arXiv subjects

Michael B. Green

Publications and source records attributed to Michael B. Green.

At least 37 records · Page 2Linked to original sources

arXiv2017

Identities between Modular Graph Forms

This paper investigates the relations between modular graph forms, which are generalizations of the modular graph functions that were introduced in earlier papers motivated by the structure of the low energy expansion of genus-one Type II superstring amplitudes. These modular graph forms are multiple sums associated with decorated Feynman graphs on the world-sheet torus. The action of standard differential operators on these modular graph forms admits an algebraic representation on the decorations. First order differential operators are used to map general non-holomorphic modular graph functions to holomorphic modular forms. This map is used to provide proofs of the identities between modular graph functions for weight less than six conjectured in earlier work, by mapping these identities to relations between holomorphic modular forms which are proven by holomorphic methods. The map is further used to exhibit the structure of identities at arbitrary weight.

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arXiv2016

Modular Graph Functions

In earlier work we studied features of non-holomorphic modular functions associated with Feynman graphs for a conformal scalar field theory on a two-dimensional torus with zero external momenta at all vertices. Such functions, which we will refer to as modular graph functions, arise, for example, in the low energy expansion of genus-one Type II superstring amplitudes. We here introduce a class of single-valued elliptic multiple polylogarithms, which are defined as elliptic functions associated with Feynman graphs with vanishing external momenta at all but two vertices. These functions depend on a coordinate, $ζ$, on the elliptic curve and reduce to modular graph functions when $ζ$ is set equal to $1$. We demonstrate that these single-valued elliptic multiple polylogarithms are linear combinations of multiple polylogarithms, and that modular graph functions are sums of single-valued elliptic multiple polylogarithms evaluated at the identity of the elliptic curve, in both cases with rational coefficients. This insight suggests the many interrelations between modular graph functions (a few of which were established in earlier papers) may be obtained as a consequence of identities involving multiple polylogarithms, and explains an earlier observation that the coefficients of the Laurent polynomial at the cusp are given by rational numbers times single-valued multiple zeta values.

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arXiv2016

Type I/heterotic duality and M-theory amplitudes

This paper investigates relationships between low-energy four-particle scattering amplitudes with external gauge particles and gravitons in the E_8 X E_8 and SO(32) heterotic string theories and the type I and type IA superstring theories by considering a variety of tree level and one-loop Feynman diagrams describing such amplitudes in eleven-dimensional supergravity in a Horava--Witten background compactified on a circle. This accounts for a number of perturbative and non-perturbative aspects of low order higher derivative terms in the low-energy expansion of string theory amplitudes, which are expected to be protected by half maximal supersymmetry from receiving corrections beyond one or two loops. It also suggests the manner in which type I/heterotic duality may be realised for certain higher derivative interactions that are not so obviously protected. For example, our considerations suggest that R**4 interactions (where R is the Riemann curvature) might receive no perturbative corrections beyond one loop by virtue of a conspiracy involving contributions from (non-BPS) Z2 D-instantons in the type I and heterotic SO(32) theories.

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arXiv2015

Proof of a modular relation between 1-, 2- and 3-loop Feynman diagrams on a torus

The coefficients of the higher-derivative terms in the low energy expansion of genus-one graviton Type II superstring scattering amplitudes are determined by integrating sums of non-holomorphic modular functions over the complex structure modulus of a torus. In the case of the four-graviton amplitude, each of these modular functions is a multiple sum associated with a Feynman diagram for a free massless scalar field on the torus. The lines in each diagram join pairs of vertex insertion points and the number of lines defines its weight $w$, which corresponds to its order in the low energy expansion. Previous results concerning the low energy expansion of the genus-one four-graviton amplitude led to a number of conjectured relations between modular functions of a given $w$, but different numbers of loops $\le w-1$. In this paper we shall prove the simplest of these conjectured relations, namely the one that arises at weight $w=4$ and expresses the three-loop modular function $D_4$ in terms of modular functions with one and two loops. As a byproduct, we prove three intriguing new holomorphic modular identities.

