This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).
We approach quaternionic analysis from the point of view of representation theory of the conformal group SL(2,H) and its Lie algebra sl(2,H). While some aspects of representation theory of compact groups were used in quaternionic analysis before, using representation theory of non-compact reductive Lie groups is entirely new. With this representation theoretic approach we intend to 1) Develop split quaternionic analysis which, in particular, will allow us to understand better a subtle aspect of representation theory such as the separation of the discrete and continuous series of unitary representations. 2) Extend the methods of split quaternionic analysis to higher dimensions in the context of (split) Clifford analysis. This theory will yield new results relating solutions of the wave equation, representation theory and index theory of Dirac operators. 3) Study instantons from the point of view of quaternionic analysis. This approach should lead to deep geometric connections with index theory of Dirac operators and also to a direct proof of the completeness of the ADHM construction of instantons. 4) Develop a general theory relating Feynman integrals, homology and representation theory. This theory should explain why many integral formulas appearing in quaternionic analysis often have the form of a Feynman integral taken over various homology cycles, typically non-compact ones. The PI will collaborate on this project with Igor Frenkel.
Quaternionic analysis can be thought of as a higher dimensional analogue of complex analysis. It was significantly developed by R.Fueter in 1930's. Since then quaternionic analysis has generated a lot of interest among mathematicians, some results were extended from complex to quaternionic analysis, there were found many applications to physics, but the theory of quaternionic analysis largely remains underdeveloped. We propose to develop quaternionic analysis from the point of view of representation theory (a branch of mathematics that studies symmetries). This approach has already been proven very fruitful and allowed to push further the parallel with complex analysis and develop a rich theory. It will provide more insight into the relationship between representation theory, geometry, quaternionic analysis, physics and PDE's. Along the way we expect to obtain new results and explicit realizations of representations of reductive Lie groups.
