We study the representation theory of the vertex operator algebras (VOAs) associated with rank-one 4d $\mathcal{N} = 3$ superconformal field theories. For the $\mathbb{Z}_3$ S-fold theory, whose VOA $\mathcal{W}_{\mathbb{Z}_3}$ has central charge $c_{\mathrm{2d}} = -15$, we use the $\mathcal{N} = 1$ Lagrangian description to obtain the unflavored Schur index in terms of Dedekind eta functions, while Wilson-loop indices yield the unflavored non-vacuum characters. These characters all solve a modular linear differential equation (MLDE) whose solution space also contains a logarithmic character. Combining flavored MLDEs from null states with Zhu's associative algebra and a free-field realization, we study four highest-weight modules of $\mathcal{W}_{\mathbb{Z}_3}$ and their flavored characters in closed form. A parallel analysis applies to the $\mathcal{N} = 3$ theories obtained by gauging a discrete $\mathbb{Z}_n$ flavor subgroup of $\mathcal{N} = 4$ $U(1)$ and $SU(2)$ super-Yang--Mills, for which we also obtain closed-form Schur indices and a new free-field realization of the VOA of the $\mathbb{Z}_4$ quotient.
We study the quantized six-vertex (Q6V) model on arbitrary admissible diagrams at $q=\varepsilon$, where $\varepsilon$ is a primitive $N$th root of unity with $N$ odd. For the commuting layer transfer matrices $\mathbb{T}(y)$ with mixed boundary conditions, we establish the quantum Frobenius property \[ \mathbb{T}(y)\m...
Rei Inoue, A. Kuniba, Yuji Terashima et al.· 0 citations
We compute the homological Euler characteristics of $\mathrm{SL}_4(\mathbb{Z})$ and $\mathrm{GL}_4(\mathbb{Z})$ with coefficients in arbitrary irreducible rational highest-weight representations. Applying Wall's formula, we combine the orbifold Euler characteristics of centralizers of torsion elements with traces compu...
We solve the Bethe Ansatz equations (BAEs) to evaluate the superconformal index of four-dimensional $\mathcal{N}=4$ super-Yang--Mills with rank-two gauge algebra, at equal angular momentum fugacities $p=q$. The solutions for $A_2$ are known, and those for $D_2\cong A_1\oplus A_1$ follow readily from the (equally known)...
We determine the complete ordered solution set of the Fermat-type matrix equation $X^n+Y^n=Z^n$ in $\mathrm{GL}_2(\mathbb{Z})$ for every integer $n\ge3$, expressed in terms of simultaneous integral-conjugacy orbits while keeping the variable symmetries of the equation separate. The equation is solvable if and only if $...
Line operators of 4d $\mathcal{N}=2$ theories of class $\mathcal{S}$ of type $A_{N-1}$ in a half Omega background provide a physical model for $SL_N$ skein algebras. In this paper we extend this correspondence to $GL_N$ skein algebras and prove that the $GL_2$ skein algebra of the punctured torus is isomorphic to the $...
Four-dimensional $\mathcal{N}=3$ superconformal field theories (SCFTs) are an exotic class of quantum field theories that admit no conventional Lagrangian description. In this Letter, we construct an $\mathcal{N}=1$ supersymmetric gauge theory and argue that its infrared fixed point exhibits supersymmetry enhancement t...
Hong-Liang Jiang· 0 citations
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