Topological boundary conditions in abelian Chern-Simons theory
aa r X i v : . [ h e p - t h ] S e p Topological boundary conditions in abelianChern-Simons theory
Anton Kapustin
California Institute of Technology, Pasadena, CA 91125, U.S.A.
Natalia Saulina
Perimeter Institute, Waterloo, Canada
September 23, 2010
Abstract
We study topological boundary conditions in abelian Chern-Simons theory and lineoperators confined to such boundaries. From the mathematical point of view, their re-lationships are described by a certain 2-category associated to an even integer-valuedsymmetric bilinear form (the matrix of Chern-Simons couplings). We argue that bound-ary conditions correspond to Lagrangian subgroups in the finite abelian group classifyingbulk line operators (the discriminant group). We describe properties of boundary lineoperators; in particular we compute the boundary associator. We also study codimen-sion one defects (surface operators) in abelian Chern-Simons theories. As an application,we obtain a classification of such theories up to isomorphism, in general agreement withthe work of Belov and Moore. pi-strings-195 Introduction
Topological field theories provide an interesting playground for studying the properties ofpath-integrals. Recently it became clear that a complete understanding of TFT necessitatesa study of boundary conditions and defects of various dimensions. The relationships betweentopological defects are most conveniently described using the machinery of higher categorytheory, see e.g. [1] for an informal explanation of this. The goal of this paper is to describesome of this machinery in the simplest nontrivial example of a 3d TFT: abelian Chern-Simonstheory. Any 3d TFT associates a number to a closed oriented 3-manifold, a vector space toa Riemann surface, a category to a circle, and a 2-category to a point. In the case of Chern-Simons theory the first three of these are well understood; our goal (which is partially achievedin this paper) is to describe the last one.The case of abelian Chern-Simons theory is particularly attractive because its action iscompletely determined by a simple datum: an even integral symmetric bilinear form K . Thusfrom a mathematical point of view the path-integral for Chern-Simons theory can be viewedas a black box which produces a 2-category from such a form. Abelian Chern-Simons theoriesare also known to describe some phases of actual physical systems (Fractional Quantum Hallphases). We will see that not all abelian Chern-Simons theories admit topological boundaryconditions; in particular, the set of boundary conditions appears to be nonempty only if K has zero signature. For such K basic boundary conditions on the classical level are labeledby Lagrangian subgroups of the gauge group T , where “Lagrangian” means isotropic andcoisotropic with respect to an indefinite metric on T given by the bilinear form K . On thequantum level the answer is similar, but the role of T is played by a certain finite abeliangroup D (the discriminant group), and the role of K is played by a Q / Z -valued quadraticform q on D . Both of these are determined by K . Objects of the 2-category of boundaryconditions are labeled by Lagrangian subgroups of the group D .Different symmetric bilinear forms may give rise to identical pairs ( D , q ). We will seethat many other properties of the theory depend only on ( D , q ), and this suggests thatmany abelian Chern-Simons theories which are distinct on the classical level are isomorphicquantum-mechanically. We will argue below that isomorphism classes of quantum abelianChern-Simons theories correspond to stable equivalence classes of even integer-valued sym-metric bilinear forms. This is in general agreement with the work of Belov and Moore [2]; wediscuss the precise relationship below. Our approach to establishing an isomorphism betweenTFTs is based on the notion of a “duality wall” [3, 4]. This is a surface operator whichseparates two isomorphic theories; it is distinguished among all such surface operators by theproperty of being invertible.Since abelian Chern-Simons theory is a free theory, there are no perturbative quantumcorrections, and one might naively think that quantum effects are relatively trivial. We willsee that this expectation is wrong because of nonperturbative quantum effects due to so-calledmonopole operators. The effects of these operators are important even for bulk observables(Wilson lines); for example, we will see that they lead to a nontrivial associator in the OperatorProduct of Wilson lines. The mathematical counterpart of this fact is well-known and goesback to the work of Joyal and Street [5], but as far as we know until now there was no physicalderivation of these results. We stress that we only study topological boundary conditions. It is well-known that any Chern-Simonstheory admits a boundary which carries a chiral WZW model; however, these degrees of freedom are nottopological (the partition function of Chern-Simons theory coupled to such boundary degrees of freedomdepends on the conformal class of the metric on the boundary).
The gauge group of an abelian Chern-Simons theory is a torus which we will think of asa quotient of an n -dimensional vector space by a subgroup Λ ≃ Z n . We will denote thistorus T Λ . In general, for any finite-rank free abelian group Γ we will denote by T Γ thetorus (Γ ⊗ R ) / (2 π Γ). The gauge field is a connection on a principal T Λ -bundle over anoriented 3-manifold M ; locally, on a trivializing patch U ⊂ M , it is represented by a 1-form A = A µ dx µ taking values in the vector space t Λ = Λ ⊗ R , the Lie algebra of T Λ . Under agauge transformation α : U → T Λ it transforms as A A + dα. The action in Euclidean signature is schematically S CS = i π Z M K ( A, dA ) , where K is a symmetric bilinear form on Λ ⊗ R . It depends only on the smooth structure on M . In order for exp( − S ) to be well-defined (i.e. independent of the choice of trivializationof the T Λ -bundle), K must be integer-valued and even on Λ. That is, for any λ, λ ′ ∈ Λ wemust have K ( λ, λ ′ ) ∈ Z , K ( λ, λ ) ∈ Z . A more careful analysis shows that one can make sense of the Chern-Simons theory even if K is odd, provided we endow M with a spin structure. Such a variant of the theory is calledspin Chern-Simons theory; it has been studied in detail by Belov and Moore [2]. We will notconsider spin Chern-Simons theories in this paper.Following the mathematical terminology, we will call a free abelian group Λ ≃ Z n equippedwith an integer-valued symmetric bilinear form K a lattice of rank n . We will say that a latticeis even if K is even. Thus classical abelian Chern-Simons theories are labeled by even lattices.2he basic observables in abelian Chern-Simons theory are Wilson loops. Let X be anelement of the dual lattice Λ ∗ = Hom(Λ , Z ), i.e. a linear function on Λ ⊗ R which takesinteger values on Λ. Let L be an oriented closed curve in M . The Wilson loop with charge X and support L is W X ( L ) = exp (cid:18) − i Z L X ( A ) (cid:19) . In other words, X defines a unitary one-dimensional representation of T Λ , X ( A ) is the con-nection on the associated line bundle, and W X ( L ) is the holonomy of this connection along L . Chern-Simons theory also admits observables localized on surfaces (surface operators); wewill discuss them in section 7. Any 3d TFT associates an inner-product vector space H Σ to a Riemann surface Σ. Themapping class group of Σ (i.e. the quotient of the group of diffeomorphisms of Σ by itsidentity component) acts projectively on H Σ . In the case of abelian Chern-Simons theory H Σ is obtained by geometric quantization of the moduli space of flat T Λ -connections on Σ. Thelatter space is a torus with a symplectic form ω = 14 π Z Σ K ( δA, δA ) . Its quantization is the space of holomorphic sections of a line bundle L whose curvature is ω .For a genus g Riemann surface Σ , it has dimension | det K | g . As discussed in [2], the action ofthe mapping class group of Σ g on H Σ factors through the group Sp (2 g, Z ). Explicit formulasfor the action of Sp (2 g, Z ) on H Σ are given in [2]; we will not need them in this paper. In any 3d TFT observables localized on closed curves form an additive C -linear categorywhich we will call the category of line operators. Given a pair of line operators, the spaceof morphisms between them is the space of local operators inserted at their joining point.Equivalently, the space of morphisms is the vector space which 3d TFT attaches to a 2-spherewith two marked points, the points being marked by line operators.In the case of abelian Chern-Simons theory line operators are Wilson lines W X , X ∈ Λ ∗ ,and their direct sums. A Wilson line is localized on a one-dimensional submanifold of M which may have boundary points. The only local operator which can be inserted into a givenWilson loop W X is the identity operator (times an arbitrary complex number). Thus thespace of endomorphisms of W X is C , for any X . Local operators sitting at the joining point of two distinct Wilson lines W X and W X ′ aremore interesting. Such an operator must carry electric charge X ′ − X . To construct such anoperator we need to consider an insertion of a Dirac monopole. Recall that a Dirac monopole The category of bulk line operators is in fact semi-simple, and Wilson lines are simple objects in thiscategory.
3s characterized by the topological class of the gauge field on a small 2-sphere centered on it.The topology of a T Λ bundle on S is described by its first Chern class m = 12 π Z S dA ∈ Λ . In Chern-Simons theory such an operator is not gauge-invariant: under a gauge transformation α : M → T Λ the exponential of minus the Chern-Simons action is multiplied by a phaseexp( − iK ( α ( p ) , m )) , where p is the insertion point of the monopole operator. This means that the electric chargeof a Dirac monopole is Km with m ∈ Λ. Such an operator can be inserted at the joiningpoint of W X and W X ′ provided X ′ − X = Km . Thus Hom( W X , W X ′ ) = C if X ′ − X = Km for some m ∈ Λ, and zero otherwise.A nonzero morphism between W X and W X + Km is invertible (its inverse is the Diracmonopole with magnetic charge − m ). Thus line operators W X and W X + Km are isomor-phic as objects of the category of line operators. It is convenient to identify isomorphicobjects, thereby passing to an equivalent category whose objects are isomorphism classes ofWilson line observables. The nice thing about abelian Chern-Simons theory is that after suchidentification the set of objects is finite. Namely, it is a finite abelian group D = Λ ∗ / Im K ,where Im K is the image of Λ in Λ ∗ under the map K : Λ → Λ ∗ . The group D is called thediscriminant group of the lattice (Λ , K ). From now on we will regard the charge X as anelement of the quotient group D = Λ ∗ / Im K .If the inverse of K is integral, then Im K = Λ ∗ and the group D is trivial. Such alattice (Λ , K ) is called self-dual in the physical literature and unimodular in the mathematicalliterature. It can be shown that the signature of an even unimodular lattice is divisible by8. An example of a positive-definite even unimodular lattice is the root lattice of E ; inview of the above the corresponding Chern-Simons theory has trivial discriminant group andtherefore all Wilson line observables are isomorphic to the trivial one. Another example ofan even unimodular lattice is Z with an indefinite form K = (cid:18) (cid:19) . The corresponding lattice (Λ , K ) is often denoted U in the mathematical literature. Thecorresponding Chern-Simons theory has gauge group U (1) × U (1) and an action i π Z M ( adb + bda ) . In this theory all Wilson lines are trivial as well.To get an example of a Chern-Simons theory with a nontrivial category of line operatorswe can take G = U (1) and let K = k ∈ Z . Then D = Z k . The category of line operators in any 3d TFT is a semi-simle braided monoidal category. Very roughly, a monoidal category is a category with an associative tensor product; a braided In fact, a ribbon category, see e.g. [5, 6, 7] for a definition. A ⊗ B and B ⊗ A for any two objects A and B . Precise definitions can be found in appendixA. For abelian Chern-Simons theory the braided monoidal structure can be described veryexplicitly [7]. The tensor product of objects arises from the Operator Product of Wilson lines,which implies that W X ⊗ W Y = W X + Y , X, Y ∈ D . The definition of monoidal structure also involves an associator which is an isomorphism a ( X, Y, Z ) : ( W X ⊗ W Y ) ⊗ W Z → W X ⊗ ( W Y ⊗ W Z ) . This isomorphism must satisfy the pentagon identity, see appendix A. In the case of abelianChern-Simons theory the associator is merely a nonzero complex number depending on
X, Y and Z , but it is nevertheless fairly nontrivial. To see how this nontrivial associator comesabout, let us first look at the braiding. The braiding is an isomorphism W X ⊗ W Y → W Y ⊗ W X , X, Y ∈ D , satisfying a compatibility condition with the associator (the hexagon identities, see appendixA). In the case of abelian Chern-Simons theory the braiding isomorphism originates from theAharonov-Bohm phase which arises when two Wilson lines arranged on a line exchange order.This phase is exp( − iπK − ( X, Y )) . Here
X, Y ∈ Λ ∗ are the electric charges of the two Wilson lines, and K − is the Q -valuedbilinear form on Λ ∗ whose matrix is the inverse of that of K .It will be convenient to use additive notation for Aharonov-Bohm phases, i.e. we willthink of them as elements of Q / Z rather than of U (1) and will write φ instead of exp(2 πiφ ).Thus we will write the braiding phase as s ( X, Y ) = − K − ( X, Y ) . (1)The key observation is that this phase is not invariant under X X + Km but may shift by1 / D × D . To fix the ambiguity, one has to chooseparticular representatives in Λ ∗ for all elements in D = Λ ∗ / Im K .It is this innocent-looking procedure that generates a nontrivial associator. The simplestway to see this is to look at the hexagon identities (appendix A) relating the associator andthe braiding. The braiding phase s ( X, Y ) is a Q / Z -valued function on D × D , and the hexagonidentities (35,36) say that the failure of s to be bilinear is expressed through the associator.The braiding phase (1) is bilinear when regarded as a function on Λ ∗ × Λ ∗ , so if we do notidentify Wilson lines whose charges differ by an element of Im K , it is consistent to set theassociator to zero. But to regard it as a function on D × D , we need to choose particularrepresentatives in Λ ∗ for all elements of D , and then the braiding (1) is no longer bilinear,and the associator must be nontrivial.Let us now compute the associator for bulk Wilson lines in Chern-Simons theory. Considerfirst a simple example where D = Z M for some M . Let e ∈ Λ ∗ be a generator of D , thenwe can uniquely lift an element X ∈ D to an element of Λ ∗ of the form Ae , where A is aninteger from 0 to M −
