arXiv subjects

Matthias Wilhelm

Publications and source records attributed to Matthias Wilhelm.

At least 37 records · Page 2Linked to original sources

arXiv2022

Solving the Hagedorn temperature of AdS5/CFT4 via the Quantum Spectral Curve: Chemical potentials and deformations

We describe how to calculate the Hagedorn temperature of $\mathcal{N}=4$ SYM theory and type IIB superstring theory on $AdS_5\times S^5$ via the Quantum Spectral Curve (QSC) -- providing further details on our previous letters 1706.03074 and 1803.04416. We solve the QSC equations perturbatively at weak 't Hooft coupling $λ$ up to seven-loop order and numerically at finite coupling, finding that the perturbative results can be expressed in terms of single-valued harmonic polylogarithms. Moreover, we generalize the QSC to describe the Hagedorn temperature in the presence of chemical potentials. Finally, we show that the Hagedorn temperature in certain deformations of $\mathcal{N}=4$ SYM theory (real-$β$ and $γ_i$ deformation) agrees with the one in $\mathcal{N}=4$ SYM theory at any value of $λ$.

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arXiv2022

The elliptic double box and symbology beyond polylogarithms

We study the elliptic double-box integral, which contributes to generic massless QFTs and is the only contribution to a particular 10-point scattering amplitude in N=4 SYM theory. Based on a Feynman parametrization, we express this integral in terms of elliptic polylogarithms. We then study its symbol, finding a rich structure and remarkable similarity with the non-elliptic case. In particular, the first entry of the symbol is expressible in terms of logarithms of dual-conformal cross-ratios, and elliptic letters only occur in the last two entries. Moreover, the symbol makes manifest a differential equation relating the double-box integral to a 6D hexagon integral, suggesting that it can be bootstrapped based on the latter integral alone.

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arXiv2021

The Hagedorn temperature of AdS5/CFT4 at finite coupling via the Quantum Spectral Curve

Building on the recently established connection between the Hagedorn temperature and integrability [Phys.Rev.Lett. 120 (2018) no.7, 071605], we show how the Quantum Spectral Curve formalism can be used to calculate the Hagedorn temperature of AdS5/CFT4 for any value of the 't Hooft coupling. We solve this finite system of finite-difference equations perturbatively at weak coupling and numerically at finite coupling. We confirm previous results at weak coupling and obtain the previously unknown three-loop Hagedorn temperature. Our finite-coupling results interpolate between weak and strong coupling and allow us to extract the first perturbative order at strong coupling. Our results indicate that the Hagedorn temperature for large 't Hooft coupling approaches that of type IIB string theory in ten-dimensional Minkowski space.

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arXiv2021

An Operator Product Expansion for Form Factors

We propose an operator product expansion for planar form factors of local operators in $\mathcal{N}=4$ SYM theory. This expansion is based on the dual conformal symmetry of these objects or, equivalently, the conformal symmetry of their dual description in terms of periodic Wilson loops. A form factor is decomposed into a sequence of known pentagon transitions and a new universal object that we call the "form factor transition". This transition is subject to a set of non-trivial bootstrap constraints, which we expect to be sufficient to fully determine it. We evaluate the form factor transition for MHV form factors of the chiral half of the stress tensor supermultiplet at leading order in perturbation theory and use it to produce OPE predictions at any loop order. We match the one-loop and two-loop predictions with data available in the literature.

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arXiv2021

An Operator Product Expansion for Form Factors II. Born level

Form factors in planar N=4 Super-Yang-Mills theory admit a type of non-perturbative operator product expansion (OPE), as we have recently shown in arXiv:2009.11297. This expansion is based on a decomposition of the dual periodic Wilson loop into elementary building blocks: the known pentagon transitions and a new object that we call form factor transition, which encodes the information about the local operator. In this paper, we compute the two-particle form factor transitions for the chiral part of the stress-tensor supermultiplet at Born level; they yield the leading contribution to the OPE. To achieve this, we explicitly construct the Gubser-Klebanov-Polyakov two-particle singlet states. The resulting transitions are then used to test the OPE against known perturbative data and to make higher-loop predictions.

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arXiv2021

A Three-Point Form Factor Through Five Loops

We bootstrap the three-point form factor of the chiral part of the stress-tensor supermultiplet in planar $\mathcal{N}=4$ super-Yang-Mills theory, obtaining new results at three, four, and five loops. Our construction employs known conditions on the first, second, and final entries of the symbol, combined with new multiple-final-entry conditions, ``extended-Steinmann-like'' conditions, and near-collinear data from the recently-developed form factor operator product expansion. Our results are expected to give the maximally transcendental parts of the $gg\to Hg$ and $H\to ggg$ amplitudes in the heavy-top limit of QCD. At two loops, the extended-Steinmann-like space of functions we describe contains all transcendental functions required for four-point amplitudes with one massive and three massless external legs, and all massless internal lines, including processes such as $gg\to Hg$ and $γ^*\to q\bar{q}g$. We expect the extended-Steinmann-like space to contain these amplitudes at higher loops as well, although not to arbitrarily high loop order. We present evidence that the planar $\mathcal{N}=4$ three-point form factor can be placed in an even smaller space of functions, with no independent $ζ$ values at weights two and three.

