Supersymmetric gauge theory, representation schemes and random matrices

Transcription

1 Supersymmetric gauge theory, representation schemes and random matrices Giovanni Felder, ETH Zurich joint work with Y. Berest, M. Müller-Lennert, S. Patotsky, A. Ramadoss and T. Willwacher MIT, 30 May / 25

2 Nekrasov partition functions Derived representation schemes Analytic properties of the partition function 2 / 25

3 Instanton partition function of N = 2 Yang Mills theory The Nekrasov instanton partition function of N = 2 supersymmetric Yang Mills theory with gauge group U(r) on R 4 in the Ω-background with parameters ɛ 1, ɛ 2 is given as a sum over r-tuples Y = (Y i ) r i=1 of Young diagrams of total size Y : Z 4D (ɛ 1, ɛ 2, a, q, λ) = Y q Y r α,β=1 1 E αβ (b)(ɛ 1 + ɛ 2 E αβ (b)), b Y α E αβ (b) = a α a β l Yβ (b)ɛ 1 + (a Yα (b) + 1)ɛ 2. a = (a 1,..., a r ) u(r) parametrizes boundary conditions of scalar fields ɛ 1, ɛ 2 equivariant parameters for the action of U(1) 2 on R 4 = C 2. a Y (b), l Y (b) arm and leg length of the box b Y. 3 / 25

4 Arm and leg length The arm and leg length of a box b in a Young diagram Y are the number of boxes to its right and below it, respectively. b a Y (b) l Y (b) 4 / 25

5 Five-dimensional supersymmetric theory Z 4D is the limit of the instanton partition function on R 4 Sλ 1 as the radius λ of the circle tends to 0: Z 5D (ɛ 1, ɛ 2, a, q, λ) = Y q Y r α,β=1 b Y α E αβ (b) = a α a β l Yβ (b)ɛ 1 + (a Yα (b) + 1)ɛ 2. (λ/2)2 sinh ( λ 2 E αβ(b) ) sinh ( λ 2 (ɛ 1 + ɛ 2 E αβ (b)) ), 5 / 25

6 Instanton count and equivariant cohomology Z 4D is the contribution of instantons to the partition function. Roughly, Z 4D = q M n 1 r,n n=0 is the generating function of counts of framed instantons of instanton number n = 0, 1, 2, / 25

7 Instanton count and equivariant cohomology Z 4D is the contribution of instantons to the partition function. Roughly, Z 4D = q M n 1 r,n n=0 is the generating function of counts of framed instantons of instanton number n = 0, 1, 2,.... The integral over moduli space M r,n makes sense if we consider 1 as an equivariant differential form for T = U(1) 2 U(1) r where U(1) 2 acts on R 4 = C 2 via (z 1, z 2 ) (e iφ 1 z 1, e iφ 2 z 2 ) and U(1) r is the Cartan torus of the group U(r) of constant gauge transformations. 6 / 25

8 Instanton count and equivariant cohomology Z 4D is the contribution of instantons to the partition function. Roughly, Z 4D = q M n 1 r,n n=0 is the generating function of counts of framed instantons of instanton number n = 0, 1, 2,.... The integral over moduli space M r,n makes sense if we consider 1 as an equivariant differential form for T = U(1) 2 U(1) r where U(1) 2 acts on R 4 = C 2 via (z 1, z 2 ) (e iφ 1 z 1, e iφ 2 z 2 ) and U(1) r is the Cartan torus of the group U(r) of constant gauge transformations. The integral is then computed/defined by the localization formula: fixed points are labeled by r-tuples of partitions. 6 / 25

9 Instanton count in the 5D theory and equivariant K-theory In the five-dimensional case equivariant cohomology is replaced by equivariant K-theory: Z 5D = n v n p [O Mn,r ] K T (pt)[[v]], v = qλ 2r e rλ(ɛ 1+ɛ 2 ). is the generating function of the direct image by the map to a point of the class of the trivial line bundle. 7 / 25