1. That is, we can label Wilson lines by integers from 0 to M − W A , W B , W C with charges A, B, C ∈ [0 , M −
1] extended in thetime direction and placed along the x -axis in R in this order. Whenever the merger of twoWilson lines produces a Wilson line with charge U e ∈ Λ ∗ with U ≥ M , we must convert U to the range [0 , M −
1] by attaching a semi-infinite Wilson line terminating on a monopoleoperator with a magnetic charge K − M e . We will move these semi-infinite Wilson lines tothe left of the left-most Wilson line W X . Then it is easy to see that the difference betweentwo different ways of computing the triple product W A ⊗ W B ⊗ W C is the braiding of W A and the semi-infinite Wilson line arising from the merger of W B and W C . Thus the associator a ( A, B, C ) = exp(2 πih ( A, B, C )) is a phase given by h ( A, B, C ) = 12 A (cid:20) B + CM (cid:21) M K − ( e, e ) , where [ w ] stands for integral part of w. Note that the bilinear form
M K − on Λ ∗ is integral,so from the multiplicative viewpoint the associator is at most a sign. A straightforwardcomputation best left to the reader shows that h ( A, B, C ) satisfies the pentagon identity.The braiding in this case is s ( A, B ) = − AB K − ( e, e ) . It is easy to verify that the hexagon identities are also satisfied.The general case is surprisingly subtle. Suppose the discriminant group is D ≃ N M i =1 Z M i . Let e , . . . , e N ∈ Λ ∗ be generators of D . This means that any element of Λ ∗ can be written ina unique way as a linear combination of e i with integral coefficients plus an element in Im K ,and in addition M i e i ∈ Im K for all i . Then we can uniquely lift X ∈ D to an element of Λ ∗ of the form A e + · · · + A N e N , where A i is an integer in the range [0 , M i − ~A = ( A , . . . , A N ) with all components in the standard range. The group operation on suchvectors will be denoted ~A ⊙ ~B and is given explicitly by( ~A ⊙ ~B ) i = ( A i + B i ) mod M i . To compute the braiding and the associator, we first need to define more precisely Wilsonlines and their tensor product. Let us choose an order on the set of generators of D : e < e < . . . < e N − < e N . We define a Wilson line W ~A as the ordered product of Wilson lines W A e , . . . , W A N e N . Thismakes sense if we think of a Wilson line as a thin ribbon oriented in a particular direction.For brevity we will refer to W A i e i as constituents of W ~A . It is easy to show that for odd M the form M K − is even, so the associator can be nontrivial only foreven M . ~A = W A e . . . W A N e N Naively, the tensor product of two Wilson lines W ~A , W ~B is defined by placing them side byside along the chosen direction and fusing. The expected answer is W ~A ⊙ ~B , but to see this weneed to modify the naive definition in two ways. First, we need to rearrange the constituentsto put them in the standard order. Second, in general the i th component of ~A + ~B is outsideof the standard range [0 , M i − A i + B i = ( A i ⊙ B i ) + M i (cid:20) A i + B i M i (cid:21) , and move the Wilson lines I ( A i ,B i ) with charges e i M i (cid:20) A i + B i M i (cid:21) to the left of all other constituent Wilson lines. The net result is the Wilson line W ~A ⊙ ~B together with the Wilson line I ( ~A, ~B ) with the charge X i e i M i (cid:20) A i + B i M i (cid:21) (2)to the left of it. Since (2) lies in Im K , the Wilson line I ( ~A, ~B ) is isomorphic to the trivial one.The figure below illustrates the tensor multiplication of W ~A and W ~B for N = 2 .W A W A W B W B e − iπA B g −−−−−−−→ I ( A ,B ) W A ⊙ B I ( A ,B ) W A ⊙ B e − iπM [ A B M ] ( A ⊙ B ) g −−−−−−−−−−−−−−−−→ I ( ~A, ~B ) W ~A ⊙ ~B It should be noted that there is a certain arbitrariness in this definition. We are free toredefine the tensor product W ~A ⊗ W ~B → W ~A ⊙ ~B by an arbitrary invertible endomorphism of W ~A ⊙ ~B . Since the endomorphism algebra of anyWilson line is C , we may regard this endomorphism as a phase k ( ~A, ~B ). This modifies boththe braiding and the associator in a fairly obvious way, see appendix A for details. Forexample the braiding is modified as follows: s ( ~A, ~B ) → s ( ~A, ~B ) − k ( ~A, ~B ) + k ( ~B, ~A ) .
7e will say that such a redefinition modifies the associator and the braiding by a cobound-ary. Braided monoidal categories whose associator and braiding differ by a coboundary areregarded as equivalent.We are now ready to compute the braiding and the associator. Naively, braiding is aparticular isomorphism of W ~A ⊗ W ~B and W ~B ⊗ W ~A arising from moving the second Wilsoncounterclockwise around the first one. The corresponding Aharonov-Bohm phase is s ( ~A, ~B ) = − X i,j A i B j g ij , where we denoted g ij = K − ( e i , e j ). However, since our definition of the tensor product ismore complicated than mere fusion, one needs to correct this expression. The tensor productoperation involves rearranging the constituents of Wilson lines and then splitting off Wilsonlines isomorphic to the trivial one and moving them to the left of all other constituents. TheAharonov-Bohm phase associated with the first step of this operation is ψ ( ~A, ~B ) = − X i>j A i B j g ij . The Aharonov-Bohm phase associated with the second step of the operation is ψ ( ~A, ~B ) = − X i>j M i (cid:20) A i + B i M i (cid:21) ( A j ⊙ B j ) g ij . The phase from both steps is ψ ( ~A, ~B ) = ψ ( ~A, ~B ) + ψ ( ~A, ~B ) . The braiding is an invertible endomorphism of W ~A ⊙ ~B corresponding to the following manip-ulations: first we reverse the above two steps to get from W ~A ⊙ ~B to a composite of W ~A and W ~B , then we exchange the order of W ~A and W ~B by moving W ~B counterclockwise, and finallywe reassemble I ( ~A, ~B ) W ~A ⊙ ~B . I ( ~A, ~B ) W ~A ⊙ ~B e − iπψ ( ~A, ~B ) W ~A W ~B e iπs ( ~A, ~B ) W ~B W ~A e iπψ ( ~B, ~A ) I ( ~A, ~B ) W ~A ⊙ ~B The corresponding Aharonov-Bohm phase is s ( ~A, ~B ) = s ( ~A, ~B ) − ψ ( ~A, ~B ) + ψ ( ~B, ~A ) . Note that the phase ψ drops out of this expression because it is symmetric in ~A and ~B .Therefore the braiding phase is s ( ~A, ~B ) = − X i
10 local diffeomorphism-invariant constraint on the t Λ -valued 1-form A k requires it to liein some subspace of t Λ . The corresponding submanifold is Lagrangian if and only if thissubspace is isotropic with respect to the symmetric bilinear form K and its dimension is halfthe dimension of t Λ . Such a subspace exists if and only if the signature of K vanishes. Forsuch K the dimension of isotropic subspaces ranges from 0 to dim t Λ , so we will call anisotropic subspace of dimension dim t Λ a Lagrangian subspace.Note that the well-known boundary condition for Chern-Simons theory leading to a chiralboson on the boundary is A ¯ z = 0, where z is a local complex coordinate on Σ. This conditionalso defines a Lagrangian submanifold in the space of A k , but it depends on a choice ofcomplex structure on Σ and is not a topological boundary condition.The condition that A k lies in a Lagrangian subspace of t Λ means that the gauge groupon the boundary is broken down to a Lagrangian subgroup. Such a subgroup of T Λ may beconnected or disconnected. If it is connected, then it is a torus T . If it is disconnected, thenit is isomorphic to a product of a finite abelian group and a Lagrangian torus.We will first study the case of a connected boundary gauge group. The case when theboundary gauge group is a product of a torus and a finite abelian group is more complicatedand is considered in section 6.A torus is a divisible abelian group and therefore we can always decompose T Λ into aproduct of T and another torus. Thus if we denote Λ = H ( T , Z ), then Λ is a subgroup ofΛ, and moreover Λ is a direct summand of Λ. That is, the exact sequence of abelian groups0 → Λ → Λ → Λ / Λ → . splits. We will denote by P the injective map from Λ to Λ; if we choose generators for Λ and Λ, P is represented by an integral matrix of size 2 n × n , where 2 n is the rank of Λ. Thetransposed matrix P t can be regarded as a surjective map from Λ ∗ to Λ ∗ . The fact that T Λ is isotropic is equivalent to the matrix identity P t KP = 0 . To proceed further we need to fix a gauge. We introduce a Lagrange multiplier field B takingvalues in the dual t ∗ Λ of the Lie algebra t Λ = Λ ⊗ R of T Λ , as well as the Faddeev-Popov ghostfield C and the anti-ghost field ¯ C taking values in t Λ and t ∗ Λ respectively. The BRST operator Q acts as δ Q A = dC, δ Q C = 0 , δ Q ¯ C = B, δ Q B = 0 . (4)We will use the Lorenz gauge d ⋆ A = 0, where ⋆ is the Hodge star operator with respectto some Riemannian metric on M . On a closed 3-manifold M , the gauge-fixed action hasthe form S CS + GF = S CS + Z M δ Q (cid:16) ( ¯ C, d ⋆ A ) (cid:17) = Z M i π K ( A, dA ) + ( d ¯ C, ⋆dC ) + (
B, d ⋆ A ) ! . (5)Here ( · , · ) denotes the pairing between an element of t ∗ Λ and t Λ . This action is obviouslyBRST-invariant.Now suppose M has a boundary ∂M = Σ. Since the boundary gauge group is a subgroupof T Λ , the boundary value of the field C is restricted accordingly: it takes values in the11ubspace t Λ . To determine the rest of boundary conditions, we consider the boundary termin the variation of the gauge-fixed action: δ bdry S CS + GF = i π Z Σ K (cid:0) A k , δA k (cid:1) + Z Σ vol Σ (cid:16) ( B, δA ⊥ ) + ( δ ¯ C, ∂ ⊥ C ) + ( ∂ ⊥ ¯ C, δC ) (cid:17) . Here and below the subscript ⊥ denotes the component of a 1-form orthogonal to the bound-ary.Requiring the vanishing of these boundary terms tell us first of all that ∂ ⊥ ¯ C takes valuesin the subspace of t ∗ Λ which annihilates t Λ , i.e. in t ∗ Λ / Λ . The boundary value of ¯ C mustlie in a complementary subspace of t ∗ Λ which we denote H , and then ∂ ⊥ C must lie in theannihilator of H . Finally, BRST-invariance requires B to lie in H , and then A ⊥ must lie inin the annihilator of H .A choice of a decomposition t ∗ Λ ≃ t ∗ Λ / Λ ⊕ H may be regarded as a choice of a splitting ofan exact sequence of vector spaces0 → t ∗ Λ / Λ → t ∗ Λ → t ∗ Λ → . (6)That is, we may describe H as the image of an injective map R : t ∗ Λ → t ∗ Λ such that P t R = I . Note that R is a map of vector spaces and does not necessarily arise froma splitting of the exact sequence of the abelian groups0 → (Λ / Λ ) ∗ → Λ ∗ → Λ ∗ → . As discussed above, this sequence always splits, so we could, if we wished, choose R to be asplitting of this exact sequence of free abelian groups. That is, we could choose R to be anintegral solution of the matrix equation P t R = I . However, there is no reason to do this, sofor the time being R will remain a map of vector spaces. We will see later that there are somenatural integrality constraints on R , but these constraints do not reduce to the statementthat the matrix representing R is integral.To write all these boundary conditions in a compact form, let us define the projector Q from t Λ to H ∗ and the complementary projector I − Q : Q = I − P R t , I − Q = P R t . Then ker Q = t Λ , ker( I − Q ) = H ∗ , ker Q t = H, ker( I − Q t ) = t ∗ Λ / Λ . These equalities are equivalent to the matrix identities QP = 0 , R t Q = 0 . The boundary conditions can now be written as follows: QA k = 0 , ( I − Q ) A ⊥ = 0 , Q t B = 0 , QC = 0 , Q t ¯ C = 0 (7)( I − Q ) ∂ ⊥ C = 0 , ( I − Q ) t ∂ ⊥ ¯ C = 0 (8)Note that to prove BRST-invariance of these boundary conditions we have to use theequation of motion for A ⊥ which implies ∂ ⊥ B | Σ ∼ KdA k . Indeed, the BRST variation of thesecond equation in (8) gives: ( I − Q ) t ∂ ⊥ B | Σ = 0which is true due to P t KP = 0 . .3 Constraints on the splitting We will now discuss some natural constraints on the splitting R which arise from consideringquantization of charges in the presence of a boundary. Let the space-time be a half-space x >
0, with the boundary located at x = 0. Consider an electric charge X ∈ Λ ∗ whoseworldline is given by x = 0 , x = L . We will regard x as time. We would like to solvefor the static field created by this charge using the method of images. Let ˆ X be the imagecharge. Consider a point P ∗ on the boundary with spatial coordinates x = a, x = 0, let ρ = √ L + a be the distance from this point to the charge, and let α be the polar angle ofthe charge measured from the point P ∗ . (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) X u P ∗ u ˆ X AAAAAAAA α -6 x x Then the gauge field at P ∗ is given by A = K − ( X + ˆ X ) cos αρ , A = K − ( ˆ X − X ) sin αρ (9)Imposing the boundary conditions (7) leads to( I − Q ) K − ( X + ˆ X ) = 0 , QK − ( ˆ X − X ) = 0The sum of these two equations gives the image charge:ˆ X = K (2 Q − I ) K − X. (10)It seems natural to require the image charge to be integral. Indeed, instead of consideringa Wilson line parallel to the boundary, we may consider a Wilson line piercing the boundary.Then the image Wilson line will also pierce the boundary and we would need to require itscharge to be integral. Therefore the following matrix must be integral:2 KQK − ∈ Mat(2 n, Z ) . (11)A further integrality constraint emerges if we consider a monopole operator of charge m ∈ Λ on which a Wilson line of charge Km terminates. This Wilson line is isomorphic tothe trivial one, therefore the image Wilson line should also be isomorphic to the trivial one.That is, the electric charge of the image Wilson line must be of the form K ˆ m for some ˆ m ∈ Λ.This implies 2 Q ∈ Mat(2 n, Z ) . (12)Thus the compatibility of the method of images with the quantization of charges seems torequire that the constraints (11,12) be satisfied. Note that for a given Lagrangian subgroup13 ⊂ Λ it is not obvious that a splitting R exists such that both constraints are satisfied. Forexample, we can always choose R to be integral; then Q is also integral, thereby satisfying(12). However, the constraint (11) is still nontrivial and may rule out some subgroups Λ which would otherwise be allowed.We will see later that the physical properties of boundary conditions are actually inde-pendent of Q and all formulas make perfect sense for an arbitrary splitting Q not satisfyingany integrality constraints. We therefore believe that the above integrality constraints on Q are an artifact of the method of images, and probably there is a better approach which avoidsthem altogether. Let us describe a concrete example of a topological boundary condition in an abelian Chern-Simons theory. We will make use of it in sections 5 and 6. Let the gauge group be T Λ = U (1) m and the coupling matrix K be block-diagonal K = (cid:18) K m − K m (cid:19) , K m ∈ M at ( m, Z ) . (13)An obvious choice of a connected Lagrangian subgroup in T Λ is the torus corresponding tothe sublattice Λ ⊂ Λ consisting of elements of the form (cid:18) λλ (cid:19) , λ ∈ Λ . The corresponding matrix P has the form P = (cid:18) I m I m (cid:19) where I m is the m × m identity matrix.The most general splitting R satisfying P t R = I m is given by R = 12 (cid:18) I m + r I m − r (cid:19) (14)where r is an m × m real matrix. This implies2 Q = (cid:18) I m − r t − ( I m − r t ) − ( I m + r t ) I m + r t (cid:19) , KQK − = (cid:18) I m − K m r t K − m I m − K m r t K − m I m + K m r t K − m I m + K m r t K − m (cid:19) . The two integrality constraints (11) and (12) in this case are: r ∈ M at ( m, Z ) , K − m r K m ∈ M at ( m, Z ) . The discriminant group is a direct sum D = n M γ =1 Z M γ ⊕ n M γ =1 Z M γ where M γ are determined from K m . In the simplest special case K m = 2 N I m with N ∈ Z thelattice Λ is isomorphic to Z m , the sublattice Λ is isomorphic to Z m , and the discriminantgroup is D ≃ Z ⊕ m N . In section 6.2 we will consider an example of a boundary condition in a Chern-Simonstheory with a non-block-diagonal K . 14 Boundary line operators