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arXiv2021

Direct Integration for Multi-leg Amplitudes: Tips, Tricks, and When They Fail

Direct hyperlogarithmic integration offers a strong alternative to differential equation methods for Feynman integration, particularly for multi-particle diagrams. We review a variety of results by the authors in which this method, employed with some care, can compute diagrams of up to eight particles and four loops. We also highlight situations in which this method fails due to an algebraic obstruction. In a large number of cases the obstruction can be associated with a Calabi-Yau manifold.

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arXiv2020

From Hagedorn to Lee-Yang: Partition functions of $\mathcal{N}=4$ SYM theory at finite $N$

We study the thermodynamics of the maximally supersymmetric Yang-Mills theory with gauge group U(N) on R x S^3, dual to type IIB superstring theory on AdS_5 x S^5. While both theories are well-known to exhibit Hagedorn behavior at infinite N, we find evidence that this is replaced by Lee-Yang behavior at large but finite N: the zeros of the partition function condense into two arcs in the complex temperature plane that pinch the real axis at the temperature of the confinement-deconfinement transition. Concretely, we demonstrate this for the free theory via exact calculations of the (unrefined and refined) partition functions at N<=7 for the su(2) sector containing two complex scalars, as well as at N<=5 for the su(2|3) sector containing 3 complex scalars and 2 fermions. In order to obtain these explicit results, we use a Molien-Weyl formula for arbitrary field content, utilizing the equivalence of the partition function with what is known to mathematicians as the Poincare series of trace algebras of generic matrices. Via this Molien-Weyl formula, we also generate exact results for larger sectors.

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arXiv2020

Embedding Feynman Integral (Calabi-Yau) Geometries in Weighted Projective Space

It has recently been demonstrated that Feynman integrals relevant to a wide range of perturbative quantum field theories involve periods of Calabi-Yaus of arbitrarily large dimension. While the number of Calabi-Yau manifolds of dimension three or higher is considerable (if not infinite), those relevant to most known examples come from a very simple class: degree-$2k$ hypersurfaces in $k$-dimensional weighted projective space $\mathbb{WP}^{1,\ldots,1,k}$. In this work, we describe some of the basic properties of these spaces and identify additional examples of Feynman integrals that give rise to hypersurfaces of this type. Details of these examples at three and four loops are included as ancillary files to this work.

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arXiv2019

A Quantum Framework for AdS/dCFT through Fuzzy Spherical Harmonics on $S^4$

We consider a non-supersymmetric domain-wall version of $\mathcal{N} = 4$ SYM theory where five out of the six scalar fields have non-zero classical values on one side of a wall of codimension one. The classical fields have commutators which constitute an irreducible representation of the Lie algebra $\mathfrak{so}(5)$ leading to a highly non-trivial mixing between color and flavor components of the quantum fields. Making use of fuzzy spherical harmonics on $S^4$, we explicitly solve the mixing problem and derive not only the spectrum of excitations at the quantum level but also the propagators of the original fields needed for perturbative quantum computations. As an application, we derive the one-loop one-point function of a chiral primary and find complete agreement with a supergravity prediction of the same quantity in a double-scaling limit which involves a limit of large instanton number in the dual D3-D7 probe-brane setup.

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arXiv2019

Rooting Out Letters: Octagonal Symbol Alphabets and Algebraic Number Theory

It is widely expected that NMHV amplitudes in planar, maximally supersymmetric Yang-Mills theory require symbol letters that are not rationally expressible in terms of momentum-twistor (or cluster) variables starting at two loops for eight particles. Recent advances in loop integration technology have made this an `experimentally testable' hypothesis: compute the amplitude at some kinematic point, and see if algebraic symbol letters arise. We demonstrate the feasibility of such a test by directly integrating the most difficult of the two-loop topologies required. This integral, together with its rotated image, suffices to determine the simplest NMHV component amplitude: the unique component finite at this order. Although each of these integrals involve algebraic symbol alphabets, the combination contributing to this amplitude is---surprisingly---rational. We describe the steps involved in this analysis, which requires several novel tricks of loop integration and also a considerable degree of algebraic number theory. We find dramatic and unusual simplifications, in which the two symbols initially expressed as almost ten million terms in over two thousand letters combine in a form that can be written in five thousand terms and twenty-five letters.

