Matrix Models, Check-operators & Quantum Spectral Curves Andrei - - PowerPoint PPT Presentation

Matrix Models, Check-operators & Quantum Spectral Curves

Andrei Mironov

P.N.Lebedev Physics Institute and ITEP

2nd French Russian Conference Random Geometry and Physics,

Institut Henri Poincar´ e, Paris, October 17-21 2016

Plan Introduction Multiple solutions to the Virasoro constraints Check-operators Seiberg-Witten (SW) like solutions and integrable properties Quantum curves Quantum curves from degenerate conformal blocks Modular kernels in CFT

A.Mironov (LPI/ITEP) Matrix Models,Check-operators, and Quantum Spectral Curves 2016 2 / 23

Old Story: Matrix Models: Virasoro constraints = loop equations = Ward identities + integrability New Story: Check Operators: The space of solutions to the loop equations + quantized Whitham flows An application: Strings: matrix model networks as a tool to study topological strings and Nekrasov functions. Algebraically: conformal blocks of Virasoro/W and Ding-Iohara-Miki algebras Quantum field theory: supersymmetric quiver gauge theories of Seiberg-Witten type

A.Mironov (LPI/ITEP) Matrix Models,Check-operators, and Quantum Spectral Curves 2016 3 / 23

Two-parametric deformations of Seiberg-Witten (SW) systems: Gauge theory Nekrasov function quantum SW system Seiberg-Witten system Integrable system conformal matrix models, KP hierarchy ↓ ǫ2 → 0 quantum many-body integrable system, = ǫ1 ↓ ǫ1 → 0 classical finite-dimensional integrable system Spectral curves degenerate conf. block equation Schr¨

• dinger (Baxter) equation

Spectral curve + Whitham flows

A.Mironov (LPI/ITEP) Matrix Models,Check-operators, and Quantum Spectral Curves 2016 4 / 23

Matrix integrals and check-operators

Simplest example: the Hermitean matrix integral Z =

• dM exp [ TrV (M) ]

where we parameterize V (M) =

• k=0

tkM k Loop equations = Virasoro constraints: LnZ = 0, n ≥ −1 Ln =

• tk

∂ ∂tk+n +

• a+b=n

∂2 ∂ta∂tb ∂Z ∂t0 = NZ

A.Mironov (LPI/ITEP) Matrix Models,Check-operators, and Quantum Spectral Curves 2016 5 / 23

Solutions as formal series

There is no solution such that Z is a power series! Gaussian case (t2 → t2 − α): Z =

• dM exp
• −αTrM 2 + TrV (M)
• = c0 + c1t1 + c(1)

2 t2 1 + c(2) 2 t2 + . . .

The coefficients are constructed from

• dM exp
• −αTrM 2

M k ci ∼ α−i/2 which is evident by dimensional argument.

A.Mironov (LPI/ITEP) Matrix Models,Check-operators, and Quantum Spectral Curves 2016 6 / 23

More general (Dijkgraaf-Vafa) case: Z =

• dM exp [ TrW(M) + TrV (M) ]

W(M) = n TkM k, tk − → Tk + tk in the Virasoro constraints. The coefficients are constructed from Z =

• dM exp [ TrW(M) ] M k

i.e. there are combinations of Tk’s in denominators. How many solutions? Solutions are parameterized by an arbitrary function of n − 2 variables Tk. Two of them are fixed by L0Z = 0, L−1Z = 0 Thus, for n = 2 (Gaussian case) there is a unique solution.

A.Mironov (LPI/ITEP) Matrix Models,Check-operators, and Quantum Spectral Curves 2016 7 / 23

More general (Dijkgraaf-Vafa) case: Z =

• dM exp [ TrW(M) + TrV (M) ]

W(M) = n TkM k, tk − → Tk + tk in the Virasoro constraints. The coefficients are constructed from Z =

• dM exp [ TrW(M) ] M k

i.e. there are combinations of Tk’s in denominators. How many solutions? Solutions are parameterized by an arbitrary function of n − 2 variables Tk. Two of them are fixed by L0Z = 0, L−1Z = 0 Thus, for n = 2 (Gaussian case) there is a unique solution.

