Narrow Results

We continue our study of zero-dimensional field theories in which the fields take values in a strong homotopy Lie algebra. In the first part, we review in detail how higher Chern–Simons theories arise in the AKSZ-formalism. These theories form a universal starting point for the construction of $L_{\infty }$-algebra models. We then show how to describe superconformal field theories and how to perform dimensional reductions in this context. In the second part, we demonstrate that Nambu–Poisson and multisymplectic manifolds are closely related via their Heisenberg algebras. As a byproduct of our discussion, we find central Lie $p$-algebra extensions of $𝔰𝔬(p+2)$. Finally, we study a number of $L_{\infty }$-algebra models which are physically interesting and which exhibit quantized multisymplectic manifolds as vacuum solutions.

We review the BV formalism in the context of $0$-dimensional gauge theories. For a gauge theory $(X_0,S_0)$ with an affine configuration space $X_0$, we describe an algorithm to construct a corresponding extended theory $(\tilde{X},\tilde{S})$, obtained by introducing ghost and anti-ghost fields, with $\tilde{S}$ a solution of the classical master equation in ${{𝒪}}_{\tilde{X}}$. This construction is the first step to define the (gauge-fixed) BRST cohomology complex associated to $(\tilde{X},\tilde{S})$, which encodes many interesting information on the initial gauge theory $(X_0,S_0)$. The second part of this article is devoted to the application of this method to a matrix model endowed with a $U(2)$-gauge symmetry, explicitly determining the corresponding $\tilde{X}$ and the general solution $\tilde{S}$ of the classical master equation for the model.

Differential reformulations of field theories are often used for explicit computations. We derive a one-matrix differential formulation of two-matrix models, with the help of which it is possible to diagonalize the one- and two-matrix models using a formula by Itzykson and Zuber that allows diagonalizing differential operators with respect to matrix elements of Hermitian matrices. We detail the equivalence between the expressions obtained by diagonalizing the partition function in differential or integral formulation, which is not manifest at first glance. For one-matrix models, this requires transforming certain derivatives to variables. In the case of two-matrix models, the same computation leads to a new determinant formulation of the partition function, and we discuss potential applications to new orthogonal polynomials methods.