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arXiv2015

On the modular structure of the genus-one Type II superstring low energy expansion

The analytic contribution to the low energy expansion of Type II string amplitudes at genus-one is a power series in space-time derivatives with coefficients that are determined by integrals of modular functions over the complex structure modulus of the world-sheet torus. These modular functions are associated with world-sheet vacuum Feynman diagrams and given by multiple sums over the discrete momenta on the torus. In this paper we exhibit exact differential and algebraic relations for a certain infinite class of such modular functions by showing that they satisfy Laplace eigenvalue equations with inhomogeneous terms that are polynomial in non-holomorphic Eisenstein series. Furthermore, we argue that the set of modular functions that contribute to the coefficients of interactions up to order D**10 R*4 are linear sums of functions in this class and quadratic polynomials in Eisenstein series and odd Riemann zeta values. Integration over the complex structure results in coefficients of the low energy expansion that are rational numbers multiplying monomials in odd Riemann zeta values.

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arXiv2014

Matching the $D^6 {\cal R}^4$ 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.

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arXiv2014

Zhang-Kawazumi Invariants and Superstring Amplitudes

Invariance of Type IIB superstring theory under SL(2,Z) or S-duality implies dependence on the complex coupling T through real and complex modular forms in T. Their structure may be understood explicitly in an expansion of superstring corrections to Einstein's equations of gravity, in powers of derivatives D and curvature R. The perturbative loop expansion in the string coupling for the 4-string amplitude governs corrections of the form D^{2p} R^4 for all p. We show that, at two-loop order, the D^6 R^4 term is proportional to the integral of a modular invariant introduced by Zhang and Kawazumi in number theory and related to the Faltings delta-invariant studied for genus-two by Bost. The structure of two-loop superstring amplitudes for p>3 leads to higher invariants, which generalize Zhang--Kawazumi invariants at genus two. An explicit formula is derived for the unique higher invariant associated with order D^8 R^4. In an attempt to compare the prediction for the D^6 R^4 correction from superstring perturbation theory with the one produced by S-duality and supersymmetry of Type IIB, various reformulations of the invariant are given. This comparison with string theory leads to a predicted value for the integral of the Zhang-Kawazumi invariant over the moduli space of genus-two surfaces.

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arXiv2013

String theory dualities and supergravity divergences

We demonstrate how duality invariance of the low energy expansion of the four-graviton amplitude in type II string theory determines the precise coefficients of multiloop logarithmic ultraviolet divergences of maximal supergravity in various dimensions. This is illustrated by the explicit moduli-dependence of terms of the form D^{2k}R^4, with k<= 3, in the effective action. Furthermore, we show that in the supergravity limit the perturbative contributions are swamped by an accumulation of non-perturbative effects of zero-action instantons.

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arXiv2013

Small representations, string instantons, and Fourier modes of Eisenstein series (with an appendix by D. Ciubotaru and P. Trapa)

This paper concerns some novel features of maximal parabolic Eisenstein series at certain special values of their analytic parameter s. These series arise as coefficients in the R4 and D4R4 interactions in the low energy expansion of scattering amplitudes in maximally supersymmetric string theory reduced to D=10-d dimensions on a torus T^d, d<8. For each d these amplitudes are automorphic functions on the rank d+1 symmetry group E_d+1. Of particular significance is the orbit content of the Fourier modes of these series when expanded in three different parabolic subgroups, corresponding to certain limits of string theory. This is of interest in the classification of a variety of instantons that correspond to minimal or next-to-minimal BPS orbits. In the limit of decompactification from D to D+1 dimensions many such instantons are related to charged 1/2-BPS or 1/4-BPS black holes with euclidean world-lines wrapped around the large dimension. In a different limit the instantons give nonperturbative corrections to string perturbation theory, while in a third limit they describe nonperturbative contributions in eleven-dimensional supergravity. A proof is given that these three distinct Fourier expansions have certain vanishing coefficients that are expected from string theory. In particular, the Eisenstein series for these special values of s have markedly fewer Fourier coefficients than typical ones. The corresponding mathematics involves showing that the wavefront sets of the Eisenstein series are supported on only certain coadjoint nilpotent orbits - just the minimal and trivial orbits in the 1/2-BPS case, and just the next-to-minimal, minimal and trivial orbits in the 1/4-BPS case. Thus as a byproduct we demonstrate that the next-to-minimal representations occur automorphically for E6, E7, and E8, and hence the first two nontrivial low energy coefficients are exotic theta-functions.

