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Philippe Mathieu

Publications and source records attributed to Philippe Mathieu.

15 recordsLinked to original sources
arXiv2024

$\mathrm{U}(1)^{n}$ Chern-Simons theory: partition function, reciprocity formula and Chern-Simons duality

The $\mathrm{U}(1)$ Chern-Simons theory can be extended to a topological $\mathrm{U}(1)^n$ theory by taking a combination of Chern-Simons and BF actions, the mixing being achieved with the help of a collection of integer coupling constants. Based on the Deligne-Beilinson cohomology, a partition function can then be computed for such a $\mathrm{U}(1)^n$ Chern-Simons theory. This partition function is clearly a topological invariant of the closed oriented $3$-manifold on which the theory is defined. Then, by applying a reciprocity formula a new expression of this invariant is obtained which should be a Reshetikhin-Turaev invariant. Finally, a duality between $\mathrm{U}(1)^n$ Chern-Simons theories is demonstrated.

math-ph↗
arXiv2023

Virchow: A Million-Slide Digital Pathology Foundation Model

The use of artificial intelligence to enable precision medicine and decision support systems through the analysis of pathology images has the potential to revolutionize the diagnosis and treatment of cancer. Such applications will depend on models' abilities to capture the diverse patterns observed in pathology images. To address this challenge, we present Virchow, a foundation model for computational pathology. Using self-supervised learning empowered by the DINOv2 algorithm, Virchow is a vision transformer model with 632 million parameters trained on 1.5 million hematoxylin and eosin stained whole slide images from diverse tissue and specimen types, which is orders of magnitude more data than previous works. The Virchow model enables the development of a pan-cancer detection system with 0.949 overall specimen-level AUC across 17 different cancer types, while also achieving 0.937 AUC on 7 rare cancer types. The Virchow model sets the state-of-the-art on the internal and external image tile level benchmarks and slide level biomarker prediction tasks. The gains in performance highlight the importance of training on massive pathology image datasets, suggesting scaling up the data and network architecture can improve the accuracy for many high-impact computational pathology applications where limited amounts of training data are available.

eess.IV↗
arXiv2023

Generalized Abelian Turaev-Viro and $\mathrm{U}\!\left(1\right)$ BF Theories

We explain how it is possible to study $\mathrm{U}\!\left(1\right)$ BF theory over a connected closed oriented smooth $3$-manifold in the formalism of path integral thanks to Deligne-Beilinson cohomology. We show how we can straightforwardly extend the definition to families of theories in any dimension. We extend then the definition of the Turaev-Viro invariant of a connected closed oriented smooth $3$-manifold in an Abelian framework to a family of invariants in any dimension. We show that those invariants can be written as discrete BF theories. We explain how the extensions of $\mathrm{U}\!\left(1\right)$ BF theory we defined can be related to the extensions of Turaev-Viro invariant we constructed.

math-ph↗
arXiv2022

Gauge theories on quantum spaces

We review the present status of gauge theories built on various quantum space-times described by noncommutative space-times. The mathematical tools and notions underlying their construction are given. Different formulations of gauge theory models on Moyal spaces as well as on quantum spaces whose coordinates form a Lie algebra are covered, with particular emphasis on some explored quantum properties. Recent attempts aiming to include gravity dynamics within a noncommutative framework are also considered.

hep-th↗
arXiv2021

Algebraic structures in $\kappa$-Poincar\'e invariant gauge theories

$\kappa$-Poincar\'e invariant gauge theories on $\kappa$-Minkowski space-time, which are noncommutative analogs of the usual $U(1)$ gauge theory, exist only in five dimensions. These are built from noncommutative twisted connections on a hermitian right module over the algebra coding the $\kappa$-Minkowski space-time. We show that twisting the action of this algebra on the hermitian module, assumed to be a copy of it, affects neither the value of the above dimension nor the noncommutative gauge group defined as the unitary automorphisms of the module leaving the hermitian structure unchanged. Only the hermiticity condition obeyed by the gauge potential becomes twisted. Similarities between the present framework and algebraic features of twisted spectral triples are exhibited.

hep-th↗
arXiv2021

Quantum instability of gauge theories on $\kappa$-Minkowski space

We consider a gauge theory on the 5-d $\kappa$-Minkowski which can be viewed as the noncommutative analog of a $U(1)$ gauge theory. We show that the Hermiticity condition obeyed by the gauge potential $A_\mu$ is necessarily twisted. Performing a BRST gauge-fixing with a Lorentz-type gauge, we carry out a first exploration of the one loop quantum properties of this gauge theory. We find that the gauge-fixed theory gives rise to a non-vanishing tadpole for the time component of the gauge potential, while there is no non-vanishing tadpole 1-point function for the spatial components of $A_\mu$. This signals that the classical vacuum of the theory is not stable against quantum fluctuations. Possible consequences regarding the symmetries of the gauge model and the fate of the tadpole in other gauges of non-covariant type are discussed.

hep-th↗
arXiv2021

Twisted BRST symmetry in gauge theories on $\kappa$-Minkowski

Algebraic properties of the BRST symmetry associated to the twisted gauge symmetry occurring in the $\kappa$-Poincar\'e invariant gauge theories on the $\kappa$-Minkowski space are investigated. We find that the BRST operation associated to the gauge invariance of the action functional can be continuously deformed together with its corresponding Leibniz rule, into a nilpotent twisted BRST operation, leading to a twisted BRST symmetry algebra which may be viewed as a noncommutative analog of the usual Yang-Mills BRST algebra.

