Gapped boundary theories in three dimensions
GGAPPED BOUNDARY THEORIES IN THREE DIMENSIONS
DANIEL S. FREED AND CONSTANTIN TELEMAN
Abstract.
We prove a theorem in 3-dimensional topological field theory: a Reshetikhin-Turaevtheory admits a nonzero boundary theory iff it is a Turaev-Viro theory. The proof immediatelyimplies a characterization of fusion categories in terms of dualizability. The main theorem appliesto physics, where it implies an obstruction to a gapped 3-dimensional quantum system admittinga gapped boundary theory. Appendices on bordism multicategories and on internal duals may beof independent interest.
Contents
1. Mathematical background and statement of main theorem 21.1. RT theories and TV theories 21.2. Definitions and terminology 41.3. Relations between RT and TV theories 51.4. Existence of boundary theories 61.5. A characterization of fusion categories 82. Proof 82.1. Bordism n -categories 82.2. A higher categorical preliminary 122.3. Proofs of Theorem A, Theorem B, and Theorem A k -morphisms in Bord n k -morphisms in Bord n, B Quantum mechanical theories bifurcate into gapped and gapless theories. The classical notion ofa local boundary condition for a partial differential equation has a quantum analog—a boundarytheory. There is a basic question: Does a gapped quantum system S admit a gapped boundarytheory? We formulate and prove a mathematical theorem (Theorem A in § Date : June 17, 2020.This material is based upon work supported by the National Science Foundation under Grant Number DMS-1611957. Parts of this work were performed at the Aspen Center for Physics, which is supported by National ScienceFoundation grant PHY-1607611. We also thank the IAS/Park City Mathematics Institute and the MathematicalSciences Research Institute, which is supported by National Science Foundation Grant 1440140. a r X i v : . [ m a t h . QA ] J un D. S. FREED AND C. TELEMAN we merely assume. The absence of a gapped boundary theory implies the presence of gapless edgemodes—conduction on the boundary—an important feature of quantum Hall systems, for example.Let F : Bord fr3 Ñ C be a 3-dimensional topological field theory, a homomorphism from a bordismmulticategory of framed manifolds to a symmetric monoidal 3-category C . We impose hypotheseson C , F to model Reshetikhin-Turaev theories [RT1, RT2, T], whose key invariant is a modulartensor category C , the value of F on a bounding framed circle. Theorem A asserts that if F admitsa nonzero boundary theory, then C is the Drinfeld center of a fusion category Φ. Conversely,given Φ there is a nonzero boundary theory built from the (regular) left Φ-module Φ. In otherwords, the class of C in the Witt group [DMNO] is a complete obstruction to the existence of anonzero boundary theory. With extra assumptions on the codomain C , we conclude (Theorem A in § F is isomorphic to the Turaev-Viro theory based on Φ.A corollary of our proof is a characterization of fusion categories (Theorem B in § C be the symmetric monoidal 2-category of finitely presentable C -linear categories, and let E p Cat C q be the Morita 3-category of tensor categories. (In this paper ‘tensor category’ means ‘algebraobject in Cat C ’.) Then Ψ P E p Cat C q is a fusion category iff Ψ is 3-dualizable and the regular leftΨ-module is 2-dualizable. The forward direction (‘only if’) is proved in [DSS].A key feature of our approach is the use of fully extended field theories. Naive analogs ofTheorem A fail in the traditional context of p , , q -theories; see Remark 1.25.Here is a brief outline of the paper. Section 1 contains background and the statements of themain theorems. The proof of Theorem A is deferred to §
2, where we begin with an informal discus-sion of bordism multicategories. The application to gapped quantum systems is the subject of §
1. Mathematical background and statement of main theorem1.1. RT theories and TV theories
In the late 1980s Witten [W1] and Reshetikhin-Turaev [RT1, RT2] introduced new invariantsof closed 3-manifolds and generalizations of the Jones invariants of knots. Witten’s starting pointis the classical Chern-Simons invariant, which he feeds into the physicists’ path integral, whereasReshetikhin-Turaev begin with an intricate algebraic structure: a quantum group. Later, quantum We do not assume that a fusion category has a simple unit: our ‘fusion’ is [EGNO]’s ‘multifusion’.
APPED BOUNDARY THEORIES IN 3D 3 groups were replaced by modular tensor categories [T], which were originally introduced in thecontext of 2-dimensional conformal field theory [MS]. These disparate approaches are reconciledin extended topological field theory [F1]. In modern terms [L1] this extended field theory is asymmetric monoidal functor(1.1) F x , , y : Bord fr x , , y ÝÑ Cat C with domain the 2-category of 3-framed F x , , y on thebounding 3-framed circle S b is the modular tensor category that defines the theory. We call (1.1)a Reshetikhin-Turaev (RT) theory.
Remark . The RT theories in the original references factor through the bordism 2-category ofmanifolds equipped with a p w , p q -structure [BHMV], that is, an orientation and a trivializationof the first Pontrjagin class p . There is a unique isomorphism class of p w , p q -structures on acircle, so no distinction between bounding and nonbounding circles. “Spin Chern-Simons theories”require a trivialization of the second Stiefel-Whitney class w as well, so they use the full 3-framing.For those theories the codomain should include Z { Z -gradings; see Remark 1.26.A fully extended topological field theory has domain Bord fr3 “ Bord fr x , , , y , the 3-category of3-framed bordisms of dimension ď
3. There is no canonical codomain for these theories, so for nowwe posit an arbitrary symmetric monoidal 3-category C . The cobordism hypothesis —conjecturedby Baez-Dolan [BD], proved by Hopkins-Lurie in 2 dimensions and by Lurie [L1] in all dimensions;see also [AF]—asserts that a fully extended theory(1.3) F : Bord fr3 ÝÑ C is determined by its value F p`q on a positively oriented 3-framed point. Furthermore, a object of C determines a unique theory (1.3), up to a contractible space of choices. A symmetricmonoidal 3-category C has a fully dualizable part C fd Ă C whose objects are 3-dualizable and whosemorphisms have all adjoints. We say C has duals if C “ C fd . A functor (1.3) factors through C fd .Given a general RT theory (1.1) it is still an open problem to construct C and an extension (1.3),or even better to construct a single C which works for all RT theories. (However, see [He] for aspecial case in the framework of bicommutant categories.)There is a subclass of RT theories, the Turaev-Viro (TV) theories [TV], which are fully ex-tended. Let Fus be the symmetric monoidal 3-category whose objects are fusion categories; seeDefinition 1.9.
Theorem 1.4 (Douglas-Schommer-Pries-Snyder [DSS]) . Fus has duals, i.e.,
Fus “ Fus fd . The version in [MS] uses a central charge in Q { Z , whereas the version standard in mathematics, which we use,only has a central charge in Q { Z . In this paper we use discrete categories: for example Cat C is a p , q -category (as opposed to a more general p8 , q -category). Some of our exposition in this section applies to p8 , n q -categories though we just write ‘ n -categories’. A of a 3-manifold is a global parallelism, a trivialization of its tangent bundle. For a manifold M ofdimension k ă R ´ k ‘ T M Ñ M . D. S. FREED AND C. TELEMAN
In particular, a fusion category Φ is 3-dualizable in Fus. The cobordism hypothesis implies thatthere is a fully extended topological field theory(1.5) T Φ : Bord fr3 ÝÑ Fus , unique up to equivalence, whose value on a chosen framed point ` is T Φ p`q “ Φ. A TV theory is afully extended theory with codomain Fus; its truncation to Bord fr x , , y is an RT theory. Examplesinclude 3-dimensional gauge theory for a finite group G , in which case Φ is the fusion categoryof finite rank complex vector bundles over G with convolution product; there is also a versiontwisted by a cocycle for a class in H p G ; Q { Z q , as in [DW]. Special toral Chern-Simons theoriesare also TV theories. We remark that for a spherical fusion category Φ, the theory T Φ descendsto the bordism multicategory of oriented manifolds. Conjecturally, every fusion category admits aspherical structure. The definitions and terminology for abelian categories are standard; see [EGNO, §
1] for example.For tensor categories there is tremendous variation in the literature, so we spell out our usage here.The term ‘modular tensor category’ is standard; see [EGNO, § Definition 1.6. (i) Cat C is the symmetric monoidal 2-category defined as follows. Its objects are finitelycocomplete C -linear categories. 1-morphisms in Cat C are right exact C -linear functors—functors that preserve finite colimits—and 2-morphisms are natural transformations. Thesymmetric monoidal structure is the Deligne-Kelly tensor product (cid:2) ; see [K, D, Fr].(ii) A tensor category is an algebra object in Cat C .(iii) E p Cat C q is the symmetric monoidal 3-category defined as follows. Its objects are tensorcategories. A 1-morphism M : A Ñ B is an object M P Cat C equipped with the structureof a p B, A q -module category, i.e., a left p A (cid:2) B op q -module category. A 2-morphism M Ñ M is a 1-morphism in Cat C which respects the bimodule structure. A 3-morphism is a naturaltransformation of functors. Remark . (1) We do not assume that a tensor category has internal duals (rigidity). See Appendix B fora discussion of internal duals in tensor categories. Also, we do not assume that the tensorunit of a tensor category is a simple object.(2) The algebraic theory of tensor categories is exposited in the text Etingof-Gelaki-Nikshych-Ostrik [EGNO] (where rigidity and simple unit are included in the definition of ‘tensorcategory’). The theory of Morita higher categories, such as E p Cat C q , is developed byHaugseng [H], Johnson-Freyd-Scheimbauer [JS], Gwilliam-Scheimbauer [GS], among others.Douglas-Schommer-Pries-Snyder [DSS] define a 3-category of tensor categories which is asubcategory of E p Cat C q ; in particular, they assume rigidity. Tensor categories in an infinitesetting are explored in [BJS], and in an -setting in [L2]. The symbol ‘Rex’ is sometimes used in place of ‘Cat C ’; it emphasizes the right exactness of 1-morphisms. APPED BOUNDARY THEORIES IN 3D 5 (3) The symmetric monoidal structure on E p Cat C q is Deligne-Kelly tensor product, as in Cat C .Composition of 1-morphisms in E p Cat C q is the relative Deligne-Kelly tensor product: ten-sor product of module categories over a tensor category. Its existence is discussed in [BZBJ,Remark 3.2.1] and [JS, Example 8.10].(4) For a 3-category C set Ω C “ End C p q , the endomorphism 2-category of the tensor unitobject. There is a canonical identification(1.8) Ω E p Cat C q “ Cat C . The 3-category Fus of fusion categories is introduced in [DSS].
Definition 1.9. (i) FSCat is the full subcategory of Cat C whose objects are finite semisimple abelian categories.(ii) A fusion category is a finite semisimple rigid tensor category.(iii) Fus is the symmetric monoidal 3-category subcategory of E p Cat C q whose objects are fusioncategories and whose 1-morphisms are finite semisimple abelian bimodule categories. Remark . (1) FSCat Ă Cat C is closed under Deligne-Kelly tensor product [Fr, § “ FSCat . A companion result to Theorem 1.4 asserts that the symmetric monoidal 2-category FSCat hasduals. We also need the following result.
Theorem 1.12.
