Multiplication of differential operators in terms of connections using Lie-Rinehart algebras
Montag, 8.11.21, 16:15-17:15, Hörsaal II, Albertstr. 23b
The multiplication of two differential operators in an open set of \(\bmathbb{R}^n\)\nis explicitly known in terms of their standard symbols: this is a motivating point\nfor the theory of deformation quantization. On a differentiable manifold equipped\nwith a connection ∇ in the tangent bundle the same formula --where partial\nderivatives are replaced with symmetrized covariant derivatives-- will be wrong in\ngeneral, and one has to correct it by terms containing torsion and curvature and\ntheir covariant derivatives. In our work with my PhD student Hamilton Menezes de Araujo\nwe shall give an 'explicit formula' of the corrected formula in the more algebraic framework\nof Lie-Rinehart algebras L (G.S.Rinehart, 1963, which are now being used in the study\nof singular foliations) over a commutative unital K-algebra A (where the ground ring K should\ncontain the rational numbers as a subring). L generalizes the\nLie algebra of vector fields (more generally Lie algebroids) and A the algebra of\nsmooth functions in differential geometry. The enveloping algebra of L over A introduced\nby Rinehart plays the rôle of the differential operator algebra. The arising combinatorial problems\ncan conveniently be treated in terms of the fibrewise shuffle comultiplication in the free algebra\ngenerated by L over A and the associated convolution products. The torsion and curvature\nterms arise in a morphism of Lie-Rinehart algebras Z from the free Lie algebra generated\nby L over A equipped with a Lie-Rinehart bracket isomorphic to the one on M.Kapranov's\n'free path Lie algebroid' (2007) to L which are related to the (infinitesimal)\nholonomy of the connection. Z is obtained by a simple explicit linear recursion.\nThe framework allows to discuss `family theorems' by replacing the ground ring K\nbut the smooth function algebra of the base of a fibered manifold.
Nu invariants of extra twisted connected sums
Montag, 15.11.21, 16:15-17:15, Hörsaal II, Albertstr. 23b
The \(\bnu\) invariant is an invariant of \(G_2\)-structures on closed 7-manifolds. It can be computed in examples and has been used to show that for some closed spin 7-manifolds, the moduli space of \(G_2\)-metrics is not connected.\n\nIn this talk, we will present the computation for extra twisted connected sums and show how to obtain a tractable formula in the end.
Elliptic Genera and \(G_2\)-manifolds
Montag, 29.11.21, 16:15-17:15, Hörsaal II, Albertstr. 23b
In 1988 Witten showed that the universal elliptic genus of a manifold \(M\) can be interpreted as the index of a twisted Dirac operator on the the loop space of \(M\). Furthermore he discovered, that the index of this Dirac operator has similar modular properties, if one restricts to string manifolds. The resulting modular form is now called the Witten genus.\n\nIn my talk I will give an introduction to modular forms and I will formaly derive the Witten genus from the index theorem.\n\nIf we compare the Witten genus with the elliptic genus in dimension \(8\), there occur characteristic classes, which are connected with the Nu-invariant of \(G_2\)-manifolds.