Langlands duality from modular duality

Transcription

1 Langlands duality from modular duality Jörg Teschner DESY Hamburg

2 Motivation There is an interesting class of N = 2, SU(2) gauge theories G C associated to a Riemann surface C (Gaiotto), in particular C Lagrangian of G C. The theories G C generalize theories studied by Seiberg and Witten. Low energy theory described in terms of prepotential F C (a), a = (a 1,..., a 3g 3+n ), which can be calculated from spectral curve of the SU(2)-Hitchin system associated to C. Gauge theory instanton calculus (Moore, Nekrasov, Shatashvili) defines a natural deformation Z C (a; ɛ 1, ɛ 2 ) of F C (a), F C (a) = lim ɛ 1ɛ 2 Z C (a; ɛ 1, ɛ 2 ). ɛ 1,ɛ 2 0 Amazing observation (Alday, Gaiotto, Tachikawa): exp(z C (a; ɛ 1, ɛ 2 )) = Conformal blocks of Liouville theory. What has Liouville theory to do with the Hitchin system?

3 Simplest example of a Hitchin system: Gaudin model Case C = P 1 \ {z 1,..., z n }. Consider θ(y) = In this case transfer matrix: q = tr(θ 2 ) = DOF: J r sl(2) C, r = 1,..., n modulo SL(2, C). ( ) J 0 J + J J 0 = n ( δ r (y z r ) 2 + H ) r y z r A Poisson bracket can be introduced as n J r y z r,, H r = s r J a r J b s z r z s η ab. { J 0 r, J ± s } = ±δ rs J ± s, { J + r, J s } = 2δ rs J 0 s. The H r commute w.r.t. the Poisson structure Integrability.

4 Quantum SL(2, C)-Gaudin model Cconsider the tensor product of n principal series representations P j of SL(2, C). It corresponds to the tensor product of representations of the Lie algebra sl(2, C) generated by differential operators Jr a acting on functions Ψ(x 1, x 1,..., x n, x n ) as J r = xr, J 0 r = x r xr j r, J + r = x 2 r xr + 2j r x r, and complex conjugate operators J a r. Casimir parameterized via j r as j r (j r + 1). Let H r s r J rs z r z s, Hr s r J rs z r z s, where the differential operator J rs is defined as J rs := η aa J a r J a s := J 0 r J 0 s ( J + r J s + J r J + s ), while J rs is the complex conjugate of J rs. The Gaudin Hamiltonians are mutually commuting, [ H r, H s ] = 0, [ H r, H s ] = 0, [ H r, H s ] = 0.

5 Separation of variables (SOV) for the Gaudin model I Diagonalize J by means of the Fourier transformation Ψ(µ 1,..., µ n ) = 1 π n d 2 x 1... d 2 x n n µ r 2j r+2 e µ rx r µ r x r Ψ(x 1,..., x n ). The generators J a r are mapped to the differential operators D a r, D r = µ r, D 0 r = µ r µr, D + r = µ r 2 µ r j r(j r + 1) µ r, Define variables to y 1,..., y n 2, u related to the variables µ 1,..., µ n via n i=1 n 2 µ i j=1 = u (t y j) t z n i i=1 (t z i).

6 Separation of variables (SOV) for the Gaudin model II It was shown by Sklyanin that the eigenvalue equations H r Ψ = E r Ψ are transformed into Baxter equations ( 2 y k + t(y k ))χ(y k ) = 0, t(y) n ( jr (j r + 1) (y k z r ) 2 E ) r y k z r. The dependence with respect to the variables y k has completely separated. Solutions to the Gaudin-eigenvalue equations H r Ψ = E r Ψ can therefore be constructed in factorized form Ψ = n 2 k=1 χ k (y k ; q).

7 Generalization: Hitchin s integrable system Phase space: M H Moduli space of Higgs pairs (E, θ), { E = (V, A 0,1 ) holomorphic SU(2)-bundle on C θ H 0 (C, End(E) Ω 1 C). }, modulo complex gauge transformations. Associate to (E, θ) the quadratic differential q = tr(θ 2 ). Expanding q with respect to a basis {q 1,..., q 3g 3+n } of the 3g 3 + n-dimensional space of quadratic differentials, q = 3g 3+n H r q r, defines functions H r, r = 1,..., 3g 3 + n on M H (C) called Hitchin s Hamiltonians.

8 (Pre-) Quantization of Hitchin s integrable system The main results of Beilinson-Drinfeld on the geometric Langlands correspondence can roughly be reformulated in the language from integrable models: There exist differential operators D r, r = 1,..., 3g 3 + n which commute with each other and have H r as their leading symbol (classical limit). To each solution of the eigenvalue equations D r Ψ = E r Ψ there corresponds an oper, a second order differential operator locally of the form ɛ 2 2 y + t(y) ( Baxter equation), which behaves under change of coordinates on C as t(y) (y (w)) 2 t(y(w)) 1 {y, w}, 2 y {y, w} y 3 2 ( ) y 2. y There exist natural operations H(y) ( Hecke functors ) on the spaces of solutions to the eigenvalue equations which act as H(y) : Ψ ψ(y)ψ, (ɛ 2 2 y + t(y))ψ(y) = 0. Hecke functors Q-operators. (cf. A. Gerasimov s talk and O. Foda, to appear).