Line operators on a particular boundary are objects of a monoidal category. Morphisms inthis category arise from local operators sitting at the joining point of two line operators,while the monoidal structure arises from the fusion of line operators. Fusion is associative,in a suitable sense, but is not commutative, in general. The most obvious boundary lineoperators are boundary Wilson lines and their direct sums. We will argue below that in factthere are no other boundary line operators.Since the boundary gauge group T Λ is a subgroup of the bulk gauge group T Λ , thegroup of boundary charges Hom( T Λ , U (1)) = Λ ∗ is a quotient of the group of bulk chargesHom( T Λ , U (1)) = Λ ∗ . This also means that a boundary Wilson line can be regarded as aresult of fusing a bulk Wilson line with the boundary.A bulk Wilson line can end on the boundary if its endpoint is not charged with respectto the boundary gauge group, or if its charge can be screened by a monopole operator. Thismeans that the charge of such a Wilson line must lie in the subgroup (Λ / Λ ) ∗ ⊕ Im K of Λ ∗ . Ifwe identify Wilson lines which are isomorphic in the category of bulk line operators, we needto quotient this subgroup by Im K . Thus, once we take screening by monopole operators intoaccount, Wilson lines ending on the boundary have charges in the subgroup L ⊂ D where L = ((Λ / Λ ) ∗ ⊕ Im K ) / Im K. Monopole operators also affect the classification of boundary Wilson lines. Morphismsbetween different boundary Wilson lines arise from monopole operators which carry electriccharge Km for some m ∈ Λ. Here we should regard Km not as an element of Λ ∗ , but asan element of its quotient Λ ∗ . In other words, two boundary Wilson lines are isomorphic ifand only if the difference of their charges lies in the sublattice Im P t K of Λ ∗ . (We follow theconventions of the previous section and denote by P t the surjective map from Λ ∗ to Λ ∗ .) Thusisomorphism classes of boundary Wilson lines are labeled by elements of the finite abeliangroup D = Λ ∗ / Im P t K which we will call the boundary discriminant group. We can write D a bit differently by noting that P t K annihilates Λ (by virtue of P t KP = 0), and thereforecan be regarded as a map K from Λ / Λ to Λ ∗ . Then D = Λ ∗ / Im K .We have seen above that properties of bulk line operators are determined by the discrim-inant group D and the Q / Z -valued quadratic form q on it. We will argue in section 7 thatfor fixed signature these data determine the isomorphism class of the abelian Chern-Simonstheory. Thus it should be possible to describe properties of boundary line operators in termsof D and q (the signature is zero for topological boundary conditions to exist). To do this,note that the boundary discriminant group D can also be written as D = D / L . where L is the subgroup of D classifying Wilson lines which can end on the boundary. Thusas far as the classification of boundary Wilson lines is concerned, the important object is thesubgroup L . We will see later that the boundary associator also depends only on D , q , and L .Recall that on the classical level the boundary condition is determined by a choice of asublattice Λ ⊂ Λ such that the subspace Λ Q = Λ ⊗ Q is a Lagrangian subspace of Λ Q = Λ ⊗ Q .This means that the metric K vanishes when restricted to Λ Q (i.e. Λ Q is isotropic), and thatthe orthogonal complement of Λ Q in Λ Q is contained in Λ Q (i.e. Λ Q is coisotropic).15n the quantum level the relevant data are the discriminant group D and the quadraticform q , while the boundary condition is described by a subgroup L . Let us show that L is aLagrangian subgroup of D , in the sense that it is both isotropic and coisotropic with respectto the quadratic form q . To show that it is isotropic, note that an element X ∈ L can be liftedto an element ˜ X ∈ Λ ∗ which annihilates Λ plus an element in Im K . The element in Im K does not affect the quadratic form q and may be ignored. The element ˜ X ∈ Λ ∗ is conormalto the subspace Λ ⊗ Q of Λ ⊗ Q . Since by assumption Λ ⊗ Q is a Lagrangian subspace, ˜ X can be written as Kλ for some λ ∈ Λ ⊗ Q . But then q ( X ) = 12 K − ( ˜ X, ˜ X ) = 12 K ( λ, λ ) = 0 . so L is isotropic.To show that L is coisotropic, consider an element Y ∈ D such that g ( X, Y ) = 0 for all X ∈ L . This implies that once we lift Y to ˜ Y ∈ Λ ∗ , it will satisfy K − ( ˜ X, ˜ Y ) ∈ Z , ∀ ˜ X ∈ (Λ / Λ ) ∗ . Now recall that Λ splits into a direct sum of Λ and Λ / Λ . Once we choose the splitting, wecan decompose K − ˜ Y into a component in Λ ⊗ Q and a component in (Λ / Λ ) ⊗ Q . Theabove condition says that the latter component actually belong to the lattice (Λ / Λ ) whilethe former component is unconstrained. Thus K − ˜ Y lies in the set (Λ / Λ ) ⊕ (Λ ⊗ Q ), andtherefore ˜ Y = Kλ + KP r, where λ ∈ Λ / Λ and r ∈ Λ ⊗ Q . The first term lies in Im K , while the second term lies in(Λ / Λ ) ∗ . Hence Y ∈ L , which means that L is coisotropic.It is tempting to conjecture that one can associate a boundary condition to any Lagrangiansubgroup of D . Some further evidence for this will be presented in section 6, where wewill construct more general boundary conditions and will see that they also correspond toLagrangian subgroups in D .Physical quantities should not change if one replaces a boundary Wilson line with anotherone which is isomorphic to it. Let us verify this for a simple observable which is analogousto the Hopf link in the bulk theory. Consider a boundary Wilson line with charge z ∈ Λ ∗ located at x = 0 and a bulk Wilson line with charge X ∈ Λ ∗ whose shape is a semi-circle inthe ( x , x ) plane with the center at the origin. (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) z u X -6 x x Since both ends of the bulk Wilson line are on the boundary, X must satisfy P t X = 0(unless we insert monopole operators on the endpoints). We will call this configuration of16ilson lines a half-link. Its expectation value can be computed as follows. Lifting z ∈ Λ ∗ to an element Z ∈ Λ ∗ , we may regard the boundary Wilson line as a limit of a bulk Wilsonline with charge Z . When we fuse it with the boundary, it collides with its image charge K (2 Q − K − Z , so the field created by a boundary charge can be computed as the field ofa bulk charge 2 KQK − Z : A = 2 QK − Zdθ where θ is the angular coordinate in the ( x , x ) plane. The expectation value of the half-linkcan be computed as the Aharonov-Bohm phase of the charge X moving along the half-circleparameterized by θ ∈ [ − π/ , π/ − i Z X ( A )) = exp( − πi ( X, QK − Z )) . It follows from P t X = 0 that Q t X = X , so the above phase can also be written asexp( − πiK − ( X, Z )) . Note first of all that it is independent of the lift of z to Z . Indeed, different lifts differ byelements of ker P t . Since t Λ is Lagrangian, any element in ker P t can be written as KP f forsome f ∈ t Λ . But then the Aharonov-Bohm phases differ by a factorexp( − πiK − ( X, KP f )) = exp( − πi ( P t X, f )) = 1 . Second, if we shift either X or Z by an element of of the form Km , m ∈ Λ, the Aharonov-Bohm phase also does not change. This shows that replacing a Wilson line (either bulk orboundary) with an isomorphic one does not change the expectation value of the half-link. Wecan make it explicit if we regard X as taking values in L = (Λ / Λ ) ∗ / Im K t and z as takingvalues in D = Λ ∗ / Im K = D / L . Then the expectation value of the half-link isexp( − πi g ( X, Z )) , where Z is a lift of z to D . The precise choice of the lift is unimportant, since the bilinearform g : D × D → Q / Z vanishes when restricted to L × L . Fusing two boundary Wilson lines with charges x, y ∈ D = D / L gives a boundary Wilson linewith charge x + y . In order to completely specify the monoidal structure on the category ofboundary line operators we also need to determine the associator morphism. Note that whilethe isomorphism class of the resulting Wilson line does not change if we exchange x and y ,there is no natural isomorphism between W x ⊗ W y and W y ⊗ W x . That is, the monoidal categoryof boundary Wilson lines does not have a natural symmetric or even braided structure.On the other hand, a certain analog of braiding arises if we consider the relationshipbetween bulk and boundary Wilson lines. Consider a bulk Wilson line with charge X ∈ D .Fusing it with the boundary gives a boundary Wilson line whose charge (an element of D / L ) isthe image of X under the natural projection π : D → D / L . We will denote this charge π ( X ).Given a boundary Wilson line with charge y ∈ D / L we may consider fusing W y with W π ( X ) in two different orders. In this case there is a natural isomorphism between W y ⊗ W π ( X ) and W π ( X ) ⊗ W y arising from the fact that one can change the order of boundary Wilson lines by17oving one of them to the bulk. Of course, this natural isomorphism depends on X ∈ D , notjust its image π ( X ) in D / L . We will call this natural isomorphism a semi-braiding betweenthe bulk Wilson line W X and the boundary Wilson line W y .Let us explain the mathematical meaning of the semi-braiding. In the theory of monoidalcategories there is a notion of a Drinfeld center. It is defined by analogy with the center ofan associative algebra. The Drinfeld center of a monoidal category C is a braided monoidalcategory Z ( C ) whose objects are pairs ( a, χ a ), where a is an object of C and χ a is a familyof isomorphisms b ⊗ a → a ⊗ b for all objects b of C . These isomorphisms must satisfy somecompatibility constraints with the tensor product in the category C , see appendix B for details.It turns out that the semi-braiding defines a map from the set of bulk line operators to theset of objects of the Drinfeld center of the category of boundary line operators. In fact, itcan be shown that the map on objects can be extended to a strong monoidal functor fromthe category of bulk line operators to the Drinfeld center of the category of boundary lineoperators. An interested reader is referred to appendix B for mathematical details.Let us compute the semi-braiding isomorphism between the bulk Wilson line W X , X ∈ D and a boundary Wilson line W y , y ∈ D . To write down a concrete formula, one needs to chooseisomorphisms W y ⊗ W y ′ ≃ W y + y ′ for all boundary Wilson lines, so that the semi-braidingbecomes an isomorphism from W π ( X )+ y to itself, i.e. a complex number. An importantingredient in the definition of this isomorphism and in the further construction of the semi-braiding is the Aharonov-Bohm phase arising from the counterclockwise transport of charge X along a semi-circle centered at the boundary charge y . (We have arranged Wilson lines sothat both W X and W y are along x direction and W y sits at x = x = 0). The computationof this phase is similar to the computation of the expectation value of the half-link, exceptthat the bulk charge X is not assumed to be annihilated by P t . We use the method of imagesto replace the computation in the presence of a boundary with the computation without theboundary but with image charges included. (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) Y s ˆY s X s ˆX s -6 x x The boundary charge y ∈ Λ ∗ is thus lifted to a bulk charge Y ∈ Λ ∗ and moved slightlyaway from the boundary. Its image charge is ˆ Y = K (2 Q − K − Y . The image charge for thebulk charge X is ˆ X = K (2 Q − K − X . The Aharonov-Bohm phase is a sum of two terms:the phase acquired by the charge X while moving counterclockwise in the static field createdby Y + ˆ Y and the phase acquired by the charge Y due to a time-like gauge field created bymoving charges X and ˆ X . The latter phase is the same as the phase that the charge Y wouldacquire if it moved counterclockwise in the static field created by the charge X − ˆ X . The firstphase is exp (cid:18) − iπ K − ( X, Y + ˆ Y ) (cid:19) . (cid:18) − iπ K − ( X − ˆ X, Y ) (cid:19) . Their product is exp ( − πi ( X, ˆ g Y )) , ˆ g = K − + QK − − K − Q t . Note that this Aharonov-Bohm phase is independent of the choice of lifting y ∈ Λ ∗ to Y ∈ Λ ∗ .Indeed, any two such lifts differ by an element of the form KP r for some r ∈ t Λ . But Q t KP r = KP r , and QK − KP r = 0, so the phase is unchanged if we shift Y by KP r .Now we are ready to define the product of boundary Wilson lines. Let bulk and boundarydiscriminant groups be D = Q Ni =1 Z M i and D = Q nγ =1 Z m γ respectively. Let v < v . . . < v n be an ordered set of generators of D . For all γ ∈ { , . . . , n } we choose a lift of v γ ∈ D to e γ ∈ D . We also choose generators e n +1 , . . . , e N for the subgroup L . We therefore get anordered set of generators e < . . . < e N for D . In order to define the product of boundaryWilson lines we need to lift e i to Λ ∗ . We denote these lifts ˜ e i , i = 1 , . . . , N and write˜ v γ = ˜ e γ , γ = 1 , . . . , n. We can always choose the lift of the remaining N − n generators so that ˜ e n +1 , . . . , ˜ e N lie inthe Lagrangian sublattice (Λ / Λ ) ∗ , i.e. there exist r ˆ i ∈ Λ ⊗ Q , ˆ i ∈ { n + 1 , . . . , N } such that˜ e ˆ j = KP r ˆ j ˆ j = n + 1 , . . . , N. (15)Note also that since by assumption v γ generates Z m γ , we have m γ v γ = 0 in D , and therefore m γ e γ ∈ L , and therefore m γ ˜ e γ ∈ (Λ / Λ ) ∗ ⊕ Im K . Explicitly, this means that there exist f γ ∈ Λ ⊗ Q such that m γ ˜ e γ − KP f γ ∈ Im K, γ = 1 , . . . , n.