The 2-dualizable objects of
Cat C are the finite semisimple abelian categories. There are many versions of this theorem, such as [Se, Ti]; see [BDSV, Appendix] for a survey. Forthe category Cat C of finitely cocomplete C -linear categories, see [BJSS, Remark 3.5]. Let(1.13) T : Bord fr3 ÝÑ Fusbe a TV theory. Then the associated modular tensor category T p S b q , the value of T on the bounding3-framed circle, is the Drinfeld center of T p`q . It is the modular tensor category of the associatedRT theory, which is the truncation(1.14) T x , , y : Bord fr x , , y ÝÑ Cat C . D. S. FREED AND C. TELEMAN
Also, the value T p S n q on the nonbounding 3-framed circle is the Drinfeld cocenter of T p`q , amodule category over T p S b q . (Notice that T p S n q is equivalent to T p S b q if T p`q is spherical.) See[DSS, § F x , , y p S b q . Remark . The double | F | “ F F _ of an RT theory F is the truncation of a TV theory, as wenow explain. ( F _ is the dual theory to F in the symmetric monoidal category of theories.) Let C be a modular tensor category. Suppose F : Bord fr3 Ñ C is an extension of an RT theory (1.1)with F p S b q “ C , where we assume the hypotheses on C in Theorem A below. Use the cobordismhypothesis to define theories(1.16) F _ , | F | : Bord fr3 ÝÑ C which are characterized by(1.17) F _ p`q “ F p`q _ | F | p`q “ F p`q b F p`q _ “ F p S q . Here F p`q _ is the dual object to F p`q P C . The theory F _ may be defined as the composition of F with the involution Bord fr3 Ñ Bord fr3 which reverses the first frame vector. Using this descriptionidentify the braided fusion category F _ p S b q with C rev , which is the same underlying fusion cate-gory C equipped with the inverse braiding. It follows that | F | p S b q – C (cid:2) C rev , which by [EGNO,Proposition 8.20.12] is braided tensor equivalent to the Drinfeld center of C . Let(1.18) F : Bord fr3 ÝÑ C be a 3-dimensional 3-framed topological field theory, as in (1.3). Lurie [L1, Example 4.3.22] definesan extended bordism 3-category Bord fr3 , B and an inclusion Bord fr3 Ñ Bord fr3 , B . (See § A.3 for adefinition of objects in Bord fr3 , B .) A boundary theory for F is an extension(1.19) r F : Bord fr3 , B Ñ C of F to a symmetric monoidal functor. The cobordism hypothesis with singularities implies that r F isdetermined by a 3-dualizable object F p`q P C together with a 2-dualizable 1-morphism 1 Ñ F p`q .To isolate the data of the boundary theory, let(1.20) τ ď F : Bord fr2 ÝÑ C Objects of the Drinfeld cocenter of a fusion category C are pairs p x, γ q in which x is an object of C and thefunctor γ : x b ´ Ñ ´ b x ˚˚ is a twisted half-braiding. APPED BOUNDARY THEORIES IN 3D 7 be the truncation of F to 2-framed 0-, 1-, and 2-dimensional bordisms. It is a once categorifiedtopological field theory . Then the data of a boundary theory for F is a natural transformation (1.21) β : 1 ÝÑ τ ď F of symmetric monoidal functors on Bord fr2 . The precise meaning of ‘natural transformation’ inmulticategories is elaborated in [JS], where the cobordism hypothesis with singularities [L1, § Theorem A.
Let C be a symmetric monoidal 3-category whose fully dualizable part C fd containsthe 3-category Fus of fusion categories as a full subcategory. Let F : Bord fr3 Ñ C be a 3-framedtopological field theory such that (a) F p S q is isomorphic in C to a fusion category, and (b) F p S b q is invertible as an object in the 4-category E p Ω C q of braided tensor categories.Assume F extends to r F : Bord fr3 , B Ñ C such that the associated boundary theory β : 1 Ñ τ ď F isnonzero. Then F p S b q is braided tensor equivalent to the Drinfeld center of a fusion category Φ ,which may be taken to be End C fd p β q . The hypothesis that Fus is a full subcategory of C fd ensures that Theorem A applies to TV theories.That hypothesis implies that(1.22) Ω C fd “ FSCat;see (1.8). Since FSCat Ă Cat C , the x , , y truncation F x , , y of F has the form (1.1). Hypothe-ses (a) and (b) express that F x , , y is an RT theory. A modular tensor category is invertible asan object in the 4-category of braided tensor categories [S-P, BJSS]; hypothesis (b) captures thiscentral feature of RT theories. It remains to explain what it means that β is nonzero . Observethat the value of β on a closed 1-manifold is an object in a finite semisimple complex linear abeliancategory, and the value of β on a closed 2-manifold is a vector in a finite dimensional complexvector space. We require that β take a nonzero value on some closed 1- or 2-manifold. Remark . The unit in our fusion category Φ may not be simple. We prove (Corollary C.2) thatΦ is Morita equivalent to a fusion category Φ with simple unit. The category Φ is canonical interms of r F “ p F, β q , whereas Φ is only determined up to Morita equivalence. Remark . At first glance it might seem that the hypotheses of Theorem A, which lead to (1.22),are too restrictive: we might rather have nonsemisimple categories be possible values of F x , , y .However, this is ruled out by Theorem 1.12 and a dimensional reduction argument, such as in theproof of Lemma 2.28.We prove Theorem A in § § There are also boundary theories τ ď F Ñ
1. They are extension of F to a variation of Bord fr3 , B . D. S. FREED AND C. TELEMAN
Theorem A . In the context of Theorem A assume in addition that (i) for all x, y P C the 2-category C p x, y q of 1-morphisms is finitely cocomplete, and (ii) composition of 1-morphisms in C isfinitely cocontinuous. Then F is isomorphic to the Turaev-Viro theory T Φ , and the boundary theoryis determined by the regular left module category Φ .Remark . If F is isomorphic to the tensor unit theory, then for any Turaev-Viro representa-tive T Φ the fusion category Φ is endomorphisms of a finite semisimple abelian category C . Underthe isomorphism T Φ „ ÝÝÑ
1, executed via the Morita trivialization of End p C q , the regular left Φ-module goes over to the finite semisimple abelian category C . Note that the p , q part of the p , , q -theory based on C assigns a semisimple commutative algebra to S b .If from the beginning we work with p , , q -theories (1.1), then the conclusion of Theorem A can fail. For example, consider a theory with β p S b q “ C r x s{p x q , a nonsemisimple commutativealgebra. This theory is not fully extendable, so does not arise as in previous paragraph. Remark . There is a generalization of RT theories (1.1) with codomain a 2-category of “super”complex linear categories. Some developments in the theory of these categories—which are eitherenriched over the category sVect C of super vector spaces or are a module category over sVect C —especially for fusion supercategories, may be found in [GK, BE, U]. Our main theorems generalizeto allow supercategories in the codomain, a topic we hope to return to elsewhere. En route to proving Theorem A, we prove the following characterization of fusion categories.
Theorem B.
Let Ψ P E p Cat C q be a tensor category. Then Ψ is a fusion category iff (i) Ψ is 3-dualizable in E p Cat C q , and (ii) Ψ as a left Ψ -module is 2-dualizable as a 1-morphism in E p Cat C q . The forward direction, proved in [DSS], is stated as Theorem 1.4. We prove the converse in § Remark . Let A “ C r x s{p x q be the non-semisimple algebra of dual numbers. Thetensor category Ψ of finite dimensional A - A bimodules is Morita equivalent to Vect C , so is 3-dualizable, but it is not a fusion category: for example, it does not have internal duals. Thisexample illustrates that ‘fusion’ is not a Morita invariant notion. The dualizability in Theorem B(i)is Morita invariant, whereas the regular module in Theorem B(ii) is not. For example, under theMorita equivalence which sends Ψ to Vect C , the regular left module Ψ Ψ is sent to the linear categoryof A -modules, which is not the regular left module over Vect C . We regard Theorem A as a Moritainvariant variant of Theorem B.
2. Proof2.1. Bordism n -categories As a preliminary, we recall features of bordism multicategories and explain how they are encodedin the pictures we draw. In Appendix A we give a formal and precise account valid in all dimensions;
APPED BOUNDARY THEORIES IN 3D 9 the heuristic exposition here is focused on the low dimensional cases of interest. See [BM, CS, AF]for complete constructions of the bordism multicategory.
Figure 1.
A 2-morphism C > C Y ÝÝÑ ∅ in Bord Arrows of time.
Begin with Bord , the 2-category of unoriented bordisms of dimension ď ∅ C ÝÝÑ ∅ of the empty 0-manifold is a closed 1-manifold. In the bordism 2-category its tangent bundle is stabilized to the rank 2 vector bundle R ‘ T C Ñ C , where R Ñ C is the trivial bundle of rank 1 with a chosen orientation. This orientation is called an “arrow oftime”. In a bordism C > C Y ÝÝÑ ∅ , as depicted in Figure 1, the arrows of time distinguish incomingand outgoing boundary components. An object P in Bord is a finite set of points with stabilizedtangent bundle R ‘ R Ñ P and chosen orientations on each of the trivial line bundles. Figure 2 isa 1-morphism P > P C ÝÝÑ ∅ . Note that the manifolds P , P have two ordered arrows of time; theordering is depicted in our figures by the number of arrowheads. (In our conventions the double-headed arrow has index ´ ´ C ; that is, correspondingtrivial summands augmenting the tangent bundle are identified at B C . Figure 2.
A 1-morphism P > P C ÝÝÑ ∅ in Bord Remark . In a geometric bordism multicategory of manifolds of dimension ď n , a manifold ofdimension k ă n is embedded in successive germs of a p k ` q -, p k ` q -, . . . , and n -manifold. Foran p n ´ q -dimensional manifold C , the arrow of time is an orientation of the normal bundle to C in the germ. The additional arrows of time in higher codimension orient normal bundles to lowerstrata at corners. In a topological bordism multicategory the germs can be and usually are replacedby a stabilization of the tangent bundle; see (A.12). Constancy data.
A 2-morphism in Bord is a compact 2-manifold Y with corners and arrowsof time together with constancy data [CS, Definition 5.1]. Namely, if Y “ ˚ Y > ˚ Y ´ > ˚ Y ´ is thepartition into boundaries and corners, so dim ˚ Y ´ k “ ´ k , then there is a free involution specifiedon ˚ Y ´ with quotient P and an embedding(2.2) r , s ˆ P ã Ñ Y ´
10 D. S. FREED AND C. TELEMAN
Figure 3. (a) a legal 2-morphism and (b) not a 2-morphism such that the arrows of time are constant along the image of r , s ˆ t p u for all p P P . ThusFigure 3(a) is legal: the dashed vertical edges comprise the image of the constancy embedding (2.2).The constancy gives rise to an interpretation of Figure 3(a) as a 2-morphism(2.3) P > P C (cid:39) (cid:39) C > C (cid:55) (cid:55) (cid:75) (cid:83) Y ∅ , essentially by collapsing the dashed edges. On the other hand, Figure 3(b) is not allowed becausethere is no embedding r , s ˆ t P i u ã Ñ Y ´ for which the specified arrows of time are constant.A formal specification of constancy data is (A.8), a consequence of Definition A.4. Tangential structures.
The main point to emphasize is that in a bordism n -category ofmanifolds of dimension ď n , the tangential structure is on the stabilized rank n tangent bundle. Inparticular, Figure 4 is a valid 1-morphism in Bord fr2 , the bordism 2-category of manifolds.In the figure the arrows of time are notated as earlier. The 2-frame f , f is depicted as a long linesegment ( f ) followed by a short line segment p f q . There is no relationship imposed between thearrows of time and the tangential structure (2-framing). Figure 4.
A 2-framed 1-morphism
Remark . The relative positions of the framing and the arrows of time does have significance,for example in Figure 5 below.
Two conventions.
We depict objects and morphisms in Bord fr2 as manifolds with cornersembedded in the Euclidean plane—the plane of the paper/screen—together with arrows of time,constancy data, and orthonormal 2-framings. The first convention is that we identify two suchwhich are related by either a translation or a translation composed with a reflection about eithera horizontal or a vertical line. (If any such identification exists it is unique, since it preserves
APPED BOUNDARY THEORIES IN 3D 11 the 2-framings.) Take the standard points ` , ´ to be those depicted in Figure 5. Although ourpictures lie in Bord fr2 , our arguments in the proof are for Bord fr3 . (Only in the proof of Lemma 2.30do we use truly 3-dimensional pictures.) The second convention, then, is that we implicitly inflateorthonormal 2-framings to orthonormal 3-framings by adjoining the unit vector pointing into thepaper/screen as the first frame vector f . The additional arrow of time points in the same directionas f . Figure 5.
The standard points
Remark . In topological field theory, which is modeled on Wick-rotated field theory, there is nonotion of time versus space. Nonetheless, our conventions are motivated by regarding the last framevector as “timelike” and the remaining frame vectors as “spacelike”. In the conventions establishedin Appendix A, frame vectors are labeled by codimension, so the timelike direction is labeled ´ and 0 in Bord . (Note the shift in Remark A.13(2).) Duals and adjoints.
