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SUMMARY:Vertex algebra and Lie algebra structures on the homology of modul
 i spaces - Dominic Joyce (University of Oxford)
DTSTART:20240618T101500Z
DTEND:20240618T111500Z
UID:TALK217369@talks.cam.ac.uk
DESCRIPTION:Let A be a C-linear additive category coming from geometry\, e
 .g. an abelian category or derived category of coherent sheaves on a smoot
 h projective scheme\, or of representations of a quiver with relations. Th
 en we can form two moduli stacks of objects in A: the usual moduli stack M
 \, and the "projective linear" moduli stack M^{pl}\, in which we rigidify 
 M by quotienting out isotropy groups by multiples of the identity. There i
 s a morphism \\pi : M &mdash\;> M^{pl} which is a [*/G_m]-fibration.&nbsp\
 ;&nbsp\; We form the homology H_*(M\,Q)\, H_*(M^{pl}\,Q)\, by which we mea
 n the ordinary homology of the topological classifying space of the stacks
 .&nbsp\; I will explain how under very weak assumptions (basically we need
  total Ext groups Ext^*(E\,F) to be finite-dimensional for all E\,F in A) 
 we can give H_*(M\,Q) the structure of a graded vertex algebra\, and H_*(M
 ^{pl}\,Q) the structure of a graded Lie algebra (which is that constructed
  from the vertex algebra H_*(M\,Q) by Borcherds).&nbsp\;\nThese vertex and
  Lie algebras appear naturally in the study of enumerative invariants (e.g
 . of semistable coherent sheaves on projective surfaces\, or Fano 4-folds)
 \, and this was how I found them.&nbsp\;&nbsp\;\nThe vertex algebras can b
 e computed explicitly in a lot of cases\, and usually turn out to be some 
 form of super-lattice vertex algebra. In particular\, by work of my studen
 t Jacob Gross\, if A = D^b coh(X) for X a projective curve\, surface\, or 
 smooth toric variety\, then we can write down H_*(M\,Q)\, H_*(M^{pl}\,Q) w
 ith their vertex and Lie algebra structures completely explicitly (the for
 mulae are complicated).&nbsp\;\nThe vertex algebra construction admits a l
 ot of interesting generalizations. For example\, there is a version for eq
 uivariant homology if a group G acts on A\, M\, M^{pl}. For complex-orient
 ed generalized homology theories\, such as K-theory\, one can define a var
 iant of vertex algebra defined using the formal group law attached to the 
 generalized homology theory. Work by my student Chenjing Bu defines "twist
 ed" representations of a vertex algebra on homology of moduli spaces of se
 lf-dual objects in a self-dual abelian category &nbsp\;(e.g. vector bundle
 s with orthogonal or symplectic structures). For &ldquo\;odd Calabi-Yau" c
 ategories A there is a variant which produces a vertex Lie algebra.&nbsp\;
LOCATION:External
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