arXiv subjects

Eric D'Hoker

Publications and source records attributed to Eric D'Hoker.

At least 55 records · Page 3Linked to original sources

arXiv2018

Warped $AdS_6\times S^2$ in Type IIB supergravity II: Global solutions and five-brane webs

Motivated by the construction of holographic duals to five-dimensional superconformal quantum field theories, we obtain global solutions to Type IIB supergravity invariant under the superalgebra $F(4)$ on a space-time of the form $AdS_6 \times S^2$ warped over a two-dimensional Riemann surface $Σ$. In earlier work, the general local solutions were expressed in terms of two locally holomorphic functions $\mathcal A_\pm$ on $Σ$ and global solutions were sketched when $Σ$ is a disk. In the present paper, the physical regularity conditions on the supergravity fields required for global solutions are implemented on $\mathcal A_\pm$ for arbitrary $Σ$. Global solutions exist only when $Σ$ has a non-empty boundary $\partial Σ$. The differentials $\partial \mathcal A_\pm$ are allowed to have poles only on $\partial Σ$ and each pole corresponds to a semi-infinite $(p,q)$ five-brane. The construction for the disk is carried out in detail and the conditions for the existence of global solutions are articulated for surfaces with more than one boundary and higher genus.

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arXiv2018

Holographic duals for five-dimensional superconformal quantum field theories

We construct global solutions to Type IIB supergravity with 16 residual supersymmetries whose space-time is $AdS_6 \times S^2$ warped over a Riemann surface. Families of solutions are labeled by an arbitrary number $L\geq 3$ of asymptotic regions, in each of which the supergravity fields match those of a $(p,q)$ five-brane, and may therefore be viewed as near-horizon limits of fully localized intersections of five-branes in Type IIB string theory. These solutions provide compelling candidates for holographic duals to a large class of five-dimensional superconformal quantum field theories which arise as non-trivial UV fixed points of perturbatively non-renormalizable Yang-Mills theories, thereby making them more directly accessible to quantitative analysis.

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arXiv2018

Warped $AdS_6\times S^2$ in Type IIB supergravity I: Local solutions

We investigate the existence of solutions with 16 residual supersymmetries to Type IIB supergravity on a space-time of the form $AdS_6 \times S^2$ warped over a two-dimensional Riemann surface $Σ$. The $SO(2,5) \times SO(3)$ isometry extends to invariance under the exceptional Lie superalgebra $F(4)$. In the present paper, we construct the general Ansatz compatible with these symmetries, derive the corresponding reduced BPS equations, and obtain their complete local solution in terms of two locally holomorphic functions ${\cal A}_\pm$ on $Σ$, subject to certain positivity and regularity conditions. Globally, $({\cal A}_+, {\cal A}_-)$ are allowed to be multiple-valued on $Σ$ and be holomorphic sections of a holomorphic bundle over $Σ$ with structure group contained in $SU(1,1) \times { C}$. Globally regular solutions are expected to provide the near-horizon geometry of $(p,q)$ 5-brane and 7-brane webs which are holographic duals to five-dimensional conformal field theories. A preliminary analysis of the positivity and regularity conditions will be presented here, leaving the construction of globally regular solutions to a subsequent paper.

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arXiv2017

$AdS_2 \times S^6$ versus $AdS_6 \times S^2$ in Type IIB supergravity

We obtain the complete local solutions with 16 supersymmetries to Type IIB supergravity on a space-time of the form $AdS_{2}\times S^{6}$ warped over a Riemann surface $Σ$ in terms of two locally holmorphic functions on $Σ$. We construct the general Ansatz for the bosonic supergravity fields and supersymmetry generators compatible with the $SO(2,1)\oplus SO(7)$ isometry algebra of space-time, which extends to the corresponding real form of the exceptional Lie superalgebra $F(4)$. We reduce the BPS equations to this Ansatz, obtain their general local solutions, and show that these local solutions solve the full Type IIB supergravity field equations and Bianchi identities. We contrast the $AdS_{2}\times S^{6}$ solution with the closely related $AdS_6\times S^2$ case and present our results for both in parallel. Finally, we present a preliminary analysis of positivity and regularity conditions for $AdS_{2}\times S^{6}$, but postpone the construction of globally regular solutions to a subsequent paper.

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arXiv2017

Hierarchy of Modular Graph Identities

The low energy expansion of Type II superstring amplitudes at genus one is organized in terms of modular graph functions associated with Feynman graphs of a conformal scalar field on the torus. In earlier work, surprising identities between two-loop graphs at all weights, and between higher-loop graphs of weights four and five were constructed. In the present paper, these results are generalized in two complementary directions. First, all identities at weight six and all dihedral identities at weight seven are obtained and proven. Whenever the Laurent polynomial at the cusp is available, the form of these identities confirms the pattern by which the vanishing of the Laurent polynomial governs the full modular identity. Second, the family of modular graph functions is extended to include all graphs with derivative couplings and worldsheet fermions. These extended families of modular graph functions are shown to obey a hierarchy of inhomogeneous Laplace eigenvalue equations. The eigenvalues for the extended family of dihedral modular graph functions are calculated analytically for the simplest infinite sub-families and obtained by Maple for successively more complicated sub-families. The spectrum is shown to consist solely of eigenvalues $s(s-1)$ for positive integers $s$ bounded by the weight, with multiplicities which exhibit rich representation-theoretic patterns.