A topological bordism n -category or p8 , n q -category has all duals andadjoints. Duals are formed by reversing an arrow of time. This can be done at any depth, whichreflects the O p q ˆ n -action discussed in [L1, Remark 4.4.10] and [BS-P, § fr2 , depicted in Figure 5, are dual by reversing the double-headed arrow oftime. Right and left adjoints of morphisms are constructed by a more complicated prescriptionwhich we specify in § A.2.5.
Coloring with a boundary theory.
We now briefly describe a bordism 2-category Bord fr2 , B intowhich Bord fr2 embeds. (Use the second convention of § fr3 , B .) A sketch of this construction appears in [L1, Example4.3.22]; details for Bord fr n, B , n P Z ě , are provided in § A.3.
Remark . One key point is that a boundary theory (1.21) for an n -dimensional theory is essen-tially an p n ´ q -dimensional theory. This explains why at a boundary component “colored” witha boundary theory there is one fewer arrow of time and a constraint on the tangential structure,as we explain below. Figure 6.
A 1-morphism in Bord fr2 , B The 0-morphisms in Bord fr2 , B are the same as 0-morphisms in Bord fr2 : a finite set of 2-framedpoints equipped with arrows of time. There are new 1-morphisms, such as the one depicted inFigure 6. The two boundary points are colored and the extra arrow of time at colored boundarypoints is replaced by the condition that the first frame vector be an inward normal. (Recall that This is appropriate for a boundary theory 1 Ñ τ ď F ; for a boundary theory τ ď F Ñ we number by codimension—see Remark 2.5—so the arrows are labeled ´ , ´ Y “ ˚ Y > ˚ Y ´ > ˚ Y ´ , there is a submanifold with boundary˚ B ´ > ˚ B ´ Ă ˚ Y ´ > ˚ Y ´ that is colored with the boundary condition; in Figure 7 we have ˚ B ´ “ ˚ Y ´ ,and ˚ Y ´ z ˚ B ´ consists of three open line segments. There are no arrows of time on ˚ B ´ and thefirst frame vector on ˚ B ´ is the inward normal. The constancy data (2.2), vacuous for Figure 7, isas in § Y ´ z ˚ B ´ replacing ˚ Y ´ . (Figure 10 below illustrates the constancy data.) Figure 7.
A 2-morphism in Bord fr2 , B Remark . There does not exist a 2-morphism in Bord fr2 from which Figure 7 is obtained bycoloring a subset of the boundary (with inward arrow of time).
Remark . Intuitively, the colored boundary components include a time direction—they aretimelike—which motivates the convention about which frame vector is constrained; see Remark 2.5.
Internal homs.
Let M be a (weak) 2-category. A 1-morphism x f ÝÑ y in M defines afunctor M p z, x q f ˝´ ÝÝÝÑ M p z, y q for any z P M . We call its right adjoint, if it exists, the rightinternal hom functor:(2.9) M p z, x q f ˝´ (cid:47) (cid:47) M p z, y q Hom R p f, ´q (cid:111) (cid:111) . The left internal hom is defined as a right adjoint for right composition for any w P M :(2.10) M p y, w q ´˝ f (cid:47) (cid:47) M p x, w q Hom L p f, ´q (cid:111) (cid:111) . See [W] and [MaSi, §
16] for discussions. We thank Emily Riehl for correspondence and for pointing us to [W]. The nonstandard terminology and notationare our responsibility. The name ‘right internal hom’ is apt if z “ x , and ‘left internal hom’ is apt if w “ y . APPED BOUNDARY THEORIES IN 3D 13 If M is a symmetric monoidal 2-category, and the 1-morphism f has right and left adjoints f R , f L in M , then there are natural isomorphisms(2.11) Hom R p f, g q – f R ˝ g P M p z, x q , g : z Ñ y, Hom L p f, h q – h ˝ f L P M p y, w q , h : x Ñ w. If h “ g “ f , hence z “ x and w “ y , then we use the notations(2.12) End R p f q : “ Hom R p f, f q – f R ˝ f P M p x, x q , End L p f q : “ Hom L p f, f q – f ˝ f L P M p y, y q . The 1-morphism End R p f q is an algebra object in the 1-category M p x, x q , and End L p f q is analgebra object in M p y, y q . The unit of End R p f q is the unit 1 Ñ f R ˝ f of the adjunction, and themultiplication uses the counit f ˝ f R Ñ R p f q ˝ End R p f q “ f R ˝ p f ˝ f R q ˝ f ÝÑ f R ˝ f “ End R p f q . The formulas for End L p f q are similar. Remark . This discussion applies to n -categories, n ě
3, by taking 2-categorical slices.
Remark . A symmetric monoidal functor preserves internal homs, internal endomorphisms,and the composition laws.
Internal homs in
Fus and E p Cat C q . Recall from Definition 1.6 that Fus is a symmetricmonoidal 3-category whose objects are fusion categories. Our convention, different than somereferences, is that a p B, A q -module category M is a 1-morphism M : A Ñ B in Fus. This conventionrenders tensor products and compositions in the same order. The following results are generalizedin [BJS, § Proposition 2.16 ([DSS, § . Let
A, B P Fus and M : A Ñ B a finite semisimple p B, A q -bimodule category. Then M has right and left adjoints, and we can take them to be (2.17) M R “ Hom B p M, B q ,M L “ Hom A p M, A q . Corollary 2.18.
If also
C, D P Fus and N : C Ñ B , P : A Ñ D are finite semisimple bimodulecategories, then (2.19) Hom R p M, N q “
Hom B p M, N q , Hom L p M, P q “
Hom A p M, P q . In particular, (2.20) End R p M q “ End B p M q , End L p M q “ End A p M q . Furthermore, the algebra structure on
End R p M q ( and End L p M q ) is composition of module functorendomorphisms. Proof.
All but the final assertion follow from Proposition 2.16 and [DSS, Proposition 2.4.10]. Forthe algebra structure on End R p M q , the unit 1 Ñ M R ˝ M is the p A, A q -bimodule map(2.21) A ÝÑ Hom B p M, B q (cid:2) B M – End B p M q which maps a P A to right multiplication by a ; in particular 1 P A maps to the identity endo-morphism of M . Since the counit M ˝ M R Ñ R p M q End B p M q (cid:2) A End B p M q – Hom B p M, B q (cid:2) B ` M (cid:2) A Hom B p M, B q ˘ (cid:2) B M ÝÑ Hom B p M, B q (cid:2) B M – End B p M q (2.22)is the usual composition of endomorphisms of M . (cid:3) We prove a partial converse to Proposition 2.16 in E p Cat C q , which for convenience we state for right adjoints only. Proposition 2.23.
Suppose
A, B P E p Cat C q and M : A Ñ B has a right adjoint N . Then N – Hom B p M, B q as p A, B q -bimodule categories.Proof. The adjunction can be expressed as isomorphisms(2.24) Hom A p X, N (cid:2) B Y q „ ÝÝÑ
Hom B p M (cid:2) A X, Y q which are functorial in left A -module categories X and right A -module categories Y . Choose X “ A and Y “ B to obtain the desired isomorphism N „ ÝÝÑ
Hom B p M, B q . The isomorphism intertwinesthe right A -action on X and the right B -action on Y . (cid:3) Figure 8.
Duality of ` and ´ : evaluation and coevaluation Figure 9.
The 1-morphisms whose r F -images are β p`q and β R p`q To begin, in § C such that Ω C fd Ă Cat C and a symmetric monoidal functor r F : Bord fr3 , B Ñ C . The restriction of r F to Bord fr3 is denoted F ,and we define β as in (1.21). APPED BOUNDARY THEORIES IN 3D 15
Figure 10.
The unit 1 Ñ b R ` ˝ b ` and counit b ` ˝ b R ` Ñ The fusion category Φ and Theorem B. Diagrams for the extended theory r F : Bord fr3 , B Ñ C ,are drawn according to the rules of § § A.3 for more detail. Figure 9 depicts a 1-morphism b ` : ∅ Ñ ` in Bord fr2 , B Ă Bord fr3 , B and its right adjoint b R ` : ` Ñ ∅ , the latter constructed accordingto § A.2.5. The r F -image of b ` is β p`q : 1 Ñ F p`q . The cobordism hypothesis with singularities[L1, § β R : τ ď F Ñ F p`q and β R p`q : “ r F p b R ` q , hence the verification that β R is right adjoint to β proceeds by producing aunit and counit (Figure 10) for an adjunction between b ` and b R ` in Bord fr2 , B , and then using their r F -images to exhibit β R p`q as the right adjoint of β p`q in C . Figure 11.
The category Φ is the r F -image of End R p b ` q “ b R ` ˝ b ` Definition 2.25.
Set(2.26) Φ “ r F ` End R p b ` q ˘ P Ω C fd Ă Cat C . See Figure 11 for a depiction of End R p b ` q . Also, note we can write Φ “ End R ` β p`q ˘ . By virtueof being End R of a 1-morphism, Φ is an algebra object in Cat C , i.e., Φ is a tensor category. Thecomposition law (2.13) is the r F -image of the 2-morphism depicted in Figure 12. Proposition 2.27. Φ is a fusion category. The proof of Proposition 2.27 is broken up into Lemma 2.28 and Lemma 2.30.
Lemma 2.28. Φ is a finite semisimple abelian category. The stronger hypothesis of Theorem A, that Fus Ă C fd is a full subcategory, immediately impliesLemma 2.28 in view of (1.22). Proof.
The 1-morphism in Bord fr2 , B depicted in Figure 13 has left boundary colored with β and rightboundary colored with β R . It evaluates under p F, β, β R q to Φ. Define the dimensional reduction (2.29) F : Bord fr2 ÝÑ FSCat
Figure 12.
The monoidal structure of Φ is the r F -image of this 2-morphism of F as the theory whose value on any object or morphism in Bord fr2 is the value of F on itsCartesian product with the 1-morphism ∅ Ñ ∅ in Figure 13. Since the 2-framing of the latteris induced from a 1-framing, the Cartesian product is naturally equipped with a 3-framing. Thelemma now follows since F p`q is a 2-dualizable category, hence is finite semisimple abelian byTheorem 1.12. (cid:3) Figure 13.
A 1-framed bordism with boundary theories β on the left and β R on the right Lemma 2.30. Φ is a rigid monoidal category. That is, Φ has internal left and right duals [EGNO, § Proof.
As a corollary of Lemma 2.28, the dual Φ _ to Φ in FSCat is its opposite category. We proverigidity by verifying the hypotheses of Theorem B.1. Figure 14.
The right adjoint to the unit in Figure 10
The unit of Φ is the r F -image of the first 2-morphism in Figure 10. Its right adjoint has r F -imagethe counit ε of (B.3). We implement the prescription of § A.2.5 to compute it. The main concern isthe 3-framing which results on the boundary of the hemidisk; it necessarily extends to the interior,and since π SO “ APPED BOUNDARY THEORIES IN 3D 17
Recall ( § f , f , f ; that f is depicted as long, f asshort; and that in all previous pictures, such as Figure 10, the vectors f , f lie in the plane ofthe paper/screen and f is perpendicular to that plane and points into the paper/screen. Now,in Figure 14, the vectors f , f rotate in the plane perpendicular to f as we descend from theincoming boundary. The long dotted line in indicates that f points into the paper/screen; thelong solid line in indicates that f points out of the paper/screen. Compose this right adjointwith the multiplication depicted in Figure 12 to compute the 2-morphism in Bord fr3 whose r F -imageis the pairing B of Appendix B. It and the 2-morphisms obtained from it by duality (B.5) aredepicted in Figure 15. The Frobenius condition of Definition B.7 is satisfied since the latter two2-morphisms are invertible in Bord fr3 . Figure 15.
Multiplication followed by the counit and its duals
The right adjoint bordism to Figure 12, computed following § A.2.5, is depicted in Figure 16;its r F -image is the comultiplication ∆ on Φ. Again it suffices to compute the framing on theboundary. Notice that the half-turns in the framing cancel on the vertical colored edges, whereasthey cohere into a full turn on the colored half-circle. The second condition in Theorem B.1, thatthe comultiplication is a bimodule map, follows immediately from Figure 17. (cid:3) Figure 16.
