Fishnets: Conformal Field Theories and Feynman Graphs

2 Sept 2024, 09:00 → 6 Sept 2024, 18:00 Europe/Berlin

Wegelerstr. 10 - Seminar Room 2.019 - 53115 Bonn (Bethe Center)

Fishnets: Conformal Field Theories and Feynman Graphs

Poster_Fishnets.jpg

Poster_Fishnets.jpg

Poster_Fishnets.pdf

Poster_Fishnets.pdf

ScheduleBetheForumFishnets.pdf

Monday 2 September

10:00 Registration & Coffee

1 Thermodynamic limits of fishnet graphs with various boundary conditions

Speaker: Benjamin Basso

Basso.pdf

12:00 Lunch Break

2 Double scaling limit of rectangular fishnets

Large fishnet Feynman graphs have different thermodynamical limits for periodic and open boundary conditions. A Coulomb-gas integral representation of $m\times n$ rectangular fishnets has been found by Basso and Dixon. Generically the free energy per site depends only on the aspect ratio $m/n$ and not on the two cross ratios. The bulk of the fishnet becomes sensitive to the positions of the four points in the spacetime only if some of the edges become light-like. Here I give the analytic solution in the double scaling limit where all four edges approach the light cone in a controlled way, generalising the result by Basso, Dixon, Kosower, Krajenbrink and Zhong.

Speaker: Ivan Kostov

Kostov.pdf

14:30 Coffee Break

3 Hexagons in the Z2 orbifold of N=4 SYM and in the fishnet theory

We consider three-point functions of protected scalar operators in the Z2 orbifold of N=4 SYM. We explain how the hexagonalisation procedure applies to these structure constants and allows to rederive results that were previously obtained through supersymmetric localisation. This setting is simple enough to permit us to test higher order terms of the hexagon conjecture. Our proposal for these higher order terms is inspired by rigorous computations of hexagon corrections in the fishnet theory.

Speaker: Gwenaël Ferrando

Ferrando.pdf

17:00 Reception

Tuesday 3 September

4 Fishnet Integrals in Two Dimensions

In this talk, I want to discuss the interplay between fishnet integrals in two dimensions, their associated geometry and their Yangian and permutation symmetry. I will extend previous observations for square-fishnet integrals also to hexagonal ones. In particular, I want to show how the Yangian and permutation symmetry of fishnet integrals fixes them and what this means on the level of the associated geometries. For hexagonal fishnet integrals, the star-triangle identity gives even rise to different geometric interpretations. An introduction into necessary concepts to understand the geometry of fishnet integrals, particularly into Calabi-Yau varieties, will be given in the talk.

Speaker: Christoph Nega

Nega.pdf

10:30 Coffee Break

5 Long Range Asymptotic Baxter-Bethe Ansatz for N=4 BFKL

The Balitsky-Fadin-Kuraev-Lipatov regime of N=4 is another corner where one can
make a contact with fishnet theory.
We demonstrate that the Balitsky-Fadin-Kuraev-Lipatov regime of maximally
supersymmetric Yang-Mills theory can be explicitly solved up to the L+1 order in
weak coupling by uncovering a novel long-range asymptotic Baxter-Bethe ansatz
for trajectories with L scalar fields. The set of equations we have found is
reminiscent of the Beisert-Eden-Staudacher equations for local operators but
instead applies to non-local operators corresponding to the horizontal Regge
trajectories. We also verify and give new predictions for the light-ray operator
spectrum by resummation of the leading singularities in our result.

Speaker: Nikolay Gromov

Gromov.pptx

12:00 Lunch Break

6 Recursive Structure of Four-Point Fishnet Integrals

We consider conformal four-point fishnet integrals to investigate how much of their mathematical structure in two spacetime dimensions carries over to higher dimensions. In particular, we discuss recursions in the loop order and spacetime dimension. This results e.g. in new expressions for conformal ladder integrals with generic propagator powers in all even dimensions and allows us to lift results on 2d fishnet integrals with underlying Calabi-Yau geometry to higher dimensions. Moreover, we demonstrate that the Basso-Dixon generalizations of these integrals obey different variants of the Toda equations of motion, thus establishing a connection to classical integrability and the family of so-called tau-functions. We then show that all of these integrals can be written in a double copy form that combines holomorphic and anti-holomorphic building blocks. Here integrals in higher dimensions are constructed from an intersection pairing of two-dimensional "periods'' together with their derivatives.

Speaker: Sven Stawinski

Stawinski.pdf

14:30 Coffee Break

7 Feynman Integrals and Hypergeometric Functions: Recent Results and Mathematica Implementations

Hypergeometric function theory and Feynman Integral calculus go hand in hand. A
series of recent investigations that culminated in the construction of several Mathematica packages that are based on Mellin-Barnes techniques, Method of Regions, hypergeometric function theory, etc., is reviewed in this talk to encourage the community to explore the use of these packages. Two-loop integrals appearing in chiral perturbation theory are a typical example that will be used for purposes of illustration. The talk will be easily accessible to early Ph.D. students in elementary particle physics and field theory.

