On finite symmetries and their gauging in two dimensions

Journal of High Energy Physics, Mar 2018

Lakshya Bhardwaj, Yuji Tachikawa

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On finite symmetries and their gauging in two dimensions

Accepted: March On nite symmetries and their gauging in two Lakshya Bhardwaj 0 1 3 Yuji Tachikawa 0 1 2 0 Kashiwa , Chiba 277-8583 , Japan 1 Waterloo , Ontario, N2L 2Y5 Canada 2 Kavli Institute for the Physics and Mathematics of the Universe, University of Tokyo 3 Perimeter Institute for Theoretical Physics It is well-known that if we gauge a Zn symmetry in two dimensions, a dual Zn symmetry appears, such that re-gauging this dual Zn symmetry leads back to the original theory. We describe how this can be generalized to non-Abelian groups, by enlarging the concept of symmetries from those de ned by groups to those de ned by unitary fusion categories. We will see that this generalization is also useful when studying what happens when a non-anomalous subgroup of an anomalous nite group is gauged: for example, the gauged theory can have non-Abelian group symmetry even when the original symmetry is an Abelian group. We then discuss the axiomatization of two-dimensional topological quan- Anyons; Discrete Symmetries; Global Symmetries; Topological Field Theories - tum eld theories whose symmetry is given by a category. We see explicitly that the gauged version is a topological quantum eld theory with a new symmetry given by a dual category. 3 Symmetries as categories in two dimensions HJEP03(218)9 Basic notions of symmetry categories Comments More notions of symmetry categories Groups and representations of groups as symmetry categories 4 Gaugings and symmetry categories Module and bimodule categories Duality of C(G) and Rep(G) Gauging by an algebra object 4.4 Symmetries of the gauged theory from bimodules for the algebra object Gauging of C(G) to get Rep(G) and vice versa Gaugings and module categories 4.7 (Re-)gauging and its e ect on the symmetry category 4.8 The e ect of the gauging on Hilbert space on S1 1 Introduction 2 Re-gauging of nite group gauge theories Abelian case Non-Abelian case { i { 5 More examples of symmetry categories and their gauging Symmetry category with two simple lines Symmetry category of SU(2) WZW models and other RCFTs Gauging a subgroup of a possibly-anomalous group Integral symmetry categories of total dimension 6 Integral symmetry categories of total dimension 8 Tambara-Yamagami categories 6 2d TFT with C symmetry and their gauging 2d TFTs without symmetry TFT with C symmetry on a cylinder TFT with C symmetry on a general geometry 6.4 Gauged TFT with the dual symmetry 7 Conclusions A Group cohomology Introduction Let us start by considering a two-dimensional theory T with Zn symmetry. We can gauge it to get the gauged theory T =Zn. This gauged theory is known to have a new Zn symmetry, and re-gauging it gives us back the original theory: T =Zn=Zn = T [1]. It is also well-known that this phenomenon generalizes to any arbitrary nite Abelian group G. That is, gauging a theory T with G symmetry results in a theory T =G with a new nite Abelian group symmetry G^ such that gauging the new theory by the new symmetry takes us back to the original theory: T =G=G^ = T . One natural question arises: can it be generalized to higher dimensions? Yes, according to [2], where the generalized concept of p-form symmetries the Hamiltonian of T =G. When G is non-Abelian, we will argue that the information of G in T =G is captured by operators Ui which still commute with the Hamiltonian but these operators are now in general non-unitary. These operators can be constructed by wrapping a Wilson line for G along the spatial circle. Hence, Wilson lines should be thought of as generalized symmetries for the theory T =G. In fact, we will also argue that there is a natural notion of gauging this symmetry formed by Wilson lines such that gauging T =G results back in the original theory T . This raises the following question: how do we specify a generalized symmetry that a theory can admit? In this paper, we give an answer to this question: a general nite symmetry of a two-dimensional theory is speci ed by a structure which is known to mathematicians in the name of unitary fusion categories. We prefer to call it symmetry categories.1 For the gauged theory T =G for possibly non-Abelian group G, the Wilson line operators form Rep(G), which is a symmetry category formed by the representations of G. Similarly, a general symmetry category C physically corresponds to more general line operators of T . We also discuss how a symmetry category C can be gauged. It turns out that there is no canonical way of gauging a generic symmetry category. Pick one way M of gauging the symmetry C of a theory T . Denote the gauged theory by T =M and its symmetry category by C0. It then turns out that there exists a dual way M0 of gauging C0 such that 1We do not claim that this is the ultimate concept for the 0-form nite symmetry in two dimensions; of all, we will see that symmetry categories C capture symmetries together with their anomalies. Then, the machinery we spell out allows us to compute what is the symmetry of the gauge theory T =H when we gauge a subgroup H of a possibly anomalous avor symmetry G. For example, if we gauge a non-anomalous Z2 subgroup of Z2 Z2 Z2 with a suitable choice of the anomaly, we can get non-anomalous non-Abelian symmetry D8 and Q8, the dihedral group and the quaternion group of order 8.2 In general, the symmetry C of the gauged theory is neither a group nor Rep(G) for a nite group, and we need the concept of symmetry categories to describe it. There are also vast number of symmetry categories not related to nite groups, formed by topological line operators of two-dimensional rational conformal eld theories (RCFTs). In particular, any unitary modular tensor category, or equivalently any Moore-Seiberg data, can be thought of as a symmetry category, by forgetting the braiding. We should emphasize here that this generalization of the concept of symmetry from that de ned by groups to that de ned by categories was already done long ago by other authors, belonging to three somewhat independent lines of studies, namely in the study of the subfactors and operator algebraic quantum eld theories, in the study of representation theory, and in the study of RCFTs. Each of the communities produced vast number of papers, and not all of them can be cited here. We recommend to the readers textbooks by Bischo , Longo, Kawahigashi and Rehren [6] and Etingof, Gelaki, Nikshych and Ostrik [7] from the rst two communities and the articles by Carqueville and Runkel [8] and by Brunner, Carqueville and Plencner [9] from the third community as the starting points. Our rst aim in this paper is then to summarize the content of these past works in a way hopefully more accessible to other researchers of quantum eld theory, including the authors of this paper themselves, emphasizing the point of view related to the modern study of symmetry protected topological phases. What we explain in this rst part of the paper is not new, except possibly the way of the presentation, and can all be found in the literature in a scattered form. Our second aim is to axiomatize two-dimensional topological quantum eld theories (TFTs) whose symmetry is given by a symmetry category C. This is a generalization of the 2Recently in [5], Gaiotto, Kapustin, Komargodski and Seiberg performed an impressive study of the phase structure of thermal 4d su(2) Yang-Mills theory. One important step in the analysis is the symmetry structure of the thermal system, which is essentially three-dimensional. As a dimensional reduction from 4d, the system has a Z2 Z2 0-form symmetry and a Z2 1-form symmetry, with a mixed anomaly. Then the authors gauged the Z2 1-form symmetry, and found that the total 0-form symmetry is now D8. This D8 was then used very e ectively to study the phase diagram, but that part of their paper does not directly concern us here. Their analysis of turning an anomalous Abelian symmetry by gauging a non-anomalous subgroup into a non-Abelian symmetry is a 3d analogue of what we explain in 2d. See their section 4.2, appendix B and appendix C. Clearly an important direction to pursue is to generalize their and our constructions to arbitrary combinations of possibly-higher-form symmetries in arbitrary spacetime dimensions, but that is outside of the scope of this paper. { 2 { circles together with arbitrary network of line operators from C . The rest of the paper is organized as follows. First in section 2, as a preliminary, we recall how gauging of a nite Abelian symmetry G can be undone by gauging the new nite Abelian symmetry G^, and then brie y discuss how this can be generalized to non-Abelian symmetries G, by regarding Rep(G) as a symmetry. This e ort of generalizing the story to a non-Abelian group makes the possibility and the necessity of a further generalization to symmetry categories manifest. We exploit this possibility and describe the generalization in detail in subsequent sections. Second, we have two sections that form the core of the paper. In section 3, we introduce the notion of symmetry categories, and discuss how we can regard as symmetry categories both a nite group G with an anomaly and the collection Rep(G) of representations of G. We then explain in section 4 that physically distinct gaugings of a given symmetry category C correspond to indecomposable module categories M of C, and we describe how to obtain the new symmetry C0 of the theory T =M for a given theory T with a symmetry C. Third, in section 5, we give various examples illustrating the notions introduced up to this point. Examples include the form of new symmetry categories C0 when we gauge a non-anomalous subgroup H of an anomalous nite group G, and the symmetry categories of RCFTs. Fourth, in section 6, we move on to the discussion of the axioms of two-dimensional TFTs whose symmetry is given by a symmetry category C. We also construct the gauged TFTs T =M given an original TFT T with a symmetry category C and a gauging speci ed by its module category M. Sections 5 and 6 can be read independently. Finally, we conclude with a brief discussion of what remains to be done in section 7. We have an appendix A where we review basic notions of group cohomology used in the paper. Before proceeding, we note that we assume that the space-time is oriented throughout the paper. We also emphasize that all the arguments we give, except in section 6, apply to non-topological non-conformal 2d theories. 