We also have M i ˜ e i ∈ Im K, i = 1 , . . . , N.
Recall (see section 3) that we think of a bulk Wilson line W ~A with charge ~A = A e + . . . + A N e N as the ordered product of bulk Wilson lines W A e , . . . , W A N e N . Similarly, wedefine a boundary Wilson line W ~a with charge ~a = a v + . . . + a n v n as the ordered productof boundary Wilson lines W a v , . . . , W a n v n . For brevity we will refer to W a i v i as constituentsof W ~a . When fusing W ~a with W ~b we first need to rearrange the constituents to put them intothe standard order. This is possible since we chose the lift of v γ to e γ and hence we can liftthe boundary Wilson line W b γ v γ to the bulk Wilson line W b γ e γ and move it counterclockwisearound W a β v β for γ < β . Second, in general the γ -th component of of ~a + ~b is outside thestandard range [0 , m γ − a γ + b γ = ( a γ ⊙ b γ ) + m γ (cid:20) a γ + b γ m γ (cid:21) where brackets denote the integral part, and move the ‘trivial’ boundary lines I ( a γ ,b γ ) withcharges v γ m γ (cid:20) a γ + b γ m γ (cid:21)
19o the left of all other constituent boundary lines. The net result is a boundary Wilson line W ~a ⊙ ~b together with the trivial boundary Wilson line I ( ~a,~b ) with charge n X γ =1 v γ m γ (cid:20) a γ + b γ m γ (cid:21) to the left of W ~a ⊙ ~b . In going from W ~a W ~b to W ~a ⊙ ~b we get a phase given by (in additive notation)ˆ ψ ( ~a,~b ) = − X α<γ a γ b α ˆ g αγ − X α>γ m α (cid:20) a α + b α m α (cid:21) ( a γ ⊙ b γ )ˆ g αγ where ˆ g αγ = (˜ v α , ˆ g ˜ v γ ) . This is illustrated for n = 2 below. (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) s W b s W b s W a s W a e − iπb a ˆ g (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) s W a ⊙ b s I ( a ,b ) s W a ⊙ b s I ( a ,b ) e − iπm [ a b m ] ( a ⊙ b ) ˆ g (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) s W ~a ⊙ ~b s I ( ~a,~b ) Note also that fusing a bulk Wilson line W ~B with the boundary gives a product of boundaryWilson lines I ~B W π ( ~B ) where I B α is a trivial Wilson line with charge v α m α ˜ B α and we denote π ( ~B ) α = B α − m α ˜ B α , ˜ B α = (cid:20) B α m α (cid:21) . Now we are ready to compute the semi-braiding isomorphism c ( sb ) ~a ~B : W ~a ⊙ π ( ~B ) W π ( ~B ) ⊙ ~a which is defined (see cartoon below) by inverting the first and the second arrows and followingthe other three in the direction indicated. (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) s I ~B s I ( a,b ) s W ~a ⊙ π ( ~B ) e i π ( ˆ ψ ( ~a,~b )+ β ( a, ˜ B ) ) ←−−−−−−−−−−− (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) s W π ( ~B ) s I ~B s W ~a ← (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) s W ~a s W ~B e i πρ ( ~a ; ~B ) −−−−−→ (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) s W ~a s W ~B → (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) s I ~B s W π ( ~B ) s W ~a e i π ˆ ψ ( ~b,~a ) −−−−−→ (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) s W π ( ~B ) ⊙ ~a s I ( ~a,~b ) s I ~B Thus the semi-braiding isomorphism is the phase c ( sb ) ~a ~B = e πis sb ( ~a ; ~B ) where s sb ( ~a ; ~B ) = s (0) sb ( ~a ; ~B ) + ˆ ψ ( ~b, ~a ) − ˆ ψ ( ~a,~b ) , s (0) sb ( ~a ; ~B ) = ρ ( ~a ; ~B ) − β ( ~a, ~ ˜ B ) , ~b = π ( ~B ) . (16)20he terms with ˆ ψ arise from our definition of the tensor products of boundary Wilson lines.On the other hand ρ ( ~a ; ~B ) is a phase which arises when the bulk line W ~B is moved counter-clockwise to the left of the boundary Wilson line W ~a : ρ ( ~a ; ~B ) = − N X j =1 n X γ =1 B j a γ (˜ e j , ˆ g ˜ v γ )Finally, β ( a, ˜ B ) is a phase which arises in going from W ~a I ~B W π ( ~B ) to I ~B W ~a W π ( ~B ) by moving W ~a clockwise to the right of I ~B : β ( a ; ˜ B ) = 12 X α,γ a γ m α ˜ B α ˆ g γα . We can further simplify this using (15) and(
KP r ˆ j , ˆ g ˜ v γ ) = 2( r ˆ j , P t ˜ v γ ) . The final answer for the semi-braiding is s sb ( ~a ; ~B ) = − n X γ =1 b γ a γ (˜ v γ , K − ˜ v γ ) − X α>γ b α a γ (˜ v α , K − ˜ v γ ) (17) − N X ˆ j = n +1 n X γ =1 B ˆ j a γ ( r ˆ j , P t ˜ v γ ) − X α,γ a γ m α ˜ B α (˜ v γ , K − ˜ v α ) . Note that the dependence on the projector Q dropped out of the final expression.Now let us compute the boundary associator: a ( bry ) a,b,c : (cid:0) W ~a ⊗ W ~b (cid:1) ⊗ W ~c W ~a ⊗ (cid:0) W ~b ⊗ W ~c (cid:1) . The picture below shows the two ways to get to I ( ~a,~b,~c ) W ~a ⊙ ~b ⊙ ~c where the “trivial” boundaryWilson line I ( a δ ,b δ ,c δ ) has charge m δ h a δ + b δ + c δ m δ i . (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) s W ~c s W ~b s W ~a e i π ˆ ψ ( ~a,~b ) −−−−−→ (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) s W ~c s W ~a ⊙ ~b s I ( ~a,~b ) e πi (cid:0) ˆ ψ ( ~a ⊙ ~b,~c )+ h (cid:1) −−−−−−−−−−→ (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) s W ~a ⊙ ~b ⊙ ~c s I ( ~a,~b,~c ) e πi (cid:0) ˆ ψ ( ~a,~b ⊙ ~c )+ h (cid:1) ←−−−−−−−−−− (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) s W ~b ⊙ ~c s I ( ~b,~c ) s W ~a e i π ˆ ψ ( ~b,~c ) ←−−−−− (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) s W ~c s W ~b s W ~a Here the phase h arises from the rearrangement of “trivial” boundary Wilson lines fromthe configuration I ( ~a,~b ) I ( ~a ⊙ ~b,~c ) to I ( ~a,~b,~c ) by moving I ( a β ⊙ b β ,c β ) counterclockwise to the left of I ( a γ ,b γ ) for β < γ : h ( ~a,~b, ~c ) = − X γ>β m γ (cid:20) a γ + b γ m γ (cid:21) m β (cid:20) ( a β ⊙ b β ) + c β m β (cid:21) ˆ g βγ . h arises by first moving W ~a clockwise to the right of I ( ~b,~c ) and thenrearranging trivial lines from I ( ~b,~c ) I ( ~a,~b ⊙ ~c ) to I ( ~a,~b,~c ) by moving I ( a β ,b β ⊙ c β ) counterclockwise tothe left of I ( b γ ,c γ ) for β < γ : h ( ~a,~b, ~c ) = 12 X γ,β a γ m β (cid:20) b β + c β m β (cid:21) ˆ g γβ − X γ>β m γ (cid:20) b γ + c γ m γ (cid:21) m β (cid:20) ( b β ⊙ c β ) + a β m β (cid:21) ˆ g βγ . In this way we find that the boundary associator is a phase a ( bry ) a,b,c = e πih bry ( a,b,c ) where h bry ( a, b, c ) = h − h + ˆ ψ ( a, b ⊙ c ) − ˆ ψ ( a ⊙ b, c ) + ˆ ψ ( b, c ) − ˆ ψ ( a, b ) . (18)The last four terms in this formula are a coboundary, i.e. they can be removed by redefin-ing the boundary tensor product of objects W ~a and W ~b by a phase ˆ ψ ( a, b ) . Explicitly, thiscoboundary term is12 X γ>β a γ (cid:20) b β + c β m β (cid:21) m β ˆ g βγ + 12 X γ<β a γ (cid:20) b β + c β m β (cid:21) m β ˆ g βγ − X γ<β c γ (cid:20) a β + b β m β (cid:21) m β (˜ v β , K − ˜ v γ )+ 12 X γ>β m γ (cid:20) a γ + b γ m γ (cid:21) m β (cid:20) ( a β ⊙ b β ) + c β m β (cid:21) ˆ g γβ − X γ>β m γ (cid:20) b γ + c γ m γ (cid:21) m β (cid:20) ( b β ⊙ c β ) + a β m β (cid:21) ˆ g γβ Putting all these formulas together we get the boundary associator: h bry ( a, b, c ) = 12 X δ a δ m δ (cid:20) b δ + c δ m δ (cid:21) g δδ − X γ<β c γ m β (cid:20) a β + b β m β (cid:21) g γβ (19)+ X γ = β a γ m β (cid:20) b β + c β m β (cid:21) g γβ . A straightforward computation shows that the boundary associator (19) satisfies the pen-tagon identity. It is also independent of the projector Q which parameterizes the choice ofthe splitting.The semi-braiding and the boundary associator are not unrelated: they satisfy the twoboundary hexagon identities described in appendix B. One of them expresses the fact thatthe pair ( W π ( B ) , s sb ( · , B )) is a well-defined object of the Drinfeld center of the category ofboundary Wilson lines, and the other one says that the map which sends the object W B tothe object of the Drinfeld center ( W π ( B ) , s sb ( · , B )) respects the monoidal structure on thetwo categories. One can check that the semi-braiding and the boundary associator computedabove satisfy the boundary hexagon identities.It is important to realize that the boundary associator and the semi-braiding depend onthe exact way the boundary tensor product is defined. If we redefine the product of W ~a and W ~b by a phase k ( a, b ), the boundary associator and the semi-braiding change as follows: h ′ bry ( a, b, c ) = h bry ( a, b, c ) + k ( a, b ⊙ c ) − k ( a ⊙ b, c ) + k ( b, c ) − k ( a, b ) , (20) s ′ sb ( a, B ) = s sb ( a, B ) + k ( π ( B ) , a ) − k ( a, π ( B )) . (21)We will say that ( h ′ bry , s ′ sb ) differs from ( h bry , s sb ) by a coboundary k ( a, b ). The new boundaryassociator still satisfies the pentagon identity, and the new boundary associator and the semi-braiding together still satisfy the boundary hexagon identities. Such a redefinition of the22oundary tensor product replaces the monoidal category of boundary Wilson lines with anequivalent one and should be regarded as physically trivial. Note that the semi-braiding (16)and the boundary associator (18) differ from h (0) bry = h − h and s (0) sb by a coboundary ˆ ψ ( ~a,~b ) . This coboundary makes h bry and s sb independent of the projector Q. This independence isless obvious if we use h (0) bry and s (0) sb because under a change of Q they change by a coboundary.It is useful to note that the boundary associator can be further simplified by adding acoboundary corresponding to k ( a, b ) = X γ>β a γ b β (˜ v γ , K − ˜ v β ) . The new boundary associator is h ′ bry ( ~a,~b, ~c ) = 12 X δ a δ m δ (cid:20) b δ + c δ m δ (cid:21) (˜ v δ , K − ˜ v δ ) + X γ<β a γ m β (cid:20) b β + c β m β (cid:21) (˜ v γ , K − ˜ v β ) . (22)The new semi-braiding is s ′ sb ( ~a ; ~B ) = − n X γ =1 b γ a γ (˜ v γ , K − ˜ v γ ) − X γ>α b α a γ (˜ v α , K − ˜ v γ ) (23) − N X ˆ j = n +1 n X γ =1 B ˆ j a γ ( r ˆ j , P t ˜ v γ ) − X α,γ a γ m α ˜ B α (˜ v γ , K − ˜ v α ) . Finally let us discuss how the boundary associator depends on various arbitrary choices wehave made. The formulas obviously depend on the ordering of the generators v , . . . , v n . Asin the bulk case, it is easy to check that changing the order modifies the boundary associatorand the semi-braiding by a coboundary. We also chose a lift of the generators of D to D .We claim that changing a lift also adds a coboundary. Indeed, changing a lift amounts to areplacement ˜ v γ ˜ v γ + KP f γ , f γ ∈ Λ ⊗ Q , KP f γ ∈ Λ ∗ . Simultaneously we need to perform a shift X ˆ j B ˆ j r ˆ j X ˆ j B ˆ j r ˆ j − X γ b γ f γ so that P Nj =1 B j e j is unchanged. Then a short computation shows that this transformationadds to ( h ′ bry , s ′ sb ) a coboundary corresponding to k ( a, b ) = − X α<γ a α b γ (˜ v α , P f γ ) . Alternatively, it is easy to check that ( h (0) bry , s (0) sb ), which differs from ( h ′ bry , s ′ sb ) by a coboundary,is unchanged when we do the above transformation. To summarize, the boundary associatorand the semi-braiding do not depend on arbitrary choices, up to a coboundary.Despite using the “cohomological” terminology, we have not explained what cohomologytheory is relevant here. It is easy to supply such a theory if we ignore the semi-braiding. Thenthe boundary associator phase defines a 3-cochain in the standard complex which computes23he cohomology of the group D = D / L with values in the trivial module Q / Z . The pen-tagon identity says that this 3-cochain is a 3-cocycle, and identifying boundary associatorsdiffering by a coboundary is exactly the same as identifying cohomologous 3-cocycles. Ourcomputations in this section show how to associate a canonical element if H ( D / L , Q / Z ) toany Lagrangian subgroup L of a finite abelian group D equipped with a quadratic function q . In fact, it is easy to see that in all our computations we never used the fact that L iscoisotropic, so the formulas are valid for an arbitrary isotropic subgroup of ( D , q ). We discussthe relevance of the coisotropic condition in section 8. Let us illustrate the above discussion by two simple examples. The first one involves thetopological boundary condition introduced in section 4.4. The bulk gauge group is U (1) m and the block-diagonal Chern-Simons coupling matrix is given by (13). The bulk discriminantgroup D consists of pairs ( x , y ) where x and y are elements of Z m defined up to addition ofelements of the form K λ , where λ is an arbitrary element of Z m . Elements of D label bulkWilson lines. The boundary gauge group is the diagonal U (1) m subgroup. The charge of aWilson line ending on the boundary must be in the kernel of P t , therefore it must have theform ( x , − x ) , where x ∈ Z m and is again defined up to addition of K λ . Elements of this formdefine the subgroup L in D . The boundary discriminant group D = D / L is isomorphic to L ,and in