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arXiv2013

Multiparticle one-loop amplitudes and S-duality in closed superstring theory

Explicit expressions for one-loop five supergraviton scattering amplitudes in both type II superstring theories are determined by making use of the pure spinor formalism. The type IIB amplitude can be expressed in terms of a doubling of ten-dimensional super Yang--Mills tree amplitude, while the type IIA amplitude has additional pieces that cannot be expressed in that manner. We evaluate the coefficients of terms in the analytic part of the low energy expansion of the amplitude, which correspond to a series of terms in an effective action of the schematic form D^{2k}R^5 for 0\le k \le 5 (where R is the Riemann curvature). Comparison with earlier analyses of the tree amplitudes and of the four-particle one-loop amplitude leads to an interesting extension of the action of SL(2,Z) S-duality on the moduli-dependent coefficients in the type IIB theory. We also investigate closed-string five-particle amplitudes that violate conservation of the U(1) R-symmetry charge -- processes that are forbidden in supergravity. The coefficients of their low energy expansion are shown to agree with S-duality systematics. A less detailed analysis is also given of the six-point function, resulting in the vanishing of the analytic parts of the R^6 and D^4 R^6 interactions in the ten-dimensional effective action, but not in lower dimensions.

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arXiv2011

Eisenstein series for higher-rank groups and string theory amplitudes

Scattering amplitudes of superstring theory are strongly constrained by the requirement that they be invariant under dualities generated by discrete subgroups, E_n(Z), of simply-laced Lie groups in the E_n series (n<= 8). In particular, expanding the four-supergraviton amplitude at low energy gives a series of higher derivative corrections to Einstein's theory, with coefficients that are automorphic functions with a rich dependence on the moduli. Boundary conditions supplied by string and supergravity perturbation theory, together with a chain of relations between successive groups in the E_n series, constrain the constant terms of these coefficients in three distinct parabolic subgroups. Using this information we are able to determine the expressions for the first two higher derivative interactions (which are BPS-protected) in terms of specific Eisenstein series. Further, we determine key features of the coefficient of the third term in the low energy expansion of the four-supergraviton amplitude (which is also BPS-protected) in the E_8 case. This is an automorphic function that satisfies an inhomogeneous Laplace equation and has constant terms in certain parabolic subgroups that contain information about all the preceding terms.

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arXiv2010

5 loops in 24/5 dimensions

A first quantised approach to loop amplitudes based on the pure spinor particle is applied to the systematics of four-particle amplitudes in maximally supersymmetric field theories. Counting of fermionic zero modes allows the identification of momentum factors multiplying R**4 in the case of supergravity (and F**4 in the Yang--Mills case) thereby making manifest their ultraviolet properties as a function of dimension, D. For L=2,3,4 loops the leading supergravity divergence is in D=4+6/L dimensions and proportional to d*2L R**4, in line with earlier field theory calculations. However, at five loops there is a radical change in the systematics, suggesting the presence of a contribution with an explicit L=5 logarithmic ultraviolet divergence when D=24/5 that is proportional to d*8 R**4. We further argue that d*8 R**4 should receive contributions from all loops, which would imply that N=8 supergravity (with D=4) is not protected by supersymmetry from a seven-loop divergence.

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arXiv2010

Automorphic properties of low energy string amplitudes in various dimensions

This paper explores the moduli-dependent coefficients of higher derivative interactions that appear in the low-energy expansion of the four-graviton amplitude of maximally supersymmetric string theory compactified on a d-torus. These automorphic functions are determined for terms up to order D^6R^4 and various values of d by imposing a variety of consistency conditions. They satisfy Laplace eigenvalue equations with or without source terms, whose solutions are given in terms of Eisenstein series, or more general automorphic functions, for certain parabolic subgroups of the relevant U-duality groups. The ultraviolet divergences of the corresponding supergravity field theory limits are encoded in various logarithms, although the string theory expressions are finite. This analysis includes intriguing representations of SL(d) and SO(d,d) Eisenstein series in terms of toroidally compactified one and two-loop string and supergravity amplitudes.