9 Liouville theory I Liouville theory is a CFT, central charge c = 1+6Q 2, Q := b+b 1. It is characterized by the correlation functions of n primary fields e 2α rφ(z r, z r ) denoted as e 2α n φ(z n, z n) e 2α 1φ(z 1, z 1 ) C q. C q is a family of Riemann surfaces parameterized by q = (q 1,..., q 3g 3+n ). The dimension r of the primary field e 2α rφ(z r, z r ) is r αr := α r (Q α r ). The correlation functions can be represented in a holomorphically factorized form e 2α n φ(z n, z n) e 2α 1φ(z 1, z 1 ) C q = dµ(a) Fq σ (a) 2. The conformal blocks F σ q (a) e 2α nφ(z n) e 2α 1φ(z 1 ) a are objects that are defined from the representation theory of the Virasoro algebra.

10 Liouville theory II Consider insertions of degenerate fields like On,l a e 2α nφ(z n) e 2α 1φ(z 1 ) e 1 b φ(y l) e 1 b φ(y 1) a The conformal blocks satisfy the BPZ equations with differential operators D BPZ y k D BPZ y k = b 2 2 y 2 k + D BPZ y k O n,l = 0, k = 1,..., l, being for g = 0 given as n ( ) r (y k z r ) y k z r z r l k =1 k k ( 3b (y k y k ) 2 1 y k y k y k ).

11 Solutions to KZ equations from Liouville theory Claim: (Stoyanovsky; Ribault, J.T.) Ansatz G(µ z) := u δ ( n i=1 µ i) Θn (y z) F(y z), yields a solution to the KZ-equations (k + 2) z r Φ(x z) = H r Φ(x z). from any given solution F(y z) to the BPZ-equations. Θ n (y z) is defined as Θ n (y z) = r<s n z 1 2b 2 rs k<l n 2 y 1 2b 2 kl n n 2 k=1 (z r y k ) 1 2b 2. (1) Variables µ 1,..., µ n related to y 1,..., y n 2, u via n n 2 µ r = u k=1 (t y k) t z n r (t z r). Parameters are related via b 2 = (k + 2) 1, α r α(j r ) := b(j r + 1) + b 1.

12 Critical level limit Consider G(µ z) := u δ ( n i=1 µ i) Θn (y z) F(y z), for k 2 b. On the one hand note that one may solve in the form (k + 2) z r G(x z) = H r G(x z). G(x z) exp( b 2 S(z))Ψ(µ z)(1 + O(b 2 )), provided that Ψ(x z) is a solution to the Gaudin eigenvalue equations H r Ψ = E r Ψ with E r given in terms of S(z) by E r = zr S(z). On the other hand: Modular duality of Liouville theory (invariance under b b 1 ) limit b is equivalent to classical limit: F(y z) exp( b 2 S(z)) n 2 k=1 ψ k (y k ), where ( 2 y + t(y)ψ k (y) = 0. Geometric Langlands correspondence

13 Quantization conditions from Yang s potential It is not easy to define quantization conditions for the Hitchin systems. Proposal (Nekrasov-Shatashvili; cf. K.Kozlowski, J.T. for Toda-example): For algebraically integrable systems it is natural to formulate the quantization conditions in terms of Yang s potential W C (a), Function W C : B C, B: Space of eigenvalue of the Hamiltonians, a = (a 1,..., a N ), a special system of coordinates on B. Quantization conditions can then be formulated as a r W C (a) = 2πin r, r = 1,..., N. For SL(2, C)-Gaudin/Hitchin it is also natural to consider complex quantization where Re(a r ) = πim r, a r Re(W C (a)) = πin r, r = 1,..., N.

14 Yang s potential for Hitchin systems Define H r (a, z) as the accessory parameters which gives the oper P the monodromy ρ P : π 1 (C) PSL(2, C) such that 2 cosh a r 2 := tr(ρ P (γ r )), for curves γ 1,..., γ n 3 that constitute a cut system. We claim that the function W(a, z) which does the job can be defined by the equations H r (a, z) = z r W(a, z), with H r defined by the expansion q(y) = n ( δ r integrability condition is H s = H r. z r z s (y z r ) 2 + H r y z r ) for g = 0. This follows from W(a, z): Semiclassical Liouville conformal block. Note The W(a, z) W ɛ1 (a, z) = lim ɛ2 0 ɛ 2Z C (a; ɛ 1, ɛ 2 ), ɛ 1.

15 This is part of a larger picture [arxiv ] Isomonodromic deformations (A) ɛ2 (C) ɛ1 Hitchin system Liouville theory WZNW-model (B) ɛ1 (D) ɛ2 quantized Hitchin systems where the arrows may be schematically characterized as follows: (A) Hyperkähler rotation within the Hitchin moduli space M H (C). (B) Quantization [Beilinson-Drinfeld], [Nekrasov-Shatashvili] (C) Quantization [arxiv ] (D) This arrow may be called quantum hyperkähler rotation.