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We revisit the subject of perturbatively quantizing the nonlinear sigma model in two dimensions from a rigorous, mathematical point of view. Our main contribution is to make precise the cohomological problem of eliminating potential…

We study the supersymmetric extensions of the $O(3)$ $\sigma$-model in $1+1$ and $2+1$ dimensions. We show that it is possible to construct non-equivalent supersymmetric versions of a given model sharing the same bosonic sector and free…

Let X be a smooth complex projective variety, and let Y in X be a smooth very ample hypersurface such that -K_Y is nef. Using the technique of relative Gromov-Witten invariants, we give a new short and geometric proof of (a version of) the…

It is well known that Hermitian and non-Hermitian models exhibit distinct physics and require different theoretical tools. In this work, we propose a unified generating-function framework for both classes with generic boundary conditions…

For a prime p>3, we compute a finite generating set for the SL_2(F_p)-invariants of the third symmetric power representation. The proof relies on the construction of an infinite SAGBI basis and uses the Hilbert series calculation of Hughes…

We have derived the perimeter generating function of a model of punctured staircase polygons in which the internal staircase polygon is rotated by a 90degree angle with respect to the outer staircase polygon. In one approach we calculated a…

An afinne-invariant view of generating functions of symplectic transformations of an affine symplectic space is discussed. More generally, it works for symmetric symplectic spaces. The note is completely elementary, but it yields some nice…

We present an expression for the generating function of correlation functions of the sine-Gordon integrable field theory on a cylinder, with compact space. This is derived from the Destri-De Vega integrable lattice regularization of the…

Let $T$ be an underlying space with a non-atomic measure $\sigma$ on it. In [{\it Comm.\ Math.\ Phys.}\ {\bf 292} (2009), 99--129] the Meixner class of non-commutative generalized stochastic processes with freely independent values,…

In this work we discuss the phase structure of a deformed supersymmetric nonlinear sigma model in a three-dimensional space-time. The deformation is introduced by a term that breaks supersymmetry explicitly, through imposing a slightly…

Given an operad P with a finite Groebner basis of relations, we study the generating functions for the dimensions of its graded components P(n). Under moderate assumptions on the relations we prove that the exponential generating function…

We give closed form expressions for the numbers of multi-rooted plane trees with specified degrees of root vertices. This results in an infinite number of integer sequences some of which are known to have an alternative interpretation. We…

We define new families of noncommutative symmetric functions and quasi-symmetric functions depending on two matrices of parameters, and more generally on parameters associated with paths in a binary tree. Appropriate specializations of both…

We study the pressure of the "edge-triangle model", which is equivalent to the cumulant generating function of triangles in the Erd\"os-R\'enyi random graph. By analyzing finite graphs of increasing volume, as well as the graphon…

We show that the noncommutativity of space-time destroys the renormalizability of the 1/N expansion of the O(N) Gross-Neveu model. A similar statement holds for the noncommutative nonlinear sigma model. However, we show that, up to the…

In this paper, we count acyclic and strongly connected uniform directed labeled hypergraphs. For these combinatorial structures, we introduce a specific generating function allowing us to recover and generalize some results on the number of…

We prove an explicit integral formula for computing the product of two shifted Riemann zeta functions everywhere in the complex plane. We show that this formula implies the existence of infinite families of exact exponential sum identities…

We construct the non-minimal linear representations of the N=4 Extended Supersymmetry in one-dimension. They act on 8 bosonic and 8 fermionic fields. Inequivalent representations are specified by the mass-dimension of the fields and the…

Four-dimensional N-extended superconformal symmetry and correlation functions of quasi-primary superfields are studied within the superspace formalism. A superconformal Killing equation is derived and its solutions are classified in terms…

We construct novel conformal sigma models in three dimensions. Nonlinear sigma models in three dimensions are nonrenormalizable in perturbation theory. We use Wilsonian renormalization group equation method to find the fixed points.…