arXiv subjects

Eric D'Hoker

Publications and source records attributed to Eric D'Hoker.

At least 37 records · Page 2Linked to original sources

arXiv2022

Two-loop superstring five-point amplitudes III, construction via the RNS formulation: even spin structures

The contribution from even spin structures to the genus-two amplitude for five massless external NS states in Type II and Heterotic superstrings is evaluated from first principles in the RNS formulation. Using chiral splitting with the help of loop momenta this problem reduces to the evaluation of the corresponding chiral amplitude, which is carried out using the same techniques that were used for the genus-two amplitude with four external NS states. The results agree with the parity-even NS components of a construction using chiral splitting and pure spinors given in earlier companion papers arXiv:2006.05270 and arXiv:2008.08687.

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arXiv2022

Snowmass White Paper: String Perturbation Theory

The purpose of this White Paper is to review recent progress towards elucidating and evaluating string amplitudes, relating them to quantum field theory amplitudes, applying their predictions to string dualities, exploring their connection with gravitational physics, and deepening our understanding of their mathematical structure. We also present a selection of targets for future research.

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arXiv2021

Integrating three-loop modular graph functions and transcendentality of string amplitudes

Modular graph functions (MGFs) are $\mathrm{SL}(2,\mathbb{Z})$-invariant functions on the Poincaré upper half-plane associated with Feynman graphs of a conformal scalar field on a torus. The low-energy expansion of genus-one superstring amplitudes involves suitably regularized integrals of MGFs over the fundamental domain for $\mathrm{SL}(2,\mathbb{Z})$. In earlier work, these integrals were evaluated for all MGFs up to two loops and for higher loops up to weight six. These results led to the conjectured uniform transcendentality of the genus-one four-graviton amplitude in Type II superstring theory. In this paper, we explicitly evaluate the integrals of several infinite families of three-loop MGFs and investigate their transcendental structure. Up to weight seven, the structure of the integral of each individual MGF is consistent with the uniform transcendentality of string amplitudes. Starting at weight eight, the transcendental weights obtained for the integrals of individual MGFs are no longer consistent with the uniform transcendentality of string amplitudes. However, in all the cases we examine, the violations of uniform transcendentality take on a special form given by the integrals of triple products of non-holomorphic Eisenstein series. If Type II superstring amplitudes do exhibit uniform transcendentality, then the special combinations of MGFs which enter the amplitudes must be such that these integrals of triple products of Eisenstein series precisely cancel one another. Whether this indeed is the case poses a novel challenge to the conjectured uniform transcendentality of genus-one string amplitudes.

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arXiv2021

Identities among higher genus modular graph tensors

Higher genus modular graph tensors map Feynman graphs to functions on the Torelli space of genus-$h$ compact Riemann surfaces which transform as tensors under the modular group $Sp(2h , \mathbb Z)$, thereby generalizing a construction of Kawazumi. An infinite family of algebraic identities between one-loop and tree-level modular graph tensors are proven for arbitrary genus and arbitrary tensorial rank. We also derive a family of identities that apply to modular graph tensors of higher loop order.

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arXiv2021

Exploring transcendentality in superstring amplitudes

It is well known that the low energy expansion of tree-level superstring scattering amplitudes satisfies a suitably defined version of maximum 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 ${\cal M}_R$ surrounding the cusp and its complement ${\cal M}_L$, and an exact expression is obtained for the contribution to the amplitude from ${\cal M}_R$. The low-energy expansion of the ${\cal M}_R$ contribution is proven to be free of irreducible multiple zeta-values to all orders. The contribution to the amplitude from ${\cal M}_L$ is computed in terms of modular graph functions up to order $D^{12} {\cal R}^4$ in the low-energy expansion, and general arguments are used beyond this order to conjecture the transcendentality properties of the ${\cal M}_L$ contributions. Maximal transcendentality of the full amplitude holds provided we assign a non-zero weight to certain harmonic sums, an assumption which is familiar from transcendentality assignments in quantum field theory amplitudes.

