Introduc)on to Topological and Conformal Field Theory

Transcription

1 Introduc)on to Topological and Conformal Field Theory Robbert Dijkgraaf Ins$tute for Advanced Study Progress in Theore4cal Physics 2015 New Insights Into Quantum Ma;er

2 Topological Field Theory Describes the (approximate) ground states of quantum mader. Skeleton of more physical quantum field theories such as 2d CFT. Exact results for sectors of (susy) QFT. Toy model to study geometrical proper4es of QFT. Connects to deep mathema4cs.

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4 Defining Quantum Field Theory Algebra x 1 O 1 (x 1 )...O n (x n ) Geometry x n local operators sca;ering amplitudes cut & paste topological indices defect operators

5 QFT in dimension n + 1 Closed space manifold of dim n: Hilbert space of states! X n!h X!X Space of wave func4ons Examples! Ψ[ϕ X ] on field space! X =!n, T n, S n,! n 1 S 1,

6 Hilbert Space Hilbert space of states sa4sfy certain axioms H =! * H X = H X H! X Y = H X H Y!X!Y

7 QFT in dimension n + 1 Space- 4me manifold of dim n+1: evolu4on operator! Φ M :!H X H Y + orienta4on - orienta4on Composi4on law (curng and gluing)! Φ N!M = Φ N!Φ M

8 Standard space 4me Hamiltonian picture! M = X [0,t]! t!x! Φ t = e th :!H X H X! Φ t 1 +!t 2 = Φ t1 Φ t2

9 Par))on func)ons Closed manifold Boundary states! Φ M :!!! Φ M :! H Y Φ M! Φ! M = dϕ!e S[ϕ ] Φ M = M H Y! Φ M [ϕ Y ]= ϕ Y dϕ!e S! Φ M = M 2 M 1

10 Trace (par4al) trace! Tr H X Φ M Par44on func4on! Φ M = Tr!e th!m = X S 1

11 Hilbert space of a point Par44on func4on Quantum Mechanics! Φ t = e th :H!H!S 1! Φ S 1 = Tr!e th

12 Supersymmetric QM Maps!x µ (t)!coordinates!on!σ!+!fermions Wave func4ons = differen4al forms Ψ = α µ1 µ k (x)dx µ 1 dx µ k H Σ = Ω * (Σ) H = Δ= dd + d d * * ( ) Ground states = harmonic forms, topological QM Harm * (Σ) H * (Σ) Par44on func4on = WiDen index Tr ( 1) F = Tr (( 1) deg e th ) = Euler(Σ)

13 Topological Field Theory in 1+1 Hilbert space of circle: finite dimensional Special states! S1!H,!!basis!!φ i, i = 1,,N! φ 0 = 0 = 1 H! 0 =! 0 H *

14 Topological Field Theory in 1+1 Bilinear form η : H H! ( ) η! ij = η φ i,φ j Non- degenerate, inverse! η 1 = Pair of pants: mul4plica4on c :H H H φ! i φ j = c k ij φ k

15 Algebra of states Hilbert space = commuta4ve, associa4ve algebra α β α β! c k k = c ij ji ( α β ) γ = α ( β γ )! c n c l ij nk = c n l jk c ni

16 Example: Z N Gauge Field Holonomy around circle! φ k = e2πik/n, k = 0,1,,N 1! φ k φ l! φ k+l

17 Frobenius Algebras Unit! 1 H,!!!1 α = α Frobenius algebra! c ijk = c ij n η nk = c jki = ( ) = η( α,β γ ) η α β,γ ( ) = α β 0 η α,β! If semi- simple (no idempotents ), basis!α n = 0! e i! η ( e,e ) = δ i j ij, e i e j = λ i δ ij e i

18 Par))on Func)ons Compute par44on func4on in genus g Φ g =! Par44on func4on in genus 1!i!i i! λ i λ i 2g 2! S1 S 1 : Φ 1 = Tr!1!=!dim!H!i

19 2d Landau- Ginzburg Models 2d sigma model with superpoten4al W(x,,x )! 1 n!!h =![x,,x ]/(dw ) 1 n W = 0 x! i semi- simple = massive model = non- deg cri4cal pts

20 Supersymmetric Sigma Model ϕ : X Σ, + fermions Ground states = cohomology classes α,β H * ( X ) α β α β Not semi- simple. For example, ground states for CP N are 1,x,x 2,,x N 1, x N = 1

21 Classical Intersec)on Product dual cycles α β γ = Σ = α β γ Σ

22 Quantum Cohomology Add two- dimensional instantons of degree d rat curves degree d = e dt Σ for CP N x N +1 = 0 x N +1 = e t

23 Boundaries: Extended TFT Consider spaces with boundaries, interval I!a!b I H a,b, a,b C! C = category of boundary condi4ons!b!b!c!a!a!c! H ab H bc H ac!a!b

24 Boundaries: Extended TFT No longer commuta4ve!a!b!c!b!c!a But s4ll associa4ve!b!a!c!d

25 Extended TFT in n + 1 dim Codim 0. closed space- 4me manifold: par44on func4on! M n+1!φ M!!M Codim 1. space- like boundary: Hilbert space! X = M! X n!h X!Vect Codim 2. corners: category of boundary condi4ons! B = X Codim 3...! Bn 1!C B!Cat

