Konstantin Alkalaev

A bstract: We establish a correspondence between conformal partial waves on flat, thermal, and defect backgrounds using the shadow formalism. We demonstrate that scalar one-point thermal blocks can be systematically obtained from their four-point flat-space and two-point defect counterparts by considering specific operator configurations. This framework allows us to derive the thermal Casimir equation as a diagonal reduction of the flat-space Casimir system without introducing chemical potentials. We further show that defect two-point blocks with a spin- l exchange operator correspond to thermal one-point blocks for an external spin- l operator.

We show that Feynman diagrams in AdS$_2$ space can be decomposed into infinite series of matrix elements of Wilson line network operators. The case of the 3-point scalar Feynman diagram with endpoints in the bulk is studied in detail. The resulting decomposition is similar to the conformal block decomposition of Witten diagrams, i.e. it comprises a single-trace term and infinite sums of double-trace terms. We derive a number of AdS propagator identities which relate the standard bulk-to-bulk propagators with the modified bulk-to-bulk propagators of two different types responsible for extracting single-trace and double-trace terms.

A bstract: We explicitly construct a class of multivariate generalized hypergeometric series which is conjectured in our previous paper [1] to calculate multipoint one-loop parametric conformal integrals in D dimensions. Our approach is based on a simple diagrammatic algorithm which systematically builds both arguments and series coefficients in terms of a convex polygon which is part of the Baxter lattice. The examples of the box, pentagon, and hexagon integrals are considered in detail.

We propose a systematic approach to calculating n-point one-loop parametric conformal integrals in D dimensions which we call the reconstruction procedure. It relies on decomposing a conformal integral over basis functions which are generated from a set of master functions by acting with the cyclic group ℤ _n . In order to identify the master functions we introduce a bipartite Mellin-Barnes representation by means of splitting a given conformal integral into two additive parts, one of which can be evaluated explicitly in terms of multivariate generalized hypergeometric series. For the box and pentagon integrals (i.e. n = 4, 5) we show that a computable part of the bipartite representation contains all master functions. In particular, this allows us to evaluate the parametric pentagon integral as a sum of ten basis functions generated from two master functions by the cyclic group ℤ₅. The resulting expression can be tested in two ways. First, when one of propagator powers is set to zero, the pentagon integral is reduced to the known box integral, which is also rederived through the reconstruction procedure. Second, going to the non-parametric case, we reproduce the known expression for the pentagon integral given in terms of logarithms derived earlier within the geometric approach to calculating conformal integrals. We conclude by considering the hexagon integral (n = 6) for which we show that those basis functions which follow from the computable part of the bipartite representation are not enough and more basis functions are required. In the second part of our project we will describe a method of constructing a complete set of master/basis functions in the n-point case.

A bstract: We find a holographic reconstruction formula for gravitational Wilson line network operators in AdS ₂ evaluated between Ishibashi states of the algebra sl (2 , ℝ). It is given in integral form where the integrand is the global conformal block multiplied by a smearing function which is the product of the scalar bulk-to-boundary propagators. The integral can be explicitly calculated as multidimensional series of which arguments are rational functions of endpoint coordinates. In the case of two and three endpoints the resulting expressions allow one to establish a number of relations between the gravitational Wilson line networks and Witten diagrams for massive scalar fields in AdS ₂ .

Using the shadow formalism we find global conformal blocks of torus CFT₂. It is shown that n-point torus blocks in the “necklace” channel (a loop with n legs) are expressed in terms of a hypergeometric-type function which we refer to as the necklace function.