functions Anton Nedelin String Theory and Quantum Gravity, Ascona 6 July 2017 based on works with M. Zabzine and F.Nieri and 1605.

Transcription

1 3d functon Unverstà degl stud d Mlano-Bcocca and INFN, sezone d Mlano-Bcocca Strng Theory and Quantum Gravty, Ascona 6 July 2017 based on works wth M. Zabzne and F.Ner and

2 Content 3d functon n hermtan matrx model from physcs representaton algebra Free feld representaton 3d generatng 2

3 Ward denttes 3d functon Hermtan matrx model wth Djkgraaf, Verlnde, Verlnde 91 most general polynomal potental Morozov, Mronov 91 N Z N ({t}) = d x (x x j ) 2 t k x k k 0 e Invarant under Ward Identtes:,j =1 <j x x + ɛ n x n+1 n k=0 x k x n k j + k 0 kt k x k+n = 0, n > 0 : ˆLn Z N ({t}) = 0 for n > 0 ˆL n = n k=0 2 t k t n k + kt k k=0 t k+n [ˆLn, ˆL m ] = (n m) ˆL n+m Clue to the ntegrablty of the matrx model! Morozov 93, 95 3

4 3d functon Path ntegral of SUSY gauge theory Z = Dφ e S(φ) Supersymmetrc localzaton Pestun, 07 Matrx ntegral Z = d N x Z 1 loop ({x}) e 1 g 2 Rch varety of new matrx models comng from varous gauge theores (dfferent multplet content, space-tme dmensons, manfolds etc.) x 2 Can we generalze constructon to the localzaton nspred matrx models? 4

5 Generatng 3d functon Typcal localzaton matrx model Z = d N x Z 1 loop ({x}) e 1 g 2 1-loop determnant f (x x j ) j Consder Wlson loop: ( W R = d N x s R {e 2πx } ) Z 1 loop ({x}) e 1 Construct WL generatng functon: x 2 Yang-Mlls couplng or CS level. Gaussan potental. We need general potental k t k x k! g 2 x 2, Z ({t}) R s R (x) - Schur pol. s R ({ˆt}) W R Z ({t}) = d N x Z 1 loop ({x}) e 1 g 2 x 2 + t k e 2πkx k>0 t n = k ˆt n k n For g 2 (k CS = 0 n 3d) t s exactly what we need! 5

6 Ward Identtes 3d functon How to generalze constructon? Frst way: Choose dfferent bass of transformatons x f ({x}) Example: A.N., M.Zabzne, 15 Matrx model vector multplet on S 3 S 1 N dz Z N ({t}) = ( ) ( ) ts z θ z zj s! z s Wlson loop z j ; q θ z ; q es=0 =1 <j on S 1 fber ( Transformatons generated by the operator D q x n+1... ) Constrants: Tn q Z N ({t}) = 0 Tn q2 Z N ({t}) = 0 [ T q n, Tm q ] ) ) = q n m ([n] q [m] q ([2] q T q2 n+m T q n+m Proper choce of bass: =Dfferental Constrants Hard (a lot of guessng and playng wth expressons) 6

7 free bosons 3d functon Hermtan matrx model N Z N ({t}) = d x (x x j ) 2 e =1 <j Hesenberg algebra representaton a n n t n, a n 2 t n, 0 1 k 0 Crucal property of screenng currents: [ˆLn, S(w)] State n free bosons theory t k x k d N x S(x ) 0 Screenng current S(x) : e n 0 x n a n n : e Q x P = d ] dw [ˆLn O(w), dw S(w) = 0 operators: ˆLn = 1 : a 4 n k a k : n Hence we obtan! k Z k=0 2 + t k t n k k=0 ˆL n Z N ({t}) = 0 n > 0 kt k t k+n 7