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arXiv2019

Insights into the Mind of a Trojan Designer: The Challenge to Integrate a Trojan into the Bitstream

The threat of inserting hardware Trojans during the design, production, or in-field poses a danger for integrated circuits in real-world applications. A particular critical case of hardware Trojans is the malicious manipulation of third-party FPGA configurations. In addition to attack vectors during the design process, FPGAs can be infiltrated in a non-invasive manner after shipment through alterations of the bitstream. First, we present an improved methodology for bitstream file format reversing. Second, we introduce a novel idea for Trojan insertion.

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arXiv2019

A (Bounded) Bestiary of Feynman Integral Calabi-Yau Geometries

We define the rigidity of a Feynman integral to be the smallest dimension over which it is non-polylogarithmic. We argue that massless Feynman integrals in four dimensions have a rigidity bounded by 2(L-1) at L loops, and we show that this bound may be saturated for integrals that we call marginal: those with (L+1)D/2 propagators in (even) D dimensions. We show that marginal Feynman integrals in D dimensions generically involve Calabi-Yau geometries, and we give examples of finite four-dimensional Feynman integrals in massless $ϕ^4$ theory that saturate our predicted bound in rigidity at all loop orders.

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arXiv2019

A Quantum Check of Non-Supersymmetric AdS/dCFT

Via a challenging field-theory computation, we confirm a supergravity prediction for the non-supersymmetric D3-D7 probe-brane system with probe geometry AdS_4 x S^2 x S^2, stabilized by fluxes. Supergravity predicts, in a certain double-scaling limit, the value of the one-point functions of chiral primaries of the dual defect version of N=4 SYM theory, where the fluxes translate into SO(3) x SO(3)-symmetric, Lie-algebra-valued vacuum expectation values for all six scalar fields. Using a generalization of the technique based on fuzzy spherical harmonics developed for the related D3-D5 probe-brane system, we diagonalize the resulting mass matrix of the field theory. Subsequently, we calculate the planar one-loop correction to the vacuum expectation values of the scalars in dimensional reduction and find that it is UV finite and non-vanishing. We then proceed to calculating the one-loop correction to the planar one-point function of any single-trace scalar operator and explicitly evaluate this correction for a 1/2-BPS operator of length L at two leading orders in the double-scaling limit, finding exact agreement with the supergravity prediction.

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arXiv2018

Rationalizing Loop Integration

We show that direct Feynman-parametric loop integration is possible for a large class of planar multi-loop integrals. Much of this follows from the existence of manifestly dual-conformal Feynman-parametric representations of planar loop integrals, and the fact that many of the algebraic roots associated with (e.g. Landau) leading singularities are automatically rationalized in momentum-twistor space---facilitating direct integration via partial fractioning. We describe how momentum twistors may be chosen non-redundantly to parameterize particular integrals, and how strategic choices of coordinates can be used to expose kinematic limits of interest. We illustrate the power of these ideas with many concrete cases studied through four loops and involving as many as eight particles. Detailed examples are included as ancillary files to this work's submission to the arXiv.

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arXiv2018

Traintracks Through Calabi-Yaus: Amplitudes Beyond Elliptic Polylogarithms

We describe a family of finite, four-dimensional, $L$-loop Feynman integrals that involve weight-$(L+1)$ hyperlogarithms integrated over $(L-1)$-dimensional elliptically fibered varieties we conjecture to be Calabi-Yau. At three loops, we identify the relevant K3 explicitly; and we provide strong evidence that the four-loop integral involves a Calabi-Yau threefold. These integrals are necessary for the representation of amplitudes in many theories---from massless $φ^4$ theory to integrable theories including maximally supersymmetric Yang-Mills theory in the planar limit---a fact we demonstrate.

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arXiv2018

The Hagedorn temperature of AdS5/CFT4 via integrability

We establish a framework for calculating the Hagedorn temperature of AdS5/CFT4 via integrability. Concretely, we derive the thermodynamic Bethe ansatz equations that yield the Hagedorn temperature of planar N=4 super Yang-Mills theory at any value of the 't Hooft coupling. We solve these equations perturbatively at weak coupling via the associated Y-system, confirming the known results at tree-level and one-loop order as well as deriving the previously unknown two-loop Hagedorn temperature. Finally, we comment on solving the equations at finite coupling.

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arXiv2018

The Elliptic Double-Box Integral: Massless Amplitudes Beyond Polylogarithms

We derive an analytic representation of the ten-particle, two-loop double-box integral as an elliptic integral over weight-three polylogarithms. To obtain this form, we first derive a four-fold, rational (Feynman-)parametric representation for the integral, expressed directly in terms of dual-conformally invariant cross-ratios; from this, the desired form is easily obtained. The essential features of this integral are illustrated by means of a simplified toy model, and we attach the relevant expressions for both integrals in ancillary files. We propose a normalization for such integrals that renders all of their polylogarithmic degenerations pure, and we discuss the need for a new 'symbology' of iterated elliptic/polylogarithmic integrals in order to bring them to a more canonical form.

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