Letters in Mathematical Physics

We prove that the complex manifold of the superposition Eguchi-Hanson metric plus the pseudo-Fubini-Study metric is equal to the total space of the holomorphic line bundle of degree −n on the Riemann sphere. The apparent singularities of the metric can be resolved only if the Eguchi-Hanson parameter satisfies a 4=4(n−2)2(n+1)/3Λ2, n≥3. We give a geometrical explanation of the fact that we need n≥3. Finally, we generalize the metric of Gegenberg and Das to obtain a triaxial vacuum metric.

We conjecture a closed form expression for the simplest class of multiplicity-free quantum 6j-symbols for $${U_q(\mathfrak{sl}_N)}$$ . The expression is a natural generalization of the quantum 6j-symbols for $${U_q(\mathfrak{sl}_2)}$$ obtained by Kirillov and Reshetikhin. Our conjectured form enables computation of colored HOMFLY polynomials for various knots and links carrying arbitrary symmetric representations.

Given complex numbers m1, I1 and nonnegative integers m2, I2, such that m1+m2 = I1+ I2, we define I2-dimensional hypergeometric integrals Ia,b(z; m1, m2, I1, I2), a,b = 0,. . . ,min)(m2,I2), depending on a complex parameter z. We show that Ia,b(z;m1, m2,I1, I2) = Ia,b(z;I1, I2,m1,m2), thus establishing an equality of I2 and m2-dimensional integrals. This identity allows us to study asymptotics of the integrals with respect to their dimension in some examples. The identity is based on the ( $$\cal{g l}$$ k, $$\cal{g l}$$ k,) duality for the KZ and dynamical differential equations.

It is shown that the linear transport operator for a slab with periodic boundary conditions, or for infinite medium, can be analyzed within the Friedrichs' model of perturbation for non-selfadjoint operators. This leads to rephrasing Case's generalized eigenfunction expansion in terms of Hilbert space operators only.

The author's recent results on an asymptotic description of the Schlesinger equation are generalized to the Jimbo–Miwa–Môri–Sato (JMMS) equation. As in the case of the Schlesinger equation, the JMMS equation is reformulated to include a small parameter ε. By the method of multi-scale analysis, the isomonodromic problem is approximated by slow modulations of an isospectral problem. A modulation equation of this slow dynamics is proposed and shown to possess a number of properties similar to the Seiberg–Witten solutions of low energy supersymmetric gauge theories.

We give a birational realization of affine Weyl group of type A (1) m−1 × A (1) n−1. We apply this representation to construct some discrete integrable systems and discrete Painlevé equations. Our construction has a combinatorial counterpart through the ultra-discretization procedure.

We present a simple and general method for constructing Wick-ordered entire functions of free fields with an indefinite metric, based on using an appropriate generalization of the Paley–Wiener–Schwartz theorem.

Motivated by mathematical structures which arise in string vacua and gauge theories with $${{\mathcal{N}=2}}$$ supersymmetry, we study the properties of certain generalized theta series which appear as Fourier coefficients of functions on a twisted torus. In Calabi–Yau string vacua, such theta series encode instanton corrections from k Neveu–Schwarz five-branes. The theta series are determined by vector-valued wave-functions, and in this work we obtain the transformation of these wave-functions induced by Kontsevich–Soibelman symplectomorphisms. This effectively provides a quantum version of these transformations, where the quantization parameter is inversely proportional to the five-brane charge k. Consistency with wall-crossing implies a new five-term relation for Faddeev’s quantum dilogarithm $${\Phi_b}$$ at b = 1, which we prove. By allowing the torus to be non-commutative, we obtain a more general five-term relation valid for arbitrary b and k, which may be relevant for the physics of five-branes at finite chemical potential for angular momentum.