Theoretical Physics Seminar: Davide Passaro

Title: Reversing the orientation: new frameworks for the computation of Z-hat invariants for positive Seifert manifolds

Abstract: Z-hat invariants are topological invariants for three manifolds that have deep physical meaning and are intimately related to Cern-Simons theory. Current computational methods only allow for the computation of Z-hat invariants only on a subset of Seifert manifolds which satisfy a negativity property. On such manifolds the Z-hat invariant are Quantum Modular Forms. CP symmetry on Chern-Simons theory suggests a relation between Z-hat invariant on pairs of negative and positive Seifert manifolds related by orientation reversal. A direct application of the Chern-Simons relation on known Z-hat however leads to diverging series. By leveraging the quantum modularity of the Z-hat invariant however it is possible to understand the Chern-Simons relation, and additionaly to predict the specific quantum modular properties of the proposed Z-hat invariant on positive Seifert manifolds. This prediction is known as the False-Mock conjecture. We find that, in specific cases where we can either regularize the diverging series or we can re-express the Z-hat invariant using inverted Habiro series that the relation can be applied the False-Mock conjecture is satisfied.