Geometry of Conformal Field Theory

Transcription

1 Geometry of Conformal Field Theory Yoshitake HASHIMOTO (Tokyo City University) 2010/07/10 (Sat.) AKB Differential Geometry Seminar Based on a joint work with A. Tsuchiya (IPMU)

2 Contents 0. Introduction 1. Harish-Chandra pairs and D-modules 2. Families of stable curves 3. Vertex algebras and chiral algebras 1

3 0. Introduction Conformal Field Theory Representation thoery of -dim. Lie alg. affine Lie alg., Virasoro alg. + Moduli of Riemann surfaces, Moduli of G-bundles over Riemann surfaces 2

4 CFT and topology Tsuchiya-Kanie (1988): Jones representation Kohno, Drinfeld: quantum groups Witten: Chern-Simons gauge theory Categorification of Tsuchiya-Kanie? Jones-Witten theory = Khovanov homology 3

5 CFT and 4-dim. gauge theory Nakajima (1990 s) Instanton moduli and affine Lie alg. Alday-Gaiotto-Tachikawa (2009) Instanton moduli and Virasoro alg. 4

6 Geometric Langlands correspondence Non-abelian Class Field Theory Wakimoto modules, screening operators Kapustin-Witten (2006) 4-dim. gauge theory 5

7 References 1. E. Frenkel and D. Ben-Zvi, Vertex Algebras and Algebraic Curves, 2nd. ed., AMS, K. Ueno, Conformal Field Theory with Gauge Symmetry, AMS,

8 1. Harish-Chandra pairs and D-modules Representation theory of Lie algebras Representations of a Lie algebra g = Lie(G) induced by those of the Lie group G can be understood geometrically. More generally, we study representations of g whose restrictions on Lie(K) g are induced by those of K. 7

9 Harish-Chandra pair (g, K) Example G: Lie group, g = Lie(G), K G Def. If K: Lie group, g: Lie algebra with a K-action with K-equivariant embedding Lie(K) g, (g, K) is called Harish-Chandra pair. 8

10 (g, K)-modules Representation theory of HC pairs Def. V : (g, K)-module (1) V : K-module (2) g End(V ): K-equivariant To study (g, K)-modules geometrically, we needs the language of sheaf 9

11 Structure sheaf X: complex manifold Example U X open set O X (U) = {holomorphic functions on U} O X : U O X (U) structure sheaf of X 10

12 Definition of sheaf Def. F : U F(U) sheaf on X ρ 12 = ρ U1 U 2 : F(U 2 ) F(U 1 ), ρ 13 = ρ 12 ρ 23 F( U α ) F(U α ) F(U α U β ) exact F(U) set, abelian group, C-vector space, C-algebra,... F(U) = Γ(U, F) F x = lim U x F(U) stalk at x X 11

13 Examples of sheaves (1) V : C-vector space V X : connected U V constant sheaf (2) F : O X -module F(U): O X (U)-module + compatibility (3) Θ X (U) = {holomorphic vector fields on U} (4) Ω q X (U) = {holomorphic q-forms on U} 12

14 Sheaf D X D X (U) End C (O X (U)) the C-algebra generated by O X (U), Θ X (U) O X left D X -module K X = Ω dim(x) X canonical sheaf, right D X -module L fξ (ω) = L ξ (fω) (f O X, ξ Θ X, ω K X ) 13

15 Vector bundles with flat connections (E, ): vector bundle with a flat connection over X E: sheaf of local sections of E Θ X ξ ( ξ : E E), [ξ, η] = [ ξ, η ] = E: left D X -module M: left D X -module, F p M F p+1 M, ξ (F p M) F p+1 (M) 14

16 Harish-Chandra pair (g, K) (over C) K: Lie group, g: Lie algebra with a K-action, Lie(K) g: K-equivariant embedding Def. V : (g, K)-module V : K-module, g End C (V ): K-equivariant 15

17 (g, K)-action Z: complex manifold with a K-action Lie(K) g Θ a Z K-equivariant, compatible For example, Z is a K-invariant open subset of a G-space. Setting π : Z S principal K-bundle a : O Z g Θ Z surjective 16

18 Localization functor : (g, K)-mod D S -mod (V ) = (π (D Z Ug V )) K the left adjoint of Γ : D S -mod (g, K)-mod Γ(M) = Γ(Z, π M) Hom DS ( (V ), M) = Hom g (V, Γ(M)) 17

19 Terminology (1) a : g Θ Z induces Ug D Z (2) π : Z S, F: sheaf on Z, M: sheaf on S, (π F)(U) = F(π 1 (U)) direct image (π 1 M) z = M π(z) inverse image π M = π 1 M π 1 O S O Z 18

