 0-form symmetries
In the case that there is no 1-form symmetry, and the 2-group symmetry simply defines an ordinary 0-form symmetry , we expect to recover the usual classifying space of . In this case, the 2-group corresponds to the quadruplet , to use the topological crossed module notation, and indeed . To see this from the nerve construction, we first view as a category, which is very simple in this case: with only identity morphisms at each element. So there are no non-degenerate -simplices when , while the set of 0-simplices is . Hence, the geometric realisation of the nerve is simply the group itself, and its classifying space is , as claimed.
Pure 1-form symmetries
The case of a âpure 1-form symmetryâ, in which there is no 0-form symmetry at all, corresponds to 2-groups of the form , where is the (abelian) 1-form symmetry group. The corresponding classifying space is bartels2006higher
[TABLE]
which we often denote simply , which coincides with the well-known classifying space of an abelian gerbe brylinski2007loop . For example, when , is an EilenbergâMaclane space .
An example: pure 1-form symmetry.
For an explicit example that illustrates how to actually calculate the geometric realization of the nerve of such a , we consider the simplest case where . Viewing this as a category as in (2.1), we have that consists of only one element because is the trivial group, denoted simply by , while the group is just isomorphic to . Diagrammatically, this structure can be represented as
[TABLE]
The nerve of is then a simplicial set built out of these objects and morphisms, whose low dimensional components are given as follows.
[TABLE]
This is enough information for us to work out the CW complex cell structure for its geometric realisation, which recall is the topological space that we denote . The 0-cell is just a point. The 1-cell comes from the union of the 0-cell and an interval, with the gluing rule given by the 1-simplex
{\bullet}$$\scriptstyle{-1}
i.e. identifying both ends of the interval with the 0-cell. Hence, the 1-cell has the topology of a 1-sphere . Similarly, the form of the 2-simplex, written more suggestively as
[TABLE]
tells us that the 2-cell is constructed by identifying the 1-cell as the boundary of a 2-ball, with antipodal points on the boundary identified. This gives the topology of the 2-dimensional real projective space . Building inductively in this way, one can show that is topologically . On the other hand, we know that this is the same as the classifying space of , i.e. . Therefore, the classifying space of the 1-form symmetry bundle is .
This argument can be generalised to any 2-group of the form . For any group , it is proven that the geometrisation of the nerve is the topological space Segal1968 , and the classifying space of -2-bundles is thus , coinciding with the known classification of -gerbes when is abelian.
2-groups with a trivial map
Moving up in complexity, let us now consider a 2-group with both the constituent 1-groups and being non-trivial, but with at least one of the maps in the crossed module definition being trivial. To wit, consider a particular 2-group , written using the notation of (2.2), where the map is trivial i.e. is the identity automorphism for any .
An example: and .
Probably the simplest example of a 2-group of this kind is one where and are both . Since is trivial, is automatically trivial. If and do not interact at all, meaning that the map is also trivial (the module is âuncrossedâ), then the 2-group factorises as a product of a 0-form and a 1-form symmetry:
[TABLE]
whose classifying space is just .
Since must be a homomorphism, the only other option for is the identity map . Setting , then as a category and , whose action can be fully captured in the following diagram:
[TABLE]
Since the morphisms shown in the diagram are all the morphisms in this category, we can identify the morphisms and as the identity morphisms at the objects and , respectively.
The components of the nerve in low dimensions are given by
[TABLE]
From these data, it is easy to see that the 0-cell is just a pair of points, the 1-cell is a circle ,
{\bullet_{-1}}$${\bullet_{1}}$$\scriptstyle{(-1,-1)}$$\scriptstyle{(-1,1)}
, and the 2-cell is a 2-sphere , constructed by joining
{\bullet_{-1}}$${\bullet_{1}}
and
{\bullet_{1}}$${\bullet_{-1}}
as two âhemispheresâ, with the 1-cell at the equator. Continuing ad infinitum we obtain . But the infinite sphere is contractible, which means it is homotopy equivalent to a point. Hence, the classifying space for must be trivial.
This result should not be a surprise because the crossed module under consideration, namely with an isomorphism, is equivalent to the trivial 2-group because both and are trivial!
3.1.2 General as a fibration
Our discussion so far has been mostly pedagogical; the idea was to acquaint the reader with some basic computations of the classsifying spaces of 2-groups. Thankfully, more powerful tools are available to us for the computations that we will be interested in. First and foremost, for any 2-group there is a fibration666The extensions encoded in (3.3) are in fact classified by  schommer2011central , the SegalâMitchison group cohomology in degree 3 (a degree that usually classifies 2-step extensions).
[TABLE]
Note that, because the 1-form symmetry is always an abelian group, is itself a (topological) group, and so can be defined using the ordinary definition of the classifying space of a group. We will see in the next Subsection that this fibration is sufficient for our purpose of anomaly classification.
3.2 Cobordism description of 2-group anomalies
In order to classify anomalies associated to a 2-group via cobordism, most of the structure contained in as a 2-group can be discarded. As discussed in the previous Subsection, -2-bundles are classified by the topological space , where is the geometric realization of the nerve (henceforth just âthe nerveâ) of , regardless of other information contained in . Thus, we can use the nerve , itself a topological group, to define a tangential structure on spacetime just as we would for an ordinary 0-form symmetry. For instance, if we are interested in a fermionic theory with internal 2-group symmetry (with no mixing with the spacetime symmetry), we can take the symmetry type to be777If we do not require a spin structure, we could use SO instead of Spin in defining the symmetry type â as we do in §4.
[TABLE]
Directly applying FreedâHopkinsâ classification Freed:2016rqq of invertible, reflection-positive field theories to this symmetry type, suggests that the cobordism group
[TABLE]
correctly classifies anomalies in -dimensional fermionic theories with the 2-group symmetry . Our detection of anomaly theories is therefore distilled into computing the spin bordism groups of , for which we can use the same methods as for an ordinary symmetry.
In particular, our general strategy will be to build up the spin bordism groups of by using the âdefining fibrationâ described above, , as follows. We first use the Serre spectral sequence (SSS) Serre1951HomologieSD to compute the cohomology of from the fibration. Then, one can use this newly computed cohomology to compute the spin bordism groups of , for example via the Adams spectral sequence (ASS) Adams:1958 . Alternatively, we can convert the result to homology using the universal coefficient theorem, and use it as an input of the AtiyahâHirzebruch spectral sequence (AHSS) AHSS1961:ProcSymp61
[TABLE]
for another fibration . Kapustin and Thorngren already used the SSS associated to (3.3) in Kapustin:2013uxa to compute the cohomology of . Moreover, this approach has been used to calculate bordism groups relevant to 2-group symmetries with non-trivial Postnikov class; Wan and Wang did so using the ASS in Ref. Wan:2018bns , while Lee and Tachikawa used the AHSS in Ref Lee:2020ewl . We usually adopt the AHSS-based approach in this paper.
4 Maxwell revisited
As a (rather lengthy) warm-up example, let us discuss Maxwell theory. We consider a 4d gauge theory without matter, and variants thereof in which we couple the action to various TQFTs without changing the underlying symmetry structure. The action for vanilla Maxwell theory is
[TABLE]
where denotes the dynamical gauge field and denotes the Hodge dual, assuming a metric on .
This theory has two 1-form global symmetries Gaiotto:2014kfa , referred to as âelectricâ and âmagneticâ 1-form symmetries. Since each is associated with a continuous group, one can write down the corresponding conserved 2-form currents, which are
[TABLE]
The former is conserved () by Maxwellâs equations , and the latter by (and using ).888For Maxwell theory in dimensions, is always a 2-form current and so the electric symmetry is always 1-form, while is more generally a -form, and so the magnetic symmetry is generally a -form symmetry.
It was already observed in Gaiotto:2014kfa that there is a mixed ât Hooft anomaly between the two 1-form symmetries, which is a local anomaly that can be represented by an anomaly polynomial , where and are background 2-form gauge fields for the electric and magnetic 1-form symmetries respectively. This kind of anomaly and its consequences concerning various quantum numbers are discussed in Ref. Hsin:2019fhf ; Brennan:2022tyl (see also Wang-Senthil:2016a ). In Ref. Jian:2020qab , a variety of 5d SPT phases that have Maxwell-like theories on the 4d boundary, protected by electric and magnetic 1-form symmetries, were derived. Here we show how these results are simply captured by the cobordism classification.
