Correlator correspondences for Gaiotto-Rapčák dualities and first order formulation of coset models

We derive correspondences of correlation functions among dual conformal field theories in two dimensions by developing a “first order formulation” of coset models. We examine several examples, and the most fundamental one may be a conjectural equivalence between a coset (SL(n)k ⊗SL(n)−1)/SL(n)k−1 and $$ \mathfrak{sl}(n) $$ sl n Toda field theory with generic level k. Among others, we also complete the derivation of higher rank FZZ-duality involving a coset SL(n + 1)k /(SL(n)k ⊗ U(1)), which could be done only for n = 2, 3 in our previous paper. One obstacle in the previous work was our poor understanding of a first order formulation of coset models. In this paper, we establish such a formulation using the BRST formalism. With our better understanding, we successfully derive correlator correspondences of dual models including the examples mentioned above. The dualities may be regarded as conformal field theory realizations of some of the Gaiotto-Rapčák dualities of corner vertex operator algebras.

Correlator correspondences for subregular $$ \mathcal{W} $$-algebras and principal $$ \mathcal{W} $$-superalgebras

We examine a strong/weak duality between a Heisenberg coset of a theory with $$ \mathfrak{sl} $$ sl n subregular $$ \mathcal{W} $$ W -algebra symmetry and a theory with a $$ \mathfrak{sl} $$ sl n|1-structure. In a previous work, two of the current authors provided a path integral derivation of correlator correspondences for a series of generalized Fateev-Zamolodchikov-Zamolodchikov (FZZ-)duality. In this paper, we derive correlator correspondences in a similar way but for a different series of generalized duality. This work is a part of the project to realize the duality of corner vertex operator algebras proposed by Gaiotto and Rapčák and partly proven by Linshaw and one of us in terms of two dimensional conformal field theory. We also examine another type of duality involving an additional pair of fermions, which is a natural generalization of the fermionic FZZ-duality. The generalization should be important since a principal $$ \mathcal{W} $$ W -superalgebra appears as its symmetry and the properties of the superalgebra are less understood than bosonic counterparts.

q-deformation of corner vertex operator algebras by Miura transformation

Recently, Gaiotto and Rapcak proposed a generalization of WN algebra by considering the symmetry at the corner of the brane intersection (corner vertex operator algebra). The algebra, denoted as YL,M,N, is characterized by three non-negative integers L, M, N. It has a manifest triality automorphism which interchanges L, M, N, and can be obtained as a reduction of W1+∞ algebra with a “pit” in the plane partition representation. Later, Prochazka and Rapcak proposed a representation of YL,M,N in terms of L + M + N free bosons by a generalization of Miura transformation, where they use the fractional power differential operators.In this paper, we derive a q-deformation of the Miura transformation. It gives a free field representation for q-deformed YL,M,N, which is obtained as a reduction of the quantum toroidal algebra. We find that the q-deformed version has a “simpler” structure than the original one because of the Miki duality in the quantum toroidal algebra. For instance, one can find a direct correspondence between the operators obtained by the Miura transformation and those of the quantum toroidal algebra. Furthermore, we can show that the both algebras share the same screening operators.

Higher rank FZZ-dualities

We examine strong/weak dualities in two dimensional conformal field theories by generalizing the Fateev-Zamolodchikov-Zamolodchikov (FZZ-)duality between Witten’s cigar model described by the $$ \mathfrak{sl}(2)/\mathfrak{u}(1) $$ sl 2 / u 1 coset and sine-Liouville theory. In a previous work, a proof of the FZZ-duality was provided by applying the reduction method from $$ \mathfrak{sl}(2) $$ sl 2 Wess-Zumino-Novikov-Witten model to Liouville field theory and the self-duality of Liouville field theory. In this paper, we work with the coset model of the type $$ \mathfrak{sl}\left(N+1\right)/\left(\mathfrak{sl}(N)\times \mathfrak{u}(1)\right) $$ sl N + 1 / sl N × u 1 and investigate the equivalence to a theory with an $$ \mathfrak{sl}\left(N+\left.1\right|N\right) $$ sl N + 1 N structure. We derive the duality explicitly for N = 2, 3 by applying recent works on the reduction method extended for $$ \mathfrak{sl}(N) $$ sl N and the self-duality of Toda field theory. Our results can be regarded as a conformal field theoretic derivation of the duality of the Gaiotto-Rapčák corner vertex operator algebras Y0,N,N+1[ψ] and YN,0,N+1[ψ−1].

Characterization of elastic-plastic corner deformation and stress fields

Purpose The purpose of this paper is to present the characteristics of elastic-plastic deformation and stress fields at the intersection of a crack front and the free surface of a three-dimensional body, referred to as corner fields. Design/methodology/approach The structures of elastic-plastic corner deformation field were assessed experimentally by looking at the corner border displacement and strain fields on the surface of a compact tension (CT) specimen using digital image correlation method. For assessment and verification purposes, the results were compared with the fields predicted through finite element analysis. The latter method was used further to assess the corner stress field. Findings The characteristics of displacement, strain and stress fields in the vicinity of a corner vertex in a finite geometry CT specimen in a strain hardening condition are independent of load and geometry. One of the distinctive features that becomes evident in this study is that the stress state at the corner vertex at θ=0° is a simple uniaxial tension. Originality/value This paper provides some insights on the structure of elastic-plastic corner fields that could optimistically be served as a fundamental framework towards the development of analytical solutions for elastic-plastic corner fields.

Free boundary Darcy flows with surface tension: Analytical and numerical study

We study surface tension effects for two-dimensional Darcy flow with a free boundary in a corner between two non-parallel walls. The analytic solution is based on two governing expressions constructed in an auxiliary parameter domain, namely a complex velocity and a derivative of the complex potential. These expressions admit a general solution for the problem in a corner geometry for the flow generated by a source/sink at the corner vertex or at infinity. We derive an integral equation in terms of the velocity modulus and angle at the free surface, determined by the dynamic boundary condition. A numerical procedure, used to solve the obtained system of equations, and numerical results concerning the effect of surface tension on the time evolution of the free boundary, are discussed.