Remarks on Chern-Simons Theory. Dan Freed University of Texas at Austin

Transcription

1 Remarks on Chern-Simons Theory Dan Freed University of Texas at Austin 1

2 MSRI:

3 Classical Chern-Simons 3

4 Quantum Chern-Simons Witten (1989): Integrate over space of connections obtain a topological invariant of a closed oriented 3-manifold X G = SU(n) P = X G A Ω 1 (X; g) S(A) = X A da 1 6 A A A, basic inner product F k (X) = F X e iks(a) da, k Z F X is the space (stack) of connections on X Warning: This path integral is only a motivating heuristic 4

5 F k (X) = F X e iks(a) da, k Z First sign of trouble: F(X) depends on orientation + another topological structure (2-framing, p 1 -structure) Extend to compact manifolds with boundary: path integral with boundary conditions on the fields (connections) Obtain invariants of knots and links: Jones polynomial HOMFLYPT = Hoste, Ocneanu, Millett, Freyd, Lickorish, Yetter, Pzytycki, Traczyk 5

6 F k (X) = F X e iks(a) da, k Z Question: How can we make mathematics out of the path integral heuristic? Focus on topological case. Plan: Axiomatization: define a topological quantum field theory (TQFT) Constructions: generators and relations vs. a priori State a theorem inspired by this physics General observations about geometry-physics interaction 6

7 Path Integrals: Formal Structure Fix a dimension n 1 F X S X : F R X fields on an n-manifold X action functional X n Y n 1 (Y F X p in p out F Y F Y Need measures ) n 1 to define X = Y Y L 2 (F Y ) (p out ) e is X (p in ) L 2 (F Y ) linearization of correspondence diagram 7

8 Path Integrals X n L 2 (F Y ) (p out ) e is X (p in ) L 2 (F Y ) Y n 1 (Y ) n 1 Closed manifold of dimension F(-) n complex # n-1 Hilbert space 8

9 Path Integrals: Multiplicativity Disjoint union: F X X F X F X S X X (φ φ )=S X (φ)+s X (φ ) Gluing bordisms: Cartesian square F X X X X F X F X Y Y Y Y Y Y F Y F Y F Y Need measures which are consistent under gluing to define the various pushforward maps (path integrals) 9

10 Axiomatization: Definition Witten, Segal, Atiyah,... Bord n Vect C bordism category of compact n-manifolds category of finite dim l complex vector spaces Definition: A TQFT is a monoidal functor F : Bord n Vect C Closed manifold of dimension F(-) n element of C n-1 C-module 10

11 Structure: n=2 on oriented manifolds A C A A A A C C A A A Frobenius algebra: associative (commutative) algebra with identity and nondegenerate trace 11

12 Constructions: Generators & Relations Theorem: There is an equivalence 2d TQFTs commutative Frobenius algebras This is a folk theorem: Dijkgraaf, Abrams,... Bordism category of 2-manifolds generated by elementary bordisms: 3-manifolds are more complicated... 12

13 Extended TQFT Extend in two ways: families of manifolds manifolds of lower dimension Before: 1-2 theory. Now: theory, theory, etc. Closed manifold of dimension n F(-) element of C n-1 C-module n-2 C-linear category 13

14 Extended TQFT Extend in two ways: families of manifolds manifolds of lower dimension Before: 1-2 theory. Now: theory, theory, etc. Closed manifold of dimension F(-) Category # n element of C -1 n-1 C-module 0 n-2 C-linear category 1 14

15 Extended TQFT Extend in two ways: families of manifolds manifolds of lower dimension Before: 1-2 theory. Now: theory, theory, etc. Closed manifold of dimension F(-) Category # n element of C -1 n-1 C-module 0 n-2 C-linear category 1 n-3 linear 2-category 2 15

16 Definition: The category number of a mathematician is the largest integer n such that he/she can ponder n-categories for a half hour without developing a migraine. 16

17 Extended TQFT Chern-Simons: path integral (heuristic) gives a 2-3 theory extension to theory? extension to theory? 17

18 Chern-Simons as a Theory 1-2 TQFTs commutative Frobenius algebras Theorem: There is an equivalence TQFTs modular tensor categories This is due to Reshetikhin-Turaev (related work: Moore- Seiberg, Kontsevich, Walker,...) The modular tensor category is constructed from a quantum group or loop group and is F (S 1 ) Bordism category: oriented manifolds with p 1 -structure Unitary TQFTs unitary modular tensor categories These are semisimple theories, somewhat special The Chern-Simons invariant is nowhere in sight 18