4.1 Cobordism classification of 1-form anomalies
Maxwell theory has two global 1-form symmetries, and no 0-form global symmetries. The classifying space of this global symmetry structure is that of an abelian -gerbe with . From (3.2),
[TABLE]
Since there are no fermions, we do not need a spin structure, and so can define pure Maxwell theory on any smooth, orientable 4d spacetime manifold.999We do not account for any time-reversal symmetry, that would enable us to pass from oriented to unoriented bordism. The 5d invertible field theories with this symmetry type are therefore classified by the generalized cohomology group
[TABLE]
where this sequence splits, but not canonically. In terms of anomaly theories, the factor on the right of this SES captures local anomalies, and the factor on the left captures global anomalies; the abelian group that detects all anomalies is isomorphic to the direct product of the two.
Local anomalies
In Appendix B.1 we compute that , and so
[TABLE]
corresponding to the integral of the degree-6 anomaly polynomial on a generator of . This detects the local mixed ât Hooft anomaly between the two 1-form symmetries discussed in Gaiotto:2014kfa .
Global anomalies
In Appendix B.1 we also compute that . Thus, the group classifying global anomalies is
[TABLE]
In terms of characteristic classes, these three factors of correspond to
[TABLE]
where denote the second and third StiefelâWhitney classes of the tangent bundle, and is a unique generator of (c.f. Appendix A.1) which can be thought of as a mod 2 reduction of an integral cohomology class represented by .
4.2 Phases of non-spin Maxwell theory
The invariant corresponds to a global gravitational anomaly seen on non-spin manifolds, related to that in Wang:2018qoy . The bordism invariants correspond loosely to ât Hooft anomalies for each of the 1-form symmetries, that prevents them from being gauged on certain (non-spin) gravitational backgrounds. We discuss these in more detail next.
The phase
To see the global anomaly that is captured by the ,101010For simplicity, we consider the case where the 1-form symmetry is electric. The corresponding story for the magnetic 1-form symmetry can be obtained by applying the electric-magnetic duality. we must go beyond the vanilla Maxwell theory described by (4.1), and couple 4d Maxwell to a topological term. The modified action is
[TABLE]
where denotes an integral cohomology class and is mod reduction, and where we shall write for both the first Chern class of the gauge bundle and its mod reduction. The topological term couples a background magnetic 2-form gauge field to the part of the magnetic 1-form symmetry, and then equates it to the second StiefelâWhitney class of the tangent bundle. The electric 1-form symmetry shifting , where is a closed 1-form, remains intact.
This theory describes fermionic-monopole electrodynamics; the topological term forces all monopoles to become fermionic Thorngren:2014pza ; Ang:2019txy . To see this, following Wang:2018qoy , consider adding a magnetic monopole via an ât Hooft line operator of charge along that we excise from with the boundary condition that on a small 2-sphere around . The theory is now defined on the complement of in the presence of the ât Hooft operator, which is a manifold with boundary . For the integral to make sense on a manifold with boundary, we need a trivialisation of the integrand on .111111This is because for an -manifold with boundary , the fundamental homology class is in . The integration of a cohomology class on is in fact a pairing between a cohomology class and the fundamental class , so when we write for a cohomology class , we need to first find a class in that âcorrespondsâ to to make sense of the pairing. This is always possible because the long exact sequence in cohomology
implies that any class can always be lifted to a class . The choice of this lift is the choice of trivialisation of on . Now, since is non-trivial on due to the non-zero monopole charge, we have to trivialise the factor. A trivialisation of the second StiefelâWhitney class is nothing but a spin structure. Since there is a unique spin structure on , a spin structure on is the same as a spin structure along . Therefore, to define the additional phase, we must choose a spin structure along the monopole worldline: the monopole is a fermion.
In order to see the ât Hooft anomaly afflicting , we now couple the theory to a background electric 2-form gauge field , and promote the 1-form global symmetry transformation above to a âlocalâ transformation by relaxing the condition that the 1-form parameter has to be closed. In fact, does not strictly need to be a 1-form on â more precisely, we now consider shifting the gauge field by any connection . For a connection with non-zero curvature , the action (4.7) shifts under , by
[TABLE]
which encodes the ât Hooft anomaly associated with certain âlarge 2-form gauge transformationsâ, and on certain gravitational backgrounds.
For example, take to be . As usual, we can parametrize with three complex coordinates , , and , such that and with the equivalence for . Define the 2-form , which is just the volume form on the submanifold defined by . Its cohomology class can be taken as a generator for , and likewise can be taken as a generator for . We also have that . If we shift by a connection with , the shift in the action is
[TABLE]
where here denotes the pairing between mod 2 homology and cohomology, thus realising the -valued global anomaly for odd values of .
As usual, there is a dual description of this anomaly in terms of the 5d SPT phase that captures it via inflow to the 4d boundary. If were nullbordant, the phase would here be where recall , for a 5-manifold such that and to which the background fields are extended. To see this, first note that one could cancel the anomalous shift (4.8) by adding a âcounter-termâ where is a closed 2-form constructed from ,121212To construct , first note that there is an integral lift of because for any orientable 4-manifold. (One sees this from the long exact sequence in cohomology associated to the coefficient sequence , for which the Bockstein connecting homomorphism where ). Next, given the integral class , construct any complex line bundle over with , and take to be the curvature 2-form of that bundle.
recalling that the gauge transformation for is . However, this â4d actionâ is not properly quantised. (It is âhalf-quantisedâ, precisely because there is a anomaly.) One must instead write it as the 5d action . Thus the fermionic-monopole electrodynamics has exactly the right anomaly to be a boundary state of the SPT phase given by (see also Ref. Jian:2020qab ).131313This mixed anomaly can also be interpreted as a remnant of the mixed anomaly between the electric and magnetic 1-form symmetry when only the subgroup of the electric 1-form symmetry is coupled to a background field .
Bordism generator for the anomaly.
Of course, it is important to note that is not nullbordant in , given the signature is a 4d bordism invariant. Given we saw the anomaly explicitly on , one cannot in fact realise it via the counterterm because there is no 5-manifold bounded by . But as is well-known Freed:2004yc , the phase of the partition function on such a non-nullbordant manifold suffers from an ambiguity, since it can always be shifted by a choice of generalized theta angle, i.e. a coupling to a non-trivial TQFT corresponding to an element in . Thus, one can fix to a reference phase, and then is uniquely defined on any other 4-manifold that is bordant to .
This phase can be calculated by constructing a 5d mapping torus by taking a cylinder that interpolates between and , and gluing its ends to make a closed 5-manifold. This will be a representative of the generator of the factor in that we have claimed is dual to .
In more detail, take and let denote any reference choice of background 2-form gauge field for the electric 1-form symmetry on . Next take the product manifold with a product metric. Over the interval one implements (for example linearly) the âlargeâ 2-form gauge transformation , .141414We refer to this as a âlargeâ gauge transformation because the gauge parameter is closed but not exact, having a âwinding numberâ of . Since and are gauge-equivalent, the manifold can be glued at 0 and to make a closed 5-manifold with -2-bundle:
[TABLE]
where .
To verify that this mapping torus can be taken as a generator of the bordism group , it is enough to evaluate the bordism invariant âanomaly theoryâ on and find a non-trivial value for the phase. Physically, evaluating computes the phase accrued by the partition function on upon undergoing the 2-form gauge transformation . Doing the computation, we have that , where is the non-trivial element of obtained by pulling back along the projection , and is the non-trivial element of obtained by pulling back the generator of along . For odd, we have that , the unique non-trivial element of . Thus for odd, and so the anomaly theory evaluates to on this mapping torus, while it is trivial for even.
The phase
An identical account can be given for the magnetic 1-form symmetry, if one replaces the topological coupling in (4.7) by , where denotes the first Chern class of the electromagnetic dual of the gauge field. In that case, the 4d theory suffers from a ât Hooft anomaly afflicting the magnetic 1-form symmetry, which obstructs from being gauged on non-spin manifolds such as . The 5d anomaly theory is .
Physically, this same anomaly can be understood from a different perspective. Suppose one couples vanilla Maxwell theory (4.1) to a charge-1 fermion, defined on all orientable 4-manifolds using a structure. The electric 1-form symmetry is explicitly broken, but one can still probe anomalies involving the magnetic 1-form symmetry and the gravitational background. The magnetic 1-form symmetry remains intact, even though we effectively have a âcharge-â monopole due to the constraint that follows the definition of , because this monopole is not dynamical but rather acts as a constraint on the field configurations that are summed in the path integral. The symmetry type is now , with , for which the cobordism group .151515The two factors of arise simply because we have included the â gauge symmetryâ in our definition of the symmetry type, because it is now entangled with the spacetime symmetry. If one computed the reduced bordism these factors would go, leaving only the -valued ât Hooft anomaly for the magnetic 1-form symmetry. Of course, all the anomalies involving the electric 1-form symmetry are absent, as is the anomaly because the requirement trivialises , but the global anomaly remains.
