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Bruno Le Floch

Publications and source records attributed to Bruno Le Floch.

At least 37 records · Page 2Linked to original sources
arXiv2022

Cyclic spacetimes through singularity scattering maps. The laws of quiescent bounces

For spacetimes containing quiescent singularity hypersurfaces we propose a general notion of junction conditions based on a prescribed singularity scattering map, as we call it, and we introduce the notion of a cyclic spacetime (also called a multiverse) consisting of spacetime domains bounded by spacelike or timelike singularity hypersurfaces, across which our scattering map is applied. A local existence theory is established here while, in a companion paper, we construct plane-symmetric cyclic spacetimes. We study the singularity data space consisting of the suitably rescaled metric, extrinsic curvature, and matter fields which can be prescribed on each side of the singularity, and for the class of so-called quiescent singularities we establish restrictions that a singularity scattering map must satisfy. We obtain a full characterization of all scattering maps that are covariant and ultralocal, in a sense we define and, in particular, we distinguish between, on the one hand, three laws of bouncing cosmology of universal nature and, on the other hand, model-dependent junction conditions. The theory proposed in this paper applies to spacelike and timelike hypersurfaces and without symmetry restriction. It encompasses bouncing-cosmology scenarios, both in string theory and in loop quantum cosmology, and puts strong restrictions on their possible explicit realizations.

gr-qc↗
arXiv2022

A slow review of the AGT correspondence

Starting with a gentle approach to the AGT correspondence from its 6d origin, these notes provide a wide (albeit shallow) survey of the literature on numerous extensions of the correspondence up to early 2020. This is an extended writeup of the lectures given at the Winter School YRISW 2020, to appear in a special issue of JPhysA. Class S is a wide class of 4d $N=2$ supersymmetric gauge theories (ranging from super-QCD to non-Lagrangian theories) obtained by twisted compactification of 6d $N=(2,0)$ superconformal theories on a Riemann surface. This 6d construction yields the Coulomb branch and Seiberg-Witten geometry of class S theories, geometrizes S-duality, and leads to the AGT correspondence, which states that many observables of class S theories are equal to 2d CFT correlators. For instance, the four-sphere partition function of a 4d $N=2$ SU(2) superconformal quiver theory is equal to a Liouville CFT correlator of primary operators. Extensions of the AGT correspondence abound: asymptotically-free gauge theories and Argyres-Douglas theories correspond to irregular CFT operators, quivers with higher-rank gauge groups and non-Lagrangian tinkertoys such as $T_N$ correspond to Toda CFT correlators, and nonlocal operators (Wilson-'t Hooft loops, surface operators, domain walls) correspond to Verlinde networks, degenerate primary operators, braiding and fusion kernels, and Riemann surfaces with boundaries.

hep-th↗
arXiv2021

Cyclic spacetimes through singularity scattering maps. Plane-symmetric gravitational collisions

We study the plane-symmetric collision of two gravitational waves and describe the global spacetime geometry generated by this collision. To this end, we formulate the characteristic initial value problem for the Einstein equations, when Goursat data describing the incoming waves are prescribed on two null hypersurfaces. We then construct a global solution representing a cyclic spacetime based on junction conditions associated with a prescribed singularity scattering map (a notion recently introduced by the authors). From a mathematical analysis standpoint, this amounts to a detailed analysis of the Goursat and Fuchsian initial value problems associated with singular hyperbolic equations, when junction conditions at interfaces are prescribed. In our construction, we introduce a partition into monotonicity diamonds (that is, suitable spacetime domains) and we construct the solution by concatenating domains across interfaces of timelike, null, or spacelike type.

gr-qc↗
arXiv2021

Universal scattering laws for quiescent bouncing cosmology

Cosmological bounces occur in many gravity theories. We define singularity scattering maps relating large scale geometries before and after the bounce (assuming no BKL oscillations) and encoding microscopic details of the theory. By classifying all suitably local maps we uncover three universal laws: scaling of Kasner exponents, canonical transformation of matter, directional metric scaling. These are indeed obeyed by Bianchi I bounces in string theory, loop quantum cosmology and modified matter models; our classification then determines how inhomogeneities and anisotropies traverse bounces, and precisely extracts model-dependent degrees of freedom.

gr-qc↗
arXiv2020

On the global evolution of self-gravitating matter. Nonlinear interactions in Gowdy symmetry

We are interested in the evolution of a compressible fluid under its self-generated gravitational field. Assuming here Gowdy symmetry, we investigate the algebraic structure of the Euler equations satisfied by the mass density and velocity field. We exhibit several interaction functionals that provide us with a uniform control on weak solutions in suitable Sobolev norms or in bounded variation. These functionals allow us to study the local regularity and nonlinear stability properties of weakly regular fluid flows governed by the Euler-Gowdy system. In particular for the Gowdy equations, we prove that a spurious matter field arises under weak convergence, and we establish the nonlinear stability of weak solutions.

