arXiv subjects

Bruno Le Floch

Publications and source records attributed to Bruno Le Floch.

At least 19 recordsLinked to original sources

arXiv2026

Finite-energy spacetimes with torus symmetry. Cauchy stability of Einstein-Euler and Einstein-Navier-Stokes areal flows

We establish finite-energy Cauchy stability for vacuum and matter spacetimes with $T^2$ symmetry on $T^3$, under general constitutive equations satisfying hyperbolicity and mild asymptotic conditions at vacuum and large mass-energy density. First, using the area of the symmetry orbits as a time function, we introduce the JKL formulation of Einstein areal flows}, a first-order evolution-constraint system coupling a wave-map structure for the geometry to hyperbolic matter balance laws and weighted transport equations for the twists and the momentum tangent to the symmetry orbits. Second, we also define two classes of hyperbolic Einstein-Navier-Stokes models, constructed from (as we call them) a Navier-Stokes potential and a relaxation rate map. The proposed particle-production model has divergence-free matter stress and non-negative particle-number production, while the proposed dissipated-energy model conserves particle number and dissipates fluid mass-energy, while an auxiliary stress restores the total stress-energy conservation required by the contracted Bianchi identity. For regular finite-energy (Einstein, Euler, Navier-Stokes) flows, we establish maximum and invariant-domain principles, geometric and energy monotonicity formulas, spacelike and timelike energy estimates, and total-variation estimates. Third, together, these estimates yield finite-energy Cauchy stability in the future expanding regime and, in the contracting regime, until the spacelike volume collapses. They are independent of the viscosity (or relaxation) mechanisms and cover vacuum, unbounded mass-energy density, and arbitrary finite rapidity. They also extend to weakly regular Einstein-Euler areal flows satisfying a reference mathematical entropy inequality.

math.AP↗

arXiv2026

Unleash $Q$! Cohomology, Localization, and Interpolation in Parisi-Sourlas Supersymmetry

Parisi-Sourlas supersymmetric models are known to undergo dimensional reduction; that is, their physics is captured by models in two fewer dimensions. In this work, we revisit dimensional reduction, providing new arguments and reformulating existing proofs in terms of the cohomology of a supercharge $Q$. We obtain three main results. First, we show that the recently developed picture of dimensional reduction via decoupling of operators admits a natural explanation in terms of $Q$-exactness. Second, we provide a new proof of dimensional reduction using the supersymmetric localization argument. Third, we revisit Cardy's "interpolation" proof, which is reminiscent of localization but does not rely on saddle-point methods, and show that it can be understood as a consequence of deforming the action by a $Q$-exact term. Finally, we show that while existing nonperturbative proofs of dimensional reduction apply only to scalar Lagrangians, our formulation of Cardy's argument extends to any theory with Parisi-Sourlas supersymmetry.

hep-th↗

arXiv2026

Optimal localization for the Einstein constraints

We establish the existence of asymptotically Euclidean solutions, or initial data sets, to Einstein's vacuum constraints exhibiting both gravitational shielding and arbitrarily low decay. We resolve a conjecture of Carlotto and Schoen on gluing two solutions across an asymptotically conical domain: in the interpolation region we establish optimal estimates at and beyond harmonic decay, together with ADM invariant estimates. Starting from a seed data set that asymptotically solves the constraints, we project it to an exact solution, whose difference from the seed inherits the natural optimal radial decay rate. The seed-to-solution projection operator composes the linearized constraint operator with its formal adjoint. Its harmonic kernel at infinity generates energy-momentum modulators, which define relative invariants equal to the differences of the usual ADM invariants whenever those exist. We introduce harmonic, radial, and shell stability conditions on the angular gluing function; they ensure the positivity and radial monotonicity of the functionals governing optimal decay. A follow-up paper proves these conditions from weighted Poincaré, Korn, and Hardy inequalities, leading to verifiable conditions on the localization function. As particular cases, our theory applies to solutions without localization and for solutions in the small aperture limit. Our analysis of the linearized scalar curvature operator relies crucially on a new fourth-order curvature functional, and also applies to gluing scalar-flat metrics.