The phase
The final -valued global anomaly that is detected by (4.6) corresponds to the 5d SPT phase . To exhibit this anomaly, one starts from vanilla Maxwell theory and couples as a background gauge field to both the electric and magnetic 1-form symmetries. This SPT phase is purely gravitational and has been extensively analysed in the literature. Its boundary states include the theory of all-fermion electrodynamics Kravec:2014aza ; Wang:2018qoy and a fermionic theory with an emergent gauge symmetry Wang:2018qoy .
4.3 Scalar QED and 1-form anomaly interplay
Now let us couple Maxwell theory, defined as before with an SO structure, to a charge-2 boson. This coupling to matter breaks the electric 1-form symmetry down to a discrete remnant,
[TABLE]
while the full magnetic 1-form symmetry is preserved. This version of scalar QED therefore furnishes us with two global 1-form symmetries, one discrete and one continuous, and no 0-form symmetries. The classifying space of the global symmetry is therefore
[TABLE]
and the 5d invertible field theories with this symmetry type are classified by the sequence (4.4) but with replaced by . In Appendix B.2 we compute the relevant bordism groups to stitch together this generalized cohomology group.
Local anomalies:
We compute from Appendix B.2 that
[TABLE]
and so there are no local anomalies; clearly, the mixed local ât Hooft anomaly ââ can no longer be realised now that one of the 1-form symmetries is broken to a discrete group, because the associated 2-form gauge field now has zero curvature.
Global anomalies:
On the other hand, we find that . The group classifying global anomalies is now
[TABLE]
and we notice that there is an extra factor of corresponding to an extra global anomaly. At the level of characteristic classes, we can represent these four global anomalies in terms of products of StiefelâWhitney classes and cohomology classes of that are in degree-5. There are five of them in total:
[TABLE]
where is the unique generator of (see Appendix A.1). However, once pulled back to an orientable 5-manifold , Wuâs relation Wu1950 tells us that , so it is not an independent bordism invariant, and we end up with four invariants, as expected from the bordism group computation.
The anomaly interplay
Given an appropriate map between two spectra and , there is an induced pullback map between the corresponding cobordism theories, , that can be used to relate anomalies between the two theories. This idea of âanomaly interplayâ has recently been used, for example, to relate local anomalies in 4d gauge theory to Wittenâs anomaly Davighi:2020bvi , to study anomalies in symmetries in 2d Grigoletto:2021zyv ; Grigoletto:2021oho with applications to bootstrapping conformal field theories, and to derive anomalies in non-abelian finite group symmetries in 4d Davighi:2022icj . Physically, the idea of âpulling back anomaliesâ from one symmetry to another is not new, but goes back to Elitzur and Nairâs analysis of global anomalies Elitzur:1984kr , following Witten WITTEN1983422 . In all these cases the symmetry type takes the form , where is a 0-form symmetry. In that case, a map of spectra is induced by any group homomorphism .
In the present case, where we have theories with 1-form global symmetries, it is straightforward to adapt this notion of anomaly interplay (which is just pullback in cobordism). Again, the crucial fact we use is that the nerves associated with the 1-form symmetries are themselves just ordinary (topological) groups, between which we can define group homomorphisms. Letting denote the subgroup embedding, there is an associated map between symmetry types, and an induced pullback map between the cobordism theories,
[TABLE]
Note that, since is a contravariant functor, the map between anomaly theories goes in the opposite direction to the subgroup embedding that we started with. Moreover, there is a pullback diagram for the whole short exact sequence characterizing , which encodes the notion of âanomaly interplayâ:
[TABLE]
This anomaly interplay diagram can be used to track ât Hooft anomalies in the 1-form global symmetries, through the âintegrating inâ of a charge-2 boson.
Since all the generators of the cobordism groups can be represented by characteristic classes, we can represent the pullback by its action on the characteristic classes. The non-trivial action of is encoded in
[TABLE]
Trivially, maps and to themselves.
The most interesting pullback relation is Eq. (4.17), which says that a -valued local anomaly pulls back to a -valued global anomaly. This is somewhat analogous to the interplay studied in Davighi:2020bvi ; Davighi:2020uab between local anomalies and global anomalies for 4d 0-form symmetries, where the maps there corresponded to pulling back exponentiated -invariants.
To see how the interplay works in this example involving 1-form symmetries, which is arguably simpler than the story for chiral fermion anomalies, we follow the general methodology set out in Ref. Davighi:2020uab . To wit, we start with a 5-manifold representative of a class in that is dual to (i.e. on which evaluates to 1 mod 2). We choose
[TABLE]
equipped with a background 2-bundle that has support only on , and a background 2-bundle that has support only on . Specifically, the 2-form connection has non-trivial 2-holonomy round , viz 1 mod 2. The 2-form connection on can be written , , where is the volume form on such that , is any connection on the factor, and parametrizes . The important thing is the flux relation
[TABLE]
Recalling , integrating gives mod 2. When is odd, is not nullbordant, and the background fields cannot be simultaneously extended to any 6-manifold that bounds.
Now, to show that this global anomaly is the pullback under of the local anomaly , we simply use the subgroup embedding to embed the 2-connection in an 2-connection . One can take
[TABLE]
The 5-manifold (4.19) equipped with these structures and can be regarded as the pushforward in bordism of the that we started with, . This is nullbordant in ; it can be realised as the boundary of a six-manifold , where is one half of a 3-sphere that is bounded by , to which the electric 2-form connection (4.21) can now be extended with .161616We emphasize that is no longer a flat connection when extended into the bulk, even though it restricts to a flat connection on the boundary of . Using Stokesâ theorem, we thus have
[TABLE]
obtaining the same phase from as we did from .
On the other hand, we only need to see that the characteristic class reduces to the class when we restrict to a flat 2-bundle with mod 2 2-holonomy, in order to show that pulls back to . To see this most clearly, it is best to use the language of cochains instead of differential forms. By identifying with , we represent the 2-form gauge field by a real 2-cochain . If the gauge field is flat, must be trivial as a cochain valued in . In other words, is an integral 3-cochain, whose cohomology class can be identified with . To use this flat 2-cochain to describe a 2-form gauge field, we further impose that it must have mod 2 2-holonomy, i.e. must be half-integral-valued, not just any real cochain. Then is an integral cochain whose mod 2 reduction defines a cohomology class in that coincides with . As , where is the Bockstein homomorphism in the long exact sequence for cohomology
[TABLE]
induced by the short exact sequence , it can be represented by the mod 2 reduction of . But this is exactly the same as which represents . Therefore, the mod 2 reduction of is when we embed the 1-form symmetry inside the 1-form symmetry.
There will be no further examples of anomaly interplay in the present paper; in particular, we do not consider any examples with non-trivially fibred 2-groups (although see footnote 28 for some comments along these lines).
5 QED anomalies revisited
The Maxwell examples in the previous Section exhibited only 1-form global symmetries. In this Section, we move on to theories with both 0-form and 1-form global symmetries that are fused together in a non-trivial 2-group structure.
Quantum electrodynamics (QED) in 4d with certain fermion content furnishes us with such a theory. Here both the 0-form and 1-form symmetry groups are , and the 2-group structure is non-trivial when there is a mixed ât Hooft anomaly between the global and gauged 0-form currents, as was discovered in Cordova:2018cvg . There is a further possible ât Hooft anomaly in the 2-group structure that comes from the usual cubic anomaly for the 0-form symmetry, but, rather than being a -valued local anomaly, the 2-group structure transmutes this cubic anomaly to a discrete, -valued global anomaly Cordova:2018cvg , with given by the modulus of the integral Postnikov class of the 2-group. After reviewing the physics arguments for these statements, we derive them from the cobordism perspective. Our spectral sequence calculations reproduce the order of the finite anomaly.
5.1 From the physics perspective
Consider a system of Weyl fermions coupled to a gauge group with dynamical gauge field . Assuming that a fermion with unit charge is present, the electric 1-form symmetry is broken completely. This leaves only the magnetic 1-form symmetry, with 2-form current as in (4.2), where . Given enough Weyl fermions, one can find a global 0-form symmetry that does not suffer from the ABJ anomaly. It is possible, however, that upon coupling a background gauge field to this global 0-form symmetry, there is an operator-valued mixed anomaly that is captured by the anomaly polynomial term
[TABLE]
where is the field strength for the background gauge field . It describes, via the usual descent procedure, the shift in the effective action under the background gauge transformation , with a -periodic scalar, by
[TABLE]
It was realised in Ref. Cordova:2018cvg that we should not interpret this term as an anomaly, but rather as a non-trivial 2-group structure.
