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S.:Superstring scattering amplitudes in higher genus arXiv: hepth/0803.3469
"... Abstract. In this paper we continue the program pioneered by D’Hoker and Phong, and recently advanced by Cacciatori, Dalla Piazza, and van Geemen, of finding the chiral superstring measure by constructing modular forms satisfying certain factorization constraints. We give new expressions for their p ..."
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Abstract. In this paper we continue the program pioneered by D’Hoker and Phong, and recently advanced by Cacciatori, Dalla Piazza, and van Geemen, of finding the chiral superstring measure by constructing modular forms satisfying certain factorization constraints. We give new expressions for their proposed ansätze in genera 2 and 3, respectively, which admit a straightforward generalization. We then propose an ansatz in genus 4 and verify that it satisfies the factorization constraints and gives a vanishing cosmological constant. We further conjecture a possible formula for the superstring amplitudes in any genus, subject to the condition that certain modular forms admit holomorphic roots. 1.
INFINITE DIMENSIONAL GEOMETRY AND QUANTUM FIELD THEORY OF STRINGS I. INFINITE DIMENSIONAL GEOMETRY OF SECOND QUANTIZED Free String
, 1994
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COMPACTIFIED JACOBIANS AND TORELLI MAP
"... Abstract. We compare several constructions of compactified jacobians using semistable sheaves, semistable projective curves, degenerations of abelian varieties, and combinatorics of cell decompositions and show that they are equivalent. We give a detailed description of the ”canonical compactified ..."
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Cited by 9 (0 self)
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Abstract. We compare several constructions of compactified jacobians using semistable sheaves, semistable projective curves, degenerations of abelian varieties, and combinatorics of cell decompositions and show that they are equivalent. We give a detailed description of the ”canonical compactified jacobian” in degree g − 1. Finally, we explain how Kapranov’s compactification of configuration spaces can be understood as a toric analog of the extended Torelli map. There are many papers devoted to compactifying (generalized) jacobians of curves and families of curves. Some of them are concerned primarily with existence, some provide a finer description. The approaches vary widely: some constructions use moduli of semistable rank1 sheaves, some use semistable projective curves, some use combinatorics of cell decompositions; yet others use degenerations of principally polarized abelian varieties and various notions of stable varieties. One aim of this survey is to give a definitive account in the case of nodal curves and to show, pleasingly, that in this case all of the known approaches are equivalent
”New” Veneziano amplitudes from ”old” Fermat (hyper)surfaces
, 2003
"... The history of the discovery of bosonic string theory is well documented. This theory evolved as an attempt to find a multidimensional analogue of Euler’s beta function to describe the multiparticle Veneziano amplitudes. Such an analogue had in fact been known in mathematics at least in 1922. Its ma ..."
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Cited by 3 (2 self)
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The history of the discovery of bosonic string theory is well documented. This theory evolved as an attempt to find a multidimensional analogue of Euler’s beta function to describe the multiparticle Veneziano amplitudes. Such an analogue had in fact been known in mathematics at least in 1922. Its mathematical meaning was studied subsequently from different angles by mathematicians such as Selberg, Weil and Deligne among others. The mathematical interpretation of this multidimensional beta function that was developed subsequently is markedly different from that described in physics literature. This work aims to bridge the gap between the mathematical and physical treatments. Using some results of recent publications (e.g. J.Geom.Phys.38 (2001) 81; ibid 43 (2002) 45) new topological, algebrogeometric, numbertheoretic and combinatorial treatment of the multiparticle Veneziano amplitudes is developed. As a result, an entirely new physical meaning of these amplitudes is emerging: they are periods of differential forms associated with homology cycles on Fermat (hyper)surfaces. Such (hyper)surfaces are considered as complex projective varieties of Hodge type. Although the computational formalism developed in this work resembles that used in mirror symmetry calculations, many additional results from mathematics are used along with their suitable physical interpretation. For instance, the Hodge spectrum of the Fermat (hyper)surfaces is in onetoone correspondence with the possible spectrum of particle masses. The formalism also allows us to obtain correlation functions of both conformal field theory and particle physics using the same type of the PicardFuchs equations whose solutions are being interpreted in terms of periods.
Determinantal Characterization of Canonical Curves and Combinatorial Theta Identities
, 2006
"... Canonical curves are characterized by the vanishing of combinatorial products of g + 1 determinants of the holomorphic abelian differentials. This also implies the characterization of canonical curves in terms of (g − 2)(g − 3)/2 theta identities and the explicit expression of the volume form on the ..."
