Holographic thermal correlators from supersymmetric instantons

We present an exact formula for the thermal scalar two-point function in four-dimensional holographic conformal field theories. The problem of finding it reduces to the analysis of the wave equation on the AdS-Schwarzschild background. The two-point function is computed from the connection coefficients of the Heun equation, which can be expressed in terms of the Nekrasov-Shatashvili partition function of an SU(2) supersymmetric gauge theory with four fundamental hypermultiplets. The result is amenable to numerical evaluation upon truncating the number of instantons in the convergent expansion of the partition function. We also examine it analytically in various limits. At large spin the instanton expansion of the thermal two-point function directly maps to the light-cone bootstrap analysis of the heavy-light four-point function. Using this connection, we compute the OPE data of heavy-light double-twist operators. We compare our prediction to the perturbative results available in the literature and find perfect agreement.Comment: 9 pages + appendices, 2 figures. v2: typos corrected, references adde

Black holes, Heun functions and instanton counting

In this thesis we explore and exploit the relationships between black hole physics, the mathematics of Heun’s equation, Liouville conformal field theory and 4d supersymmetric gauge theories. We study the modular properties of a class of degenerate conformal blocks, including the recently introduced irregular or confluent conformal blocks, computing their connection matrices. In a certain limit, these connection matrices descend to the connection matrices for Heun functions, which are a generalization of the hypergeometric functions. We give explicit formulae, computable in terms of the Nekrasov partition functions of a class of 4d supersymmetric gauge theories. The above study is motivated by the frequent appearance of Heun’s equation in physics, one interesting example being the perturbations of black holes. Using the connection formulae that we have obtained, we analytically solve the equations governing the perturbations of the 4d Kerr black hole and of the 5d AdS-Schwarzschild black hole

Irregular Liouville correlators and connection formulae for Heun functions

We perform a detailed study of a class of irregular correlators in Liouville Conformal Field Theory, of the related Virasoro conformal blocks with irregular singularities and of their connection formulae. Upon considering their semi-classical limit, we provide explicit expressions of the connection matrices for the Heun function and a class of its confluences. Their calculation is reduced to concrete combinatorial formulae from conformal block expansions.Comment: 61 pages, many diagrams, 2 figures, huge list of symbols, comments welcom

Holographic thermal correlators from supersymmetric instantons

We present an exact formula for the thermal scalar two-point function in four-dimensional holographic conformal field theories. The problem of finding it reduces to the analysis of the wave equation on the AdS-Schwarzschild background. The two-point function is computed from the connection coefficients of the Heun equation, which can be expressed in terms of the Nekrasov-Shatashvili partition function of an $SU(2)$ supersymmetric gauge theory with four fundamental hypermultiplets. The result is amenable to numerical evaluation upon truncating the number of instantons in the convergent expansion of the partition function. We also examine it analytically in various limits. At large spin the instanton expansion of the thermal two-point function directly maps to the light-cone bootstrap analysis of the heavy-light four-point function. Using this connection, we compute the OPE data of heavy-light double-twist operators. We compare our prediction to the perturbative results available in the literature and find perfect agreement