Shi-shyr Roan

Calabi-Yau four-folds for M- and F-theory compactifications

We investigate topological properties of Calabi-Yau four-folds and consider a wide class of explicit constructions in weighted projective spaces and, more generally, toric varieties. Divisors which lead to a non-perturbative superpotential in the effective theory have a very simple description in the toric construction. Relevant properties of them follow just by counting lattice points and can also be used to construct examples with negative Euler number. We study nets of transitions between cases with generically smooth elliptic fibres and cases with ADE gauge symmetries in the N = 1 theory due to degenerations of the fibre over codimension one loci in the base. Finally we investigate the quantum cohomology ring of this four-folds using Frobenius algebra.

Quantum Reduction for Affine Superalgebras

We extend the homological method of quantization of generalized Drinfeld–Sokolov reductions to affine superalgebras. This leads, in particular, to a unified representation theory of superconformal algebras.

Minimal resolutions of Gorenstein orbifolds in dimension three

Abstract We give the complete solution of constructing c 1 = 0 resolutions of 3-dimensional Gorenstein quotient singularities, and verify the equality between the Euler numbers of resolutions and the orbifold Euler characteristics of quotient singularities.

The structure of quotients of the Onsager algebra by closed ideals

We study the Onsager algebra from the ideal theoretic point of view. A complete classification of closed ideals and the structure of quotient algebras are obtained. We also discuss the solvable algebra aspect of the Onsager algebra through the use of formal Lie algebras.

A stochastic approach to the Euler-Poincaré number of the loop space of a developable orbifold

Abstract We define a regularised version of the de Rham operator over the free loop space. We perform a semi-classical approximation of it, such that the index of the limit operator is equal to the “orbit Euler characteristic” of physicists.

A Note on ODEs from Mirror Symmetry

We give close formulas for the counting functions of rational curves on complete intesection Calabi-Yau manifolds in terms of special solutions of generalized hypergeometric differential systems. For the one modulus cases we derive a differential equation for the Mirror map, which can be viewed as a generalization of the Schwarzian equation. We also derive a nonlinear seventh order differential equation which directly governs the Prepotential.

The Onsager algebra symmetry of τ(j)-matrices in the superintegrable chiral Potts model

We demonstrate that the τ(j)-matrices in the superintegrable chiral Potts model possess the Onsager algebra symmetry for their degenerate eigenvalues. The Fabricius–McCoy comparison of functional relations of the eight-vertex model for roots of unity and the superintegrable chiral Potts model has been carefully analysed by identifying equivalent terms in the corresponding equations, by which we extract the conjectured relation of Q-operators and all fusion matrices in the eight-vertex model corresponding to the -relation in the chiral Potts model.

The Superintegrable Chiral Potts Quantum Chain and Generalized Chebyshev Polynomials

Finite-dimensional representations of Onsager’s algebra are characterized by the zeros of truncation polynomials. The Z N-chiral Potts quantum chain hamiltonians (of which the Ising chain hamiltonian is the N = 2 case) are the main known interesting representations of Onsager’s algebra and the corresponding polynomials have been found by Baxter and Albertini, McCoy and Perk in 1987-89 considering the Yang-Baxter-integrable 2-dimensional chiral Potts model. We study the mathematical nature of these polynomials. We find that for N ≥ 3 and fixed charge Q these don’t form classical orthogonal sets because their pure recursion relations have at least N + 1-terms. However, several basic properties are very similar to those required for orthogonal polynomials. The N + 1-term recursions are of the simplest type: like for the Chebyshev polynomials the coefficients are independent of the degree. We find a remarkable partial orthogonality, for N = 3, 5 with respect to Jacobi-, and for N = 4, 6 with respect to Chebyshev weight functions. The separation properties of the zeros known from orthogonal polynomials are violated only by the extreme zero at one end of the interval

Geometric singularities and spectra of Landau-Ginzburg models

Some mathematical and physical aspects of superconformal string compactification in weighted projective space are discussed. In particular, we recast the path integral argument establishing the connection between Landau-Ginzburg conformal theories and Calabi-Yau string compactification in a geometric framework. We then prove that the naive expression for the vanishing of the first Chern class for a complete intersection (adopted from the smooth case) is sufficient to ensure that the resulting variety, which is generically singular, can be resolved to a smooth Calabi-Yau space. This justifies much analysis which has recently been expended on the study of Landau-Ginzburg models. Furthermore, we derive some simple formulae for the determination of the Witten index in these theories which are complimentary to those derived using semiclassical reasoning by Vafa. Finally, we also comment on the possible geometrical significance ofunorbifolded Landau-Ginzburg theories.

The algebraic structure of the Onsager algebra

We study the Lie algebra structure of the Onsager algebra from the ideal theoretic point of view. A structure theorem of ideals in the Onsager algebra is obtained with the connection to the finite-dimensional representations. We also discuss the solvable algebra aspect of the Onsager algebra through the formal theory.