By looking at phase transitions which occur as parameters are varied in supersymmetric gauge theories, a natural relation is found between sigma models based on Calabi-Yau hypersurfaces in weighted projective spaces and Landau-Ginzburg models. The construction permits one to recover the known correspondence between these types of models and to greatly extend it to include new classes of manifolds and also to include models with (0, 2) world-sheet supersymmetry. The construction also predicts the possibility of certain physical processes involving a change in the topology of space-time. The present paper is based on systematically exploiting one simple idea which is familiar in N = 1 supersymmetric theories in four dimensions and which we will therefore first state in that context. We consider renormalizable gauge theories constructed from vector (gauge) multiplets and charged chiral multiplets. If the

Contents 1. Introduction 2. Vanishing Theorems 3. Singularities of Pairs 4. Bertini Theorems 5. Effective Base Point Freeness 6. Construction of Singular Divisors 7. The L 2 Extension Theorem and Inversion of Adjunction 8. The Log Canonical Threshold 9. The Log Canonical Threshold and the Complex Singular Index 10. The Log Canonical Threshold and the Bernstein-Sato Polynomial 11. Rational and Canonical Singularities 1. Introduction Higher dimensional algebraic geometry has been one of the most rapidly developing research areas in the past twenty years. The first decade of its development centered around the formulation of the minimal model program and finding techniques to carry this program through. The proof of the existence of flips, given in [Mori88], completed the program in dimension three. These results, especially the progress leading up to [Mori88], are reviewed in several surveys. A very general overview is given in [Koll'ar8

We define an algebra on the space of BPS states in theories with extended supersymmetry. We show that the algebra of perturbative BPS states in toroidal compactification of the heterotic string is closely related to a generalized Kac-Moody algebra. We use D-brane theory to compare the formulation of RR-charged BPS algebras in type II compactification with the requirements of string/string duality and find that the RR charged BPS states should be regarded as cohomology classes on moduli spaces of coherent sheaves. The equivalence of the algebra of BPS states in heterotic/IIA dual pairs elucidates certain results and conjectures of Nakajima and Gritsenko & Nikulin, on geometrically defined algebras and furthermore suggests nontrivial generalizations of these algebras. In particular, to any Calabi-Yau 3-fold there are two canonically associated algebras exchanged by mirror symmetry. September

Is speculative mathematics dangerous? Recent interactions between physics and mathematics pose the question with some force: traditional mathematical norms discourage speculation, but it is the fabric of theoretical physics. In practice there can be benefits, but there can also be unpleasant and destructive consequences. Serious caution is required, and the issue should be considered before, rather than after, obvious damage occurs. With the hazards carefully in mind, we propose a framework that should allow a healthy and positive role for speculation.

We provide string theory examples where a toy model of a SUSY GUT or the MSSM is embedded in a compactification along with a gauge sector which dynamically breaks supersymmetry. We argue that by changing microscopic details of the model (such as precise choices of flux), one can arrange for the dominant mediation mechanism transmitting SUSY breaking to the Standard Model to be either gravity mediation or gauge mediation. Systematic improvement of such examples may lead to top-down models incorporating a solution to the SUSY flavor problem.

Polynomials appear in mathematics frequently, and we all know from experience that low degree polynomials are easier to deal with than high degree ones. It is, however, not clear that there is a well defined class of “low degree ” polynomials. For many questions, polynomials behave well if their degree is low enough, but the precise bound on the degree depends on the concrete problem. My interest is to investigate polynomials through their zero sets. That is, using sets of the form {(x1,..., xn)|f(x1,..., xn) = 0}. I intentionally refrain from specifying where the coordinates xi are. They could be rational, real or complex numbers, but in some cases the xi will be polynomials in a new variable t. My focus is on the polynomial f. Consider, for instance, a polynomial f: = a0 + n∑ aix k i, where ai ∈ Z \ {0}.

We provide string theory examples where a toy model of a SUSY GUT or the MSSM is embedded in a compactification along with a gauge sector which dynamically breaks supersymmetry. We argue that by changing microscopic details of the model (such as precise choices of flux), one can arrange for the dominant mediation mechanism transmitting SUSY breaking to the Standard Model to be either gravity mediation or gauge mediation. Systematic improvement of such examples may lead to top-down models incorporating a solution to the SUSY flavor problem.

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Chern numbers for singular varieties and elliptic homology By Burt Totaro A fundamental goal of algebraic geometry is to do for singular varieties whatever we can do for smooth ones. Intersection homology, for example, directly produces groups associated to any variety which have almost all the properties of the usual homology groups of a smooth variety. Minimal model theory suggests the possibility of working more indirectly by relating any singular variety to a variety which is smooth or nearly so. Here we use ideas from minimal model theory to define some characteristic numbers for singular varieties, generalizing the Chern numbers of a smooth variety. This was suggested by Goresky and MacPherson as a next natural problem after the definition of intersection homology [11]. We find that only a subspace of the Chern numbers can be defined for singular varieties. A convenient way to describe this subspace is to say that a smooth variety has a fundamental class in complex bordism, whereas a singular variety can at most have a fundamental class in a weaker homology theory, elliptic homology. We use this idea to give an algebro-geometric definition of elliptic homology: “complex bordism modulo flops equals elliptic homology.” This paper was inspired by some questions asked by Jack Morava. The descriptions of elliptic homology given by Gerald Höhn [13] were also an important

Abstract. We determine the group of conformal automorphisms of the self-dual metrics on n#CP 2 due to LeBrun for n ≥ 3, and Poon for n = 2. These metrics arise from an ansatz involving a circle bundle over hyperbolic three-space H 3 minus a finite number of points, called monopole points. We show that for n ≥ 3 connected sums, any conformal automorphism is a lift of an isometry of H 3 which preserves the set of monopole points. Furthermore, we prove that for n = 2, such lifts form a subgroup of index 2 in the full automorphism group, which we show is a semi-direct product (U(1) × U(1)) ⋉ D4, the dihedral group of order 8. Contents