fact we have D ≃ D ⊕ D .We can always write D as ⊕ nγ =1 Z m γ . Let v γ , γ = 1 , . . . , n , be an ordered set of generatorsof D . We can lift them to generators of D by writing e γ = ( v γ , , γ = 1 , . . . , n. Generators of L can be taken to have the form e n + γ = ( v γ , − v γ ) . Together they generate the whole D . Let ˜ v γ be some lift of v γ to Z m . That is, we have m γ ˜ v γ = K f γ for some f γ ∈ Z m . Then we have˜ e γ = (˜ v γ , , ˜ e n + γ = (˜ v γ , − ˜ v γ ) , γ = 1 , . . . , n. Boundary Wilson lines are labeled by vectors ~a = ( a , . . . , a n ) where a γ is an integer inthe interval [0 , m γ − h ′ bry ( ~a,~b, ~c ) = 12 X δ a δ m δ (cid:20) b δ + c δ m δ (cid:21) (˜ v δ , K − ˜ v δ ) . (24)Note that this is the same as the bulk associator in the U (1) m Chern-Simons theory with thesymmetric blinear form K . This happens because the boundary condition we are consideringcan be reinterpreted (via the folding trick, see section 7) as a trivial surface operator in the U (1) m Chern-Simons theory. Boundary Wilson lines are then reinterpreted as bulk Wilsonlines in this theory.Bulk Wilson lines are labeled by vectors ~B = ( B , . . . , B n ), where ∀ γ ∈ { , . . . , n } B γ and B n + γ are integers in the interval [0 , m γ − s ′ sb ( ~a ; ~B ) = − n X γ =1 a γ B γ (˜ v γ , K − ˜ v γ ) − X γ,β B n + β a γ (˜ v β , K − ˜ v γ ) − X γ>α a γ B α (˜ v α , K − ˜ v γ ) . (25)24ote that in this example the boundary associator is a 2-torsion, i.e it is ± U (1) × U (1) theory with K = (cid:18) − − (cid:19) . The bulk discriminant group in this example is D ≃ Z , and the quadratic function q : D → Q / Z is given by q ( x ) = − x , x ∈ Z / Z . We can take the generator of D to be ˜ e = (1 , K (cid:18) − − (cid:19) = 9 e. Consider the boundary condition defined by P = (cid:18) (cid:19) . The subgroup L ⊂ D is generated by the vector (1 , −
2) and is isomorphic to Z since (cid:18) − (cid:19) = K (cid:18) (cid:19) . The boundary discriminant group D = D / L is isomorphic to Z as well and is generated by˜ v = (1 , a ∈ { , , } . The boundary associator is h ′ bry ( a, b, c ) = − a (cid:20) b + c (cid:21) . It takes values in third roots of unity. Bulk Wilson lines are labeled by an integer B ∈ { , , . . . , } . The semi-braiding is s ′ sb ( a, B ) = 29 aB + 23 a (cid:20) B (cid:21) . So far we have been studying boundary line operators on a particular boundary. We mayalso study line operators separating two different boundary conditions; we may call themboundary-changing line operators. From the mathematical viewpoint boundary conditions ina 3d TFT are objects of a 2-category. Boundary line operators separating boundary conditions A and B are 1-morphisms from A to B , while local operators sitting at the junctions of twosuch line operators are 2-morphisms. So far we have been studying 1-morphisms from anobject to itself, and now we turn to more general 1-morphisms.In the case of abelian Chern-Simons theory boundary conditions are labeled by Lagrangiansubgroups of the discriminant group D . To describe boundary-changing line operators between Incidentally, this implies that the monoidal category of boundary line operators in this case does notadmit any braided monoidal structure. L and L it is useful to compactify our 3d TFT on an interval withboundary conditions corresponding L and L on the two ends. Boundary-changing line op-erators are in one-to-one correspondence with boundary conditions in the resulting 2d TFT.More precisely, boundary-changing line operators from L to L form a category Hom( L , L ),and this category is isomorphic to the category of D-branes in the effective 2d TFT [8, 1].Consider abelian Chern-Simons theory on M = Σ × [0 , L ] with boundary conditions( P , R ) at x = 0 and ( P , R ) at x = L. Here P i determines the boundary gauge group T Im P i , and R i is the splitting of the corresponding exact sequence (6). The splittings R i enterthe boundary conditions for A ⊥ = A , the Lagrange multiplier field B and the Faddeev-Popovghosts. It is actually easier to perform the reduction without fixing the gauge. Since at x = 0the gauge group is reduced to T Im P and at x = L it is reduced to T Im P , the 2d gauge groupwill be the isotropic subgroup A = T Im P ∩ T Im P . A is not necessarily connected, i.e. it may be a product of a torus and a finite group. ItsLie algebra is an isotropic subspace of t Λ : t = (Im P ⊗ R ) ∩ (Im P ⊗ R ) . Another way to explain this is to say that only the constant mode of A k survives the reduction,and since at x = 0 and x = L the field A k must lie in Im P ⊗ R and Im P ⊗ R , respectively,the constant zero mode must lie in the intersection of these two vector spaces.The component A gives rise to a 2d scalar ϕ = Z L dx A . In terms of this scalar the 2d effective action becomes S = i π Z K ( ϕ, dA k ) . The scalar field ϕ is not gauge-invariant: under an x -dependent gauge transformation α itshifts by α ( L ) − α (0). Since α (0) mod 2 π Λ and α ( L ) mod 2 π Λ belong to (Im P ) ⊗ R and(Im P ) ⊗ R , respectively, this means that ϕ ∈ t Λ is defined modulo arbitrary elements of thesubgroup (Im P ⊗ R ) ⊕ (Im P ⊗ R ) ⊕ π Λ . Equivalently, we may regard ϕ as living in the quotient torus T Λ / T Im P ⊕ Im P . The action S is non-degenerate because K is a non-degenerate pairing between the vectorspaces t Λ / ((Im P ⊗ R ) ⊕ (Im P ⊗ R ))and (Im P ) ⊗ R ∩ (Im P ) ⊗ R . This follows from the fact that Im P and Im P are Lagrangian subspaces of t Λ .The effective 2d TFT is thus a 2d version of the BF theory, with the periodic scalar field ϕ playing the role of B . The construction of branes in this theory is fairly obvious: one places on26he boundary a quantum-mechanical degree of freedom taking values in some representationof the gauge group A and couples it to the restriction of the gauge field. Thus branes arelabeled by elements of the abelian group Hom( A , U (1)). In the special case P = P = P thegroup A is simply the torus T Λ where Λ = Im P , so we recover the result that boundaryline operators are (direct sums of) Wilson lines for the boundary gauge group T Λ .Note that since U (1) is a divisible group, any element of Hom( A , U (1)) is a restriction ofan element of Hom( T Λ , U (1)) = Λ ∗ . That is, the group of charges of boundary-changing lineoperators is a quotient of the group of bulk charges. The kernel of the quotient map consistsof bulk charges which vanish on A . Equivalently, one might say that the kernel consists ofbulk Wilson lines which are neutral with respect to A and therefore may end on the junctionbetween the two boundaries.We should also take into account isomorphisms between boundary-changing line operatorsarising from monopole operators. That is, if wish to identify isomorphic boundary-changingline operators, we should further quotient by the subgroup Im K . If we reverse the orderof quotients, we get the following description of the set of isomorphism classes of boundary-changing line operators: it is the quotient of the bulk discriminant group D by the subgroupgenerated by L and L . This subgroup describes bulk Wilson lines which may end at thejunction of boundary conditions defined by subgroups L and L .A boundary-changing line operator from L to L can be fused “from the left” with aboundary line operator on the boundary L or “from the right” with a boundary line operatoron the boundary L . This makes the category of boundary-changing line operators a bi-module category over the monoidal categories of boundary line operators Hom( L , L ) andHom( L , L ). It is easy to see how objects map under this action: fusing a boundary lineoperator labeled by x ∈ D / L i , i = 1 , z ∈ D / ( L + L )gives a boundary-changing line operator x ◦ z = x + z ∈ D / ( L + L ). A complete descriptionof the bi-module category structure requires describing an an “associator” between ( x ◦ y ) ◦ z and x ◦ ( y ◦ z ) , where x, y ∈ D / L i and z ∈ D / ( L + L ). We leave this for future work. So far we have been assuming that the boundary gauge group is connected and therefore is atorus. More generally, the boundary gauge group may have several components each of whichis a torus. In this section we study the corresponding boundary conditions and line operatorson such boundaries.In this section it is convenient, instead of imposing boundary conditions on the gauge field A “by hand”, to introduce Lagrange multiplier fields living on the boundary whose equationsof motion enforce the desired boundary conditions. First let us see how this works in the caseof a connected boundary gauge group. Let Λ be a Lagrangian sublattice in Λ. We can tryto enforce the condition that on the boundary the holonomy of A lies in T Λ by introducinga Lagrange multiplier field ϕ living on the boundary and adding a boundary action i π Z ∂M V ( ϕ, dA ) . The field ϕ takes values in a torus T Φ whose dimension is half the rank of Λ, and Φ is afintely-generated free abelian group whose rank is half the rank of Λ. To write down the27ction we need to lift ϕ to a field valued in Φ ⊗ R . In order for the action to be well-definedthe coupling matrix V must be integral, i.e. it should be an element of the abelian groupHom(Φ , Λ ∗ ). We will denote by V t the dual homomorphism from Λ to Φ ∗ .The equation of motion for ϕ gives a constraint V t ( dA ) = 0 . Since we want the constraint to enforce the correct boundary condition for dA , we need torequire, as a minimum, ker V t = Λ . To ensure that the gauge group on the boundary is T Λ we need to impose a stronger constraintthat the boundary holonomies of A belong to the torus T Λ . The condition for this is as follows.Since ker V t = Λ , Im V annihilates Λ , i.e. Im V ⊂ ker P t , where P t is the projection fromΛ ∗ to Λ ∗ . Boundary holonomies of A belong to T Λ if and only if Im V = ker P t . Indeed,integration over ϕ enforces a constraint that contour integrals of A/ π take integral valueson Im V . On the other hand, we would like to require that contour integrals of A/ π takeintegral values on the sublattice (Λ / Λ ) ∗ of Λ ∗ , i.e. on ker P t . Clearly, for the two conditionsto be equivalent, we must have Im V = ker P t .If Im V is a proper sublattice of ker P t , then the quotient ker P t / Im V is a finite abeliangroup of order d >
1. In such a case the boundary holonomy of A does not necessarily lie inthe torus T Λ , but the d th power of the holonomy does. That is, the boundary gauge groupin this more general case is disconnected and consists of several Lagrangian tori inside thetorus T Λ . But this still ensures that δA takes values in a subspace isotropic with respect to K . Thus we can drop the constraint ker P t = Im V , thereby allowing disconnected boundarygauge groups.Let us describe the boundary gauge group in more detail. It is a subgroup of the bulkgauge group T Λ . Any such subgroup is a kernel of some homomorphism from T Λ to someother torus T Γ induced by a surjective homomorphism h from Λ to Γ. In other words, ifwe regard h as a linear map from Λ R to Γ R , then an element x ∈ Λ R / (2 π Λ) belongs to thesubgroup iff hx/ π belongs to Γ. In our case, the constraint on the holonomy x of A/ π says that V t x belongs to the lattice Φ ∗ . This means that Γ = Φ ∗ and h = V t . That is, theboundary gauge group is the kernel of the Lie group homomorphism V t T : T Λ → T Φ ∗ inducedby the homomorphism V t : Λ → Φ ∗ .So far the only constraint on V was that ker V t is a Lagrangian sublattice of Λ. Thereis one more constraint on V which arises from gauge invariance. Gauge transformations inthe bulk take values in the torus T Λ , but on the boundary they are constrained to lie in theLie subgroup ker V t T , as explained above. Under such a gauge transformations the bulk actionvaries by a boundary term − i π Z ∂M K ( f, dA ) . The gauge variation of the boundary term is i π Z ∂M V ( δϕ, dA )Note that while integration over ϕ produces a delta-functional which constrains dA to lie inΛ ⊗ R , before we integrated over ϕ the form dA can take values in Λ.28et us consider f which lies in the identity component T Λ of the boundary gauge group. Inorder to cancel the variation of the bulk term we need to let B transform as well. Comparingthe two variations we see that we must have δϕ = W T ( f ) , where the Lie group homomorphism W T : T Λ → T Φ is induced by a homomorphism W :Λ → Φ. For the total gauge variation to vanish we must have K = V W.
