In fact, it turns out that noncommutative geometry also provides a con. Noncommutative geometry and string field theory sciencedirect. The central theme of this thesis is noncommutativity in string theory. Foliation theory and atiyahsinger index theory, on singular spaces, or parametrized by singular spaces.
They play a pivotal role in the emerging field of quantum computation. The noncommutative geometry of yangmills elds 9 10. Schematically, noncommutative geometry proceeds in three steps. Nathan seiberg and edward witten jhep091999032 view the article online for updates and enhancements. Related content dbranes on the quintic ilka brunner, michael r. Pdf string theory and noncommutative geometry zengping. Alain connes, noncommutative geometry, academic press, 1995 jos e m. Manifold topology progress on the novikov conjecture, gromovlawson conjecture, etc. String theory and noncommutative geometry to cite this article. The technical background of connes program is the theory of operator algebras on hilbert space, closed in some topology. Part of the beauty of the analysis in 4 was that t duality acts within the non commutative yangmills framework. Douglas1,2 and albert schwarz1,2,3 1 institut des hautes etudes scienti.
String theory, matrix model, and noncommutative geometry. Noncommutative geometry has also been used as a framework for open string field theory 11. Noncommutative geometry of the standard model and beyond. We identify a limit in which the entire string dynamics is described by a minimally coupled supersymmetric gauge theory on a noncommutative space, and discuss the corrections away from this limit. Product of the continuum by the discrete and the symmetry breaking mechanism. Why should operator algebras be the vessel of noncommutative geometry. Concepts will be emphasized and technical details avoided contents 1. In mathematical physics, noncommutative quantum field theory or quantum field theory on noncommutative spacetime is an application of noncommutative mathematics to the spacetime of quantum field theory that is an outgrowth of noncommutative geometry and index theory in which the coordinate functions are noncommutative.
A course on noncommutative geometry in string theory. Varilly, hector figueroa, elements of noncommutative geometry, birkh auser, 20 22. School of natural sciences institute for advanced study olden lane, princeton, nj 08540 we extend earlier ideas about the appearance of noncommutative geometry in string theory with a nonzero b eld. Nuclear physics b268 1986 253294 northholland publishing company noncommutative geometry and string field theory edward witten joseph henry laboratories, princeton university, princeton, new jersey 08540, usa received 2 december 1985 an attempt is made to interpret the interactions of bosonic open strings as defining a noncommutative, associative algebra, and to formulate the classical.
The original application of noncommutative yangmills theory to string theory 2. Isbn 9155460194 this thesis describes some aspects of noncommutative geometry and conformal. Very basic noncommutative geometry vanderbilt university. Positivity in hochschild cohomology and the inequalities for the yangmills action. For example, the fact that the position and momentum operators do not commute for a quantum mechanical point particle, implies the famous he. Commutative algebraic geometry, restricts attention to spaces whose local description is via commutative rings and algebras, while noncommutative algebraic geometry allows for more general local or affine models. Spin geometry, princeton 1989 john roe, elliptic operators, topology, and asymptotic methods, crc press, 1999 spin geometry and ncg, dirac and distance. Damtp9836 dsf1698 outp9842p hepth9806099 june 1998 revised april 1999 string geometry and the noncommutative torus giovanni landi1 department of applied mathematics and th. The cycles in khomology are given by fredholm representations of the calgebra a of continuous functions. Noncommutative lattices are described at length in section 3. An introduction to twisted ktheory 9, why it appears in physics, and what it has to do with noncommutative topology. Interesting examples of noncommutative spaces abound, and they are thoroughly discussed in connes book.
Friendly introduction to the concepts of noncommutative geometry. As a formulation of mtheory, matrix theory must describe string theory when compactified on a circle. Spectral triple obtained from an algebra bundle 9 10. We identify a limit in which the entire string dynamics is described by a minimally coupled supersymmetric gauge theory on a. Models in physics notably of the quantum hall effect. Subsequently i will discuss whether such a supersymmetric theory can be given a geometrical underpinning within the framework of noncommutative geometry. At any rate, i do not think that the fundamental importance of noncommutative geometry can be overestimated. Connes theory of spectral triples and noncommutative riemannian geometry is based on the theory of dirac operators which was invented by atiyah and singer to prove the index theorem. Heisenberg and the noncommutative algebra of physical quantities 40 2.