To see why this is the case, we first couple a 2-form background gauge field , which satisfies the usual normalisation condition
[TABLE]
on closed 3-manifolds, to the magnetic 1-form symmetry via the coupling
[TABLE]
Then the potential anomaly (5.2) can be cancelled by modifying the background gauge transformation for from an ordinary 1-form gauge transformation , that is independent of the 0-form gauge transformation of , to a 2-group gauge transformation
[TABLE]
provided that we identify with .171717This is well-defined because it can be shown that is always even. Here the gauge transformation parameter is a properly normalised gauge field.
This modified transformation mixes the magnetic 1-form symmetry with the 0-form symmetry, and encodes the 2-group structure at the level of the background fields. In our quadruplet notation, the 2-group is
[TABLE]
with the Postnikov class , where we use the fact that with continuous coefficients is isomorphic to via the universal coefficient theorem. The nerve of such a 2-group is the extension
[TABLE]
The non-trivial 2-group structure can also be seen without turning on the background gauge fields explicitly. If we write for the 1-form current associated with , then the Ward identity takes the non-trivial form
[TABLE]
where are the components of the magnetic 2-form current . This Ward identity shows how the 0-form and 1-form currents are fused, realising Eq. (1.3).
The 2-group symmetry can still suffer from an ât Hooft anomaly. Recall that, when , the anomaly polynomial for our system of Weyl fermions is
[TABLE]
where is the first Chern class of the bundle, and is the first Pontryagin class of the tangent bundle, where is the curvature 2-form. Before we continue, it is important to discuss the role of these anomaly coefficients in the (co)bordism context, still for the case . NaĂŻvely, one might think that the cubic anomaly coefficient and the mixed -gravitational anomaly coefficient are the two integers that classify anomalies according to cobordism, i.e. that and can be chosen as generators of the group . However, this is not the correct identification because and are not quite independent. It can be shown that
[TABLE]
are independent integral cohomology classes.181818That is integral is evident from the definition of . To see that is integral, we observe that it is the anomaly polynomial for a Weyl fermion with charge , and then apply the AtiyahâSinger index theorem. In terms of these basis generators, we can write as
[TABLE]
Again, by the AtiyahâSinger index theorem, must be integral. Since and are integral basis generators, we can deduce that the two integers that label are
[TABLE]
Continuing, letâs switch the Postnikov class back on, and couple the background 2-form gauge field to the 1-form symmetry. We are free to add a GreenâSchwarz counter-term
[TABLE]
to the action. Under the 2-group transformation (5.5), the effective action shifts if the anomaly coefficients and are non-zero, by
[TABLE]
Since one may choose the counterterm coefficient to be any integer, one sees that is only well-defined modulo  Cordova:2018cvg . It follows that the integer that we claim classifies the anomaly is not really valued in , being well-defined only modulo . This corresponds to a global anomaly in the 2-group symmetry, that is valued in the cyclic group
[TABLE]
We emphasize that, even though is well-defined modulo (as was derived in Cordova:2018cvg ), the discrete group that classifies the global anomaly has order (and not ). The mixed gravitational anomaly remains a -valued local anomaly.
5.2 From the bordism perspective
In this Section we show how the ât Hooft anomalies afflicting the 2-group global symmetry, that we have just described, can be precisely understood using cobordism. To do so, we need to compute the generalized cohomology groups
[TABLE]
which are built from and , for each value of the Postnikov class . These abelian groups detect and classify all possible anomalies for this 2-group symmetry type.
To compute these bordism groups, we follow the general strategy outlined in §3.2. We first apply the cohomological Serre spectral sequence to the fibration
[TABLE]
to calculate the cohomology of in relevant low degrees. Then we will convert the result into homology groups, which are fed into the AtiyahâHirzebruch spectral sequence for the fibration , for which the second page is , to compute the spin bordism.
So, to begin, the page of the cohomological Serre spectral sequence for the fibration (5.17) is given by
[TABLE]
Using , given in Table 3 of Appendix A.1, we can construct the page as shown in the left-hand side of Fig. 2. In fact, what is shown there is the page, since the entries are sparse enough that there are no non-trivial differentials in the region we are interested in until page .
The differentials , , and shown on the left-hand diagram of Fig. 2 are linear in the Postnikov class (see the Appendix of Ref. Kapustin:2013uxa ). More precisely, we write
[TABLE]
using the universal coefficient theorem and the fact that . Here, we include the superscript to emphasise that this comes from our 1-form symmetry part. From this, we can write the entry in the Serre spectral sequence as . Then, the differential is given by âcontractionâ (adopting the terminology of Kapustin:2013uxa ) with the Postnikov class
[TABLE]
where we make use of the fact that the Postnikov class is a cohomology class
[TABLE]
Similar arguments apply for the differentials and . Thus, , , and map to , resulting in the page as shown, where . We can then read off the integral cohomology groups to be
[TABLE]
where the notation denotes an extension of by , viz. a group that fits in the short exact sequence . We compute the mod 2 cohomology by the same method in Appendix A.2.
Heuristically, one can also argue for the form of and as follows (c.f. Appendix B.6 of Ref. Lee:2020ewl ). Let us start with a representative of a generator of the cohomology group , and ask what it becomes when is the fibre of . Given the normalisation condition (5.3), the 3-form
[TABLE]
represents the generator of . This is true when stands on its own. However, when we pass to the 2-group and take to be the fibre of , then is not gauge-invariant under (5.5) for a general Postnikov class, and cannot be a representative of any cohomology class for . We can remedy this by modifying the definition of to
[TABLE]
The trade off is that the gauge-invariant is no longer closed; instead
[TABLE]
where is the first Chern class of the bundle, and can be taken as the generator of . The 2-group relation (5.23) implies that both the differentials and in Fig. 2 map to .191919The extra minus sign comes from the convention used to define the Postnikov class.
From this cohomological starting point, we can proceed to compute the spin bordism groups. We find it is helpful to split the discussion into the cases where is even or odd, for which the bordism group calculations are tackled using different tricks. As a warm up, we first consider the simplest case where is zero, corresponding to a 0-form and 1-form symmetry that do not mix.
5.2.1 Zero Postnikov class
We first consider the trivial toric 2-group where is a simply a product space . As is a product, we can use the KĂŒnneth theorem to determine the homology groups of from the homology groups of each factor, given by Eq. (A.2) and Table 3 in Appendix A.1. We obtain
[TABLE]
and construct the second page of the AHSS as shown in Fig. 3, with non-trivial differentials on the page indicated by coloured arrows.
The differentials on the zeroth and the first rows are the composition and , respectively, where is reduction modulo 2. To compute the action of these differentials, we need to know how the Steenrod squares act on the mod 2 cohomology ring of the product . From the mod 2 cohomology rings of and given by (A.3) and (A.4), we obtain
[TABLE]
where and are the unique generators in degree and , respectively. In our diagram, the red maps correspond to , the blue maps to being a generator of , and the magenta maps to . The differential depicted in the page from in the diagram must be non-trivial by comparing with the result computed with the Adams spectral sequence. We can then read off the bordism groups in lower degrees, which we collect below in Table 1. We piece together the cobordism group
[TABLE]
which classifies anomalies for this symmetry type. The pair of -valued local anomalies anomalies just corrresponds to the usual cubic and mixed gravitational anomalies associated with the 0-form symmetry. The presence of the 1-form symmetry here plays no role in anomaly cancellation for this dimension.
5.2.2 Even Postnikov class
Now we turn to the case where is a non-zero even integer. Continuing from the cohomology calculation above, summarised in Eq. (5.20), there are two options for the extension ; either the trivial extension or the non-trivial one , which are non-isomorphic. By comparing the mod 2 cohomology calculated from applying the universal coefficient theorem to (5.20) and the one calculated directly from the Serre spectral sequence, one can show that the correct extension is the direct product . The integral homology for is then
[TABLE]
and the mod 2 homology is
[TABLE]
The page of the AHSS is then given by Fig. 4. We observe that there are non-vanishing differentials already on the page (which will not be the case when we turn to the case of odd ). These non-trivial differentials make the spectral sequence easier to compute, as follows.
The differentials and on the second page are duals of the Steenrod square , while the differential is a mod 2 reduction followed by the dual of teichner1993signature . The Steenrod square dual to sends a unique generator in to a unique generator in (c.f. Appendix A.2), so must be the non-trivial map from to , killing off both factors. Similarly, the Steenrod square dual to acts on the unique generator of that comes from the generator of the fibre, which we will also label by . The image is a generator of that comes from of . It generates the factor in the mod 2 cohomology that is a reduction from the factor in the integral cohomology, and not the factor since it is the factor that arise from the fibreâs contribution. Therefore, both and must be non-trivial, with , as indicated in the page in Fig. 4.
