Homotopy moment maps

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• 作者：Martin Callies^a ; ^{mcallies@math.uni-bielefeld.de" class="auth_mail" title="E-mail the corresponding author} ; Yaë ; l Fré ; gier^b ; ^c ; ^d ; ^{yael.fregier@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Christopher L. Rogers^e ; ^{chrisrogers@unr.edu" class="auth_mail" title="E-mail the corresponding author} ; ^{chris.rogers.math@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Marco Zambon^f ; ^{marco.zambon@wis.kuleuven.be" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Strong homotopy Lie algebra ; Moment map ; Equivariant cohomology ; Multisymplectic geometry
• 刊名：Advances in Mathematics
• 出版年：2016
• 出版时间：5 November 2016
• 年：2016
• 卷：303
• 期：Complete
• 页码：954-1043
• 全文大小：1169 K

文摘

Associated to any manifold equipped with a closed form of degree >1 is an ‘class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816310192&_mathId=si1.gif&_user=111111111&_pii=S0001870816310192&_rdoc=1&_issn=00018708&md5=de2dd548058562983168e0ba1a19b147" title="Click to view the MathML source">L_∞class="mathContainer hidden">class="mathCode">$L_{\infty }$-algebra of observables’ which acts as a higher/homotopy analog of the Poisson algebra of functions on a symplectic manifold. In order to study Lie group actions on these manifolds, we introduce a theory of homotopy moment maps. Such a map is a class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816310192&_mathId=si1.gif&_user=111111111&_pii=S0001870816310192&_rdoc=1&_issn=00018708&md5=de2dd548058562983168e0ba1a19b147" title="Click to view the MathML source">L_∞class="mathContainer hidden">class="mathCode">$L_{\infty }$-morphism from the Lie algebra of the group into the observables which lifts the infinitesimal action. We establish the relationship between homotopy moment maps and equivariant de Rham cohomology, and analyze the obstruction theory for the existence of such maps. This allows us to easily and explicitly construct a large number of examples. These include results concerning group actions on loop spaces and moduli spaces of flat connections. Relationships are also established with previous work by others in classical field theory, algebroid theory, and dg geometry. Furthermore, we use our theory to geometrically construct various class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816310192&_mathId=si1.gif&_user=111111111&_pii=S0001870816310192&_rdoc=1&_issn=00018708&md5=de2dd548058562983168e0ba1a19b147" title="Click to view the MathML source">L_∞class="mathContainer hidden">class="mathCode">$L_{\infty }$-algebras as higher central extensions of Lie algebras, in analogy with Kostant's quantization theory. In particular, the so-called ‘string Lie 2-algebra’ arises this way.