Fusion 2-categories with no line operators are grouplike
aa r X i v : . [ m a t h . QA ] O c t FUSION 2-CATEGORIES WITH NO LINE OPERATORS ARE GROUPLIKE
THEO JOHNSON-FREYD , AND MATTHEW YU ∗ , Abstract.
We show that if C is a fusion 2-category in which the endomorphism category of theunit object is Vec or SVec , then the indecomposable objects of C form a finite group. Introduction
Just as multifusion 1-categories describe the fusion of quasiparticle excitations — 1-spacetime-dimensional objects, aka line operators — in topological phases of matter, multifusion 2-categories(first introduced in [DR18]) describe the fusion of 2-spacetime-dimensional “quasistring” excita-tions, aka surface operators. Except in very low dimensions, a typical topological phase can havequasistring excitations which are not determined by the quasiparticle excitations, and multifu-sion 2-categories are vital for the construction and classification of topological phases in mediumdimension [LKW17, LKW18, JF20].Recall that a multifusion 1-category C is fusion if the endomorphism algebra Ω C = End C (1 C )is trivial, i.e. isomorphic to C , where 1 C ∈ C denotes the monoidal unit [EGNO15]. There aretwo reasonable categorifications of this notion when C is a multifusion 2-category. The strongergeneralization, which we will call strongly fusion , is to ask that the endomorphism 1-categoryΩ C = End C (1 C ) be trivial, i.e. equivalent to Vec C . The weaker notion, which we will call merely fusion , is to ask only that Ω C = End Ω C (1 C ) be trivial, where 1 C ∈ Ω C is the identity object.A fusion 2-category is a finite semisimple monoidal 2-category that has left and right duals forobjects and a simple monoidal unit. Physically, if C describes the surface operators in a topologicalphase, then Ω C describes the line operators and Ω C describes the vertex (0-spacetime-dimensional)operators.The classification of fusion 1-categories is extremely rich [EGO04, JL09, Nat18]. The simplestexamples are the grouplike , aka pointed, fusion 1-categories, whose isomorphism classes of simpleobjects form a group G under the fusion product. These are famously classified by ordinarygroup cohomology H ( G ; U(1)). But there are many nongrouplike examples. The classification of(merely) fusion 2-categories is similarly rich, since it includes the classification of braided fusion1-categories [DR18, Construction 2.1.19]. The main result of this note shows a dramatic differencewith the strongly fusion case: Theorem A. If C is a strongly fusion -category, then the equivalence classes of indecomposableobjects of C form a finite group under the fusion product. We also address the “fermionic” case where Ω
C ∼ = SVec : Research at the Perimeter Institute is supported by the Government of Canada through Industry Canada and bythe Province of Ontario through the Ministry of Economic Development and Innovation. The Perimeter Institute isin the Haldimand Tract, land promised to the Six Nations. Dalhousie University is in Mi‘kma‘ki, the ancestral andunceded territory of the Mi‘kmaq. We are all Treaty people. Perimeter Institute for Theoretical Physics, Waterloo, Ontario . Department of Mathematics, Dalhousie University, Halifax, Nova Scotia . ∗ Corresponding author. [email protected] . Theorem B. If C is a fusion -category with Ω C ∼ = SVec , then the equivalence classes of inde-composable objects of C form a finite group, which is a central double cover of the group π C ofcomponents of C (see Definition 2.11). In particular, Theorem B asserts that the components of C do form a group. Remark.
