@article{BaerStrohmaier2016, author = {B{\"a}r, Christian and Strohmaier, Alexander}, title = {A Rigorous Geometric Derivation of the Chiral Anomaly in Curved Backgrounds}, series = {Communications in mathematical physics}, volume = {347}, journal = {Communications in mathematical physics}, publisher = {Springer}, address = {New York}, issn = {0010-3616}, doi = {10.1007/s00220-016-2664-1}, pages = {703 -- 721}, year = {2016}, abstract = {We discuss the chiral anomaly for a Weyl field in a curved background and show that a novel index theorem for the Lorentzian Dirac operator can be applied to describe the gravitational chiral anomaly. A formula for the total charge generated by the gravitational and gauge field background is derived directly in Lorentzian signature and in a mathematically rigorous manner. It contains a term identical to the integrand in the Atiyah-Singer index theorem and another term involving the.-invariant of the Cauchy hypersurfaces.}, language = {en} } @article{BaerStrohmaier2019, author = {B{\"a}r, Christian and Strohmaier, Alexander}, title = {An index theorem for Lorentzian manifolds with compact spacelike Cauchy boundary}, series = {American Journal of Mathematics}, volume = {141}, journal = {American Journal of Mathematics}, number = {5}, publisher = {Johns Hopkins Univ. Press}, address = {Baltimore}, issn = {0002-9327}, doi = {10.1353/ajm.2019.0037}, pages = {1421 -- 1455}, year = {2019}, abstract = {We show that the Dirac operator on a compact globally hyperbolic Lorentzian spacetime with spacelike Cauchy boundary is a Fredholm operator if appropriate boundary conditions are imposed. We prove that the index of this operator is given by the same expression as in the index formula of Atiyah-Patodi-Singer for Riemannian manifolds with boundary. The index is also shown to equal that of a certain operator constructed from the evolution operator and a spectral projection on the boundary. In case the metric is of product type near the boundary a Feynman parametrix is constructed.}, language = {en} } @article{BaerPfaeffle2010, author = {B{\"a}r, Christian and Pfaeffle, Frank}, title = {Asymptotic heat kernel expansion in the semi-classical limit}, issn = {0010-3616}, doi = {10.1007/s00220-009-0973-3}, year = {2010}, abstract = {Let H-h = h(2)L + V, where L is a self-adjoint Laplace type operator acting on sections of a vector bundle over a compact Riemannian manifold and V is a symmetric endomorphism field. We derive an asymptotic expansion for the heat kernel of H-h as h SE arrow 0. As a consequence we get an asymptotic expansion for the quantum partition function and we see that it is asymptotic to the classical partition function. Moreover, we show how to bound the quantum partition function for positive h by the classical partition function.}, language = {en} } @unpublished{BaerBallmann2012, author = {B{\"a}r, Christian and Ballmann, Werner}, title = {Boundary value problems for elliptic differential operators of first order}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-60023}, year = {2012}, abstract = {We study boundary value problems for linear elliptic differential operators of order one. The underlying manifold may be noncompact, but the boundary is assumed to be compact. We require a symmetry property of the principal symbol of the operator along the boundary. This is satisfied by Dirac type operators, for instance. We provide a selfcontained introduction to (nonlocal) elliptic boundary conditions, boundary regularity of solutions, and index theory. In particular, we simplify and generalize the traditional theory of elliptic boundary value problems for Dirac type operators. We also prove a related decomposition theorem, a general version of Gromov and Lawson's relative index theorem and a generalization of the cobordism theorem.}, language = {en} } @article{BaerBandara2022, author = {B{\"a}r, Christian and Bandara, Lashi}, title = {Boundary value problems for general first-order elliptic differential operators}, series = {Journal of functional analysis}, volume = {282}, journal = {Journal of functional analysis}, number = {12}, publisher = {Elsevier}, address = {Amsterdam [u.a.]}, issn = {0022-1236}, doi = {10.1016/j.jfa.2022.109445}, pages = {69}, year = {2022}, abstract = {We study boundary value problems for first-order elliptic differential operators on manifolds with compact boundary. The adapted boundary operator need not be selfadjoint and the boundary condition need not be pseudo-local.We show the equivalence of various characterisations of elliptic boundary conditions and demonstrate how the boundary conditions traditionally considered in the literature fit in our framework. The regularity of the solutions up to the boundary is proven. We show that imposing elliptic boundary conditions yields a Fredholm operator if the manifold is compact. We provide examples which are conveniently treated by our methods.