In this talk, I will explain how to calculate the $\mathbb{Z}_2$-Betti numbers of $M(\mathbb{R})$. The proof uses a motivic formula for $M$ due to Schiffmann, Mellit, and Fedorov-Soibelman-Soibelman.

In this talk I will describe an explicit correspondence between these two quantizations using a "convergence of polarizations" approach, as pioneered by Mourao, Nunes, and collaborators. This is a limiting process in which holomorphic sections (which can be seen as elements of the Kahler quantization) converge to distributional sections (which can be seen as elements of the real quantization). I will give an overview of this procedure for the case of "regular" fibres, and discuss some of the issues involved in its construction. I will also discuss work in progress to finish the story by finally tackling the case of "singular" fibres.

This is joint work with Hiroshi Konno and work in progress with Megumi Harada.

It has been understood for a while that the end of the moduli space of charge $k$ SU(2) monopoles in $R^3$ decomposes into regions where one has a glueing of monopoles of charges $k_1,..,k_s $; i.e a picture of well separated particles. This approximation is rather rough, in that comparing the metrics only gives an approximation to order 1/R, where R is the separation parameter. Any further improvement requires some form of interaction between the particles. We define spaces of quasi-monopoles, with a separate spectral curve for each charge, and an interaction through their intersection divisors. The spaces of these quasi-monopoles are hyperkahler, and approximates the monopole metric to order $e^{-cR}$. They also have torus actions, which allow a way of finalising Segal-Selby’s proof of the Sen conjecture in the coprime case.

There is a natural non-abelian generalization of ER waves where the metric is non-diagonal any more and the equations of motion become non-linear but integrable in the sense of the theory of solitons. Algebraic quantization of this model using the theory of quadratic quantum algebras of Yangian type was completed in 1990's in the works of H.Nicolai, H.Samtleben, M.Niedermeier and the speaker. However, at that time no unitary representations of these algebras were found. More recently, it was discovered by M.Reisenberger that identical algebras naturally appear (both at the Poisson and quantum level) as building blocks of the light-front approach to the full 4d gravity. Motivated by this result, in our recent work with J.Peraza and M.Reisenberger we established an isomorphism between the quadratic algebra appearing in the non-abelian quantum ER waves and the S-matrix quadratic algebra arising in the Gross-Neveu model. This opens the way to construction of unitary representations of quantum ER algebra, analysis of the energy and metric spectrum with the ultimate goal to apply this machinery to the full 4d quantum gravity.

The talk is based on earlier works with H.Nicolai (AEI Golm) and H.Samtleben (ENS Lyon), and recent work with J.Peraza (Perimeter) and M.Reisenberger (Montevideo).

First, we show that Lisiecki's horizontal polarization is the infinite time limit of the pushforward of the vertical polarization with respect to the geodesic flow for a $G$--invariant Riemannian metric.

Then we lift the geodesic flow to an intertwining unitary parallel transport on the quantum bundle that we call quantum geodesic transform (QGT). Finally we show that the QGT has a well defined limit as the geodesic time goes to infinity and that is equivalent to the FH transform.

On work in collaboration with A.C. Ferreira, J. Hilgert and J.P. Nunes

In this talk, I'll survey results (some joint with T. Baird, and some due to my student A. Davis) on the homotopy type of spaces of flat connections, contrasting the case of surfaces (where Morse Theory for the Yang-Mills functional allows one to analyze flat connections directly) with the situation for higher-dimensional manifolds. In higher dimensions, rational cohomology of $M$ and information about irreducible representations of $\pi_1 M$ can each be leveraged to produce topological information about flat connections.

Time permitting, I'll discuss some ideas for analyzing these phenomena at higher energy (that is, for connections on which the Yang-Mills functional is non-zero), based on work of Atiyah-Bott and Morrison.