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arXiv2009

Non-renormalization conditions for four-gluon scattering in supersymmetric string and field theory

The constraints imposed by maximal supersymmetry on multi-loop contributions to the scattering of four open superstrings in the U(N) theory are examined by use of the pure spinor formalism. The double-trace term k^2 t_8(tr F^2)^2 (where k represents an external momentum and F the Yang--Mills field strength) only receives contributions from L<=2 (where L is the loop number) while the single-trace term k^2 t_8(tr F^4) receives contributions from all L. We verified these statements up to L=5, but arguments based on supersymmetry suggest they extend to all L. This explains why the single-trace contributions to low energy maximally supersymmetric Yang--Mills field theory are more divergent in the ultraviolet than the double-trace contributions. We also comment further on the constraints on closed string amplitudes and their implications for ultraviolet divergences in N=8 supergravity.

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arXiv2008

Modular properties of two-loop maximal supergravity and connections with string theory

The low-momentum expansion of the two-loop four-graviton scattering amplitude in eleven-dimensional supergravity compactified on a circle and a two-torus is considered up to terms of order S^6R^4 (where S is a Mandelstam invariant and R is the linearized Weyl curvature). In the case of the toroidal compactification the coefficient of each term in the low energy expansion is generically a sum of a number of SL(2,Z)-invariant functions of the complex structure of the torus. Each such function satisfies a separate Poisson equation on moduli space with particular source terms that are bilinear in coefficients of lower order terms, consistent with qualitative arguments based on supersymmetry. Comparison is made with the low-energy expansion of type II string theories in ten and nine dimensions. Although the detailed behaviour of the string amplitude is not generally expected to be reproduced by supergravity perturbation theory to all orders, for the terms considered here we find agreement with direct results from string perturbation theory. These results point to a fascinating pattern of interrelated Poisson equations for the IIB coefficients at higher orders in the momentum expansion which may have a significance beyond the particular methods by which they were motivated.

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arXiv2008

Low energy expansion of the four-particle genus-one amplitude in type II superstring theory

A diagrammatic expansion of coefficients in the low-momentum expansion of the genus-one four-particle amplitude in type II superstring theory is developed. This is applied to determine coefficients up to order s^6R^4 (where s is a Mandelstam invariant and R^4 the linearized super-curvature), and partial results are obtained beyond that order. This involves integrating powers of the scalar propagator on a toroidal world-sheet, as well as integrating over the modulus of the torus. At any given order in s the coefficients of these terms are given by rational numbers multiplying multiple zeta values (or Euler--Zagier sums) that, up to the order studied here, reduce to products of Riemann zeta values. We are careful to disentangle the analytic pieces from logarithmic threshold terms, which involves a discussion of the conditions imposed by unitarity. We further consider the compactification of the amplitude on a circle of radius r, which results in a plethora of terms that are power-behaved in r. These coefficients provide boundary `data' that must be matched by any non-perturbative expression for the low-energy expansion of the four-graviton amplitude. The paper includes an appendix by Don Zagier.

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arXiv2007

Decoupling Supergravity from the Superstring

We consider the conditions necessary for obtaining perturbative maximal supergravity in d dimensions as a decoupling limit of type II superstring theory compactified on a (10 -- d)-torus. For dimensions d = 2 and d = 3 it is possible to define a limit in which the only finite-mass states are the 256 massless states of maximal supergravity. However, in dimensions d > 3 there are infinite towers of additional massless and finite-mass states. These correspond to Kaluza--Klein charges, wound strings, Kaluza--Klein monopoles or branes wrapping around cycles of the toroidal extra dimensions. We conclude that perturbative supergravity cannot be decoupled from string theory in dimensions d > 3. In particular, we conjecture that pure N = 8 supergravity in four dimensions is in the Swampland.

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arXiv2007

Non-renormalisation Conditions in Type II String Theory and Maximal Supergravity

This paper considers general features of the derivative expansion of Feynman diagram contributions to the four-graviton scattering amplitude in eleven-dimensional supergravity compactified on a two-torus. These are translated into statements about interactions of the form D^2k R^4 in type II superstring theories, assuming the standard M-theory/string theory duality relationships, which provide powerful constraints on the effective interactions. In the ten-dimensional IIA limit we find that there can be no perturbative contributions beyond k string loops (for k>0). Furthermore, the genus h=k contributions are determined exactly by the one-loop eleven-dimensional supergravity amplitude for all values of k. A plausible interpretation of these observations is that the sum of h-loop Feynman diagrams of maximally extended supergravity is less divergent than might be expected and could be ultraviolet finite in dimensions d < 4 + 6/h -- the same bound as for N=4 Yang--Mills.

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