Speaker: Balasubramanian Ananthanarayan

Anant.pdf

Wednesday 4 September

8 Log CFT structures in Celestial Holography

The S-Matrix in 4D asymptotically flat spacetimes is conjecturally dual to a 2D CFT living on the Celestial Sphere at null infinity. I will describe some basic features of this CFT and then focus on the appearance of log-CFT structures in the spectrum and the OPE.

Speaker: Akshay Yelleshpur Srikant

10:30 Coffee Break

9 Two-loop scattering in planar N=4 SYM theory

The doublebox, pentabox and double-pentagon Feynman integrals take on a
central role in planar N=4 SYM theory as they form a basis for two-loop scattering
in this theory. Their direct integration is challenging due to the presence of several
elliptic curves. After a brief review of the one-loop case, I will present recent
progress on the two-loop integrals, discussing their relation to higher-dimensional
one-loop integrals, their symbol structure, and connections with fishnet integrals.

Speaker: Anne Spiering

12:00 Lunch Break

10 Fishing for Bootstraps

Fishnet-type integrals arise in the strongly-deformed limit of planar N=4 super-Yang-Mills theory. What can we say about their role in scattering amplitudes and Wilson loops in the undeformed theory? Can we learn anything that would help bootstrap amplitudes in that theory? Is there any connection to antipodal duality?

Speaker: Lance Dixon

Dixon.pdf

14:30 Coffee Break

18:30 Dinner

Thursday 5 September

11 Conformal field theories from line defects, holography and the analytic bootstrap

Wilson lines are a prototypical example of defect in quantum field theory. After
reviewing the superconformal case - in which the one-dimensional defect CFT that
they define is particularly interesting - I will discuss some analytic tools that may
prove useful in this context, but are developed for generic 1d CFTs. Among them, a
representation of the four-point correlator as a Mellin amplitude and via a recently
derived dispersion relation.

Speaker: Valentina Forini

Forini.pdf

10:30 Coffee Break

12 Q-functions for fishnets and their Cosmic Galois properties

Thanks to the integrability of Fishnet Theory it is possible to obtain exact or high-order results for various observables, eg anomalous dimensions of local operators. Such exact results are particularly valuable for discovering new "theoretical phenomena" in QFT, and making (eventually defining) characterisations of them. For instance an (relatively old) observation is that the anmalous dimension of the operator tr(ϕ³) satisfies a tower of "coaction equations" to all-loop orders in coupling, which take the form of differential equations of multipe zeta values with derivatives with respect to single zetas. I will review this observation, and then show the advantages of working with the Fourier-transformed Q-functions: The coaction relations become far more obvious, and it is possible to write an extremely compact solution at any order to a simplified Baxter equation in terms of harmonic polylogarithms.

Speaker: Ömer Gürdoğan

12:00 Lunch Break

13 Integrable Staggered Fishnet sectors

In the case of either BMN operators or for insertion of magnons, the two-point functions of Fishnet theory are completely described by Integrable transfer matrices of a conformal Heisenberg magnet, in any spacetime dimension. On the other hand, the operator Tr(XY^{\dagger})^L lacks so far a description in terms of spin-chain integrability. I will illustrate this example starting from the 2D Fishnet case, arguing for a description of the Tr(XY^{\dagger})^L sector in terms of staggered spin-magnets and two-row Sklyanin transfer matrices whose integrability - ultimately based on the star-triangle identity - is realized by a Reflection Algebra instead of the quantum Yang-Baxter Equation.

Speaker: Enrico Olivucci

Olivucci.pdf

15:30 Coffee & Cake

14 Bethe Colloquium: Chiral phase transition in QCD and in solvable models

Bethe Ansatz is an elegant method to solve strongly interacting quantum
systems. It has many applications, often quite unexpected.
One such is chiral phase transition.
Chiral symmetry restoration is a spectacular phenomenon expected to happen
in QCD at high enough temperatures and densities, but mapping the
phase diagram of QCD is a very difficult problem. Similar phenomena occur
in simpler two-dimensional models, some even exactly solvable.
The phase diagram can then be explored by powerful techniques of Bethe Ansatz.

Speaker: Konstantin Zarembo

Kostya.pdf

Kostya.pptx

Friday 6 September

15 Supersymmetric diagrams and the dynamical fishnet

We will introduce the superspace formulation of the double-scaled beta-deformation of N=4 SYM, a dynamical fishnet chiral CFT. This superconformal QFT admits regular brick wall vacuum Feynman supergraphs in the planar limit. The superpropagators can be interpreted as lattice weights and the vacuum graphs as periodic partition functions, such that by the method of inversion relations the free energy in the thermodynamic limit is obtained. In the QFT context this quantity corresponds to its critical coupling. Furthermore, we consider the superspace formulation of the analog double-scaled beta-defomation of ABJM theory and find it to admit square-lattice fishnet supergraphs. The structure of the graphs allows to transfer techniques from the bi-scalar fishnet theory, among them the calculation of the critical coupling as well as all-loop results for anomalous dimensions.

Speakers: Matthias Staudacher, Moritz Kade

Staudacher:Kade.pdf

10:30 Coffee Break

16 Exploring Differential Equations for Yangian Symmetric Feynman Integrals

Speaker: Victor Mishnyakov

Mishnyakov.pdf

12:00 The End