We present significant evidence that the powerful property of Yangian invariance extends to a new large class of conformally invariant Feynman integrals. Our results apply to planar Feynman diagrams in any spacetime dimension dual to an arbitrary network of intersecting straight lines on a plane (Baxter lattice), with propagator powers determined by the geometry. We formulate Yangian symmetry in terms of a chain of Lax operators acting on the fixed coordinates around the graph, and we also extend this construction to the case of infinite-dimensional auxiliary space. Yangian invariance leads to new differential and integral equations for individual, highly nontrivial, Feynman graphs, and we present them explicitly for several examples. The graphs we consider determine correlators in the recently proposed loom fishnet CFTs. We also describe a generalization to the case with interaction vertices inside open faces of the diagram. Our construction unifies and greatly extends the known special cases of Yangian invariance to likely the most general family of integrable scalar planar graphs.

France

Vladimir Kazakov

Fedor Levkovich-Maslyuk

Russia

Victor Mishnyakov

Integrable Feynman graphs and Yangian symmetry on the loom

We introduce a comprehensive framework for analyzing finite lattice Yang-Mills theory and finite matrix models. Utilizing this framework, we examine the bootstrap approach to SU(2) Lattice Yang-Mills Theory in 2,3 and 4 dimensions. The SU(2) Makeenko-Migdal loop equations on the lattice are linear and closed exclusively on single-trace Wilson loops. This inherent linearity significantly improves the efficiency of the bootstrap approach by leveraging the problem’s convexity, permitting the inclusion of Wilson loops up to length 24. The exact upper and lower margins for the free energy per plaquette, derived from our bootstrap method, demonstrate good agreement with Monte Carlo data, achieving precision within 0 1% for the physically relevant range of couplings in both three and four dimensions. Additionally, our bootstrap data provides estimates of the string tension, in qualitative agreement with existing Monte Carlo computations.

Canada

Zechuan Zheng

Bootstrap for finite N lattice Yang-Mills theory

The Checkerboard conformal field theory is an interesting representative of a large class of non-unitary, logarithmic Fishnet CFTs (FCFT) in arbitrary dimension which have been intensively studied in the last years. Its planar Feynman graphs have the structure of a regular square lattice with checkerboard colouring. Such graphs are integrable since each coloured cell of the lattice is equal to an R-matrix in the principal series representations of the conformal group. We compute perturbatively and numerically the anomalous dimension of the shortest single-trace operator in two reductions of the Checkerboard CFT: the first one corresponds to the Fishnet limit of the twisted ABJM theory in 3D, whereas the spectrum in the second, 2D reduction contains the energy of the BFKL Pomeron. We derive an analytic expression for the Checkerboard analogues of Basso-Dixon 4-point functions, as well as for the class of Diamond-type 4-point graphs with disc topology. The properties of the latter are studied in terms of OPE for operators with open indices. We prove that the spectrum of the theory receives corrections only at even orders in the loop expansion and we conjecture such a modification of Checkerboard CFT where quantum corrections occur only with a given periodicity in the loop order.

Mikhail Alfimov

Israel

Gwenaël Ferrando

Enrico Olivucci

Checkerboard CFT

Abstract We examine the fundamental Kazakov–Migdal (FKM) model on a generic graph, whose partition function is represented by the Ihara zeta function weighted by unitary matrices. The FKM model becomes unstable in the critical strip of the Ihara zeta function. We discover a duality between small and large couplings, associated with the functional equation of the Ihara zeta function for regular graphs. Although the duality is not precise for irregular graphs, we show that the effective action in the large coupling region can be represented by a summation of all possible Wilson loops on a graph similar to that in the small coupling region. We estimate the phase structure of the FKM model in both the small and large coupling regions by comparing it with the Gross–Witten–Wadia model. We further validate the theoretical analysis through detailed numerical simulations.

Japan

So Matsuura

Kazutoshi Ohta

Phases and Duality in the Fundamental Kazakov–Migdal Model on the Graph L-functions in geometry and some applications Multipath zeta functions of graphs

We report the explicit solution for the vacuum state of the two-dimensional <math><mrow><mi>SU</mi></mrow><mo>(</mo><mi>N</mi><mo>)</mo></math> principal chiral model at large <math><mrow><mi>N</mi></mrow></math> for an arbitrary set of chemical potentials and any interaction strength, a unique result of such kind for an asymptotically free quantum field theory. The solution matches one-loop perturbative calculation at weak coupling, and in the opposite strong-coupling regime exhibits an emergent spacial dimension from the continuum limit of the <math><mrow><mi>SU</mi></mrow><mo>(</mo><mi>N</mi><mo>)</mo></math> Dynkin diagram.