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Cited by 2 (2 self)
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Canonical curves are characterized by the vanishing of combinatorial products of g + 1 determinants of the holomorphic abelian differentials. This also implies the characterization of canonical curves in terms of (g − 2)(g − 3)/2 theta identities and the explicit expression of the volume form on the moduli space of canonical curves induced by the Siegel metric. Together with the Mumford form, it naturally defines a nonholomorphic form of weight (13,13). The Mumford form itself is Petri’s theorem [1] determines the ideal of canonical curves of genus g ≥ 4 by means of relations among holomorphic differentials. As emphasized by Mumford, Petri’s relations are fundamental and should have basic applications (see pg.241 of [2]). A feature of Petri’s relations is that the holomorphic quadratic differentials he introduced depend on the choice of points. Furthermore,
Degenerations of Prym Varieties
, 2001
"... Abstract. Let (C, ι) be a stable curve with an involution. Following a classical construction, one can define its Prym variety P, which in this case turns out to be a semiabelian group variety and usually not complete. We determine the precise polarization type on the abelian part of P; define sever ..."
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Cited by 2 (0 self)
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Abstract. Let (C, ι) be a stable curve with an involution. Following a classical construction, one can define its Prym variety P, which in this case turns out to be a semiabelian group variety and usually not complete. We determine the precise polarization type on the abelian part of P; define several nice compactifications of P; give a condition for when such a compactification is essentially unique; find the indeterminacy locus of the extended Prym map from the compactified moduli of curves with a basepointfree involution to a compactified moduli of principally polarized abelian varieties; illustrate all this with numerous examples. 0
NUMBER THEORY IN PHYSICS
"... always been the case for the theory of differential equations. In the early twentieth century, with the advent of general relativity and quantum mechanics, topics such as differential and Riemannian geometry, operator algebras and functional analysis, or group theory also developed a close relation ..."
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always been the case for the theory of differential equations. In the early twentieth century, with the advent of general relativity and quantum mechanics, topics such as differential and Riemannian geometry, operator algebras and functional analysis, or group theory also developed a close relation to physics. In the past decade, mostly through the influence of string theory, algebraic geometry also began to play a major role in this interaction. Recent years have seen an increasing number of results suggesting that number theory also is beginning to play an essential part on the scene of contemporary theoretical and mathematical physics. Conversely, ideas from physics, mostly from quantum field theory and string theory, have started to influence work in number theory. In describing significant occurrences of number theory in physics, we will, on the one hand, restrict our attention to quantum physics, while, on the other hand, we will assume a somewhat extensive definition of number theory, that will allow us to include arithmetic algebraic geometry. The territory is vast and an extensive treatment would go beyond the size limits imposed by the encyclopaedia. The
Contents
, 2006
"... Abstract. We study the intersection theory on the moduli spaces of maps of npointed curves f: (C,s1,... sn) → V which are stable with respect to the weight data (a1,..., an), 0 ≤ ai ≤ 1. After describing the structure of these moduli spaces, we prove a formula describing the way each descendant ch ..."
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Abstract. We study the intersection theory on the moduli spaces of maps of npointed curves f: (C,s1,... sn) → V which are stable with respect to the weight data (a1,..., an), 0 ≤ ai ≤ 1. After describing the structure of these moduli spaces, we prove a formula describing the way each descendant changes under a wall crossing. As a corollary, we compute the weighted descendants in terms of the usual ones, i.e. for the weight data (1,..., 1), and vice versa.
New metrics on determinant of cohomology And Their applications to moduli spaces of punctured Riemann surfaces
, 1998
"... Abstract. For singular metrics, Ray and Singer’s analytic torsion formalism cannot be applied. Hence we do not have the socalled Quillen metric on determinant of cohomology with respect to a singular metric. In this paper, we introduce a new metric on determinant of cohomology by adapting a totally ..."
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Abstract. For singular metrics, Ray and Singer’s analytic torsion formalism cannot be applied. Hence we do not have the socalled Quillen metric on determinant of cohomology with respect to a singular metric. In this paper, we introduce a new metric on determinant of cohomology by adapting a totally different approach. More precisely, by strengthening results in the first paper of this series, we develop an admissible theory for compact Riemann surfaces with respect to singular volume forms, with which the arithmetic DeligneRiemannRoch isometry can be established for singular metrics. As an application, we prove the Mumford type fundamental relations for metrized determinant line bundles over moduli spaces of punctured Riemann surfaces. Moreover, using an idea of D’HokerPhong and Sarnak, we introduce a natural admssible metric associated to a punctured Riemann surface via the ArakelovPoincaré volume, a new invariant for a punctured Riemann surface. With this admissible metric, we make an intensive yet natural study on two Kähler forms on the moduli space of punctured Riemann surfaces associated to the WeilPetersson metric and the TakhtajanZograf metric (defined