arXiv subjects

Laurent Baulieu

Publications and source records attributed to Laurent Baulieu.

At least 19 recordsLinked to original sources

arXiv2026

Extended-BMS Anomalies and Flat Space Holography

We geometrically determine the BRST symmetry of a three-dimensional extended BMS field theory defined at the null boundary of asymptotically flat four-dimensional spaces. We then discuss its ghost number zero and one cocycles, independently of the four-dimensional bulk. The former determine a topological Lagrangian localized on the boundary of null infinity, while the latter classify the corresponding anomalies. This classification reproduces the eBMS anomalies known from the bulk, providing a rare non-perturbative test of flat space holography. While supertranslations are anomaly-free, superrotations admit independent central charges. This theory can thus be dual to Einstein gravity only if these central charges are dictated by the bulk.

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arXiv2026

BRST Noether Theorem and Corner Charge Bracket

We provide a proof of the BRST Noether 1.5th theorem, conjectured in [JHEP 10 (2024) 055], for a broad class of rank-1 BV theories including supergravity and 2-form gauge theories. The theorem asserts that the BRST Noether current of any BRST invariant gauge fixed Lagrangian decomposes on-shell into a sum of a BRST-exact term and a corner term that defines Noether charges. This extends the holographic consequences of Noether's second theorem to gauge fixed theories and, in particular, offers a universal gauge independent Lagrangian derivation of the invariance of the S-matrix under asymptotic symmetries. Furthermore, we show that these corner Noether charges are inherently non-integrable. To address this non-integrability, we introduce a novel charge bracket that accounts for potential symplectic flux and anomalies, providing an honest canonical representation of the asymptotic symmetry algebra. We also highlight a general origin of a BRST cocycle associated with asymptotic symmetries.

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arXiv2025

BV-BRST Noether theorem

The BRST Noether theorem, or ``Noether's 1.5 theorem'', asserts the triviality of the BRST Noether current. We provide two proofs of this theorem that are both valid without restriction on the structure of the gauge theory, extending thereby previous proofs holding in the case of gauge theories for which the solution of the master equation is linear in the antifields. We also relate explicitly the BRST Noether current to the BRST master current appearing in the master equation.

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arXiv2024

BRST Covariant Phase Space and Holographic Ward Identities

This paper develops an enlarged BRST framework to treat the large gauge transformations of a given quantum field theory. It determines the associated infinitely many Noether charges stemming from a gauge fixed and BRST invariant Lagrangian, a result that cannot be obtained from Noether's second theorem. The geometrical significance of this result is highlighted by the construction of a trigraded BRST covariant phase space, allowing a BRST invariant gauge fixing procedure. This provides an appropriate framework for determining the conserved BRST Noether current of the global BRST symmetry and the associated global Noether charges. The latter are found to be equivalent with the usual classical corner charges of large gauge transformations. It allows one to prove the gauge independence of their physical effects at the perturbative quantum level. In particular, the underlying BRST fundamental canonical relation provides the same graded symplectic brackets as in the classical covariant phase space. A unified Lagrangian Ward identity for small and large gauge transformations is built. It consistently decouples into a bulk part for small gauge transformations, which is the standard BRST--BV quantum master equation, and a boundary part for large gauge transformations. The boundary part provides a perturbation theory origin for the invariance of the Hamiltonian physical $\mathcal{S}$-matrix under asymptotic symmetries. Holographic anomalies for the boundary Ward identity are studied and found to be solutions of a codimension one Wess--Zumino consistency condition. Such solutions are studied in the context of extended BMS symmetry. Their existence clarifies the status of the $1$-loop correction to the subleading soft graviton theorem.

