Equivariant cohomology and localization of path integrals TeX Export

@book{citeulike:471868, abstract = { A topological space \$X\$ with the action of a Lie group \$G\$ has equivariant cohomology groups \$H\_G^*(X)\$, which admit various de Rham type models if \$X\$ is a smooth finite-dimensional manifold. This cohomology theory mixes the ordinary cohomology of \$X\$ with the group action in nontrivial ways. Most importantly, equivariant cohomology has a localization formula, due to Atiyah and Bott, which reduces equivariant integrals of equivariantly closed forms to integrals over the fixed point set of the action. This has many applications, the most famous being the reinterpretation and reproof of the Duistermaat-Heckman formula for the exactness of the stationary phase expansion of classical partition functions on symplectic manifolds with circle actions. <P> In the 1980s, various formal proofs of the Atiyah-Singer index theorem based on path integral techniques appeared in the physics literature. These proofs are intriguing but difficult for mathematicians to evaluate, since path integrals have resisted rigorous interpretation for over fifty years. In any case, Atiyah and Witten \ref[M. F. Atiyah, Ast\&eacute;risque No. 131 (1985), 43--59; <A HREF="/msnmain?fn=105\&fmt=doc\&r=1\&pg1=CNO\&s1=816738\&loc=fromrevtext">MR0816738 (87h:58206)</A>] realized that these formal proofs were most cleanly presented as a formal application of Atiyah-Bott/Duistermaat-Heckman localization to path integrals on the infinite-dimensional manifold of loops on \$X\$. Here "formal" means: take results valid in finite dimensions and use them without further justification in infinite dimensions. <P> Physicists view this approach as another instance of the predictive power of path integrals, so formal applications of equivariant cohomology to infinite-dimensional settings of physical interest have continued over the past fifteen years. The driving force here is the search for exactly solvable field theories (called integrable theories in the book), so that perturbative expansions of path integrals can be avoided. Such field theories are closely related to topological quantum field theories, although precise statements are hard to find. <P> The book under review is a thorough exposition of the current state of applying equivariant cohomology to quantum field theory. A large quantity of intriguing material for mathematicians, including index theory, coadjoint orbit and Bott-Borel-Weil theory, modular functors, and Mathai-Quillen formalism, is covered. However, the methods are formal, as this is a text written by a physicist in a mathematical physics series. The style may be difficult for mathematicians: calculations, often in local coordinates, take precedence over statements of theorems; there is often no distinction between a rigorous proof and a formal demonstration; and mathematical and physical concepts are introduced so rapidly that at least half the audience will probably lack the motivational background at any one time. Some details will make mathematicians nervous, such as the implicit but nontrivial switch from Lorentzian to Riemannian signature in \&\#167;4.2, or the statement in \&\#167;2.4 that "a nontrivial vector bundle can always be considered as a trivial one endowed with a nontrivial curvature". Nevertheless, if one takes the attitude that this material may make mathematical sense within the next fifty years, the book can be appreciated as a well-organized exposition of the topological content of quantum field theory from a physics viewpoint. <P> In more detail, after a brief introduction in Chapter 1, the book reviews rigorous equivariant cohomology and Atiyah-Bott localization in Chapter 2. An alternative formulation of localization due to Berline-Vergne is also discussed. In Chapter 3, applications of localization to symplectic geometry are given, including the Duistermaat-Heckman formula and Witten's proposed generalization to nonabelian group actions. <P> In Chapter 4, the book enters the formal realm of path integrals. A nice treatment of the formal proofs of the Atiyah-Singer index theorem is given, although mathematicians will prefer Atiyah's exposition. A formal version, due to Niemi-Tirkkonen, of the Duistermaat-Heckman formula for degenerate functions is used to formally derive Lefschetz-type generalizations of the index theorem. In Chapter 5, Hamiltonian systems whose path integrals can be equivariantly localized are treated. In particular, Niemi-Tirkkonen localization is used to rederive the Kirillov character formula and the Weyl character formula. The phase spaces treated here are simply connected, and Chapter 6 deals with non-simply connected phase spaces, primarily Riemann surfaces. Modular functors appear in this formal setting. Chapter 7 contains a more technical approach to not quite integrable field theories, where the perturbative expansion stops after the one-loop contributions. Chapter 8 discusses the relationship between formal equivariant localization and cohomological/topological field theories. There is a brief discussion of Mathai-Quillen formalism, where the Lagrangians of certain toplogical field theories are seen to be infinite-dimensional versions of Mathai-Quillen forms. Finally, there are brief appendices on BRST quantization and the various models of equivariant cohomology.}, address = {Berlin}, author = {Szabo, Richard J.}, citeulike-article-id = {471868}, citeulike-linkout-0 = {http://www.ams.org/mathscinet-getitem?mr=1762411}, keywords = {cohomology, equivariant, localization, path\_integrals, physics}, mrnumber = {MR1762411}, posted-at = {2006-01-20 02:23:28}, priority = {2}, publisher = {Springer-Verlag}, series = {Lecture Notes in Physics. New Series m: Monographs}, title = {Equivariant cohomology and localization of path integrals}, url = {http://www.ams.org/mathscinet-getitem?mr=1762411}, volume = {63}, year = {2000} }

