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Alday, Gaiotto, and Tachikawa conjectured relations between certain 4d N=2 supersymmetric field theories and 2d Liouville conformal field theory. We study generalizations of these relations to 4d theories with surface operators. For one type of surface operators the corresponding 2d theory is the WZW model, and for another type - the Liouville theory with insertions of extra degenerate fields. We show that these two 4d theories with surface operators exhibit an IR duality, which reflects the known relation (the so-called separation of variables) between the conformal blocks of the WZW model and the Liouville theory. Furthermore, we trace this IR duality to a brane creation construction relating systems of M5 and M2 branes in M-theory. Finally, we show that this duality may be expressed as an explicit relation between the generating functions for the changes of variables between natural sets of Darboux coordinates on the Hitchin moduli space.
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The g-function is a measure of degrees of freedom associated to a boundary of two-dimensional quantum field theories. In integrable theories, it can be computed exactly in a form of the Fredholm determinant, but it is often hard to evaluate numerically. In this paper, we derive functional equations---or equivalently integral equations of the thermodynamic Bethe ansatz (TBA) type---which directly compute the g-function in the simplest integrable theory; the sinh-Gordon theory at the self-dual point. The derivation is based on the classic result by Tracy and Widom on the relation between Fredholm determinants and TBA, which was used also in the context of topological string. We demonstrate the efficiency of our formulation through the numerical computation and compare the results in the UV limit with the Liouville CFT. As a side result, we present multiple integrals of Q-functions which we conjecture to describe a universal part of the g-function, and discuss its implication to integrable spin chains.
We initiate the study of (2,0) little string theory of ADE type using its definition in terms of IIB string compactified on an ADE singularity. As one application, we derive a 5d ADE quiver gauge theory that describes the little string compactified on a sphere with three full punctures, at low energies. As a second application, we show the partition function of this theory equals the 3-point conformal block of ADE Toda CFT, q-deformed. To establish this, we generalize the A_n triality of cite{AHS} to all ADE Lie algebras; IIB string perspective is crucial for this as well.
We compute the Schur index of Argyres-Douglas theories of type $(A_{N-1},A_{M-1})$ with surface operators inserted, via the Higgsing prescription proposed by D. Gaiotto, L. Rastelli and S. S. Razamat. These surface operators are obtained by turning on position-dependent vacuum expectation values of operators in a UV theory which can flow to the Argyres-Douglas theories. We focus on two series of $(A_{N-1},A_{M-1})$ theories; one with ${rm gcd}(N,M)=1$ and the other with $M=N(k-1)$ for an integer $kgeq 2$. For these two series of Argyres-Douglas theories, our results are identical to the characters of non-vacuum modules of the associated 2d chiral algebras, which explicitly confirms a remarkable correspondence recently discovered by C. Cordova, D. Gaiotto and S.-H. Shao.
The Maxwell operator in a 3D cylinder is considered. The coefficients are assumed to be scalar functions depending on the longitudinal variable only. Such operator is represented as a sum of countable set of matrix differential operators of first order acting in $L_2({mathbb R})$. Based on this representation we give a detailed description of the structure of the spectrum of the Maxwell operator in two particular cases: 1) in the case of coefficients stabilizing at infinity; and 2) in the case of periodic coefficients.
The definitions of para-Grassmann variables and q-oscillator algebras are recalled. Some new properties are given. We then introduce appropriate coherent states as well as their dual states. This allows us to obtain a formula for the trace of a operator expressed as a function of the creation and annihilation operators.
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