I will speak about my recent work with Zechuan Zheng where we study the SU(Nc) lattice Yang-Mills theory in the t Hooft limit Nc -> infinity, at dimensions D=2,3,4, via the numerical bootstrap method. It combines the Makeenko-Migdal loop equations, with the cut-off L on maximal length of wilson loops, and the positivity conditions on certain correlation matrices. We thus obtain rigorous upper and lower bounds on plaquette average at various couplings. The results are quickly improving with the increase of the cutoff L. In particular, for D=4 and L=16, the upper bound data in the most interesting weak coupling phase are not far from the Monte-Carlo results and they reproduce well the 3-loop perturbation theory. We also attempt to extract the information about the gluon condensate from this data. Our results suggest that bootstrap can provide a tangible alternative to, so far uncontested, Monte Carlo approach. I will also mention our bootstrap results for an "unsolvable" two-matrix model in the large N limit, where this method appears to be superior in efficiency over Monte Carlo.

Dually weighted graphs (DWG) are planar Feynman graphs bearing two sets of couplings: one set of usual couplings $t_n$ attached to the vertices of valence $n$, and another set of dual couplings $t_n^*$ attached to the faces (dual vertices) of valence $n$. Such couplings allow a deep control on possible shapes of planar graphs. For example, if one turns on only the couplings $t_4$ and $t_4^*$ the graph takes a ''fishnet form'' of a regular square lattice. The problem of counting of such graphs can be formulated as a modified hermitian one matrix model with an extra constant matrix. The partition function can be then represented in terms of the ''character expansion'' over Young tableaux, solvable by the saddle point approximation. I will review old results on DWG from my papers with M.Staudacher and Th.Wynter, including the techniques of computing Schur characters of a large Young tableau and deriving the elliptic algebraic curve for counting of planar quadrangulations. Then I will present new results from our ongoing work with F.Levkovich-Maslyuk where we count the disc quadrangulations with large, macroscopic area and boundary. This allows to extract interesting continuous limit of fluctuating 2d geometry, interpolating between the ''almost'' flat disc with a few dynamical conical defects and the disc partition function for pure 2d quantum gravity, generalizing old results for the spherical topology. ---- Part of the London Integrability Journal Club. Please register at integrability-london.weebly.com if you are a new participant. The link will be emailed on Tuesday.

I will discuss the properties of a family of four-dimensional CFTs, recently proposed by O.Gurdogan and myself, emerging as a double scaling limit of weakly coupled and strongly gamma-twisted N=4 SYM theory. These non-unitary CFTs inherit the integrability of N=4 SYM in the planar limit and present a unique opportunity of a non-perturbative study of four-dimensional conformal physics. Important physical quantities are dominated by a limited subset of Feynman graphs (such as "fishnet" graphs for the simplest, bi-scalar model). I present the results of exact calculation of some of these quantities, such as anomalous dimensions of local operators, some 3- and 4point correlation functions and scattering amplitudes, by means of spin chain techniques or the quantum spectral curve (QSC) approach originally proposed for N=4 SYM.

In the context of integrability in AdS/CFT correspondence, we construct the Green-Schwarz-Metsaev-Tseytlin supersting in the S(5) x R(1) sector, from a set of physical particles on a circle in the integrable 1+1D sigma-model with Zamolodchikov's S-matrix. We reproduce, in the limit of high density of the particles, all finite gap KMMZ solutions for classical strings and the quantum AFS string Bethe ansatz equations, as an approximation to our construction.