8 algebra 3d functon Frst natural generalzaton: q-deformed Shrash, Kubo, Awata, Odake 95 [T n, T m] = l f l(t n l T m+l T m l T n+l ) (1 q)(1 t 1 ) (p n p n )δ (1 p) n+m,0 Deformaton s parametrzed by q, t C, p = q t 1. Another parametrzaton t = q β. Structure constants f l fxed by assocatvty ( ) f l z l 1 (1 q = exp n )(1 t n ) n (1+p n ) z n l>0 n>0 lmt: q = e, 0 ) + O ( 4) T n 2δ n,0 + β 2 (ˆLn Q2 β δ n,0 ˆL n - operator wth central charge c = 1 6Q 2 β Q β = β 1 β 8

9 : free feld representaton 3d functon Hesenberg algebra: [a n, a m ] = 1 n (q n 2 q n 2 )(t n 2 t n 2 )(p n 2 + p n 2 )δ n+m,0, [P, Q] = 2 Screenng current: S q (w) : e n 0 operator: T n z n = Λ σ (z) n Z σ=±1 w n q n/2 an q n/2 : e βq w βp Λ σ (z) =: e σ z n (1+p n 0 σn ) an : q σ β 2 P p σ 2 matrx model: Z ({t}) d N w S q (w ) 0 a n (q n 2 q n 2 )t n, a n 1 n (t n 2 t n 2 )(p n 2 + p n 2 ) t n 9

10 matrx model 3d functon Ner, A.N., Zabzne 16 matrx model generatng functon: Z q ({t}) = N 0 d N w w k j (w k w 1 ; j q) (t w k w 1 1-loop for vec. + adj. multplets of mass t on D 2 ɛ S 1 ( q = e 2πɛ) Aganagc, Haouz, Kozcaz, Shakrov 13 j ; q) w κ 1 By constructon T n Z q ({t}) = 0 n > 0 T n = σ=±1 q σ β 2 p σ 2 k 0 B k ( {A (σ) k } ) B n+k ( {A (σ) n+k } ) (n+k)! k! Wlson loop along S 1 e k>0 t k j w k j FI term: κ 1 = βn βq β Bell polynomals A (σ) n = σ an n! 1+p σn 10

11 S 3 b functon: modular double 3d functon Glung two copes of N = 2 on D 2 ɛ S 1 N = 2 generatng functon on the squashed sphere S 3 b : ω2 1 z ω 2 2 z 2 2 = 1 Copy 1 Copy 2 q 1 = e 2π ω 2 ω 1 q 2 = e 2π ω 1 ω 2 t 1 = e 2πβ ω 2 ω 1 t 2 = e 2πβ ω 1 ω 2 β 1 = β β 2 = β Partton functon Z S 3({t}) = N 0 d N x b k j S 2 (x x j ω) S 2 (x x j +M ω) SL(2, Z) glung: ɛ g ɛ = ɛ 1 ɛ q 1 = e 2πɛ ; q 2 = e 2πg ɛ ; ɛ = ω 2 ω 1 ; Two Wlson loops: length L 1(2) = 2π ω 1(2) around z 2(1) = 0 2πκ 1 x ω e 1 ω 2 j=1,2 e t k,j k>0 2πk ω x ω e j T n,1 Z S 3 ({t}) = 0 b T n,2 Z S 3 ({t}) = 0 b 1-loop for vec.+adj. of mass M = β (ω 1 + ω 2 ) FI term 11

12 3d functon Constructon of s generalzed to closng algebra Resultng matrx model s generaton functon of the Wlson loops n N = 2 theory on D 2 S 1 Sb 3 Wlson loop generatng functon s annhlated by two sets of commutng operators. Generalzaton not mentoned n the talk Other spaces: Sb/Z 3 r, S 2 S 1, twsted S 2 S 1 Non-trval CS level: ncluson of vertex operators Addng fundamental hypermultplets: ncluson of vertex operators Quver theores: quver W (q,t) -algebras Gong to hgher dmensons: 4d (Lodn, Ner, Zabzne 17 ) and 5d theores? Integrablty of the? Correspondence to q-deformed ntegrable systems (Toda, Macdonald, etc.) 12

13 3d functon Thank you! 13