gr-qc↗

arXiv2026

Harmonic, radial, and shell stability of the weighted Einstein constraints on the sphere at infinity

We consider fourth-order and second-order partial differential operators localized on domains of the sphere in arbitrary dimension. These operators arise as weighted compositions of the linearized Einstein constraint operators and their adjoints, and played a key role in our resolution of the optimal localization problem in general relativity, also referred to as the gravitational shielding problem. To control the asymptotic behavior of solutions to Einstein's constraints in our companion paper (preprint arXiv:2312.17706), we introduced the notions of harmonic, radial, and shell stability. Harmonic stability controls the borderline harmonic modes, radial stability governs the radial evolution of spherical averages, and shell stability controls the coupled radial-angular evolution of solutions. In the present paper, we establish that these stability properties follow from weighted Poincaré, Korn, and Hardy inequalities. Furthermore, in arbitrary dimension, we investigate the behavior of the associated geometric constants, and conclude that the stability conditions hold for a broad class of localization functions; the theory applies to arbitrarily small localization domains, corresponding to gluing cones with arbitrarily small aperture. At the opposite extreme, our conditions also hold on the entire sphere, corresponding to the absence of localization. This completes, for gluing cones of arbitrarily small aperture in every dimension, the program initiated by A. Carlotto and R. Schoen on gravitational shielding and the construction of solutions enjoying super-harmonic decay estimates. In our proofs, we introduce Hamiltonian and momentum functionals, which we call shell functionals, and show that they enjoy monotonicity and semi-coercivity properties; their structure also suggests possible analogies with functionals arising in other curvature-related geometric problems.

math.AP↗

arXiv2026

Convergence of Nekrasov instanton sum for unitary quivers

The convergence radius of Nekrasov partition functions (as a function of instanton counting parameters) is shown to be positive for 4d $\mathcal{N}=2$ quiver gauge theories with unitary gauge groups in an open dense subset of parameters. For $U(N)$ SQCD this is established if the ratio of equivariant parameters $b^2=ε_1/ε_2$ belongs to $\mathbb{C}\setminus[0,+\infty)$ and Coulomb parameters or masses are away from a lattice of hyperplanes. For general quivers it is only established for $b^2\in\mathbb{C}\setminus\mathbb{R}$. When gauge multiplets are asymptotically free, the radius is infinite, whereas in the (mass-deformed) conformal case the radius admits a positive lower bound that only depends on $b^2$. The proof relies on the expression of the partition function as a sum over tuples of partitions, and a proof of absolute convergence based on combinatorial inequalities on products of (co)hook lengths. Through the AGT correspondence this implies that large classes of Virasoro and W-algebra conformal blocks on the sphere or torus have positive convergence radius, for generic dimensions and complex central charges.

hep-th↗

arXiv2026

Convergence of Nekrasov instanton sum with adjoint matter

The Nekrasov instanton partition function of the 4d $\mathcal{N}=2^*$ $U(N)$ gauge theory (a mass deformation of 4d $\mathcal{N}=4$ super-Yang-Mills theory), which is a generating series of equivariant integrals over instanton moduli spaces, is given by a sum over colored partitions weighted by a counting parameter $\mathfrak{q}$. This note proves convergence of the series in the unit disk $|\mathfrak{q}|<1$ for generic parameters. Specifically, the absolute convergence radius of this sum is determined, assuming that mass and Coulomb branch parameters avoid some lattice. If the ratio $b^2=ε_1/ε_2$ of equivariant parameters is in $\mathbb{C}\setminus[0,+\infty)$, the radius is $1$, as expected. If $b^2$ is non-negative, three cases arise: the radius is finite if $b^2$ has finite exponential type (a generalization of Brjuno numbers), namely there exists $C>0$ such that $|b^2-p/q|>\exp(-Cq)$ for all integers $p,q\neq 0$; the series diverges if $b^2$ is super-exponentially well approximable by rationals; and if $b^2$ is rational some terms are singular. The AGT correspondence translates these results to convergence of torus one-point conformal blocks of the Virasoro and $W_N$ algebras with non-real $b$, within the unit disk. For the Virasoro algebra this corresponds to a central charge in $\mathbb{C}\setminus[25,+\infty)$.