2 2.1 Re-gauging of nite group gauge theories Abelian case Let us start by reminding ourselves the following well-known fact [1]: Let T be a 2d theory with avor symmetry given by an Abelian group G. Let us assume that G is non-anomalous and can be gauged, and denote the resulting theory by T =G. Then this theory has the Pontrjagin dual of G, such that T =G=G^ = T . avor symmetry G^, which is the Recall the de nition of the Pontrjagin dual G^ of an Abelian group G. As a set, it is given by G^ = f : G ! U(1) j is an irreducible representation g: (2.1) { 3 { Note that is automatically one-dimensional. Therefore the product of two irreducible representations is again an irreducible representation, which makes G^ into a group. G and G^ are isomorphic as a group but it is useful to keep the distinction because there is no canonical isomorphism between them. In the literature on 2d theories, gauging of a nite group G theory is more commonly called as orbifolding by G, and the fact above is often stated as follows: a G-orbifold has a dual G^ symmetry assigning charges to twisted sectors, and orbifolding again by this dual G^ symmetry we get the original theory back. This dual G^ symmetry is also known as the quantum symmetry in the literature. This fact can be easily shown as follows. Let ZT [M; A] denote the partition function HJEP03(218)9 of T on M with the external background G gauge eld A. Here A can be thought of as taking values in H1(M; G). Then the partition function of the gauged theory T =G on M is given by ZT=G[M ] / P A ZT [M; A]. Here and in the following we would be cavalier on the overall normalization of the partition functions. More generally, with the background gauge eld B for the dual G^ symmetry, the partition function is given by ZT=G[M; B] / X ei(B;A)ZT [M; A] A (2.2) (2.3) (2.4) where B 2 H1(M; G^) and ei(B;A) is obtained by the intersection pairing ei( ; ) : H1(M; G^) H1(M; G) ! H2(M; U(1)) ' U(1): The equation (2.2) says that the partition function of T =G is essentially the discrete Fourier transform of that of T , and therefore we dually have T = T =G=G^: ZT [M; A] / X ei(A;B)ZT=G[M; B]: B This statement was generalized to higher dimensions in e.g. [2]: Let T be a d-dimensional theory with p-form avor symmetry given by an Abelian group G. Let us assume that G is non-anomalous and can be gauged, and denote the resulting theory by T =G. Then this theory has the dual (d 2 p)form avor symmetry G^, such that T =G=G^ = T . B 2 Hd 1 p(M; G^). 2.2 Non-Abelian case The derivation is entirely analogous to the 2d case, except that now A 2 Hp+1(M; G) and The facts reviewed above means that the nite Abelian gauge theory T =G still has the full information of the original theory T , which can be extracted by gauging the dual symmetry G^. It is natural to ask if this is also possible when we have a non-Abelian symmetry G, which we assume to be an ordinary 0-form symmetry. { 4 { This is indeed possible3 by suitably restating the derivation above, but we will see that we need to extend the concept of what we mean by symmetry. To show this, we rst massage (2.4) in a suitable form which admits a straightforward generalization. Let us consider the case of ZT [M; A = 0] for illustration. By Poincare duality, B can also be represented as an element of H1(M; G^). Then, (2.4) can be rewritten as ZT [M ] / X g^1; ;g^n ZT=G[M; g^1; ; g^n] where i 2 f1; ; ng labels generators of H1(M ) and g^i is an element of G^ associated to the cycle labeled by i. Each summand on the right hand side, ZT=G[M; g^1; ; g^n], is then the expectation value of Wilson loops in representations labeled by g^i placed along the cycle i. Now, we can sum the G^ elements for each i separately to obtain where W reg denotes the insertion of a Wilson line in the regular representation along the cycle i. This is because, for an abelian G, the regular representation is just the sum of representations corresponding to elements g^ of G^. The relation (2.6) says that by inserting all possible Wilson lines on all possible cycles, we are putting the delta function for the original gauge eld A. We now note that the relation (2.6) holds for a non-Abelian G as well, if we insert W reg not only for the generators of H1(M ) but for the generators of 1(M ). This can be seen by the fact that tr g in the i regular representation is nonzero if and only if g is the identity. i The identity (2.6) means that the ungauged theory T can be recovered from the gauged theory T =G by inserting line operators W reg in an appropriate manner. This is analogous to the construction of the gauged theory T from the ungauged theory T by inserting line operators representing the G symmetry in an appropriate manner. Given the importance of Wilson lines in recovering the information of the ungauged theory, we assign the status of dual symmetry to Wilson lines. Let us phrase it another way. When G is Abelian, the dual (d 2)-form G^ symmetries can be represented by 1-cycles labeled by elements of G^, forming a group. When G is nonAbelian, the dual symmetry can still be represented by 1-cycles labeled by representations Rep(G) of G. We can still multiply lines, corresponding to the tensor product of the representations, and this operation reduces to group multiplication of G^ in the abelian case. But Rep(G) is not a group if G is non-Abelian. Therefore this is not a avor symmetry group, it is rather a avor symmetry something. We summarize this observation as follows: Let T be a d-dimensional theory with 0-form avor symmetry given by a possibly non-Abelian group G. Let us assume that G is non-anomalous and can be gauged, and denote the resulting theory by T =G. Then this theory has Rep(G) as the dual (d 2)-form avor symmetry `something', such that T =G=Rep(G) = T . 3That this is possible was already shown for two-dimensional theories in [3], as an example of a much more general story, which we will also review in the forthcoming sections. Here we describe the construction in an elementary language. { 5 { Symmetries as categories in two dimensions Any nite group, possibly non-Abelian, can be the 0-form symmetry group of a theory. In addition to this, we saw in the last section that Rep(G), the representations of G, can also be the (d 2)-form symmetry something of a d-dimensional theory. We do not yet have a general understanding of what should be this something in general d dimensions for general combinations of various p-form symmetries. However, at least for d=2 and p=0, we already have a clear concept for this something in the literature, which includes both groups G and representations of groups Rep(G) and much more. In this section we explain what it is. 3.1 Basic notions of symmetry categories In two dimensions, a 0-form nite symmetry element can be represented by a line operator with a label a. Inserting this line operator on a space-like slice S corresponds to acting on the Hilbert space associated to S by a possibly non-unitary operator Ua corresponding to the symmetry element. Moreover, Ua commutes with the Hamiltonian H associated to any foliation of the two-dimensional manifold. In addition, Ua cannot change under a continuous deformation of its path. Therefore the line operator under consideration is automatically topological. Topological line operators, in general, form a structure which mathematicians call a tensor category. We want to restrict our attention to topological line operators describing a nite symmetry. Such topological line operators form a structure which mathematicians call a unitary fusion category. We call it instead as a symmetry category, to emphasize its role as the nite symmetry of unitary two-dimensional quantum eld theory. We start by stating the slogan, and then ll in the details: Finite avor symmetries of 2d theories are characterized by symmetry categories. 