10 Instanton count in the 5D theory and equivariant K-theory In the five-dimensional case equivariant cohomology is replaced by equivariant K-theory: Z 5D = n v n p [O Mn,r ] K T (pt)[[v]], v = qλ 2r e rλ(ɛ 1+ɛ 2 ). is the generating function of the direct image by the map to a point of the class of the trivial line bundle. T acts on M r,n and thus H i (M r,n, O) is a representation of T. Let ch T V K T (pt) = Z[ ˆT ] = Z[q ±1 1, q±2 2, u±1 1,..., u±1 r ] denote the character of a finite dimensional virtual representation V. Then Z 5D = 2rn v n ch T ( 1) i H i (M r,n, O). n=0 i=0 7 / 25

11 Instanton count in the 5D theory and equivariant K-theory The cohomology groups H i (M r,n, O) are infinite dimensional with finite dimensional weight spaces for the action of U(1) 2. Then Z 5D takes value in a completion of K T (pt). In fact Z 5D Z[u 1 ±1,..., u±1 r ][[q 1, q 2, v]] 8 / 25

12 ADHM equations Atiyah, Drinfeld, Hitchin, Manin (ADHM) gave a description of the moduli space of framed instantons (torsion free sheaves on CP 2 with trivialization at infinity) of instanton number n for the group U(r) in terms of linear algebra data: M r,n = T (Mat n,n (C) Mat n,r (C))//GL n 9 / 25

13 ADHM equations Atiyah, Drinfeld, Hitchin, Manin (ADHM) gave a description of the moduli space of framed instantons (torsion free sheaves on CP 2 with trivialization at infinity) of instanton number n for the group U(r) in terms of linear algebra data: M r,n = T (Mat n,n (C) Mat n,r (C))//GL n The hamiltonian quotient is the GIT quotient µ 1 (0)/GL n for the moment map µ(x, I, Y, J) = [X, Y ] + IJ. 9 / 25

14 ADHM equations Atiyah, Drinfeld, Hitchin, Manin (ADHM) gave a description of the moduli space of framed instantons (torsion free sheaves on CP 2 with trivialization at infinity) of instanton number n for the group U(r) in terms of linear algebra data: M r,n = T (Mat n,n (C) Mat n,r (C))//GL n The hamiltonian quotient is the GIT quotient µ 1 (0)/GL n for the moment map µ(x, I, Y, J) = [X, Y ] + IJ. GIT means that we restrict to the four-tuples (X, I, Y, J) obeying the stability condition: there is no non-trivial proper subspace of C n containing I (C r ) that is invariant under X and Y. 9 / 25

15 Gauge theory on S 4, AGT correspondence The square of the absolute value of the Nekrasov partition function Z 4D 2 (or Z 5D 2 ) appears in the integrand over the Coulomb parameters of the partition function of N = 2 supersymmetric gauge theory on S 4 (or S 4 S 1 ) with ellipsoidal metric with half-axes ɛ 1, ɛ 2 (Pestun). 10 / 25

16 Gauge theory on S 4, AGT correspondence The square of the absolute value of the Nekrasov partition function Z 4D 2 (or Z 5D 2 ) appears in the integrand over the Coulomb parameters of the partition function of N = 2 supersymmetric gauge theory on S 4 (or S 4 S 1 ) with ellipsoidal metric with half-axes ɛ 1, ɛ 2 (Pestun). Partition functions for gauge theory with matter fields are obtained by replacing the trivial bundle by suitable vector bundles (or their Chern characters in the 4D case). 10 / 25

17 Gauge theory on S 4, AGT correspondence The square of the absolute value of the Nekrasov partition function Z 4D 2 (or Z 5D 2 ) appears in the integrand over the Coulomb parameters of the partition function of N = 2 supersymmetric gauge theory on S 4 (or S 4 S 1 ) with ellipsoidal metric with half-axes ɛ 1, ɛ 2 (Pestun). Partition functions for gauge theory with matter fields are obtained by replacing the trivial bundle by suitable vector bundles (or their Chern characters in the 4D case). By the AGT (Alday Gaiotto Tachikawa) correspondence, Nekrasov instanton partition functions are related to conformal blocks of Liouville or Toda theories, or their q-deformations for the 5D theory. 10 / 25