Just as grouplike fusion 1-categories in which the simple objects form a group G areclassified by H ( G ; U(1)), the strongly fusion 2-categories with simple objects G are classified byH ( G ; U(1)) [DR18, Remark 2.1.17]. In the fermionic case, if one additionally assumes that theactions of End(1 C ) ∼ = SVec on End( X ) given by tensoring on the left and on the right agree, thenone can show that the options with π C = G are classified by “extended group supercohomology”SH ( G ) defined in [WG18]. There is a canonical map SH ( G ) → H ( G ; Z ) which takes anextended supercohomology class to its Majorana layer ; the group of simple objects in C is thecorresponding central extension Z .G . Although in general Majorana layers of supercohomologyclasses have no reason to be trivial, we were unable to find an example where the extension Z .G did not split.The outline of our paper is as follows. Section 2 reviews the definition of multifusion 2-categoryfrom [DR18]. In particular, we recall their notion of “component” of a semisimple 2-category in § § §
4, respectively.In future work, we will use these theorems to give a complete classification of 5-spacetime-dimensional topological orders.2.
Semisimple and multifusion 2-categories
The definition and basic theory of semisimple and multifusion 2-categories were first introducedin [DR18]. Since this theory is new, we take this section to review the main features.Recall that a 2-category C is C -linear if all hom-sets of 2-morphisms are vector spaces over C ,and both 1- and 2-categorical compositions of 2-morphisms are bilinear. Definition 2.1.
An object in a linear 2-category is decomposible if it is equivalent to a direct sumof nonzero objects, and indecomposable if it is nonzero and not decomposable.
Remark.
We will slightly abuse the language and use the terms “simple” and “indecomposable”interchangeably. A simple object X in a 2-category is one such that any faithful 1-morphism A ֒ → X is either 0 or an equivalence. In finite semisimple 2-categories all indecomposable objectsare simple [DR18].In particular the objects which we consider in the 2-category will only be sums of finitely manysimple objects, and decompositions are unique up to permutations. In our goal to define a semisim-ple 2-category, we present some definitions for the higher categorical generalization of the notionof idempotent splitting and idempotent complete for 1-categories, also discussed in [GJF19]. Definition 2.2.
A 2-category C is locally idempotent complete if for all objects A, B ∈ C , the1-category hom C ( A, B ) is idempotent complete. It is locally finite semisimple if hom C ( A, B ) isfurthermore a finite semisimple C -linear category (i.e. an abelian C -linear category with finitelymany isomorphism classes of simple object and in which every object decomposes as a finite directsum of simple objects). In what follows, we will assume C is a locally idempotent complete 2-category. Definition 2.3. A separable monad is a unital algebra object p ∈ hom C ( A, A ) , for a simple object A , whose multiplication m : p ◦ p → p admits a section as a p - p bimodule. Definition 2.4. A (unital) condensation in a -category C is an adjunction f ⊣ g ≡ ( f : A ⇆ B : g, η : 1 A → g ◦ f, ǫ : f ◦ g → B ) which is separable in the sense that its counit ǫ admits a section, USION 2-CATEGORIES WITH NO LINE OPERATORS ARE GROUPLIKE 3 i.e. if there exists a -morphsism φ : 1 B → f ◦ g which is the right inverse of ǫ . When there is sucha separable adjuction, we will write “ A ֓ → B ,” and say that A condenses onto B . Definition 2.5.
A separable monad p is separably split if there exists a separable adjunction f ⊣ g and g ◦ f ∼ = p . A separable splitting is a choice of this isomorphism. Proposition 2.6 ([DR18, Proposition 1.3.4]) . A separable monad in C which admits a separablesplitting, admits a unique up-to-equivalence separable splitting. Admitting a separable splitting implies that the adjunction f ⊣ g admits A as an Eilenberg-Moore object. In 1-categories, this is can be seen as module decomposition by forming a projectorfrom an idempotent. The subtlety in 2-categories is that now there is no “orthogonal complement”to the projector, as in 1-categories. Definition 2.7.
A 2-category C is if it is locally idempotent complete andevery separable monad splits. Remark.