The right adjoint to Figure 12
Remark . In the context of Theorem A, identify the category Φ with its collection of objectsHom
Fus p , Φ q “ Hom C fd p , Φ q . By duality,(2.32) Hom C fd p , Φ q – Hom C fd ` , β R p`q ˝ β p`q ˘ – Hom C fd ` β p`q , β p`q ˘ , which explains the last statement in Theorem A. The numbering 0,1,2 corresponds to the numbering ´ , ´ , ´ Figure 17.
Comultiplication is a bimodule map
Proof of Theorem B.
As pointed out earlier, the forward direction follows from Theorem 1.4. Forthe converse, first apply the cobordism hypothesis to construct F : Bord fr3 Ñ E p Cat C q with F p`q “ Ψ, and then apply the cobordism hypothesis with singularities to construct an exten-sion r F : Bord fr3 , B Ñ E p Cat C q whose associated boundary theory β : 1 Ñ τ ď F has β p`q “ Ψ Ψ,the regular left Ψ-module. The category defined in Definition 2.25 is Φ “ End R p Ψ Ψ q . By Propo-sition 2.23 the right adjoint to Ψ Ψ is Hom Ψ p Ψ , Ψ q , which may be identified with Ψ Ψ , the regularright Ψ-module. Hence Φ “ End R p Ψ Ψ q – Ψ (cid:2) Ψ Ψ – Ψ. Conclude using Proposition 2.27. (cid:3)
A basic Morita equivalence.
The ` point and ´ point (Figure 5) are duals in Bord fr2 . Chooseduality data as the evaluation e : ` >´ Ñ ∅ and coevaluation c : ∅ Ñ ´ > ` ´ ` id b c ÝÝÝÝÑ ` > ´ > ` e b id ÝÝÝÑ ` ¯ „ ÝÝÑ ´ ` id ÝÝÑ ` ¯ that proves that p´ , c, e q are duality data for ` is the 2-morphism of Figure 18. Figure 18.
The S-diagram (2.33)
The 1-morphism e has right and left adjoints e R , e L : ∅ Ñ ´ > ` depicted in Figure 19. Theirconstruction follows the general prescription in § A.2.5; see especially Figure 25. In Bord fr2 theyare distinct and distinct from coevaluation: e R ‰ c ‰ e L . In Bord fr3 we have e L – e R since π SO – Z { Z with generator a full rotation of the frame f , f , f in the f - f plane. In Bord fr3 we have an isomorphism(2.34) S b – e ˝ e L “ End L p e q illustrated in Figure 20. APPED BOUNDARY THEORIES IN 3D 19
Figure 19.
The right and left adjoints to e Remark . In Bord fr3 the nonbounding 3-framed circle S n satisfies the isomorphism(2.36) S n – e ˝ c. Figure 20.
The isomorphism S b – e ˝ e L in Bord fr2 Define the 2-framed 0-sphere as S “ ` > ´ . Note e : S Ñ ∅ . Proposition 2.37.
Let F satisfy the hypotheses of Theorem A. Let Ξ be a fusion category whichis isomorphic to F p S q . Then Ξ is Morita equivalent to F p S b q . Recall that a fusion category is indecomposable if it is not a nontrivial direct sum. As a prelimi-nary we prove the following.
Lemma 2.38. Ξ is an indecomposable fusion category.Proof. As in Remark 1.15 introduce the double theory(2.39) | F | : Bord fr3 ÝÑ Fuscharacterized by | F | p`q “ Ξ – F p S q “ F p`q b F p´q – F p`q b F p`q _ . Then by Hypothesis (b)of Theorem A, deduce that | F | p S b q – F p S b q (cid:2) F p S b q rev is invertible as a braided tensor category.(Recall that the reverse of a braided tensor category is the same underlying tensor category equippedwith the inverse braiding.) On the other hand, | F | p S b q is the Drinfeld center of | F | p`q – Ξ. Sincethe Drinfeld center of the direct sum of tensor categories is the direct sum of the Drinfeld centers,and a nontrivial direct sum is not invertible, it follows that Ξ is indecomposable. (cid:3)
Proof of Proposition 2.37.
Define(2.40) M : “ Ξ „ ÝÝÑ F p S q F p e q ÝÝÝÝÑ , where 1 “ F p ∅ q “ Vect C P Fus Ă C is the tensor unit. Since Fus Ă C is a full subcategory, M is a1-morphism in Fus. By (2.34) and (2.20) we have the categorical equivalences(2.41) F p S b q » cat F ` End L p e q ˘ » cat End L ` F p e q ˘ » cat End Ξ p M q . The last assertion of Corollary 2.18 implies that (2.41) is an equivalence of tensor categories.To conclude the proof of Proposition 2.37 we must show that M is a faithful right Ξ-module.According to the remark after [EGNO, Definition 7.12.9], this follows from Lemma 2.38 since F p e q isnonzero by virtue of being part of the duality data between F p`q and F p´q . (cid:3) The theory T Φ and Theorem A. Since Φ is a fusion category, as in (1.5) the cobordismhypothesis produces(2.42) T Φ : Bord fr3 ÝÑ Fus , a theory of Turaev-Viro type with T Φ p`q “ Φ. Then T Φ p S b q is the Drinfeld center(2.43) Z ` Φ ˘ “ End T p S q ` Φ ˘ , where T Φ p S q “ T Φ p`q (cid:2) T Φ p´q » Φ (cid:2) Φ mo . (Here Φ mo is the monoidal opposite to the monoidalcategory Φ, which is its dual in Fus.) We identify(2.44) T Φ – End R p β q as algebra objects in the 2-category of symmetric monoidal functors Bord fr2 Ñ Ω C “ FSCat.
Figure 21.
The bordisms b ´ and b ˘ Let b ´ : ∅ Ñ ´ be the dual to b R ` , obtained by reversing the double-headed arrow; see § b ˘ “ b ` > b ´ . Then the proof of [JS, Proposition 7.10] implies that β p´q – r F p b ´ q .Define the composition(2.45) N : 1 r F p b ˘ q ÝÝÝÝÝÑ F p S q „ ÝÝÑ Ξ . Notice r F p b ˘ q “ β p S q . As in the proof of Proposition 2.37, N is a 1-morphism in Fus. Apply (2.44)and (2.20) to deduce an equivalence of tensor categories(2.46) T Φ p S q – End R p N q – End Ξ p N q . APPED BOUNDARY THEORIES IN 3D 21
Lemma 2.47.
The left Ξ -module category N provides a Morita equivalence Ξ „ ÝÝÑ T Φ p S q .Proof. As in the proof of Proposition 2.37, it suffices to show that N is nonzero. But β p S q “ β p`q b β p´q , so if N “ β p`q “
0, and then the cobordism hypothesis wouldimply β “
0, which contradicts the hypothesis in Theorem A that β is nonzero. (cid:3) Figure 22.
The equivalence (2.49)
Lemma 2.48.
There is an equivalence of categories (2.49) Φ » cat M (cid:2) Ξ N. Proof.
Compare Figure 11 and Figure 22 and apply r F to deduce (2.49). (cid:3) Since the ` Ξ , T Φ p S q ˘ -bimodule N is invertible—by Lemma 2.47 it induces a Morita equivalence—from Lemma 2.48 we deduce that tensoring with id N induces a tensor equivalence(2.50) α : End Ξ p M q (cid:2) id N ÝÝÝÝÝÑ
End T Φ p S q ` Φ ˘ . By (2.41) and (2.43) this is a tensor equivalence(2.51) α : F p S b q ÝÑ T Φ p S b q , since T Φ p S b q is the Drinfeld center of Φ. To complete the proof of Theorem A we need the following. Lemma 2.52. α is a braided equivalence.Proof. We have already proved that α is a tensor equivalence; it remains to verify the condition that α preserve the braiding. Lemma 2.47 states that the ` Ξ , T Φ p S q ˘ -bimodule N induces anisomorphism Ξ Ñ T Φ p S q in Fus. By the cobordism hypothesis this induces an isomorphism(2.53) θ : F d „ ÝÝÑ T d Φ of topological field theories Bord fr3 Ñ Fus, where F d p`q “ Ξ and T d Φ p`q “ T Φ p S q – Φ (cid:2) Φ mo .( F d is essentially the double theory of Remark 1.15.) Let r F d : Bord fr3 , B Ñ Fus be the extensionof F d with boundary theory β d characterized by(2.54) β d p`q : 1 β p S q ÝÝÝÝÝÑ F p S q „ ÝÝÑ Ξ . Our conventions are that N is a 1-morphism T Φ p S q Ñ Ξ, but in this proof we use categories of right modulesrather than left modules, and so the convention applies oppositely.
