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As the interaction of mathematics and theoretical physics continues to intensify, the theories developed in mathematics are being applied to physics, and conversely. This book centers around the theory of primitive forms which currently plays an active and key role in topological field theory (theoretical physics), but was originally developed as a mathematical notion to define a "good period mapping" for a family of analytic structures.

The invited papers in this volume are expository in nature by participants of the Taniguchi Symposium on "Topological Field Theory, Primitive Forms and Related Topics" and the RIMS Symposium bearing the same title, both held in Kyoto. The papers reflect the broad research of some of the world's leading mathematical physicists, and should serve as an excellent resource for researchers as well as graduate students of both disciplines.



Degenerate Double Affine Hecke Algebra and Conformal Field Theory

We introduce a class of induced representations of the degenerate double affine Hecke algebra H of \(g{l_N}\)(ℂ) and analyze their structure mainly by means of intertwiners of H. We also construct them from \(\hat s{l_m}\)(ℂ)-modules using Knizhnik-Zamolodchikov connections in the conformai field theory. This construction provides a natural quotient of induced modules, which turns out to be the unique irreducible one under a certain condition. Some conjectural formulas are presented for the symmetric part of these quotients.
Tomoyuki Arakawa, Takeshi Suzuki, Akihiro Tsuchiya

Vertex Algebras

In this paper we try to define the higher-dimensional analogues of vertex algebras. In other words we define algebras which we hope have the same relation to higher-dimensional quantum field theories that vertex algebras have to one-dimensional quantum field theories (or to “chiral halves” of two-dimensional conformai field theories).
Richard E. Borcherds

Extensions of Conformal Modules

In this paper we classify extensions between irreducible finite conformal modules over the Virasoro algebra, over the current algebras and over their semidirect sum.
Shun-Jen Cheng, Victor G. Kac, Minoru Wakimoto

String Duality and a New Description of the E 6 Singularity

We discuss a new type of Landau-Ginzburg potential for the E 6 singularity of the form \(W = const + \left( {{Q_1}\left( x \right) + {P_1}\left( x \right)\sqrt {{P_2}\left( x \right)} } \right)/{x^3}\) which featured in a recent study of heterotic/type II string duality. Here Q1, P1 and P2 are polynomials of degree 15, 10 and 10, respectively. We study the properties of the potential in detail and show that it gives a new and consistent description of the E 6 singularity.
Tohru Eguchi

A Mirror Theorem for Toric Complete Intersections

We prove a generalized mirror conjecture for non-negative complete intersections in symplectic toric manifolds. Namely, we express solutions of the PDE system describing quantum cohomology of such a manifold in terms of suitable hypergeometric functions.
Alexander Givental

Precious Siegel Modular Forms of Genus Two

We give a review of the recent results concerning Siegel modular forms with respect to the paramodular groups of genus 2 and their applications to Algebraic Geometry and Physics. Some facts mentioned below have not been published before.
Valeri Gritsenko

Non-Abelian Conifold Transitions and N = 4 Dualities in Three Dimensions

We show how the Higgs mechanism for non-abelian N = 2 gauge theories in four dimensions is geometrically realized in the context of type II strings as transitions among compactifications of Calabi-Yau threefolds. We use this result and T-duality of a further compacitification on a circle to derive N = 4, d = 3 dual field theories. This reduces dualities for N = 4 gauge systems in three dimensions to perturbative symmetries of string theory. Moreover, we find that the dual of a gauge system always exists but may or may not correspond to a lagrangian system. In particular, we verify a conjecture of Intriligator and Seiberg that an ordinary gauge system is dual to compacitification of exceptional tensionless string theory down to three dimensions.
Kentaro Hori, Hirosi Ooguri, Cumrun Vafa

GKZ Systems, Gröbner Fans, and Moduli Spaces of Calabi-Yau Hypersurfaces

We present a detailed analysis of the GKZ (Gel’fand, Kapranov and Zelevinski) hypergeometric systems in the context of mirror symmetry of Calabi-Yau hypersurfaces in toric varieties. As an application, we will derive a concise formula for the prepotential about large complex structure limits.
Shinobu Hosono

Semisimple Holonomic D-Modules

There is a theory of Weil sheaves of Pierre Deligne (and BBG [1]) in characteristic p and a theory of mixed Hodge modules of Morihiko Saito [3] in characteristic 0. Weil sheaves or Hodge modules satisfy the following properties. In the statements, we write a pure perverse sheaf instead of a pure perverse Weil sheaf or a Hodge module.
Masaki Kashiwara

K3 Surfaces, Igusa Cusp Forms, and String Theory

It has recently become apparent that the elliptic genera of K3 surfaces (and their symmetric products) are intimately related to the Igusa cusp form of weight ten. In this contribution, I survey this connection with an emphasis on string theoretic viewpoints.
Toshiya Kawai

Hodge Strings and Elements of K. Saito’s Theory of Primitive Form

The Hodge strings construction of solutions to associativity equations is proposed. From the topological string theory point of view, this construction formalizes the integration over the position of the marked point procedure for computation of amplitudes. From the mathematical point of view the Hodge strings construction is just a composition of elements of harmonic theory (known among physicists as a t-part of t − t* equations [CV, D2]) and the K. Saito construction of flat coordinates (starting from flat connection with a spectral parameter).
We also show how elements of the Saito theory of primitive form appear naturally in the Landau-Ginzburg version of harmonic theory if we consider the holomorphic pieces of germs of harmonic forms at the singularity.
A. Losev

Summary of the Theory of Primitive Forms

In this article we review the theory of primitive forms associated to the semi- universal deformation of an isolated critical point of a holomorphic function basically following the original paper [18] as elementary as possible.
Atsushi Matsuo

Affine Hecke Algebras and Macdonald Polynomials

This article is intended to be a survey on recent developments in symmetric and nonsymmetric Macdonald polynomials from the viewpoint of affine and double affine Hecke algebras. We also include an application of (double) affine Hecke algebras to the description of symmetries in Macdonald polynomials. In this article, we consider the Macdonald polynomials of type A n-1 exclusively to simplify the presentation, while a considerable part of the statements below has been extended to arbitrary root systems.
Masatoshi Noumi

Duality for Regular Systems of Weights: A Précis

The theory of a regular system of weights was originally developed in order to understand the flat structure in the period maps for primitive forms [S9]. In the present work, the theory is applied to understand the self-duality of ADE and the strange duality of Arnold. Beyond the original purpose, the theory yields further a class of dual weight systems, which seem to have close connection with the Conway group and to be interesting to be studied yet. On the other hand, the duality of weight systems has an interpretation in terms of the Dedekind eta function. But its meaning is not yet clear.
Kyoji Saito

Flat Structure and the Prepotential for the Elliptic Root System of Type D4(1,1)

By Saito’s theory of the primitive form [S-2], the flat structure and the prepotential F are defined on S, where XS is a semi-universal deformation of an isolated singularity. The prepotential F is defined as a generating function of the structure constants of the function ring of the critical set of XS. Also the prepotential F satisfies WDVV equations coming from the associativity of the function ring of the critical set (see also Matsuo[M]).
Ikuo Satake

Generalized Dynkin Diagrams and Root Systems and Their Folding

Graphs which generalize the simple or affine Dynkin diagrams are introduced. Each diagram defines a bilinear form on a root system and thus a reflection group. We present some properties of these groups and of their natural “Coxeter element.” The folding of these graphs and groups is also discussed, using the theory of C-algebras.
Jean-Bernard Zuber
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