gr-qc↗
arXiv2020

M2-brane surface operators and gauge theory dualities in Toda

We give a microscopic two dimensional ${\cal N}=(2,2)$ gauge theory description of arbitrary M2-branes ending on $N_f$ M5-branes wrapping a punctured Riemann surface. These realize surface operators in four dimensional ${\cal N}=2$ field theories. We show that the expectation value of these surface operators on the sphere is captured by a Toda CFT correlation function in the presence of an additional degenerate vertex operator labelled by a representation ${\cal R}$ of $SU(N_f)$, which also labels M2-branes ending on M5-branes. We prove that symmetries of Toda CFT correlators provide a geometric realization of dualities between two dimensional gauge theories, including ${\cal N}=(2,2)$ analogues of Seiberg and Kutasov--Schwimmer dualities. As a bonus, we find new explicit conformal blocks, braiding matrices, and fusion rules in Toda CFT.

hep-th↗
arXiv2020

Intersecting Surface Defects and Two-Dimensional CFT

We initiate the study of intersecting surface operators/defects in four-dimensional quantum field theories (QFTs). We characterize these defects by coupled 4d/2d/0d theories constructed by coupling the degrees of freedom localized at a point and on intersecting surfaces in spacetime to each other and to the four-dimensional QFT. We construct supersymmetric intersecting surface defects preserving just two supercharges in N = 2 gauge theories. These defects are amenable to exact analysis by localization of the partition function of the underlying 4d/2d/0d QFT. We identify the 4d/2d/0d QFTs that describe intersecting surface operators in N = 2 gauge theories realized by intersecting M2-branes ending on N M5-branes wrapping a Riemann surface. We conjecture and provide evidence for an explicit equivalence between the squashed four-sphere partition function of these intersecting defects and correlation functions in Liouville/Toda CFT with the insertion of arbitrary degenerate vertex operators, which are labeled by representations of SU(N).

hep-th↗
arXiv2020

S-duality wall of SQCD from Toda braiding

Exact field theory dualities can be implemented by duality domain walls such that passing any operator through the interface maps it to the dual operator. This paper describes the S-duality wall of four-dimensional ${\cal N}=2$ SU(N) SQCD with 2N hypermultiplets in terms of fields on the defect, namely three-dimensional ${\cal N}=2$ SQCD with gauge group U(N-1) and 2N flavours, with a monopole superpotential. The theory is self-dual under a duality found by Benini, Benvenuti and Pasquetti, in the same way that T[SU(N)] (the S-duality wall of ${\cal N}=4$ super Yang-Mills) is self-mirror. The domain-wall theory can also be realized as a limit of a USp(2N-2) gauge theory; it reduces to known results for N=2. The theory is found through the AGT correspondence by determining the braiding kernel of two semi-degenerate vertex operators in Toda CFT.

hep-th↗
arXiv2019

Compensated compactness and corrector stress tensor for the Einstein equations in T2 symmetry

We consider the Einstein equations in T2 symmetry, either for vacuum spacetimes or coupled to the Euler equations for a compressible fluid, and we introduce the notion of T2 areal flows on T3 with finite total energy. By uncovering a hidden structure of the Einstein equations, we establish a compensated compactness framework and solve the global evolution problem for vacuum spacetimes as well as for self-gravitating compressible fluids. We study the stability and instability of such flows and prove that, when the initial data are well-prepared, any family of T2 areal flows is sequentially compact in a natural topology. In order to handle general initial data we propose a relaxed notion of T2 areal flows endowed with a corrector stress tensor (as we call it) which is a bounded measure generated by geometric oscillations and concentrations propagating at the speed of light. This generalizes a result for vacuum spacetimes in: Le Floch B. and LeFloch P.G., Arch. Rational Mech. Anal. 233 (2019), 45-86. In addition, we determine the global geometry of the corresponding future Cauchy developments and we prove that the area of the T2 orbits generically approaches infinity in the future-expanding regime. In the future-contracting regime, the volume of the T3 spacelike slices approaches zero and, for generic initial data, the area of the orbits of symmetry approaches zero in Gowdy symmetric matter spacetimes and in T2 vacuum spacetimes.

gr-qc↗
arXiv2019

Marginal Deformations of 3d N=4 Linear Quiver Theories

We study superconformal deformations of the $T_ρ^{\hatρ}[SU(N)]$ theories of Gaiotto-Hanany-Witten, paying special attention to mixed-branch operators with both electrically- and magnetically-charged fields. We explain why all marginal ${\cal N}=2$ operators of an ${\cal N}=4$ CFT$_3$ can be extracted unambiguously from the superconformal index. Computing the index at the appropriate order we show that the mixed moduli in $T_ρ^{\hatρ}[SU(N)]$ theories are double-string operators transforming in the (Adjoint, Adjoint) representation of the electric and magnetic flavour groups, up to some overcounting for quivers with abelian gauge nodes. We comment on the holographic interpretation of the results, arguing in particular that gauged supergravities can capture the entire moduli space if, in addition to the (classical) parameters of the background solution, one takes also into account the (quantization) moduli of boundary conditions.

