Week 05.06.2023 – 11.06.2023

We will discuss BPS objects in M theory compactified on a Calabi-Yau three fold X. From the microscopic point of view such degeneracies are encoded in the partition function of the topological strings on X through the Gopakumar-Vafa formula. For the first part of the talk, as an example we will focus on quintic, and discuss how Gopakumar-Vafa invariants can be calculated systematically from the knowledge of boundary condition on the moduli space together with holomorphic ambiguity equation and mirror symmetry. When the entropy thus obtained is plotted against the left moving angular momentum for fixed M2 brane charge, there is a clear transition point at a critical angular momentum. Comparison of the the curve with leading order results from supergravity in 5d shows a large deviation. We will explain the conceptual origin of such deviations using Ooguri-Strominger-Vafa conjecture in 4d string theory though 4d-5d lift. In particular we will observe that the curve is well approximated by the (suitably corrected) entropy of BMPV blackhole for smaller angular momentum and for larger angular momentum by the (suitably corrected) entropy of a particular EEMR blackring. We will show that these observations remain valid on a class of one parameter Calabi-Yau three folds.

The p-adic section conjecture is a long standing conjecture of Grothendieck about curves of high genus over p-adic fields, linking the p-adic points of a curve to sections of a short exact sequence of étale fundamental groups. A powerful way of interpreting the section conjecture is as a fixed point statement, and this interpretation makes the statement look like many other theorems in algebraic topology. For this talk, we'll first introduce the framing of the section conjecture as a fixed point statement, and then show this interpretation allows us to give an alternate proof of part of a result of Pop and Stix towards the section conjecture. This new proof generalises to other fields, and the new fields allow us to extend the original result to a larger class of varieties.