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arXiv2023

BRST BMS4 Symmetry and its Cocycles from Horizontality Conditions

The BRST structure of the extended Bondi-Metzner-Sachs symmetry group of asymptotically flat manifolds is investigated using the recently introduced framework of the Beltrami field parametrization of four-dimensional metrics. The latter identifies geometrically the two physical degrees of freedom of the graviton as fundamental fields. The graded BRST BMS4 nilpotent differential operator relies on four horizontality conditions giving a Lagrangian reformulation of the asymptotic BMS4 symmetry. A series of cocycles is found which indicate the possibility of anomalies for three-dimensional Lagrangian theories to be built in the null boundaries of asymptotically flat spaces from the principle of BRST BMS4 invariance.

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arXiv2022

Unimodular Gauge and ADM Gravity Path Integral

This paper proposes a definition of gravitational observables and of their path integral formula within the framework of ADM foliation and the choice of unimodular gauge classes. The method enforces a BRST invariant gauge fixing of the lapse and shift fields. It yields the quantum level extension of the known classical property that the conformal classes of internal metrics of constant Lorentz time leafs define the gravitational physical degrees of freedom.

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arXiv2021

Leaf of Leaf Foliation and Beltrami Parametrization in $d>2$ dimensional Gravity

This work shows the existence of a d>2 dimensional covariant "Beltrami vielbein" that generalizes the d=2 situation. Its definition relies on a sub-foliation Σ^{ADM}_{d-1}=Σ_{d-3}\timesΣ_2 of the Arnowit--Deser--Misner leaves of d-dimensional Lorentzian manifolds {\cal M}_d. Σ_2 stands for the sub-foliating randomly varying Riemann surfaces in {\cal M}_d. The "Beltrami d -bein" associated to any given generic vielbein of {\cal M}_d is systematically determined by a covariant gauge fixing of the Lorentz~symmetry of the latter. It is parametrized by \frac{d(d+1)}2 independent fields belonging to different categories. Each one has a specific interpretation. The Weyl invariant field sector of the Beltrami d-bein selects the \frac{d(d-3)}{2} physical local degrees of freedom of d>2 dimensional gravity. The components of the Beltrami d-bein are in a one to one correspondence with those of the associated Beltrami d-dimensional metric. The Beltrami parametrization of the Spin connection and of the Einstein action delivers interesting expressions. Its use might easier the search of new Ricci flat solutions classified by the genus of the sub-manifold Σ_2. A gravitational "physical gauge" choice is introduced that takes advantage of the geometrical specificities of the Beltrami parametrization. Further restrictions simplify the expression of the Beltrami vielbein when {\cal M}_d has a given spatial holonomy. This point is exemplified in the case of d=8 Lorentzian spaces with G_2\subset SO(1,7) holonomy. The Lorentzian results presented in this paper can be extended to the Euclidean case.

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arXiv2020

Unimodular Gauge in Perturbative Gravity and Supergravity

This paper explains the Unimodular gauge fixing of gravity and supergravity in the framework of a perturbative BRST construction. The unphysical sector contains additional BRST-exact quartets to suppress possible ambiguities and impose both the Unimodular gauge fixing condition on the metric and a gauge condition for the reparametrization symmetry of the unimodular part of the metric. The Unimodular gauge choice of the metric must be completed by a $γ$-Traceless gauge condition for the Rarita--Schwinger field in the case of supergravity. This gives an interesting new class of gauges for gravity and supergravity.

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arXiv2020

Second Order Langevin Equation and Definition of Quantum Gravity By Stochastic Quantisation

Euclidean quantum gravity might be defined by stochastic quantisation that is governed by a higher order Langevin equation rather than a first order stochastic equation. In a transitory phase where the Lorentz time cannot be defined, the parameter that orders the evolution of quantum gravity phenomena is the stochastic time. This changes the definition of causality in the period of primordial cosmology. The prediction of stochastically quantised gravity is that there will a transition from an oscillating quantum phase to a semi-classical one, when the Lorentz time emerges. The end of the transition, as it can be observed from now and described by inflation models, is a diluted Universe, following the inflation phenomenological evolution. It is filled at the beginning with scattered classical primordial black holes. The smallest ones will quickly decay in matter, with a standard quantum field theory evolution till our period. The stable heavier black holes will remain, forming a good fraction of the dark matter and the large black holes observed in the galaxies. In a theoretically related way, this framework suggests the possibility of a gravitational parton content for "point-like" particles, in the same five dimensional quantum field theory context as in the primordial cosmology, with a (+----) signature for the 5d metrics. The very precise and explicit result expressed in this paper is actually far more modest than its motivation. We compute explicitly the meaning of a second order Langevin equation in zero dimensions and define precisely what is second order stochastic quantisation in a soluble case.