A.Mironov (LPI/ITEP) Matrix Models,Check-operators, and Quantum Spectral Curves 2016 7 / 23

Technical tools: loop equations Generating functions of correlators = resolvents: ρ(1)(z) =

• Tr

1 z − M

• = 1

Z ˆ ∇zZ = ˆ ∇zF ρ(2)(z1, z2) =

• Tr

1 z1 − M Tr 1 z2 − M

• c

= ˆ ∇z1 ˆ ∇z2F . . . where ˆ ∇z =

• k

1 zk+1 ∂ ∂tk , Z = exp F Loop equations: [T(z)Z]− = 0, T(z) ≡ Ln zn+2 ρ(1)(z)2 + ˆ ∇zρ(1)(z) + W ′(z)ρ(1)(z) +

• W ′(z)ρ(1)(z)
• +
• polynomial of degree n−2

+

• V ′(z)ρ(1)(z)
• −
• =0 as tk→0

= 0

A.Mironov (LPI/ITEP) Matrix Models,Check-operators, and Quantum Spectral Curves 2016 8 / 23

Check-operator: acting on the space of solutions. fn−2(z) =

• W ′(z)ρ(1)(z)
• + ≡ ˇ

RzF, ˇ Rz = −

• a,b

(a + b + 2)Ta+b+2za ∂ ∂Tb Classical spectral curve: Genus expansion: tk, Tk → 1 g tk, 1 g Tk, Z = exp 1 g2 F

• ,

F =

• k

g2kFk Leading term of the genus expansion: ρ(1)(z) = −W ′(z) + y(z) 2 where y(z)2 ≡ W ′(z)2 − 4fn−2(z) determines the classical spectral curve.

A.Mironov (LPI/ITEP) Matrix Models,Check-operators, and Quantum Spectral Curves 2016 9 / 23

Examples: Gaussian case: n = 2, f0(z) = const, y2 = z2 − const: semi-circle distribution. W3-case: n = 3, f1(z) is a linear function, the spectral curve is a torus, the space of solutions is described by a function of one variable. Meaning: typical integration in the matrix (eigenvalue) integral

• dxeW3(x)

implies two independent choices of contour (Airy functions). N eigenvalues in the integrand part into two groups (two contours), N1 and N2, N1 + N2 = N. This describes the two-cut solution (torus). Arbitrary function of one variable corresponds to one arbitrary variable, fraction N1/N2.

A.Mironov (LPI/ITEP) Matrix Models,Check-operators, and Quantum Spectral Curves 2016 10 / 23

Summary of general properties:

• 1. Any solution is unambiguously labeled by an arbitrary function of n − 2 T-variables: the bare

F(0)(T).

• 2. Solutions of the Virasoro constraints (or loop equations) are constructed from F(0)(T) by an

evolution operator ˆ U(T, t) that does not depend on F(0)(T) : Z(T, t) = ˆ U(T, t)eF(0)(T )

• 3. The evolution operator can be completely expressed in terms of the unique operator

ˇ y ≡

• W ′(x)2 − 4 ˇ

R(x), ˇ R(x) ≡ −

• a,b=0

(a + b + 2)Ta+b+2xa ∂ ∂Tb its derivatives and W ′(x).

A.Mironov (LPI/ITEP) Matrix Models,Check-operators, and Quantum Spectral Curves 2016 11 / 23

Consequence: ρ(1)(z) = ˆ ∇z(t)F = ˇ ∇z(T)F The main check-operator ˇ ∇z is expressed through y, its derivatives and W ′(x). Important: [ ˆ ∇z1, ˆ ∇z2] = 0, but [ ˇ ∇z1, ˇ ∇z2] = 0. Main property: [

• Ai

dz ˇ ∇z,

• Bj

dz ˇ ∇z] = δij The curve: y2 = W ′2(x) − 4 ˇ R(x)F(0).

A.Mironov (LPI/ITEP) Matrix Models,Check-operators, and Quantum Spectral Curves 2016 12 / 23

DV/SW system:

Choose the basis of eigenfunctions

• Ai dz ˇ

∇zZa = aiZa, i.e.