3.1.1 Objects The objects in a symmetry category C correspond to topological line operators generating the symmetry. More precisely, a theory T admitting the symmetry C will admit topological line operators labeled by the objects of C. Henceforth, we will drop the adjective topological in front of line operators. For any line operator labeled by an object a, we have a partition function of T with the line inserted along an oriented path C, which we can denote as h a(C) i (3.1) where the dots stand for additional operators inserted away from C. 3.1.2 Morphisms The morphisms in a symmetry category C correspond to topological local point operators which can be inserted between two lines. More precisely, consider two labels a and b, and a path C such that up to a point p 2 C we have the label a and from the point p we have { 6 { HJEP03(218)9 3.1.3 Existence of a trivial line C contains an object 1, which labels the trivial line of T . We have h 1(C) i = h i : 3.1.4 Additive structure Given two objects a and b, there is a new object a We abbreviate a a by 2a, a a Hilbert space obtained by wrapping a 3.1.5 Tensor structure h (a b)(C) i = h a(C) i + h b(C) i : a by 3a, etc. The linear operator Ua b acting on the b on a circle is then given by Ua + Ub. b in C. In terms of lines in T , we have (3.2) (3.3) (3.4) a m b = a a⊗b m b the label b. Then, we can insert a possibly-line-changing topological operator labeled by m at p. We call such m a morphism from a to b, and denote this statement interchangeably as either HJEP03(218)9 m : a ! b or m 2 Hom(a; b): The set Hom(a; b) is taken to be a complex vector space and it labels (a subspace of) topological local operators between line operators corresponding to a and b in T . From now on, we would drop the adjective topological in front of local operators. Given two objects a and b, we have an object a b in C. This corresponds to considering two parallel-running line operators a and b as one line operator. The linear operator Ua b acting on the Hilbert space obtained by wrapping a b on a circle is then given by UaUb. The trivial object 1 acts as an identity for this tensor operation. That is, there exist canonical isomorphisms a 1 ' a and 1 a ' a for each object a. We can always nd an equivalent category in which these isomorphisms are trivial, that is a 1 = 1 a = a. Hence, we can assume that these isomorphisms have been made trivial in C. Henceforth, the unit object will also be referred to as the identity object. Consider three lines C1;2;3 meeting at a point p, with C1;2 incoming and C3 outgoing. We can put the label a, b, c on C1;2;3, respectively. We demand that the operators we can put at the junction point p is given by m 2 Hom(a b; c). This label a b corresponds to a composite line as can be seen by the following topological deformation shown in gure 1. { 7 { The de nition of a b here includes a choice of the implicit junction operator where the lines labeled by a, b and a b meet. In this paper, whenever we draw gures with such implicit junction operators, we always choose the operator to be the one labeled by the identity morphism id : a b ! a Simplicity of the identity, semisimplicity, and niteness The simple objects a 2 C are objects for which Hom(a; a) is one-dimensional. In general, for any object x, there is always a canonical identity morphism from x to x which labels the identity operator on the line labeled by x. For a simple object a, the existence of the identity morphism implies that there is a natural isomorphism Hom(a; a) ' C as an algebra. We assume for simplicity that the identity object 1 is simple. We also assume that every object x has a decomposition as a nite sum x = M Naa a (3.5) (3.6) where Na is a nonnegative integer and a is simple. In other words, every object x is semisimple. Finally, we assume niteness, that is the number of isomorphism classes of simple objects is nite. Below, we will be somewhat cavalier on the distinction between simple objects and isomorphism classes of simple objects. 3.1.7 Associativity structure The data in a symmetry category C includes certain isomorphisms implementing associativity of objects a;b;c 2 Hom((a b) c; a (b c)) which we call associators.4 Fusion matrices F for the Moore-Seiberg data, and the (quantum) 6j symbols for the (quantum) groups are used in the literature to capture the data of associators. The associator a;b;c corresponds to a local operator which implements the process of exchanging line b from the vicinity of a to the vicinity of c. See gure 2. They satisfy the pentagon identity which states the equality of following two morphisms ((a b) c) d ! (a = ((a (b b) c)) c) d ! a d ! (a ((b b) c) (c d) ! a d) ! a (b (b (c (c d)) d)) (3.7) where each side of the equation stands for the composition of the corresponding associators. The pentagon identity ensures that exchanging two middle lines b, c between two outer lines a, d in two di erent ways is the same, see gure 3. 4Since we can and do choose the identity morphisms 1 a ! a and a 1 ! a to be trivial, the associator a;b;c is also trivial when any of a, b, c is trivial. { 8 { α a b c a b c HJEP03(218)9 a a b the local operator labeled by the associator morphism is obtained by squeezing the region between (a b) c and a (b c) shown above to a point. tensoring of four lines lead to the same result. R a = = R a = L a a a a a a 1 1 HJEP03(218)9 For every object a, C contains a dual object a . The line labeled by the dual object has the property that h a(C) i = h a (C~) i : Here, C~ denotes the same path C but with a reverse orientation, and the morphisms attached at the junctions on C~ need to be changed appropriately as we explain below at the end of this subsubsection. We require that the dual of the dual is naturally isomorphic to the original object: (a ) ' a. The dual operation also changes the order of the tensoring: L a a (3.8) (3.9) (3.10) (3.11) (3.12) (3.13) (a b) = b a : We demand that there are evaluation morphisms and co-evaluation morphisms5 aR : a a ! 1; aL : a a ! 1 aR : 1 ! a a; aL : 1 ! a a : aR = aL (pa 1) = aL (1 pa 1) aR = (1 pa 1) aL = (pa 1) L a gure 4. we require These label local operators corresponding to the process of folding a line operator a. See We note that aR and aL are not necessarily equal. However, we require as part of de nition of dual structure that they are related as follows where pa is an isomorphism from a to a and pa is an isomorphism from a to a . Similarly, 5We use the convention that when something is denoted by x, co-something is denoted by x. This usage is unconventional, in particular for the case of coproduct for which is de nitely the standard notation, but it reduces the amount of notations that one has to remember. = a a HJEP03(218)9 category C. condition with the associator The data of pa and pa is referred to in the literature as a pivotal structure on the fusion The evaluation and co-evaluation morphisms have to satisfy the following consistency ( aR 1) a;a ;a 1 (1 R a ) = 1 as morphisms from a to a. This ensures that a line with two opposite folds in the right direction can be unfolded as shown in gure 5. A similar identity is satis ed by aL and L a which ensures that two opposite folds in the left direction can be unfolded. Using evaluation and co-evaluation morphisms of di erent parity we can construct loops of lines (dimCC a)id : 1 (dimC a)id : 1 R !a a L !a a a a L !a 1; R !a 1: These are morphisms from 1 to 1 and hence they are proportional to the identity morphism. The proportionality factors de ne two numbers: the counter-clockwise dimension dimCC a of a and the clockwise dimension dimC a of a. See gure 6. Since we can replace the label a by a at the cost of ipping the orientation of line, we must have Indeed, this follows from (3.12) and (3.13). In fact, it turns out that we can further argue that dimCC a = dimC a ; dimCC a = dimC a: dimCC a = dimC a dim a: To see this, place a small counter-clockwise loop of line a around the \north pole" on the sphere. Let there be no other insertions anywhere on the sphere. This evaluates to (3.14) (3.15) (3.16) (3.17) (3.18) (3.19) The loop, if it contains no other operators in it, can then be shrunk and the partition function with the loop is equal to dimCC a = dim a times the partition function without the loop, with all other insertions unchanged. dimCC a ZS2 where ZS2 is the partition function on sphere. Now we can move the line such that it looks like a small clockwise loop around the \south pole" on the sphere. This evaluates to dimC a ZS2 . Equating the two expressions we nd (3.19).6 This is a further constraint on C. If the fusion category C satis es (3.19), then C is called a spherical fusion category in the literature. Since aR and aL are not necessarily equal, we have to specify whether a folding of line a to the right should be read as the morphism aR or the morphism aL . Similarly there is a speci cation of aR vs. aL . This issue can be dealt with in two ways, which are technically equivalent but have a rather di erent avor. One method. One perspective is to regard that a line is always labeled by the pair (the local orientation, an object in C). Then, a pair ("; a) and (#; a ) are isomorphic but not actually the same. We note that this distinction needs to be made even when a ' a . Then we make the rule that when a vertical line is labeled by ("; a) up to some point and then labeled by (#; a ) from that point, we insert the pivotal structure pa 2 Hom(a; a) at that point. This approach would be preferred by those who have no trouble with adding local orientations as a new datum to a topological line operator. Another method. and between aR vs. Another perspective is to think that the change between aR and aL aL is canceled by changing the nearby morphisms. This method might be preferred by those who do not want to add local orientation as a new datum to a topological line operator. We emphasize that, in this approach, the operation of exchanging a by a with a reversed orientation does not change the local operators at the junctions. Instead it changes the way the local operators at the junctions are read as morphisms in the associated symmetry category C. 6The authors thank Shu-Heng Shao for discussion related to this point. a b HJEP03(218)9 speci ed as aL . This changes the morphism from m on the left side to (pa 1 side. These two diagrams provide two di erent categorical representations of the same physical con guration. The following moves are su cient to specify what happens in any situation: R 1. Consider a morphism a equal to aL 1 a;a ;c (1 2. Consider a morphism aR equal to aR c;a ;a (n 1) : b 1 a;a ;c (1 n) : a b ! c where n = (pa 1 1) m : b ! a b ! c where m : b ! a c. This is c;a ;a (m 1) : b a ! c where m : b ! c a . This is a ! c where n = (pa 1) m : b ! c a . 