Defects in topological field theory: from categorical tools to applications in physics and representation theory

Transcription

1 Defects in topological field theory: from categorical tools to applications in physics and representation theory Christoph Schweigert Mathematics Department Hamburg University based on work with Jürgen Fuchs and Gregor Schaumann July 25, 2016

2 Overview 1 Introduction 1: TFT Some applications of TFTs Definition of topological field theories Extended topological field theories Topological field theories with defects 2 Introduction 2: reprn theory Finite tensor categories and (bi)module categories Motivation 1: Brauer-Picard groups Motivation 2: Frobenius-Schur indicators 3 Categories for generalized Wilson lines Labelling 2-cells / surface defects Labelling 1-cells / generalized Wilson lines Balancings and monads Categories for 1-manifolds Comparison for Dijkgraaf-Witten theories 4 Functors for surfaces Transmission functors Functors for general surfaces 5 Outlook

3 First introduction: TFT - motivations from physics Distinguished subspaces ( ground states ) of state spaces of quantum systems: Extended supersymmetries: BPS states Extra structure: parameter dependent ring structures Topological phases of matter: Spaces of ground states for different topologies (instead of a local order parameter of a symmetry breaking mechanism) Extra structure: representation of mapping class groups, e.g. braid groups Observables in topological theories: Wilson lines ( invariants of ribbon graphs), Wilson networks (couplings of Wilson lines). Associated to stratifications of submanifolds ( defects ). Applications of distinguished subspaces Quantum codes (mapping class group quantum gates) Conformal blocks for two-dimensional (conformal) field theories (CFTs) Remark: Important CFTs (e.g. critical percolation) are based on non semi-simple categories.

4 Definition of TFT: topological field theories in two dimensions Definition (Cobordism category) Objects: closed oriented 1-manifolds. Morphisms: spans S M S with M oriented 2-manifold with boundary M = S S, up to diffeomorphism relative boundary. Monoidal product is disjoint union. Definition (Two-dimensional topological field theory) An (oriented) topological field theory is a symmetric monoidal functor tft : cob2,1 or vect. (It is a representation of cob2,1.) or Remark Topological fact: cob2,1 or is the free symmetric monoidal category on a commutative Frobenius object. Corollary: tft(s 1 ) is a commutative Frobenius algebra. Any such Frobenius algebra gives a two-dimensional TFT.

5 Extended three-dimensional topological field theories Wilson lines/ Knots and links are part of the picture: Definition (2-vector spaces) Denote by 2-vect the symmetric monoidal bicategory Objects: finitely semisimple k-linear abelian categories. 1-Morphisms: k-linear functors, 2-morphisms: k-linear natural transformations. The monoidal product is given by the Deligne product. Definition (Cobordism bicategory) Objects: closed oriented 1-manifolds. 1-Morphisms: spans S M S with M oriented 2-manifold with boundary M = S S. 2-Morphisms: 3-manifolds with corners up to diffeomorphisms The monoidal product is given by disjoint union. Definition (Extended topological field theory) A extended oriented topological field theory is a symmetric monoidal 2-functor tft : cob or 3,2,1 2-vect.

6 Evaluation of a TFT Definition (Extended topological field theory) A extended oriented topological field theory is a symmetric monoidal 2-functor tft : cob or 3,2,1 2-vect. Examples: 1-morphism 2-morphism gives tensor product : tft(s 1 ) tft(s 1 ) tft(s 1 ) gives braiding opp Braided monoidal category (precise statement later)

7 Topological field theories with defects Consider a larger symmetric monoidal bicategory cob,or 3,2,1 : Objects: 1-manifolds with marked points 1-Morphisms: 2-manifolds with boundary 2-morphisms: 3-manifolds with corner Definition (TFT with defects) An (oriented) TFT with defects is a symmetric monoidal functor tft : cob,or 3,2,1 2-vect. Goal: construct TFTs with defects! Important class of examples: The modular category tft(s 1 ) = Z(A) is the Drinfeld center of a spherical fusion category A. Remarks State-sum construction [Turaev-Viro] Includes Dijkgraaf-Witten theories (finite gauge theories)

8 Two applications of TFT with defects: quantum codes and CFT 1. Some problems with quantum codes: Realistic samples have simple topology (disc). Dimension of distinguished subspace sensitive to genus (Verlinde formula). Relevant representations of mapping class groups too small to admit no universal quantum gates. Possible way out: two-layer systems with twist defects Effectively conformal blocks at higher genus Increased dimension Representations of braid groups admitting universal gates 2. Holographic construction of CFT correlators (including ultimately non-ssi CFTs)

9 Introduction 2: motivations from representation theory General setting: categories with finiteness properties. k a field, here k = C. Definition (Finite category) A k-linear category C is finite, if 1 C has finite-dimensional spaces of morphisms. 2 Every object of C has finite length. 3 C has enough projectives. 4 There are finitely many isomorphism classes of simple objects. Remarks 1 Linear category A finite A = A-mod with A finite-dimensional k-algebra. 2 Right/left exact functors are (co-)representable Definition (Finite tensor category) A finite tensor category is a finite rigid monoidal linear category. In particular, the tensor product is exact in each argument.