Here K : Λ → Λ ∗ is a restriction of K to Λ .We can write this constraint in a more convenient form by noting that for any x ∈ Λ K ( x ) annihilates Λ . Similarly, for any y ∈ Φ V ( y ) annihilates Λ . Hence we can interpret K as a homomorphism from Λ to (Λ / Λ ) ∗ , and we can interpret V as a homomorphismfrom Φ to (Λ / Λ ) ∗ . Then the equation K = V W becomes an equation for three integralsquare matrices.For a fixed K and Λ we may look for possible Φ , V and W by trying to factorize K asa product of two integral square matrices V and W . A slightly different way to phrase thisis as follows. K gives an embedding of Λ into (Λ / Λ ) ∗ (as a sublattice of maximal rank).Finding V and W such that K = V W is equivalent to finding a sublattice Φ of (Λ / Λ ) ∗ whichcontains K (Λ ). There are two obvious solutions. We can take Φ = (Λ / Λ ) ∗ , V = 1 and W = K . Or we can take Φ = Λ , V = K and W = 1. We will call them the minimal and themaximal solutions. The boundary gauge group for the minimal solution is the torus T Λ . Forthe maximal solution the boundary gauge group is the kernel of the Lie group homomorphism T Λ → T Λ ∗ induced by the homomorphism π ◦ K : Λ → Λ ∗ . It is disconnected in general, withthe identity component being T Λ . The minimal and maximal solutions coincide if and onlyif K = 1. Now let us analyze which bulk Wilson lines may terminate on the boundary. Let us start witha simple case of an abelian gauge theory with gauge group G = U (1) × U (1) and Chern-Simonslevels 2 N and − N . That is, we take the matrix K to be K = (cid:18) N − N (cid:19) . We can identify both Λ and Λ ∗ with Z ⊕ Z . We choose Λ to consist of elements of Z ⊕ Z ofthe form ( n, n ), so that the torus T Λ is the diagonal subgroup of U (1) × U (1). The quotient(Λ / Λ ) ∗ is one-dimensional and can be identified with the sublattice of Z ⊕ Z of the form( n, − n ). The image of K can be identified with the sublattice consisting of the elements of theform (2 N, − N ). The matrices V and W are one-by-one in this case and satisfy V W = 2 N .Thus choices of Φ are in 1-1 correspondence with divisors of 2 N . If v is such a divisor, thenwe set V = v and W = 2 N/v = w . The minimal solution is v = 1 , w = 2 N , the maximalsolution is v = 2 N, w = 1.The boundary gauge group is a subgroup of U (1) × U (1) defined by the condition v ( x − y ) =0, where x and y take values in R / π Z . The solution of this constraint is y = x + 2 πℓv , ℓ ∈ Z /v Z U (1) × Z v .The identity component of the gauge group is U (1) ≃ R / π Z . Under the correspondinggauge transformations the boundary scalar ϕ transforms as ϕ ϕ + wf, f ∈ C ∞ ( ∂M, R / π Z ) . We still have not determined the transformation law of ϕ under the discrete part of theboundary gauge group. Consider a bulk gauge transformation whose value on the boundaryis given by ( f , f ) = (0 , πℓ/v )The variation of the bulk action under such a transformation is2 N iℓv Z ∂M dA = iwℓ Z ∂M dA . Since R dA is quantized in units of 2 π , the exponential of this term is 1. Now let us lookat the boundary action. The most general transformation law for ϕ under the discrete gaugetransformations is ϕ ϕ + 2 πmℓ/v, where ℓ ∈ Z /v Z is the parameter of gauge transformation and m determines the choice of thetransformation law for ϕ . It is easy to see that the variation of the boundary action is alsoan integral multiple of 2 πi and so no constraint on m arises from gauge-invariance. We willsee below that m is constrained by locality considerations. This will lead to an additionalconstraint on the choice of v and w .A Wilson line can terminate on the boundary if and only if the charge of the end-point ofthe Wilson line can be screened by a local operator sitting on the boundary. Recall also thatthere are bulk monopole operators which carry bulk electric charges of the form (2 N Z , N Z ),so we may regard bulk electric charges as defined modulo 2 N . On the boundary we haveadditional local operators built from ϕ . There are both order and disorder operators of thiskind. The order operators have the form e iνϕ , ν ∈ Z . We will call the integer ν the momentum. The charge of such an operator with respect to theidentity component of the boundary gauge group is νw . A disorder operator is defined by thecondition that as one goes around the insertion point ϕ winds µ times: ϕ ϕ + 2 πµ, µ ∈ Z . We will call µ the winding number. In the presence of a disorder operator dϕ is a closed1-form with a singularity at the insertion point such that d ( dϕ ) = 2 πµδ ( x − x ). Thevariation of the boundary action with respect to the identity component of the boundarygauge group vanishes. Hence the charge of the end-point of a Wilson line has to be screenedby the order operator alone. This means that such a Wilson line must have a charge of theform X = ( X , X ) with X + X = 0 mod w . For the minimal solution v = 1 , w = 2 N thisconstraint reduces to X + X = 0 mod 2 N , i.e. trivial charge with respect to the boundarygauge group, modulo screening by bulk monopoles. This agrees with what we obtained insection 4.1 without introducing the boundary scalar ϕ .30or v = 1 the boundary gauge group also has a factor Z v , and one needs to require thatthe Z v charges of the endpoint of the Wilson line and the order operator cancel. If the Z v charge of ϕ is m mod v , then this condition is X = νm + µv where ν ∈ Z is defined by the relation X + X = νw, and µ is an arbitrary integer.So far m was left undetermined. We will now argue that diffeomorphism invariance ofthe boundary condition on the quantum level requires 2 m = w mod 2 v . Consider M ofthe form S × I . The boundary of this manifold has two connected components, so let usconsider a Wilson line which begins at one boundary components and ends on the other one.On the quantum level Wilson lines have to be regularized by thickening them into ribbons;this is called a choice of framing. Topological correlators for bulk Wilson loops depend onthis choice: twisting a ribbon of charge X by a full turn multiplies the correlator by [9]exp( iπK − ( X, X )) = exp(2 πi q ( X )) . (26)But in the situation described above such a twist can be undone by a rotation of the boundary S . Hence we must require that the correlator be independent of the choice of framing. Inthe simple example we are considering this gives the following condition: X − X = 0 mod 4 N. Expressing X and X in terms of ν and µ we get ν w ( w − m ) − µνvw = 0 mod 4 N. Since vw = 2 N , this implies w − m = 0 mod 2 v. Therefore w must be even and m = w/ v . Since w is even and v = 2 N/w , we see that v must actually be a divisor of N , not just a divisor of 2 N .With this choice of m the constraints on X , X may be rewritten in a more symmetricway: X + X = 0 mod w, X − X = 0 mod 2 v. These constraints ensure that the endpoints of Wilson lines are mutually local. Indeed, theAharonov-Bohm phase arising from transporting a charge X around a charge Y isexp( − πi g ( X, Y )) = exp( − πi ( q ( X + Y ) − q ( X ) − q ( Y ))) . (27)Since our constraints ensure that the phase (26) is trivial for all Wilson lines ending on theboundary, the phase (27) is also trivial.Let us discuss another example of a generalized boundary condition in a U (1) × U (1)Chern-Simons theory, with a nondiagonal form K : K = (cid:18) − − (cid:19) . P = (cid:18) (cid:19) We can solve the condition K = V W by letting V t = v (1 , − , W = 4 v , where v divides 4. On the boundary a gauge transformation ( f , f ) must satisfy v ( f − f ) ∈ π Z . The general solution has the form f = 2 f + 2 παv α = 0 , . . . , v − . That is, the boundary gauge group is isomorphic to U (1) × Z v . Let the charge of a Wilsonline ending on the boundary be X = ( X , X ) , then the screening of the U (1) charge on theboundary requires 2 X + X = W ν, ν ∈ Z . (28)The locality condition in this case reads X + 2 X X = 0 mod 8 . (29)The screening of the Z v charge on the boundary requires X = mν + µv (30)for some integer µ . Now expressing X , X in terms of µ, ν we get from (29): W ( W − m ) = 0 mod 8 . So we conclude 4 v − m = 0 mod 2 v This equation determines the allowed values of v and the corresponding discrete charge m ofthe field ϕ : • v = 2 , m = 1 mod 2 • v = 1 , m = 0In the case v = 2 , m = 1 the charge X = ( X , X ) ∈ Λ ∗ of a Wilson line ending on theboundary must satisfy X = 0 mod 4 . After taking into account identifications due to monopoles, we may regard both X and X as integers modulo 4. Then the condition on ( X , X ) becomes simply X = 0.On the other hand, in the case v = 1 , m = 2 the charge of the Wilson line ending on theboundary must satisfy X + 2 X = 0 . D = Z × Z , with the quadraticfunction q ( X ) = −
18 ( X + 2 X X )Both in the case v = 1 and in the case v = 2 charges of Wilson lines ending on the boundarygenerate a subgroup L of D isomorphic to Z . It is easy to check that in both cases L is Lagrangian with respect to q . This lends support to the proposal that an arbitraryLagrangian subgroup of ( D , q ) defines a valid topological boundary condition.Let us indicate how to extend this analysis to a general abelian Chern-Simons theory. Anorder operator on the boundary has the form e iν ( ϕ ) , where ν is an element of the lattice Φ ∗ = Hom(Φ , Z ) = Hom( T Φ , U (1)). The charge ofsuch an operator with respect to the identity component of the boundary gauge group T Λ is W t ( ν ) ∈ Λ ∗ . Thus the charge X of a Wilson line endpoint can be screened by such an orderoperator only if P t X lies in the lattice Im W t . In general Im W t is a sublattice of maximalrank in Λ ∗ , so this is a nontrivial condition.We also need to require that the charge of the Wilson line with respect to the discrete partof the boundary gauge group be screened by the order operator. The transformation law of ϕ under the boundary gauge group ker V t T is specified by a homomorphism ˜ W : ker V t T → T Φ .The restriction of ˜ W to the torus T Λ must coincide with the homomorphism W T : T Λ → T Φ induced by W : Λ → Φ. Given a choice of ˜ W , the charge cancelation condition which takesinto account the discrete charge is as follows: if we regard the Wilson line charge X as anelement of Hom( T Λ , U (1)), then its restriction to ker V t T must have the form ν ◦ ˜ W for some ν ∈ Hom( T Φ , U (1)).The choice of ˜ W is constrained by the requirement that for any Wilson line ending on theboundary we must have exp(2 πi q ( X )) = 1 . This condition also implies that Wilson lines ending on the boundary are mutually local andensures that the subgroup L ⊂ D spanned by their charges is isotropic.Boundary Wilson lines are restrictions of bulk Wilson lines. As before, charges of boundaryWilson lines take values in the quotient group D / L . The tensor product of Wilson linescorresponds to the group operation in D / L . The computation of the boundary associator andthe semi-braiding proceeds in the same way as in section 5.2, the only difference being thatnow certain boundary Wilson lines are trivial because they can be screened by order operatorsexp( iνϕ ) on the boundary. The formulas of section 5.2 for the boundary associator and thesemi-braiding remain valid. A surface operator in a 3d TFT is a defect of codimension one. Such defects can be regardedas objects of a monoidal 2-category [1]. Objects of this 2-category are surface operators Isotropicity of L is equivalent to the locality constraint on the charges of Wilson lines ending on theboundary. The fact that L is coisotropic is less trivial, and we do not know of a general proof. A , we may construct a boundary condition ina 3d TFT A × A ∗ , where A ∗ denotes the parity reversal of A . This is accomplished by“folding” the worldvolume at the location of the surface operator. Using this folding trick,the 2-category of surface operators in the theory A may be identified with the 2-category ofboundary conditions in the theory A × A ∗ .If A is an abelian Chern-Simons theory corresponding to the bilinear form K , then A × A ∗ is the abelian Chern-Simons theory corresponding to the bilinear form K ⊕ ( − K ). Thus theresults obtained so far allow us to classify surface operators in an arbitrary abelian Chern-Simons theory and determine the 2-category structure on them. The monoidal structure is anew ingredient which requires a separate study.Note that the form K ⊕ ( − K ) on the lattice Λ ⊕ Λ has signature zero, so there is noobstruction for the existence of topological boundary conditions in such a theory. In fact,in any such theory there are two obvious choices of a Lagrangian sublattice in Λ ⊕ Λ: thediagonal one Λ + = { ( λ, λ ) | λ ∈ Λ } and the anti-diagonal one Λ − = { ( λ, − λ ) | λ ∈ Λ } . The diagonal boundary condition corresponds to the “trivial” or “invisible” surface operatorwhose insertion is equivalent to no surface operator at all. Such a surface operator is theidentity object in the monoidal 2-category of surface operators. The anti-diagonal surfaceoperator is characterized by the fact that the gauge field changes its sign when one goesacross the insertion surface.The boundary condition for the U (1) m theory considered in sections 4.4 and 5.3 