• Ai dz ˇ

∇zFa =

• Ai dzρ(1)(z) = ai, then
• Bi

dzρ(1)(z) = ∂F ∂ai Ni are associated with ai =

• Ai

ρ(1)(z)dz Integrable properties: ZN(t) is a τ-function of the Toda chain (as a formal series) ZDV (T, Ni) is the SW system; it satisfies the Whitham hierarchy in the planar limit, Ti are Whitham flows.

A.Mironov (LPI/ITEP) Matrix Models,Check-operators, and Quantum Spectral Curves 2016 13 / 23

Quantum spectral curve:

The Backer-Akhiezer function: ΨBA(z) = eV (z)/2Ψ(z) where Ψ(z) = Z

• tk −

1 kzk

• Z(t)

= 1 Z(t)zNe

z dξ ˆ ∇ξ = det(zM)

and < ... > means the matrix model average. From the Virasoro constraints the quantum spectral curve:

• ∂2

z + V ′(z)∂z + ˇ

Rz

• Ψ(z) = 0

In the limit, ∂ log Ψ(z) = ρ(1)(z) it turns into the classical spectral curve (planar loop equation): ρ(1)(z)2 + V ′(z)ρ(1)(z) + ˇ RzF = 0 The equation for the Baker-Akhiezer function looks as

• ∂2

z − 1

2V ′′(z) + 1 4V ′(z)2 − 1 2[ ˇ RzV (z)] + ˇ Rz

• ΨBA(z) = 0

ˇ Rz contains the derivatives w.r.t. the Whitham times.

A.Mironov (LPI/ITEP) Matrix Models,Check-operators, and Quantum Spectral Curves 2016 14 / 23

AGT and conformal blocks: quantum spectral curve

Conformal block: G(x, ∆; ∆i, c), ∆ = (Q − α)α, c = 1 + 6Q2, Q = b − 1/b, Vα(z) = eiαφ(z). Degenerate field: (b2L2

−1 − L−2)V1/2b(z) is a primary field, i.e. V1/2b(z) is degenerate at the second level. The equation

for the 5-point block with the degenerate field at z:       b2z(z − 1)∂2

z + (2z − 1)∂z − q(q − 1)

z − q ∂q + rational function of q

• check-operator

      G5(z|0, q, 1, ∞) = 0 This is the quantum spectral curve, q is a counterpart of Tk.

A.Mironov (LPI/ITEP) Matrix Models,Check-operators, and Quantum Spectral Curves 2016 15 / 23

Comment.

In the limit when all ∆i → ∞, this equation is reduced to the non-stationary Schr¨

• dinger SU(2)

periodic Toda chain equation:

• ∂2

z − 2Λ2 cosh z + 1

4 ∂ ∂Λ

• GT oda

5

= 0 where Λ is the limit of rescaled q. log Λ is known to play the role of the first Whitham time in the Seiberg-Witten theory.

A.Mironov (LPI/ITEP) Matrix Models,Check-operators, and Quantum Spectral Curves 2016 16 / 23

Conformal matrix model.

This quantum curve is for the conformal matrix model: G4(0, q, 1, ∞) = q2α1α2(1 − q)2α2α3

i

dui∆2b2(u)u2bα1

i

(1 − ui)2bα3(q − ui)2bα2 where ∆(u) is the Van-der-Monde determinant and α, α4 are fixed from the conditions: N1 contours [0, q] with bN1 = α − α1 − α2 N2 contours [0, 1] with bN2 = Q − α − α3 − α4 N1 and N2 are associated with the Dijkgraaf-Vafa Ni. Since G4 =

• Vα1(0)Vα2(q)Vα3(1)Vα4(∞)

q Vb(u)du N1 1 Vb(u)du N2

CF T

the degenerate block G5 = V1/2b(z) . . .CF T , < V1/2b(z)Vb(u) >CF T = z − u, G5 =< det(z − ui) >, the equation for G5 is exactly the quantum spectral curve.

