Based on prototypical example of Al.Zamolodchikov's recursion relations for the four point conformal block and using recently proposed Alday-Gaiotto-Tachikawa (AGT) conjecture, recursion relations are derived for the generalized prepotential of ${\cal N}=2$ SYM with $f=0,1,2,3,4$ (anti) fundamental or an adjoint hypermultiplets. In all cases the large expectation value limit is derived explicitly. A precise relationship between generic 1-point conformal block on torus and specific 4-point conformal block on sphere is established. In view of AGT conjecture this translates into a relation between partition functions with an adjoint and 4 fundamental hypermultiplets.
Fields and Strings Laboratory, Institute of Physics Ecole Polytechnique Fédérale de Lausanne（洛桑联邦理工学院物理研究所弦场实验室）；Scuola Normale Superiore and INFN（比萨高等师范学院和意大利国家核物理研究所）
We study the renormalization group flow equations for correlation functions of weakly coupled quantum field theories in AdS. Taking the limit where the external points approach the conformal boundary, we obtain a flow of conformally invariant correlation functions. We solve the flow for one- and two-point functions and show that the corrections to the conformal dimensions can be obtained as an integral over the Mellin amplitude of the four-point function. We also derive the flow of the Mellin amplitude for higher $n$-point functions. We then consider the flows at tree level and one loop (in Ad…
We study multipoint correlators of protected scalars on the Maldacena-Wilson line in $\mathcal{N}=4$ SYM. Working at weak coupling in the planar limit, we derive an explicit recursion relation that captures next-to-leading order correlators with an arbitrary number of insertions of the fundamental scalar field. By pinching fundamental scalars together, we can build composite protected operators with higher values of the R-charge. Our result then encompasses arbitrary $n$-point correlators of protected operators with arbitrary weight. As a demonstration of our method, we give explicit formulae …
This thesis is divided into two parts, where in the first part we investigate the computation of Virasoro 1-point blocks on the torus in the framework of Zamolodchikov's recursion relation. It is widely accepted that this recursion relation contains unphysical poles in the central charge $c$. At each order we conjecture how the pole free expressions depend on the internal and external conformal dimensions and central charge, and propose how to compute it numerically. In this thesis, we have calculated the pole free expression up to order 4. In the second part we introduce a conformal field the…
We introduce the software blocks_3d for computing four-point conformal blocks of operators with arbitrary Lorentz representations in 3d CFTs. It uses Zamolodchikov-like recursion relations to numerically compute derivatives of blocks around a crossing-symmetric configuration. It is implemented as a heavily optimized, multithreaded, C++ application. We give performance benchmarks for correlators containing scalars, fermions, and stress tensors. As an example application, we recompute bootstrap bounds on four-point functions of fermions and study whether a previously observed sharp jump can be e…
We point out that an infinite class of Witten diagrams is invariant under a Yangian symmetry. These diagrams are building blocks of holographic correlators and are related by a web of differential recursion relations. We show that Yangian invariance is equivalent to the consistency conditions of the recursion relations.
We consider $U(N)$ $\mathcal N=4$ super Yang-Mills theory and discuss how to extract the strong coupling limit of non-planar corrections to observables involving the $\frac{1}{2}$-BPS Wilson loop. Our approach is based on a suitable saddle point treatment of the Eynard-Orantin topological recursion in the Gaussian matrix model. Working directly at strong coupling we avoid the usual procedure of first computing observables at finite planar coupling $λ$, order by order in $1/N$, and then taking the $λ\gg 1$ limit. In the proposed approach, matrix model multi-point resolvents take a simplified fo…
We show that time intervals of width $Δτ$ in 3-dimensional conformal field theories (CFT$_3$) on the Lorentzian cylinder admit an infinite dimensional symmetry enhancement in the limit $Δτ\rightarrow 0$. The associated vector fields are approximate solutions to the conformal Killing equations in the strip labelled by a function and a conformal Killing vector on the sphere. An Inonu-Wigner contraction yields a set of symmetry generators obeying the extended BMS$_4$ algebra. We analyze the shadow stress tensor Ward identities in CFT$_d$ on the Lorentzian cylinder with all operator insertions in …
In this paper we investigate different ways of deriving the A-cycle period as a series in instanton counting parameter $q$ for ${\cal N}=2$ SYM with up to four antifundamental hypermultiplets in NS limit of $Ω$ background. We propose a new method for calculating the period and demonstrate its efficiency by explicit calculations. The new way of doing instanton counting is more advantageous compared to known standard techniques and allows to reach substantially higher order terms with less effort. This approach is applied for the pure case as well as for the case with several hypermultiplets. We…
We derive parametric integral representations for the general $n$-point function of scalar operators in momentum-space conformal field theory. Recently, this was shown to be expressible as a generalised Feynman integral with the topology of an $(n-1)$-simplex, featuring an arbitrary function of momentum-space cross ratios. Here, we show all graph polynomials for this integral can be expressed in terms of the first and second minors of the Laplacian matrix for the simplex. Computing the effective resistance between nodes of the corresponding electrical network, an inverse parametrisation is fou…
With a view to understanding extended-BMS symmetries in the framework of the $AdS_4/CFT_3$ correspondence, asymptotically AdS geometries are constructed with null impulsive shockwaves involving a discontinuity in superrotation parameters. The holographic dual is proposed to be a two-dimensional Euclidean defect conformal field localized on a particular timeslice in a three-dimensional conformal field theory on de Sitter spacetime. The defect conformal field theory generates a natural action of the Virasoro algebra. The large radius of curvature limit $\ell\to\infty$ yields spacetimes with nont…