hep-th↗
arXiv2021

Boundary electromagnetic duality from homological edge modes

Recent years have seen a renewed interest in using `edge modes' to extend the pre-symplectic structure of gauge theory on manifolds with boundaries. Here we further the investigation undertaken in \cite{FP2018} by using the formalism of homotopy pullback and Deligne-Beilinson cohomology to describe an electromagnetic (EM) duality on the boundary of $M=B^{3}\times\mathbb{R}$. Upon breaking a generalized global symmetry, the duality is implemented by a BF-like topological boundary term. We then introduce Wilson line singularities on $\partial M$ and show that these induce the existence of dual edge modes, which we identify as connections over a $\left(-1\right)$-gerbe. We derive the pre-symplectic structure that yields the central charge in \cite{FP2018} and show that the central charge is related to a non-trivial class of the $\left(-1\right)$-gerbe.

math-ph↗
arXiv2020

Single Extra Dimension from $\kappa$-Poincar\'e and Gauge Invariance

We show that $\kappa$-Poincar\'e invariant gauge theories on $\kappa$-Minkowski space with physically acceptable commutative (low energy) limit must be 5-d. The gauge invariance requirement of the action fixes the dimension of the $\kappa$-Minkowski space to $d=5$ and selects the unique twisted differential calculus with which the construction can be achieved. We characterize a BRST symmetry related to the 5-d noncommutative gauge invariance though the definition of a nilpotent operation, which is used to construct a gauge-fixed action. We also consider standard scenarios assuming (compactification of) flat extra dimension, for which the 5-d deformation parameter $\kappa$ can be viewed as the bulk 5-d Planck mass. We study physical properties of the resulting 4-d effective theories. Recent data from collider experiments require $\kappa\gtrsim\mathcal{O}(10^{13})\ \text{GeV}$. The use of standard test of in-vacuo dispersion relations of Gamma Ray Burst photons increases this lower bound by 4 orders of magnitude. The robustness of this bound is discussed in the light of possible new features of noncommutative causal structures.

hep-th↗
arXiv2020

Gauge theories on $\kappa$-Minkowski spaces: Twist and modular operators

We discuss the construction of $\kappa$-Poincar\'e invariant actions for gauge theories on $\kappa$-Minkowski spaces. We consider various classes of untwisted and (bi)twisted differential calculi. Starting from a natural class of noncommutative differential calculi based on a particular type of twisted derivations belonging to the algebra of deformed translations, combined with a twisted extension of the notion of connection, we prove an algebraic relation between the various twists and the classical dimension d of the $\kappa$-Minkowski space(-time) ensuring the gauge invariance of the candidate actions for gauge theories. We show that within a natural differential calculus based on a distinguished set of twisted derivations, d=5 is the unique value for the classical dimension at which the gauge action supports both the gauge invariance and the $\kappa$-Poincar\'e invariance. Within standard (untwisted) differential calculi, we show that the full gauge invariance cannot be achieved, although an invariance under a group of transformations constrained by the modular (Tomita) operator stemming from the $\kappa$-Poincar\'e invariance still holds.

hep-th↗
arXiv2019

Homological perspective on edge modes in linear Yang-Mills and Chern-Simons theory

We provide an elegant homological construction of the extended phase space for linear Yang-Mills theory on an oriented and time-oriented Lorentzian manifold $M$ with a time-like boundary $\partial M$ that was proposed by Donnelly and Freidel [JHEP 1609, 102 (2016)]. This explains and formalizes many of the rather ad hoc constructions for edge modes appearing in the theoretical physics literature. Our construction also applies to linear Chern-Simons theory, in which case we obtain the extended phase space introduced by Geiller [Nucl. Phys. B 924, 312 (2017)].

hep-th↗
arXiv2017

Geometric aspects of interpolating gauge fixing in Chern-Simons Theory

In this article we investigate an interpolating gauge fixing procedure in $(4l+3)$-dimensional abelian Chern-Simons theory. We show that this interpolating gauge is related to the covariant gauge in a constant anisotropic metric. We compute the corresponding propagators involved in various expressions of the linking number in various gauges. We comment on the geometric interpretations of these expressions, clarifying how to pass from one interpretation to another.

hep-th↗
arXiv2017

Flat connections in three-manifolds and classical Chern-Simons invariant

A general method for the construction of smooth flat connections on 3-manifolds is introduced. The procedure is strictly connected with the deduction of the fundamental group of a manifold M by means of a Heegaard splitting presentation of M. For any given matrix representation of the fundamental group of M, a corresponding flat connection A on M is specified. It is shown that the associated classical Chern-Simons invariant assumes then a canonical form which is given by the sum of two contributions: the first term is determined by the intersections of the curves in the Heegaard diagram, and the second term is the volume of a region in the representation group which is determined by the representation of pi_1(M) and by the Heegaard gluing homeomorphism. Examples of flat connections in topologically nontrivial manifolds are presented and the computations of the associated classical Chern-Simons invariants are illustrated.

hep-th↗
arXiv2017

Exact computations in topological abelian Chern-Simons and BF theories

We introduce Deligne cohomology that classifies U(1) fibre bundles over 3-manifolds endowed with connections. We show how the structure of Deligne cohomology classes provides a way to perform exact (non-perturbative) computations in U(1) Chern-Simons theory (resp. BF theory) at the level of functional integrals. The partition functions (and observables) of these theories are strongly related to topological invariants well-known by the mathematicians.

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