18 A special case of the AGT correspondence in 5D: Gaiotto states Deformed Virasoro algebra [T n, T m ] = l=1 r l (T n l T m+l T m l T n+l ) (1 q 1)(1 q 2 ) 1 q 1 q 2 (q n 1q n 2 q n 1 q n 2 )δ m+n,0. r l x l = exp n 1 l 0 (1 q1 n)(1 qn 2 ) x n 1 + q1 nqn 2 n 11 / 25

19 A special case of the AGT correspondence in 5D: Gaiotto states Deformed Virasoro algebra [T n, T m ] = l=1 r l (T n l T m+l T m l T n+l ) (1 q 1)(1 q 2 ) 1 q 1 q 2 (q n 1q n 2 q n 1 q n 2 )δ m+n,0. r l x l = exp n 1 l 0 (1 q1 n)(1 qn 2 ) x n 1 + q1 nqn 2 n The Verma module M h is generated by h with relations T n h = δ n,0 h h, for n 0. It has a grading M h = n=0 M h,n by eigenspaces of T 0 to the eigenvalues h + n. It is orthogonal for the Shapovalov bilinear form on M h so that S( h, h ) = 1 and S(T n x, y) = S(x, T n y). 11 / 25

20 A special case of the AGT correspondence in 5D: Gaiotto states A Gaiotto state is a formal power series G = n=0 ξn G n with coefficients G n M h,n such that T 1 G = ξ G, T j G = 0, j / 25

21 A special case of the AGT correspondence in 5D: Gaiotto states A Gaiotto state is a formal power series G = n=0 ξn G n with coefficients G n M h,n such that T 1 G = ξ G, T j G = 0, j 2. Its norm squared S( G, G ) is a degenerate limit of a conformal block and coincides by the AGT correspondence to the Nekrasov partition functions of the N = 2 SUSY Yang Mills theory. 12 / 25

22 A special case of the AGT correspondence in 5D: Gaiotto states A Gaiotto state is a formal power series G = n=0 ξn G n with coefficients G n M h,n such that T 1 G = ξ G, T j G = 0, j 2. Its norm squared S( G, G ) is a degenerate limit of a conformal block and coincides by the AGT correspondence to the Nekrasov partition functions of the N = 2 SUSY Yang Mills theory. We will see that the norm squared, which is a priori a formal power series in ξ has in fact a finite radius of convergence. 12 / 25

23 Representation schemes Let S be an associative unital algebra over C and S A an algebra over S (the basic example is S = C). Let V be a finite dimensional S-module, and let ρ: S End V be the associated representation. 13 / 25

24 Representation schemes Let S be an associative unital algebra over C and S A an algebra over S (the basic example is S = C). Let V be a finite dimensional S-module, and let ρ: S End V be the associated representation. The (relative) representation scheme R V (S\A) is the space of representations A End(V ) restricting to ρ on S. It is an affine algebraic scheme. The character scheme is R V (S/A)/ Aut S (V ). 13 / 25

25 Representation schemes Let S be an associative unital algebra over C and S A an algebra over S (the basic example is S = C). Let V be a finite dimensional S-module, and let ρ: S End V be the associated representation. The (relative) representation scheme R V (S\A) is the space of representations A End(V ) restricting to ρ on S. It is an affine algebraic scheme. The character scheme is R V (S/A)/ Aut S (V ). Example: S = C, V = C n. R V (C\A) parametrizes the representations of A in n n matrices. The character scheme parametrizes equivalence classes of n-dimensional representations. 13 / 25

26 Representation schemes Example A path algebra of a quiver with vertex set I, S subalgebra generated by idempotents (e i ) i I. R V (S\A) parametrizes representations of the quiver on (Im ρ(e i )) i I. 14 / 25

27 Representation schemes Example A path algebra of a quiver with vertex set I, S subalgebra generated by idempotents (e i ) i I. R V (S\A) parametrizes representations of the quiver on (Im ρ(e i )) i I. Example µ 1 (0) is the representation scheme of the path algebra of n r on V = C r C n with ADHM relations XY YX + IJ = 0 on the generators. 14 / 25

28 Derived representation schemes and representation homology The assignment A O(R V (S\A)) extends to a well-defined functor from the category DGA S of differential graded (dg) S-algebras to the category CDGA C of commutative dg algebra. 15 / 25