3. Consider a morphism (1 m) is equal to (1 is equal to (n n) 1) a ;a;b 1 b;a;a 4. Consider a morphism (m 1) a ;a;b aL : b ! a 1 b;a;a aL : b ! c aR : b ! a aR : b ! c c where m : a a where m : b c where n = m (pa 1 1) : a a where n = m (1 pa) : b b ! c. This b ! c. a ! c. This a ! c. These moves follow from (3.12) and (3.13). We draw a picture of the rst move in gure 7. The other three moves are also described by similar pictures. 3.1.9 Unitary structure The unitary structure requires an existence of a conjugate-linear involution sending m 2 Hom(a; b) to my 2 Hom(b; a), generalizing the Hermitian conjugate in the standard We require that the evaluation and the coevaluation morphisms are related by this linear algebra. conjugate operation: aR = ( aL)y; aL = ( aR)y: (3.20) We further require my m 2 Hom(a; a) to be positive semi-de nite in the following sense: since we assumed the semisimplicity and the niteness of the number of simple objects, Hom(a; a) can naturally be identi ed with a direct sum of a matrix algebra. Then we require my m to have non-negative eigenvalues. The above positivity condition requires dim a > 0 for all a. As we will see in section 3.3.4 below, the unitarity implies sphericity. 3.2 We have several comments: Note that category theorists do not like unitary structures, since it is speci c to the base eld C while they would like to keep everything usable for arbitrary base eld. For this reason they often distinguish various concepts of operations and various structures satis ed by them, such as rigid structure, pivotal structure, spherical structure and pseudo-unitary structure.7 If we consider unitary 2d quantum eld theories (or more precisely its Wick-rotated versions which are re ection-positive), the unitary structure is the most natural one. Operator algebraic quantum eld theorists in fact work in this setting, since for them the existence of the positive-de nite inner product on the Hilbert space is paramount. Unfortunately their papers often phrase purely categorical results in the operator algebra theoretic language, which makes them somewhat harder for outsiders to digest. From this point of view their review article [6] is very helpful, where a concise translation between terminologies of two di erent schools is given. Every property given above, except the simplicity of identity, semisimplicity and niteness, is a straightforward expression of how topological lines and the junction operators associated to symmetries should behave. We impose the simplicity of identity, semisimplicity and niteness to make the situation tractable. When the semisimplicity is dropped, the category is called a nite tensor category; when the simplicity of identity is dropped, the category is called a nite multi-fusion category; when both are dropped, it is called a nite multi-tensor category. When niteness is dropped, we simply drop the adjective \ nite". Indeed, if we consider all topological lines in a given 2d theory and all topological operators on topological lines, they might not in general form a unitary fusion category. Rather, our point of view is that we take a subset of topological lines and subspaces of topological operators on the lines so that they form a unitary fusion category, and then it can be thought of as a symmetry of the 2d theory. A very similar categorical structure was introduced by Moore and Seiberg [11] in the analysis of 2d RCFTs and 3d TFTs. In the category theory they are now called unitary modular tensor categories. In fact, the unitary modular tensor categories are also unitary fusion categories, where the latter description is obtained by forgetting the braiding. 7The rigid structure posits the existence of the left dual a and the right dual a, satisfying various conditions. It can be shown that isomorphisms a ' a. In our description, the pivotal structure relates aL and aR . A pivotal structure is a ' a , and a ' a. The pivotal structure is a collection of largest eigenvalue of (Na)bc for all simple a. For the de nition of Nacb, see section 3.3.3. called spherical if dim a = dim a for all a. A spherical structure is called pseudo-unitary if jdim aj is the 3.3 Before discussing examples, it is useful to set up a few more notions: `Homomorphisms' between symmetry categories In the case of two groups G1 and G2, we have the concept of homomorphisms ' : G1 ! G2, preserving the group multiplication. Similarly, we can talk about symmetry functors ' : C1 ! C2 between two symmetry categories, together with the data specifying how the structures listed above are mapped. Among them are isomorphisms a;b 2 Hom('(a) '(b); '(a b)) which tell us how the tensor structure of C1 is mapped into the tensor structure of C2. For example, the morphisms a;b map the associator of C1 to the associator of C2. Two symmetry functors '; '0 : C1 ! C2 are considered equivalent when there is a set of isomorphisms a 2 Hom('(a); '0(a)) a b a;b = 0a;b( a b): such that categories. When a symmetry functor has an inverse, it is called an equivalence between symmetry (3.21) HJEP03(218)9 (3.22) (3.23) 3.3.2 Products of symmetry categories In the case of two groups G1 and G2, their product G1 G2 is also a group. Similarly, given two symmetry categories C1 and C2, we denote their product as C1 C2, whose simple objects are given by a1 a2 where a1;2 are simple objects of C1;2, respectively. This product is called Deligne's tensor product of categories. 3.3.3 Fusion rule of unitary fusion categories A symmetry category comes with a lot of structures. Sometimes it is useful to forget about most of them as follows. For each isomorphism class of simple objects a, introduce a symbol [a], and de ne their multiplication by [a][b] := P c Nacb[c] when a b = L c Nacbc. This makes non-negative integral linear combinations of [a]'s into an algebra over Z+ with a speci c given basis. We call this algebra R(C) the fusion ring of the symmetry category C. In the case of modular tensor categories, this algebra is also called the Verlinde algebra. We would often call this algebra as just the fusion rule of C. Let n be the number of isomorphism classes of simple objects. Then we can regard (Na)bc as n n matrices and [a] 7! Na is the adjoint representation of the fusion ring. 3.3.4 Determination of dimensions of objects The dimensions are xed by Nacb. To see this, consider the n-dimensional vector v := (dim a)a where a runs over the isomorphism classes of simple objects. Its entries are positive real numbers thanks to the unitarity. Furthermore, v is the simultaneous eigenvector of all Na's with eigenvalues dim a. Then by the Perron-Frobenius theorem, dim a is the largest eigenvalue of the matrix Na, which is guaranteed to be positive. The argument above applies both to dimC and dimCC, and therefore the sphericity is implied by the unitarity. Unitarity also guarantees dim a = dim a . We de ne the total dimension of the symmetry category C by the following formula: dim C = X(dim a)2: a Here the sum runs over the isomorphism classes of simple objects. (3.24) (3.25) HJEP03(218)9 Groups and representations of groups as symmetry categories Symmetry categories C(G; ) As an example, let us recast an ordinary group G as a symmetry category. We rst regard each element g 2 G as a simple object denoted by the same letter in the category. We g g0 := gg0; g := g 1: Taking g1;g2;g3 to be the identity maps, they clearly form a unitary fusion category, which we denote by C(G). More generally, the pentagon identity among g1;g2;g3 says that is a 3-cocycle on G valued in U(1).8 Denote the resulting fusion category by C(G; ). In the literature it is often denoted VecG. We clearly have dim g = 1. Thus the total dimension of this symmetry category is the order of the group. When 1 and 2 di er by a coboundary of a 2-cochain , we can construct an equivalence of categories between C(G; 1) and C(G; 2) using the functor speci ed using the same in (3.21). This means that in the de nition of C(G; ), one can regard 2 H3(G; U(1)). It is also clear that any unitary fusion category whose simple objects are all invertible can be made to be of this form. Summarizing, A symmetry category C whose simple lines are all invertible is equivalent to C(G; ) where G is a nite group and is an element in H3(G; U(1)). 