arXiv subjects

A. G. Tsuchiya

Publications and source records attributed to A. G. Tsuchiya.

4 recordsLinked to original sources

arXiv2026

On theta function expressions of cyclic products of fermion correlation functions in genus two

In arXiv:2211.09069, significant progress was made in decomposing simple products of fermion correlation functions, and in summing over spin structures of superstring amplitudes in genus two under cyclic constraints. In this manuscript we consider part of the same subject using a framework in which one of the branch points of the genus two curve is fixed at infinity. This framework is a direct generalization of the popular one in the case of genus one. We address some of the issues that remained unresolved in our previous paper arXiv:2209.14633. We show that the spin structures of the simple products of fermion correlation functions with cyclic conditions depend only on the Pe-function values at the half-periods of the genus two surface, for any number of factors in the products. Similar to the genus one case, we can provide basis functions to decompose the product. Consequently, the trilinear relations found in arXiv:2211.09069 can be derived from the known set of differential equations of genus two Pe-functions by setting the variables equal to the half-periods of the non-singular and even spin structures, as is the case for genus one. Based on these considerations, we present a procedure for expressing the results of decomposed formulae in terms of the unique genus two theta function for two, three, and four point cases, and discuss a realistic approach for calculating six point function. A general formula for the expression of the results in terms of the theta function for the product of an arbitrary number of the fermion correlation functions is not yet derivable.

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arXiv2023

On a formula of spin sums, Eisenstein-Kronecker series in higher genus Riemann surfaces

We discuss a decomposition formula of simple products of fermion correlation functions with cyclic constrains and its applications to spin sums of super string amplitudes. Based on some facts which are noted or derived in this paper, we propose a candidate of the form of this decomposition formula for some of higher genus cases which includes genus two case. Although we had to use several conjectures and assumptions due to unsolved mathematical difficulties, the method described in the text may be an efficient way to obtain the decomposition formula in higher genus cases. In particular, for those cases, we propose a concrete method to sum over non singular even spin structures for the product of arbitrary number of the fermion correlation functions with cyclic constraints in super string amplitudes. We also propose an explicit generalization of Eisenstein-Kronecker series to the higher genus cases in the process of considerations above.

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arXiv2017

On new theta identities of fermion correlation functions on genus g Riemann surfaces

Theta identities on genus g Riemann surfaces which decompose simple products of fermion correlation functions with a constraint on their variables are considered. This type of theta identities is, in a sense, dual to Fay s formula, by which it is possible to sum over spin structures of certain part of superstring amplitudes in NSR formalism without using Fay s formula nor Riemann s theta formula in much simpler, more transparent way. Also, such identities will help to cast correlation functions among arbitrary numbers of Kac-Moody currents in a closed form. As for genus 1, the identities are reported before in ref[1] [2]. Based on some notes on genus 1 case which were not reported in ref[1] [2] and relating those to the results of the Dolan Goddard method ref[3] on describing Kac-Moody currents in a closed form, we propose an idea of generalizing genus 1 identities to the case of genus g surfaces. This is not a complete derivation of the higher genus formula due to difficulties of investigating singular part of derivatives of genus g Weierstrass Pe functions. Mathematical issues remained unsolved for genus g >1 are described in the text.

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arXiv2015

On the pole structures of the disconnected part of hyper elliptic g loop M point super string amplitudes

Structures of the disconnected part of higher genus superstring amplitudes restricted to the hyper elliptic cases are investigated in the NSR formalism, based on the DHoker Phong and recent results. A set of equations, which we can regard as a basic tool to sum over the spin structures of any of g loop, M point amplitudes systematically, is shown by using a classical result of Abelian functions. We discuss structures of g loop, M point massless external boson superstring amplitudes by assuming that the spin structure dependence of any of the disconnected amplitudes is only on one kind of constants, the genus g Weierstrass Pe function valued at the summation of g number of half periods chosen out of 2g+1 half periods. This is a natural generalization of the case of genus 1. This assumption will be validated by a conjectured theorem which states that the spin structure dependent part of any string amplitude will be naturally decomposed into two parts. One is composed of manifestly modular invariant functions of positions of inserting operators, and the other is the polynomial of Pe function constants related to the moduli of Riemann surfaces only. It is shown that this is actually the case for any M for g=1, and M=1,2,3 for any g. Due to a technical problem, our consideration is at present restricted to the case that g(g+1)divided by 2 is odd. Example calculations are shown for the genus 2 by the method described here. In particular, our method correctly reproduces biholomorphic 1 form of DHoker Phong result as for the four point amplitudes of the disconnected parts.

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