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1,246
KodairaSpencer theory of gravity and exact results for quantum string amplitudes
 Commun. Math. Phys
, 1994
"... 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 particu ..."
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Cited by 376 (49 self)
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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.
The number of faces of simplicial convex polytopes
 Advances in Math. 35
, 1980
"... Let P be a simplicial convex dpolytope with fi = fi(P) faces of dimension i. The vector f(P) = (f., fi,..., fdel) is called the fvector 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 ..."
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Cited by 111 (1 self)
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Let P be a simplicial convex dpolytope with fi = fi(P) faces of dimension i. The vector f(P) = (f., fi,..., fdel) is called the fvector 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 fvector of some simplicial convex dpolytope. 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 ddefine polytope P with f(P) = (f., fi,..., f&, where we set fel = 1. The vector h(P) = (h, , h,,..., hd) is called the hvector of P 181. The DehnSommerville 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+,‘) + (n61; 1) +... + (7;;). Also define Oci> = 0. Let us say that a vector (k, , k,,..., Kd) of integers is an Mvector (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 hvector of a simplicial convex dpolytope if and only if h, = 1,
A holomorphic Casson invariant for CalabiYau 3folds, and bundles on K3 fibrations
 J. Differential Geom
, 2000
"... We briefly review the formal picture in which a CalabiYau nfold is the complex analogue of an oriented real nmanifold, 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 CalabiYau 3fol ..."
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Cited by 105 (5 self)
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We briefly review the formal picture in which a CalabiYau nfold is the complex analogue of an oriented real nmanifold, 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 CalabiYau 3fold. 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 GromovWitten – like invariants of Fano 3folds. It also allows us to define the holomorphic Casson invariant of a CalabiYau 3fold 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 GromovWitten invariants of the “Mukaidual ” 3fold for others. As an example the invariant is shown to distinguish Gross ’ diffeomorphic 3folds. Finally the Mukaidual 3fold is shown to be CalabiYau and its cohomology is related to that of X. 1
The spectrum of BPS branes on a noncompact CalabiYau
, 2000
"... We begin the study of the spectrum of BPS branes and its variation on lines of marginal stability on OIP2(−3), a CalabiYau ALE space asymptotic to C 3 /Z3. We show how to get the complete spectrum near the large volume limit and near the orbifold point, and find a striking similarity between the de ..."
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Cited by 97 (10 self)
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We begin the study of the spectrum of BPS branes and its variation on lines of marginal stability on OIP2(−3), a CalabiYau ALE space asymptotic to C 3 /Z3. We show how to get the complete spectrum near the large volume limit and near the orbifold point, and find a striking similarity between the descriptions of holomorphic bundles and BPS branes in these two limits. We use these results to develop a general picture of the spectrum. We also suggest a generalization of some of the ideas to the quintic CalabiYau.
Split States, Entropy Enigmas, Holes and Halos
, 2007
"... We investigate degeneracies of BPS states of Dbranes on compact CalabiYau manifolds. We develop a factorization formula for BPS indices using attractor flow trees associated to multicentered black hole bound states. This enables us to study background dependence of the BPS spectrum, to compute e ..."
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Cited by 93 (11 self)
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We investigate degeneracies of BPS states of Dbranes on compact CalabiYau manifolds. We develop a factorization formula for BPS indices using attractor flow trees associated to multicentered black hole bound states. This enables us to study background dependence of the BPS spectrum, to compute explicitly exact indices of various nontrivial Dbrane systems, and to clarify the subtle relation of DonaldsonThomas invariants to BPS indices of stable D6D2D0 states, realized in supergravity as “hole halos. ” We introduce a convergent generating function for D4 indices in the large CY volume limit, and prove it can be written as a modular average of its polar part, generalizing the fareytail expansion of the elliptic genus. We show polar states are “split ” D6antiD6 bound states, and that the partition function factorizes accordingly, leading to a refined version of the OSV conjecture. This differs from the original conjecture in several aspects. In particular we obtain a nontrivial measure factor g −2 top e−K and find factorization requires a cutoff. We show that the main factor determining the cutoff and therefore the error is the existence of “swing states ” — D6 states which exist at large radius but do not form stable D6antiD6 bound states. We point out a likely breakdown of the OSV conjecture at small gtop (in the large background CY volume limit), due to the surprising phenomenon that for sufficiently large background Kähler moduli, a charge ΛΓ supporting single centered black holes of entropy ∼ Λ2S(Γ) also admits twocentered BPS black hole realizations whose entropy grows like Λ3 when Λ → ∞.
Packing Lines, Planes, etc.: Packings in Grassmannian Spaces
, 1996
"... We address the question: How should N ndimensional subspaces of mdimensional 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 th ..."
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Cited by 86 (10 self)
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We address the question: How should N ndimensional subspaces of mdimensional 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 mspace 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.
Toeplitz Quantization Of Kähler Manifolds And gl(N), N → ∞ Limits
"... For general compact Kähler manifolds it is shown that both Toeplitz quantization and geometric quantization lead to a welldefined (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 s ..."
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Cited by 84 (10 self)
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For general compact Kähler manifolds it is shown that both Toeplitz quantization and geometric quantization lead to a welldefined (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 finitedimensional matrix algebras gl(N), N → ∞.
Stable pairs, linear systems and the Verlinde formula
 Invent. Math
, 1994
"... 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 rout ..."
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Cited by 84 (8 self)
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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 wellknown theory of stable
Categorical mirror symmetry: the elliptic curve
 Adv. Theor. Math. Phys
, 1998
"... 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 ..."
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Cited by 81 (10 self)
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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.