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arXiv2021

Revisiting the Multi-Monopole Point of $SU(N)$ $\mathcal{N} = 2$ Gauge Theory in Four Dimensions

Motivated by applications to soft supersymmetry breaking, we revisit the expansion of the Seiberg-Witten solution around the multi-monopole point on the Coulomb branch of pure $SU(N)$ $\mathcal{N}=2$ gauge theory in four dimensions. At this point $N-1$ mutually local magnetic monopoles become massless simultaneously, and in a suitable duality frame the gauge couplings logarithmically run to zero. We explicitly calculate the leading threshold corrections to this logarithmic running from the Seiberg-Witten solution by adapting a method previously introduced by D'Hoker and Phong. We compare our computation to existing results in the literature; this includes results specific to $SU(2)$ and $SU(3)$ gauge theories, the large-$N$ results of Douglas and Shenker, as well as results obtained by appealing to integrable systems or topological strings. We find broad agreement, while also clarifying some lingering inconsistencies. Finally, we explicitly extend the results of Douglas and Shenker to finite $N$, finding exact agreement with our first calculation.

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arXiv2021

Elliptic modular graph forms I: Identities and generating series

Elliptic modular graph functions and forms (eMGFs) are defined for arbitrary graphs as natural generalizations of modular graph functions and forms obtained by including the character of an Abelian group in their Kronecker--Eisenstein series. The simplest examples of eMGFs are given by the Green function for a massless scalar field on the torus and the Zagier single-valued elliptic polylogarithms. More complicated eMGFs are produced by the non-separating degeneration of a higher genus surface to a genus one surface with punctures. eMGFs may equivalently be represented by multiple integrals over the torus of combinations of coefficients of the Kronecker--Eisenstein series, and may be assembled into generating series. These relations are exploited to derive holomorphic subgraph reduction formulas, as well as algebraic and differential identities between eMGFs and their generating series.

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arXiv2021

Asymptotics of the $D^8 R^4$ genus-two string invariant

We continue our investigation of the modular graph functions and string invariants that arise at genus-two as coefficients of low energy effective interactions in Type II superstring theory. In previous work, the non-separating degeneration of a genus-two modular graph function of weight $w$ was shown to be given by a Laurent polynomial in the degeneration parameter $t$ of degree $(w,w)$. The coefficients of this polynomial generalize genus-one modular graph functions, up to terms which are exponentially suppressed in $t$ as $t \to \infty$. In this paper, we evaluate this expansion explicitly for the modular graph functions associated with the $D^8 {\cal R}^4$ effective interaction for which the Laurent polynomial has degree $(2,2)$. We also prove that the separating degeneration is given by a polynomial in the degeneration parameter $\ln (|v|)$ up to contributions which are power-behaved in $v$ as $v \to 0$. We further extract the complete, or tropical, degeneration and compare it with the independent calculation of the integrand of the sum of Feynman diagrams that contributes to two-loop type II supergravity expanded to the same order in the low energy expansion. We find that the tropical limit of the string theory integrand reproduces the supergravity integrand as its leading term, but also includes sub-leading terms proportional to odd zeta values that are absent in supergravity and can be ascribed to higher-derivative stringy interactions.

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arXiv2021

Modular graph functions and odd cuspidal functions -- Fourier and Poincaré series

Modular graph functions are $SL(2,{\mathbb Z})$-invariant functions associated with Feynman graphs of a two-dimensional conformal field theory on a torus of modulus $τ$. For one-loop graphs they reduce to real analytic Eisenstein series. We obtain the Fourier series, including the constant and non-constant Fourier modes, of all two-loop modular graph functions, as well as their Poincaré series with respect to $Γ_\infty \backslash PSL(2,{\mathbb Z})$. The Fourier and Poincaré series provide the tools to compute the Petersson inner product of two-loop modular graph functions using Rankin-Selberg-Zagier methods. Modular graph functions which are odd under $τ\to - \bar τ$ are cuspidal functions, with exponential decay near the cusp, and exist starting at two loops. Holomorphic subgraph reduction and the sieve algorithm, developed in earlier work, are used to give a lower bound on the dimension of the space $\mathfrak{A}_w$ of odd two-loop modular graph functions of weight $w$. For $w \leq 11$ the bound is saturated and we exhibit a basis for $\mathfrak{A}_w$.