Before continuing, we pause here to emphasize the importance of using (spin) cobordism, rather than just cohomology, to study ât Hooft anomalies in these theories. At the level of cohomology, the existence of a non-trivial -valued cohomology class might suggest there is a non-trivial SPT phase on 5-manifolds , or equivalently a non-trivial anomaly theory for the corresponding 4d theory, with partition function
[TABLE]
where is pulled back to . But, one can use Wuâs relation to trade the Steenrod square operation for a StiefelâWhitney class Wu1950 , viz.
[TABLE]
If we now restrict to being a spin manifold, then is trivial and we immediately learn that this SPT phase is trivial. Of course, this âtrivialisationâ of the SPT phase corresponding to the cohomology class in is automatically captured by the spectral sequence computation for spin bordism, by the non-triviality of the map on turning from the second to the third page of the AHSS.
Continuing with the bordism computation, we now find that the entries in the range stabilise on this page, whence we can read off the spin bordism groups for 2-group symmetry with even Postnikov class up to degree-5, which are listed in Table 1. There is still an undetermined group extension in but we will argue from the physics point of view in Section 6 that it must be the non-trivial extension . In summary, our results are those given in Table 1.
5.2.3 Odd Postnikov class
We now turn to the case where is odd, for which the spin bordism computation turns out to be rather more difficult, technically. Firstly, when is odd, the extension is unambiguously (which is, for odd , isomorphic to ). The integral homology of is as written in Eq. (5.27), and the mod 2 homology groups are now
[TABLE]
by the universal coefficient theorem. Now we feed these results into the AHSS for the trivial point fibration . The has no non-trivial differentials in the range of interest.
The page is shown in Fig. 5, with a single potentially non-trivial differential in the range of interest, . If this differential were trivial, then would equal , and would equal . The factor of in would correspond to the global anomaly that we described in the previous Section, which we know to be valued in as in (5.15). The extra factor of would presumably correspond to a further global anomaly that we have not seen so far by physics arguments. On the other hand, if this differential were the non-trivial map, then would equal , agreeing precisely with the physics account, and would just equal .
It turns out that this all-important differential is non-trivial. But being a differential on the third page, there are no straightforwardly-applicable formulae (analogous to the formulae in terms of Steenrod squares that are available on the second page) that we can use to compute it directly. Indeed, our usual AHSS plus ASS techniques are not sufficient to constrain this differential. This differential can nonetheless be evaluated using different arguments,202020We are very grateful to Arun Debray for sharing this ingenious argument to compute this differential. Appendix C is written by Arun Debray. which we present in Appendix C. The gist of the argument is as follows.
The central character is a long exact sequence in bordism groups,212121See Appendix E of Ref Debray:2023yrs for a related theorem. The proof of the theorem used in this paper, and that of Ref Debray:2023yrs , will appear in future work DebrayNEW . analogous to the Gysin long exact sequence in ordinary homology, of the form
[TABLE]
Here is the pullback bundle (which is rank 2) of the tautological line bundle along the quotient map , is the associated sphere bundle of , and is the Thom spectrum of the virtual bundle. This long exact sequence in bordism is extremely powerful: by proving that the groups and both lack free and 2-torsion summands, one learns that the group in the middle, which is our bordism group of interest, also lacks 2-torsion. The factor discussed above is therefore absent (from which we learn that it must be killed by the differential which is therefore non-trivial).
Putting things together, we can thus extract all the spin bordism groups for 2-group symmetry , with odd Postnikov class , up to degree-5.
5.2.4 The free part
When is non-trivial, regardless of its parity, we can also show that the free part of is given by coming from the entry . First, observe that the free part of the , if there is one at all, can only come from this entry as any other entry from the diagonal is either trivial or pure torsion. Next, we need to show that is non-trivial. The only differential that could kill it along the way is . But this differential must be trivial because is pure torsion (as , given by in (5.27), is pure torsion), and if is pure torsion. Therefore, there is one free factor in the diagonal with , and we obtain
[TABLE]
From these results, we piece together the cobordism group that classifies anomalies for this global 2-group symmetry, valid for even and odd :
[TABLE]
In terms of the anomaly coefficients, the space of anomaly theories is classified by
[TABLE]
in agreement with the results of the previous Subsection. To reiterate, the local anomaly associated with (i.e. whose coefficient is the sum of charges cubed) becomes a global anomaly when the 0-form and 1-form symmetries are fused into a 2-group defined so that the mixed ât Hooft anomaly () vanishes. The mixed gravitational anomaly associated with remains a -valued local anomaly.
The result for the spin bordism group in degree-3 in Table 1 is also interesting, implying the presence of an novel global anomaly structure for the corresponding symmetry in two dimensions. We discuss this from a physics perspective in §6.
6 Abelian 2-group enhancement in two dimensions
In this Section, we consider 2d avatars of the 4d 2-group anomalies that we have been discussing. In this lower-dimensional version we will find some interesting differences.
In §5.2, we computed the spin bordism groups for the 2-group symmetry . In particular, we obtained
[TABLE]
where . With the fourth bordism group given by , we can put things together to get the cobordism group
[TABLE]
This is similar to the result for that was pertinent to anomalies in 4d, except for the fact that the global anomaly is here âtwice as fineâ as the anomaly in 4d.222222The -valued local anomaly here is simply the gravitational anomaly associated with in the degree-4 anomaly polynomial, which can always be cancelled by adding neutral fermions and so plays no further role in our discussion. That extra division by 2 will correspond to a subtle new global anomaly associated with the spin structure, which will be our main interest in this Section.
Before we discuss the global anomaly in more depth, let us first discuss the physics interpretation of a 2-group symmetry in two dimensions, which is rather different to the 4d case. Recall that in 4d, the 1-form symmetry was identified with the magnetic 1-form symmetry, with 2-form current proportional to . But in general spacetime dimensions, the magnetic symmetry is a -form symmetry, and so there is no such symmetry in 2d. There is, however, a trivially conserved â1-form symmetryâ whose 2-form current is simply , the volume form. This symmetry does not act on any line operators in the theory and so one should not think of it as a physical 1-form symmetry â but it will play a role in what follows.
Now suppose there is a global 0-form symmetry with background gauge field with curvature . If we first cancel the pure gravitational anomaly by adding neutral fermions, the local anomalies in are captured by the degree-4 anomaly polynomial
[TABLE]
The usual ât Hooftian interpretation of would be that, if one tries to gauge , the anomaly âbreaksâ the symmetry in the quantum theory. However, Sharpe re-interprets as indicating a weaker 2-group symmetry structure in the 2d quantum theory Sharpe:2015mja , corresponding to the extension . In other words, in 2d one can invoke the auxiliary 1-form symmetry to trade an anomaly in a 0-form symmetry for a 2-group structure. This is of course just a simpler version of the 4d situation described in §5.1, in which a mixed ât Hooft anomaly corresponding to a term was re-interpreted as signalling a weaker 2-group symmetry.
But in this Section we will see that, in the case that one needs a spin structure to define the field theory, it is not always possible to completely absorb the ât Hooft anomaly associated with by a well-defined 2-group structure. Rather, there can be a residual -valued anomaly left over, which one should interpret as an ât Hooft anomaly in the 2-group symmetry itself.
6.1 âSpin structure anomaliesâ in two dimensions
To see how this works, let us now be more precise and set . Letting denote the charges of a set of (say, left-handed) chiral fermions, the anomaly polynomial is
[TABLE]
where is the first Chern class of the bundle. By inflow, one can use the associated ChernâSimons form to compute the variation of the effective action under the background gauge transformation ,
[TABLE]
Now, let us try to implement the philosophy above, and trivialise this anomaly by promoting to a 2-group global symmetry by fusing with . We turn on a background 2-form gauge field that couples to the trivially conserved current , via the coupling
[TABLE]
because .
We consider the and symmetries to be fused via the toric 2-group structure , where is the Postnikov class. This prescribes the by-now-familiar transformation (5.5) on the background gauge fields, under which
[TABLE]
We see that the local anomaly associated with the term in is completely removed iff
[TABLE]
But, since the Postnikov class is neccessarily integral, this is possible only when is an even integer. Thus, there is an order 2 anomaly remaining when is odd.