hep-th↗

arXiv2026

Stability and instability of torus-symmetric Einstein spacetimes with square-integrable connection

We study the global evolution problem for the Einstein equations under T2 symmetry on T3, allowing vacuum, scalar-field, and compressible-fluid matter models, governed by a general equation of state including isothermal and polytropic fluids. Under this symmetry, we obtain the first non-perturbative, global existence and stability theory with connection coefficients being merely square-integrable, which allows both impulsive gravitational waves and shock waves. In areal gauge, we introduce new fluid and geometric variables and reformulate the Einstein-Euler system as a first-order system of nonlinear balance laws with constraints and an entropy structure. The resulting formulation exhibits hyperbolicity, null forms, entropy currents, div-curl structure, maximum principles, and spacetime estimates. This leads to a notion of tame Einstein-Euler flow for which the essential geometric and fluid variables are square-integrable (finite energy), and the secondary variables are absolutely continuous (or, more generally, of bounded variation). In this non-perturbative and weak regularity setting, the equations remain meaningful even when the Weyl curvature concentrates into Dirac masses along timelike hypersurfaces, and the Ricci curvature remains only integrable. Our main results are a global existence theorem for areal foliations, a nonlinear stability theorem for well-prepared initial data, and a nonlinear instability theorem for geometrically oscillatory data, the latter producing measure corrections to the stress energy tensor. In the future-contracting regime, the areal foliation reaches a geometric singularity where the volume of T3 spatial slices degenerates to zero. The areal function reaches zero generically in the non-vacuum Gowdy-symmetric and vacuum torus-symmetric cases. In the future-expanding regime, the areal foliation is complete.

gr-qc↗

arXiv2026

Scattering laws for interfaces in self-gravitating matter flows

We consider the evolution of self-gravitating matter fields that may undergo phase transitions, and we connect ideas from phase transition dynamics with concepts from bouncing cosmology. Our framework introduces scattering maps prescribed on two classes of hypersurfaces: a gravitational singularity hypersurface and a fluid-discontinuity hypersurface. By analyzing the causal structures induced by the light cone and the acoustic cone, we formulate a local evolution problem for the Einstein-Euler system in the presence of such interfaces. We explain how suitable scattering relations must supplement the field equations in order to ensure uniqueness and thus yield a complete macroscopic description of the evolution. This viewpoint builds on a theory developed in collaboration with G. Veneziano for quiescent (velocity-dominated) singularities in solutions of the Einstein equations coupled to a scalar field, where the passage across the singular hypersurface is encoded by a singularity scattering map. The guiding question is to identify junction prescriptions that are compatible with the Einstein and Euler equations, in particular with the propagation of constraints. The outcome is a rigid set of universal relations, together with a family of model-dependent parameters. Under physically motivated requirements (general covariance, causality, constraint compatibility, and ultra-locality), we aim to classify admissible scattering relations arising from microscopic physics and characterizing, at the macroscopic level, the dynamics of a fluid coupled to Einstein gravity.

