Scattering amplitudes in quantum field theories have intricate analytic properties as functions of the energies and momenta of the scattered particles. In perturbation theory, their singularities are governed by a set of nonlinear polynomial equations, known as Landau equations, for each individual Feynman diagram. The singularity locus of the associated Feynman integral is made precise with the notion of the Landau discriminant, which characterizes when the Landau equations admit a solution. In order to compute this discriminant, we present approaches from classical elimination theory, as well as a numerical algorithm based on homotopy continuation. These methods allow us to compute Landau discriminants of various Feynman diagrams up to 3 loops, which were previously out of reach. For instance, the Landau discriminant of the envelope diagram is a reducible surface of degree 45 in the three-dimensional space of kinematic invariants. We investigate geometric properties of the Landau discriminant, such as irreducibility, dimension and degree.

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Recently methods of quantum statistical mechanics have been fruitfully applied to the study of phases of quantum lattice systems at zero temperature. For example, a rigorous definition of a Short-Range Entangled phase of matter has been given and a classification of such phases in one spatial dimension has been achieved. I will discuss some of these developments, focusing on the topology and geometry of the space of Short-Range Entangled states. According to a conjecture of A. Kitaev, these spaces form a loop spectrum in the sense of homotopy theory. This conjecture implies that to any family of Short-Range entangled states in one dimension one can associate a gerbe on the parameter space. I will show how to construct such a gerbe. Thе curvature of this gerbe is a closed 3-form with quantized periods and can be regarded as a higher-dimensional generalization of the curvature of the Berry connection.

https://us02web.zoom.us/j/81778962715?pwd=QnpNS2ErSnBCTWRYUHphd1VMMysyZz09

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Metals are an interesting class of gapless quantum many-body systems. Many metals are described by the famous « Fermi liquid theory » at low temperatures, but there are also many metallic materials for which Fermi liquid theory is an inadequate description. In this talk, I will argue that a productive way to think about certain properties of metals, beyond Fermi liquid theory, is in the language of emergent symmetries and anomalies, thus importing ideas originally developed in the context of gapped topological phases of matter and their boundaries. From this point of view, I will show how to derive a vast generalization of Luttinger’s theorem, the result that relates the volume enclosed by the Fermi surface of a Fermi liquid to the microscopic charge density. From this one can derive a number of consequences, including strong constraints on the emergent symmetry group of compressible metals. I also discuss implications for electrical resistivity.

 

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https://us02web.zoom.us/j/85119690238?pwd=MC96eVpLZTN6aXhCdHZpTVJYSzFTZz09

ID de réunion : 851 1969 0238
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I will explain how the conformal bootstrap can be adapted to place rigorous bounds on the spectra of automorphic forms on locally symmetric spaces. A locally symmetric space is of the form HG/K, where G is a non-compact semisimple Lie group, K the maximal compact subgroup of G, and H a discrete subgroup of G. If we take G = SL(2,R), then spaces of this form are precisely hyperbolic surfaces and hyperbolic 2-orbifolds. Automorphic forms then come in two types: modular forms, and eigenfunctions of the hyperbolic Laplacian, also known as Maass forms. The bootstrap constraints arise from the associativity of function multiplication on the space HG, and are very similar to the usual correlator bootstrap equations, with G playing the role of the conformal group. For G=SL(2,R), I will use this method to prove upper bounds on the lowest positive eigenvalue of the Laplacian on all closed hyperbolic surfaces of a fixed genus. The bounds at genus 2 and genus 3 are very nearly saturated by the Bolza surface and the Klein quartic. This is based on upcoming work with P. Kravchuk and S. Pal.

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https://us02web.zoom.us/j/82068794423?pwd=TjZ3V0hhTFl5MEtCdEM4Lys1UHlEQT09

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