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arXiv2019

Weyl Symmetry in Stochastic Quantum Gravity

We propose that the gauge principle of d-dimensional Euclidean quantum gravity is Weyl invariance in its stochastic (d+1)-dimensional bulk. Observables are defined as depending only on conformal classes of d-dimensional metrics. We work with the second order stochastic quantization of Einstein equations in a (d+1)-dimensional bulk. There, the evolution is governed by the stochastic time, which foliates the bulk into Euclidean d-dimensional leaves. The internal metric of each leaf can be parametrized by its unimodular part and conformal factor. Additional bulk metric components are the ADM stochastic lapse and a stochastic shift. The Langevin equation determines the acceleration of the leaf as the sum of a quantum noise, a drift force proportional to Einstein equations and a viscous first order force. Using Weyl covariant decomposition, this Langevin equation splits into irreducible stochastic equations, one for the unimodular part of the metric and one for its conformal factor. For the first order Langevin equation, the unphysical fields are the conformal factor, which is a classical spectator, and the stochastic lapse and shift. These fields can be gauge-fixed in a BRST invariant way in function of the initial data of the process. One gets observables that are covariant with respect to internal reparametrization in each leaf, and invariant under arbitrary reparametrization of the stochastic time. The interpretation of physical observable at finite stochastic time is encoded in a transitory (d+1)-dimensional phase where the Lorentz time cannot be defined. The latter emerges in the infinite stochastic time limit by an abrupt phase transition from quantum to classical gravity.

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arXiv2018

On Stochastic Quantisation of Supersymmetric Theories

We explain how stochastic TQFT supersymmetry can be made compatible with space supersymmetry. Taking the case of N=2 supersymmetric quantum mechanics, (the proof would be the same for the Wess-Zumino model), we determine the kernels that ensure the convergence of the stochastic process toward the standard path integral, under the condition that they are covariant under supersymmetry. They depend on a massive parameter M that can be chosen at will and modifies the course of the stochastic evolution, but the infinite stochastic time limit of the correlation functions is in fact independent on the choice of M.

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arXiv2018

An Angular Dependent Supersymmetric Quantum Mechanics with a ${\bf Z}_2$-invariant Potential

We generalize the conformally invariant topological quantum mechanics of a particle propagating on a punctured plane by introducing a potential that breaks both the rotational and the conformal invariance down to a ${\bf Z}_2$ angular-dependent discrete symmetry. We derive a topological quantum mechanics whose localization gauge functions give interesting self-dual equations. The model contains an order parameter and exhibits a spontaneous symmetry breaking with two ground states above a critical scale. Unlike the ordinary $O(2)$-invariant Higgs potential, an angular-dependence is found and saddle points, instead of local maxima, appear, posing subtle questions about the existence of instantons. The supersymmetric quantum mechanical model is constructed in both the path integral and the operatorial frameworks.