29 Derived representation schemes and representation homology The assignment A O(R V (S\A)) extends to a well-defined functor from the category DGA S of differential graded (dg) S-algebras to the category CDGA C of commutative dg algebra. The derived representation scheme of A is obtained by replacing A be a weakly equivalent cofibrant object QA DGA S and applying the representation functor: DRep V (S\A) := O(R V (S\QA)) CDGA C It is well-defined up to quasi-isomorphism. 15 / 25

30 Derived representation schemes and representation homology The assignment A O(R V (S\A)) extends to a well-defined functor from the category DGA S of differential graded (dg) S-algebras to the category CDGA C of commutative dg algebra. The derived representation scheme of A is obtained by replacing A be a weakly equivalent cofibrant object QA DGA S and applying the representation functor: DRep V (S\A) := O(R V (S\QA)) CDGA C It is well-defined up to quasi-isomorphism. The representation homology of A relative to V is the graded algebra H (S\A, V ) = H (DRep V (S\A)) 15 / 25

31 Derived representation scheme for ADHM relations The path algebra of the quiver X Y J e 1 e 2 I is generated by X, Y, I, J and the idempotents e 1, e / 25

32 Derived representation scheme for ADHM relations The path algebra of the quiver X Y J e 1 e 2 I is generated by X, Y, I, J and the idempotents e 1, e 2. A cofibrant replacement QA of the path algebra with ADHM relation has an additional generator Θ of degree 1 whose differential enforces the relation on homology: dθ = XY YX IJ, dx = dy = di = dj = / 25

33 Derived representation scheme for ADHM relations DRep V (S\A) is the free graded commutative algebra generated by matrix entries with induced differential C[x αβ, y αβ, i αµ, j µβ, θ αβ α, β = 1,..., n, µ = 1,... r] dθ αβ = γ (x αγ y γβ y αγ x γβ ) + µ i αµ j µβ. 17 / 25

34 Derived representation scheme for ADHM relations DRep V (S\A) is the free graded commutative algebra generated by matrix entries with induced differential C[x αβ, y αβ, i αµ, j µβ, θ αβ α, β = 1,..., n, µ = 1,... r] dθ αβ = γ (x αγ y γβ y αγ x γβ ) + µ i αµ j µβ. The torus (C ) 2 acts by rescaling of X, Y. Also GL n GL r acts on the derived representation scheme. In particular we have an action of T = U(1) 2 U(1) r on the GL n -invariants of representation homology. 17 / 25

35 Derived representation scheme for ADHM relations Observation (Y. Berest, G.F., A. Ramadoss, S. Patotsky, T. Willwacher) The Nekrasov partition function on R 4 S 1 coincides with the generating function of the weighted Euler characteristics of the representation homology of the ADHM quiver: Z 5D = n=0 v n i ( 1) i ch T H i (S\A, V ) GLn, V = C n C r. 18 / 25

36 Integral formula for the partition function The calculation of the weighted Euler characteristic of H i (S\A, V ) GLn leads to the integral formula for the partition function (due to Nekrasov, Shatashvili,... ). Z(v) = v n Z n, n=0 ( ) 1 1 q 1 q n 2 Z n = n!(2πi) n (1 q 1 )(1 q 2 ) n r 1 (1 u α /z j )(1 q 1 q 2 z j /u α ) z j =1 j=1 α=1 j k (z j z k )(z j q 1 q 2 z k ) (z j q 1 z k )(z j q 2 z k ) more suitable to study analytic properties. The generators of the character group ˆT are identified with the Ω-background and Coulomb parameters via q i = e λɛ i, u α = e λaα. n j=1 dz j z j. 19 / 25

37 Analytic properties Range of parameters of interest Gauge theory on S 4 S 1 : ɛ 1, ɛ 2 > 0 and a i 1R. Exponential variables 0 < q 1, q 2 < 1, u i = 1. AGT correspondence for Liouville theory. Central charge c = 1 + 6(b + b 1 ) 2 (1, ), ɛ 1 = b, ɛ 2 = b 1. Strongly coupled Liouville theory: 1 < c < 25, ɛ 1 = ɛ 2 S 1. Weakly coupled Liouville theory: c > 25, ɛ 1, ɛ 2 > / 25

