arXiv subjects
Publications and source records attributed to Minoru Wakimoto.
For the affine Lie algebra $C_2^{(1)}$ we study non-principal and non-coprincipal admissible modules of integer level and their quantum Hamiltonian reduction, and show that they have $\Gamma_0(2)$-modular invariance.
We prove that the span of normalized characters of subprincipal admissible modules over an affine Lie algebra of subprincipal admissible level $k$ is $SL_2(\mathbf{Z})$-invariant and find the explicit modular transformation formula.
We find modular transformations of normalized characters for the following $W$-algebras: (a) $W^{min}_k(\frak{g})$, where $\frak{g}=D_n \, (n \geq 4)$, or $E_6$, $E_7$, $E_8$, and $k$ is a negative integer $\geq -2$, or $\geq -\frac{h^{\vee}}{6}-1$, respectively; (b) quantum Hamiltonian reduction of the $\hat{\frak{g}}$-module $L(k\Lambda_0)$, where $\frak{g}$ is a simple Lie algebra, $f$ is its non-zero nilpotent element, and $k$ is a principal admissible level with the denominator $u > \theta(x)$, where $2x$ is the Dynkin characteristic of $f$ and $\theta$ is the highest root of $\frak{g}$. We prove that these vertex algebras are modular invariant. A conformal vertex algebra is called modular invariant if its character $tr_V q^{L_0-c/24}$ converges to a holomorphic modular function in the complex upper half-plane on a congruence subgroup. We find explicit formulas for their characters. Modular invariance of $V$ is important since, in particular, conjecturally it implies that $V$ is simple, and that $V$ is rational, provided that it is lisse.
In this paper we show that, for the Deligne exceptional series representations of negative integer level of affine Lie algebras, the quantum Hamiltonian reduction vanishes except for the cases where the nilpotent element is conjugate to $e_{-\theta}$ or $e_{-\theta_s}$.
In this paper we compute the characters of certain non-irreducible N=4 superconformal modules which are different from the ones treated in our previous paper, and study their relation with characters of N=2 superconformal modules. Also, for these non-irreducible N=4 modules, we deduce the expression of characters in terms of string functions.
In this paper, we study modular transformation properties of a certain class of functions with indefinite quadratic forms.
In this paper we study the characters of N=3 superconformal modules by using the Zwegers' theory on modification of mock theta functions.
In this paper we study the N=4 superconformal modules obtained from the quantum Hamiltonian reduction of principal admissible representations of the affine Lie superalgebra $\hat{A}(1,1)$, and show that there exists a series of N=4 superconformal modules whose characters are modular functions and written explicitly by the Mumford's theta functions.
In the explicit formula for the signed mock theta functions $Φ^{(-)[m,s]}$ obtained from the coroot lattice of $D(2,1;a)$, functions with indefinite quadratic forms naturally take place. We compute their modular transformation properties by applying the Zwegers' modification theory of mock theta functions and show that the $\mathbf{C}$-linear span of these functions is $SL_2(\mathbf{Z})$-invariant.
In this paper we obtain explicit formulas for mock theta functions $Φ^{[m,s]}(τ, z_1, z_2,t)$ $(m \in \frac12 \mathbf{N}, s \in \frac12 \mathbf{Z})$ by using the coroot lattice of the Lie superalgebra $D(2,1,a)$ and the Kac-Peterson's identity. As its application, we study the branching functions of tensor products of N=3 modules and prove the formula conjectured in the previous paper.
In this paper we consider the simplest class of Appell's functions, and obtain their explicit formulas and modular transformation properties and asymptotic behaviors.
In this paper, we compute the Zwegers's modification of the mock theta functions $Φ^{[m,0] \, \ast}$ and study the modular transformation properties of the indefinite modular forms which appear in the explicit formula for the modified functions $\tildeΦ^{[m,0] \, \ast}$.
We obtain an explicit formula for the mock theta function $Φ^{[m,s]}$ in the case when either $m$ or $s$ is a half of an odd integer by using the coroot lattice of $D(2,1;a)$. This enables us, together with the recurrence formula for $Φ^{[m,s]}$, to know $Φ^{[m,s]}$ for all $m$ and $s$. As its application, we deduce explicit formulas for the character and the supercharacter of N=3 modules obtained from the quantum Hamiltonian reduction of $\hat{osp}(3|2)$-modules.
In this paper we study the branching functions of tensor products of N=3 superconformal modules.
We find an explicit formula for the conformal vector of any quantum affine $W$-algebra in its free field realization.
We develop the notions of multiplicative Lie conformal and Poisson vertex algebras, local and non-local, and their connections to the theory of integrable differential-difference Hamiltonian equations. We establish relations of these notions to $q$-deformed $W$-algebras and lattice Poisson algebras. We introduce the notion of Adler type pseudodifference operators and apply them to integrability of differential-difference Hamiltonian equations.
We introduce the notion of a multiplicative Poisson $λ$-bracket, which plays the same role in the theory of Hamiltonian differential-difference equations as the usual Poisson $λ$-bracket plays in the theory of Hamiltonian PDE. We classify multiplicative Poisson $λ$-brackets in one difference variable up to order 5. Applying the Lenard-Magri scheme to a compatible pair of multiplicative Poisson $λ$-brackets of order 1 and 2, we establish integrability of some differential-difference equations, generalizing the Volterra chain.
We prove a character formula for the irreducible modules from the category $\mathcal{O}$ over the simple affine vertex algebra of type $A_n$ and $C_n$ $(n \geq 2)$ of level $k=-1$. We also give a conjectured character formula for types $D_4$, $E_6$, $E_7$, $E_8$ and levels $k=-1, \cdots, -b$, where $b=2,3,4,6$ respectively.