corre-sponds to the trivial surface operator in the U (1) m theory. Note that while we have a wholefamily of such operators differing by a choice of the splitting R , they all appear to be iso-morphic, since their physical properties (e.g. the associator for the line operators and thesemi-braiding) do not seem to depend on R . To demonstrate this more formally, one needs toconstruct an invertible morphism between surface operators corresponding to different choicesof R . From the physical viewpoint such a morphism and its inverse are boundary-changingline operators between the corresponding boundary conditions in the folded theory A × A ∗ .Since we have not studied the composition of line operators sitting at the junction of differentboundary conditions, we leave this for future work. In this section we will apply what we have learned so far to the problem of classificationof abelian Chern-Simons theories on the quantum level. It was noticed in [10] that certainabelian Chern-Simons theories with a nontrivial K are nevertheless trivial on the quantumlevel (isomorphic to the trivial theory). The classification problem for abelian Chern-Simonstheories was studied in detail in [2], both for even and odd lattices. According to [2], twoChern-Simons theories are considered equivalent if their spaces of states on any Riemannsurface are isomorphic as projective representations of the mapping class group. For evenlattices the main result of [2] is that equivalence classes of abelian Chern-Simons theories areclassified by the following data: 34 Signature σ of the bilinear form K modulo 24; • The discriminant group D = Λ ∗ /K (Λ); • The quadratic function q : D → Q / Z derived from K .The analog of this for odd lattices is a bit more complicated; we will not discuss this casesince it corresponds to spin Chern-Simons theories rather than regular 3d TFTs.An example of a pair of even Chern-Simons theories which are equivalent on the quantumlevel but not classically is given by positive-definite even unimodular lattices Γ and Γ ⊕ Γ , where Γ is the weight lattice of Spin (32) / Z , and Γ is the root lattice of the Lie algebra E .These lattices are not linearly equivalent (i.e. there is no integral linear transformation M with det M = ± σ = 16 and trivial D and q . The classification scheme proposed in [2] identifies 3d TFTs which have isomorphic spacesof states on an arbitrary Riemann surface. It leaves open a possibility that other observables(e.g. line operators or surface operators) may distinguish theories equivalent in this sense. Abetter definition of equivalence which ensures that all topological observables coincide is basedon the notion of a duality wall [3, 4]. If two theories are truly equivalent, there should be acodimension-1 topological defect between them which implements the duality transformationon the fields. This defect enables one to map any observable in one theory to an observablein the other theory, so that correlators are preserved. The map is defined by inserting thecodimension-1 defect on the boundary of a tubular neighborhood of the observable that onewishes to dualize. This is the reason for the name “duality wall”. In three dimensions aduality wall is a surface operator which has an inverse. From the mathematical viewpoint, 3dTFTs form a 3-category, and a duality wall is an invertible morphism between two objects ofthis 3-category.We will now show that a duality wall exists between any two abelian Chern-Simons theorieswith even K for which the following data coincide: • Signature σ of the bilinear form K ; • The discriminant group D ; • The quadratic function q : D → Q / Z derived from K .Note that this is almost the same data as in [2], except that signatures are required to beidentical rather than equal modulo 24. This is the best result one could possibly hope for.Indeed, since the properties of bulk line operators are described by D and q , Chern-Simonstheories for which these data are different cannot be equivalent. Further, if the signaturesare not identical, then no topological surface operator (invertible or not) can be constructedbetween such theories. Indeed, after performing the folding trick, the problem of findinga surface operator between two theories with bilinear forms K , K becomes equivalent tothe problem of finding a topological boundary condition for a theory with a bilinear form K ⊕ ( − K ). If K and K have different signatures, then K ⊕ ( − K ) has nonzero signature,and therefore does not admit any topological boundary conditions, as argued above. In otherwords, in the 3-category of 3d TFTs there are no morphisms between abelian Chern-Simonstheories with different signatures.To illustrate the difference between the classification scheme of [2] and that based onduality walls consider the theory based on a positive-definite even unimodular lattice Γ ⊕ . and Λ haveisomorphic discriminant-bilinear forms g , g if and only if they are stably equivalent, i.e.Λ ⊕ S ∼ = Λ ⊕ S (31)where S and S are even unimodular lattices. The symbol ∼ = in (31) stands for linearequivalence.If Λ and Λ have the same signature, S and S also have the same signature. Since we canalways replace S with S ′ = S ⊕ ( −S ) and S with S ′ = S ⊕ ( −S ), we may assume withoutloss of generality that S and S have signature zero. Another well-known mathematical factis that any even unimodular lattice with zero signature is isomorphic to a sum of severalcopies of the rank-two lattice U , whose symmetric bilinear form is K U = (cid:18) (cid:19) . Puttingall this together, we see that if Λ and Λ have the same σ , D , and q , then there exists anintegral matrix M with det ( M ) = ± K ⊕ K ⊕ m U = M T ( K ⊕ K ⊕ m U ) M (32)This implies that the theories defined by the bilinear forms K ′ = K ⊕ K ⊕ m U and K ′ = K ⊕ K ⊕ m U are equivalent and related by the change of variables in the path-integral whichsets A ′ = M A ′ . The duality wall between them is the surface operator defined by thecondition that the restriction of A ′ is equal to the restriction of M A ′ .To complete the proof it is sufficient to show that for any even symmetric bilinear form K and any integer m the Chern-Simons theories corresponding to K and K ⊕ K ⊕ mU areequivalent. In fact, it is sufficient to show this for the case K = 0 and m = 1: given a dualitywall between K U and the trivial theory one may construct a duality wall between K ⊕ K ⊕ mU and K by combining m copies of the former duality wall with the “invisible” wall between K and K .The fact that the Chern-Simons theory corresponding to K U is trivial was explained in[10]. We would like to demonstrate this more formally by exhibiting a duality wall betweenthe theory K U and the trivial theory. Such a wall is a boundary condition for the theory K U .An obvious choice of a Lagrangian subgroup in U (1) × U (1) is given by P = (cid:18) (cid:19) . This means that the restriction of the gauge field to the boundary has the form ( A , A is an arbitrary U (1) gauge field. The boundary gauge group is U (1). We can take thesplitting R to be R = P . Let us denote by O ∅ ; U the resulting surface operator between thetheory K U and the trivial one. We need to show that it is invertible. The inverse is thesame surface operator, but parity-reversed; we will denote it O U ; ∅ . We need to show that thesurface operators O ∅ ; U and O U ; ∅ satisfy O ∅ ; U O U ; ∅ = I ∅ , O U ; ∅ O ∅ ; U = I U , (33)36here I ∅ i the invisible surface operator in the trivial theory, and I U is the invisible surfaceoperator in the theory K U .To prove the first equality in (33) we need to study the Chern-Simons theory defined by K U on an interval [0 , L ] times Σ, where Σ is an arbitrary Riemann surface. The boundaryconditions are given by P on both ends of the interval. We use the fact that Chern-Simonstheory on M = Σ × [0 , L ] with such boundary conditions reduces to the U (1) BF -type theorywith the action S = i π Z Σ ϕ dA, (34)where ϕ is a periodic scalar with period 2 π . This reduction was explained in section 5.4. Sucha theory is trivial for any Σ (its partition function is one), which proves the first equality in(33).To prove the second equality, we need to consider the theory K U on an arbitrary 3-manifold M and with an insertion of O U ; ∅ O ∅ ; U along a Riemann surface Σ embedded into M . Thismeans that we excise a region H = Σ × [0 , L ] and insert O U ; ∅ at Σ × { } and O ∅ ; U at Σ × { L } .The theory K U assigns to Σ a one-dimensional Hilbert space L [10]. The boundary of the3-manifold M \ H is a disjoint sum of two copies of Σ with opposite orientation, so the theory K U attaches to it a vector in the space L ⊗L ∗ = End( L ). Let us denote this vector Φ( M ; H ).The partition function of the theory K U on M is given by the inner product h I Σ | Φ( M , H ) i , where I Σ is the identity element in End( L ) (it represents the invisible surface operator insertedat Σ). On the other hand, the partition function of the theory K U on M with an insertionof O U ; ∅ O ∅ ; U along Σ is given by the inner product h Ψ Σ ⊗ Ψ ∗ Σ | Φ( M , H ) i , where Ψ Σ ∈ L is the boundary state corresponding to the boundary condition O U ; ∅ . Sincethe Hilbert space L is one-dimensional, the state Ψ Σ ⊗ Ψ ∗ Σ is equal to I Σ times the norm ofΨ Σ . The latter norm is equal to the partition function of the BF-theory (34) on Σ, i.e. 1.Hence the partition function of the theory K U on M coincides with the partition function ofthe same theory with an insertion of O U ; ∅ O ∅ ; U . This proves the second equality in (33). In this paper we studied topological boundary conditions in an arbitrary abelian Chern-Simonstheory and argued that they are classified by subgroups of the discriminant group D whichare Lagrangian with respect to the quadratic function q : D → Q / Z . Given such a Lagrangiansubgroup L , boundary line operators are labeled by elements of the finite group D / L . Themonoidal category of boundary line operators is a twisted version of the category of D / L -graded vector spaces, where the twist is given by a certain canonical element in H ( D / L , Q / Z )which we explicitly described. Furthermore, we showed that for any Lagrangian L there is amonoidal functor (which we called the semi-braiding) from the category of bulk line operatorsto the Drinfeld center of the category of boundary line operators.These results raise the question how to describe the full 2-category of boundary conditionsfor an abelian Chern-Simons theory. By analogy with the Rozansky-Witten model [8, 12] onemay propose the following conjecture. Let L be a Lagrangian subgroup of D , and let C L be37he corresponding monoidal category of boundary line operators. To any other boundarycondition one may attach a module category over C L , namely, the category of line operatorssitting at the junction of the chosen boundary condition and L . Axioms of 3d TFT implythis map extends to a functor from the 2-category of boundary conditions to the 2-categoryof module categories over C L . It is natural to conjecture that this functor is an equivalence.In other words, different choices of L give 2-Morita-equivalent monoidal categories C L . TheDrinfeld centers of all these monoidal categories are equivalent, and presumably the semi-braiding functors establishes an equivalence between these Drinfeld centers and the braidedmonoidal category of bulk line operators. Note that if we take L be isotropic rather thanLagrangian, then the formula for the element in H ( D / L , Q / Z ) still makes sense, so themonoidal category of D / L -graded vectors spaces twisted by this cocycle is well-defined, butits Drinfeld center is “too large” and is not equivalent to the category of bulk line operators.Presumably the role of the coisotropic condition is to ensure that the Drinfeld center of thecategory C L is of the right size.In this paper we only briefly discussed surface operators in abelian Chern-Simons theory.It would be interesting to understand how to fuse surface operators, i.e. to understand themonoidal structure on the 2-category of surface operators. It would be even more interesting toextend the analysis to nonabelian Chern-Simons theories. Nonabelian Chern-Simons theorieswith a simple gauge group do not seem to admit topological boundary conditions, but theyhave interesting surface operators whose properties are not understood.Recently higher categorical structures in abelian Chern-Simons theory have been studiedfrom a rather different standpoint by Freed, Hopkins, Lurie and Teleman [13]. These authorsconsider the situation when the signature of K is not necessarily zero, and the theory itselfis regarded as an “anomalous” 3d TFT which sits on a boundary of a 4d TFT. It would bevery interesting to understand the relationship between our work and [13] in the case when K has zero signature. Acknowledgments
We would like to thank D. Freed, A. Kitaev, J. Lurie, G. Moore, V. Ostrik, and L. Rozanskyfor useful discussions and advice. We are grateful to the Aspen Center for Physics for anexcellent working atmosphere. This work was supported in part by the DOE grant DE-FG02-92ER40701.