A.Mironov (LPI/ITEP) Matrix Models,Check-operators, and Quantum Spectral Curves 2016 17 / 23

Modular kernels in CFT

Modular kernel G(x, a; ai, b) =

• da′K(a, a′; ai, b)G(1 − x, a′; ai, b)

ai = αi − Q/2. Modular kernel (Ponsot, Teschner, 1999) K(a, a′; ai, b) ∼

• dx

4

• i=1

Sb(ξi) Sb(ζi) ξi, ζi are linear functions of all parameters and x. Representation of G(x, a; ai, b) as a β-ensemble with β = b2 K(a, a′; ai, b) = e4πiaa′

A.Mironov (LPI/ITEP) Matrix Models,Check-operators, and Quantum Spectral Curves 2016 18 / 23

1-point toric conformal block G(τ, a; µ) = 1 + q ∆ext(1 − ∆ext) 2∆ + 1

• + O(q2)

with ∆ext = µ(Q − µ), µ is the adjoint hypermultiplet mass. G(τ, a; µ) =

• da′K(a, a′; µ)G(−τ −1, a′; µ)

Modular kernel due to Ponsot, Teschner: K(a, a′; µ) =

• dξ

Sb(ξ + µ/2 − a′)Sb(ξ + µ/2 + a′) Sb(ξ + Q − µ/2 − a′)Sb(ξ + Q − µ/2 + a′)e4πiaξ Modular kernel from β-ensemble: G(τ, a; µ) = 1 N(a)Z(τ, a; µ), N(a) = Γb(2a + µ)Γb(2a + Q − µ) Γb(2a)Γb(2a + Q) Z(τ, a; µ) =

• da′e2πiaa′Z(−τ −1, a′; µ)

A.Mironov (LPI/ITEP) Matrix Models,Check-operators, and Quantum Spectral Curves 2016 19 / 23

Quantum oscillator A = ei ˆ

P ,

B = ei ˆ

Q

ˆ AZa(Q) = eiaZa(Q), ˆ B ˜ Za(Q) = eia′ ˜ Za′(Q) Za(Q) =

• da′eiaa′/ ˜

Za′(Q) Check-operators ˇ A = eia, ˇ B = e∂a ˇ AaK(a, a′) = ˇ Ba′K(a, a′)

A.Mironov (LPI/ITEP) Matrix Models,Check-operators, and Quantum Spectral Curves 2016 20 / 23

We expect the conformal block to be an eigenfunction of some LA: LAG = λG, LBG = Λ(∂λ)G Claim Lγ = eb

• γ ˇ

∇

Since [LA, LB] = 1, we obtain that K(a, a′; µ) is Fourier!!! Subtlety ˇ ∇ has two branches, i.e. there are ˇ ∇±! G is globally defined but Z(a) is not! There are two branches at a > 0 and a < 0. Thus, Lγ =

• 1

N(a)eb

• γ ˇ

∇+N(a) +

1 N(−a)e−b

• γ ˇ

∇−N(−a)

• A.Mironov

(LPI/ITEP) Matrix Models,Check-operators, and Quantum Spectral Curves 2016 21 / 23

One can realize check-operators in the space of eigenvalues:

• A

dz ˇ ∇±

z → ±2πia,

• B

dz ˇ ∇±

z → ±1

2∂a Thus, one obtains LB = Γ(2ab)Γ(bQ + 2ab) Γ(bµ + 2ab)Γ(b(Q − µ) + 2ab)e

b 2 ∂a + (a → −a)

Since L′

A = cos 2πba the equations ˇ

AaK(a, a′) = ˇ Ba′K(a, a′) for the modular kernel becomes 1 2 sin 2πb(a − µ/2) sin 2πba e− b

2 ∂a + sin 2πb(a + µ/2)

sin 2πba e

b 2 ∂a

• K(a, a′) = cos 2πba′ K(a, a′)

At large a only one exponential survives giving the pure exponential kernel. The solution of the full equation is given by K(a, a′; µ) =

• dξC1(ξ)C2(a′)

Sb(ξ + µ/2 − a′)Sb(ξ + µ/2 + a′) Sb(ξ + Q − µ/2 − a′)Sb(ξ + Q − µ/2 + a′)e4πiaξ which coincides with the Ponsot-Teschner formula at C1 = C2 = 1.

A.Mironov (LPI/ITEP) Matrix Models,Check-operators, and Quantum Spectral Curves 2016 22 / 23

Thank you for your attention!

A.Mironov (LPI/ITEP) Matrix Models,Check-operators, and Quantum Spectral Curves 2016 23 / 23