3.4.2 C(G; ) and the anomaly As is by now familiar, this cohomology class avor symmetry in two dimensions [12, 13]. One way to see it is as follows [2]: insert a network of lines with trivalent junctions between them on the spacetime manifold . Let the lines be labeled by simple objects of C(G; ), that is by group elements. And let every junction of the form g g0 ! gg0 be labeled by the identity morphism gg0 ! gg0. Such a con guration can also be thought of as reproducing the e ects of a background connection on which has holonomies given by g on crossing transversely a line labeled by g. Now, consider a local region looking like the left hand side of gure 8. Move the lines such 8As already noted in footnote 4, in our convention g1;g2;g3 is trivial whenever any of g1;2;3 is the identity. Such a cocycle is called normalized. It is a well-known fact in group theory that group cohomology can be computed by restricting every cochains involved to be normalized. 2 H3(G; U(1)) speci es the anomaly of G HJEP03(218)9 topological line operators. If the symmetry is anomalous, they lead to di erent partition functions. that now it looks like the right hand side of gure 8. This changes the partition function by (g; g0; g00). The new background connection is just a gauge transform of the original background connection. Hence, we see that precisely captures the anomaly in the avor symmetry. Morally, this means the following: A symmetry category C includes the speci cation of its anomaly. Fixing a group G, the set of its anomalies forms an Abelian group. Notice that, in our language, C(G; ) for di erent have the same fusion ring R = Z+G. Thus, we can ask the following more general question: does the set of symmetry categories C with the same fusion ring R form an Abelian group? The answer is that we need a coproduct on R. To see this, let us recall why the anomaly of a avor symmetry forms an Abelian group from the perspective of quantum eld theory. In general, given two theories T1;2, we can consider the product theory T1 T2 which is just two decoupled theories considered as one. When Ti has avor symmetry group Gi, the product T1 T2 has avor symmetry group G1 G2. When G1 = G2 = G, we can take the diagonal G subgroup of G G and regard T1 T2 to have avor symmetry G. Now, when Ti has the anomaly i, we de ne the anomaly of T1 T2 to be the sum 1 + 2. This abstractly de nes the addition operation on the anomaly. The crucial step that does not directly generalize to symmetry categories is the existence of the diagonal subgroup G G G. In order to de ne the addition operation on the set of fusion categories sharing the same fusion rule R, similarly we need a coproduct R ! R 3.4.3 R. C(G; ) and the G-SPT phases Next, xing a 3-cocycle , let us ask what is the autoequivalence of C(G; ), that is, the self equivalence that preserves the structure as a symmetry category. Pick an autoequivalence ' : g ! '(g) with the associated g;h 2 Hom('(g)'(h); '(gh)). Clearly ' is an automorphism of G. Fixing ' to be the identity, needs to be a 2-cocycle so that it does not change . Furthermore, two such 's are considered equivalent when h h gh g they di er by a 2-coboundary, due to (3.23). Therefore can be thought of as taking values in H2(G; U(1)). Summarizing, Autoequivalence of C(G; 0) is the semidirect product Aut(G) n H2(G; U(1)). The Aut(G) part is clear: it just amounts to renaming the topological lines associated to the group operation. How should we think of H2(G; U(1))? It is telling us that instead of choosing the identity operator as the implicit junction operator for g g0 ! gg0 as done in gure 1, we can choose g;g0 times the identity. This will not change the associator but will change the partition function associated to a background connection on . This corresponds to coupling a two-dimensional theory with C(G; ) symmetry with a twodimensional bosonic symmetry protected topological (SPT) phase, which is speci ed by the 2-cocycle protected by the avor symmetry G. The 2-cocycle is also known as a discrete torsion of G. to two trivalent junctions. As an example, consider a torus with holonomies g; h around two 1-cycles. They can be represented using the topological lines as in gure 9. There, we resolved the intersection We now change the operators at the two junctions to g;h and g 1;h 1 given by the values of the 2-cocycle. In total the phase of the partition function changes by cg;h = g;h= h;g (3.26) which is the standard relation between the discrete torsion phase c on the torus and the 2-cocycle [14]. We can thus generalize as follows: Autoequivalences of a symmetry category C generalize the notion of renaming and multiplying by SPT-phases for a group symmetry. We need to keep in mind however that the phases introduced by in the general case do not have an interpretation of multiplying a SPT phase protected by C, since the product of two theories with symmetry C has symmetry C C but is not guaranteed to have symmetry C, as already discussed above. 3.4.4 Rep(G) as symmetry category Next, let us discuss Rep(G) for a nite group G. Its structure as a symmetry category is straightforward: the objects are representations of G, the morphism space Hom(R; S) (X ; ; X 0; 0 ; X 00; 00 ) = 00( ( ; 0)); (X ; ; X 0; 0 ; Y 00; 00 ) = 00( ( ; 0)); (X ; ; Y 0; 0 ; X 00; 00 ) = ( ; 00); (Y ; ; X 0; 0 ; X 00; 00 ) = ( 0 00 1)( ( 0; 00)); (X ; ; Y 0; 0 ; Y 00; 00 ) = 00( ( ; ( 0) 1 00)); (Y ; ; X 0; 0 ; Y 00; 00 ) = ( 0; ( 0) 1( 00) 1); (Y ; ; Y 0; 0 ; X 00; 00 ) = ( 0( 00) 1)( ( ( 0) 1; 00)); (Y ; ; Y 0; 0 ; Y 00; 00 ) = sgn( ) ( ( 0) 1; 0( 00) 1): As a check of the computation, we can directly con rm that these de ne a 3-cocycle on G. In the eight cases Rep(D8), Rep(Q8), KP, TY and S we discussed above, we always have H = Z2 and the resulting group G = (Z2 Z2) o Z2 is D8. To see this, regard Z2 Z2 as the group of ipping the coordinates x and y of R2 generated by (5.36) (5.37) (5.38) (5.39) (5.40) (5.41) (5.42) (5.43) (5.44) (5.45) (5.46) (6.1) can be computed using the projections given above and the associators (5.22), (5.23) and (5.24) of the original category. We nd respectively, and Z2 acting on Z2 Z2 to be the exchange of x and y given by (x; y) 7! ( x; y); (x; y) 7! (x; y) (x; y) 7! (y; x): Dually, with a suitably chosen on D8 and gauging the Z2 subgroup ipping the x coordinate, we get the four symmetry categories given above. 6 6.1 2d TFT with C symmetry and their gauging 2d TFTs without symmetry As a warm-up, let us recall the structure of 2d TFTs without any symmetry. We follow the exposition in [10] closely, see in particular their appendix A. We start with a vector space V of states on S1 and one wants to de ne a consistent transition amplitude Z : V m ! V n corresponding to a given topological surface with m incoming circles and n outgoing circles. We need four basic maps I, I, M , M corresponding to four basic geometries given in gure 25. First, we construct maps IM : V V ! C and M I : C ! V V as in gure 26. This inner product must be non-degenerate because it just corresponds to a cylinder geometry which pairs a state on one circle with a dual state on the other circle. Using it, we can I : V ! C I : C ! V M : V V ! V M : V ! V identify V and V . Then, I is an adjoint of I and M is an adjoint of M . Therefore, to every property involving M , we can write down a corresponding property involving M , and similarly for statements about I and I. This allows us to reduce the number of independent statements we need to write down roughly by half; we do not repeat these adjoint statements below. We consider M as giving a product on V . There is no order on the two incoming circles of a pair of pants and hence the product is commutative, see gure 27. We can also see that M is associative from gure 28 and that I is a unit of the multiplication M from gure 29. Also, by composing these inner products with the product, we see that the product is invariant under permuting three legs, see gure 30. After these preparations, let us associate a map Z : V m ! V n to a surface with m incoming circles and n outgoing circles. We pick a time coordinate t : that at t = 0 we start with m initial circles and at t = 1 we nish with n ! [0; 1] such nal circles. As time goes from 0 to 1, the number of circles generically stay constant but can either increase or decrease by one unit at speci c times 0 = t0 < t1 < t2 < < tp = 1. Cut once in each interval (ti; ti+1). This divides into p pieces. The geometry of each piece contains some cylinders, which correspond to trivial transition amplitude, and exactly one non-trivial geometry out of the four non-trivial cases shown in gure 25. This gives us an expression for Z in terms of the four maps I; I; M; M . However, one could choose a di erent time t0 which starts with same m initial circles and ends with same n nal circles. In general, this would lead to a di erent cutting of and a di erent compositions of four maps I; I; M; M . We need to make sure that they agree. We can continuously deform the time function t to obtain the time function t0. The critical points ti will move under this deformation and will cross each other. It is also possible for two critical points to meet and annihilate each other or for two critical points to pop out of nowhere. We therefore need to ensure that Z remains invariant when ti and ti+1 cross each other, and when two critical points are created or annihilated. For this, we just need to ensure that the two-step composition from the cut between ti 1 and ti to the cut between ti+1 and ti+2 remains invariant under these processes. All