We develop techniques to compute higher loop string amplitudes for twisted N = 2 theories with ĉ = 3 (i.e. the critical case). An important ingredient is the discovery of an anomaly at every genus in decoupling of BRST trivial states, captured to all orders by a master anomaly equation. In a particular realization of the N = 2 theories, the resulting string field theory is equivalent to a topological theory in six dimensions, the Kodaira– Spencer theory, which may be viewed as the closed string analog of the Chern–Simon theory. Using the mirror map this leads to computation of the ‘number ’ of holomorphic curves of higher genus curves in Calabi–Yau manifolds. It is shown that topological amplitudes can also be reinterpreted as computing corrections to superpotential terms appearing in the effective 4d theory resulting from compactification of standard 10d superstrings on the corresponding N = 2 theory. Relations with c = 1 strings are also pointed out.

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Let P be a simplicial convex d-polytope with fi = fi(P) faces of dimension i. The vector f(P) = (f., fi,..., fdel) is called the f-vector of P. In 1971 McMullen [6; 7, p. 1791 conjectured that a certain condition on a vector f = (f., fi,..., fd...J of integers was necessary and sufficient for f to be the f-vector of some simplicial convex d-polytope. Billera and Lee [l] proved the sufficiency of McMullen’s condition. In this paper we prove necessity. Thus McMullen’s conjecture is completely verified. First we describe McMullen’s condition. Given a simplicial convex d-define polytope P with f(P) = (f., fi,..., f&, where we set fel = 1. The vector h(P) = (h, , h,,..., hd) is called the h-vector of P 181. The Dehn-Sommerville equations, which hold for any simplicial convex polytope, are equivalent to the statement that hi = hdpi, 0 < i,< d [7, Sect. 5.11. If k and i are positive integers, then k can be written uniquely in the form h = (7) + (;“-;) +.. ’ + (q), where ni> ni_r>...> nj> j> 1. Following [6, 8, 91, define h’i ’ = (y+,‘) + (n6-1; 1) +... + (7;;). Also define Oci> = 0. Let us say that a vector (k, , k,,..., Kd) of integers is an M-vector (after F. S. Macaulay) if k, = 1 and 0 < k,.+r < kii ’ for 1 < i, ( d- 1. McMullen’s conjecture may now be stated as follows: A sequence (h, , h,,..., hd) of integers is the h-vector of a simplicial convex d-polytope if and only if h, = 1,

We briefly review the formal picture in which a Calabi-Yau n-fold is the complex analogue of an oriented real n-manifold, and a Fano with a fixed smooth anticanonical divisor is the analogue of a manifold with boundary, motivating a holomorphic Casson invariant counting bundles on a Calabi-Yau 3-fold. We develop the deformation theory necessary to obtain the virtual moduli cycles of [LT], [BF] in moduli spaces of stable sheaves whose higher obstruction groups vanish. This gives, for instance, virtual moduli cycles in Hilbert schemes of curves in P 3, and Donaldson – and Gromov-Witten – like invariants of Fano 3-folds. It also allows us to define the holomorphic Casson invariant of a Calabi-Yau 3-fold X, prove it is deformation invariant, and compute it explicitly in some examples. Then we calculate moduli spaces of sheaves on a general K3 fibration X, enabling us to compute the invariant for some ranks and Chern classes, and equate it to Gromov-Witten invariants of the “Mukai-dual ” 3-fold for others. As an example the invariant is shown to distinguish Gross ’ diffeomorphic 3-folds. Finally the Mukai-dual 3-fold is shown to be Calabi-Yau and its cohomology is related to that of X. 1

Let X be a smooth projective complex curve of genus g ≥ 2, let Λ→X be a line bundle of degree d> 0, and let (E, φ) be a pair consisting of a vector bundle E →X such that Λ 2 E = Λ and a section φ ∈ H 0 (E) − 0. This paper will study the moduli theory of such pairs. However, it is by no means a routine generalization of the well-known theory of stable

We describe an isomorphism of categories conjectured by Kontsevich. If M and ˜ M are mirror pairs then the conjectural equivalence is between the derived category of coherent sheaves on M and a suitable version of Fukaya’s category of Lagrangian submanifolds on ˜ M. We prove this equivalence when M is an elliptic curve and ˜ M is its dual curve, exhibiting the dictionary in detail.

For general compact Kähler manifolds it is shown that both Toeplitz quantization and geometric quantization lead to a well-defined (by operator norm estimates) classical limit. This generalizes earlier results of the authors and Klimek and Lesniewski obtained for the torus and higher genus Riemann surfaces, respectively. We thereby arrive at an approximation of the Poisson algebra by a sequence of finite-dimensional matrix algebras gl(N), N → ∞.

We address the question: How should N n-dimensional subspaces of m-dimensional Euclidean space be arranged so that they are as far apart as possible? The results of extensive computations for modest values of N; n; m are described, as well as a reformulation of the problem that was suggested by these computations. The reformulation gives a way to describe n- dimensional subspaces of m-space as points on a sphere in dimension (m \Gamma 1)(m+2), which provides a (usually) lowerdimensional representation than the Pl ucker embedding, and leads to a proof that many of the new packings are optimal. The results have applications to the graphical display of multidimensional data via Asimov's grand tour method.

(i) Topology of embedded surfaces. Let X be a smooth, simply-connected 4-manifold, and ξ a 2-dimensional homology class in X. One of the features of topology in dimension 4 is the fact that, although one may always represent ξ as the fundamental class of some smoothly

this paper is to describe an attempt to understand and generalize a recent formula of Deninger [1997] by means of systematic numerical experiment. This conjectural formula,