Top ological insulators display two remarkable properties. Firstly\, they are genuine thermodynamic phases\, i.e. they are separated by sharp phase boun daries where Anderson’s localization length diverges. Secondly\, when tw o distinct topological phases are interfaced\, wave propagation is enabled along the interface\, which cannot be suppressed by disorder. In the firs t part of the talk\, I will exemplify these phenomena with exactly solvabl e models\, of which one with disorder\, as well as with numerical simulati ons. In the second part of the talk\, I will show how index theorems gener ated with Alain Connes’ quantized calculus explain both remarkable prope rties mentioned above.

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I will describe a new class of noncommutative field theories\ , building on many older works in the literature\, which possess 'braided gauge symmetries'. Their construction is motivated by recent attempts to r elieve the constraints imposed by conventional star-gauge symmetries and t heir tension with twisted diffeomorphisms\, and by the modern perspective on classical field theories based on homotopy algebras. I will review all of the necessary background\, focusing on the case of diffeomorphism invar iant theories for illustration. As an example\, I will show how these cons iderations lead to a new theory of noncommutative gravity in four dimensio ns within the Einstein-Cartan-Palatini formalism.

\n LOCATION:https://researchseminars.org/talk/NCGandPH/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Raimar Wulkenhaar DTSTART;VALUE=DATE-TIME:20210531T150000Z DTEND;VALUE=DATE-TIME:20210531T160000Z DTSTAMP;VALUE=DATE-TIME:20240715T182723Z UID:NCGandPH/5 DESCRIPTION:Title: From noncommutative field theory towards topological recurrence.\nby Raimar Wulkenhaar as part of Noncommutative Geometry and Physics\n\n\nAbst ract\nFinite-dimensional approximations of noncommutati ve quantum field\ntheories are matrix models. They often show rich mathema tical\nstructures: many of them are exactly solvable or even related to\ni ntegrability\, or they generate numbers of interest in enumerative or\nalg ebraic geometry. For many matrix models\, it was possible to prove that\n they are governed by a universal combinatorial structure called\nTopologi cal Recursion. The probably most beautiful example is\nKontsevich's matrix Airy function which computes intersection numbers on\nthe moduli space of stable complex curves. The Kontsevich model arises\nfrom a $\\lambda\\Phi ^3$-model on noncommutative geometry. The talk\naddresses the question whi ch 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 extens ion of topological recursion. We\nencounter numerous surprising identities on the way.

\n LOCATION:https://researchseminars.org/talk/NCGandPH/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Alessandra Frabetti (Universite’ de Lyon 1) DTSTART;VALUE=DATE-TIME:20210628T150000Z DTEND;VALUE=DATE-TIME:20210628T160000Z DTSTAMP;VALUE=DATE-TIME:20240715T182723Z UID:NCGandPH/6 DESCRIPTION:Title: Noncommutative renormalization Hopf algebras\nby Alessandra Frabetti (Universite’ de Lyon 1) as part of Noncommutative Geometry and Physics\n \n\nAbstract\nIn pQFT\, the renormalization group acts on the Lagrangian a s a group of formal diffeomorphisms in the powers of the coupling constant \, by substitution of the bare coupling and multiplication by some renorma lization factors built on the counterterms of divergent Feynman graphs.\n\ nFor scalar theories\, such groups are proalgebraic (functorial on the coe fficients algebra) and are represented by Faà di Bruno types of Hopf alge bras on graphs\, called renormalization Hopf algebras. In this talk I revi ew Connes-Kreimer's settings and comment on the improvements expected for the BPHZ formula which computes the counterterms of the graphs.\n\nFor non -scalar theories\, Feynman graphs have matrix-valued amplitudes: even if t he counterterms are scalar-valued\, the renormalization group cannot be re presented by a Hopf algebra in a functorial way\, because associativity fa ils for the composition of series with non-commutative coefficients. Both commutative and noncommutative renormalization Hopf algebras can be define d\, with different meanings. In this talk I explain in which sense the fir st ones are not functorial (hence not universal) and how the second ones r equire a functorial extension of proalgebraic groups to non-commutative al gebras which can only be done as "non-associative" groups.\n\nThe talk is based on Connes-Kreimer's results (2000)\, on joint works with Christian B rouder (2000-2006) and on the recent paper https://doi.org/10.1016/j.aim.2 019.04.053\n LOCATION:https://researchseminars.org/talk/NCGandPH/6/ END:VEVENT BEGIN:VEVENT SUMMARY:John Barrett (University of Nottingham) DTSTART;VALUE=DATE-TIME:20211025T150000Z DTEND;VALUE=DATE-TIME:20211025T160000Z DTSTAMP;VALUE=DATE-TIME:20240715T182723Z UID:NCGandPH/7 DESCRIPTION:Title: The Euclidean contour rotation in quantum gravity\nby John Barrett (U niversity of Nottingham) as part of Noncommutative Geometry and Physics\n\ n\nAbstract\nThe talk will discuss the rotation of the contour of \nfunctional integration in quantum gravity from Lorentzian geo metries to \nEuclidean geometries. In the usual framework of metric tensor s\, the \nfunctional integral does not have a good definition and so the f ormulas \nare necessarily heuristic. However\, it is hoped that these form ulas will \nprovide exact mathematical results when applied to theories th at are \nconstructed with a fundamental Planck scale cut-off.

\n LOCATION:https://researchseminars.org/talk/NCGandPH/7/ END:VEVENT BEGIN:VEVENT SUMMARY:Anna Pachoł (Queen Mary University of London) DTSTART;VALUE=DATE-TIME:20211206T160000Z DTEND;VALUE=DATE-TIME:20211206T170000Z DTSTAMP;VALUE=DATE-TIME:20240715T182723Z UID:NCGandPH/8 DESCRIPTION:Title: Twisted differential geometry and dispersion relations in κ-noncommutati ve cosmology\nby Anna Pachoł (Queen Mary University of London) as par t of Noncommutative Geometry and Physics\n\n\nAbstract\nOne of the most studied possible phenomenological effect of quantum gravi ty is the modifications in wave dispersion. Thanks to the noncommutative d eformations of wave equations in curved backgrounds we can investigate the propagation of waves in noncommutative cosmology and consider the modific ation of the dispersion relations due to noncommutativity combined with cu rvature of spacetime.\n

\nIn the talk\, I will follo w the twisted differential geometry approach\, give an overview of this fr amework and then focus on the results obtained by the Jordanian twist. The corresponding noncommutative spacetime is kappa-Minkowski considered in t he presence of Friedman-Lemaitre-Robertson-Walker (FLRW) cosmological back ground.

\n LOCATION:https://researchseminars.org/talk/NCGandPH/8/ END:VEVENT END:VCALENDAR