Appendix A. Braided monoidal categories
A monoidal category V is a category with a covariant bi-functor (tensor product) ⊗ : V ×V 7→V and a distinguished object 1 . In addition, for any three objects
U, V, W one is given anisomorphism a U,V,W : ( U ⊗ V ) ⊗ W U ⊗ ( V ⊗ W ) , and for any object U one is given a pair of isomorphisms r U : U ⊗ U, l U : 1 ⊗ U U such that the following pentagon and triangle diagrams commute:38 U ⊗ V ) ⊗ ( W ⊗ X ) a U,V,W ⊗ X ) ) SSSSSSSSSSSSSS (cid:0) ( U ⊗ V ) ⊗ W (cid:1) ⊗ X a U ⊗ V,W,X iiiiiiiiiiiiiiii a U,V,W ⊗ id X (cid:15) (cid:15) U ⊗ (cid:0) V ⊗ ( W ⊗ X ) (cid:1)(cid:0) U ⊗ ( V ⊗ W ) (cid:1) ⊗ X a U,V ⊗ W,X / / U ⊗ (cid:0) ( V ⊗ W ) ⊗ X (cid:1) id U ⊗ a V,W,X O O ( U ⊗ ⊗ V r U ⊗ id V ( ( QQQQQQQQQQQQ a U, ,V / / U ⊗ (1 ⊗ V ) id U ⊗ l V w w ppppppppppp U ⊗ V The morphism a U,V,W is called the associator. If both a U,V,W and r U , l U are identities, themonoidal category is called strict. In the cases of interest to us r U and l U are identities (i.e.we are dealing with monoidal categrories with a strict identity), but the associator may benontrivial.Let G be a finite abelian group, and let us consider the category of finite-dimensional G -graded complex vector spaces. A simple objects in this category is a one-dimensional vectorspace C X labeled by a particular element X ∈ G ; its endomorphism algebra is C for any X ∈ G . Let us define the tensor product of simple objects C X and C Y to be C X + Y . Theassociator is an element of C ∗ a X,Y,Z = exp (cid:0) πih ( X, Y, Z ) (cid:1) ∈ C ∗ , X, Y, Z ∈ G . Regarding h ( X, Y, Z ) as an C / Z -valued function on G × G × G , we may write the pentagonidentity as follows: h ( X, Y, Z + W ) + h ( X + Y, Z, W ) = h ( Y, Z, W ) + h ( X, Y + Z, W ) + h ( X, Y, Z ) . The triangle identity reads h ( X, , Z ) = 0 . We will denote such a monoidal category (
Vect G , h ). One can show that ( Vect G , h ) is equivalentto ( Vect G , h ′ ) if and only if there exists a function k : G × G → C / Z such that( h ′ − h )( X, Y, Z ) = k ( Y, Z ) − k ( X + Y, Z ) + k ( X, Y + Z ) − k ( X, Y ) . In such a case we will say that h and h ′ differ by a coboundary.A braided monoidal category is equipped with an additional structure, a family of braidingisomorphisms c U,V : U ⊗ V V ⊗ U. Braiding is required to be compatible with the tensor product, by which we mean that thefollowing two hexagon diagrams must commute:39 ⊗ ( B ⊗ C ) a − A,B,C v v mmmmmmmmmmmm c A,B ⊗ C + ( B ⊗ C ) ⊗ A ( A ⊗ B ) ⊗ C c A,B ⊗ id C ( ( QQQQQQQQQQQQ B ⊗ ( C ⊗ A ) a − B,C,A h h QQQQQQQQQQQQ ( B ⊗ A ) ⊗ C a B,A,C / / B ⊗ ( A ⊗ C ) id B ⊗ c A,C mmmmmmmmmmmm ( U ⊗ V ) ⊗ W a U,V,W v v lllllllllllll c U ⊗ V,W + W ⊗ ( U ⊗ V ) U ⊗ ( V ⊗ W ) id U ⊗ c V,W ( ( RRRRRRRRRRRRR ( W ⊗ U ) ⊗ V a W,U,V h h RRRRRRRRRRRRR U ⊗ ( W ⊗ V ) a − U,W,V / / ( U ⊗ W ) ⊗ V c U,W ⊗ id V lllllllllllll Consider again the monoidal category (
Vect G , h ). A braiding on this category is a C ∗ -valued function on G × G , c X,Y = exp (cid:0) πis ( X, Y ) (cid:1) , X, Y ∈ G and the hexagon identities become s ( X, Y + Z ) − s ( X, Y ) − s ( X, Z ) = h ( Y, X, Z ) − h ( Y, Z, X ) − h ( X, Y, Z ) , (35) s ( X + Y, Z ) − s ( Y, Z ) − s ( X, Z ) = h ( Z, X, Y ) − h ( X, Z, Y ) + h ( X, Y, Z ) . (36)Here we regard s ( X, Y ) as an C / Z -valued function.It was shown in [5] that pairs ( h, s ) and ( h ′ , s ′ ) correspond to equivalent braided monoidalcategories if their difference is a “coboundary”, i.e.( h ′ − h )( X, Y, Z ) = k ( Y, Z ) − k ( X + Y, Z ) + k ( X, Y + Z ) − k ( X, Y ) , (37)( s ′ − s )( X, Y ) = − k ( X, Y ) + k ( Y, X ) , (38)where k ( X, Y ) is an arbitrary C / Z -valued function on G × G . Appendix B. Drinfeld center, semi-braiding and the bound-ary hexagon relations
The Drinfeld center of a monoidal category C is a (braided) monoidal category Z ( C ) whoseobjects are pairs ( c, χ c ), where c is an object of C and χ c is a family of natural isomorphisms χ c ( a ) : a ⊗ c → c ⊗ a for all objects a of C . Moreover, χ c must be compatible with themonoidal structure on C in the sense that the following diagram commutes:40 a ⊗ b ) ⊗ c a a,b,c w w ooooooooooo χ c ( a ⊗ b ) + c ⊗ ( a ⊗ b ) a ⊗ ( b ⊗ c ) id a ⊗ χ c ( b ) ' ' OOOOOOOOOOO ( c ⊗ a ) ⊗ b a c,a,b g g OOOOOOOOOOO a ⊗ ( c ⊗ b ) a − a,c,b / / ( a ⊗ c ) ⊗ b χ c ( a ) ⊗ id b ooooooooooo Here a a,b,c denotes the associator in the category C .Let us specialize to the case C = ( Vect G , h ). In this case we can regard χ c as a function χ c : G → C / Z and the commutativity of the above diagram is equivalent to the equation χ c ( a + b ) = χ c ( a ) + χ c ( b ) + h ( c, a, b ) − h ( a, c, b ) + h ( a, b, c ) (39)We have mentioned above that the Drinfeld center of a monoidal category has a naturalbraided monoidal structure. Let us explain how the tensor product on objects of Z ( C ) lookslike in the case C = ( Vect G , h ). The tensor product of two simple objects ( b, χ b ) and ( c, χ c ) inthe Drinfeld center, b, c ∈ G , is defined as ( b + c, χ b,c ) with χ b,c ( a ) = χ b ( a ) + χ c ( a ) + h ( b, a, c ) − h ( b, c, a ) − h ( a, b, c ) . (40)In this paper we considered the following situation: we have a pair of monoidal categories( Vect D , h ) and ( Vect D , h ) such that D is a quotient of D , and a function s sb : D × D → C / Z .We would like to interpret this function as defining a strong monoidal functor from the formercategory to the Drinfeld center of the latter. This functor is supposed to map an object B ∈ D to an object ( b, s sb ( · ; B )) of the Drinfeld center, where b = π ( B ) and π : D → D is the quotientmap. The condition that this satisfies (39) reads s sb ( a ⊙ b ; C ) − s sb ( a ; C ) − s sb ( b ; C ) = h ( c, a, b ) − h ( a, c, b ) + h ( a, b, c ) , where we denoted the group operation by ⊙ to agree with the notation in the rest of thepaper. We will call this equation the first boundary hexagon relation.Note that this hexagon relation is satisfied if we set D = D and s sb ( b ; C ) = s ( b, c ). Thismeans that the category of bulk line operators naturally embeds into its own Drinfeld center.More generally, any braided monoidal category naturally embeds into its own Drinfeld center.If we regard a monoidal and a braided monoidal category as a categorification of an associativealgebra and a commutative associative algebra respectively, this is analogous to the statementthat a commutative algebra coincides with its own center.A strong monoidal functor must preserve the tensor product of objects. Recalling thedefinition of the tensor product (40) we conclude that the following equation must hold: s sb ( a ; B ⊙ C ) = s sb ( a ; B ) + s sb ( a ; C ) + h ( b, a, c ) − h ( b, c, a ) − h ( a, b, c ) . (41)We will this equation the second boundary hexagon relation because it is equivalent to thecommutativity of the following diagram: 41 ⊗ ( b ⊗ c ) a − ( a,b,c ) v v mmmmmmmmmmmmm c sb ( a ; B ⊗ C ) + ( b ⊗ c ) ⊗ a ( a ⊗ b ) ⊗ c c sb ( a ; B ) ⊗ id c ( ( QQQQQQQQQQQQQ b ⊗ ( c ⊗ a ) a − ( b,c,a ) h h QQQQQQQQQQQQQ ( b ⊗ a ) ⊗ c a ( b,a,c ) / / b ⊗ ( a ⊗ c ) id b ⊗ c sb ( a ; C ) mmmmmmmmmmmmm Here c sb ( a ; B ) = exp(2 πis sb ( a ; B )) and a ( a, b, c ) = exp(2 πih ( a, b, c )).Again, the equation (41) automatically holds if we set D = D and s sb ( b ; C ) = s ( b, c ),meaning that the natural embedding of the braided monoidal category ( Vect D , h ) into its ownDrinfeld center preserves tensor products.A strong monoidal functor F between two monoidal categories B and C is more thanmerely a functor which preserves tensor products. For any two objects A, B of B one must begiven an isomorphism Φ A,B : F ( A ⊗ B ) → F ( A ) ⊗ F ( B ) satisfying a compatibility relationwith the associators. One should also be given an isomorphism F ( I ) → I ′ between identityobjects satisfying a compatibility condition. In our case the latter isomorphism is the identity,and the corresponding compatibility condition is trivially satisfied. On the other hand, Φ A,B is a nontrivial phase exp(2 πiφ ( A, B )) satisfying h ( a, b, c ) − h ( A, B, C ) = φ ( A, B + C ) − φ ( A, B ) − φ ( A + B, C ) + φ ( B, C ) . (42)In our case this phase turns out to be φ ( A, B ) = 12 X γ a γ m γ (cid:20) B γ m γ (cid:21) (˜ v γ K − ˜ v γ ) + X γ>β a β m γ (cid:20) B γ m γ (cid:21) (˜ v β K − ˜ v γ ) . Finally note that the pentagon equation and the two boundary hexagon relations continueto hold if ( s sb , h ) is replaced with ( s ′ sb , h ′ ) such that s ′ sb ( a ; B ) − s sb ( a ; B ) = − k ( a, b ) + k ( b, a ) , (43) h ′ ( a, b, c ) − h ( a, b, c ) = k ( a, b + c ) − k ( a + b, c ) + k ( b, c ) − k ( a, b ) , (44)where k : D × D → C / Z is an arbitrary function. Such a transformation corresponds toreplacing ( Vect D , h ) with an equivalent monoidal category, and modifying appropriately thesemi-braiding functor (composing it with the equivalence). References [1] A. Kapustin, “Topological Field Theory, Higher Categories, and Their Applications,”arXiv:1004.2307 [math.QA].[2] D. Belov, G. Moore,“Classification of abelian spin Chern-Simons theories”, hep-th/0505235. 423] D. Gaiotto and E. Witten, “S-Duality of Boundary Conditions In N=4 Super Yang-MillsTheory,” arXiv:0807.3720 [hep-th].[4] A. Kapustin and M. 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