Requiring the unitality of p and the existence of a unit for adjunction in the 2-categorycase differs slightly from the situation in 1-categories. In 1-categories there is an equality of p = p but there is no equality of 1 and p . [GJF19] developed a nonunital version of separable monad for2-categories and showed that if C has adjoints for 1-morphisms, then the notion of 2-idempotentcompletion in Definition 2.7 and in [GJF19] agree. Definition 2.8. A C -linear -category C is finite semisimple if: it has finitely many isomorphismclasses of simple objects; it is locally finite semisimple; has adjoints for -morphisms; has directsums of objects; and is -idempotent complete. Definition 2.9. A multifusion 2-category is a monoidal finite semisimple -category in which allobjects have duals. Remark.
As noted in [DR18, Definition 2.1.6], in a fusion 2-category, left and right duals are thesame.The 1-categorical Schur’s Lemma says that in a semisimple 1-category, if two indecomposableobjects are related by a nonzero morphism, then they are isomorphic. This result fails in 2-categories, but [DR18, Proposition 1.2.19] provides the following replacement.
Proposition 2.10 (Categorical Schur’s Lemma) . If f : X → Y and g : Y → Z are nonzero -morphisms between indecomposable objects in a semisimple -category C , then gf : X → Z isnonzero.Proof. This follows from Proposition 2.12 below, since the composition of condensations is a con-densation and since condensations with nonzero target are nonzero. (cid:3)
In particular, “related by a nonzero morphism” defines an equivalence relation on the indecom-posable objects of C . (Note that, since every 1-morphism is required to have an adjoint, if there isa nonzero morphism f : X → Y , then there is a nonzero morphism f ∗ : Y → X .) Definition 2.11.
The set of components of C , denoted π C , is the set of equivalence classes ofindecomposable objects for the equivalence relation “related by a nonzero morphism.” The structure of each component is fully determined by (the endomorphism category of) anyrepresentative object. Indeed:
Proposition 2.12.
Suppose
X, Y ∈ C are simple objects connected by a nonzero -morphism f : X → Y . Then there is a condensation X ֓ → Y . In particular, Y is the image of a simplealgebra object in the fusion -category End C ( X ) .Proof. Choose g to be the right adjoint to f ; it exists because all morphisms in a semisimple 2-category are required to have adjoints. The counit ǫ : f ◦ g → Y is a nonzero 1-morphism in thesemisimple 1-category End C ( Y ) with simple target, and so has a section. (cid:3) JOHNSON-FREYD AND YU X R S b X Figure 1. Proof of Theorem A
We begin this section by developing the necessary graphical calculus in order to prove the mainresults. One important feature we will discuss is the state-operator map for fusion 2-categories andits interplay with duality. For an object X ∈ C , we denote R S b X as the wrapping of X around aboundary-framed S , see Figure 1. This integral is a map R S b : C → Ω C . This integral is an exampleof the general calculus of dualizability as in [FT20], and also arising from the cobordism hypothesis[BD95]. The boundary framing of the cylinder in Figure 1 is attained from the framing of theannulus, where the annulus framing is given by the restriction of the two-dimensional blackboardframing, see Figure 2. One could then take the framed annulus and pull the annulus into a cylinder.This results in a cylinder appropriately framed to be compatible with the state-operator map.We now describe this operation R S b algebraically. Because we are working with a fusion 2-category each object has a dual and we have a unit η X : 1 C → X ⊗ X ∗ . It corresponds to thehalf-circle with framing as in Figure 3 part (a). Also, since all 1-morphisms have adjoints, there is aright adjoint η ∗ X : X ⊗ X ∗ → C , corresponding to the framed half-circle in Figure 3 part (b). Thesetwo half-circles compose to an annulus whose framing can be continuously deformed to framing inFigure 2. All together, we find the algebraic definition: Z S b X := η ∗ X ◦ η X . The vertex operators of X are by definition the 2-morphisms 1 X ⇒ X . They are precisely theoperators that can be inserted in the interior hole in Figure 2; the blackboard framing is arranged sothat this can happen. Such an insertion may be pulled down and thought of as a map 1 C → R S b X as in Figure 4. In other words, hom(1 C , R S b X ) is the vector space of ways for the vacuum line to endon R S b X . This is the physical/geometric proof of the state-operator correspondence . Algebraically,we have: Figure 2.