Repeat the arguments of § § r F d : introduce Ξ d “ Ξ (cid:2) Ξ mo , the right Ξ d -module M d “ F d p e q , the left Ξ d -module N d “ β d p S q , and the tensor equivalence(2.55) α d : F d p S b q (cid:2) id Nd ÝÝÝÝÝÝÑ T d Φ p S b q . We have M d “ M (cid:2) M and N d “ N (cid:2) N , where the primed Ξ mo -modules are computed in thetheory whose value on ` is F p´q . Hence in the doubled theories (2.49) becomes(2.56) Φ (cid:2) Φ mo » cat p M (cid:2) M q (cid:2) Ξ (cid:2) Ξ op p N (cid:2) N q , and α d tensors with id N (cid:2) N . Since tensoring with N is the isomorphism (2.53) of theories, it followsthat the induced map on Drinfeld centers is (2.55), i.e., α d “ θ p S b q . Therefore, α d is a braided tensor functor, and then so too is its restriction to(2.57) F p S b q “ F p S b q (cid:2) Ă F p S b q (cid:2) F p S b q rev Ă F d p S b q . This completes the proof of Lemma 2.52, and so too of Theorem A. (cid:3)
For the remainder of this section we put in force the stronger hypotheses that each 2-categoryof 1-morphisms in C admits finite colimits, and that these colimits are preserved by composition of1-morphisms. This allows the relative tensor products used in the sequel. Proof of Theorem A . By the preceding it suffices to prove that F is isomorphic to T Φ , and by thecobordism hypothesis it suffices to construct an isomorphism F p`q Ñ Φ in C . Consider(2.58) Φ Φ Φ ÝÝÝÑ β p`q ÝÝÝÝÑ F p`q , where Φ Φ is the regular right Φ-module. The fusion category Φ “ β p`q R ˝ β p`q acts on Φ Φ on theleft and on β p`q on the right. Define g : Φ Ñ F p`q as the “relative tensor product”(2.59) g “ β p`q ˝ Φ Φ Φ of these actions; it is a finite colimit in C ` Φ , F p`q ˘ . Similarly, define h : F p`q Ñ Φ as(2.60) h “ Φ Φ ˝ Φ β p`q R . We claim that g and h are inverse isomorphisms. First,(2.61) h ˝ g “ Φ Φ ˝ Φ β p`q R ˝ β p`q ˝ Φ Φ Φ “ Φ Φ ˝ Φ Φ ˝ Φ Φ Φ “ Φ Φ ΦAPPED BOUNDARY THEORIES IN 3D 23 is the p Φ , Φ q -bimodule which represents id Φ . In the other direction,(2.62) g ˝ h “ β p`q ˝ Φ Φ Φ ˝ Φ Φ ˝ Φ β p`q R “ β p`q ˝ Φ β p`q R as an endomorphism of F p`q . By duality, transpose g ˝ h to a 1-morphism(2.63) p g ˝ h q T : 1 ÝÑ F p`q b F p´q “ F p S q . Then g ˝ h “ id F p`q if and only if p g ˝ h q T is the coevaluation of a duality pairing between F p`q and F p´q . Recalling the remark before (2.45), we compute(2.64) p g ˝ h q T “ β p S q ˝ Φ (cid:2) Φ mo Φ , which is the composition(2.65) 1 T Φ p c q ÝÝÝÝÝÑ T Φ p S q N ÝÝÝÑ „ Ξ ÝÝÝÑ „ F p S q . Here we use Lemma 2.47, which implies that N : T Φ p S q Ñ Ξ is an isomorphism. Also, recall that c is coevaluation in Bord fr3 (Figure 18). Hence (2.64) is the desired coevaluation map. (cid:3)
3. Application to physics
A quantum mechanical system S is gapped if its minimum energy is an eigenvalue of finitemultiplicity of the Hamiltonian, assumed bounded below, and is an isolated point of the spectrum.This notion generalizes to a relativistic quantum field theory if we understand ‘spectrum’ to meanthe spectrum of representations of the translation group of Minkowski spacetime. A basic question:(3.1) Does a gapped system S admit a gapped boundary theory?We argue heuristically that Theorems A and A gives an obstruction for certain p ` q -dimensionalsystems. We remark that the chiral WZW model is a gapless boundary theory for Chern-Simonstheory [W2], so at least for these systems a gapless boundary theory exists.We reduce (3.1) to a question in topological field theory by application of the following twoheuristic physics principles:(1) the phase of a quantum system is determined by its low energy behavior;(2) the low energy physics of a gapped quantum system is well-approximated by a topological ˚ field theory. For now we ignore the ‘ ˚ ’ in ‘topological ˚ ’. Principle (1) seems incontrovertible, though unproved,whereas (2) is more problematic. For example, certain “fracton” lattice systems seem to have nocontinuum limit. Nonetheless, (2) appears to hold in many important cases; we simply assume ithere. Applying these principles to both the bulk and boundary systems, the general problem (3.1)reduces to a question in topological field theory: Does a topological field theory F admit a nonzeroboundary theory β ? Remark . We suspect that the answer to (3.1) depends only on the phase of S , that is, its pathcomponent in a putative moduli stack of gapped systems.We now explain the ‘ ˚ ’ in ‘topological ˚ ’ by means of an example that is a main focus of interest.The starting point is a quantum field theory, though one can imagine a lattice model in its place.Namely, let S be p ` q -dimensional Yang-Mills theory with a nondegenerate Chern-Simons term.The latter gives the gauge field a mass, which means that the system is gapped. Its low energyphysics is thought to be well-approximated by a pure Chern-Simons theory Γ. Observe that S in itsWick-rotated form is a theory of manifolds equipped with an orientation and Riemannian metric. Inother words, it is a functor on a geometric bordism category of oriented Riemannian manifolds. Thenaive expectation is that Γ is a functor on the same bordism category, and this is the case. In fact,as discussed by Witten [W1, § topological field theory F and an invertible non-topological field theory α c ,where c P R is the central charge. More precisely, the pullback of Γ to the bordism category of3-framed Riemannian manifolds splits as Γ – F b α c . The theory F is topological—it does notdepend on a Riemannian metric. It is an example of an RT theory as described in §
1. The invertibledependence of Γ on the metric through α c is the ‘ ˚ ’ in ‘topological ˚ ’.The invertible theory α descends to a theory α SO1 with domain the bordism category of oriented
Riemannian manifolds. Its partition function on a closed oriented Riemannian 3-manifold X is theexponentiated η -invariant exp p πiη X { q , where η X P R { Z is the secondary invariant associated tothe signature operator [APS]. The deformation class of α SO1 is a generator of the abelian group r M T SO, Σ I Z s of invertible theories, and at least conjecturally it can be constructed using gener-alized differential cohomology. (We refer to [FH, F2] for notation and details.) The deformationclass of the lift α of α SO1 to 3-framed manifolds vanishes, since(3.3) r M T SO, Σ I Z s ÝÑ r S , Σ I Z s is the zero map. In terms of the differential cohomology construction, the equivalence class of α belongs to the subgroup of topologically trivial theories, so is defined by a universal 3-form: one-third the “gravitational Chern-Simons term”. Then for any c P R , the family α tc , ď t ď
1, is anexplicit deformation of the trivial theory to α c . Put differently, it is a “nonflat trivialization” of thetruncation τ ď α c . In other words, α c is equipped with a canonical boundary theory. Therefore,topological ˚ boundary theories for Γ correspond to topological boundary theories for F .Theorems A and A give an obstruction to the existence of a nonzero topological boundary theoryfor F : the theory F must be of Turaev-Viro type. If not, then the heuristics in this section suggest An example of a nonflat trivialization is a not-necessarily-flat section of a circle bundle with connection. Thenotion of nonflat trivialization should be part of an axiomatization of families of field theories.
APPED BOUNDARY THEORIES IN 3D 25 that there are no gapped boundary theories for Yang-Mills plus Chern-Simons, nor for a latticesystem meant to represent the same phase. It would be interesting to construct a gapped boundarytheory for Yang-Mills plus Chern-Simons in case F is of Turaev-Viro type. Remark . One implicit assumption in Principle (2) is that a gapped quantum system exhibitsrelativistic invariance in the long-range approximation. The Wick-rotated manifestation is the factthat the domain bordism category is made from manifolds whose tangential structure does notbreak O further than the subgroup SO . In particular, a 3-framing breaks relativistic invariance.Here the 3-framing is introduced to isolate the metric dependence of Γ to the invertible theory α c ;the physically relevant theory has SO -invariance. Appendix A. Bordism Multicategories
In this appendix we give the precise definitions behind the descriptions in § §
2. Complete constructions of the bordism multicategory appear in [CS, AF] amongother references. In these approaches an object or morphism is equipped with a global map to acube or stratified ball, and this data is used to define composition laws. Our limited goal here is todefine objects and morphisms in Bord n with minimal data localized at the boundary ; they too leadto composition laws, though we do not pursue the latter. Underlying a bordism is a manifoldwith corners , so we begin with a quick review in § A.1. Then in § A.2 we specify the additional datarequired for a morphism in the bordism multicategory. A similar discussion is in [CS, § n, B —in § A.3.
A.1. Manifolds with corners
There are several definitions and a long history of the subject of manifolds with corners, both ofwhich are reviewed in Joyce [J]. He develops the theory in detail, and we defer to his paper andthe references therein for details.Fix k P Z ě . A neighborhood of a point in a smooth k -manifold is modeled by an open set inreal affine space A k . Similarly, a neighborhood of a point in a k -manifold with corners is modeledby an open set in(A.1) A k ď “ ! p x , . . . , x k q P A k : x i ď ) . The usual notions of chart and atlas generalize accordingly. A point x “ p x , . . . , x k q P A k ď has depth j P Z ě if precisely j of its coordinates vanish. The depth is invariant under diffeomorphismof open sets in A k ď , so is a well-defined function(A.2) depth : M ÝÑ Z ě Nor do we specify collaring data which would give a smooth structure on compositions. on a manifold M with corners. For j P t , . . . , k u , let ˚ M ´ j Ă M denote the p k ´ j q -manifold ofpoints in M of depth j , and let M ´ j Ă M be the closure of ˚ M ´ j . If the maximum value of (A.2)is d P t , . . . , k u , we say M is a manifold with corners of depth ď d and we call d the depth of M .If d “
1, then M is a manifold with boundary . There is a canonical filtering and partition(A.3) M “ M Ą M ´ Ą ¨ ¨ ¨ Ą M ´ d “ ˚ M > ˚ M ´ > ¨ ¨ ¨ > ˚ M ´ d A face of M is the closure of a component of ˚ M ´ .The tangent space T m M to M at m P M is a k -dimensional real vector space. If m has depth j ,then there are j transverse hyperplanes H , . . . , H j Ă T m M and orientations of the lines T m M { H i :the positively oriented direction leads out of M .Variant definitions of ‘manifold with corners’ include global constraints and/or data in additionto the local normal form. For example, one might require that every point of depth j lie in j distinctfaces. The bigon in Figure 23 satisfies this condition, whereas the teardrop does not. There aremore stringent possible global specifications; see [J, Remark 2.11] and the references therein. Theextra data we introduce in § A.2 to define a morphism in a bordism multicategory endows theunderlying manifold with corners with the data/constraints to be of these more restricted types.
Figure 23.
A bigon ( a ) and a teardrop ( b ) There are two distinct notions of the boundary of a manifold with corners. For our purposes wedefine B M “ M ´ as the closed subset of points of positive depth. This is not generally a manifoldwith corners, as Figure 23 illustrates. However, there is a “blow up” which surjects onto B M andwhich is a manifold with corners; see [J, Definition 2.6]. A.2. k -morphisms in Bord n A.2.1.
The definition.
Fix n P Z ą . For k P t , . . . , n u we specify the data of a k -morphismin Bord n . (For k “ n .) Tangential structures are introduced in § A.2.4.
Definition A.4.
Fix n, k as above and suppose d P t , . . . , k u . Let X be a compact k -dimensionalmanifold with corners of depth ď d . The data of a k -morphism of depth d on X are:(i) if d ě
1, closed p k ´ d q -manifolds X ´ d , X ´ d , not both empty;(ii) if d ě
2, recursively for j “ d ´ , d ´ , . . . , p k ´ j q -manifolds X ´ j , X ´ j withcorners of depth ď d ´ j equipped with diffeomorphisms(A.5) ϕ δ ´ j : X ´p j ` q Y r , s ˆ ! X ´p j ` q > X ´p j ` q ) Y X ´p j ` q ÝÑ Bp X δ ´ j q , δ P t , u , APPED BOUNDARY THEORIES IN 3D 27 where the unions are along t u ˆ t X ´p j ` q > X ´p j ` q u and t u ˆ t X ´p j ` q > X ´p j ` q u , re-spectively;(iii) if d ě
1, a diffeomorphism(A.6) ϕ : X ´ Y r , s ˆ (cid:32) X ´ > X ´ ( Y X ´ ÝÑ B X, where the unions are along t u ˆ t X ´ > X ´ u and t u ˆ t X ´ > X ´ u , respectively; and(iv) if k ď n ´
1, locally constant functions (cid:15) ´ i : X Ñ t˘ u , i P t , . . . , n ´ k u . Figure 24.
A 2-morphism of depth 2
Remark
A.7 . (1) See Figure 24 for an example of a 2-morphism of depth 2. In that example X ´ consists oftwo points, X ´ “ ∅ is the empty 0-manifold, and X ´ « X ´ are closed intervals.(2) To interpret (A.5) for j “ d ´
1, set X δ ´p d ` q “ ∅ , δ P t , u .(3) We tell the geometric meaning of (cid:15) ´ i below.(4) In the categorical interpretation, X is a k -morphism with source and target the empty i -manifold ∅ i , i P t , . . . , k ´ d ´ u ; and source and target i -morphisms X ´p k ´ i q , X ´p k ´ i q ,respectively, i P t k ´ d, . . . , k ´ u .(5) If d ě
1, the embeddings ϕ δ ´ j , ϕ , j P t , . . . , d ´ u , δ P t , u , combine to embeddings(A.8) ψ δ ´ j : r , s j ´ ˆ X δ ´ j ÝÑ B
X, j
P t , . . . , d u , δ P t , u . For j P t , . . . , d ´ u let ˚ ψ δ ´ j denote the restriction of ψ δ ´ j to r , s j ´ ˆ ˚ X δ ´ j , and set˚ ψ δd “ ψ δd . Then B X is the disjoint union of the images of ˚ ψ δ ´ j , j P t , . . . , d u , δ P t , u .Heuristically, the bordism is “constant” on ˚ ψ δ ´ j ` r , s j ´ ˆ t x u ˘ , x P ˚ X δ ´ j .(6) The pictures in § j “ § A.2.2.
The tangent filtration.
The structure described in Definition A.4 has a tangential implica-tion. Namely, let X be a k -morphism of depth d , and suppose x P B X . Choose the unique j, δ and t , . . . , t j ´ P r , s , ˚ x P ˚ X δ ´ j such that x “ ˚ ψ δ ´ j p t , . . . , t j ´ ; ˚ x q . Then T x X has a decreasingfiltration(A.9) T x X “ T x, X Ą T x, ´ X Ą ¨ ¨ ¨ Ą T x, ´ j X “ T ˚ x ˚ X ´ j in which(A.10) T x, ´ i X “ d ˚ ψ δ ´ j ` i ´ ‘ R j ´ i ‘ T ˚ x ˚ X ´ j ˘ , i P t , . . . , j ´ u . The associated graded is a sum of real lines(A.11) L x, ´p n ´ k ` j q ‘ ¨ ¨ ¨ ‘ L x, ´p n ´ k ` q . Orient L x, ´p n ´ k ` j q so that the positive direction leads into X if δ “ X if δ “ L x, ´p n ´ k ` i q , i P t , . . . , j ´ u , sothat the positive direction points towards increasing t j ´ i . The orientations are constant over theimage of ˚ ψ δ ´ j . Moreover, Definition A.4(ii) and (iii) ensure that the orientations are consistent aswe move among the images of the various ˚ ψ δ ´ j . A.2.3.