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arXiv2020

An Alternative Method for Extracting the von Neumann Entropy from Renyi Entropies

An alternative method is presented for extracting the von Neumann entropy $-\operatorname{Tr} (ρ\ln ρ)$ from $\operatorname{Tr} (ρ^n)$ for integer $n$ in a quantum system with density matrix $ρ$. Instead of relying on direct analytic continuation in $n$, the method uses a generating function $-\operatorname{Tr} \{ ρ\ln [(1-z ρ) / (1-z)] \}$ of an auxiliary complex variable $z$. The generating function has a Taylor series that is absolutely convergent within $|z|<1$, and may be analytically continued in $z$ to $z = -\infty$ where it gives the von Neumann entropy. As an example, we use the method to calculate analytically the CFT entanglement entropy of two intervals in the small cross ratio limit, reproducing a result that Calabrese et al. obtained by direct analytic continuation in $n$. Further examples are provided by numerical calculations of the entanglement entropy of two intervals for general cross ratios, and of one interval at finite temperature and finite interval length.

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arXiv2020

Two-loop superstring five-point amplitudes I: Construction via chiral splitting and pure spinors

The full two-loop amplitudes for five massless states in Type~II and Heterotic superstrings are constructed in terms of convergent integrals over the genus-two moduli space of compact Riemann surfaces and integrals of Green functions and Abelian differentials on the surface. The construction combines elements from the BRST cohomology of the pure spinor formulation and from chiral splitting with the help of loop momenta and homology invariance. The $α' \to 0$ limit of the resulting superstring amplitude is shown to be in perfect agreement with the previously known amplitude computed in Type~II supergravity. Investigations of the $α'$ expansion of the Type~II amplitude and comparisons with predictions from S-duality are relegated to a first companion paper. A construction from first principles in the RNS formulation of the genus-two amplitude with five external NS states is relegated to a second companion paper.

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arXiv2020

Absence of irreducible multiple zeta-values in melon modular graph functions

The expansion of a modular graph function on a torus of modulus $τ$ near the cusp is given by a Laurent polynomial in $y= π\Im (τ)$ with coefficients that are rational multiples of single-valued multiple zeta-values, apart from the leading term whose coefficient is rational and exponentially suppressed terms. We prove that the coefficients of the non-leading terms in the Laurent polynomial of the modular graph function $D_N(τ)$ associated with a melon graph is free of irreducible multiple zeta-values and can be written as a polynomial in odd zeta-values with rational coefficients for arbitrary $N \geq 0$. The proof proceeds by expressing a generating function for $D_N(τ)$ in terms of an integral over the Virasoro-Shapiro closed-string tree amplitude.

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arXiv2019

Gravitational Wilson lines in AdS$_{\bf 3}$

The construction of gravitational Wilson lines in the Chern-Simons formulation of $AdS_3$ gravity in terms of composite operators in the dual boundary conformal field theory is reviewed. New evidence is presented that the Wilson line, dimensionally regularized and suitably renormalized, behaves as a bi-local operator of two conformal primaries whose dimension is predicted by SL(2,R) current algebra.

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arXiv2019

Integral of two-loop modular graph functions

The integral of an arbitrary two-loop modular graph function over the fundamental domain for $SL(2,Z)$ in the upper half plane is evaluated using recent results on the Poincaré series for these functions.

