On combinatorial expansion of the conformal blocks arising from AGT conjecture

Abstract:We study chiral deformations of N = 2 and N = 4 supersymmetric gauge theories obtained by turning on τ J tr Φ J interactions with Φ the N = 2 superfield. Using localization, we compute the deformed gauge theory partition function Z( τ |q) and the expectation value of circular Wilson loops W on a squashed four-sphere. In the case of the deformed N = 4 theory, exact formulas for Z and W are derived in terms of an underlying U(N ) interacting matrix model replacing the free Gaussian model describing the N = 4 theory. Using the AGT correspondence, the τ J -deformations are related to the insertions of commuting integrals of motion in the four-point CFT correlator and chiral correlators are expressed as τ -derivatives of the gauge theory partition function on a finite Ω-background. In the so called Nekrasov-Shatashvili limit, the entire ring of chiral relations is extracted from the -deformed Seiberg-Witten curve. As a byproduct of our analysis we show that SU(2) gauge theories on rational Ω-backgrounds are dual to CFT minimal models.

Section: The Cft Sidementioning

confidence: 99%

Wilson loops and chiral correlators on squashed spheres

Fucito

Morales

Poghossian

2015

“…These conformal blocks were in itself computed exactly [12] in terms of the Painlevé VI τ -function, capitalizing on recent developments in gauge/gravity correspondence in the form of the Alday-Gaiotto-Tachikawa (AGT) conjecture [13,14].…”

Section: Jhep08(2017)094mentioning

confidence: 99%

“…These equations are known in the literature as the Schlesinger equations. Now, the τ function for the Painlevé VI system was shown to correspond to exact c = 1 conformal blocks, and expansions near t = 0, 1 and ∞ were given in [12,38], following the proposal [13] and subsequent proof [14] of the AGT conjecture. Perhaps more tantalizingly, the same τ -function appears in the semiclassical expansion for Liouville conformal blocks [39] -see also [40].…”

Section: Jhep08(2017)094mentioning

confidence: 99%

On the Kerr-AdS/CFT correspondence

Amado

Cunha

Pallante

2017

We review the relation between four-dimensional global conformal blocks and field propagation in AdS 5 . Following the standard argument that marginal perturbations should backreact in the geometry, we turn to the study of scalar fields in the generic Kerr-AdS 5 geometry. On one hand, the result for scattering coefficients can be obtained exactly using the isomonodromy technique, giving exact expressions in terms of c = 1 chiral conformal blocks. On the other hand, one can use the analogy between the scalar field equations to the Level 2 null field Ward identity in two dimensional Liouville field theory to write approximate expressions for the same coefficients in terms of semi-classical chiral Liouville conformal blocks. Surprisingly, the conformal block thus constructed has a well-behaved interpretation in terms of Liouville vertex operators.

“…In this case, we choose coordinates (x 1 , x 2 , ψ 0 , ψ 1 ), where x µ , µ = 1, 2, represent the PCKY eigenvalues and ψ i , i = 0, 1, are the Killing parameters of the 2 associated Killing vectors. Now if we set (x 1 , x 2 , ψ 0 , ψ 1 ) ≡ (p, ir, t, φ), the metric (2.1) is written as 8) where P (p) and Q(r) are 4 th order polynomials given by [15] …”

Section: Jhep07(2014)132mentioning

confidence: 99%

“…On a more mathematical perspective, the black hole scattering is linked to the monodromy of a Fuchsian equation, [4][5][6], a fact which drew some attention of late because of its relation to conformal field theory and Liouville field theory [7,8]. A Fuchsian differential equation is one whose solutions diverge polynomially at singular points.…”

Section: Introductionmentioning

confidence: 99%

Isomonodromy, Painlevé transcendents and scattering off of black holes

Novaes

Cunha

2014

Abstract:We apply the method of isomonodromy to study the scattering of a generic Kerr-NUT-(A)dS black hole. For generic values of the charges, the problem is related to the connection problem of the Painlevé VI transcendent. We review a few facts about Painlevé VI, Garnier systems and the Hamiltonian structure of flat connections in the Riemann sphere. We then outline a method for computing the scattering amplitudes based on Hamilton-Jacobi structure of Painlevé, and discuss the implications of the generic result to black hole complementarity.