10 Module categories Definition (Module categories) Let A and B be linear monoidal categories. 1 A left A-module category is a linear category M with a bilinear functor : A M M and natural isomorphisms α : ( id M) (id A ) λ : (id A ) id M satisfying obvious pentagon and triangle axioms. We write a.m := a m. 2 Right module categories are defined analogously. 3 An A-B bimodule category is a linear category D, with the structure of a left A and right D-module category and a natural associator isomorphism (a.d).b = c.(d.b). 4 Module functors, module natural transformations defined in obvious way. Definition (Finite module categories) Let A be a finite tensor category over k. A left A-module category is finite, if the underlying category is a finite abelian category over k and the action is k-linear in each variable and right exact in the first variable.

11 Relative Deligne product Definition (Balancing, relative Deligne product) A finite tensor category, M A and AN module categories. 1 A balancing constraint for a bilinear functor F : M A A N X is a natural isomorphism F (m.a n) = F (m a.n) obeying a pentagon + unit constraint. 2 The relative Deligne product is a linear category M A N and an A-balanced functor B : M A A N M A N such that for all X, there is an adjoint equivalence Rex(M A N, X ) Rex bal (M N, X ) G G B Remarks 1 The relative Deligne product over finite tensor categories exists, unique. 2 Several realizations, e.g. M A N = Rex A(M opp, N ).

12 Braidings, Drinfeld center and relative Deligne product Definition (Half-braiding, Drinfeld center) Let A be a monoidal category. A half-braiding for V A is a natural isomorphism σ V : V V such that σ V (X Y ) = (id X σ V (Y )) (σ V (X ) id Y ) for all X, Y C. The Drinfeld center Z(A) has pairs (V, σ V ) as objects. Z(A) is a braided monoidal category. There is a forgetful functor Z(A) A. Remark Other realization of relative Deligne product [FSS]: M N B Z A(M A AN ) with B right adjoint of forgetful functor.

13 Motivation 1: Brauer-Picard groups from braided equivalences 1 Let D be a A-B-bimodule category. For objects a A and b B define C-linear functors: L a : D D R b : D D d a.d d d.b 2 Let (a, σ a) Z(A). Then L a has a natural structure of a bimodule functor: L a(d).b = (a.d).b a.(d.b) = L a(d.b) a.l a(d) = a.(a.d) (a a).d σ 1 a (a a ).d = L a(a.d). 3 Monoidal functors L : Z(A) End A B (D) and R : Z(B) End A B (D). 4 Now A = B. Fact: If D invertible, i.e. D A D = A, then.id d Z(A) End A A (D) Z(A) is a braided equivalence. Fact [ENOM 2009]: Isoclasses of invertible bimodules /A braided automorphisms of Z(A) is a group isomorphism. Goal: Understand this in terms of topological field theory!

14 Motivation 2: generalized Frobenius Schur indicators Recap V a finite-dimensional irreducible C[G]-module. 3 cases: real / pseudoreal / complex with Frobenius-Schur indicator ν FS = +1/ 1/0. Bilinear form on V either symmetric or antisymmetric. Generalization for pivotal categories: V C and X Z(C): [Kashina, Sommerhäuser, Zhu; Ng, Schauenburg] Generalized Frobenius Schur indicator: ν V,X,(n,l) := trξ V,X,(n,l). Equivariance under SL(2, Z). Congruence subgroup conjecture for doubles of fusion categories FS indicators for big finite groups ( elements) Goal: Understand equivariance in terms of topological field theory!

15 Modular tensor categories Back to three-dimensional extended TFTs: Definition (Modular tensor category) A modular tensor category C is a finite ribbon category such that the braiding is maximally non-degenerate. Various formulations exist and are equivalent [Sh]: Braided equivalence C C rev = Z(C) Coend L := C U U has non-degenerate Hopf pairing ω C Map Hom(1, L) Hom(L, 1) induced by ω C is isomorphism. C has no non-trivial transparent objects. Remark Topological fact [BDS-PV]: cob3,2,1 or is free symmetric monoidal bicategory on a single anomaly-free modular object. Corollary: tft(s 1 ) is a semisimple modular tensor category. Any such tensor category gives an extended TFT.

16 Construction of oriented TFTs with defects: labelling 2-cells / surface defects Start with a closed oriented 3-manifold + oriented polytope decomposition. 3-cells: any Turaev-Viro theory assign spherical fusion category 2-cells=surface defect: Remark Assign an A i -A f -bimodule category Transparent case: if A i = A f =: A, then A as a bimodule over itself is a distinguished label. Special case: if M, then labels for boundary 2-cells are AM vect or vectm A.