38 Analytic properties Range of parameters of interest Gauge theory on S 4 S 1 : ɛ 1, ɛ 2 > 0 and a i 1R. Exponential variables 0 < q 1, q 2 < 1, u i = 1. AGT correspondence for Liouville theory. Central charge c = 1 + 6(b + b 1 ) 2 (1, ), ɛ 1 = b, ɛ 2 = b 1. Strongly coupled Liouville theory: 1 < c < 25, ɛ 1 = ɛ 2 S 1. Weakly coupled Liouville theory: c > 25, ɛ 1, ɛ 2 > 0. Subtlety: The coefficents of the expansion of the partition functions have small denominators in these ranges: they are not defined for a dense set of parameters and they are arbitrarily small for the complement. 20 / 25

39 Analytic properties Theorem (G.F., M. Müller-Lennert) Let q 1, q 2 < 1, u α = 1. Suppose that either q 1 = q 2 or q 1, q 2 R +. Then the formal power series Z 5D (v) has convergence radius (at least) 1 and depends analytically on the parameters. 21 / 25

40 Analytic properties Theorem (G.F., M. Müller-Lennert) Let q 1, q 2 < 1, u α = 1. Suppose that either q 1 = q 2 or q 1, q 2 R +. Then the formal power series Z 5D (v) has convergence radius (at least) 1 and depends analytically on the parameters. Corollary The norm of the Gaiotto state for the deformed Virasoro algebra is analytic for ξ < q 1 q 2 1/2. 21 / 25

41 Analytic properties Theorem (G.F., M. Müller-Lennert) Let q 1, q 2 < 1, u α = 1. Suppose that either q 1 = q 2 or q 1, q 2 R +. Then the formal power series Z 5D (v) has convergence radius (at least) 1 and depends analytically on the parameters. Corollary The norm of the Gaiotto state for the deformed Virasoro algebra is analytic for ξ < q 1 q 2 1/2. The theorem amounts to an estimate of the asymptotic behaviour of the coefficients Z n = Z n (q 1, q 2, u) of the formal power series Z 5D (v): lim sup n Z 1 n matrix theory. 1. This can be done with techniques from unitary random 21 / 25

42 Random matrices We write Z n as an expectation value n r Z n = Zn 0 E n 1 (1 u α /z j )(1 q 1 q 2 z j /u α ) j=1 α=1 for a system of particles z 1,..., z n on the unit circle with Boltzmann distribution 1 exp W (z j /z k ) dz i 2πiz i j k Z 0 n for some pair potential W which is repulsive at short distances. 22 / 25

43 Equilibrium measure The estimate of E n ( ) is standard. One proves that for large n the particle configurations approach a uniform distribution on the unit circle with high probability. The asymptotic behaviour of the integral is then calculated by evaluating the integrand on this distribution. 23 / 25

44 Equilibrium measure The estimate of E n ( ) is standard. One proves that for large n the particle configurations approach a uniform distribution on the unit circle with high probability. The asymptotic behaviour of the integral is then calculated by evaluating the integrand on this distribution. The normalization factor Zn 0 (Z n at r = 0) is the weighted Euler characteristic of the representation homology H (C x, y /(xy tyx), C n ) of quantum plane, that can be computed explicitly (Berest et al) with the result ( ) v n Zn 0 1 q1 n = exp qn 2 v n (1 q1 n)(1 qn 2 ). n n=0 n=1 The right-hand side converges for v < 1 so we get that lim n Z 0 n 1 n = / 25

45 Open question The formal limit λ 0 of the estimated radius of convergence converges to the expected radius of convergence in the 4D theory. However the convergence is not uniform and we cannot deduce a result on the analyticity of Z 4D 24 / 25

46 Open question The formal limit λ 0 of the estimated radius of convergence converges to the expected radius of convergence in the 4D theory. However the convergence is not uniform and we cannot deduce a result on the analyticity of Z 4D From the point of view of random matrices the equilibrium measure in the 4D theory is no longer uniform as the two particle potential is attractive at intermediate distances. It would be interesting to describe this distribution. 24 / 25

47 Thank you for your attention 25 / 25