The vanishing of two-point functions for three-loop superstring scattering amplitudes

In this paper we show that the two-point function for the three-loop chiral superstring measure ansatz proposed by Cacciatori, Dalla Piazza, and van Geemen vanishes. Our proof uses the reformulation of ansatz in terms of even cosets, theta functions, and specifically the theory of the $\Gamma_{00}$ linear system on Jacobians introduced by van Geemen and van der Geer. At the two-loop level, where the amplitudes were computed by D'Hoker and Phong, we give a new proof of the vanishing of the two-point function (which was proven by them). We also discuss the possible approaches to proving the vanishing of the two-point function for the proposed ansatz in higher genera

Extending the Belavin-Knizhnik "wonderful formula" by the characterization of the Jacobian

A long-standing question in string theory is to find the explicit expression of the bosonic measure, a crucial issue also in determining the superstring measure. Such a measure was known up to genus three. Belavin and Knizhnik conjectured an expression for genus four which has been proved in the framework of the recently introduced vector-valued Teichmueller modular forms. It turns out that for g>3 the bosonic measure is expressed in terms of such forms. In particular, the genus four Belavin-Knizhnik "wonderful formula" has a remarkable extension to arbitrary genus whose structure is deeply related to the characterization of the Jacobian locus. Furthermore, it turns out that the bosonic string measure has an elegant geometrical interpretation as generating the quadrics in P^{g-1} characterizing the Riemann surface. All this leads to identify forms on the Siegel upper half-space that, if certain conditions related to the characterization of the Jacobian are satisfied, express the bosonic measure as a multiresidue in the Siegel upper half-space. We also suggest that it may exist a super analog on the super Siegel half-space.Comment: 15 pages. Typos corrected, refs. and comments adde

Genus four superstring measures

A main issue in superstring theory are the superstring measures. D'Hoker and Phong showed that for genus two these reduce to measures on the moduli space of curves which are determined by modular forms of weight eight and the bosonic measure. They also suggested a generalisation to higher genus. We showed that their approach works, with a minor modification, in genus three and we announced a positive result also in genus four. Here we give the modular form in genus four explicitly. Recently S. Grushevsky published this result as part of a more general approach.Comment: 7 pages. To appear in Letters in Mathematical Physic

Rigidity of SU(2,2|2)-symmetric solutions in Type IIB

We investigate the existence of half-BPS solutions in Type IIB supergravity which are invariant under the superalgebra SU(2,2|2) realized on either AdS_5 x S^2 x S^1 or AdS_5 x S^3 warped over a Riemann surface \Sigma with boundary. We prove that, in both cases, the only solution is AdS_5 x S^5 itself. We argue that this result provides evidence for the non-existence of fully back-reacted intersecting D3/D7 branes with either AdS_5 x S^2 x S^1 x \Sigma or AdS_5 x S^3 x \Sigma near-horizon limits.Comment: 55 page

An Introduction to Pure Spinor Superstring Theory

In these lecture notes presented at the 2015 Villa de Leyva Summer School, we give an introduction to superstring theory. We begin by studying the particle and superparticle in order to get a better understanding on the superstring side. Afterwards, we review the pure spinor formalism and end by computing the scattering amplitude for three gravitons at tree-level.Comment: Villa de Leyva Summer School 2015 proceedings, 28 pages, 2 figure

Dual WDVV Equations in N=2 Supersymmetric Yang-Mills Theory

This paper studies the dual form of Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations in N=2 supersymmetric Yang-Mills theory by applying a duality transformation to WDVV equations. The dual WDVV equations called in this paper are non-linear differential equations satisfied by dual prepotential and are found to have the same form with the original WDVV equations. However, in contrast with the case of weak coupling calculus, the perturbative part of dual prepotential itself does not satisfy the dual WDVV equations. Nevertheless, it is possible to show that the non-perturbative part of dual prepotential can be determined from dual WDVV equations, provided the perturbative part is given. As an example, the SU(4) case is presented. The non-perturbative dual prepotential derived in this way is consistent to the dual prepotential obtained by D'Hoker and Phong.Comment: misprints are corrected, revtex, 10 page

Classical theta constants vs. lattice theta series, and super string partition functions

Recently, various possible expressions for the vacuum-to-vacuum superstring amplitudes has been proposed at genus $g=3,4,5$. To compare the different proposals, here we will present a careful analysis of the comparison between the two main technical tools adopted to realize the proposals: the classical theta constants and the lattice theta series. We compute the relevant Fourier coefficients in order to relate the two spaces. We will prove the equivalence up to genus 4. In genus five we will show that the solutions are equivalent modulo the Schottky form and coincide if we impose the vanishing of the cosmological constant.Comment: 21 page

Four-point Functions of Lowest Weight CPOs in N=4 SYM_4 in Supergravity Approximation

We show that the recently found quartic action for the scalars from the massless graviton multiplet of type IIB supergravity compactified on AdS_5\times S^5 background coincides with the relevant part of the action of the gauged N=8 5d supergravity on AdS_5. We then use this action to compute the 4-point function of the lowest weight chiral primary operators \tr(\phi^{(i}\phi^{j)}) in N=4 SYM_4 at large $N$ and at strong `t Hooft coupling.Comment: Latex, 21p, misprints are correcte

Superstring scattering amplitudes in higher genus

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 ans\"atze 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.Comment: Minor changes; final version to appear in Comm. Math. Phy

Equidistribution Rates, Closed String Amplitudes, and the Riemann Hypothesis

We study asymptotic relations connecting unipotent averages of $Sp(2g,\mathbb{Z})$ automorphic forms to their integrals over the moduli space of principally polarized abelian varieties. We obtain reformulations of the Riemann hypothesis as a class of problems concerning the computation of the equidistribution convergence rate in those asymptotic relations. We discuss applications of our results to closed string amplitudes. Remarkably, the Riemann hypothesis can be rephrased in terms of ultraviolet relations occurring in perturbative closed string theory.Comment: 15 page