Bong H. Lian

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.

Mirror maps, modular relations and hypergeometric series II

As a continuation of [1], we study modular properties of the periods, the mirror maps and Yukawa couplings for multi-moduli Calabi-Yau varieties. In Part A of this paper, motivated by the recent work of Kachru-Vafa, we degenerate a three-moduli family of Calabi-Yau toric varieties along a codimension one subfamily which can be described by the vanishing of certain Mori coordinate, corresponding to going to the “large volume limit” in a certain direction. Then we see that the deformation space of the subfamily is the same as a certain family of K3 toric surfaces. This family can in turn be studied by further degeneration along a subfamily which in the end is described by a family of elliptic curves. The periods of the K3 family (and hence the original Calabi-Yau family) can be described by the squares of the periods of the elliptic curves. The consequences include: (1) proofs of various conjectural formulas of physicists [2][3] involving mirror maps and modular functions; (2) new identities involving multi-variable hypergeometric series and modular functions — generalizing [1]. In Part B, we study for two-moduli families the perturbation series of the mirror map and the type A Yukawa couplings near certain large volume limits. Our main tool is a new class of polynomial PDEs associated with Fuchsian PDE systems. We derive the first few terms in the perturbation series. For the case of degree 12 hypersurfaces in P 4 [6,2,2,1,1], in one limit the series of the couplings are expressed in terms of the j function. In another limit, they are expressed in terms of rational functions. The latter give explicit formulas for infinite sequences of “instanton numbers” n d .

Maximal degeneracy points of GKZ systems

Motivated by mirror symmetry, we study certain integral representations of solutions to the Gel’fand-Kapranov-Zelevinsky(GKZ) hypergeometric system. Some of these solutions arise as period integrals for Calabi-Yau manifolds in mirror symmetry. We prove that for a suitable compactification of the parameter space, there exists certain special boundary points, which we called maximal degeneracy points, at which all but one solutions become singular. 3/2/96 † email: hosono@sci.toyama-u.ac.jp ‡ email: lian@max.math.brandeis.edu ⋄ email: yau@math.harvard.edu

Picard-Fuchs equations for relative periods and Abel-Jacobi map for Calabi-Yau hypersurfaces

We study the variation of relative cohomology for a pair consisting of a smooth projective hypersurface and an algebraic subvariety in it. We construct an inhomogeneous Picard-Fuchs equation by applying a Picard-Fuchs operator to the holomorphic top form on a Calabi-Yau hypersurface in toric variety, and deriving a general formula for the

Autoequivalences of derived category of a K3 surface and monodromy transformations

d

Howe pairs in the theory of vertex algebras

-exact form on one side of the equation. We also derive a double residue formula, giving a purely algebraic way to compute the inhomogeneous Picard-Fuchs equations for the Abel-Jacobi map, which has played an important role in the recent study of D-branes (by Morrison and Walcher). Using the variation formalism, we prove that the relative periods of toric B-branes on Calabi-Yau hypersurface satisfy the enhanced GKZ-hypergeometric system proposed in physics literature (by Alim, Hecht, Mayr, and Mertens), and discuss the relations between several works in the recent study of open string mirror symmetry. We also give the general solutions to the enhanced hypergeometric system.

Chiral equivariant cohomology I

We consider autoequivalences of the bounded derived category of coherent sheaves on a K3 surface. We prove that the image of the autoequivalences has index at most two in the group of the Hodge isometries of the Mukai lattice. Motivated by homological mirror symmetry we also consider the mirror counterpart, i.e. symplectic version of it. In the case of ρ(X) = 1, we find an explicit formula which reproduces the number of Fourier-Mukai (FM) partners from the monodromy problem of the mirror K3 family. We present an explicit example in which a monodromy action does not come from an autoequivalence of the mirror side.

c = 2 Rational Toroidal Conformal Field Theories via the Gauss Product

Abstract For any vertex algebra V and any subalgebra A ⊂ V , there is a new subalgebra of V known as the commutant of A in V . This construction was introduced by Frenkel–Zhu, and is a generalization of an earlier construction due to Kac–Peterson and Goddard–Kent–Olive known as the coset construction. In this paper, we interpret the commutant as a vertex algebra notion of invariant theory. We present an approach to describing commutant algebras in an appropriate category of vertex algebras by reducing the problem to a question in commutative algebra. We give an interesting example of a Howe pair (i.e., a pair of mutual commutants) in the vertex algebra setting.

Kummer structures on a K3 surface: An old question of T. Shioda

Abstract We construct a new equivariant cohomology theory for a certain class of differential vertex algebras, which we call the chiral equivariant cohomology. A principal example of a differential vertex algebra in this class is the chiral de Rham complex of Malikov–Schechtman–Vaintrob of a manifold with a group action. The main idea in this paper is to synthesize the algebraic approach to classical equivariant cohomology due to H. Cartan, 2 with the theory of differential vertex algebras, by using an appropriate notion of invariant theory. We also construct the vertex algebra analogues of the Mathai–Quillen isomorphism, the Weil and the Cartan models for equivariant cohomology, and the Chern–Weil map. We give interesting cohomology classes in the new theory that have no classical analogues.

The Candelas-de la Ossa-Green-Parkes formula

We find a concise relation between the moduli τ,ρ of a rational Narain lattice Γ(τ,ρ) and the corresponding momentum lattices of left and right chiral algebras via the Gauss product. As a byproduct, we find an identity which counts the cardinality of a certain double coset space defined for isometries between the discriminant forms of rank two lattices.