possible types of the topology changes were enumerated carefully in appendix A of [10]. The cases are the following and their adjoints: 1. The creation or the annihilation of two critical points as shown in gure 29, or 2a. The exchange of two critical points as shown in gure 28, which we already encoun2b. the situation gure 31 where the number of intermediate circles changes from one to tered, or three, or 2c. the situation gure 32 where the A-cycle and the B-cycle of a torus is exchanged. In more detail, on one side, a circle consisting of segments a; b; c; d in this order splits to two circles consisting of a; b and c; d, which are now along the A-cycle. They then HJEP03(218)9 b c c a d = b a c d d b c a d HJEP03(218)9 on the right, the time ows from inside to the outside, and the parallel edges of the boundary need to be identi ed to form a torus. On one side, the intermediate two circles are along the A-cycle, and on the other side, they are along the B-cycle. merge into a circle consisting of four segments with the order b; a; d; c. On the other side, the two circles in the intermediate stage consists of segments b; c and d; a, and are along the B-cycle. The invariance of Z under the change 1 is the unit property itself, and the invariance under the change 2a is the associativity itself. The invariance under the change 2b can be reduced to associativity by using the cyclic invariance of the product, shown in gure 30. Finally, under the topology change 3b, the map Z is trivially invariant. In total, we have shown that a 2d TFT with no symmetry is completely de ned by a vector space V with the four maps I; I; M; M with the conditions described above. Such a vector space is known as a commutative Frobenius algebra V . 6.2 TFT with C symmetry on a cylinder Let us now move on to the discussion of TFTs with symmetry given by a symmetry category C. In this subsection we start with the simplest geometry, namely cylinders. We already discussed basics in section 4.8.1. As mentioned there, we choose a base point along each constant-time cicle, and call its trajectory the auxiliary line. Basic ingredients. We rst associate the Hibert space Va for a circle with a single insertion of a line labeled by a 2 C. We require Va b = Va Vb. We now associate a Hilbert space Va;b;c;::: for a circle with insertions of transverse lines a, b, c, . . . by fusing them in a xed particular order, starting from the closest line on the right of the base point and then toward the right: Va;b;c;::: := V( ((a b) c) ): (6.2) The case with three lines is shown in gure 33. a b space of a circle with the fused line operator. b a b Z(m) : Va ! Vb Xa;b : Va b ! Vb a Yb;a : Vb a ! Va b We have two basic operations we can perform on the cylinder, see gure 34. One is to insert a morphism m : a ! b, which de nes an operator Z(m) : Va ! Vb. Another is to move the base point to the right and to the left, which de nes morphisms Xa;b : Va b ! Vb a and Yb;a : Vb a ! Va b. Assignment of a map to a given network. With these basic operations, we can assign a map Va;b;::: ! Vc;d;::: for a cylinder equipped with an arbitrary network of lines and morphisms from the symmetry category C, where an incoming circle have insertions a, b, . . . and an outgoing circle have insertions c, d, . . . . We choose a time function t on it, and we call any time ti a critical point when either of the following happens: i) there is an insertion of a morphism on a line, ii) there is a fusion of two lines a; b into one line a b or vice versa, and iii) a line crosses an auxiliary line. Note that we do not allow the auxiliary line to bend backward in time, as part of the de nition. We order 0 = t0 < t1 < < tp 1 < tp = 1 so that the incoming circle is at t = 0 and the outgoing circle is at t = 1. Each critical point of type i) gives a factor of Z(m), that of type ii) gives a factor of Z( ) where is an appropriate associator, and that of type iii) gives a factor of X or Y . Then we de ne the map Va;b; ! Vc;d; associated with this time function t to be the composition of factors corresponding to these critical points. Consistency of the assignment. We now need to show that this assignment is consistent. There are three types of changes under which the assignment needs to be constant, namely the change of the time function t, then and a c c a d a d d b = = b a c : a b ! c d d d b HJEP03(218)9 two subnetworks give the same morphism. the change of the positions of the auxiliary line, and the change of the network in a disk region that does not change the morphism within it. The third point might need some clari cation. In the symmetry category C, a topologically di erent network can correspond to the same morphism. Then we need to ensure that if we replace a subnetwork on a cylinder accordingly, the resulting map on the Hilbert space should also be the same, see gure 35 This is not just a change in the time function, therefore we need to guarantee the invariance separately. The auxiliary line might cut though the subdiagram, as also shown in gure 35, but this does not have to be treated separately, since we can rst move the auxiliary line outside of the disk region, assuming that it is shown that the auxiliary lines can be moved. Then this third type of change can be just taken care of by assuming that we can fuse two local operators, leading to the following constraint, see gure 36: Z(n)Z(m) = Z(n m): Next, let us take care of the second type of change, where we move the auxiliary lines keeping the network and the time function xed. First, moving the auxiliary line back and forth in succession should not do anything, so we have (6.3) (6.4) Xa;b = Yb;a1; for m : a ! a0 and n : b ! b0. We can also fuse two lines before crossing the auxiliary line, see gure 40. This leads to the constraint Xb;c aZ( b;c;a)Xa;b cZ( a;b;c) = Z( c;a;b 1)Xa b;c: Finally, on the cylinder, the change in the time function itself does not do much, and possible changes are already all covered. Thus, we see that to de ne a consistent TFT therefore we have see gure 38. see gure 37. Rotating the base point all the way around should not do anything either, Then we should be able to move the morphisms across the auxiliary line, leading to two relations, as illustrated in gure 39: Xa;1 = id; Xa0;bZ(m Xa;b0 Z(1 1) = Z(1 n) = Z(n m)Xa;b; 1)Xa;b a b n a b a = b a a (6.5) (6.6) (6.7) (6.8) HJEP03(218)9 aʹ a b a b a⊗b b c ; a a b a⊗b bʹ n c a n bʹ HJEP03(218)9 with C symmetry on a cylinder, we need the data of an additive functor Z : C ! Vec with morphisms Xa;b : Va b ' Vb a satisfying (6.5), (6.6), (6.7) and (6.8). Generalized associators on the cylinder. The relations so far guarantees that we can always move the base point and change the order of the tensoring of lines in a consistent manner. For example, the relation (6.8) means that there is a single well-de ned isomorphism between V(a b) c and V(c a) b. We introduce a notation A(a b) c!(c a) b : V(a b) c ! V(c a) b (6.9) for it, and call it a generalized associator on the cylinder. We similarly introduce generalized associators for an arbitrary motion of the base point and an arbitrary rearrangement of parentheses. Each such generalized associator have multiple distinct-looking expressions in terms of sequences of Z( ), X and X 1, but they give rise to the same isomorphism. 6.3 TFT with C symmetry on a general geometry Basic data. Let us discuss now the TFT with C symmetry on a general geometry. The four basic geometries are given in gure 41. For a pair of pants, we need to join the two auxiliary lines coming from each leg into a single auxiliary line. We take the point where this happens to coincide with the critical point where two circles join to form a single circle. In what follows, we will refer to the initial two legs of a pair of pants as the initial legs and the nal leg as the product leg. We can now associate to any geometry with m initial legs and n nal legs with an arbitrarily complicated network of lines and morphisms from C a linear map as follows. We rst choose a time function t : ! [0; 1]. We call a time value ti critical when any a I : V1 ! C I : C ! V1 Ma;b : Va Vb ! Va b b M a;b : Va b ! Va Vb of the following happens: i) the topology of the constant time slice change, ii) there is a morphism, or iii) a line crosses a auxiliary line. We order the critical times so that 0 = t0 < t1 < t2 < < tp = 1. We cut once in each interval (ti; ti+1), and associate to each critical time ti one of the basic linear maps. We then compose them. We now need to guarantee that this assignment is consistent. Basic consistency conditions. Let us rst enumerate basic consistency conditions. First, we de ne the pairing of Va and Va as in gure 42: IZ( aL)Ma ;a : Va M a;a Z( aL)I : C ! Va Va ! C; Va : Then we require that this pairing is non-degenerate and can be used to identify (Va) ' Va . Under this pairing, the product Ma;b and the coproduct M b ;a are adjoint, etc. This again allows us to reduce the number of cases need to be mentioned below roughly by half. L Before proceeding, we note that we used a , aR to de ne the pairing. We can also use aR , aR to de ne a slightly di erent pairing. Exactly which pairing to be used in each situation can be determined by fully assigning orientations to every line involved in the diagram. Below, we assume that every line carries an upward orientation, unless otherwise marked in the gure. Second, a morphism can be moved across the product, see gure 43: Third, the map I de ned by the bowl geometry gives the unit, see gure 44: Ma;b(Z(m) idVb ) = Z(m idb)Ma;b: Ma;1(v I) = v; v 2 Va: a* HJEP03(218)9 (6.10) (6.11) (6.12) (6.13) (6.14) b a b = a b m a aʹ b = (6.15) (6.16) (6.17) a b⊗c up to assoc. = b a b Fourth, it is twisted commutative: Xa;bMa;b(v w) = Mb;a(w v 2 Va; w 2 Vb as illustrated in gure 45. Fifth, it is associative up to the associator: Z( a;b;c)Ma b;c(Ma;b idc) = Ma;b c(ida Mb;c); as shown in gure 46. shown in gure 47: Sixth, we want to formulate that the product is symmetric under the cyclic permutation of three circles. To do this we rst introduce a slightly generalized form of the product M(a c ;c b )!a b : Va c Vc b ! Va b a c b* a b* a b* time ows from inside to the outside. M(a c ;c b )!a b = Z(ida idb )A(a c ) (c b )!a (c c) b Ma c ;c b (6.18) where cL : c introduced in the last subsection. c ! 