17 Generalized Wilson lines Several surface defects meet in generalized Wilson lines: A decorated 1-manifold S: Decoration: 1-cells: spherical fusion categories 0-cells: bimodule categories Question: which C-linear category to associate to S?

18 Definition 1 Let A be a monoidal category, B an A-bimodule and b B. A balancing for m is a natural family (σ a : a.m m.a) a A such that (a a ).m m.(a a ) a.m.a 2 We denote by Z A(B) the category of objects with a balancing. Facts 1 Z A(A) is the Drinfeld center. 2 If A has left duals, then σ a is an isomorphism. 3 Convenient description: monad on B: Z A : B B b a A a.b.a Then Z A(B) = Z A-mod(B).

19 Categories for 1-manifolds S = Decoration: 1-cells: spherical fusion categories 0-cells: bimodule categories On the Deligne product B(S) := v S B v, we have commuting monads, i.e. distributive laws Z Ai Z Aj Z Aj Z Ai monad Z on B(S). Category for Wilson lines: Remarks C(S) := Z-mod(B(S)) 1 Transparent case: A i = A and B α = A A A, we have C = Z(A). 2 The notion contains a category valued trace and the relative Deligne product, e.g. C(S) = M 1 A1 B A2 M 2 3 Determines labels at branch points of twist defects

20 Comparison for Dijkgraaf-Witten theories Dijkgraaf-Witten theory based on a finite group G and 3-cocycle ω Z 3 (G, C ) Facts Correspond to Turaev-Viro theories based on fusion category (G vect) ω. (Indecomposable) bimodule categories are classified [O] by H G G and 2-cochain θ such that dθ = p 1 ω (p 2 ω) 1. S = tft DW (S) = [G\\G G//H, vect] τ(ω,θ) = A Gdiag -A H,θ -bimod(g-vect ω G-vect ω ) which is one realization of the category valued trace of the bimodule category.

21 Functors for surfaces: the transmission functor Extended TFT: surface with boundaries, left exact k-linear functor. Special case: to cylinder with a circular defect D = associate the transmission functor: for D A-bimodule category get functor F D : Z(A) Z(A) General principle of field theory: Invertible codimension 1 defects describe symmetries. 3d TFT of TV type: Brauer-Picard group describes symmetries TFT axioms: if D is invertible, then F D is braided. Compare two functors (F D F D) F D

22 The transmission functor for invertible defects is braided Natural isomorphism (F D F D) F D is monoidal and braided Explicit calculation for Dijkgraaf-Witten theories: D F D is the map in the description of the Brauer-Picard group

23 Functors for a general surface Assume surface Σ has only ingoing boundary components. polytope decomposition boundary circle at each vertex Goal: left exact functor ( conformal blocks ) F Σ : v Σ C(S v ) vect Idea: Theory generalizes a theory of flat connections Dynamical degrees of freedom on 1-cells (labeled by bimodule categories) Flatness condition for each 2-cell. Category B(Σ) = e Σ B # e B e = v Σ B(S v ) Spherical tensor categories associated to 2-cells give a monad on B(Σ). First step: Preblocks from state sum Insight: state sum is a coend v Σ C(S v ) vect U v Σ B(S v ) n BΣ Hom(,b e b e )

24 An example for flatness condition Disc with one insertion on the boundary Functor calculus Rex(N, M) = N opp M F n N F (n) m n m Hom N (, n) m Insertions labeled by F = Hom N (, n 1) m 1 and G = Hom N (, n 2) m 2 State sum: n N m M Hom(n 1 m 1 n 2 m 2, n m n m) = Hom N (n 2, n 1) Hom M(m 1, m 2) On the other hand Nat(F, G) = Hom(F (n), G(n)) n N = Hom(n, n N n2) Hom(n, n 1) Hom M(m 1, m 2) Thus: Preblocks are Nat(F, G)

25 Imposing the flatness condition Preblocks are End C(F, G). Blocks are defined as equalizers. End C(id M, F ) Hom(id M, Z(F )) Theorem The equalizer are module natural transformations id M F. In the absence of defects, one gets the blocks of standard TV theory for genus 0. For the cylinder with a defect along the non-contractible cycle, get the same functor as [ENOM].

26 Application to the Generalized Frobenius-Schur indicator Generalized Frobenius Schur indicator: ν V,X,(n,l) := trξ V,X,(n,l). Equivariance under SL(2, Z). To get objects both in A and in Z(A), consider a disc: Spherical fusion category A for surface Boundary labelled by A as a module category over itself n boundary insertion labeled by V A Bulk insertion X Z(A) Hom(V n, X ) Solid torus with Wilson line ν V,X,(n,l)

27 Outlook Factorization for conformal blocks. (Thus same conformal blocks as standard TV theory for any genus.) Extension to (classes of) 3-manifolds with corners. (Cf. monoidal structure on transmission functor) Existence of a surgery approach, generalizing Reshetikhin-Turaev? Applications, in particular to logarithmic conformal field theory.