Related papers: AGT correspondence, (q-)Painlev\`e equations and m…

In this note we present some results on the convergence of Nekrasov partition functions as power series in the instanton counting parameter. We focus on $U(N)$ ${\mathcal N}=2$ gauge theories in four dimensions with matter in the adjoint…

Bethe/Gauge correspondence as it is usually stated is ill-defined in five dimensions and needs a "non-perturbative" completion; a related problem also appears in three dimensions. It has been suggested that this problem, probably due to…

We study Weyl symmetry in quadrivalently glued 5-brane webs of rank $N$ $(D_4,D_4)$ conformal matter theories on a circle. We find that these theories all have affine $E_8$ Weyl symmetry in their brane webs, which indicates that they all…

It was recently conjectured that the AGT correspondence between the $U(r)$-instanton counting on $\mathbb R^4/\mathbb Z_p$ and the two-dimensional field theories with the conformal symmetry algebra $\mathcal A(r,p)$ can be considered as a…

We study conformal field theories (CFTs) on curved spaces including both orientable and unorientable manifolds possibly with boundaries. We first review conformal transformations on curved manifolds. We then compute the identity components…

BGG-equations are geometric overdetermined systems of PDEs on parabolic geometries. Normal solutions of BGG-equations are particularly interesting and we give a simple formula for the necessary and sufficient additional integrability…

The off-diagonal Bethe ansatz method is generalized to the integrable model associated with the $sp(4)$ (or $C_2$) Lie algebra. By using the fusion technique, we obtain the complete operator product identities among the fused transfer…

We discuss the recent proposal of arXiv:1608.05351 about generalization of the RTT relation to network matrix models. We show that the RTT relation in these models is modified by a nontrivial, but essentially abelian anomaly cocycle, which…

Gauge-invariant polynomial functions of matrix and tensor variables capture combinatorial structures of gauge-string duality, which can be usefully organised using finite-dimensional associative algebras. I review recent work on eigenvalue…

We review recent progress in 2D gravity coupled to $d<1$ conformal matter, based on a representation of discrete gravity in terms of random matrices. We discuss the saddle point approximation for these models, including a class of related…

We present a conformally invariant generalized form of the free particle action by connecting the wave and particle aspects through gravity. Conformal invariance breaking is introduced by choosing a particular configurat$ of dynamical…

We study the properties of the conformal blocks of the conformal field theories with Virasoro or W-extended symmetry. When the conformal blocks contain only second-order degenerate fields, the conformal blocks obey second order differential…

We present a geometric description, based on the affine Weyl group E_{6}^{(1)}, of two discrete analogues of the Painlev\'e VI equation, known as the asymmetric q-P_{V} and asymmetric d-P_{IV}. This approach allows us to describe in a…

Recent explorations of the AdS/CFT correspondence have unveiled integrable structures underlying both planar N = 4 super-Yang-Mills theory and type IIB string theory on AdS_5 x S^5. Integrability in the gauge theory emerges from the fact…

In this paper the relation between 2d topological gauge theories and Bethe Ansatz equations is reviewed. In addition we present some new results and clarifications. We hope the relations discussed here are particular examples of more…

Generic c=1 four-point conformal blocks on the Riemann sphere can be seen as the coefficients of Fourier expansion of the tau function of Painlev\'e VI equation with respect to one of its integration constants. Based on this relation, we…

We show that conformal manifolds in $d\geq 3$ conformal field theories with at least 4 supercharges are K\"ahler-Hodge, thus extending to 3d ${\cal N}=2$ and 4d ${\cal N}=1$ similar results previously derived for 4d ${\cal N}=2$ and ${\cal…

We analyze conformal blocks with multiple (semi-)degenerate field insertions in Liouville/Toda conformal field theories an show that their vector space is fully reproduced by the four-dimensional limit of open topological string amplitudes…

It is shown that a generalized Ito system of four coupled nonlinear evolution equations passes the Painleve test for integrability in five distinct cases, of which two were introduced recently by Tam, Hu and Wang. A conjecture is formulated…

Higher order Painleve equations invariant under extended affine Weyl groups $A^{(1)}_n$ are obtained through self-similarity limit of a class of pseudo-differential Lax hierarchies with symmetry inherited from the underlying generalized…