USION 2-CATEGORIES WITH NO LINE OPERATORS ARE GROUPLIKE 5 η X (a) η ∗ X (b) Figure 3. η ∗ X is by definition the universal map such that the composition with η X can be filled. The resulting framing of η ∗ X ◦ η X is homotopic to the blackboardframing of Figure 2. Lemma 3.1 (State-Operator Correspondence) . In a multifusion 2-category there is an isomorphism
End
End C ( X ) (1 X ) ∼ = hom Ω C (1 C , R S b X ) .Proof. The duality of X with X ∗ provides an equivalence of End C ( X ) ∼ = hom(1 C , X ⊗ X ∗ ). Thisequivalence identifies 1 X with η X , and so in particular End(1 X ) ∼ = End( η X ), where the left-handside is computed in End C ( X ) and the right-hand side is computed in hom(1 C , X ⊗ X ∗ ). For anyadjunctible 1-morphism f : A → B in a 2-category, End hom( A,B ) ( f ) ∼ = hom End A (1 A , f ∗ ◦ f ). Taking f = η X , with A = 1 C and B = X ⊗ X ∗ , completes the proof. (cid:3) In particular, X is simple if and only if hom Ω C (1 C , R S b X ) is one-dimensional. In the stronglyfusion case lemma 3.1 implies: Proposition 3.2.
Suppose C is strongly fusion. Then X ∈ C is indecomposable if and only if R S b X = C . (cid:3) We now consider the tensor product of two indecomposable objects X ⊗ Y mapped by the integral R S b . This represents a cylinder within a cylinder as on the left of Figure 5.In general, we see that R S b is not monoidal: a cylinder within a cylinder is not the same astwo adjacent cylinders. However, in the strongly fusion case, if X and Y are simple then we maycollapse down the inner cylinder via the state operator map into the vacuum line. We may thencollapse the outer cylinder. All together we find: Corollary 3.3.
In a strongly fusion 2-category, the tensor product of indecomposable objects isindecomposable. (cid:3) X C Figure 4.
JOHNSON-FREYD AND YU C Figure 5.
This allows us to complete the proof of Theorem A:
Proof of Theorem A. If X ∈ C is a simple object, then X ∗ is as well (since End( X ) ∼ = End( X ∗ )),and hence so is X ⊗ X ∗ (by Corollary 3.3). Since η X : 1 C → X ⊗ X ∗ is nonzero, the simple objects1 C and X ⊗ X ∗ are in the same component. However, the fact that there are no lines in the stronglyfusion case means that 1 C is the only simple object in its component. (cid:3) Remark.
We can consider working over the real numbers, which is the same as having an anti-linear involution (time-reversal). In this case, an indecomposable object is not absolutely simple,and Theorem A is no longer true. We can see this already at the level of fusion 1-categories.Consider a Z fusion 1-category with three objects { , x, x − } over C . Over the real numbers,we can exchange x and x − by the involution. There will be two objects 1 and X over the realnumbers, where X ∼ = x + x − , so that it is invariant under the involution. Schur’s lemma statesthat over the complex numbers, indecomposable means that the endomorphisms of the object isjust C . But over the real numbers, the endomorphisms are a division ring, and we have the fusion X = X + 2. 4. Proof of Theorem B
If we try to repeat the proof from § C = End(1 C ) ∼ = SVec , the first snag arises inProposition 3.2. Indecomposability of X implies that the ordinary vector space hom(1 C , R S b X ) isone-dimensional. This measures the even part of the super vector space R S b X , but says nothingabout the odd part. On the other hand, the super vector space R S b X is the superalgebra of vertexoperators on X , and so it is supercommutative because we have the freedom to move operatorsaround each other on the surface of the cylinder. Furthermore, since we are working in a semisimple2-category, this supercommutative algebra is finite dimensional and semisimple. Lemma 4.1.