26 Space closed, genus g TFT in dim! X g H g Boundaries! a 1,,a n C S 1! a 1! a n! X H ( a,,a ) g,n g 1 n! a 2

27 Ac4on of diffeomorphisms TFT in dim Representa4on on the Hilbert space ϕ : X X, ϕ Diff (X)!! ˆϕ :H X H X Torus: ac4on of! SL(2,!)! T = , S =

28 !T!S

29 2D Conformal Field Theory Conformal invariance! g µν eλ g µν Traceless stress tensor! T = 0 µ Only sensi4ve to complex structure! g µν dx µ dx ν = e λ dz 2,!!z = x + iy Chiral stress tensors: Virasoro algebra T(z)= T zz, T(z )= T zz, T = T = 0!! L,L m n T(z)= L n z n 2,! n ( )δ m, n = (n m)l + c n+m 12 m3 m

30 Category point of view Hilbert space of circle! S1!H infinite dimensional representa4on! Vir Vir Diff (S1 ) Riemann surface of genus g with n = p + q punctures! Φ :H p!h q M Finite number 3g 3 + n of moduli

31 Cylinder, plane, sphere cylinder! w = τ + iσ,!!!σ [0,2π ] τ τ plane Riemann sphere!z = e w!z = 0! z!!z =

32 Hilbert space state! i H Operators and States! i! i Local operator!z = e w! O i (0) infinite long cylinder! lim O z 0 i (z) 0 = i

33 Vacuum Vacuum State! 0 H Vacuum amplitudes = correla4ons on the sphere! 0! 0! 0 O i 1 (z 1 )!O in (z n ) 0! 0 0 = Φ S 2

34 Ac4on! S = 1 4π Free boson ϕ ϕ!d 2 z Canonical quan4za4on ϕ = 0,! ϕ(z)= α n z n 1,! ϕ(z)= α n z n 1, [α! n,α m ]= nδ n, m n n Vacuum state Fock space! α n 0 = α n 0 = 0, n 0! F 0 :!!!α n 1!α ns 0

35 Chiral algebra Stress tensor! T(z)= 1 ( 2 ϕ ) 2 Chiral current! J(z)= ϕ(z) Representa4ons: charge/momentum states α 0 p = p p, α 0 = 1! 2π Vertex operators! p = V p (0) 0, V p = eipϕ! J(z)dz! V p Fock space F :!α!α! p n ns p,!!!!!!h =!dp!f 1 p F p

36 Correlators on the sphere! V p 1 (z 1 )!!!!!!!!!!!!!!!!V pn (z n ) p i = 0! i! V p1 (z 1 )!V pn (z n ) = i,j ( z i z ) p i p j j Charge conserva4on! p 1! p 1! p 2! p 1 + p 2! p 2! p 1 + p 2

37 Homology basis Higher genus surfaces Fix loop momenta 1 2π ϕ A! I Holomorphic factoriza4on (almost)! = p I,!!!!I = 1,, g! Φ d g p Ψ ( p τ ) 2

38 Higher genus surfaces! p 1! p 2! p 1! p 2! p 1! p 2! p 1 p 2

39 Compac)fica)on Compact scalar! ϕ ϕ +2πR Quan4za4on of momentum!p = n R Winding number 1!! 2π dϕ = mr Leg/right momenta ϕ = q+ p log z + p log z +! L R p L = n! R + mr,!!p = n R R mr

40 Ra)onal Conformal Field Theory If!p R = n! R mr = 0,!!R2 = n m Let s assume! (n,m)= (k,1),!!!r2 = k, Chiral vertex operator V (z)= ei kϕ! k,!!!d = k Finite number of representa4on of extended chiral algebra! V n = einϕ/ k,!!!n = 0,1,,k 1!!! k!structure!!!!v i!v j =!V i+ j i!mod!k!i! j! i j

41 Sigma model on a group manifold g(z,z ):M G S = k! 8π WZW Model Tr ( g 1 gg 1 g)+2πks WZ Representa4on of Kac- Moody algebra at level k! J (z),!!!!a = 1,,dimG a Finite number of integrable representa4ons. For example! SU(2) k :!!!spin!j = 0, 1 2,,k 2

42 Modular Tensor Category Hilbert space decomposi4on Fusion product! H = i,j R i R j!i! j!k! N k ij number of invariant tensors Verlinde algebra Defines a 2d TFT φ i φ j =! k N ij k φ k

43 Holomorphic blocks Hilbert space decomposi4on!i! j!k!i! j!m # blocks =! p!k par44on func4on of 2d TFT!n

44 Path integral Chern- Simons Local ac4on, compact gauge group G S = k 4π Tr A da+ 2 A A A 3,!!!k!! Field equa4on δ S! δ A = k 2π F = 0! Φ = M DA!eiS A/G Large gauge transforma4ons! S S + n 2πk

45 Gauge choice, constraint Quan)za)on! A 0 = 0,!!!!F ij = 0 Configura4ons space!x! time Polariza4on! C = { flat!connections }/G! S = k 4π A d2 xdt!!ε ij!tr A j i t δ A i =!ε ij,!!!! = 2π δ A j k

46 U(1) Chern- Simons Ac4on for U(1) gauge group Quan4za4on on! X = T 2! S = k 4π A da,!!!k!!b Holonomies! p =! A, q =! A, 0 p,q 2π a b Canonical quan4za4on [p,q]= i! = 2πi! k,!!!!!p = i! d dq!a

47 But as q is periodic U(1) Chern- Simons p = n! = 2πn! k But p itself is periodic mod!2π Finite number of states p n = 2πn n,!!!!!!n = 0,1,,k 1 k H! T 2 =! k