This chimes perfectly with our computation of the bordism group in §5.2, that we quoted at the beginning of this Section, which implies that there is in general a mod global anomaly. If one imagines fixing the 2-group structure, in other words fixing an integer , then one can vary the fermion content and ask whether there is a global ât Hooft anomaly for the 2-group . We can think of the coupling term (6.5) as a GreenâSchwarz counterterm, which effectively shifts the anomaly coefficient by
[TABLE]
and so the residual anomaly is clearly
[TABLE]
This is the -valued global anomaly in the 2-group symmetry that is detected by . Even if we choose the minimal value for , a mod 2 anomaly persists iff
[TABLE]
where is the number of fermions with odd charges.
One can offer a different perspective on this anomaly by thinking about a generator for , which makes it more transparent how this mod 2 anomaly is related to the requirement of a spin structure. If we look back at the AHSS in Fig. 5, we see that the factor of that ends up in comes from the element, which stabilizes straight away from the page. Since , this suggests a generator for this factor can be taken to be a mapping torus
[TABLE]
with on the factor, corresponding to an odd-charged monopole, and the non-bounding spin structure on the , corresponding to a non-trivial element of . This would suggest that the transformation by , which counts fermion zero modes, is anomalous on with an odd-charged monopole configuration for the background gauge field.232323We do not believe, however, that this anomaly can be detected from the torus (in this case, just a circle) Hilbert space, using the methods set out in Ref. Delmastro:2021xox . This is because a system with one Weyl fermion with unit charge (together with a Weyl fermion of opposite chirality to cancel the gravitational anomaly) will have an even number of Majorana zero modes, which allows construction of a -graded Hilbert space even in the sector twisted by . (The failure to construct a -graded Hilbert space is one sign of anomalies involving the spin structure, as studied in Delmastro:2021xox .)
We can of course see this from an elementary calculation. For our set of left-handed chiral fermions with charges , the index of the Dirac operator on , which recall is the number of (LH minus RH) zero modes of the Dirac operator, is equal to
[TABLE]
by the 2d AtiyahâSinger index theorem. Thus, the total number of zero modes , which is congruent to mod 2, satisfies
[TABLE]
Choosing to be odd on , as we are free to do, we see that , which counts these zero modes, flips the sign of the partition function when
[TABLE]
For such fermion content, the 2-group symmetry and are therefore equivalently anomalous. (From the 2-group perspective, the anomalous transformation corresponds simply to the element .242424 The existence of this global -valued anomaly, despite the fact the anomalous gauge transformation is clearly connected to the identity, further evidences the fact that global anomalies do not require the existence of âlarge gauge transformationsâ captured by, say, (when restricting the spacetime topology to be a sphere). This disagreement between homotopy- and bordism-based criteria for global anomalies was discussed in Ref. Davighi:2020kok . In this case, the global anomaly arises because an otherwise local anomaly is only incompletely dealt with by the GreenâSchwarz mechanism. )
This should be contrasted with the situation in four dimensions that we examined in §5.1. In that case, the corresponding formula for the number of fermion zero modes, on say , is . But, for a spin 4-manifold, the AtiyahâSinger index theorem implies that is an even integer, and moreover cancelling the local mixed gravitational anomaly . The index and therefore the number of zero modes is then necessarily even, meaning there is no analogous anomaly in . This is why the spin bordism calculation exposes âonlyâ a mod anomaly in the 4d case, but the finer mod in 2d.252525A more mundane way to understand this difference is therefore simply as a consequence of the normalisation of the various characteristic classes on spin manifolds of the relevant dimension.
Example: the 3-4-5-0 model.
To furnish an explicit example of a theory featuring this irremovable anomaly, we consider the so-called â3-4-5-0 modelâ; that is, a gauge theory in 2d with two left-moving Weyl fermions , with charges and , and two right-moving Weyl fermions , with charges and [math]. The theory is free of both gauge and gravitational anomalies, and so we can ask about its global symmetries. There is a somewhat trivial global symmetry rotating the neutral fermion (with charge ). NaĂŻvely, there are two further global symmetries, call them and , that act on the remaining fermions, whose charges and can be chosen independently in one basis as follows:
One immediately sees that coincides with the gauge charge. Thus, the transformation can be undone by a gauge transformation. Thus, the correct global symmetry is a product with charges given by:
The corresponding ât Hooft anomaly for each factor of the global symmetry is odd as desired.
6.2 Non-spin generalisation
We have argued that the order 2 anomaly just described, for a 2d theory with 2-group symmetry , is intrinsically related to the requirement of a spin structure and the use of spin bordism. More broadly, all the 2-group anomalies that we study are sensitive to the full choice of tangential structure â this is especially the case when computing the order of a finite global anomaly.
To develop this idea further, we consider a theory defined with the same 2-group structure, but now using a Spinc structure. Recall that in dimensions the group Spin is , defined by identifying the element with the order 2 element of a symmetry, that we here take to be an auxiliary global symmetry used to define spinors; using a Spinc rather than a Spin structure allows us to define our fermionic theory on non-spin manifolds (indeed, all orientable manifolds up to and including dimension 4 admit a structure).
Having already computed the integral homology (5.27) of in §5.2, and using the well-known result for the -bordism groups of a point bahri1987eta (see the first row in Table 2), it is straightforward to use the AHSS
[TABLE]
to compute the bordism groups of interest. The page is shown in Fig. 6. Things could not be much simpler; in the region of interest, all differentials are vanishing and all the elements on up to the diagonal stabilize to the last page. The bordism groups are recorded in the second row of Table 2. There is an extension problem that we cannot resolve using the AHSS for , but it is not relevant to our discussion here.
It is curious that compared to . One might naĂŻvely expect the same classification because the anomaly coefficient is shifted by for any from the 2-group transformation of the GreenâSchwarz counter-term . The puzzle can be resolved by correctly identifying the four integers in that properly classify all perturbative anomalies, before turning on the 2-group structure.
Let be the field strength for the background gauge field of the 0-form global symmetry as before, and let denote the field strength associated to the connection.262626We use the convention for connections, typical in the physics literature (see e.g. Seiberg:2016rsg ), whereby corresponds to a properly normalised connection, and itself can be âhalf-integralâ. Consider left-moving fermions with charges under the ordinary global symmetry and with charges under the connection. There is no constraint on the , but the must all be odd integers. The anomaly polynomial associated to this matter content is given by
[TABLE]
where is the first Pontryagin class of the tangent bundle. The four anomaly coefficients that accompany each term in are not strictly independent from one another, so they cannot be taken as labels for each factor of in .
To extract a set of correct integer labels, we first choose an integral basis for the anomaly polynomial in terms of linear combinations of , , , and . One option is
[TABLE]
We know that is integral because is just the signature of which is integral (and takes a minimal value of , e.g. on ) for any orientable 4-manifold . To see the integrality of , we apply the AtiyahâSinger index theorem for a connection coupled to a single left-moving fermion with and . Next, applying the same index theorem to a single left-moving fermion with and tells us that is an integer. Finally, the class is integral by definition, and indeed it takes a minimal value of . To see this, take with such that on a subspace. Hence is the generator of , so generates the top cohomology, which evaluates to 1 on the fundamental class .
In terms of these basis elements, the anomaly polynomial can be written as , where the variables are
[TABLE]
and our arguments above tell us that each must be an integer. This is clearly the case for and . To see that is an integer, recall that are all odd. So when is odd, is even, and vice versa. So their product must be even. Similarly, to see that is an integer, we observe that and are both even and differ by 2, so one of them must be divisible by . Hence their product is divisible by .
When we turn the 2-group structure back on, the shift in anomaly coefficients induced by the GreenâSchwarz counter-term amounts to a shift
[TABLE]
It is now obvious that the local anomaly labelled by the integer is truncated down to a mod anomaly. If one first takes care to cancel the anomalies involving the connection, which amounts to choosing charges such that , , and vanish, then we find that the 2-group structure can indeed be used to completely capture the remaining ât Hooft anomaly in the ordinary 0-form global symmetry, in the sense proposed by Sharpe Sharpe:2015mja . This should be contrasted with the residual order 2 anomaly (6.7) that cannot be cancelled in this way for the theory defined with a spin structure.272727At the level of (6.7), in the case we can think of the charges for the ordinary 0-form symmetry as being âtwistedâ by the charges , which are necessarily non-zero. This makes the âeffectiveâ anomaly coefficient , analogous to appearing in (6.7), be always even.
7 Conclusion and outlook
In this paper, we explored anomalies in 2-group global symmetries in field theories in and dimensions using the cobordism classification. We compute the bordism groups and of the classifying space of the nerve of , with the tangential structure or Spin. The torsion part of the former group gives us the global anomaly while the free part of the latter gives the perturbative anomaly. Throughout the work, we focus on 2-groups whose 0-form and 1-form symmetry parts are compact, connected abelian groups ergo tori, which we refer to as âtoric 2-groupsâ.