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arXiv2019

Renormalization of gravitational Wilson lines

We continue the study of the Wilson line representation of conformal blocks in two-dimensional conformal field theory; these have an alternative interpretation as gravitational Wilson lines in the context of the AdS$_3$/CFT$_2$ correspondence. The gravitational Wilson line involves a path-ordered exponential of the stress tensor, and its expectation value can be computed perturbatively in an expansion in inverse powers of the central charge $c$. The short-distance singularities which occur in the associated stress tensor correlators require systematic regularization and renormalization prescriptions, whose consistency with conformal Ward identities presents a subtle problem. The regularization used here combines dimensional regularization and analytic continuation. Representation theoretic arguments, based on SL(2,R) current algebra, predict an exact result for the Wilson line anomalous dimension and, by building on previous work, we verify that the perturbative calculations using our regularization and renormalization prescriptions reproduce the exact result to order $1/c^3$ included. We also discuss a related, but somewhat simpler, Wilson line in Wess-Zumino-Witten models that yields current algebra conformal blocks, and we emphasize the distinction between Wilson lines constructed out of non-holomorphic and purely holomorphic currents.

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arXiv2019

Global half-BPS $AdS_2\times S^6$ solutions in Type IIB

We investigate half-BPS Type IIB supergravity solutions with spacetime geometry $AdS_2\times S^6$ warped over a Riemann surface $Σ$. The general local solution was obtained in earlier work in terms of two holomorphic functions $\mathcal A_\pm$ on $Σ$. In the first part of this paper we seek global solutions corresponding to the near-horizon behavior of $(p,q)$-string junctions. We identify the type of singularity in $\mathcal A_\pm$ needed at the boundary of $Σ$ to match the solutions locally onto the classic $(p,q)$-string solution. We construct solutions with multiple $(p,q)$-strings, but the existence of geodesically complete solutions remains unsettled. In a second part we construct multi-parameter families of non-compact globally regular and geodesically complete solutions with asymptotic regions and an $AdS_2$ throat which caps off smoothly.

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arXiv2018

Fourier series of modular graph functions

Modular graph functions associate to a graph an $SL(2,Z)$-invariant function on the upper half plane. We obtain the Fourier series of modular graph functions of arbitrary weight $w$ and two-loop order. The motivation for this work is to develop a deeper understanding of the origin of the algebraic identities between modular graph functions which have been discovered recently, and of the relation between the existence of these identities and the occurrence of cusp forms. We show that the constant Fourier mode, as a function of the modulus $τ$, consists of a Laurent polynomial in $y = π\, {\rm Im} (τ)$ of degree $(w,1-w)$, plus a contribution which decays exponentially as $y \to \infty$. The Laurent polynomial is a linear combination with rational coefficients of the top term $y^w$, and lower order terms $ζ(2k+1) y^{w-2k-1}$ for $1\leq k \leq w-1$, as well as terms $ζ(2w-2\ell-3) ζ(2\ell+1)y^{2-w}$ for $1 \leq \ell \leq w-3$. The exponential contribution is a linear combination of exponentials of $y$ and incomplete $Γ$-functions whose coefficients are Laurent polynomials in $y$ with rational coefficients.

math.NT↗

arXiv2018

Higher genus modular graph functions, string invariants, and their exact asymptotics

The concept and the construction of modular graph functions are generalized from genus-one to higher genus surfaces. The integrand of the four-graviton superstring amplitude at genus-two provides a generating function for a special class of such functions. A general method is developed for analyzing the behavior of modular graph functions under non-separating degenerations in terms of a natural real parameter $t$. For arbitrary genus, the Arakelov Green function and the Kawazumi-Zhang invariant degenerate to a Laurent polynomial in $t$ of degree $(1,1)$ in the limit $t\to\infty$. For genus two, each coefficient of the low energy expansion of the string amplitude degenerates to a Laurent polynomial of degree $(w,w)$ in $t$, where $w+2$ is the degree of homogeneity in the kinematic invariants. These results are exact to all orders in $t$, up to exponentially suppressed corrections. The non-separating degeneration of a general class of modular graph functions at arbitrary genus is sketched and similarly results in a Laurent polynomial in $t$ of bounded degree. The coefficients in the Laurent polynomial are generalized modular graph functions for a punctured Riemann surface of lower genus.

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