Related papers: Integral Van Vleck's and Kannappan's functional eq…

We provide a unified construction of the symplectic forms which arise in the solution of both N=2 supersymmetric Yang-Mills theories and soliton equations. Their phase spaces are Jacobian-type bundles over the leaves of a foliation in a…

In this paper, we develop a new method of computing three-point functions in the SU(2) sector of the $\mathcal{N}=4$ super Yang-Mills theory in the semi-classical regime at weak coupling, which closely parallels the strong coupling…

This PhD thesis explores the similarities between integrable spin chains and quantum field theories, such as Super Yang Mills. We first study integrable spin chains and build explicitly a polynomial "Backlund flow" and polynomial…

Let $f(z)=\sum_{n=1}^\infty a(n)q^n\in S^{\text{new}}_ k (\Gamma_0(N))$ be a newform with squarefree level $N$ that does not have complex multiplication. For a prime $p$, define $\theta_p\in[0,\pi]$ to be the angle for which $a(p)=2p^{( k…

We obtain the exact beta function for $N=2$ SUSY $SU(2)$ Yang-Mills theory and prove the nonperturbative Renormalization Group Equation $$ \partial_\Lambda{\cal F}(a,\Lambda)= {\Lambda\over \Lambda_0}\partial_{\Lambda_0}{\cal…

An extension of the method and results of A. Schwarz for evaluating the partition function of a quadratic functional is presented. This enables the partition functions to be evaluated for a wide class of quadratic functionals of interest in…

We consider semilinear equation of the form $-Lu=f(x,u)+\mu$, where $L$ is the operator corresponding to a transient symmetric regular Dirichlet form ${\mathcal E}$, $\mu$ is a diffuse measure with respect to the capacity associated with…

An intriguing coincidence between the partition function of super Yang-Mills theory and correlation functions of 2d Toda system has been heavily studied recently. While the partition function of gauge theory was explored by Nekrasov, the…

The quantum integrability of a class of massive perturbations of the parafermionic conformal field theories associated to compact Lie groups is established by showing that they have quantum conserved densities of scale dimension 2 and 3.…

We present an explicit method to perform similarity reduction of a Riemann-Hilbert factorization problem for a homogeneous GL (N, C) loop group and use our results to find solutions to the Painleve VI equation for N=3. The tau function of…

We develop a functional integral approach to quantum Liouville field theory completely independent of the hamiltonian approach. To this end on the sphere topology we solve the Riemann-Hilbert problem for three singularities of finite…

We consider a pair of special functions, $u_\beta$ and $v_\beta$, defined respectively as the solutions to the integral equations \begin{equation*} u(x)=1+\int^\infty_0 \frac {K(t) u(t) dt}{t+x} ~~\mbox{and}~~v(x)=1-\int^\infty_0 \frac{…

We establish estimates and representations for the remainders of Taylor approximations of the spectral action functional $V\mapsto\tau(f(H_0+V))$ on bounded self-adjoint perturbations, where $H_0$ is a self-adjoint operator with…

We present a review of the results on the associativity algebras and WDVV equations associated with the Seiberg-Witten solutions of N=2 SUSY gauge theories. It is mostly based on the integrable treatment of these solutions. We consider…

We give necessary and sufficient conditions for the existence of weak solutions to the model equation $$-\Delta_p u=\sigma \, u^q \quad \text{on} \, \, \, \R^n,$$ in the case $0<q<p-1$, where $\sigma\ge 0$ is an arbitrary locally integrable…

In both ${\cal N}=1$ and ${\cal N}=2$ supersymmetry, it is known that $\mathsf{Sp}(2n, {\mathbb R})$ is the maximal duality group of $n$ vector multiplets coupled to chiral scalar multiplets $\tau (x,\theta) $ that parametrise the Hermitian…

In the present paper we deal with the following generalization of the sine-cosine equation \begin{equation*} \int f_1(x+y-t)+f_2(x-y+t) d\mu(t)=g(x)h(y) \end{equation*} for complex valued functions $f_1$, $f_2$, $g$ and $h$ defined on a…

The integrable quantum group $spl_q(2,1)$-invariant supersymmetric t-J model with open boundaries is studied via an analytic treatment of the Bethe equations. An $su(2)$ feature is seen to hold for states at or close to half-filling. For…

We apply a functional-integral formalism for Markovian birth and death processes to determine asymptotic corrections to mean-field theory in the Malthus-Verhulst process (MVP). Expanding about the stationary mean-field solution, we identify…

Gauged Wess-Zumino-Witten theory for compact groups is considered. It is shown that this theory has fermionic BRST-like symmetry and may be exactly solved using localization approach. As an example we calculate functional integral for the…