Instanton calculus for quiver gauge theories

Transcription

1 Instanton calculus for quiver gauge theories Vasily Pestun (IAS) in collaboration with Nikita Nekrasov (SCGP) Osaka, 2012

2 Outline 4d N=2 quiver theories & classification Instanton partition function [LMNS, Nekrasov] Limit shape problem [Nekrasov-Okounkov] Cut-crossing and quiver Weyl group The spectral curve Integrable systems: G-bundles, Nahm transform and Hitchin systems

3 4d N=2 quiver theories & classification Assumptions gauge group G g = i Vert U(N i ) fundamental & bifundamental hypers c ii =2 c ij = c ij = # bifund hypers of i and j Ni f =# fund hypers of i β i 0 c ij N j N f i 0 c ij is Cartan matrix of Fin or Aff ADE Dynkin graph

4 ! Aff Ar ADE!! Aff! quivers!!!!! Aff Dr! " " " "!!! Aff E6! " " # "! " Aff E7! " # $ # "! # Aff E8 " $ % & $ # "!

5 Instanton partition function [LMNS 97,98; Nekrasov 02] Z inst T (q) = k Z Vert 0 q k M k eu T (ker /D R ) instanton charges coupling constants k = {k i i Vert} q = {e 2πıτ i i Vert} framed instanton moduli M k = i Vert M ki matter bundle R = i N N f i i i<j (N i, N j ) c ij equivariant Euler class eu T torus T = T G T F T L (a, m, ) equivariant parameters in Lie(T)

6 Complete partition function Z T (q) =Z tree T Z 1 loop T ZT inst (q) Z T (q) = k q k A/G gauge eu T (Ω 2+ ad G g )eu T ((S S + ) R) A = {Gg connections framed at infinity on R 4 }

7 Seiberg-Witten limit Z T (a, m, ; q) =exp( F T (a, m, ; q)) lim F T(a, m, ; q) =F SW (a, m; q) 1, 2 0 [Nekrasov 02] Proofs [Nekrasov-Okounkov 03] for U(N) with adj/fund matter (limit shape of partition profiles) [Nakajima-Yoshioka 03] for U(N) without matter (blow-up recursion formula) [Braverman-Etingof 04] for generic G without matter (surface operators for parabolic subgroups P of G)

8 Limit shape method [Nekrasov-Okounkov 03] relies on ADHM originally described the shape of the dominant partitions labeling -fixed points on T M inst applicable to Lagrangian theories with classical gauge groups and tensor matter examples: [Nekrasov-Shadchin 04] SO(N) and Sp(N) with adj/fund [Shadchin 05] U(N1)xU(N2) bifund and U(N1) sym/antisym

9 G = SU(N) ADHM complex Review of limit shape method Z k = M N,k eu T (ker /D R) Nakajima s lectures Losev-Marshakov-Nekrasov 03 Nekrasov-Okounkov 03 Shadchin 05 C ADHM : where V K 1 2 (B 2,B 1,J) V L K 1 2 W ( B 1,B 2,I) V Λ 2 L K 1 2 V = C k,w = C N,L C 2,K = Λ 2 L [B 1 B 2 ]+IJ =0 then E z = H 1 (C ADHM,B z ) ch T E z=0 E T = W T K 1 2 T (1 L T + Λ 2 L T )V T = W T e ı + (1 e ı 1 )(1 e ı 2 )V T = = W T 2sh ı 1 2 2shı 2 2 V T W T = N α=1 eıa α V T = k i=1 eıφ i L T = e ı 1 + e ı 2 + = 1 2 ( )

10 ch T ker /D R = L Â T (L)ch T (E R) = ch T(E z=0 R) 2sh ı 1 2 2sh ı 2 2 ch T ker /D Ei E j = (E i) T (Ej ) T 2sh ı 1 2 2sh ı 2 2 then transform ch T (M) eu T (M) using obtain d 1 ds Γ(s) exp d ds where M is any 0 1 Γ(s) T-module dt t s 1 e ıwt s=0 = log(ıw) 0 dt t s 1 ch T tm s=0 =eu T M

11 Let ρ T;M (x) be Fourier transform of ch T tm eu T ker /D Ei E j ch T tm = e ıwt ch T tm = dx ρ T;M (x)e ıxt ρ M;T (x) = δ(x w) =exp d 1 ds Γ(s) =exp 0 ρ i (x) ρ T;Ei (x) dt t s 1 dxρi (x)e ıxt dxρ j (x )e ıx t 2sh ı 1t 2 2sh ı 2t 2 dxdx ρ i (x)γ 1, 2 (x x + m ij )ρ j (x ) e ım ijt = Barnes double gamma-function γ 1, 2 (x) = d ds 1 Γ(s) 0 dt t s 1 e ixt 2sh ı 1t 2 2sh ı 2t 2

