We consider the Ising model at its critical temperature with external magnetic field ha15/8 on the square lattice with lattice spacing a. We show that the truncated two-point function in this model decays exponentially with a rate independent of a as a ↓ 0. As a consequence, we show exponential decay in the near-critical scaling limit Euclidean magnetization field. For the lattice model with a = 1, the mass (inverse correlation length) is of order h8/15 as h ↓ 0; for the Euclidean field, it equals exactly Ch8/15 for some C. Although there has been much progress in the study of critical scaling limits, results on near-critical models are far fewer due to the lack of conformal invariance away from the critical point. Our arguments combine lattice and continuum FK representations, including coupled conformal loop and measure ensembles, showing that such ensembles can be useful even in the study of near-critical scaling limits. Thus we provide the first substantial application of measure ensembles.

Let V⊆A be a conformal inclusion of vertex operator algebras and let C be a category of grading-restricted generalized V-modules that admits the vertex algebraic braided tensor category structure of Huang-Lepowsky-Zhang. We give conditions under which C inherits semisimplicity from the category of grading-restricted generalized A-modules in C, and vice versa. The most important condition is that A be a rigid V-module in C with non-zero categorical dimension, that is, we assume the index of V as a subalgebra of A is finite and non-zero. As a consequence, we show that if A is strongly rational, then V is also strongly rational under the following conditions: A contains V as a V-module direct summand, V is C_2-cofinite with a rigid tensor category of modules, and A has non-zero categorical dimension as a V-module. These results are vertex operator algebra interpretations of theorems proved for general commutative algebras in braided tensor categories. We also generalize these results to the case that A is a vertex operator superalgebra.

In a recent paper (arXiv: ) it was shown that all global energy eigenstates of asymptotically $AdS_3$ chiral gravity have non-negative energy at the linearized level. This result was questioned (arXiv: ) by Carlip, Deser, Waldron and Wise (CDWW), who work on the Poincare patch. They exhibit a linearized solution of chiral gravity and claim that it has negative energy and is smooth at the boundary. We show that the solution of CDWW is smooth only on that part of the boundary of $AdS_3$ included in the Poincare patch. Extended to global $AdS_3$, it is divergent at the boundary point not included in the Poincare patch. Hence it is consistent with the results of (arXiv: 0801.4566).

We study S duality of four dimensional N= 2 Argyres-Douglas (AD) theory engineered from 6d A_ {N-1}(2, 0) theory. We find a (p, q) sequence of SCFTs, here (p, q) is co-prime and class S theory defined on sphere corresponds to class (0, 1) theory. We represent these theories by a sphere with marked points, and S duality is interpreted as different pants decompositions of the same punctured sphere. The weakly coupled gauge theory description involves gauging AD matter which is represented by three punctured sphere.

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