gr-qc↗

arXiv2026

BPS Dendroscopy on Local $\mathbb{P}^1\times \mathbb{P}^1$

BPS states in type II string compactified on a Calabi-Yau threefold can typically be decomposed as moduli-dependent bound states of absolutely stable constituents, with a hierarchical structure labelled by attractor flow trees. This decomposition is best understood from the scattering diagram, an arrangement of real codimension-one loci (or rays) in the space of stability conditions where BPS states of given electromagnetic charge and fixed phase of the central charge exist. The consistency of the diagram when rays intersect determines all BPS indices in terms of the `attractor indices' carried by the initial rays. In this work we study the scattering diagram for a non-compact toric CY threefold known as local $\mathbb{F}_0$, namely the total space of the canonical bundle over $\mathbb{P}^1\times \mathbb{P}^1$. We first construct the scattering diagram for the quiver, valid near the orbifold point, and for the large volume slice, valid when both $\mathbb{P}^1$'s have large (and nearly equal) area. We then combine the insights gained from these simple limits to construct the scattering diagram along the physical slice of $Π$-stability conditions, which carries an action of a $\mathbb{Z}^4$ extension of the modular group $Γ_0(4)$. We sketch a proof of the Split Attractor Flow Tree Conjecture in this example, albeit for a restricted range of the central charge phase. Most arguments are similar to our early study of local $\mathbb{P}^2$ [arXiv:2210.10712], but complicated by the occurence of an extra mass parameter and ramification points on the $Π$-stability slice.

hep-th↗

arXiv2026

Implication semilattice of 990 quasigroup equational laws

In his quest to disprove a claim by Peirce that all lattices are distributive, Ernst Schröder considered 135 years ago a list of 990 equational laws on quasigroups, analogous to associativity, such as $(x // y) * z = (y // x) \backslash\backslash z$. A quasigroup is a non-associative analogue of groups, specifically a set equipped with multiplication and right/left conjugate-division operations that are compatible. Each equation of interest identifies two three-variable expressions built from these operations. I determine all $114$ equivalence classes of their conjunctions, and all implications between them. This includes as a small corner the five-element non-distributive lattice identified by Schröder.

math.RA↗

arXiv2025

The Equational Theories Project: Advancing Collaborative Mathematical Research at Scale

We report on the Equational Theories Project (ETP), an online collaborative pilot project to explore new ways to collaborate in mathematics with machine assistance. The project successfully determined all 22 028 942 edges of the implication graph between the 4694 simplest equational laws on magmas, by a combination of human-generated and automated proofs, all validated by the formal proof assistant language Lean. As a result of this project, several new constructions of magmas satisfying specific laws were discovered, and several auxiliary questions were also addressed, such as the effect of restricting attention to finite magmas.

math.RA↗

arXiv2024

Charged massive spin 2 and 3/2 propagation in a constant electromagnetic background

We present two methods for deriving the equations of motion for charged massive spin-3/2 particles. The first approach involves utilizing the Euler-Lagrange equations derived from a Lagrangian that describes the propagation of the first massive excitation of open superstrings. The second method entails enforcing the conditions that the trace vanishes and that the covariant derivative of the equations vanish. We very briefly comment about other spins.

hep-th↗

arXiv2024

Optimal shielding for Einstein gravity

To construct asymptotically-Euclidean Einstein's initial data sets, we introduce the localized seed-to-solution method, which projects from approximate to exact solutions of the Einstein constraints. The method enables us to glue together initial data sets in multiple asymptotically-conical regions, and in particular construct data sets that exhibit the gravity shielding phenomenon, specifically that are localized in a cone and exactly Euclidean outside of it. We achieve optimal shielding in the sense that the metric and extrinsic curvature { are controlled at a super-harmonic rate, regardless of how slowly they decay (even} beyond the standard ADM formalism), and the gluing domain can be a collection of arbitrarily narrow nested cones. We also uncover several notions of independent interest: silhouette functions, localized ADM modulator, and relative energy-momentum vector. An axisymmetric example is provided numerically.

gr-qc↗

arXiv2024

Adjoints in symmetric squares of Lie algebra representations

*Caveat: we learned post-factum that most of these results are not novel. We are keeping this paper for continuity reasons.* Given finite-dimensional complex representations $V$ and $V'$ of a simply-connected semisimple compact Lie group $G$, we determine the dimension of the $G$-invariant subspace of $\mathrm{adj}(G)\otimes V\otimes V'$, of $\mathrm{adj}(G)\otimes S^2 V$, and of $\mathrm{adj}(G)\otimesΛ^2 V$, where $\mathrm{adj}(G)$ is the adjoint representation. In other words we derive the multiplicity with which summands of $\mathrm{adj}(G)$ appear in a tensor product $V \otimes V'$ or (anti)symmetric square $S^2 V$ or $Λ^2 V$. We find in particular that the dimension of the $G$-invariant subspace of $\mathrm{adj}(G)\otimes S^2 V$ is larger than (resp. smaller or equal to) that of $\mathrm{adj}(G)\otimesΛ^2 V$ for a symplectic (resp. orthogonal) representation $V$.

