数学物理

We present systems of parabosons and parafermions in the context of Lie algebras, Lie superalgebras, $\mathbf{{\mathbb Z}_2\ \times {\mathbb Z}_2}$-graded Lie algebras and $\mathbf{{\mathbb Z}_2\ \times {\mathbb Z}_2}$-graded Lie…

We consider and reformulate a recent definition of multiplication between distributions. We show that this definition can be adopted, in particular, to prove biorthonormality of some distributions arising when looking to the (generalized)…

The monodromy of the $\mfsl(2)$ Casimir connection is considered. It is shown that the trace of the monodromy operator over the appropriate space of flat sections gives rise to the Jacobi theta constant and to the partial Appell-Lerch sums.

We derive Macfarlane's formula for the Thomas-Wigner angle of rotation using Clifford-algebra methods. The presentation is pedagogical and elementary, suitable for students with some basic knowledge of special relativity; no prior knowledge…

We introduce and investigate a class of $\mathfrak{gl}_{M+1}$ partition functions which is an extension of the one introduced by Foda-Manabe. We characterize the partition functions by a nested version of Izergin-Korepin analysis, and…

We propose a new technique to generate reasonable systems of partial differential equations (PDE) that could be potential candidates for depicting models in natural sciences related to quasi-linear equations. Such systems appear within…

Fermionic time-reversal-invariant insulators in two dimensions -- class AII in the Kitaev table -- come in two different topological phases. These are characterized by a $\mathbb{Z}_2$-index: the Fu-Kane-Mele index. We prove that if two…

We deal with a type I superconductor in a constant external magnetic field. We obtain the BCS-Bogoliubov gap equation with external magnetic field and apply the implicit function theorem to it. We show that there is a unique magnetic field…

We provide the first computations of colored unknots and Hopf link in $\mathbb{R}\mathbb{P}^3$ using both the topological vertex and its refinement. Our approach utilizes the toric Calabi-Yau threefold arising from the geometric transition…

The Harer-Zagier (HZ) transform maps the HOMFLY-PT polynomial into a rational function. For some special knots and links, the latter admits a simple factorised form, which is referred to as HZ factorisation. This property is preserved under…

We study the random-field Ising model on a Dyson hierarchical lattice, where the interactions decay in a power-law-like form, $J(r)\sim r^{-\alpha}$, with respect to the distance. Without a random field, the Ising model on the Dyson…

Since Bardeen-Cooper-Schrieffer theory of superconductivity is non-linear, it is difficult to study superconducting properties analytically. There is a more tractable linear criterion which determines a temperature $T_l$ below which the…

In this article we study some algebraic aspects of multicomplex numbers $\mathbb M_n$. For $n\geq 2$ a canonical representation is defined in terms of the multiplication of $n-1$ idempotent elements. This representation facilitates…

The dressing field method is a tool to reduce gauge symmetries. Here we extend it to cover the case of diffeomorphisms. The resulting framework is a systematic scheme to produce Diff(M)-invariant objects, which has a natural relational…

The group of vertical diffeomorphisms of a principal bundle forms the generalised action Lie groupoid associated to the bundle. The former is generated by the group of maps with value in the structure group, which is also the group of…

In this paper we consider density matrices operator related to non-Hermitian Hamiltonians. In particular, we analyse two natural extensions of what is usually called a density matrix operator (DM), of pure states and of the entropy…

We study the renormalized Nelson model for a scalar matter particle in a continuous confining potential interacting with a possibly massless quantized radiation field. When the radiation field is massless we impose a mild infrared…

We consider a particular class of sesquilinear forms on a {Banach quasi *-algebra} $(\A[\|.\|],\Ao[\|.\|_0])$ which we call {\em eigenstates of an element} $a\in\A$, and we deduce some of their properties. We further apply our definition to…

These are the notes on two-dimensional conformal field theory, based on a lecture course for graduate math students, given by P.M. in fall 2022 at the University of Notre Dame. These notes are intended to be substantially reworked and…

We extend the theory of topological recursion by considering Airy structures whose partition functions are highest weight vectors of particular $\mathcal{W}$-algebra representations. Such highest weight vectors arise as partition functions…