Evgeny Sobko

Denmark

Konstantin Zarembo

Large-<math><mi>N</mi></math> Principal Chiral Model in Arbitrary External Fields

We generalize local operators of the leading twist-2 of <math><mi>N</mi><mo>=</mo><mn>4</mn></math> SYM theory to the case of complex Lorentz spin j using principal series representation of <math><mi>s</mi><mi>l</mi><mo>(</mo><mn>2</mn><mo>,</mo><mi>R</mi><mo>)</mo></math>. We give the direct computation of correlation function of two such non-local operators in the BFKL regime when <math><mi>j</mi><mo>→</mo><mn>1</mn></math>. The correlator appears to have the expected conformal coordinate dependence governed by the anomalous dimension of twist-2 operator in NLO BFKL approximation predicted by Kotikov and Lipatov.

United States

Ian Balitsky

Two-point correlator of twist-2 light-ray operators in N=4 SYM in BFKL approximation

We propose a broad class of d-dimensional conformal field theories of SU(N) adjoint scalar fields generalising the 4d Fishnet CFT (FCFT) discovered by Ö. Gürdogan and one of the authors as a special limit of γ-deformed   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 SYM theory. In the planar N → ∞ limit the FCFTs are dominated by the “fishnet” planar Feynman graphs. These graphs are explicitly integrable, as was shown long ago by A. Zamolodchikov. The Zamolodchikov’s construction, based on the dual Baxter lattice (straight lines on the plane intersecting at arbitrary slopes) and the star-triangle identities, can serve as a “loom” for “weaving” the Feynman graphs of these FCFTs, with certain types of propagators, at any d. The Baxter lattice with M different slopes and any number of lines parallel to those, generates an FCFT consisting of M (M – 1) fields and a certain number of chiral vertices of different valences with distinguished couplings. These non-unitary, logarithmic CFTs enjoy certain reality properties for their spectrum due to a symmetry similar to the PT-invariance of non-hermitian hamiltonians proposed by C. Bender and S. Boettcher. We discuss in more detail the theories generated by a loom with M = 2, 3, 4, and the generalisation of the loom FCFTs for spinning fields in 4d.

The loom for general fishnet CFTs

Using the matrix-forest theorem and the Parisi-Sourlas trick we formulate and solve a one-matrix model with non-polynomial potential which provides perturbation theory for massive spinless fermions on dynamical planar graphs. This is a lattice version of 2d quantum gravity coupled to massive spinless fermions. Our model equivalently describes the ensemble of spanning forests on the same graphs. The solution is formulated in terms of an elliptic curve. We then focus on a near-critical scaling limit when both the graphs and the trees in the forests are macroscopically large. In this limit we obtain one-point scaling functions (condensates), parameterized in terms of the Lambert function. Our results provide a rare example where one can explore the flow between two gravity models — in this case, the theories of conformal matter coupled to 2d gravity with c = −2 (large trees regime) and c = 0 (small trees regime). We also compute the disc partition functions with Dirichlet and Neumann boundary conditions in the same critical limit.

Alexander Gorsky

A flow in the forest

We study the <math><mi>S</mi><mi>U</mi><mo>(</mo><mi>∞</mi><mo>)</mo></math> lattice Yang-Mills theory at the dimensions <math><mi>D</mi><mo>=</mo><mn>2</mn></math>, 3, 4 via the numerical bootstrap method. It combines the loop equations, with a cutoff <math><msub><mi>L</mi><mi>max</mi></msub></math> on the maximal length of loops, and positivity conditions on certain matrices of Wilson loop averages. Our algorithm is inspired by the pioneering paper of P. D. Anderson and M. Kruczenski [Nucl. Phys. B921, 702 (2017)] but it is significantly more efficient, as it takes into account the symmetries of the lattice theory and uses the relaxation procedure in line with our previous work on matrix bootstrap. We thus obtain rigorous upper and lower bounds on the plaquette average at various couplings and dimensions. For <math><mi>D</mi><mo>=</mo><mn>4</mn><mo>,</mo><msub><mi>L</mi><mi>max</mi></msub><mo>=</mo><mn>16</mn></math> the lower bound data appear to be close to the Monte Carlo data in the strong coupling phase and the upper bound data in the weak coupling phase reproduce well the 3-loop perturbation theory. Our results suggest that this bootstrap approach can provide a tangible alternative to the, so far uncontested, Monte Carlo approach.

Bootstrap for lattice Yang-Mills theory

Abstract In this paper, we consider an extended Kazakov–Migdal model defined on an arbitrary graph. The partition function of the model, which is expressed as the summation of all Wilson loops on the graph, turns out to be represented by the Bartholdi zeta function weighted by unitary matrices on the edges of the graph. The partition function on the cycle graph at finite N is expressed by the generating function of the generalized Catalan numbers. The partition function on an arbitrary graph can be exactly evaluated at large N, which is expressed as an infinite product of a kind of deformed Ihara zeta function. The non-zero-area Wilson loops do not contribute to the leading part of the 1/N expansion of the free energy but to the next leading. The semi-circle distribution of the eigenvalues of the scalar fields is still an exact solution of the model at large N on an arbitrary regular graph, but it reflects only zero-area Wilson loops.