1 is the evaluation morphism and A is the generalized associator The generalized product has an alternative de nition as given in gure 48, where the line c crosses three auxiliary lines. This gives an alternative expression M(a c ;c b )!a b = Z(ida b R c )A(c a) (b c)!(a b ) (c c )Mc a;b c(Xa;c Xc;b ) (6.19) and we demand the right hand sides of the equations (6.18) and (6.19) are the same. We can now formulate the cyclic symmetry of the product: M(a c ;c b )!a b and M(c b ;b a )!c a are related by the inner products, (6.20) (6.21) see gure 49. We can in fact derive this relation for general a, b, c just from the subcase when c = 1 and the relations already mentioned. We keep the general case for cosmetic reasons, since it looks more symmetric. Seventh, we need a consistency on the torus. An incoming circle consisting of four segments with lines a, b, c, d can rst split into two circles with two segments a, b and b a a c* c b* d a c* c b* b a c d d (6.22) (6.23) (6.24) V(a b) (c d) ! V(b a) (d c) V(b c) (d a) ! V(c b) (a d) action of X and the associators. c, d each and then rejoins to form a circle with four segments in the order b, a, d, c; another way this happens is that the two intermediate circles have segments b, c and d, a, see gure 50. They each determine maps (M (M )b;c;d;a : V(b c) (d a) ! V(c b) (a d) given by )a;b;c;d : V(a b) (c d) ! V(b a) (d c) and (M )a;b;c;d := Mb a;d c(Xa;b (M )b;c;d;a := Mc b;a d(Xb;c Xc;d) a b;c d; Xd;a) b c;d a: We then demand that they are equal up to the generalized associators: A(b a) (d c)!(c b) (a d)(M )a;b;c;d = (M )b;c;d;aA(a b) (c d)!(b c) (d a): Consistency in the general case. We nally nished writing down basic moves. Now we can analyze the general moves. We again have three cases: the change of the time function t on the surface , the change of the positions of the auxiliary line, and the change of the network in a disk region that does not change the morphism. Let us start by discussing the third case. This is in fact automatic once the rst two cases are taken care of, since any disk region can be put into a cylinder under a topological a b a b change, and then the auxiliary line can be moved o away from it. Then all we have to assume is that Z(m) fuses appropriately, (6.3). The change in the position of the auxiliary line can happen in the following three ways: The auxiliary line can move within a single cylinder. This was already discussed in the last subsection. When a circle with line a and a circle with line b join to form a circle, the order of a, b and the base point x in the product leg can either be x, a, b or a, b, x. The invariance under this is the twisted commutativity (6.15). A line c 2 C can cross the trivalent vertex of the auxiliary line. This move changes the number of the intersection of the line c with the three auxiliary lines in one of the two ways, 0 $ 3 or 1 $ 2. One example of the move 0 $ 3 is the equality (6.20) of the two de nitions (6.18) and (6.19) of the generalized product. The move 1 $ 2 can be deduced by combining the twisted commutativity. Finally we need to take care of the changes in the time function t. One possible change is that a morphism and a product can happen in two di erent orders. The invariance under this move is (6.13). Then there are topological changes in the cutting of the surface, which again comes in the following varieties: 1. The creation or the annihilation of two critical points does gure 44. The consistency under this change is the unit property (6.14). 2a. The exchange of two critical points does gure 46. The consistency under this change is the associativity (6.16). of the generalized product (6.21). relation (6.24). 2b. The number of intermediate circles changes from one to three. One example is drawn in gure 51. The consistency under this change can be reduced to the cyclic symmetry 2c. How the torus is decomposed is changed as in gure 50, for which we assigned a basic Summarizing, a TFT with C symmetry is captured by the data (V ; Z; X; I; I; M; M ) satisfying the various relations listed above. Namely, on the cylinder, we have (6.3), (6.5), (6.6), (6.7), (6.8), and on the general geometry, we have in addition (6.12), (6.13), (6.14), (6.15), (6.16), (6.20), (6.21), and (6.24), and nally, diagrams turned upside down correspond to adjoint linear maps. Gauged TFT with the dual symmetry Now we would like to discuss the de nition of the TFT T =A gauged by an algebra object A in terms of the ungauged TFT T . We start from the data (V ; Z; X; I; I; M; M ) for the original theory T . is given by The Hilbert space of the gauged theory was introduced in section 4.8.2. See that section for some necessary background. We use the action by A on Vp, depicted in gure 24. This P := Up;A( xR; xL) := Z(xL)Xp;AZ( xR) (6.25) where xL : A p ! p and xR : p ! p A are the morphisms de ning the A-bimodule structure on p. As we already discussed, P turns out to be a projector, and we de ne Wp to be the projection P Vp. Now, we de ne the data (W ; Z~; X~ ; I~; I~; M~ ; M~ ) for T =A in terms of corresponding data for T . The morphism map Z~. We de ne the new Z~ to be the restriction of the old Z. We need to check that if the initial state lay in Wp Vp then the nal state also necessarily lies in Wq, for a bimodule morphism p ! q. This can be checked by gluing a cylinder on top of the nal state which corresponds to the action of A. The wrapped A line can then be taken across any bimodule morphism until it wraps the p line at the start of the cobordism. But then the wrapped A line has no e ect and can be removed because the initial state we started with is invariant under the action of A. The base point-change map X~ . We now want to de ne the new X~p;q : Wp Aq ! Wq Ap. We use and (see section 4.4) to de ne X~p;q = Z( )Xp;qZ( ). This is well de ned because a wrapped A line at the end of the cobordism can be moved to an A line propagating between a and b at the start of the cobordism which can be removed because of the de nition of A. The unit and counit maps I~ and I~. Now the new unit morphism I~ would be a map from C to WA the de nition of A. Similarly, ~I = IZ(v) where v : A ! 1 is the co-unit morphism in the de nition of A. These are well-de ned as can be shown by manipulations similar to those VA. We de ne it as I~ = Z(u)I where u : 1 ! A is the unit morphism in we are now going to perform for the de nition of M~ . The product and coproduct maps M~ and M~ . The new M~ is de ned analogously as M~ p;q = Z( )Mp;q. This M~ can be shown to be well de ned as a map Wp Wq ! Wp Aq by a series of manipulations using a lot of properties of A. See gure 52. We represent a pair of pants as a 3-punctured plane for ease of illustration. The lower punctures correspond to input legs and the upper puncture corresponds to the product leg of the pair of pants. Unlabeled lines correspond to A. To explain various manipulations, let us refer to manipulations involving =i as \step i". =4 p p p p A q p A q q q q =1 =3 =5 p p p p A q p A q q q q main text. p, and the right action on p A q is de ned by the right action on q. In step 2, we introduce an A line wrapping the leg carrying q. We can do that because the input state in that leg is invariant under the action of A. In step 3, we rst use the fact that q is an A-bimodule and then use the fact that q is a right A-module. In step 4, we use the fact that q is a left module for A. Finally, in step 5, we rst use the fact that q is a bimodule, then the fact that the tensor product is A, and then the fact that p is a bimodule. Ultimately, we can simply remove both the A lines because the input states are invariant under the action of A. We de ne M~ as the adjoint of M~ . To complete the de nition of T =A, we have to check that the operations de ned above satisfy the various conditions that we described in the last section. Most of them just concern some trivial topological manipulations of lines and are manifestly satis ed. Some others, such as (6.14), can be checked using manipulations similar to the ones we have been doing in this sub-section. Yet some others, like the complicated relation (6.20) and (6.24)), require us to simplify a lot of and , but this can be done. This completes the de nition of T =A. 7 Conclusions In this paper we reviewed the notion of unitary fusion categories, or symmetry categories as we prefer to call them, and how they formalize the generalized notion of nite symmetries of a two-dimensional system. We studied various explicit examples of such symmetry