20 Coinvariants V : (g, K)-module a : O Z g Θ Z surjective V coinv = O Z V/Ker(a) O Z V V coinv = DZ Ug V D Z -module 19

21 CFT case (1) O = C[[z]], K = C((z)) = O[z 1 ] Der(K) = C((z)) z, Der 0 (O) = zc[[z]] z Aut(O) = {z a 1 z + a 2 z 2 + a 1 0} (Der(K), Aut(O)): Harish-Chandra pair C: compact Riemann surface, p C H 1 (C, Θ C ) = Θ C (C {p})\der(k p )/Der 0 (O p ) 20

22 CFT case (2) G: reductive algebraic group, g = Lie(G) (g(k), G(O)): Harish-Chandra pair P C: principal G(C)-bundle H 1 (C, g P ) = Γ(C {p}, g P )\g(k p )/g(o p ) 21

23 CFT case D S -module structure on (V ) Knizhnik-Zamoldchikov connection 22

24 2. Families of stable curves Stable curves Let g, n be non-negative integers such that 2g 2 + n > 0, and I = {1, 2,..., n}. An n-pointed, or I-pointed stable curve of genus g over a scheme B is a proper flat morphism π : C B together with n sections s I = (s i : B C) i I such that 23

25 (i) The geometric fibers C b of π at b B are reduced and connected curves with at most ordinary double points. (ii) C b is smooth at s i (b). (iii) s i (b) s j (b) for i j. (iv) Each non-singular rational component of C b has at least 3 points which are sections or intersections with other components. (v) dim H 1 (C b, O Cb ) = g. 24

26 Stabilization Let I = {1, 2,..., n}, I + = I {n + 1}. An I-pointed stable curve X = (π : C B, s I = (s i ) i I ) and another section s n+1 : B C are given. A pair (X +, f) of an I + - pointed stable curve X + = (π + : C + B, s + I + = (s + i ) i I + ) and a morphism f : C + C over B is called a stabilization if 25

27 (i) fs + i = s i (i I + ) (ii) There are two possible cases for a geometric fiber C b + : a) If C b is smooth at s n+1 (b) and s n+1 (b) s i (b) (i I), f b : C b + C b is an isomorphism. b) If not, there is a rational component E of C b + such that s + n+1 (b) E and f b(e) = {s n+1 (b)} and f b : C b + E C b {s n+1 (b)} is an isomorphism. 26

28 Remark The stabilization (X +, f) is unique up to isomorphisms for X, s n+1. 27

29 Tower For an I-pointed stable curve X = (π : C B, s I = (s i ) i I ), the first projection on the fiber product X 2 = (p 1 : C B C C, π s I = (π s i ) i I ) is an I-pointed stable curve over C. 28

30 We define an I + -pointed stable curve X (2) = (π (2) : C (2) C, s (2) I + ) over C as the stabilization of X 2 for the diagonal section : C C B C. Similarly we define (n + k 1)-pointed stable curve X (k) = (π (k) : C (k) C (k 1), s (k) I (k)) over C (k 1) as X (k) = (X (k 1) ) (2). 29

31 Regular family An I-pointed stable curve (π : C B, s I ) is a regular family if (1) C, B: non-singular (2) The image of π of singular points in the fibers is a set of normal-crossing divisors in B. 30

32 Proposition For a regular family (π : C B, s I ), the variety C (2) is also non-singular. 31

33 Deformation theory Let (π : C B, s I = (s i ) i I ) be a regular family. We put S I = i I s i(b), π Ω B = π 1 Ω B OB O C, F = Hom OC (, O C ( S I )). We apply a left exact functor π F to the short exact sequence 0 π Ω B Ω C Ω C/B 0. 32

34 Since π is proper, π F (π Ω B ) = π Hom OC (π Ω B, O C ( S I )) = Hom OB (Ω B, O B ) = Θ B. We obtain an exact sequence 0 π Θ C/B ( S I ) π Θ C ( S I ) Θ B ρb R 1 (π F )Ω C/B R 1 (π F )Ω C 0. The map ρ B is called a Kodaira-Spencer map. 33

35 3. Vertex algebras and chiral algebras Taylor expansions O z1,..., z n = C[[z 1,..., z n ]] P z1, z 2 = (z 1 z 2 ) 1 O z2 [(z 1 z 2 ) 1 ] We have a decomposition of O z2 -modules, O z1, z 2 [(z 1 z 2 ) 1 ] = P z1, z 2 O z1, z 2. 34