As a warm-up example, we first looked at Maxwellâs theory in 4d, on a general orientable manifold (not necessarily spin). The 2-group symmetry in this theory contains only the 1-form symmetry part, being a product of the electric 1-form symmetry and the magnetic 1-form symmetry. The familiar mixed anomaly Gaiotto:2014kfa between these two factor can be given in terms of a bordism invariant in the free part of . There are also three global anomalies, probed by the torsion subgroup of , which can only be seen on non-spin manifolds and can be interpreted as mixed anomalies between the 1-form symmetries and gravity. We also studied anomalies when one of the factor breaks explicitly to its cyclic subgroup due to the presence of a charged scalar, and related them to anomalies in pure Maxwellâs theory via anomaly interplay.
When both the 0-form and 1-form symmetry constituents of are , with the Postnikov class given by , we used a variety of techniques to prove that the relevant spin bordism groups are
[TABLE]
The free part of the sixth bordism group signals the existence of a perturbative anomaly in 4d, which is just the mixed anomaly between the 0-form symmetry and gravity. On the other hand, the perturbative cubic anomaly for 0-form symmetry that would be there were to vanish now transmutes into a discrete global anomaly captured by the torsion part of the 5th bordism group. These cobordism computations are in perfect agreement with the physics argument put forward by Córdova, Dumitrescu, and Intriligator Cordova:2018cvg .
We next discussed the fate of anomalies for a 0-form symmetry in 2d, for both spin and non-spin manifolds. We revisited the construction, originally due to Sharpe Sharpe:2015mja , whereby this 0-form symmetry is enhanced to a 2-group symmetry by mixing it with a 1-form symmetry whose current is given by the top form on the underlying manifold. When the anomaly coefficient for the pure anomaly is even, we found that, by choosing the Postnikov class of the enhanced 2-group to be , the anomaly can be fully absorbed away. On the other hand, in the presence of the spin structure, when is odd and restricting to spin-manifolds, the most one can do is reduce the anomaly down to an order 2 discrete anomaly, that we intrepret as an anomaly in the spin structure.
There are many future directions to pursue that are natural extensions of this work. One obvious alley is to look at more general classes of 2-group beyond our âtoricâ assumption, e.g. 2-groups whose 0-form symmetry is a non-abelian Lie group, and/or 2-groups whose 0-form or 1-form symmetry is a finite group Benini:2018reh ; DelZotto:2020sop ; Apruzzi:2021vcu ; Bhardwaj:2021wif ; Apruzzi:2021mlh ; Lee:2021crt ; DelZotto:2022fnw ; DelZotto:2022joo ; Bhardwaj:2022scy ; Carta:2022fxc . One can rigorously analyse anomalies in these 2-group symmetries using the cobordism classification. Doing so involves an extension of the mathematical machinery developed in this paper, and would offer a non-trivial check on other methods of seeing the anomalies that have been explored recently in the literature Bhardwaj:2022dyt . One can also study how these anomalies relate to the anomalies studied in this paper through a generalised version of anomaly interplay.282828A version of anomaly interplay valid for theories with 2-group symmetry can be sketched as follows. In principle, one ought to start by defining a smooth 2-homorphism between a pair of 2-groups and , which would induce a map between the corresponding bordism spectra. But our description of tangential structures in §3.2 suggests a shortcut: it is good enough to start with an ordinary (1-)homomorphism between the topological groups , discarding all information not captured by the nerves. This induces a map between cobordism groups going the other way. These ideas will be developed in future work.
Another avenue to pursue concerns a higher-dimensional generalisation of the 2d 2-groups discussed in § 6, as follows. We saw that one can cancel (almost) all pure anomalies in 2d by enhancing the 0-form symmetry to a 2-group by making use of the top-form current. One then wonders whether there is an analogous story for âtop-groupâ symmetry structures in higher dimensions, e.g. 4-group symmetry in 4d, in which a âtopological 3-form symmetryâ with is fused with the ordinary 0-form symmetry to kill the cubic anomaly. Moreover, the remnant order 2 anomaly in the 2d case is a direct result of the non-trivial role played by the spin structure. It remains to be seen whether such an interplay between âtop-groupâ symmetries and spin-structure anomalies persists in higher dimensions.
For all these anomalies, there is of course a dual interpretation concerning symmetry protected topological (SPT) phases of matter that are protected by these generalised symmetry types, which are rigorously classified by cobordism. Furthermore, it is conceivable that by extending the bordism computations to more generalised symmetries one could uncover novel anomalies/phases, and suggest corresponding new constraints on the dynamics of theories with such symmetry.
Acknowledgements.
We are deeply grateful to Arun Debray for several stimulating discussions, and for supplying a mathematical argument that enabled us to complete this paper. We thank Pietro Benetti Genolini, Philip Boyle Smith, and Ben Gripaios for helpful discussions. We are also grateful to Iñaki GarcĂa Etxebarria and Sakura SchĂ€fer-Nameki for asking questions that partly motivated this project. This work was partially carried out while both authors were at DAMTP, University of Cambridge, in which time NL was supported by David Tongâs Simons Investigator Award, and we were both supported by the STFC consolidated grant ST/P000681/1. JD has received funding from the European Research Council (ERC) under the European Unionâs Horizon 2020 research and innovation programme under grant agreement 833280 (FLAY), and by the Swiss National Science Foundation (SNF) under contract 200020-204428. NL is supported by the Royal Society of London and by the STFC consolidated grant in âParticles, Strings and Cosmologyâ number ST/T000708/1.
Appendix A Additional cohomology computations
In this Appendix, we collect results and computations for various cohomology groups.
A.1 Cohomology groups of EilenbergâMaclane spaces
In this Subsection, we collect classic results for cohomology groups of various EilenbergâMaclane spaces . The mod 2 cohomology results below are obtained by Serre in the classic paper Serre1953aP .
The mod 2 cohomology ring of an EilenbergâMaclane space is given by
[TABLE]
where is the unique generator in .
The classifying space of is a space. We have
[TABLE]
where is the unique generator in called the universal first Chern class. The mod 2 version is given simply by
[TABLE]
where is now the mod 2 reduction of the universal first Chern class. The action of the Steenrod squares are given by , which follows directly from the axioms.
The mod 2 cohomology ring of an EilenbergâMaclane space is given by
[TABLE]
where is the unique generator in .
Integral cohomology groups of can be obtained by applying the universal coefficient theorem to the homology groups computed in Ref. breen2014derived . The resulting cohomology groups are shown in Table 3 below.
A.2 Mod 2 cohomology ring of the classifying space of an abelian 2-group
In this Subsection, we use the Serre spectral sequence to calculate the mod 2 cohomlogy ring of when is the abelian 2-group studied in §5.2 with the Postnikov class .
The cohomological Serre spectral sequence induced by the fibration is
[TABLE]
The entries on the page can be computed from the results given in §A.1. There is no non-trivial differentials on the or pages in the range of degrees we are interested in, and the entries in this range propagate to the page, shown in Fig. 7. As explained in §5.2, the differentials and on the page are given by contraction with the Postnikov class . Thus, they are non-trivial if and only if is odd. In this case, the resulting page is shown on the right-hand side of Fig. 7. When is even, the entries stabilise already on the The entries in the range of interest stabilise on this page. On the other hand, when is even, the entries stabilise already on the page. In either case, we can read off the mod 2 cohomology groups to be
On the pages of the Serre spectral sequence, there are non-trivial actions on the mod 2 cohomology rings of the fibre and the base that converge to actions on (Theorem 6.15 of McCleary2000aB ). In particular, in the even case, we have
[TABLE]
and
[TABLE]
where , , , and , are now taken to be the unique generators for , , respectively.
Appendix B Additional bordism calculations
B.1 Maxwell
Here we compute the bordism groups for the symmetry type
[TABLE]
relevant to âfree Maxwell theoryâ, i.e. gauge group with no dynamical matter, which has intact electric and magnetic 1-form global symmetries, discussed in §4. Since , the classifying space for a 1-form symmetry is an EilenbergâMaclane space whose cohomology is known. Using the KĂŒnneth theorem we find the integral homology of , as recorded in Table 5. We also need the oriented bordism groups of a point 10.2307/1970136 , which are recorded in Table 6.
We next write down the AHSS for oriented bordism associated to the trivial fibration of by a point, for which the second page is given by
[TABLE]
This is shown in Fig. 8.
