Topological insulators display two remarkable properties. Firstly\, th ey are genuine thermodynamic phases\, i.e. they are separated by sharp pha se boundaries where Anderson’s localization length diverges. Secondly\, when two distinct topological phases are interfaced\, wave propagation is enabled along the interface\, which cannot be suppressed by disorder. In t he first part of the talk\, I will exemplify these phenomena with exactly solvable models\, of which one with disorder\, as well as with numerical s imulations. In the second part of the talk\, I will show how index theorem s generated with Alain Connes’ quantized calculus explain both remarkabl e properties mentioned above.
\n\nPLEASE NOTE THE CHANGE IN TIME (!)I will describe a new class of noncommutative field th eories\, building on many older works in the literature\, which possess 'b raided gauge symmetries'. Their construction is motivated by recent attemp ts to relieve the constraints imposed by conventional star-gauge symmetrie s and their tension with twisted diffeomorphisms\, and by the modern persp ective on classical field theories based on homotopy algebras. I will revi ew all of the necessary background\, focusing on the case of diffeomorphis m invariant theories for illustration. As an example\, I will show how the se considerations lead to a new theory of noncommutative gravity in four d imensions within the Einstein-Cartan-Palatini formalism.
\n LOCATION:https://stable.researchseminars.org/talk/NCGandPH/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Raimar Wulkenhaar DTSTART:20210531T150000Z DTEND:20210531T160000Z DTSTAMP:20260404T111324Z UID:NCGandPH/5 DESCRIPTION:Title: From noncommutative field theory towards topological recurrence.\nby Raimar Wulkenhaar as part of Noncommutative Geometry and Physics\n\ n\nAbstract\nFinite-dimensional approximations of nonco mmutative quantum field\ntheories are matrix models. They often show rich mathematical\nstructures: many of them are exactly solvable or even relate d to\nintegrability\, or they generate numbers of interest in enumerative or\nalgebraic geometry. For many matrix models\, it was possible to prove that\n they are governed by a universal combinatorial structure called\nT opological Recursion. The probably most beautiful example is\nKontsevich's matrix Airy function which computes intersection numbers on\nthe moduli s pace of stable complex curves. The Kontsevich model arises\nfrom a $\\lamb da\\Phi^3$-model on noncommutative geometry. The talk\naddresses the quest ion which structures are produced when replacing\n$\\lambda\\Phi^3$ by $\\ lambda\\Phi^4$. The final answer will be that\n$\\lambda \\Phi^4$ obeys an extension of topological recursion. We\nencounter numerous surprising ide ntities on the way.
\n LOCATION:https://stable.researchseminars.org/talk/NCGandPH/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Alessandra Frabetti (Universite’ de Lyon 1) DTSTART:20210628T150000Z DTEND:20210628T160000Z DTSTAMP:20260404T111324Z UID:NCGandPH/6 DESCRIPTION:Title: Noncommutative renormalization Hopf algebras\nby Alessandra Fr abetti (Universite’ de Lyon 1) as part of Noncommutative Geometry and Ph ysics\n\n\nAbstract\nIn pQFT\, the renormalization group acts on the Lagra ngian as a group of formal diffeomorphisms in the powers of the coupling c onstant\, by substitution of the bare coupling and multiplication by some renormalization factors built on the counterterms of divergent Feynman gra phs.\n\nFor scalar theories\, such groups are proalgebraic (functorial on the coefficients algebra) and are represented by Faà di Bruno types of Ho pf algebras on graphs\, called renormalization Hopf algebras. In this talk I review Connes-Kreimer's settings and comment on the improvements expect ed for the BPHZ formula which computes the counterterms of the graphs.\n\n For non-scalar theories\, Feynman graphs have matrix-valued amplitudes: ev en if the counterterms are scalar-valued\, the renormalization group canno t be represented by a Hopf algebra in a functorial way\, because associati vity fails for the composition of series with non-commutative coefficients . Both commutative and noncommutative renormalization Hopf algebras can be defined\, with different meanings. In this talk I explain in which sense the first ones are not functorial (hence not universal) and how the second ones require a functorial extension of proalgebraic groups to non-commuta tive algebras which can only be done as "non-associative" groups.\n\nThe t alk is based on Connes-Kreimer's results (2000)\, on joint works with Chri stian Brouder (2000-2006) and on the recent paper https://doi.org/10.1016/ j.aim.2019.04.053\n LOCATION:https://stable.researchseminars.org/talk/NCGandPH/6/ END:VEVENT BEGIN:VEVENT SUMMARY:John Barrett (University of Nottingham) DTSTART:20211025T150000Z DTEND:20211025T160000Z DTSTAMP:20260404T111324Z UID:NCGandPH/7 DESCRIPTION:Title: The Euclidean contour rotation in quantum gravity\nby John Bar rett (University of Nottingham) as part of Noncommutative Geometry and Phy sics\n\n\nAbstract\nThe talk will discuss the rotation of the contour of \nfunctional integration in quantum gravity from Lorentz ian geometries to \nEuclidean geometries. In the usual framework of metric tensors\, the \nfunctional integral does not have a good definition and s o the formulas \nare necessarily heuristic. However\, it is hoped that the se formulas will \nprovide exact mathematical results when applied to theo ries that are \nconstructed with a fundamental Planck scale cut-off.
\n LOCATION:https://stable.researchseminars.org/talk/NCGandPH/7/ END:VEVENT BEGIN:VEVENT SUMMARY:Anna Pachoł (Queen Mary University of London) DTSTART:20211206T160000Z DTEND:20211206T170000Z DTSTAMP:20260404T111324Z UID:NCGandPH/8 DESCRIPTION:Title: Twisted differential geometry and dispersion relations in κ-nonco mmutative cosmology\nby Anna Pachoł (Queen Mary University of London) as part of Noncommutative Geometry and Physics\n\n\nAbstract\nOne of the most studied possible phenomenological effect of quantu m gravity is the modifications in wave dispersion. Thanks to the noncommut ative deformations of wave equations in curved backgrounds we can investig ate the propagation of waves in noncommutative cosmology and consider the modification of the dispersion relations due to noncommutativity combined with curvature of spacetime.\n
\nIn the talk\, I wil l follow the twisted differential geometry approach\, give an overview of this framework and then focus on the results obtained by the Jordanian twi st. The corresponding noncommutative spacetime is kappa-Minkowski consider ed in the presence of Friedman-Lemaitre-Robertson-Walker (FLRW) cosmologic al background.
\n LOCATION:https://stable.researchseminars.org/talk/NCGandPH/8/ END:VEVENT END:VCALENDAR