19 Extended TQFT Theories which go down to a point (0-1, 0-1-2, , etc.) are the most local so should have a simple structure Much current work on theories: Elliptic cohomology (Stolz-Teichner) Structure (generator and relations) theorems (Moore-Segal; Costello; Lurie-Hopkins) Applies to string topology (Chas-Sullivan) 19

20 Chern-Simons as a Theory What is F (point)? Should be a 2-category. Try to realize as 2-category of modules over a monoidal 1-category Unknown for general theories, but will describe for case when G is a finite group. Starting data is λ H 4 (BG; Z) Deriving F (S 1 ) from F (point): The Drinfeld Center Z(C) of a monoidal category C is the braided monoidal category whose objects are pairs (A, θ) of objects A in C and natural isomorphisms θ : A A. 20

21 Reduction of CS to a 1-2 Theory Dimensional reduction: F (M) =F (S 1 M) Closed manifold of dimension F(-) F (-) 3 element of C 2 C-module element of Z 1 C-linear category Z-module Note integrality of dimensional reduction: Frobenius ring Some thought about this transition suggests K-theory (refinement of Hochshild homology) 21

22 Loop groups and K-Theory Joint work with Michael Hopkins and Constantin Teleman Closed manifold of dimension F(-) F (-) 2 C-module element of Z 1 C-linear category Z-module Positive energy representations of loop groups Twisted equivariant K-theory of G Theorem (FHT): There is an isomorphism of rings Dirac family Φ : R τ σ (LG) K τ G(G) 22

23 A Priori Construction of F X Y Y Path integral: F X p in p out F Y F Y F : infinite dimensional stack of G-connections L 2 (F Y ) (p out ) e is X (p in ) L 2 (F Y ) Need consistent measures to define path integral Topology: p in M X p out K(M Y ) (p out ) (p in ) K(M Y ) M Y M Y M : finite dimensional stack of flat G-connections Need consistent orientations to define pushforward 23

24 A Priori Construction of F M S 1 = G /G K(G /G) = K G (G) p in M X p out M Y M Y K(M Y ) (p out ) (p in ) K(M Y ) Theorem (FHT): There exist universal, hence consistent, orientations. Summary: Generators and relations construction of Chern-Simons theory (quantum groups) A priori construction of the 1-2 dimensional reduction of Chern-Simons (twisted K-theory) Classical Chern-Simons invariant has disappeared from the discussion 24

25 Chern-Simons in Quantum Physics Among the many possibilities we mention: Large N duality in string theory relates quantum Chern-Simons invariants to Gromov-Witten invariants (Gopakumar-Vafa) Quantum Chern-Simons theory and other theories are at the heart of proposed topological quantum computers (Freedman et. al.) 25

26 Geometry/QFT-Strings Interaction More spectacular impact on geometry/topology elsewhere: 4d gauge theory (4-manifolds, geometric Langlands) and 2d conformal field theory (mirror symmetry, etc.) New links in math (here reps of loop groups and K-theory) Deep mathematics, bidirectional influence, big success! But...little beyond formal aspects of QFT (and string theory) has been absorbed into geometry and topology Over past 25 years good understanding of scale-invariant part of the physics: topological and conformal Perhaps more focus needed now on geometric aspects of scale-dependence in the physics 26

27 How Little We Know Our axiomatization does not capture a basic feature of the path integral: stationary phase approximation F k (X) = F X e iks(a) da, k Z The discrete parameter k is, where is Planck s constant 1/ The semi-classical limit 0 can be computed in terms of classical topological invariants. So we obtain a prediction for the asymptotic behavior of F k (X) as k 27

28 How Little We Know F k (X) 1 2 e 3πi/4 e i(k+2)s X (A) e (2πi/4)I X (A) τ X (A) A M o X flat connections M o X S X (A) I X (A) τ X (A) irreducible flat connections, assumed isolated classical Chern-Simons invariant spectral flow (Atiyah-Patodi-Singer) Reidemeister torsion LHS: loop groups or quantum groups RHS: topological invariants of flat connections 28

29 How Little We Know F k (X) 1 2 e 3πi/4 e i(k+2)s X (A) e (2πi/4)I X (A) τ X (A) A M o X flat connections 29