Matching the (DR4)-R-6 interaction at two-loops

The coefficient of the $D^6 {\cal R}^4$ interaction in the low energy expansion of the two-loop four-graviton amplitude in type II superstring theory is known to be proportional to the integral of the Zhang-Kawazumi (ZK) invariant over the moduli space of genus-two Riemann surfaces. We demonstrate that the ZK invariant is an eigenfunction with eigenvalue 5 of the Laplace-Beltrami operator in the interior of moduli space. Exploiting this result, we evaluate the integral of the ZK invariant explicitly, finding agreement with the value of the two-loop $D^6 {\cal R}^4$ interaction predicted on the basis of S-duality and supersymmetry. A review of the current understanding of the $D^{2p} {\cal R}^4$ interactions in type II superstring theory compactified on a torus $T^d$ with $p \leq 3$ and $d \leq 4$ is included.Comment: 40 pages, various typos and coefficients corrected in version

Infrared divergences and harmonic anomalies in the two-loop superstring effective action

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are creditedArticle funded by SCOAP3. This research is partially supported by STFC (Grant ST/L000415/1, String theory, gauge theory & duality

Constraints on Automorphic Forms of Higher Derivative Terms from Compactification

By dimensionally reducing the higher derivative corrections of ten-dimensional IIB theory on a torus we deduce constraints on the E_{n+1} automorphic forms that occur in d=10-n dimensions. In particular we argue that these automorphic forms involve the representation of E_{n+1} with fundamental weight \lambda^{n+1}, which is also the representation to which the string charges in d dimensions belong. We also consider a similar calculation for the reduction of higher derivative terms in eleven-dimensional M-theory.Comment: Minor corrections, to appear in JHE

BPS Saturated Amplitudes and Non-perturbative String Theory

The study of the special F^4 and R^4 in the effective action for the Spin(32)/Z_2 and type II strings shed some light on D-brane calculus and on instanton contributions counting. The D-instanton case is discussed separately

Higher derivative type II string effective actions, automorphic forms and E11

By dimensionally reducing the ten-dimensional higher derivative type IIA string theory effective action we place constraints on the automorphic forms that appear in the effective action in lower dimensions. We propose a number of properties of such automorphic forms and consider the prospects that E11 can play a role in the formulation of the higher derivative string theory effective action.Comment: 34 page

Eisenstein series for infinite-dimensional U-duality groups

We consider Eisenstein series appearing as coefficients of curvature corrections in the low-energy expansion of type II string theory four-graviton scattering amplitudes. We define these Eisenstein series over all groups in the E_n series of string duality groups, and in particular for the infinite-dimensional Kac-Moody groups E9, E10 and E11. We show that, remarkably, the so-called constant term of Kac-Moody-Eisenstein series contains only a finite number of terms for particular choices of a parameter appearing in the definition of the series. This resonates with the idea that the constant term of the Eisenstein series encodes perturbative string corrections in BPS-protected sectors allowing only a finite number of corrections. We underpin our findings with an extensive discussion of physical degeneration limits in D<3 space-time dimensions.Comment: 69 pages. v2: Added references and small additions, to be published in JHE

M-Theory and Maximally Supersymmetric Gauge Theories

In this informal review for non-specalists we discuss the construction of maximally supersymmetric gauge theories that arise on the worldvolumes branes in String Theory and M-Theory. Particular focus is made on the relatively recent construction of M2-brane worldvolume theories. In a formal sense, the existence of these quantum field theories can be viewed as predictions of M-Theory. Their construction is therefore a reinforcement of the ideas underlying String Theory and M-Theory. We also briefly discuss the six-dimensional conformal field theory that is expected to arise on M5-branes. The construction of this theory is not only an important open problem for M-Theory but also a significant challenge to our current understanding of quantum field theory more generally.Comment: Minor corrections and an added acknowledgement. Accepted for publication in Annual Review

Exploring transcendentality in superstring amplitudes

Abstract It is well known that the low energy expansion of tree-level superstring scattering amplitudes satisfies a suitably defined version of uniform transcendentality. In this paper it is argued that there is a natural extension of this definition that applies to the genus-one four-graviton Type II superstring amplitude to all orders in the low-energy expansion. To obtain this result, the integral over the genus-one moduli space is partitioned into a region $$\mathrm{\mathcal{M}}$$ ℳ R surrounding the cusp and its complement $$\mathrm{\mathcal{M}}$$ ℳ L , and an exact expression is obtained for the contribution to the amplitude from $$\mathrm{\mathcal{M}}$$ ℳ R . The low-energy expansion of the $$\mathrm{\mathcal{M}}$$ ℳ R contribution is proven to be free of irreducible multiple zeta-values to all orders. The contribution to the amplitude from $$\mathrm{\mathcal{M}}$$ ℳ L is computed in terms of modular graph functions up to order D 12 $$\mathrm{\mathcal{R}}$$ ℛ 4 in the low-energy expansion, and general arguments are used beyond this order to conjecture the transcendentality properties of the $$\mathrm{\mathcal{M}}$$ ℳ L contributions. Uniform transcendentality of the full amplitude holds provided we assign a non-zero weight to certain harmonic sum functions, an assumption which is familiar from transcendentality assignments in quantum field theory amplitudes.</jats:p

The D^{2k} R^4 Invariants of N=8 Supergravity

The existence of a linearized SUSY invariant for N=8 supergravity whose gravitational components are usually called R^4 was established long ago by on-shell superspace arguments. Superspace and string theory methods have also established analogous higher dimensional D^{2k} R^4 invariants. However, very little is known about the SUSY completions of these operators which involve other fields of the theory. In this paper we find the detailed component expansion of the linearized R^4 invariant starting from the corresponding superamplitude which generates all component matrix elements of the operator. It is then quite straightforward to extend results to the entire set of D^{2k} R^4 operators.Comment: 17 page