We show that Feynman diagrams in AdS <inf>2</inf> 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.

Russia

Konstantin Alkalaev

Vladimir Khiteev

Wilson network decomposition of AdS Feynman diagrams in two dimensions

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 $\textit{D}$ 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.

Semyon Mandrygin

Multipoint conformal integrals in $\textit{D}$ dimensions. Part II. Polygons and basis functions

We propose a systematic approach to calculating $\textit{n}$-point one-loop parametric conformal integrals in $\textit{D}$ 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 . 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.

Multipoint conformal integrals in $\textit{D}$ dimensions. Part I. Bipartite Mellin-Barnes representation and reconstruction

We find a holographic reconstruction formula for gravitational Wilson line network operators in AdS evaluated between Ishibashi states of the algebra $\textit{sl}$(2$\textit{,}$ 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 .

Holographic reconstruction for AdS Wilson line networks and scalar Witten diagrams

We consider α-heavy conformal operators in CFT$_{2}$ which dimensions grow as <math><mi>h</mi><mo>=</mo><mi>O</mi><mo>(</mo><msup><mrow><mi>c</mi></mrow><mrow><mi>α</mi></mrow></msup><mo>)</mo></math> with α being non-negative rational number and conjecture that the large-c asymptotics of the respective 4-point Virasoro conformal block is exponentiated similar to the standard case of <math><mi>α</mi><mo>=</mo><mn>1</mn></math>. It is shown that the leading exponent is given by a Puiseux polynomial which is a linear combination of power functions in the central charge with fractional powers decreasing from α to 0 according to some pattern. Our analysis is limited by considering the first six explicit coefficients of the Virasoro block function in the coordinate. For simplicity, external primary operators are chosen to be of equal conformal dimensions that, therefore, includes the case of the vacuum conformal block. The consideration is also extended to the 4-point <math><msub><mrow><mi>W</mi></mrow><mrow><mn>3</mn></mrow></msub></math> conformal block of four semi-degenerate operators, in which case the exponentiation hypothesis works the same way. Here, only the first three block coefficients can be treated analytically.

Israel

Pavel Litvinov

A note on the large- Conformal Block Asymptotics and -heavy Operators

We develop the thermal shadow formalism to study the conformal blocks decomposition in D-dimensional conformal field theory on   <math> <msubsup> <mi>S</mi> <mi>β</mi> <mn>1</mn> </msubsup> <mo>×</mo> <msup> <mi>S</mi> <mrow> <mi>D</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </math>  $$ {\mathbbm{S}}_{\beta}^1\times {\mathbbm{S}}^{D-1} $$ , where the temperature is T = β −1. It is demonstrated that both the 1-point thermal (T ≠ 0) conformal blocks and the 4-point plane (T = 0) conformal blocks are defined by the same 4-point conformal integral. It is shown that up to power prefactors the 1-point thermal conformal block is given by the fourth Appell function.

One-point thermal conformal blocks from four-point conformal integrals

We develop the relation between gravitational Wilson line networks, defined as a particular product of Wilson line operators averaged over the cap states, and conformal correlators in the context of the AdS <inf>2</inf>/CFT <inf>1</inf> correspondence. The <italic>n</italic>-point <math altimg='si1.svg'><mi>s</mi><mi>l</mi><mo stretchy='false'>(</mo><mn>2</mn><mo>,</mo><mi mathvariant='double-struck'>R</mi><mo stretchy='false'>)</mo></math> comb channel global conformal block in CFT <inf>1</inf> is explicitly calculated by means of the extrapolate dictionary relation from the gravitational Wilson line network with <italic>n</italic> boundary endpoints stretched in AdS <inf>2</inf>. Remarkably, the Wilson line calculation directly yields the conformal block in a particularly simple form which up to the leg factor is given by the comb function of cross-ratios. It is also found that the comb channel structure constants are expressed in terms of factorials and triangle functions of conformal weights which form determines fusion rules for a given 3-valent vertex. We obtain analytic expressions for the Wilson line matrix elements in AdS <inf>2</inf> which are building blocks of the Wilson line networks. We analyze general cap states and specify those which lead to asymptotic values of the Wilson line networks interpreted as boundary correlators of CFT <inf>1</inf> primary operators. The cases of (in)finite-dimensional <math altimg='si1.svg'><mi>s</mi><mi>l</mi><mo stretchy='false'>(</mo><mn>2</mn><mo>,</mo><mi mathvariant='double-struck'>R</mi><mo stretchy='false'>)</mo></math> modules carried by Wilson lines are treated on equal footing that boils down to consideration of singular submodules and their contributions to the Wilson line matrix elements.

Andrey Kanoda

Wilson networks in AdS and global conformal blocks

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.

Torus shadow formalism and exact global conformal blocks

We consider the conformal block decomposition in arbitrary exchange channels of a two-dimensional conformal field theory on a torus. The channels are described by diagrams built of a closed loop with external legs (a necklace sub-diagram) and trivalent vertices forming trivalent trees attached to the necklace. Then, the n-point torus conformal block in any channel can be obtained by acting with a number of OPE operators on the k-point torus block in the necklace channel at k = 1, …, n. Focusing on the necklace channel, we go to the large-c regime, where the Virasoro algebra truncates to the sl(2, ℝ) subalgebra, and obtain the system of the Casimir equations for the respective k-point global conformal block. In the plane limit, when the torus modular parameter q → 0, we explicitly find the Casimir equations on a plane which define the (k + 2)-point global conformal block in the comb channel. Finally, we formulate the general scheme to find Casimir equations for global torus blocks in arbitrary channels.

Mikhail Pavlov

Torus conformal blocks and Casimir equations in the necklace channel

We study the boundary effective action of the colored version of the Jackiw-Teitelboim (JT) gravity. We derive the boundary action, which is the color generalization of the Schwarzian action, from the su(N, N) BF formulation of the colored JT gravity. Using different types of the SU(N, N) group decompositions both the zero and finite temperature cases are elaborated. We provide the semi-classical perturbative analysis of the boundary action and discuss the instability of the spin-1 mode and its implication for the quantum chaos. A rainbow-AdS2 geometry is introduced where the color gauge symmetry is spontaneously broken.