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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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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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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.
Degenerations of Prym Varieties
, 2002
"... 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. In this paper we study the question whether there are “good” compactifications of P in ana ..."
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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. In this paper we study the question whether there are “good” compactifications of P in analogy to compactified Jacobians. The answer to this question depends on whether we consider degenerations of principally polarized Prym varieties or degenerations with the induced (nonprincipal) polarization. We describe degeneration data of such degenerations. The main application of our theory lies in the case of degenerations of principally polarized Prym varieties where we ask whether such a degeneration depends on a given oneparameter family containing (C, ι) or not. This
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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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,
Biswas,I.: Theta functions and Szego kernels
"... Abstract. We study relations between two fundamental constructions associated to vector bundles on a smooth complex projective curve: the theta function (a section of a line bundle on the moduli space of vector bundles) and the Szegö kernel (a section of a vector bundle on the square of the curve). ..."
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Abstract. We study relations between two fundamental constructions associated to vector bundles on a smooth complex projective curve: the theta function (a section of a line bundle on the moduli space of vector bundles) and the Szegö kernel (a section of a vector bundle on the square of the curve). Two types of relations are demonstrated. First, we establish a higher–rank version of the prime form, describing the pullback of determinant line bundles by difference maps, and show the theta function pulls back to the determinant of the Szegö kernel. Next, we prove that the expansion of the Szegö kernel at the diagonal gives the logarithmic derivative of the theta function over the moduli space of bundles for a fixed, or moving, curve. In particular, we recover the identification of the space of connections on the theta line bundle with moduli space of flat vector bundles, when the curve is fixed. When the curve varies, we identify this space of connections with the moduli space of extended connections, which we introduce. 1.
The Singular Locus of the Theta Divisor and Quadrics through a Canonical Curve
, 710
"... A section K on a genus g canonical curve C is identified as the key tool to prove new results on the geometry of the singular locus Θs of the theta divisor. The K divisor is characterized by the condition of linear dependence of a set of quadrics containing C and naturally associated to a degree g e ..."
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A section K on a genus g canonical curve C is identified as the key tool to prove new results on the geometry of the singular locus Θs of the theta divisor. The K divisor is characterized by the condition of linear dependence of a set of quadrics containing C and naturally associated to a degree g effective divisor on C. K counts the number of intersections of special varieties on the Jacobian torus defined in terms of Θs. It also identifies sections of line bundles on the moduli space of algebraic curves, closely related to the Mumford isomorphism, whose zero loci characterize special varieties in the framework of the AndreottiMayer approach to the Schottky problem, a result which also reproduces the only previously known case g = 4. This new approach, based on the combinatorics of determinantal relations for twofold products of holomorphic abelian differentials, sheds light on basic structures, and leads to the explicit expressions, in terms of theta functions, of the canonical basis of the abelian holomorphic differentials and of the constant defining the Mumford form. Furthermore, the metric on the moduli space of canonical curves, induced by the Siegel metric, which is shown to be equivalent to the KodairaSpencer map of the square of the Bergman reproducing kernel, is explicitly expressed in terms of the Riemann period matrix only, a result previously known for the trivial cases g = 2 and
UNIVERSAL VECTOR BUNDLE OVER THE REALS
, 909
"... Abstract. Let XR be a geometrically irreducible smooth projective curve, defined over R, such that XR does not have any real points. Let X = XR ×R C be the complex curve. We show that there is a universal real algebraic line bundle over XR × Pic d (XR) if and only if χ(L) is odd for L ∈ Pic d (XR). ..."
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Abstract. Let XR be a geometrically irreducible smooth projective curve, defined over R, such that XR does not have any real points. Let X = XR ×R C be the complex curve. We show that there is a universal real algebraic line bundle over XR × Pic d (XR) if and only if χ(L) is odd for L ∈ Pic d (XR). There is a universal quaternionic algebraic line bundle over X × Pic d (X) if and only if the degree d is odd. Take integers r and d such that r ≥ 2, and d is coprime to r. Let MXR (r, d) (respectively, MX(r, d)) be the moduli space of stable vector bundles over XR (respectively, X) of rank r and degree d. We prove that there is a universal real algebraic vector bundle over XR × MXR (r, d) if and only if χ(E) is odd for E ∈ MXR (r, d). There is a universal quaternionic vector bundle over X × MX(r, d) if and only if the degree d is odd, where E ∈ MX(r, d). The cases where XR is geometrically reducible or XR has real points are also investigated. 1.