Introduction to M(atrix) theory and noncommutative geometry

By A. Konechny, A. Schwarz. 2000

198 citations recorded by the source corpus.

Abstract 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| 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 (Sections 10–12) devoted to solitons and instantons on noncommutative Euclidean space is almost independent of the first part.

Subfields: Mathematical Physics, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/07ea98fe8f70b1e1a085f1d71adcc5d9c5c75f71

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Citations in the corpus

M(atrix) Theory: Matrix Quantum Mechanics as a Fundamental Theory (2001, background): “The first clue that a similar correspondence might exist for D3-branes was the demonstration by Klebanov (1997) that the leading term in a semiclassical calculation of the absorption cross-section of a dilaton s-wave by a system of many 3-branes is precisely reproduced by the (3+1)-dimensional…”
Noncommutative field theory (2001, background): “Such a deformed algebra will be denoted by M θ , so that M 0 = A(M ).”
D-branes and K-theory in 2D topological field theory (2006, background): “It might come as a surprise that the cross-product algebra of spacetime A(X) ×G is not the appropriate Frobenius algebra for G-equivariant topological field theory, in view of the occurrence of the crossed-product algebra as a central concept in the theory of D-branes on orbifolds developed in [49][”
Komaba Lectures on Noncommutative Solitons and D-Branes (2001, background): “More mathematically sophisticated reviews of solitons in noncommutative gauge theory which cover many topics not discussed here can found in [1,2].”
Exact Solution of Quantum Field Theory on Noncommutative Phase Spaces (2003, background): “Despite the enormous amount of activity centered around the study of noncommutative field theories (see [1]–[3] for reviews), some of the most fundamental questions surrounding them have yet to be answered to full satisfaction.”
Dirichlet Branes and Mirror Symmetry (2009, background): “It might come as a surprise that the crossed-product algebra of spacetime A(X) ×G is not the appropriate Frobenius algebra for G-equivariant topological field theory, in view of the occurrence of the crossed-product algebra as a central concept in the theory of D-branes on orbifolds developed in [14”