Celebratio Mathematica — Schwarz

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Works connected to Alain Connes

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A. Connes and A. Schwarz: “Matrix Vieta theorem revisited,” Lett. Math. Phys. 39 : 4 (1997), pp. 349–353. MR 1449580 Zbl 0874.15010 article

A. Connes, M. R. Douglas, and A. Schwarz: “Noncommutative geometry and matrix theory: Compactification on tori,” J. High Energy Phys. 1998 : 2 (1998). article no. 3, 35 pages. MR 1613978 Zbl 1018.81052 ArXiv hep-th/9711162 article

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A. Schwarz: “Noncommutative supergeometry, duality and deformations,” Nuclear Phys. B 650 : 3 (2003), pp. 475–496. To Alain Connes on his 55th birthday. MR 1952770 Zbl 1006.81083 ArXiv hep-th/0210271 article

Following Lett. Math. Phys. 50 (1999) 309, we introduce a notion of \( \mathcal{Q} \)-algebra that can be considered as a generalization of the notion of \( \mathcal{Q} \)-manifold (a supermanifold equipped with an odd vector field obeying \( \{Q,Q\} = 0 \)). We develop the theory of connections on modules over \( \mathcal{Q} \)-algebras and prove a general duality theorem for gauge theories on such modules. This theorem containing as a simplest case \( SO(d,d,\mathbb{Z}) \)-duality of gauge theories on noncommutative tori can be applied also in more complicated situations. We show that \( \mathcal{Q} \)-algebras appear naturally in Fedosov construction of formal deformation of commutative algebras of functions and that similar \( \mathcal{Q} \)-algebras can be constructed also in the case when the deformation parameter is not formal.