FORMALITY AND KONTSEVICH–DUFLO THEOREM

Open Access
- Author:
- Liao, Hsuan Yi
- Graduate Program:
- Mathematics
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- April 10, 2018
- Committee Members:
- Ping Xu, Dissertation Advisor/Co-Advisor
Ping Xu, Committee Chair/Co-Chair
Adrian Ocneanu, Committee Member
Nigel David Higson, Committee Member
Martin Bojowald, Outside Member
Mathieu Philippe Stienon, Dissertation Advisor/Co-Advisor - Keywords:
- deformation quantization
differential geometry
mathematical physics
formality theorem
Duflo theorem
Todd class
Lie algebroid - Abstract:
- Kontsevich’s formality theorem states that there exists an L-infinity quasi-isomorphism from the dgla of polyvector fields on a smooth manifold M to the dgla of polydifferential operators on M, which extends the classical Hochschild-Kostant-Rosenberg map. The construction of Kontsevich formality morphism involves Fedosov type resolutions of the dglas of polyvector fields and polydifferential opartors on smooth manifolds. We introduce, for every Z-graded manifold, a formal exponential map defined in a purely algebraic way and study its properties. As an application, we give a simple new construction of a Fedosov type resolution of the algebra of smooth functions of Z-graded manifolds and we extend the Emmrich-Weinstein theorem to the context of Z-graded manifolds. We also extend Kontsevich’s formality theorem to Lie pairs, a framework which includes a range of diverse geometric contexts such as complex manifolds, foliations, and g-manifolds (that is, manifolds endowed with an action of a Lie algebra g). On a Lie pair, we define the L-infinity algebra of polyvector fields and the L-infinity algebra of polydifferential operators. Their corresponding cohomology groups are Gerstenhaber algebras. We establish the following formality theorem for Lie pairs: there exists an L-infinity quasi-isomorphism from the L-infinity algebra of polyvector fields to the L-infinity algebra of polydifferential operators whose first Taylor coefficient is equal to the Hochschild-Kostant-Rosenberg map twisted by the square root of a Todd cocycle. As a consequence, we prove a Kontsevich-Duflo type theorem for Lie pairs: the Hochschild-Kostant-Rosenberg map twisted by the square root of the Todd class of the Lie pair (L,A) is an isomorphism of Gerstenhaber algebras from the cohomology of polyvector fields to the cohomology of polydifferential operators. As applications, we establish formality theorems and Kontsevich-Duflo type theorems for complex manifolds, foliations, and g-manifolds. In the case of complex manifolds, we recover the Kontsevich-Duflo theorem of complex geometry.