categories, some of which are related rather directly to nite groups and some of which are not. We then studied how a symmetry category can be gauged and be re-gauged back. We also de ned 2d topological quantum eld theories admitting a symmetry given by a symmetry category. Many questions remain. Here we mention just two. The rst is how to generalize the constructions discussed in this paper to higher dimensions. In a sense, this is a merger of the generalized symmetry in two dimensions in the sense of this paper, and of the generalized symmetry in the sense of [2]. That there should be something that combines both is clear: even in general spacetime dimension d, the 0-form symmetries can be any non-Abelian group G, possibly with an anomaly specied by Hd(G; U(1)) in the bosonic case and by subtler objects in the fermionic case. Then the (d 2)-form symmetry needs to be extended at least to allow Rep(G). When d = 3, it seems that the notion of 1-form symmetries needs to be extended at least to include modular tensor categories, with an action of the 0-form symmetry group G with an anomaly. What should be the notion in d = 4 and higher? The second is to actually construct two-dimensional systems T for a given symmetry category C. For any group G without an anomaly, there is the trivial theory where the Hilbert space is always one-dimensional. How about the other cases? We can roughly classify symmetry categories C as follows, depending on the simplest possible theories T that have C as a symmetry: 1. The simplest C-symmetric theories have one-dimensional Hilbert space. These would be C-symmetry protected topological (SPT) phases. 2. The simplest C-symmetric theories have nite-dimensional Hilbert space. These would be C-symmetry enriched topological (SET) phases. 3. The simplest C-symmetric theories have in nite-dimensional Hilbert space. Taking the low energy limit, these would be C-symmetric conformal eld theories (CFTs). 4. There is no C-symmetric theory. Clearly this classi cation forms a hierarchy, and it would be nice if we have a uniform construction that tells easily which stage of the above classi cation a given symmetry category C belongs to. There is a recent paper in this general direction [36], where a construction of 2d theory starting from any given symmetry category C was discussed. We hope to see more developments in the future. Actually, there are various symmetry categories C constructed in the subfactor theory, e.g. what is called the Haagerup fusion category, for which no C-symmetric theory is known. If a theory symmetric under the Haagerup fusion category could be constructed, it would be considered as a huge breakthrough. Acknowledgments The authors would like to thank Mikhail Kapranov for bringing the crucial reference [7] to the authors' attention. LB is also grateful to Kevin Costello, Davide Gaiotto and Theo Johnson-Freyd for helpful discussions. The authors thank Nils Carqueville and Eric Sharpe for comments on, and Marcel Bischo , Kentaro Hori and Shu-Heng Shao for pointing out erroneous claims in an earlier version of the manuscript. The work of LB is partially supported by the Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development and Innovation. The work of YT is partially supported in part by JSPS KAKENHI Grant-in-Aid (Wakate-A), No.17H04837 and JSPS KAKENHI Grant-in-Aid (Kiban-S), No.16H06335 and by WPI Initiative, MEXT, Japan at IPMU, the University of Tokyo. The di erential squares to zero: d2 = 0. Then we de ne the group cohomology Hi(G; A) as the cohomology of this di erential. Explicitly, the rst few di erentials are given by df (g; h) = gf (h) f (gh) + f (g); df (a; b; c) = af (b; c) f (ab; c) + f (a; bc) f (a; b); df (x; y; z; w) = xf (y; z; w) f (xy; z; w) + f (x; yz; w) f (x; y; zw) + f (x; y; z): Some points on notation. It does not lead to any loss of generality if we assume that every cochain/cocycle/coboundary is normalized, i.e. it is zero whenever at least one of gi = 1. See e.g. [13]. We have assumed throughout the paper that every cochain is normalized. We are often interested in Hi(G; U(1)) for i = 2; 3 where the action of G on U(1) is taken to be trivial. Henceforth, we will assume the trivial action whenever we write U(1). It is also convenient to treat U(1) elements as phases and in this case the + sign in above de nitions should be replaced by the usual multiplication of phases. For instance we have, ogy valued in U(1). df (g; h) = f (h)f (g) f (gh) : We have used the product notation throughout the paper in the context of group cohomolIn this appendix, we collect various standard facts about group cohomology. De nition. Given a nite group G and its module A, we de ne n-cochains Cn(G; A) as functions Gn ! A. The di erential is given by df (g1; : : : ; gn+1) = g1f (g2; : : : ; gn+1) n i=1 + X( 1)if (g1; : : : ; gigi+1; : : : ; gn+1) + ( 1)n+1f (g1; : : : ; gn): (A.1) (A.2) (A.3) (A.5) (A.6) (A.7) Pull-back. Recall that given a map M1 ! M2 between two manifolds, one can pull-back n-forms on M2 to n-forms on M1. The analogous statement in group cohomology is that Hi(G; A). Explicitly, let h : G ! G0, then h~ : Hi(G0; A) ! Hi(G; A) is given by given a group homomorphism G ! G0, we obtain a module homomorphism Hi(G0; A) ! h~( )(g1; ; gi) = (h(g1); ; h(gi)) : Cup product. when A is a ring. If There is an operation called cup product Ci(G;A) Cj(G;A) ! Ci+j(G;A) 2 Hi(G; A) and 2 Hj(G; A), then the cup product is de ned as ( [ )(g1; ; gi+j) = (g1; ; gi) (gi+1; ; gi+j): It can be easily checked that this product descends to a product on cohomologies. One-dimensional representations of G. Let us ask what is the meaning of H1(G;U(1)). The 1-cochains are maps from G to U(1) and imposing the cocycle condition turns them into group homomorphisms. Hence H1(G; U(1)) is the group formed by one-dimensional representations of G. In particular, when G is a nite Abelian group, then H1(G; U(1)) ' G^ Projective representations of G. Now, let us ask what is the meaning of 2 H2(G;U(1)). We want to interpret the (g1; g2) as the phases de ning a projective representation of G. The cocyle condition reads (g1; g2) (g1g2; g3) = (g2; g3) (g1; g2g3) which is the associativity condition such phases are required to satisfy. Such a cocycle can be shifted by a coboundary of the form (A.8) (A.9) (A.10) (A.11) d (g1; g2) = (g1) (g2) 0 ! H ! G ! K ! 0 which corresponds to rephasing of the group action on the projective representation. Thus, we see that H2(G; U(1)) classi es the phases encountered in projective representations upto rephasing. The usual representations correspond to the trivial element of H2(G; U(1)). Crossed products and extensions of groups. Consider an Abelian group H and a (possibly non-Abelian) group K. Consider an action of K on H and use it to de ne H2(K; H). An element 2 H2(K; H) can be used to de ne a group extension G of K by H, that is there is a short exact sequence and G is called the -cross product of K and H and it is written as G = H o K. Explicitly, the group G as a set is the direct product H K with the group multiplication given as follows (h1; k1)(h2; k2) = (h1 + (k1 . h2) + (k1; k2); k1k2): Here, . denotes the action of K on H via inner automorphism in G. The reader can verify that the associativity of the group multiplication is ensured by the cocycle condition on . Shifting by a coboundary changes G upto isomorphism. Hence, group extensions of K by an Abelian group H are classi ed by a group action of K on H along with an element in H2(K; H) de ned using the group action. Bicharacters on G. Consider an Abelian group G and a trivial module A of G. Given a cohomology element in H2(G; A) represented by a cocycle (g; h), one can form (g; h) = (g; h) (h; g) which is an antisymmetric function on G. This is indeed well de ned because adding a coboundary to doesn't change . When A is U(1) this is a bicharacter, and there's a bijection between an antisymmetric bicharacter on G and H2(G; U(1)). A useful isomorphism. Recall the familiar statement that any closed n-form is exact locally. In group cohomology, the analogous statement is that Hi(G; R) = 1; that is Hi(G; R) is trivial. We can use this to obtain Hi+1(G; Z) ' Hi(G; U(1)). This follows from the long exact sequence associated to the sequence 0 ! Z ! R ! U(1) ! 0. Explicit group cohomologies. H (Zn; Z) = Z[x2]=(nx2). H2((Zn)k; U(1)) = (Zn)k(k 1)=2. H3((Zn)k; U(1)) = (Zn)k+k(k 1)=2+k(k 1)(k 2)=6, with generators given by particular, H3(Zn; U(1)) = Zn and its generator has the cocycle where a; b; c = f0; 1; : : : ; n 1g and hai is the mod n function to f0; : : : ; n (i)(a; b; c) = e2 iai(bi+ci hbi+cii)=n2 ; (i;j)(a; b; c) = e2 iai(bj+cj hbj+cji)=n2 ; (i;j;k)(a; b; c) = e2 iaibjck=n (a; b; c) = e2 ia(b+c hb+ci)=n2 : (A.12) (A.13) (A.14) (A.15) 1g. In (A.16) For Dm the dihedral group with m elements we have [37], H (Dm; Z) = Z[a2; b2; c3; d4]=(2a2; 2b2; 2c3; md4; (b2)2 + a2b2 + (m2=4)d4): (A.17) In particular, H2(D2n+1; U(1)) = 0, H3(D2n+1; U(1)) = Z4n+2; H2(D2n; U(1)) = Z2 Z2, H3(D2n; U(1)) = Z2 Z2 Z2n. The explicit generators can be found in [38]. For Q8, the quaternion group, we have [39] H (Q8; Z) = Z[A2; B2; C4]=(2A2; 2B2; 8C4; A22; B22; A2B2 4C4): (A.18) Open Access. 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Lakshya Bhardwaj, Yuji Tachikawa. On finite symmetries and their gauging in two dimensions, Journal of High Energy Physics, 2018, 189, DOI: 10.1007/JHEP03(2018)189