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arXiv2016

Early Universes with Effective Discrete Time

The mechanism for triggering the universe inflation could be that at very early periods the time variable was discrete instead of smooth. Alternatively, and perhaps equivalently, it could be the consequence that the metrics of the early universe was a strongly concentrated gravitational coherent state with very high frequency oscillations, allowing local pair creations by a generalisation to gravity of the Schwinger mechanism, perhaps by creation of black holes of masses superior to the Planck scale. The lattice spacing between two clicks in the discrete time picture corresponds to the inverse frequency of the gravitational coherent state in the other picture. In both cases, a much lower time than the Planck time might represent a new fundamental scale, giving new type of physics. To make possible a concrete estimation of the pair production probability, we propose that the oscillating coherent state metrics that defines this very early geometry minimises the Einstein gravity action coupled to interacting 1-,2- and 3-forms. Part of our intuition relies on a condensed matter analogy with laser-induced superconductivity. An independent section suggests that the new physical time scale we introduce for the Markovian discret time of the pre-inflation epoch can be identified with the stochastic time one uses for standard quantisation in the post-inflation limit. In the pre-inflation phase its discreteness is the observable driving force for the universe evolution, while in the post inflation phase it becomes unobservable and it can can effectively replaced by the continuous time coordinate of standard quantum field theory.

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arXiv2016

Chains of topological oscillators with instantons and calculable topological observables in topological quantum mechanics

We extend to a possibly infinite chain the conformally invariant mechanical system that was introduced earlier as a toy model for understanding the topological Yang-Mills theory. It gives a topological quantum model that has interesting and computable zero modes and topological invariants. It confirms the recent conjecture by several authors that supersymmetric quantum mechanics may provide useful tools for understanding robotic mechanical systems (Vitelli et al.) and condensed matter properties (Kane et al.), where trajectories of effective models are allowed or not by the conservation of topological indices. The absences of ground state and mass gaps are special features of such systems.

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arXiv2012

Twisted N=1, d=4 supergravity and its symmetries

We display the construction of a twisted superalgebra for the N=1 Euclidian supergravity on 4-manifolds with an almost complex structure. It acts on a representation of twisted supersymmetry made of forms with odd and even statistics and it is covariant under a SU(2)\subset SO(4) Lorentz invariance of the manifold's tangent-space. It contains 4 twisted supersymmetry generators, one nilpotent scalar, one vector and one pseudo-scalar. The superalgebra closes on the twisted fields of supergravity in its new minimal set of auxiliary fields. Its couplings to the twisted Wess and Zumino and vector multiplets are also determined.

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arXiv2012

Twisted Superalgebras and Cohomologies of the N=2 Superconformal Quantum Mechanics

We prove that the invariance of the N=2 superconformal quantum mechanics is controlled by subalgebras of a given twisted superalgebra made of 6 fermionic (nilpotent) generators and 6 bosonic generators (including a central charge). The superconformal quantum mechanics actions are invariant under this quite large twisted superalgebra. On the other hand, they are fully determined by a subalgebra with only 2 fermionic and 2 bosonic (the central charge and the ghost number) generators. The invariant actions are Q_i-exact (i=1,2,...,6), with a Q_{i'}-exact (i'\neq i) antecedent for all 6 fermionic generators. It follows that the superconformal quantum mechanics actions with Calogero potentials are uniquely determined even if, in its bosonic sector, the twisted superalgebra does not contain the one-dimensional conformal algebra sl(2), but only its Borel subalgebra. The general coordinate covariance of the non-linear sigma-model for the N=2 supersymmetric quantum mechanics in a curved target space is fully implied only by its worldline invariance under a pair of the 6 twisted supersymmetries. The transformation connecting the ordinary and twisted formulations of the N=2 superconformal quantum mechanics is explicitly presented.

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arXiv2012

Minimizing gauge-functional for 2-d gravity and string theory

We show the existence of a minimizing procedure for selecting a unique representative on the orbit of any given Riemann surface that contributes to the string partition function. As it must, the procedure reduces the string path integral to a final integration over a particular fundamental domain, selected by the choice of the minimizing functional. This construction somehow demystifies the Gribov question.

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arXiv2011

One-dimensional structures behind twisted and untwisted superYang-Mills theory

We give a one-dimensional interpretation of the four-dimensional twisted N=1 superYang-Mills theory on a Kaehler manifold by performing an appropriate dimensional reduction. We prove the existence of a 6-generator superalgebra, which does not possess any invariant Lagrangian but contains two different subalgebras that determine the twisted and untwisted formulations of the N=1 superYang-Mills theory.

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