In this seminar, we set ourselves the ambitious goal of studying the main characteristic features of Dirac operators, presenting the celebrated Atiyah-Singer index theorem and examining a proof of this fundamental result in physical terms, using supersymmetric quantum mechanics. Finally, if time permits, we plan also to have a look at a recent formulation of the index theorem in Lorentzian setting.
This topics will be discussed together, in weekly meetings, where each time one of the participants will present a talk based on a limited selection of references.
The first four talks are meant to provide an essential, but comprehensive, account of the theory of Dirac operators on Riemannian and Lorentzian manifolds, culminating in the formulation of the Atiyah-Singer index theorem. These talks, mostly based on [2], represent quite a challenge as, even without dealing with the demonstrations in detail, the material to cover is very substantial.
The subsequent two talks will be devoted to the development of the basic notions of supersymmetric quantum mechanics and to the study of a "physically motivated" proof of the Atiyah-Singer index theorem, following essentially [8].
Finally, in the last three talks, we will look at how Dirac operators behave on Lorentzian manifolds and how the notion of index theorem, proposed in this setting in [3], leads to a geometrical description of the chiral anomaly [4].
Meetings will take place simultaneously in person in Hannover and Potsdam and also on Zoom, to allow for interaction between the participants in different places. Therefore, online participation of single individuals is possible (and welcome!)
The format of the seminar allows for some flexibility. This means that the program can also be modified en route and eventually adapted to better encompass the interests of the participants.
Time: Wednesdays, 16:30-18, Berlin Time (GMT+1).
Speaker: Alessandro Pietro Contini (Leibniz Universität Hannover)
References: Chapter 4 in [2] (see [1] and [9] for more) and Section 1 in [10].
Introduction: We discuss basics of Dirac operators on (spin) manifolds and outline the relations between these, supersymmetric quantum mechanics, and the Atiyah-Singer index theorem.
Section 4.1: Invariant polynomials, relation to curvature, definition of the Chern-Weil class, invariance w.r.t. the connection, k-th Chern class and properties.
Section 4.2: Additive and multiplicative characteristic classes.
Section 4.3: Definition and properties of the Pontryagin class.
Speaker: Alessandro Pietro Contini (Leibniz Universität Hannover)
References: Chapter 4 in [2]. Complementary material in [8], Chapter 11.
Section 4.1: Invariant polynomials, relation to curvature, definition of the Chern-Weil class, invariance w.r.t. the connection, k-th Chern class and properties.
Section 4.2: Additive and multiplicative characteristic classes. Splitting principle.
Section 4.3: Definition and properties of the Pontryagin class. The Atiyah-Dirac genus.
Notes of the talk available: here
Speaker: Romeo Segnan Dalmasso
References: Chapter 2 in [2] (see [7] and [8] for more).
Section 2.1: Definition of Clifford algebras, existence, uniqueness and other properties.
Section 2.2: Definition of the groups Pin and Spin (in general signature), properties of Spin.
Section 2.3: Spinors, spinor representation and Clifford multiplication in the case of dimension even; spinors, spinor representation and Clifford multiplication in the case of dimension odd.
Section 2.4: Spin structures, spin manifolds and equivalence of spin structures; spinor bundles, Clifford multiplication; spinor connection and relation to curvature.
Section 11.6 in [8]: Cech cohomology associated with a cover. First and second Stiefel-Whitney class.
Speaker: Alessandro Pietro Contini (Leibniz Universität Hannover)
References: Chapter 1, Section 2.5 and Chapter 6 in [2] (see [7] for more).
Section 1.1: Definition of differential operator and of principal symbol; formal adjoint of an operator and properties.
Section 1.2: Sobolev spaces on compact manifolds, embedding and extension theorems.
Section 1.3: Definition of Laplace- and Dirac-type operators; Hodge Laplacian, Bochner formula, Hodge star operator, codifferential; twisting of first order operators.
Section 1.4: Some tool from the analysis of Dirac-type operators, elliptic estimates, formal self-adjointedness, spectral decomposition.
Section 2.5: Definition of the classical (twisted) Dirac operator, formal self-adjointedness, relation to Ricci and scalar curvature.
Sections 6.1, 6.2, 6.3, 6.4: Definition of all the ingredients of the (twisted) Dirac operator on Lorentzian manifolds (completely analogous to the Riemannian case).
Speaker: Alberto Richtsfeld (University of Potsdam)
References: Chapter 3 in [2] (see [5] for more).
Section 3.1: The heat equation and the (true) heat kernel of a Laplace-type operator.
Section 3.2: The formal heat kernel and some asymptotics properties.
Section 3.4: The index of a Dirac-type operator, preliminary version of the Atiyah-Singer index theorem, homotopy invariance of the index, multiplicity of the index for coverings.
Speaker: Jørgen Lye (Leibniz Universität Hannover)
References: Section 12.9 in [8] (see [1], [6] and [9] for more).
Section 12.9.1: Clifford algebras and fermions
Section 12.9.2: Supersymmetric quantum mechanics in flat space
Section 12.9.3: Supersymmetric quantum mechanics in a general manifold
Notes of the talk available: here
Speaker: Jørgen Lye (Leibniz Universität Hannover)
References: Section 12.10 in [8] (see [1], [6] and [9] for more).
Section 12.10.1: Index of the Dirac operator
Section 12.10.2: Path integral and index theorem
Notes of the talk available: here
Speaker: Rebecca Roero (University of Potsdam)
References: Sections 1, 2, 3 in [3].
Speaker: Lennart Ronge (University of Potsdam)
References: Sections 4, 5, 6, 7 in [3].
Speaker: Onirban Islam (University of Potsdam)
References: [4]
Alvarez-Gaumé, L. "Supersymmetry and the Atiyah-Singer index theorem", Commun. Math. Phys. 90, 161-173 (1983).
Bär, C. "Spin geometry", available online at this link.
Bär, C. and Strohmaier, A. "An index theorem for Lorentzian manifolds with compact spacelike Cauchy boundary." American Journal of Mathematics 141 (5), 1421-1455 (2019).
Bär, C. and Strohmaier, A. "A Rigorous Geometric Derivation of the Chiral Anomaly in Curved Backgrounds". Commun. Math. Phys. 347, 703–721 (2016).
Gilkey, P.B. "Invariance Theory: The Heat Equation and the Atiyah-Singer Index Theorem" (1st ed.). 1994 CRC Press.
Hanisch, F. and Ludewig, M. "A Rigorous Construction of the Supersymmetric Path Integral Associated to a Compact Spin Manifold". Commun. Math. Phys. 391, 1209-1239 (2022), available online at this link.
Lawson, H. B. and Michelsohn, M.-L. "Spin Geometry" (PMS-38). Princeton University Press, 1989.
Nakahara, M. "Geometry, Topology and Physics" (2nd ed.). 2003 CRC Press.
Witten, E. "Constraints on supersymmetry breaking", Nuclear Physics B202, (1982).
Friedan, D. and Windey, P. "Supersymmetric derivation of the Atiyah-Singer index and the chiral anomaly", Nuclear Physics B 235 (1984).