We want to read off the fifth and sixth bordism groups, which receive contributions from the (blue) and (orange) diagonals respectively. For the sixth bordism group, there is a single factor to consider. The only possible differential is , whose kernel is obviously isomorphic to , meaning that . Thus,
[TABLE]
For the fifth bordism group, the element stabilizes to the last page, because the only possible differential is which is therefore the zero map. For the element, it is again the map that is important. We can in fact deduce that this differential must be zero simply because it hits the zeroth column which records the bordism groups of a point, and since there is a canonical split for connected, where the second factor is the reduced bordism of , the factor must stabilize to the last page.292929Equivalently, we could have used the AHSS for reduced bordism to get the same result. This also makes it obvious that there can be no non-trivial extension, and so we can read off
[TABLE]
It is easy to read off the lower-degree bordism groups too, and we summarize the results in Table 7.
B.2 Scalar QED with charge-2 boson
Next we compute the bordism groups for
[TABLE]
relevant to QED with a charge-2 boson, which has the effect of breaking the 1-form symmetry down to a discrete subgroup. Using the KĂŒnneth theorem we find the integral homology of , as recorded in Table 8.
We next write down the AHSS for oriented bordism associated to the trivial fibration , for which the second page is given by (see Fig. 9),
[TABLE]
We want to read off and from the stabilization of elements on the and diagonals respectively. For the fifth bordism group, very similar arguments to those of B.1 tell us that and , and the canonical split again tells us that the extension problem is trivial, giving:
[TABLE]
For , there is a potentially non-vanishing differential on the fifth page, . Without knowing this differential, one cannot read off â the most we can conclude is that it equals , , or . In any case, we have that is pure torsion, which is all thatâs relevant for discussing the corresponding anomalies in 4d.
Appendix C Computation of a key differential
This Appendix is written by Arun Debray.
In this appendix, we finish the computation from §5.2.3, showing that if is a -group with 0- and 1-form parts both and -invariant equal to times the generator of , where is odd, then . After the calculation in §5.2.3, the only ambiguity is in the -torsion, which could be either [math] or , depending on the value of a in the Atiyah-Hirzebruch spectral sequence in Fig. 5. This differential resisted several of our standard techniques to address it, as did an analogous differential in the Adams spectral sequence computing the -completion of .
Here we finish the computation of the -torsion subgroup of in a different way, resolving the differentials only implicitly. Specifically, we fit into a long exact sequence (C.2) and compute enough other terms in the long exact sequence to determine . This long exact sequence is a version of the Gysin sequence for spin bordism, and one of its homomorphisms has a geometric interpretation as a Smith homomorphism, a map between bordism groups defined by sending a manifold to a smooth representative of the Poincaré dual to a characteristic class of . The general version of (C.2) was written down in Debray:2023yrs to study Smith homomorphisms and their applications to anomalies of quantum field theories; see there and Cordova:2019wpi ; Hason:2020yqf for more on Smith homomorphisms and their role in physics.
Theorem C.1** (Debray:2023yrs ).**
Let be a vector bundle of rank over a CW complex , be the sphere bundle of , be the Thom spectrum of the rank-zero virtual vector bundle , and be a generalised homology theory. Then there is a long exact sequence
[TABLE]
The map is the Smith homomorphism, but we will not need that fact; see Debray:2023yrs for more information.
Proof.
We will show that the Thom space of is the homotopy cofibre of , so there is the Puppe long exact sequence . Once this is established, (C.1) follows: , and induces an isomorphism for any generalised homology theory and shifts generalised homology down in grading by , yielding (C.1).303030The long exact sequence (C.1), and indeed this whole appendix, could have been done with , but once we apply bordism, it is conceptually helpful to replace by â is a bordism group of -dimensional manifolds, and is a bordism group of -manifolds. The degree shift must occur somewhere, and we believe our choice is clearest.
The homotopy cofibre of an inclusion of CW complexes such that the image of is a union of cells is the quotient ; in general, one can compute the homotopy cofibre of a map between CW complexes by changing and by a homotopy so that is indeed a cellular inclusion. For from the theorem statement, replace with the inclusion of into the disc bundle of ; the map is a homotopy equivalence and , and the CW structures on and the standard CW structure on can be used to put CW structures on and such that the image of is a union of cells. Thus the homotopy cofibre of is the quotient , which is by definition . â
We are interested in , and specifically in -torsion. To avoid worrying about torsion for other primes, we localise at , meaning we tensor with the ring of rational numbers whose denominators in lowest terms are odd. If is a finitely generated abelian group, so that is a direct sum of a free abelian group and cyclic groups of prime-power order, localising at has the effect of replacing with , preserving all -torsion, and sending all odd-prime-power torsion to zero. Since we are trying to determine the -torsion subgroup of , which is a finitely generated abelian group, we lose no relevant information by localising at . Moreover, since is a flat ring, tensoring with preserves long exact sequences (i.e. -localisation is exact), so we can still apply (C.1) after -localising.
For any -group , the classifying space has a CW structure (e.g. because it is the geometric realization of a simplicial space), so we can invoke Theorem C.1 for . Quotienting by its 1-form symmetry defines a map to the 0-form part of , which on classifying spaces induces a map . Let be the pullback of the tautological complex line bundle by . Then Theorem C.1, together with exactness of -localisation, gives us a long exact sequence
[TABLE]
We will show in Theorem C.3 that and both vanish; exactness then implies as well, meaning has no -torsion, which is what we want to prove in this appendix.
Before doing so, we first need to figure out what is:
Lemma C.2**.**
The sphere bundle is homotopy equivalent to , and this homotopy equivalence identifies the bundle map with the map given by inclusion of the 1-form symmetry.
Proof.
Let be the classifying map for . Then the sphere bundle of is the pullback of the sphere bundle of the tautological line bundle , i.e. there is a homotopy pullback square
[TABLE]
The sphere bundle of a complex line bundle is the associated principal -bundle; for , this is the tautological bundle , meaning is contractible. In general, a homotopy pullback in which one of the two legs is contractible is the homotopy fibre (denoted ) of the other leg, meaning the map can be identified with the canonical map . Recall that arose from a short exact sequence of topological groups
[TABLE]
by applying the classifying space functor; specifically, , , and , and . The classifying space functor turns short exact sequences of groups into fibre sequences of spaces, meaning that for (C.4) the homotopy fibre of is homotopy equivalent to , and this homotopy equivalence identifies the canonical map with . For our specific choices of , , and , this tells us that up to homotopy equivalence, the map is the map as claimed in the theorem statement. â
Lemma C.3**.**
* and both vanish.*
Proof.
We check both of these using the Adams spectral sequence; the relevant differentials and extension questions are trivial for degree reasons, so these computations are straightforward analogues of Beaudry and Campbellâs computations in Ref. beaudry2018guide . We direct the reader to beaudry2018guide for more on our proof strategy and notation. The Adams spectral sequence computes -completed spin bordism, not -localised spin bordism, but because and are finitely generated abelian groups, this distinction is not important: the -localisation of a finitely generated abelian group vanishes if and only if the -completion of vanishes, if and only if lacks both free and -torsion summands.
For , we need as input the -module structure on in degrees and below. This was computed by Serre (Serre1953aP, , §10): is a polynomial algebra on generators in degree , in degree , and other generators in degrees too high to matter to us. Using this, one can compute that if
denotes the -module , then there is an isomorphism of -modules
[TABLE]
where is concentrated in degrees and above, hence is irrelevant for our computation. We draw this decomposition in Fig. 10, left. Liulevicius (Liu62, , Theorem 3) computes \operatorname*{Ext}_{\mathcal{A}(1)}({\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}\mathbb{Z}/2}) and BeaudryâCampbell (beaudry2018guide, , Figure 29) compute \operatorname*{Ext}_{\mathcal{A}(1)}({\color[rgb]{0,0.44,0.57}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.44,0.57}\pgfsys@color@cmyk@stroke{0.98}{0.13}{0}{0.43}\pgfsys@color@cmyk@fill{0.98}{0.13}{0}{0.43}\raisebox{0.0pt}{\rotatebox{180.0}{Q}}}), so we can draw the -page of the Adams spectral sequence in Fig. 10, right. The -page vanishes in topological degree , i.e. for , so the -page must also vanish in that degree, and we conclude .
For , using the description of the -module structure on in low degrees from §A.2 and the way StiefelâWhitney classes twist the Steenrod squares of a Thom spectrum (see (beaudry2018guide, , §3.3)), we obtain an isomorphism of -modules
[TABLE]
where and is concentrated in degrees and above, hence is irrelevant to our computation. We draw this isomorphism in Fig. 11, left.
is computed in (beaudry2018guide, , Example 4.5.6 and Figure 22), so we can draw the -page of the Adams spectral sequence in Fig. 11, right. The -page is empty in topological degree , so the -page is also empty, and we conclude . â
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