For example, by a celebrated theorem of gelfand and naimark 91, one knows. Its development, mainly due to alain connes, is providing an increasing number of applications and deeper insights for instance in foliations, k theory, index theory, number theory but also in quantum physics of elementary particles. Undeniably however the connection with string theory gave a strong impulse to the study of noncommutative geometry and to field theories on. Noncommutative geometry, gauge theory and renormalization. Comprehensive summaries of uppsala dissertations from the faculty of science and technology 1004. A particular and most prominent realization of the program of noncommutative geometry has been lead by alain connes. An important connection between ktheory, topology, geometric group theory and noncommutative geometry is given by the isomorphism conjectures, such as those due to baumconnes, bostconnes and farrelljones, which predict certain topological formulas for various kinds of ktheory of a crossed product, in both the c algebraic. After all, the gelfandnaimark theorem is not about operators, and, according to. We extend earlier ideas about the appearance of noncommutative geometry in string theory with a nonzero bfield.
String theory and noncommutative geometry iopscience. Noncommutative algebraic geometry is a branch of mathematics, and more specifically a direction in noncommutative geometry, that studies the geometric properties of formal duals of noncommutative algebraic objects such as rings as well as geometric objects derived from them e. Noncommutative geometry builds on, and vastly extends, this fundamental duality between classical geometry and commutative algebras. This is for instance the origin of the role of noncommutative geometry in twisted ktheory.
Mar 11, 20 maxim kontsevich wallcrossing and geometry at infinity of betti moduli spaces duration. Noncommutative geometry ncg is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of spaces that are locally presented by noncommutative algebras of functions possibly in some generalized sense. Noncommutative geometry has also been used as a framework for open string. Aug 07, 2017 in quantum mechanics, when two operators do not commute, it leads to an uncertainty relation between them. We explore in detail how noncommutative structures can emerge in case of the interacting bosonic string and even in the. The metric aspect of noncommutative geometry 34 chapter 1. What is the relationship between noncommutative geometry and. Enumerative geometry and string theory pdf free download. The noncommutative geometry of electrodynamics 127 9. Pdf string theory and noncommutative geometry semantic. Dbranes and k theory edward witten string primer enrique alvarez and patrick meessen.
Noncommutative geometry is one of the most deep and vital research subjects of presentday mathematics. Some applications of noncommutative topology and noncommutative geometry that have appeared in the recent physics literature. Connes theory, which is generally known as noncommutative geometry, is a rapidly growing new area of mathematics that interacts with and contributes to many disciplines in mathematics and physics. Noncommutative geometry lecture 1 elliptic operators and index problems nigel higson penn state university july, 2002. The point of departure is the brst approach to string field theory. Noncommutative geometry is the study of noncommutative algebras as if they were algebras of functions on spaces, like the commutative algebras associated to a ne algebraic varieties, di erentiable manifolds, topological spaces, and measure spaces. Noncommutative algebraic geometry is the study of spaces represented or defined in terms of algebras, or categories. Institut des hautes etudes scientifiques ihes 2,669 views 1. Ktheory and noncommutative geometry lecture 1 elliptic. String theory and noncommutative gauge theory semantic scholar. Strings, conformal field theory and noncommutative geometry. This is the first installment of a paper in three parts, where we use noncommutative geometry to study the space of commensurability classes of qlattices and we show that the arithmetic.
Maxim kontsevich wallcrossing and geometry at infinity of betti moduli spaces duration. What is the significance of noncommutative geometry in. Techniques from noncommutative geometry have been used to constructs models of gauge theory on these noncommutative lattices 4, 5. For other thoughts about applications of noncommutative geometry in physics, see e. Pdf string theory and noncommutative geometry nathan. After a general introduction to the concept of supersymmetry, i will explain why it can provide promising candidates for viable theories beyond the standard model. First of all, anyone who does quantum theory is doing noncommutative. In this book, we discuss several types of geometric objects in the usual sense of. Friendly introduction to the concepts of noncommutative geometry notes pour le s. Part of the beauty of the analysis in 4 was that tduality acts within the noncommutative yangmills framework, rather than, as one might expect, mixing the modes of noncommutative yangmills theory with string winding states and other stringy ex. An introduction to noncommutative spaces and their geometry. Grassmann variables, grassmann integration and pfa ans 7 notes 8 chapter 10. Le boismarie, buressuryvette, 91440 france 2 department of physics and astronomy rutgers university piscataway, nj 088550849 usa 3 department of mathematics.
A basic example is the group ring of a discrete group. It comes from the application of the noncommutative geometry ncg techniques pioneered by a. I guess my point with all of this is that all the esoteric machinery of ncg seems less artificial when viewed through the lens of index theory. Statistical state of a macroscopic system and quantum statistical mechanics 45 3.
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