Seminar series
Date
Mon, 07 May 2018
12:45
Location
L3
Speaker
Dominic Joyce
Organisation
Oxford



 Suppose A is a nice abelian category (such as coherent sheaves coh(X) on a smooth complex projective variety X, or representations mod-CQ of a quiver Q) or T is a nice triangulated category (such as D^bcoh(X) or D^bmod-CQ) over C. Let M be the moduli stack of objects in A or T. Consider the homology H_*(M) over some ring R.
  Given a little extra data on M, for which there are natural choices in our examples, I will explain how to define the structure of a graded vertex algebra on H_*(M). By a standard construction, one can then define a graded Lie algebra from the vertex algebra; roughly speaking, this is a Lie algebra structure on the homology H_*(M^{pl}) of a "projective linear” version M^{pl} of the moduli stack M.
  For example, if we take T = D^bmod-CQ, the vertex algebra H_*(M) is the lattice vertex algebra attached to the dimension vector lattice Z^{Q_0} of Q with the symmetrized intersection form. The degree zero part of the graded Lie algebra contains the associated Kac-Moody algebra.
  The construction appears to be new, but is connected with a lot of work in Geometric Representation Theory, to do with Ringel-Hall-type algebras and their representations, such as the results of Grojnowski-Nakajima on Hilbert schemes. The vertex algebra construction is enormously general, and applies in huge classes of examples. There is a differential-geometric version too.
  The question I am hoping someone in the audience will answer is this: what is the physical interpretation of these vertex algebras?
  It is in some sense an "even Calabi-Yau” construction: when applied to coh(X) or D^bcoh(X), it is most natural for X a Calabi-Yau 2-fold or Calabi-Yau 4-fold, and is essentially trivial for X a Calabi-Yau 3-fold. I discovered it when I was investigating wall-crossing for Donaldson-Thomas type invariants for Calabi-Yau 4-folds. So perhaps one should look for an explanation in the physics of Calabi-Yau 2-folds or 4-folds, with M the moduli space of boundary conditions for the associated SCFT.

 
 
Please contact us with feedback and comments about this page. Last updated on 03 Apr 2022 01:32.