12 Assume cycle m ij =0 (always true for all Fin and Aff DE quivers b/c there are no cycles) Absorb bi-fund masses to U(1) Coloumb moduli F = i πıτ i Z T = k dx x 2 ρ(x) i,j k(x) 1 2 γ 1 2 (x) [Dρ] k e F k(x) = x2 2 (log x 3 2 ) as i 0 Critical density (with constraints): d 2 dx 2 δf δρ =0 dx dx ρ i (x)k(x x )c ij ρ j (x ) I iα dxρ i (x) =1 (no fundamentals) I iα dx xρ i (x) =a iα I i dx x 2 ρ i (x) =a i k

13 critical density solves (no fundamentals) ψ j (x + i0) + ψ j (x i0) = 2πıτ i c ij ψ j (x), j=i x I i where potentials ψ j (x) = k xx (x x )ρ(x )dx k xx (x) = log x, i 0 in term of exp-potentials y i (x) =expψ i (x) critical equations with fundamentals y i (x + i0)y i (x i0) = Q(x) j=i y c ij j (x) involve matter polynomials N F i Q i (x) =q i f=1 (x + m if )

14 Cut-crossing and quiver Weyl group cut-crossing jump ψ i (x + i0) ψ i (x i0) from the critical equations: s i : ψ j (x) T i (x) = log Q i (x) ψ j (x) k ψ k(x)c kj + T j (x), j = i ψ j (x), j = i The cut-crossing transformations generate the quiver Weyl group Wq! quiver Cartan t q {ψ i (x)} t(x) = t(x)+ i α i ψ i (x) t(x) = i Λ i T i (x). s i : t t α i (t)α i. si - simple root reflections

15 Wq invariants Define canonical Wq invariant functions equivalently χ i (t) =e Λ i( t) χ i = e Λ w i (t) w W/W i w W/W i w(y i )=y i +... is Weyl orbit of i-th fundamental weight (also we can use characters) y i = x N i + O(1), x All terms y j(x) Q j (x) contain integer powers j of yi(x) and positive integer powers of Qi(x) in combinations such that each term is majorated by O(x Ni ) as x χ i = c i (q)x N i + O(1)

16 Solution to the quiver limit shape problem is given by the system of analytic equations on the exp-potentials yi(x) {χ i (y; Q) =P i (x), deg P i = N i } This defines the curve Σ C x (C y i )n, n = #gauge groups (χ 1 =) y 1 + Q 1 y 2 y 1 + Q 1 Q 2 1 y 2 = P 1 (χ 2 =) y 2 + Q 2 y 1 y 2 + Q 1 Q 2 1 y 2 = P 2

17 Example: A2 quiver canonical Weyl invariant system (χ 1 =) y 1 + Q 1 y 2 y 1 + Q 1 Q 2 1 y 2 = P 1 (χ 2 =) y 2 + Q 2 y 1 y 2 + Q 1 Q 2 1 y 2 = P 2 implies Seiberg-Witten curve equation y 3 1 P 1 (x)y P 2 (x)q 1 (x)y 1 Q 1 (x) 2 Q 2 (x) =0

18 Spectral curve in terms of characteristic polynomial for Fin quivers To each irrep R of Gq we can associate the spectral polynomial and the algebraic curve: det R (y e t )=0 The coefficients of powers of y in the expansion det R (y e t )= dim R k=0 y dim R k ( 1) k tr Λ k Re t are expressed polynomially in terms of invariants χ i and, hence, P i (x) and Q i (x)

19 Aff quivers, theta-functions and Gq bundles on elliptic curve E t = C /q Z elliptic curve q = r i=0 qa i i a i Dynkin marks Ĝ q orbits on ˆt q Gq-flat connections on E G C q polystable degree ADE theta-functions zero holomorphic bundles on E WP a 0,...,a r

20 Algebraic integrable system associated to affine ADE quiver M =Bun 0,ss G q,n (C x E t, E) B = Maps N (C x, Bun 0,ss G q (E)) Pť dim C M =2Nh dim C B = Nh The fibers are Lagrangian tori for holomorphic symplectic structure on M

21 Affine AD quivers, Nahm transform and Hitchin systems holomorphic SU(r+1) / SO(2r) / Sp(2r) -bundles of instanton charge N on C E Nahm transform ADHM Higgs bundles on with (ADHM dual) SU(N) / Sp(2N) / SO(N) structure group E SL(N) / Sp(2N) / SO(N) Hitchin system on elliptic curve (with r+1 / 2r / 2r punctures ) E

22 Thank you!