}, language = {en} } @unpublished{BaerGinoux2012, author = {B{\"a}r, Christian and Ginoux, Nicolas}, title = {Classical and quantum fields on Lorentzian manifolds}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-59973}, year = {2012}, abstract = {We construct bosonic and fermionic locally covariant quantum fields theories on curved backgrounds for large classes of fields. We investigate the quantum field and n-point functions induced by suitable states.}, language = {en} } @article{BaerBecker2014, author = {B{\"a}r, Christian and Becker, Christian}, title = {Differential characters and geometric chains}, series = {Lecture notes in mathematics : a collection of informal reports and seminars}, volume = {2112}, journal = {Lecture notes in mathematics : a collection of informal reports and seminars}, publisher = {Springer}, address = {Berlin}, isbn = {978-3-319-07034-6; 978-3-319-07033-9}, issn = {0075-8434}, doi = {10.1007/978-3-319-07034-6_1}, pages = {1 -- 90}, year = {2014}, abstract = {We study Cheeger-Simons differential characters and provide geometric descriptions of the ring structure and of the fiber integration map. The uniqueness of differential cohomology (up to unique natural transformation) is proved by deriving an explicit formula for any natural transformation between a differential cohomology theory and the model given by differential characters. Fiber integration for fibers with boundary is treated in the context of relative differential characters. As applications we treat higher-dimensional holonomy, parallel transport, and transgression.}, language = {en} } @article{BaerGouduchonMoroianu2005, author = {B{\"a}r, Christian and Gouduchon, Paul and Moroianu, Andrei}, title = {Generalized Cylinders in Semi-Riemannian and Spin Geometry}, year = {2005}, abstract = {We use a construction which we call generalized cylinders to give a new proof of the fundamental theorem of hypersurface theory. It has the advantage of being very simple and the result directly extends to semi-Riemannian manifolds and to embeddings into spaces of constant curvature. We also give a new way to identify spinors for different metrics and to derive the variation formula for the Dirac operator. Moreover, we show that generalized Killing spinors for Codazzi tensors are restrictions of parallel spinors. Finally, we study the space of Lorentzian metrics and give a criterion when two Lorentzian metrics on a manifold can be joined in a natural manner by a 1-parameter family of such metrics.}, language = {en} } @article{Baer2015, author = {B{\"a}r, Christian}, title = {Geometrically formal 4-manifolds with nonnegative sectional curvature}, series = {Communications in analysis and geometry}, volume = {23}, journal = {Communications in analysis and geometry}, number = {3}, publisher = {International Press of Boston}, address = {Somerville}, issn = {1019-8385}, pages = {479 -- 497}, year = {2015}, abstract = {A Riemannian manifold is called geometrically formal if the wedge product of any two harmonic forms is again harmonic. We classify geometrically formal compact 4-manifolds with nonnegative sectional curvature. If the sectional curvature is strictly positive, the manifold must be homeomorphic to S-4 or diffeomorphic to CP2. This conclusion stills holds true if the sectional curvature is strictly positive and we relax the condition of geometric formality to the requirement that the length of harmonic 2-forms is not too nonconstant. In particular, the Hopf conjecture on S-2 x S-2 holds in this class of manifolds.}, language = {en} } @article{Baer2015, author = {B{\"a}r, Christian}, title = {Green-Hyperbolic Operators on Globally Hyperbolic Spacetimes}, series = {Communications in mathematical physics}, volume = {333}, journal = {Communications in mathematical physics}, number = {3}, publisher = {Springer}, address = {New York}, issn = {0010-3616}, doi = {10.1007/s00220-014-2097-7}, pages = {1585 -- 1615}, year = {2015}, abstract = {Green-hyperbolic operators are linear differential operators acting on sections of a vector bundle over a Lorentzian manifold which possess advanced and retarded Green's operators. The most prominent examples are wave operators and Dirac-type operators. This paper is devoted to a systematic study of this class of differential operators. For instance, we show that this class is closed under taking restrictions to suitable subregions of the manifold, under composition, under taking "square roots", and under the direct sum construction. Symmetric hyperbolic systems are studied in detail.}, language = {en} }