Vertex Algebras et al. - Some Literature
Posted by Urs Schreiber
I need to better understand the theory of vertex algebras. For my own personal convenience, here is a selection of some links.
A good entry point seems to be
Edward Frenkel
Vertex Algebras and Algebraic Curves
Séminaire Bourbaki
52eme année, 1999-2000, no 875
which is a summary of the book
E. Frenkel & D. Ben-Zvi
Vertex Algebras and Algebraic Curves
Mathematical Surveys and Monographs (vol. 88)
AMS (2004)
.
The main point of this entry is that these particular two texts are available online.
This also discusses (in section 6.6 of the first item) the chiral deRham complex, a sheaf of vertex (super)algebras on any smooth scheme (which I recently mentioned here), and points to other stuff that I long to understand, like the application of the chiral deRham complex in the computation of elliptic genera of Calabi-Yau spaces in
L. Borisov, A. Libgober
Elliptic genera of toric varieties and applications to mirror symmetry
Invent. Math 140 (2000) 453-485
or to mirror symmetry in
R. Borcherds
Vertex algebras, Kac-Moody algebras and the monster
Proc. Nat. Acad. Sci. USA 83 (1986) 3068-3071
,
which is actually where vertex algebras (in their mathematical incarnation) have been defined first.
One aspect emphasized by Frenkel is the possibility to formulate vertex algebras in a coordinate independent way. The coordinate independent conception of the operator product expansion is apparently discussed in the book
A. Beilison & V. Drinfeld
Chiral Algebras
Colloquium Publications (2004)
.
There is of course much more literature. This might however suffice for a start…
Re: Vertex Algebras et al. - Some Literature
Factorization algebras give an alternative formulation of vertex algebras that requires very little space, and in that sense may be more intuitive. Ben-Zvi has called this formulation the correct definition of vertex algebra.
You can find the definition and further discussion in Chapter 3, Section 5 of Chiral Algebras, or in the last chapter of Vertex Algebras and Algebraic Curves.
A factorication algebra is simply a sheaf of vector spaces on the space of finite subsets of a curve, together with natural isomorphisms for disjoint and . There are some additional conditions like “no local sections supported on diagonals” and ” quasicoherent”, that make the category of these algebras equivalent to the category of chiral algebras. If your curve is a very small disc, you get a traditional vertex algebra.