hep-th↗
arXiv2019

KdV charges in $T\bar{T}$ theories and new models with super-Hagedorn behavior

Two-dimensional CFTs and integrable models have an infinite set of conserved KdV higher spin currents. These currents can be argued to remain conserved under the $T\bar{T}$ deformation and its generalizations. We determine the flow equations the KdV charges obey under the $T\bar{T}$ deformation: they behave as probes "riding the Burgers flow" of the energy eigenvalues. We also study a Lorentz-breaking $T_{s+1}\bar{T}$ deformation built from a KdV current and the stress tensor, and find a super-Hagedorn growth of the density of states.

hep-th↗
arXiv2019

Action of Weyl group on zero weight space

For any simple complex Lie group we classify irreducible finite-dimensional representations $ρ$ for which the longest element $w_0$ of the Weyl group acts nontrivially on the zero weight space. Among irreducible representations that have zero among their weights, $w_0$ acts by $\pm$Id if and only if the highest weight of $ρ$ is a multiple of a fundamental weight, with a coefficient less than a bound that depends on the group and on the fundamental weight.

math.RT↗
arXiv2019

The range of geometrical frustration in lattice spin models

The concept of geometrical frustration in condensed matter physics refers to the fact that a system has a locally preferred structure with an energy density lower than the infinite ground state. This notion is however often used in a qualitative sense only. In this article, we discuss a quantitative definition of geometrical frustration in the context of lattice models of binary spins. To this aim, we introduce the framework of local energy landscapes, within which frustration can be quantified as the discrepancy between the energy of locally preferred structures and the ground state. Our definition is scale-dependent and involves an optimization over a gauge class of equivalent local energy landscapes, related to one another by local energy displacements. This ensures that frustration depends only on the physical Hamiltonian and its range, and not on unphysical choices in how it is written. Our framework shows that a number of popular frustrated models, including the antiferromagnetic Ising model on a triangular lattice, only have finite-range frustration: geometrical incompatibilities are local and can be eliminated by an exact coarse-graining of the local energies.

cond-mat.stat-mech↗
arXiv2019

Solving a family of $T\bar{T}$-like theories

We deform two-dimensional quantum field theories by antisymmetric combinations of their conserved currents that generalize Smirnov and Zamolodchikov's $T\bar{T}$ deformation. We obtain that energy levels on a circle obey a transport equation analogous to the Burgers equation found in the $T\bar{T}$ case. This equation relates charges at any value of the deformation parameter to charges in the presence of a (generalized) Wilson line. We determine the initial data and solve the transport equations for antisymmetric combinations of flavor symmetry currents and the stress tensor starting from conformal field theories. Among the theories we solve is a conformal field theory deformed by $J\bar{T}$ and $T\bar{T}$ simultaneously. We check our answer against results from AdS/CFT.

hep-th↗
arXiv2018

AGT/Z$_2$

We relate Liouville/Toda CFT correlators on Riemann surfaces with boundaries and cross-cap states to supersymmetric observables in four-dimensional N=2 gauge theories. Our construction naturally involves four-dimensional theories with fields defined on different Z$_2$ quotients of the sphere (hemisphere and projective space) but nevertheless interacting with each other. The six-dimensional origin is a Z$_2$ quotient of the setup giving rise to the usual AGT correspondence. To test the correspondence, we work out the RP$^4$ partition function of four-dimensional N=2 theories by combining a 3d lens space and a 4d hemisphere partition functions. The same technique reproduces known RP$^2$ partition functions in a form that lets us easily check two-dimensional Seiberg-like dualities on this nonorientable space. As a bonus we work out boundary and cross-cap wavefunctions in Toda CFT.

hep-th↗
arXiv2016

Exact Results in D=2 Supersymmetric Gauge Theories

We compute exactly the partition function of two dimensional N=(2,2) gauge theories on S^2 and show that it admits two dual descriptions: either as an integral over the Coulomb branch or as a sum over vortex and anti-vortex excitations on the Higgs branches of the theory. We further demonstrate that correlation functions in two dimensional Liouville/Toda CFT compute the S^2 partition function for a class of N=(2,2) gauge theories, thereby uncovering novel modular properties in two dimensional gauge theories. Some of these gauge theories flow in the infrared to Calabi-Yau sigma models - such as the conifold - and the topology changing flop transition is realized as crossing symmetry in Liouville/Toda CFT. Evidence for Seiberg duality in two dimensions is exhibited by demonstrating that the partition function of conjectured Seiberg dual pairs are the same.

hep-th↗
arXiv2016

Supersymmetric localization in two dimensions

This is an introductory review to localization techniques in supersymmetric two-dimensional gauge theories. In particular we describe how to construct Lagrangians of N=(2,2) theories on curved spaces, and how to compute their partition functions and certain correlators on the sphere, the hemisphere and other curved backgrounds. We also describe how to evaluate the partition function of N=(0,2) theories on the torus, known as the elliptic genus. Finally we summarize some of the applications, in particular to probe mirror symmetry and other non-perturbative dualities.

hep-th↗