TY - BOOK ID - citeulike:471868 L3 - citeulike-article-id:471868 N2 - A topological space $X$ with the action of a Lie group $G$ has equivariant cohomology groups $H_G^*(X)$, which admit various de Rham type models if $X$ is a smooth finite-dimensional manifold. This cohomology theory mixes the ordinary cohomology of $X$ with the group action in nontrivial ways. Most importantly, equivariant cohomology has a localization formula, due to Atiyah and Bott, which reduces equivariant integrals of equivariantly closed forms to integrals over the fixed point set of the action. This has many applications, the most famous being the reinterpretation and reproof of the Duistermaat-Heckman formula for the exactness of the stationary phase expansion of classical partition functions on symplectic manifolds with circle actions. <P> In the 1980s, various formal proofs of the Atiyah-Singer index theorem based on path integral techniques appeared in the physics literature. These proofs are intriguing but difficult for mathematicians to evaluate, since path integrals have resisted rigorous interpretation for over fifty years. In any case, Atiyah and Witten \ref[M. F. Atiyah, Ast&eacute;risque No. 131 (1985), 43--59; <A HREF="/msnmain?fn=105&fmt=doc&r=1&pg1=CNO&s1=816738&loc=fromrevtext">MR0816738 (87h:58206)</A>] realized that these formal proofs were most cleanly presented as a formal application of Atiyah-Bott/Duistermaat-Heckman localization to path integrals on the infinite-dimensional manifold of loops on $X$. Here "formal" means: take results valid in finite dimensions and use them without further justification in infinite dimensions. <P> Physicists view this approach as another instance of the predictive power of path integrals, so formal applications of equivariant cohomology to infinite-dimensional settings of physical interest have continued over the past fifteen years. The driving force here is the search for exactly solvable field theories (called integrable theories in the book), so that perturbative expansions of path integrals can be avoided. Such field theories are closely related to topological quantum field theories, although precise statements are hard to find. <P> The book under review is a thorough exposition of the current state of applying equivariant cohomology to quantum field theory. A large quantity of intriguing material for mathematicians, including index theory, coadjoint orbit and Bott-Borel-Weil theory, modular functors, and Mathai-Quillen formalism, is covered. However, the methods are formal, as this is a text written by a physicist in a mathematical physics series. The style may be difficult for mathematicians: calculations, often in local coordinates, take precedence over statements of theorems; there is often no distinction between a rigorous proof and a formal demonstration; and mathematical and physical concepts are introduced so rapidly that at least half the audience will probably lack the motivational background at any one time. Some details will make mathematicians nervous, such as the implicit but nontrivial switch from Lorentzian to Riemannian signature in &#167;4.2, or the statement in &#167;2.4 that "a nontrivial vector bundle can always be considered as a trivial one endowed with a nontrivial curvature". Nevertheless, if one takes the attitude that this material may make mathematical sense within the next fifty years, the book can be appreciated as a well-organized exposition of the topological content of quantum field theory from a physics viewpoint. <P> In more detail, after a brief introduction in Chapter 1, the book reviews rigorous equivariant cohomology and Atiyah-Bott localization in Chapter 2. An alternative formulation of localization due to Berline-Vergne is also discussed. In Chapter 3, applications of localization to symplectic geometry are given, including the Duistermaat-Heckman formula and Witten's proposed generalization to nonabelian group actions. <P> In Chapter 4, the book enters the formal realm of path integrals. A nice treatment of the formal proofs of the Atiyah-Singer index theorem is given, although mathematicians will prefer Atiyah's exposition. A formal version, due to Niemi-Tirkkonen, of the Duistermaat-Heckman formula for degenerate functions is used to formally derive Lefschetz-type generalizations of the index theorem. In Chapter 5, Hamiltonian systems whose path integrals can be equivariantly localized are treated. In particular, Niemi-Tirkkonen localization is used to rederive the Kirillov character formula and the Weyl character formula. The phase spaces treated here are simply connected, and Chapter 6 deals with non-simply connected phase spaces, primarily Riemann surfaces. Modular functors appear in this formal setting. Chapter 7 contains a more technical approach to not quite integrable field theories, where the perturbative expansion stops after the one-loop contributions. Chapter 8 discusses the relationship between formal equivariant localization and cohomological/topological field theories. There is a brief discussion of Mathai-Quillen formalism, where the Lagrangians of certain toplogical field theories are seen to be infinite-dimensional versions of Mathai-Quillen forms. Finally, there are brief appendices on BRST quantization and the various models of equivariant cohomology. SE - Lecture Notes in Physics. New Series m: Monographs AD - Berlin TI - Equivariant cohomology and localization of path integrals PB - Springer-Verlag VL - 63 KW - cohomology KW - equivariant KW - localization KW - path_integrals KW - physics AU - Szabo, Richard PY - 2000/// UR - http://www.ams.org/mathscinet-getitem?mr=1762411 ER -