Workshop: Non-commutative geometry, scattering theory and the Witten index

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Program

On January 31 - February 1 there will be four introductory talks discussing the background to the problems described below. These will be accessible to advanced graduate students and postdoctoral fellows. There will be a program of advanced research seminars developed by the organisers for February 2 - February 4.

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Scientific background

Over twenty years ago Bolle, Gesztesy, Grosse, Schweiger, and Simon showed that under certain conditions on the Hamiltonian of a supersymmetric quantum system, there is a regularized Witten index and it equals minus the jump in the Krein spectral shift function from quantum scattering theory. The latter was shown to be precisely the spectral flow in [1]. There is also a remarkable parallel between the resolution of I.M. Singer's problem on representing spectral flow as an integral of a one form in the paper [3] and Witten index computations. Both involve relative trace compatible perturbations.

Great progress in the interaction between index theory, spectral flow and the Krien spectral shift function using the new method of double operator integration has just been made [4]. This is also sufficient to understand the Witten index in the Fredholm case. The issue unresolved at this time is the meaning of the quantum mechanical Witten index in the non-Fredholm case. It is real valued and so not a Fredholm index in the usual sense. However it may be a semifinite index and may be computable using the methods of non-unital semifinite noncommutative geometry [2].

The initial and important step is to prove non-unital index theorems using the techniques of noncommutative geometry and this is now well under way in a preprint being written by Carey, Gayral, Rennie, and Sukochev. They have found a conjectural symbol algebra for generalised Toeplitz operators in the non-unital setting [2] and are using this to derive a non-unital local index formula. The second step is to compare the Witten index with (still conjectural) non-unital McKean-Singer formulae and a known non-unital spectral flow formula under study by Carey and Rennie.

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The timing of the workshop

These investigations are likely to be more solid by the time of the proposed workshop. It thus seems to be an optimal moment to have some key researchers from the field meet to exchange ideas and to initiate collaborations. At the same time it will provide an opportunity for researchers, particularly junior fellows, to be exposed to these ideas.

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References

[1]   N. Azamov, A.L. Carey, and F.A. Sukochev, The spectral shift function and spectral flow, Comm. Math. Phys. 276, no.1, 51-91; and N.A. Azamov, A.L. Carey, P.G. Dodds, and F.A. Sukochev, Operator integrals, spectral shift, and spectral flow, Canad. J. Math. 61, 241-263.

[2]   A.L. Carey, V. Gayral, A. Rennie, and F. Sukochev, Integration on locally compact noncommutative spaces, arXiv:0912.2817.

[3]   A.L. Carey, F. A. Sukochev, and D. Potapov, Spectral flow is the integral of one forms on the Banach manifold of self adjoint Fredholm operators, Advances in Mathematics 222 (2009), 1809-1849.

[4]   F. Gesztesy, Y. Latushkin, K.A. Makarov, F. Sukochev, and Y. Tomolov, The index formula and spectral shift function for relatively trace class perturbations, arXiv:1004.1582.

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