The inflated tangent bundle of a k -morphism. Define the “inflated tangent bundle”(A.12) r T X “ R ‘ ¨ ¨ ¨ ‘ R looooomooooon n ´ k times ‘ T X ÝÑ X, where R Ñ X is the constant line bundle with fiber R . The (cid:15) ´ i in Definition A.4(iv) are orientationsof these trivial line bundles: (cid:15) ´ i “ (cid:15) ´ i “ ´ (cid:15) ´ i are the “arrows of time” discussed in § Remark
A.13 . (1) The index i in (cid:15) ´ i and L x, ´ i is the codimension of the relevant morphism.(2) For the 2-morphisms of various depths depicted in Figure 24 and in §
2, in Bord the single-headed arrows correspond to codimension i “ i “
2. In Bord the indexing should be shifted to ´ ´
2, and ´
1, thoughin § ´ ´
1, and 0 instead.(3) Fix i P t , . . . , n u . Then the i th duality discussed in § (cid:15) ´ i or theorientation of L x, ´ i , at most one of which is defined for each k -morphism.(4) In Figure 24 the arrows of time on X ´ , drawn on the upper right of the figure, only havemeaning relative to a fixed standard horizontal/vertical basis of R , the tangent space tothe plane of the paper or screen; then we deduce the values of (cid:15) ´ , (cid:15) ´ . Similarly for thesingle-headed arrow of time in X δ ´ , δ P t , u on the right hand side of Figure 24. Definition A.14.
Equip each X δ ´ j , j P t , . . . , d u , δ P t , u , with the structure of a p k ´ j q -morphism of depth d ´ j as follows. Set (cid:15) ´p n ´ k ` j q “ (cid:15) ´p n ´ k ` j ´ q “ ¨ ¨ ¨ “ (cid:15) ´p n ´ k ` q “ (cid:15) ´p n ´ k q , . . . , (cid:15) ´ to agree with the values for X . APPED BOUNDARY THEORIES IN 3D 29
A.2.4.
Tangential structures.
Let ρ n : X n Ñ B GL n R be a continuous map. The choice of classify-ing map X Ñ B GL n R for the inflated tangent bundle r T X Ñ X of a k -dimensional manifold X withcorners, k ď n , is a contractible choice we assume given. Then a tangential structure of type ρ n isa lift of that classifying map to X n . We can use rigid models instead, such as for orientations, spinstructures, or n -framings. An isomorphism X Ñ X of manifolds with corners is a diffeomorphismΦ : X Ñ X together with a linear isomorphism r T X Ñ Φ ˚ r T X and a homotopy of the tangentialstructure on X to the pullback of the tangential structure on X . If rigid models are employed, thehomotopy may be replaced by a more rigid alternative, which may be a combination of conditionsand data.There is a variant of Definition A.4 for tangential structures of type ρ n : each manifold X δ ´ j with corners is equipped with a tangential structure and the diffeomorphisms ϕ δ ´ j , ϕ are lifted toisomorphisms in the sense of the previous paragraph. The tangential structure on r , s ˆ Y is takento be that on Y , extended to be constant along the r , s -direction. An isomorphism Φ : X Ñ X of k -morphisms of depth d is an isomorphism X Ñ X of manifolds with corners and tangentialstructures, required to commute with (cid:15) i , (cid:15) ´ i ; and a collection of isomorphisms p X q δ ´ j Ñ X δ ´ j ofmanifolds with corners and tangential structures, compatible with p ϕ q δ ´ j , ϕ δ ´ j and ϕ , ϕ . A.2.5.
Duals and adjoints.
An object in Bord n is a finite set of points X and arrow-of-timefunctions (cid:15) ´ i : X Ñ t˘ u , i P t , . . . , n u ; the tangential structure, if present, is on the trivialvector bundle X ˆ R n . The dual object X _ consists of the same data, but with (cid:15) ´ n replacedwith ´ (cid:15) ´ n . Evaluation and coevaluation morphisms are constructed from r , s ˆ X ; see Figure 8.More generally, a closed k -manifold X adorned with functions (cid:15) ´ i : X Ñ t , u , i P t , . . . , n ´ k u ,is an object in Ω k Bord n . Its dual replaces (cid:15) ´p n ´ k q with ´ (cid:15) ´p n ´ k q .If 1 ď k ď n ´
1, then a k -morphism X in Bord n has both a right adjoint X R and a left adjoint X L .We specify data for these objects X A , where A “ R for the right adjoint and A “ L for the leftadjoint. Define X A “ X as a manifold with corners. Set p X A q δ ´ j “ X δ ´ j for j P t , . . . , d u , δ P t , u ;and set (cid:15) A ´ i “ (cid:15) ´ i for i P t , . . . , n ´ k u . Reverse the arrows of time on the codimension one strata:set p X A q δ ´ “ X ´ δ ´ for δ P t , u . Construct diffeomorphisms p ϕ A q δ ´ j , ϕ A from the correspondingdiffeomorphisms in the data of X . For unoriented bordisms that is a complete specification of X A ;in particular, right and left adjoints agree. If a tangential structure is present, define a tangentialstructure on X A according to the following procedure. Choose collar neighborhoods(A.15) c p X A q ´ « r , q ˆ p X A q ´ c p X A q ´ « p´ , s ˆ p X A q ´ and let t be the coordinate in the intervals r , q , p´ , s . In the collar the tangent bundle splitsoff a trivial line bundle:(A.16) T ` c p X A q δ ´ ˘ – R ´p n ´ k ` q ‘ T p X A q δ ´ . The triple consisting of X A , a unit, and a counit, is unique up to unique isomorphism in an appropriate 2-category truncation of Bord n . Here we define X A and only give an indication of the construction of the unit andcounit of the adjunctions. Orient the summand R ´p n ´ k ` q according to the opposite of the orientation of L ´p n ´ k ` q in (A.11),with which it is identified at t “
0. In other words, orient it according to the arrow of time in X A .Let R ´p n ´ k q denote the trivial summand in the inflated tangent bundle (A.12) that corresponds tocodimension n ´ k ; it is oriented according to (cid:15) ´p n ´ k q . Let(A.17) V “ R ´p n ´ k ` q ‘ R ´p n ´ k q with its direct sum orientation; V is a direct summand of the inflated tangent bundle r T ` c p X A q δ ´ ˘ inthe collar. Transport the tangential structure from X to X A as follows. At t “ ´ id ‘ id on (A.16): flip the sign on R ´p n ´ k ` q . Moving in the collars (A.15)in X A from t “ t “ p´ q δ { V which begins at id V andends at ´ id V and turns (A.18) ! counterclockwiseclockwise ) according as A “ ! RL ) . For | t | ě { X to X A via the hyperplane reflection in the extended tangent bundle which flips the sign on R ´p n ´ k q . Figure 25.
Right and left adjoints of evaluation in Bord fr2
Figure 25, a reworking of Figure 8, illustrates the right and left adjoints of the evaluation map e : ` >´ Ñ ∅ in Bord fr2 . In these figures the single-headed red arrows indicate the orientation (cid:15) ´ and the double-headed red arrows the orientation of L ´ , determined by whether a boundarycomponent is incoming or outgoing. The counterclockwise vs. clockwise specification (A.18) can bechecked in four adjunctions: e A as an adjoint of e and e as an adjoint of e A , each for A “ R, L . Remark
A.19 . A useful isomorphic representative of X A is obtained via the identity diffeomor-phism of X A , lifted to the inflated tangent bundle r T X A as the hyperplane reflection which is ´ idon R ´p n ´ k q . This is illustrated by the diffeomorphisms in Figure 25, under which both the framingsand arrows of time have been transported. Counterclockwise rotation turns the positive direction in the first summand of (A.17) towards the positivedirection in the second summand.
APPED BOUNDARY THEORIES IN 3D 31
Figure 26.
The first move towards a counit X ˝ X R Ñ id The units and counits of the adjunctions may be constructed in two stages, which we now sketch.The first step for the counit X ˝ X R Ñ id is illustrated in Figure 26. Glue p X R q ´ to X ´ “ p X R q ´ by adjoining a cylinder; then the tangential structures are such that we can push along a 2-diskin the direction of the vector space V in (A.17) to construct a p k ` q -dimensional bordism whicheliminates the cylinder and the collared neighborhoods of those boundary components. For thesecond stage, choose a Morse function f : X R z c p X R q ´ Ñ r , q with f ´ p q “ p X R q ´ , and use2 ´ f as a Morse function on X z cX ´ . Do surgeries to cancel corresponding critical points and soproduce the desired p k ` q -dimensional bordism to the identity k -morphism (cylinder) on p X R q ´ .See Figure 10 for an example of a unit and counit, though with trivial second stage. A.3. k -morphisms in Bord n, B A.3.1.
The definition.
The bordism multicategory with boundary theory Bord n, B is an extensionof Bord n . The boundary B X of a k -morphism of depth d has a distinguished subset B ´ , the“colored” subset of § Definition A.20.
Fix n P Z ą , k P t , . . . , n u and d P t , . . . , k u . Let X be a compact k -dimensional manifold with corners of depth ď d . The data of a k -morphism of depth d in Bord n, B on X are:(i) if d ě
2, closed p k ´ d q -manifolds X ´ d , X ´ d , B ´ d , B ´ d , not all empty;(ii) if d ě
3, recursively for j “ d ´ , d ´ , . . . , p k ´ j q -manifolds X ´ j , X ´ j , B ´ j , B ´ j with corners of depth ď d ´ j equipped with diffeomorphisms(A.21) ϕ δ ´ j : X ´p j ` q Y r , s ˆ ! X ´p j ` q > X ´p j ` q ) Y X ´p j ` q Y B δ ´p j ` q ÝÑ Bp X δ ´ j q ,β δ ´ j : B ´p j ` q Y r , s ˆ ! B ´p j ` q > B ´p j ` q ) Y B ´p j ` q ÝÑ Bp B δ ´ j q , for δ P t , u ;(iii) if d ě
1, compact p k ´ q -manifolds X ´ , X ´ , B ´ with corners of depth ď d ´ ϕ δ ´ : X ´ Y r , s ˆ (cid:32) X ´ > X ´ ( Y X ´ Y B δ ´ ÝÑ Bp X δ ´ q β ´ : B ´ Y r , s ˆ (cid:32) B ´ > B ´ ( Y B ´ ÝÑ Bp B ´ q , for δ P t , u ;(iv) if d ě
1, a diffeomorphism(A.23) ϕ : X ´ Y r , s ˆ (cid:32) X ´ > X ´ ( Y X ´ Y B ´ ÝÑ B X ; and(v) if k ď n ´
1, locally constant functions (cid:15) ´ i : X Ñ t˘ u , i P t , . . . , n ´ k u . Figure 27.
A 2-morphism of depth 2 and a 3-morphism of depth 3
Remark
A.24 . As in Remark A.7(2), set X δ ´p d ` q “ B δ ´p d ` q “ B δ ´p d ` q “ ∅ , δ P t , u . Thecategorical interpretation of the bordism described in Remark A.7(3) is still valid. The embed-dings (A.8) still exist, but now the embeddings ˚ ψ δ ´ j do not cover B X . Rather, the embeddings β δ ´ j , β ´ , j P t , . . . , d ´ u , δ P t , u , combine to embeddings(A.25) γ δ ´ j : r , s j ´ ˆ B δ ´ j ÝÑ B ´ , j P t , . . . , d u , δ P t , u . Let ˚ γ δ ´ j be the restriction of γ δ ´ j to r , s j ´ ˆ ˚ B δ ´ j . Then B X is the disjoint union of three sets:(i) the images of ˚ ψ δ ´ j , j P t , . . . , d u , δ P t , u ; (ii) the images of ˚ γ δ ´ j , j P t , . . . , d ´ u , δ P t , u ;and (iii) the p k ´ q -manifold ˚ B ´ . A.3.2.