The only finite-dimensional semisimple supercommutative superalgebra A with one-dimensional bosonic part is C .Proof. If x ∈ A is a nonzero odd element, then it is nilpotent (since x = xx = − xx and so x = 0).Thus the principle ideal generated by x is proper. On the other hand, it is not a direct summand of A as an A -module because projection operators are bosonic, and so the only projection operatorsin A are 0 and 1. This contradicts the semisimplicty of A . (cid:3) This implies the fermionic versions of Proposition 3.2 and Corollary 3.3.To complete the proof of Theorem B, it suffices to observe that if X ∈ C is indecomposable,then, since the unit map η X : 1 C → X ⊗ X ∗ is nonzero, X ⊗ X ∗ is an indecomposable object in theidentity component of C , and so invertible, and thus X is invertible. Indeed, since Ω C ∼ = SVec , by
USION 2-CATEGORIES WITH NO LINE OPERATORS ARE GROUPLIKE 7
Proposition 2.12 there are precisely two simple objects in the identity component of C , correspondingto the two simple superalgebras C and Cliff(1), and Cliff(1) is famously Morita-invertible. Remark.
In fact, X ⊗ X ∗ is always trivial, and never the nontrivial simple objected Cliff(1).Indeed, it is a general fact of monoidal higher categories that if an object is invertible, then itsinverse is its dual. One can also see this directly by running the proof of Proposition 2.12 for thenonzero 1-morphism f = η X . Then g = η ∗ X , and the simple algebra in question is the composition p = gf = η ∗ X ◦ η X = R S b X = C . References [BD95] John C. Baez and James Dolan. Higher-dimensional algebra and topological quantum field theory.
Journalof Mathematical Physics , 36(11):6073–6105, Nov 1995.[DR18] Christopher L. Douglas and David J. Reutter. Fusion 2-categories and a state-sum invariant for 4-manifolds, 2018.[EGNO15] P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik.
Tensor Categories . Mathematical Surveys and Mono-graphs. American Mathematical Society, 2015.[EGO04] Pavel Etingof, Shlomo Gelaki, and Viktor Ostrik. Classification of fusion categories of dimension pq.
International Mathematics Research Notices , 2004(57):3041–3056, 2004.[FT20] Daniel S. Freed and Constantin Teleman. Gapped boundary theories in three dimensions, 2020.[GJF19] Davide Gaiotto and Theo Johnson-Freyd. Condensations in higher categories, 2019.[JF20] Theo Johnson-Freyd. On the classification of topological orders, 2020.[JL09] David Jordan and Eric Larson. On the classification of certain fusion categories.
Journal of Noncommu-tative Geometry , page 481–499, 2009.[LKW17] Tian Lan, Liang Kong, and Xiao-Gang Wen. Classification of (2+1)-dimensional topological order andsymmetry-protected topological order for bosonic and fermionic systems with on-site symmetries.
PhysicalReview B , 95(23), Jun 2017.[LKW18] Tian Lan, Liang Kong, and Xiao-Gang Wen. Classification of (3 + 1)D bosonic topological orders: Thecase when pointlike excitations are all bosons.
Phys. Rev. X , 8:021074, Jun 2018.[Nat18] Sonia Natale. On the classification of fusion categories. In
Proceedings of the International Congressof Mathematicians—Rio de Janeiro 2018. Vol. II. Invited lectures , pages 173–200. World Sci. Publ.,Hackensack, NJ, 2018.[WG18] Qing-Rui Wang and Zheng-Cheng Gu. Towards a Complete Classification of Symmetry-Protected Topo-logical Phases for Interacting Fermions in Three Dimensions and a General Group SupercohomologyTheory.