36 The first projection µ 0 : O z1, z 2 [(z 1 z 2 ) 1 ] P z1, z 2 is given by µ 0 ( f(z1 )g(z 2 ) (z 1 z 2 ) n ) for n N, f(z), g(z) O z. = n 1 k=0 1 k! k z 2 f(z 2 ) g(z 2 ) (z 1 z 2 ) n k P z1, z 2 = Oz1, z 2 [(z 1 z 2 ) 1 ]/O z1, z 2 is an O z1, z 2 -module. 35

37 Vertex algebra A vertex algebra (V, Y, 0 ) is a collection of data: a complex vector space V a linear map Y : V Hom(V, V ((z))) written as Y (a, z) = n Z a (n) z n 1 (a V, a (n) = Res z=0 (Y (a, z)z n dz) End(V )) a vector 0 V (called vacuum vector) satisfying the following conditions: 36

38 (Locality) Y (a, z), Y (b, w) are mutually local for any a, b V. In other words, there exists a linear map Y 2 : V V Hom(V, V [[z, w]][z 1, w 1 ])[(z w) 1 ] such that Y (a, z)y (b, w) = ε Y 2 (a, b; z, w), ε (z w) 1 = w n z n 1, n 0 Y (b, w)y (a, z) = ε 0 Y 2 (a, b; z, w), ε 0 (z w) 1 = w n z n 1 n<0 37

39 (Vacuum) Y (a, z) 0 V [[z]], Y (a, z) 0 z=0 = a (a V ). In other words, a ( 1) = a, a (n) 0 = 0 (n 0). (Translation) There exists a linear map T : V V such that T 0 = 0, [T, Y (a, z)] = z Y (a, z) (a V ). 38

40 D-module Θ z1,..., z n = n i=1 O z1,..., z n zi D z1,..., z n the C-algebra generated by O z1,..., z n, Θ z1,..., z n. The ring O z1,..., z n is a left D z1,..., z n -module. The projection µ 0 : O z1, z 2 [(z 1 z 2 ) 1 ] P z1, z 2 is a homomorphisms of left D z1, z 2 -modules. 39

41 The Lie algebra Θ z1,..., z n acts on ω z1,..., z n = O z1,..., z n dz 1 dz n by Lie derivative. It makes ω z1,..., z n a right D z1,..., z n -module. The map µ R 0 : O z 1, z 2 [(z 1 z 2 ) 1 ]dz 1 dz 2 P z1, z 2 dz 1 dz 2 induced by µ 0 is a homomorphism of right D z1, z 2 -modules. 40

42 Jacobi identity O = O z1, z 2, z 2 = C[[z 1, z 2, z 3 ]][z 1 1, z 1 2, z 1 3 ] t ij = (z i z j ) 1. Since t ij t jk = t ij t ik + t ik t jk for distinct i, j, k, we deduce O[t ij, t ik, t jk ] = O[t ij, t ik ] + O[t ik, t jk ]. 41

43 Hence the natural maps O[t ij, t ik ] O[t ik ] O[t ij, t ik, t jk ] O[t ik, t jk ] are isomorphisms. Let O[t ij, t ik ] O[t ij ] + O[t ik ] O[t ij, t ik, t jk ] O[t ik, t jk ] + O[t ij ] µ 1[23] : O[t 12, t 13, t 23 ] O[t 12, t 13, t 23 ] O[t 12, t 13 ] + O[t 23 ] be the projection. 42

44 µ [12]3, µ 2[13] : O[t 12, t 13, t 23 ] O[t 12, t 13, t 23 ] O[t 12, t 13 ] + O[t 23 ] are the compositions O[t 12, t 13, t 23 ] O[t 12, t 13, t 23 ] O[t 13, t 23 ] + O[t 12 ] O[t 12, t 23 ] O[t 12 ] + O[t 23 ] O[t 12, t 13, t 23 ] O[t 12, t 13, t 23 ] O[t 12, t 23 ] + O[t 13 ] O[t 13, t 23 ] O[t 13 ] + O[t 23 ] O[t 12, t 13, t 23 ] O[t 12, t 13 ] + O[t 23 ], O[t 12, t 13, t 23 ] O[t 12, t 13 ] + O[t 23 ] 43

45 µ 1[23] = µ [12]3, µ 2[13] = 0 on O[t 12, t 23 ], µ 1[23] = µ 2[13], µ [12]3 = 0 on O[t 13, t 23 ]. Therefore µ 1[23] = µ [12]3 + µ 2[13]. 44

46 Chiral algebra C: compact Riemann surface A: right D C -module A A( )! A: D C -module hom unit: Ω C A compatible with µ Ω skew-symmetry: µ = σ 12 µ σ 12 Jacobi identity: µ 1[23] = µ [12]3 + µ 2[13] 45