The tangent filtration.
For points x P B X in the image of one of the ˚ ψ δ ´ j , the filtra-tion (A.9) and orientation of the lines (A.11) apply. If x P B X lies in the image of some ˚ γ δ ´ j ,choose t , . . . , t j ´ P r , s , ˚ b P ˚ B δ ´ j such that x “ ˚ γ δ ´ j p t , . . . , t j ´ ;˚ b q . Then T x X has a decreasingfiltration(A.26) T x X “ T x, X Ą T x, ´ X Ą ¨ ¨ ¨ Ą T x, ´p j ´ q X Ą T x, ´ j X “ T ˚ x ˚ B ´ j in which(A.27) T x, ´ X “ T x B ´ ,T x, ´ i X “ d ˚ γ δ ´ j ` i ´ ‘ R j ´ i ‘ T ˚ x ˚ B ´ j ˘ , i P t , . . . , j u . APPED BOUNDARY THEORIES IN 3D 33
The associated graded is a sum of real lines(A.28) L x, ´p n ´ k ` j q ‘ ¨ ¨ ¨ ‘ L x, ´p n ´ k ` q . Orient L x, ´p n ´ k ` j q so that the positive direction leads into X . (This is for a boundary theory1 Ñ τ ď F ; for boundary theories τ ď F Ñ L x, ´p n ´ k ` j ´ q so that the positive direction leads into B ´ if δ “ B ´ if δ “
1. Orient L x, ´p n ´ k ` j q , i P t , . . . , j ´ u , so that the positive direction points towards increasing t j ´ ´ i .These orientations—arrows of time—can be omitted (as in §
2) since they can be deduced from thearrows of time on the rest of B X . A.3.3.
Tangential structures.
Finally, any n -dimensional tangential structure is constrained atpoints x P B ´ Ă B X : it reduces to an p n ´ q -dimensional tangential structure on T x, ´ X “ T x B ´ . Appendix B. Internal Duals
We describe here an abstract notion of internal duals, generalizing from a tensor category (Defi-nition B.14) to an algebra object in a 2-category (Theorem B.18). In particular, we show that ourTFT F with nonzero boundary condition β leads to a fusion category Φ “ End R ` β p`q ˘ (Defini-tion 2.25). Since our knowledge of Φ comes from TFT calculus, we must avoid unpictorial internalstructures (for example, the use of contravariant functors such as x ÞÑ x ˚ ) in describing internalduality. The main application is Theorem B.1.
A tensor category which is dual to its opposite and satisfies the Frobenius conditionof Definition B.7 and the bimodule property of Proposition B.13 has internal left and right duals.
For our Φ, these conditions are checked in Lemma 2.30.We refer to [BJS] for another discussion of rigidity and dualizability.
Remark
B.2 . In the setting of TFT, the conditions separate neatly into a Frobenius-bimodulecondition and an adjunction condition, reflecting two different geometric properties of a TFT withboundary generated by an algebra object and its regular boundary conditions. The logic of ourapplication to Φ compels a different path; we will return to the more natural statements in a futurepaper.Let Φ , Φ _ be a dual pair of objects in a symmetric monoidal 2-category p M , (cid:2) q . We are mostlyinterested in the categorical case when Φ , Φ _ are an opposite couple of linear categories paired byHom, and can even restrict to semisimple categories, but the algebra below is agnostic about that,unless we explicitly flag it. It is convenient to denote the duality pairing Φ _ (cid:2) Φ Ñ by writing x ξ | y y , as in the categorical case, when y P Φ, ξ P Φ _ “ Φ op . This convention is symmetric undersimultaneous swapping of the arguments and of Φ with Φ _ . When checking identities, conversion to the formalism of arrows is straightforward. Equalities stand for canonical isomorphisms of1-morphisms.Assume given an E structure on Φ, with strict unit η : Ñ Φ and multiplication ∇ : Φ (cid:2) Φ Ñ Φ.When Φ is a category and when no confusion ensues, we also write x ¨ y for ∇ p x, y q and 1 for thetensor unit. The dual object Φ _ is a Φ-Φ bimodule. This bimodule is invertible, if Φ is 2-dualizableas an algebra object, and represents then the Serre autofunctor of the (category of modules overthe) E object Φ.We shall not adopt the a priori assumption of 2-dualizability here; however, we will require that η and ∇ have right adjoints ε : Φ Ñ and ∆ : Φ Ñ Φ (cid:2) Φ. This condition is always met in thecategorical case, with explicit formulas for ε and for the dual functor ∆ _ : Φ _ (cid:2) Φ _ Ñ Φ _ :(B.3) ε p z q “ Hom Φ p , z q ; ∆ _ p x op (cid:2) y op q “ ∇ p x, y q op , for x, y, z P Φ . With this structure, Φ _ becomes a tensor category with unit 1 op “ ε _ p q . More generally, the dualobject Φ _ is an E object with the same features as Φ: the dual arrow ∇ _ defines a co-multiplicationwhich is right adjoint to the multiplication ∆ _ , and the latter has unit ε _ , with right adjoint η _ . Remark
B.4 . This interpretation of the dual right adjoint of ∇ holds for any functor ϕ : X Ñ Y between categories which are in duality with their opposites: namely, ϕ op : X op Ñ Y op is ϕ op “p ϕ R q _ “ p ϕ _ q L . In particular, adjoints exist. Recall also that, when X, Y are 2-dualizable, with(additive) Serre automorphisms S ` X , S ` Y , the left and right adjoints of ϕ are related by S ` Y ˝ ϕ L “ ϕ R ˝ S ` X . Commuting duals with adjoints will therefore bring out additive Serre functors.Define now the pairing B : Φ (cid:2) Φ Ñ as B “ ε ˝ ∇ . When Φ is a category, B p x, y q “ Hom Φ p , x ¨ y q , for two general objects x, y . From B , we define a dual pair of functors, by dualizingseparately with respect to each variable:(B.5) f, f _ : Φ Ñ Φ _ , f p x q : “ B p x, ´q , f _ p y q : “ B p´ , y q Proposition B.6. f is a right, and f _ a left Φ -module morphism.Proof. x f p x ¨ y q | z y “ B p x ¨ y, z q “ ε p x ¨ y ¨ z q “ B p x, y ¨ z q “ x f p x q | y ¨ z y “ x f p x q .y | z y , so that f p x ¨ y q “ f p x q .y , and similarly for f _ . (cid:3) Definition B.7.
We say that Φ satisfies the (non-symmetric) adjoint Frobenius condition when B is a perfect pairing: that is, f and f _ are isomorphisms. If so, we define the Serre automorphismof Φ as S b “ p f _ q ´ ˝ f. Proposition B.8.
Assume that Φ satisfies the Frobenius condition. The following natural isomor-phisms apply: (i) B p x, y q “ B p y, S b x q . In particular, symmetry of B is equivalent to a trivialization of S b . (ii) f ˝ η “ f _ ˝ η “ ε _ . For a category, f p q “ f _ p q “ op . (iii) As functors Φ Ñ Φ _ , we have ε _ . p q “ S b p q .ε _ . In the categorical case, op .x “ S b p x q . op . At any rate, we can reduce to the categorical case by passing to the functors on M represented by Φ , Φ _ . APPED BOUNDARY THEORIES IN 3D 35 (iv) S b is naturally a tensor automorphism of Φ , and twisting the Φ -action by S b induces theSerre autofunctor M ÞÑ Φ _ (cid:2) Φ M on the -category of left Φ -modules.Remark B.9 . Promoting S b to a tensor functor means equipping it with isomorphisms, compatiblewith the associativity and unit laws on Φ, S b ˝ ∇ – ∇ ˝ p S b ˆ S b q , S b ˝ η – η ;while not evident from the expression p f _ q ´ ˝ f , they do follow from Parts (i)-(iii), as in theproof below. On the other hand, reduction of Φ _ (cid:2) Φ to a tensor automorphism of Φ is a formalconsequence of the isomorphy of f _ . Proof.
Parts (i)-(iii) are immediate from the properties of f, f _ , B ; thus, B p x, y q “ x f p x q | y y “ x f _ ˝ S b p x q | y y “ x f p y q | S b p x qy for (i) , x f p q | x y “ B p , x q “ ε p x q “ B p x, q “ x f _ p q | x y for (ii), S b p x q . op “ f _ ` S b p x q ˘ “ f p x q “ op .x for (iii) . Multiplicativity of S b now follows: f _ ` S b p xy q ˘ “ S b p xy q . op “ op .xy “ S b p x q . op .y “ S b p x q ¨ S b p y q . op “ f _ ` S b p x q ¨ S b p y q ˘ , using categorical notation for simplicity. To complete (iv), consider the following diagram of rightΦ-modules, with left multiplication in the bottom row:Φ (cid:2) Φ ∇ (cid:47) (cid:47) S b (cid:2) f (cid:15) (cid:15) Φ f (cid:15) (cid:15) Φ (cid:2) Φ _ (cid:47) (cid:47) Φ _ We claim this commutes naturally. Assuming this, let us interpret the diagram: the right verticalarrow f gives an isomorphism of the identity with the Serre autofunctor on Φ-modules, while theleft arrow exhibits the necessary intertwining twist by S b in the left Φ-action.Exploiting the right Φ-module structure, it suffices to check commutativity on Φ (cid:2) η , when thisbecomes the isomorphism S b p x q .f p q “ f p q .x “ f p x q , from (ii) and (iii). (cid:3) Remark
B.10 . The Serre functor S b above need not agree with the additive Serre automorphism S ` Φ of Remark B.4, which is independently defined whenever the object Φ P M is 2-dualizable. However,the two will agree for a fusion category Φ, because of its 3-dualizability. See also Remark B.19below for a general relation between the two.The isomorphisms f, f _ allow us to transport the structure tensors η, ∇ , ∆ , ε to a matchingstructure on Φ _ , denoted by overbars. Choosing either f or f _ results in isomorphic structureson Φ _ , because all structure tensors commute with S b . Dualizing them gives a new structure ¯ ε _ , ¯∆ _ , ¯ ∇ _ , ¯ η _ on Φ. We get the following diagram, in which the bottom row maps are related tothe top row maps by duality and adjunction using uniform rules, ε “ η R , ∆ “ ∇ R , ¯ ε “ ¯ η R , ¯∆ “ ¯ ∇ R and all ensuing relations:(B.11) Φ (cid:2) Φ ∇ (cid:42) (cid:42) Φ ∆ (cid:106) (cid:106) ε (cid:41) (cid:41) η (cid:106) (cid:106) f (cid:40) (cid:40) f _ (cid:118) (cid:118) Φ (cid:2) Φ ∆ _ (cid:42) (cid:42) Φ ∇ _ (cid:105) (cid:105) η _ (cid:41) (cid:41) ε _ (cid:106) (cid:106) Φ _ (cid:2) Φ _ ∆ _ (cid:43) (cid:43) Φ _ ∇ _ (cid:106) (cid:106) η _ (cid:41) (cid:41) ε _ (cid:106) (cid:106) Φ _ (cid:2) Φ _ ∇ (cid:43) (cid:43) Φ _ ∆ (cid:106) (cid:106) ε (cid:41) (cid:41) η (cid:106) (cid:106) The dual corners are related by the morphism f _ . Because ¯ η “ f ˝ η , etc., we find from Proposi-tion B.8.ii that Proposition B.12.
In the diagram above, units and traces match in each row: ¯ η “ ε _ , ¯ ε “ η _ . (cid:3) Proposition B.13.
