Topological Invariants from Higher Categories

Invariants of Higher-Dimensional Knots and Topological Quantum Field Theories

The present work can be divided into two main parts: i) the first deals with the functionalintegral quantization via BV formalism of BF theories in any dimension, and ii) the second with (isotopy-) invariants of higher-dimensional knots in Euclidean spaces. We briefly comment on the motivations and the results of part i) and ii). BF theory in 3 dimensions (with the addition of a so-called “cosm...

Topological Recursion Relations and Gromov-witten Invariants in Higher Genus

We state and prove a topological recursion relation that expresses any genus-g Gromov-Witten invariant of a projective manifold with at least a (3g − 1)-st power of a cotangent line class in terms of invariants with fewer cotangent line classes. For projective spaces, we prove that these relations together with the Virasoro conditions are sufficient to calculate the full GromovWitten potential....

Topological Field Theory, Higher Categories, and Their Applications

It has been common wisdom among mathematicians that Extended Topological Field Theory in dimensions higher than two is naturally formulated in terms of n-categories with n > 1. Recently the physical meaning of these higher categorical structures has been recognized and concrete examples of Extended TFTs have been constructed. Some of these examples, like the Rozansky-Witten model, are of geomet...

From Operator Categories to Topological Operads

In this paper it is shown that an assortment of operads near to many topologists’ hearts enjoy (homotopy) universal properties of an expressly combinatorial nature. These include the operads An and En. The main idea that makes this possible is the new notion of an operator category, which controls the homotopy types of these operads in a strong sense.

Higher Operads, Higher Categories