We establish and analyze a new relationship between the matrices describing an arbitrary component of a spin $s$, where $2s\in \mathbb{Z}^+$, and the matrices of $\mathbb{C}P^{2s}$ two-dimensional Euclidean sigma models. The spin matrices are constructed from the rank-1 Hermitian projectors of the sigma models or from the antihermitian immersion functions of their soliton surfaces in the $\mathfrak{su}(2s+1)$ algebra. For the spin matrices which can be represented as a linear combination of the generalized Pauli matrices, we find the dynamics equation satisfied by its coefficients. The equation proves to be identical to the stationary equation of a two-dimensional Heisenberg model. We show that the same holds for the matrices congruent to the generalized Pauli ones by any coordinate-independent unitary linear transformation. These properties open the possibility for new interpretations of the spins and also for application of the methods known from the theory of sigma models to the situations described by the Heisenberg model, from statistical mechanics to quantum computing.
In these notes we study the duality between sigma-models and Toda QFT's. We claim that $\mathfrak{gl}(n|n)$ affine Toda field theory behaves in the strong coupling limit as $η-$deformed $\mathbb{CP}(n-1)$ sigma-model plus a free field.
The main objective of this paper is to establish a new connection between the Hermitian rank-1 projector solutions of the Euclidean $\mathbb{C}P^{2S}$ sigma model in two dimensions and the particular hypergeometric orthogonal polynomials called Krawtchouk polynomials. We show that any such projector solutions of the $\mathbb{C}P^{2S}$ model, defined on the Riemann sphere and having a finite action, can be explicitly parametrised in terms of these polynomials. We apply these results to the analysis of surfaces associated with $\mathbb{C}P^{2S}$ models defined using the generalised Weierstrass f…
In this paper we investigate a novel connection between the effective theory of M2-branes on $(\mathbb{C}^2/\mathbb{Z}_2\times \mathbb{C}^2/\mathbb{Z}_2)/\mathbb{Z}_k$ and the $\mathfrak{q}$-deformed Painlevé equations, by proposing that the grand canonical partition function of the corresponding four-nodes circular quiver $\mathcal{N}=4$ Chern-Simons matter theory solves the $\mathfrak{q}$-Painlevé VI equation. We analyse how this describes the moduli space of the topological string on local $\text{dP}_5$ and, via geometric engineering, five dimensional $N_f=4$ $\text{SU}(2)$ $\mathcal{N}=1$ …
Generalizations of the AGT correspondence between 4D $\mathcal{N}=2$ $SU(2)$ supersymmetric gauge theory on ${\mathbb {C}}^2$ with $Ω$-deformation and 2D Liouville conformal field theory include a correspondence between 4D $\mathcal{N}=2$ $SU(N)$ supersymmetric gauge theories, $N = 2, 3, \ldots$, on ${\mathbb {C}}^2/{\mathbb {Z}}_n$, $n = 2, 3, \ldots$, with $Ω$-deformation and 2D conformal field theories with $\mathcal{W}^{\, para}_{N, n}$ ($n$-th parafermion $\mathcal{W}_N$) symmetry and $\widehat{\mathfrak{sl}}(n)_N$ symmetry. In this work, we trivialize the factor with $\mathcal{W}^{\, par…
A class of representations of the $\mathbb{Z}_2\times\mathbb{Z}_2$-graded special linear Lie superalgebra $\mathfrak{sl}(m_1+1,m_2|n_1,n_2)$ and quantum statistics
For $\mathbb N^*:=\mathbb N \setminus \{0\}$, we consider the collection $\mathfrak M(N)$ of all the $N$ rows, for which, for $n=1,\cdots,N$, the $n-th$ row consists of an increasing sequence $(a_j^n)_j$ of real numbers. For $\mathfrak A \in \mathfrak M(N)$, we define its spectrum $σ(\mathfrak A)$ by $σ(\mathfrak A)=\{λ\in \mathbb R \;|\; λ=\sum_{n=1}^Na_{j_n}^n\}\,,$ where $(j_1,j_2,\dots,j_N)\in (\mathbb N^*)^N$. This spectrum is discrete and consists of an infinite sequence that can be ordered as a strictly increasing sequence $λ_k(\mathfrak A)$. For $λ\in σ(\mathfrak A)$ we denote by $m(λ,…
This paper aims to show that a certain moduli space $\textsf{T}$, which arises from the so-called Dwork family of Calabi-Yau $n$-folds, carries a special complex Lie algebra containing a copy of $\mathfrak{sl}_2(\mathbb{C})$. In order to achieve this goal, we introduce an algebraic group $\sf G$ acting from the right on $\textsf{T}$ and describe its Lie algebra ${\rm Lie}({\sf G})$. We observe that ${\rm Lie}({\sf G})$ is isomorphic to a Lie subalgebra of the space of the vector fields on $\textsf{T}$. In this way, it turns out that ${\rm Lie}({\sf G})$ and the modular vector field ${\sf R}$ g…
An enhanced Leibniz algebra is an algebraic struture that arises in the context of particular higher gauge theories describing self-interacting gerbes. It consists of a Leibniz algebra $(\mathbb{V},[ \cdot, \cdot ])$, a bilinear form on $\mathbb{V}$ with values in another vector space $\mathbb{W}$, and a map $t \colon \mathbb{W} \to \mathbb{V}$, satisfying altogether four compatibility relations. Our structure theorem asserts that an enhanced Leibniz algebra is uniquely determined by the underlying Leibniz algebra $(\mathbb{V},[ \cdot, \cdot ])$, an appropriate abelian ideal ${\mathfrak i}$ in…