Under the Frobenius assumption, the following conditions are equivalent: (i)
The co-product ∆ is a Φ - Φ bimodule map (for the outer Φ -actions on the two Φ -factors). (ii) The multiplication ∆ _ is a Φ - Φ bimodule map (for the inner Φ -actions on the two Φ _ -factors). (iii) The two structures on Φ in the top row of (B.11) are transpose-isomorphic. (iv) The two structures on Φ _ in the bottom row of (B.11) are transpose-isomorphic.Proof. Parts (i) and (ii) are equivalent by duality, (iii) and (iv) are so via the isomorphisms inducedby f . Note further that the diagonal arrow is compatible with the Φ-Φ bimodule structures: on theright, because f is a right module map, an on the left, because we could equally well have used theleft module isomorphism f _ instead. In light of the matching units, which are free generators ofΦ _ over Φ, the bimodule condition determines the multiplication maps and forces the agreementof the remaining structure maps on each row. (cid:3) Definition B.14.
When Φ is a category, the internal right and left duals ˚ x, x ˚ of an object x P Φare the objects characterized (up to unique isomorphism, if they exist) by the functorial (in y, z )identities(B.15) Hom p y ¨ x, z q “ Hom p y, x ˚ ¨ z q , Hom p y ¨ x, z q “ Hom p y, z ¨ ˚ x q . It turns out that the conditions in (B.13) force the existence of internal duals and their expressionin terms of f _ and f . To see this, we first give an abstract formulation.Dualizing the product ∇ in the second argument gives the left multiplication map λ : Φ Ñ Φ (cid:2) Φ _ .In the categorical case, λ p x q represents the left multiplication by x P Φ. Similarly, for the firstargument we get the right multiplication map ρ : Φ Ñ Φ (cid:2) Φ _ . Repeating this for ∆ _ leads tothe two maps λ , ρ : Φ _ Ñ Φ _ (cid:2) Φ. In the abusive but readable argument notation, with Greekarguments living in Φ _ ,(B.16) x λ p ξ q | y (cid:2) ξ y “ x ∆ _ p ξ , ξ q | y y , x ρ p ξ q | y (cid:2) ξ y “ x ∆ _ p ξ , ξ q | y y . The maps λ , ρ will be the abstract versions of the ‘tensoring with duals’ x op ÞÑ p z ÞÑ x ˚ ¨ z q , x op ÞÑ p z ÞÑ z ¨ ˚ x q . APPED BOUNDARY THEORIES IN 3D 37
Remark
B.17 . λ , ρ are related to the op-conjugates λ op , ρ op : Φ _ Ñ Φ _ (cid:2) Φ as follows: λ – p Id (cid:2) S ` Φ ´ q ˝ λ op , ρ – p Id (cid:2) S ` Φ ´ q ˝ ρ op The source of the additive Serre correction S ` Φ is described in Remark B.4.Denote by τ the symmetry Φ (cid:2) Φ _ Ñ Φ _ (cid:2) Φ. Theorem B.18.
The equivalent conditions of Proposition B.13 are also equivalent to: (i) λ – τ ˝ λ ˝ f ´ . (ii) ρ – τ ˝ ρ ˝ p f _ q ´ . (iii) In the categorical case: Φ has internal left and right duals.Proof. We check the two sides by pairing against a triple of arguments p ξ , y, ξ q P Φ _ ˆ Φ ˆ Φ _ ,leaving to the reader the unenviable task of convert this to identities between morphisms, dualsand adjoints. Having written out the left sides in (B.16) above, we start with the right side of (i): x τ ˝ λ ˝ f ´ p ξ q | y (cid:2) ξ y “ x λ ˝ f ´ p ξ q | ξ (cid:2) y y “ x ξ | f ´ p ξ q ¨ y y “ x ξ .f ´ p ξ q | y y where the middle line is the definition of λ , while dot represents the right multiplication action of f ´ ξ upon ξ P Φ _ . Agreement with λ is then equivalent to ξ .f ´ p ξ q “ ∆ _ p ξ , ξ q ;but using the right module property of f , we have ξ .f ´ p ξ q “ f “ f ´ p ξ q ¨ f ´ p ξ q ‰ , thus reaching Condition (iv) in Proposition B.13.Similarly, for the right side of (ii), x τ ˝ ρ ˝ p f _ q ´ p ξ q | y (cid:2) ξ y “ x ρ ˝ p f _ q ´ p ξ q | ξ (cid:2) y y “ x ξ | y ¨ p f _ q ´ p ξ qy “ xp f _ q ´ p ξ q .ξ | y y and identity (ii) is equivalent to p f _ q ´ p ξ q .ξ “ ∆ _ p ξ , ξ q , which follows as before, this time from the left-module property of f _ .Finally, for Part (iii) we must convert the identities into the recognizable form (B.14). For this,we let ξ , be opposites of objects x , P Φ; then, ∆ _ p ξ , ξ q is the opposite object to x ¨ x , andwe can rewrite x ξ | f ´ p ξ q ¨ y y “ Hom Φ ` x , f ´ p ξ q ¨ y ˘ , x ξ | y ¨ p f _ q ´ p ξ qy “ Hom Φ ` x , y ¨ p f _ q ´ p ξ q ˘ , x ∆ _ p ξ , ξ q | y y “ Hom Φ p x ¨ x , y q exhibiting f ´ p ξ q as x ˚ and p f _ q ´ p ξ q as ˚ x in Definition B.14. (cid:3) Remark
B.19 . Under the assumptions of (B.13), and if, in addition, Φ is 2-dualizable, the additiveSerre functor S ` Φ is related to S b :(B.20) S ` p x ¨ y q “ S b p x q ¨ S ` p y q “ S ` p x q ¨ S b´ p y q . In particular, we have S ` p x q “ S b p x q ¨ S ` p q “ S ` p q ¨ S b´ p x q , and, as the functor S ` is invertible, S ` p q must be a unit. In the categorical case, S b p x q “ x ˚˚ ,and the relations follow by applying Serre duality to the adjunction relations in Definition B.14.One instance of (B.20) is when S ` “ S b “ S b´ , which happens in the case of fusion cate-gories [DSS], but that is not the only option. Thus, if Φ is the derived category of bounded coherentsheaves on a projective manifold, with the obvious internal duals, the multiplicative Serre functor S b is the identity, while the functor S ` is tensoring with the canonical line bundle. Appendix C. Complete reducibility of fusion categories
A fusion category whose unit is simple cannot be decomposed as a direct sum, even after passingto a Morita equivalent model: otherwise, we would split the unit. The following converse followseasily from several statements in [EGNO], but we give a complete proof, at the price of rehashingsome basic facts. Throughout, Φ will denote a fusion category.
Theorem C.1 (Complete Reducibility) . Φ is Morita equivalent to a direct sum of fusion categorieswith simple unit. Corollary C.2. Φ is Morita equivalent to a fusion category Φ with simple unit if and only if theDrinfeld center of Φ is invertible. We prove Corollary C.2 at the end of the appendix.We break up the proof of Theorem C.1 into small steps. Let “ ř i p i be the decomposition ofthe unit of Φ into simple objects. Call an object x self-adjoint if it is isomorphic with x ˚ . Lemma C.3.
Each p i is a self-adjoint projector: p ˚ i – p i , p i – p i , End p p i q “ C . In addition, p i p j “ if i ‰ j .Proof. We have p i “ p i ¨ “ ř j p i p j , so p i p j “ j , when it equals p i . Onthe other hand, p j “ ¨ p j “ ř k p k p j , but the sum contains p i p j “ p i , so p i “ p j , proving themultiplicative claims. Further, ˚ “ , and p ˚ i p i ‰ p , p ˚ i p i q “ End p p i q ‰
0, so wemust have p ˚ i – p i . (cid:3) Lemma C.3 gives a “matrix decomposition” of Φ asΦ – à i,j p i ¨ Φ ¨ p j “ : à i,j Φ ij , APPED BOUNDARY THEORIES IN 3D 39 with fusion categories Φ ii having simple units p i on the diagonal, Φ ii -Φ jj bimodule categories Φ ij (identified with Φ opji under internal duality), and multiplication compatible with matrix calculus:Φ ij ˆ Φ jl Ñ Φ il , Φ ij ¨ Φ kl “ j ‰ k. The equivalence classes of indices generated by the condition Φ ij ‰ indecomposable if a singleblock is present. We claim that an indecomposable Φ is Morita equivalent to any of its diagonalentries, selecting Φ for our argument.The equivalence is induced by the first row and the first column of Φ: the Φ -Φ bimodule R : “ À i Φ i and the Φ-Φ bimodule C : “ À j Φ j . We check it in the following two lemmata. Lemma C.4.
The multiplication map R (cid:2) Φ C Ñ Φ is an equivalence of Φ - Φ bimodulecategories.Proof. We have Φ (cid:2) Φ Φ “ Φ, and splitting the left factor Φ into its rows R i and the right factorinto its columns C j gives a direct sum decomposition of Φ as R i (cid:2) Φ C j . Examining the action of theprojectors p k , on R i on the left and on C j on the right, identifies this with the Φ ij decompositionof Φ. (cid:3) Lemma C.5.
The multiplication map µ : C (cid:2) Φ R Ñ Φ is an equivalence of Φ - Φ bimodulecategories. The proof of this direction requires some preliminary facts.
Lemma C.6 (Linearity of adjoints) . The adjoints ϕ L , ϕ R of an Φ -linear map ϕ : M Ñ N betweenright or left finite semisimple Φ -module categories have a natural Φ -linear structure.Proof. Choosing left modules and the right adjoint, we write a functorial isomorphismHom M ` m, ϕ R p x.n q ˘ “ Hom M ` m, x.ϕ R p n q ˘ by rewriting the left side asHom N p ϕ p m q ; x.n q “ Hom N p ˚ x.ϕ p m q ; n q “ Hom M p ϕ p ˚ x.m q , n q “ Hom M ` ˚ x.m, ϕ R p n q ˘ and finish by moving x back to the right. The other cases are similar. (cid:3) Lemma C.7. If N “ Φ above, then ϕ ˝ ϕ L p q is self-adjoint in Φ .Proof. Let h p x, y q : “ dim Hom p x ; y q . In our semisimple case, h p x, y q “ h p y, x q , as we are onlycounting multiplicities of simple objects. Moreover, x is determined up to isomorphism by itsmultiplicities, so x is self-adjoint iff h p x, y q “ h p x ˚ , y q for all y ; the latter is also h p x, y ˚ q . We showthis for x “ ϕ ˝ ϕ L p q : h p x, y ˚ q “ h p y ¨ x, q “ h p ϕ ˝ ϕ L p y q , q “ h p , ϕ ˝ ϕ L p y qq“ h p ϕ L p q , ϕ L p y qq “ h p ϕ L p y q , ϕ L p qq “ h p y, ϕ ˝ ϕ L p qq “ h p x, y q . (cid:3) We only use semisimplicity here to ensure the existence of adjoints.
Lemma C.8.
In Lemma C.3, every self-adjoint projector (cid:36) is isomorphic to a sum of distinct p i .Proof. Let (cid:36) “ p ` x , where p collects all the p i appearing in (cid:36) . Writing the relation (cid:36) – (cid:36) as p ` x – p ` p ¨ x ` x ¨ p ` x , we see that each p i appears at most once, otherwise its multiplicity in p exceeds the one in p .Moreover, an isomorphism x – x ˚ gives an identification Hom p , x q “ End p x q , while Hom p , x q “ p x q “ x “ (cid:3) Proof of Lemma C.5.
Writing B for the Φ-Φ bimodule category C (cid:2) Φ R , Lemma C.4 implies that B (cid:2) Φ B – B , µ -compatibly with the identification Φ (cid:2) Φ Φ “ Φ. The left adjoint µ L is also abimodule map, by Lemma C.6, and is similarly idempotent, µ L (cid:2) µ L – µ L . Then, µ ˝ µ L is anidempotent bimodule endomorphism of Φ. It is the multiplication by the object p : “ µ ˝ µ L p q —on the left, or on the right — which must then be a projector in Φ. Moreover, p self-adjoint byLemma C.7. Lemma C.8 identifies it as a sum of p i . If p fl , it would split the image Φ ¨ p – p ¨ Φas a block of Φ, contradicting indecomposability.It follows that µ ˝ µ L – Id Φ , splitting B into Φ and a complementary bimodule. But the relation B (cid:2) Φ B – B can only hold if this complement is zero, so µ is an equivalence. (cid:3) Proof of Corollary C.2.
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