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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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arXiv2017

Warped $AdS_6\times S^2$ in Type IIB supergravity III: Global solutions with seven-branes

We extend our previous construction of global solutions to Type IIB supergravity that are invariant under the superalgebra $F(4)$ and are realized on a spacetime of the form $AdS_6 \times S^2$ warped over a Riemann surface $Σ$ by allowing the supergravity fields to have non-trivial $SL(2,{\mathbb R})$ monodromy at isolated punctures on $Σ$. We obtain explicit solutions for the case where $Σ$ is a disc, and the monodromy generators are parabolic elements of $SL(2,{\mathbb R})$ physically corresponding to the monodromy allowed in Type IIB string theory. On the boundary of $Σ$ the solutions exhibit singularities at isolated points which correspond to semi-infinite five-branes, as is familiar from the global solutions without monodromy. In the interior of $Σ$, the solutions are everywhere regular, except at the punctures where $SL(2,{\mathbb R})$ monodromy resides and which physically correspond to the locations of $[p,q]$ seven-branes. The solutions have a compelling physical interpretation corresponding to fully localized five-brane intersections with additional seven-branes, and provide candidate holographic duals to the five-dimensional superconformal field theories realized on such intersections.

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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

Emergent Super-Virasoro on Magnetic Branes

The low energy limit of the stress tensor, gauge current, and supercurrent two-point correlators are calculated in the background of the supersymmetric magnetic brane solution to gauged five-dimensional supergravity constructed by Almuhairi and Polchinski. The resulting correlators provide evidence for the emergence of an N=2 super-Virasoro algebra of right-movers, in addition to a bosonic Virasoro algebra and a $U(1) \oplus U(1)$-current algebra of left-movers (or the parity transform of left- and right-movers depending on the sign of the magnetic field), in the holographically dual strongly interacting two-dimensional effective field theory of the lowest Landau level.

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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

Higher Order Deformations of Complex Structures

Deformations of complex structures by finite Beltrami differentials are considered on general Riemann surfaces. Exact formulas to any fixed order are derived for the corresponding deformations of the period matrix, Green's functions, and correlation functions in conformal field theories with vanishing total central charge. The stress tensor is shown to give a simple representation of these deformations valid to all orders. Such deformation formulas naturally enter into the evaluation of superstring amplitudes at two-loop order with Ramond punctures, and at higher loop order, in the supergravity formulation of the RNS superstring.

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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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arXiv2015

Exact M-Theory Solutions, Integrable Systems, and Superalgebras

In this paper, an overview is presented of the recent construction of fully back-reacted half-BPS solutions in 11-dimensional supergravity which correspond to near-horizon geometries of M2 branes ending on, or intersecting with, M5 and M5$'$ branes along a self-dual string. These solutions have space-time manifold ${\rm AdS}_3 \times S^3 \times S^3$ warped over a Riemann surface $Σ$, and are invariant under the exceptional Lie superalgebra $D(2,1;γ) \oplus D(2,1;γ)$, where $γ$ is a real continuous parameter and $|γ|$ is governed by the ratio of the number of M5 and M5$'$ branes. The construction proceeds by mapping the reduced BPS equations onto an integrable field theory on $Σ$ which is of the Liouville sine-Gordon type. Families of regular solutions are distinguished by the sign of $γ$, and include a two-parameter Janus solution for $γ>0$, and self-dual strings on M5 as well as asymptotically ${\rm AdS}_4/{\mathbb Z}_2$ solutions for $γ<0$.

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arXiv2015

The Super Period Matrix with Ramond Punctures in the supergravity formulation

In a very recent preprint, Witten showed how to construct a $g|r \, \times \, g|r$ super period matrix for super Riemann surfaces of genus $g$ with $2r$ Ramond punctures, which is symmetric in the ${\bf Z}_2$ graded sense. He also showed how it can be applied to analyze supersymmetry breaking in string compactifications which are supersymmetric at tree-level. Witten's construction is in the purely holomorphic formulation of super Riemann surfaces. In this paper, a construction is given in the formulation of two-dimensional supergravity. The variations of the super period matrix with respect to supermoduli deformations are also given, as well as an explicit illustration of how the super period matrix with two Ramond punctures would emerge from a degeneration of the super period matrix without punctures in higher genus.

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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

Holographic Entropy and Calabi's Diastasis

The entanglement entropy for interfaces and junctions of two-dimensional CFTs is evaluated on holographically dual half-BPS solutions to six-dimensional Type 4b supergravity with m anti-symmetric tensor supermultiplets. It is shown that the moduli space for an N-junction solution projects to N points in the Kaehler manifold SO(2,m)/( SO(2) x SO(m)). For N=2 the interface entropy is expressed in terms of the central charge and Calabi's diastasis function on SO(2,m)/(SO(2) x SO(m)), thereby lending support from holography to a proposal of Bachas, Brunner, Douglas, and Rastelli. For N=3, the entanglement entropy for a 3-junction decomposes into a sum of diastasis functions between pairs, weighed by combinations of the three central charges, provided the flux charges are all parallel to one another or, more generally, provided the space of flux charges is orthogonal to the space of unattracted scalars. Under similar assumptions for N>3, the entanglement entropy for the N-junction solves a variational problem whose data consist of the N central charges, and the diastasis function evaluated between pairs of N asymptotic AdS_3 x S^3 regions.

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arXiv2014

Topics in Two-Loop Superstring Perturbation Theory

In this contribution to the Proceedings of the Conference on Analysis, Complex Geometry, and Mathematical Physics, an expository overview of superstring perturbation theory to two loop order is presented to an audience of mathematicians and physicists. Recent results on perturbative supersymmetry breaking effects in Heterotic string theory compactified on Z_2 \times Z_2 Calabi-Yau orbifolds, and the calculation of the two-loop vacuum energy in these theories are discussed in detail, and the appearance of a new modular identity with respect to Sp(4,Z)/Z_4 is reviewed.

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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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