math.RT↗

arXiv2024

BPS Dendroscopy on Local $P^2$

The spectrum of BPS states in type IIA string theory compactified on a Calabi-Yau threefold famously jumps across codimension-one walls in complexified Kähler moduli space, leading to an intricate chamber structure. The Split Attractor Flow Conjecture posits that the BPS index $Ω_z(γ)$ for given charge $γ$ and moduli $z$ can be reconstructed from the attractor indices $Ω_*(γ_i)$ counting BPS states of charge $γ_i$ in their respective attractor chamber, by summing over a finite set of decorated rooted flow trees known as attractor flow trees. If correct, this provides a classification (or dendroscopy) of the BPS spectrum into different topologies of nested BPS bound states, each having a simple chamber structure. Here we investigate this conjecture for the simplest, albeit non-compact, Calabi-Yau threefold, namely the canonical bundle over the projective plane $P^2$. Since the Kähler moduli space has complex dimension one and the attractor flow preserves the argument of the central charge, attractor flow trees coincide with scattering sequences of rays in a two-dimensional slice of the scattering diagram in the space of stability conditions on the derived category of compactly supported coherent sheaves on $K_{P^2}$. We combine previous results on the scattering diagram of $K_{P^2}$ in the large volume slice with new results near the orbifold point $\mathbb{C}^3/\mathbb{Z}_3$, and prove that the Split Attractor Flow Conjecture holds true on the physical slice of $Π$-stability conditions. In particular, while there is an infinite set of initial rays related by the group $Γ_1(3)$ of auto-equivalences, only a finite number of possible decompositions $γ=\sum_iγ_i$ contribute to the index $Ω_z(γ)$ for any $γ$ and $z$, with constituents $γ_i$ related by spectral flow to the fractional branes at the orbifold point.

hep-th↗

arXiv2023

B-brane transport in anomalous (2,2) models and localization

We study how B-branes in two-dimensional N=(2,2) anomalous models behave as we vary the energy scale and bulk parameters in the quantum Kähler moduli space. We focus on (2,2) theories defined by abelian gauged linear sigma models (GLSM). Guided by the hemisphere partition function we find how B-branes split in arbitrary phases into components on the Higgs branch and other branches: this generalizes the band restriction rules of Herbst-Hori-Page to (abelian) anomalous models. Secondly, we address divergences in non-compact models, through the central example of GLSMs for Hirzebruch-Jung resolutions of cyclic surface singularities. For a brane with compact support we explain how to regularize and compute the hemisphere partition function and extract its Higgs branch component, which we match in the zero-instanton sector to the geometric central charge of the brane. To this aim, we clarify the definition of zero-instanton geometric central charge for objects in the derived category of a non-compact toric orbifold.

hep-th↗

arXiv2023

Mixed moduli in 3d N=4 higher-genus quivers

We analyze exactly marginal deformations of 3d N=4 Lagrangian gauge theories, especially mixed-branch operators with both electric and magnetic charges. These mixed-branch moduli can either belong to products of electric and magnetic current supermultiplets, or be single-trace (non-factorizable). Apart from some exceptional quivers that have additional moduli, 3d N=4 theories described by genus g quivers with nonabelian unitary gauge groups have exactly g single-trace mixed moduli, which preserve the global flavour symmetries. This partly explains why only linear and circular quivers have known AdS_4 supergravity duals. Indeed, for g>1, AdS_4 gauged supergravities cannot capture the entire g-dimensional moduli space even if one takes into account the quantization moduli of boundary conditions. Likewise, in a general Lagrangian theory, we establish (using the superconformal index) that the number of single-trace mixed moduli is bounded below by the genus of a graph encoding how nonabelian gauge groups act on hypermultiplets.

hep-th↗

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↗