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SUMMARY:A Ringel-Hall type construction of vertex algebras - Dominic Joyce
 \, Oxford
DTSTART:20181107T141500Z
DTEND:20181107T151500Z
UID:TALK110494@talks.cam.ac.uk
CONTACT:Mark Gross
DESCRIPTION:Vertex algebras” are complicated algebraic structures coming
  from Physics\,\nwhich also play an important role in Mathematics in areas
  such as monstrous\nmoonshine and geometric Langlands.\nI will explain a n
 ew geometric construction of vertex algebras\, which seems to\nbe unknown.
  The construction applies in many situations in algebraic geometry\,\ndiff
 erential geometry\, and representation theory\, and produces vast numbers 
 of\nnew examples. It is also easy to generalize the construction in severa
 l ways to\nproduce different types of vertex algebra\, quantum vertex alge
 bras\,\nrepresentations of vertex algebras\, … It seems to be related to
  work by\nGrojnowski\, Nakajima and others\, which produces representation
 s of interesting\ninfinite-dimensional Lie algebras on the homology of mod
 uli schemes such as\nHilbert schemes.\nSuppose A is a nice abelian categor
 y (such as coherent sheaves coh(X) on a\nsmooth complex projective variety
  X\, or representations mod-CQ of a quiver Q)\nor T is a nice triangulated
  category (such as D^b coh(X) or D^bmod-CQ) over C.\nLet M be the moduli s
 tack of objects in A or T\, as an Artin stack or higher\nstack. Consider t
 he homology H_*(M) over some ring R.\nGiven a little extra data on M\, for
  which there are natural choices in our\nexamples\, I will explain how to 
 define the structure of a graded vertex algebra\non H_*(M). By a standard 
 construction\, one can then define a graded Lie algebra\nfrom the vertex a
 lgebra\; roughly speaking\, this is a Lie algebra structure on\nthe homolo
 gy H_*(M^{pl}) of a “projective linear” version M^{pl} of the moduli\n
 stack M.\nFor example\, if we take T = D^bmod-CQ\, the vertex algebra H_*(
 M) is the lattice\nvertex algebra attached to the dimension vector lattice
  Z^{Q_0} of Q with the\nsymmetrized intersection form. The degree zero par
 t of the graded Lie algebra\ncontains the associated Kac-Moody algebra.\nT
 here is also a differential-geometric version\, involving putting a vertex
 \nalgebra structure on homology of moduli stacks of connections on a compa
 ct\nmanifold  X equipped with an elliptic complex E.
LOCATION:CMS MR13
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