Celebratio Mathematica

A. Konechny and A. Schwarz: “On \( (k\oplus l|g) \)-dimensional supermanifolds,” pp. 201–206 in Supersymmetry and quantum field theory (Kharkov, Ukraine, 5–7 January 1997). Edited by J. Wess and V. P. Akulov. Lecture Notes in Physics 509. 1998. Proceedings of the D. Volkov Memorial Seminar. MR 1677320 Zbl 0937.58001 incollection

Dmitri V. Volkov	Related
Vladimir P. Akulov	Related
Julius Wess	Related
Anatoly Vladimirovich Konechny	Related

A. Konechny and A. Schwarz: “BPS states on non-commutative tori and duality,” Nucl. Phys., B 550 : 3 (1999), pp. 561–584. MR 1693262 Zbl 0949.58004 ArXiv hep-th/9811159 article

A. Konechny and A. Schwarz: “\( 1/4 \)-BPS states on noncommutative tori,” J. High Energy Phys. 1999 : 9 (1999). article no. 30, 15 pages. MR 1720696 Zbl 0957.81086 ArXiv hep-th/9907008 article

A. Konechny and A. Schwarz: “Moduli spaces of maximally supersymmetric solutions on non-commutative tori and non-commutative orbifolds,” J. High Energy Phys. 2000 : 9 (2000), pp. article no. 005, 24 pages. MR 1789106 Zbl 0989.81623 ArXiv hep-th/0005174 article

A maximally supersymmetric configuration of super Yang–Mills living on a non-commutative torus corresponds to a constant curvature connection. On a non-commutative toroidal orbifold there is an additional constraint that the connection be equivariant. We study moduli spaces of (equivariant) constant curvature connections on non-commutative even-dimensional tori and on toroidal orbifolds. As an illustration we work out the cases of \( \mathbb{Z}_2 \) and \( \mathbb{Z}_4 \) orbifolds in detail. The results we obtain agree with a commutative picture describing systems of branes wrapped on cycles of the torus and branes stuck at exceptional orbifold points.

A. Konechny and A. Schwarz: “Compactification of M(atrix) theory on noncommutative toroidal orbifolds,” Nuclear Phys. B 591 : 3 (2000), pp. 667–684. MR 1797572 Zbl 1042.81580 ArXiv hep-th/9912185 article

A. Konechny and A. Schwarz: “Theory of \( (k\oplus l|q) \)-dimensional supermanifolds,” Selecta Math. (N.S.) 6 : 4 (2000), pp. 471–486. MR 1847384 Zbl 1001.58001 article

A. Konechny and A. Schwarz: “Introduction to M(atrix) theory and noncommutative geometry,” Phys. Rep. 360 : 5–6 (2002), pp. 353–465. MR 1892926 Zbl 0985.81126 ArXiv hep-th/0012145 article

Noncommutative geometry is based on an idea that an associative algebra can be regarded as “an algebra of functions on a noncommutative space”. The major contribution to noncommutative geometry was made by A. Connes, who, in particular, analyzed Yang–Mills theories on noncommutative spaces, using important notions that were introduced in his papers (connection, Chern character, etc). It was found recently that Yang–Mills theories on noncommutative spaces appear naturally in string/M-theory; the notions and results of noncommutative geometry were applied very successfully to the problems of physics.

In this paper we give a mostly self-contained review of some aspects of M(atrix) theory, of Connes’ noncommutative geometry and of applications of noncommutative geometry to M(atrix) theory. The topics include introduction to BFSS and IKKT matrix models, compactifications on noncommutative tori, a review of basic notions of noncommutative geometry with a detailed discussion of noncommutative tori, Morita equivalence and \( SO(d,d|\mathbb{Z}) \)-duality, an elementary discussion of noncommutative orbifolds, noncommutative solitons and instantons. The review is primarily intended for physicists who would like to learn some basic techniques of noncommutative geometry and how they can be applied in string theory and to mathematicians who would like to learn about some new problems arising in theoretical physics.

The second part of the review devoted to solitons and instantons on noncommutative Euclidean space is almost independent of the first part.

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