### 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

$\to$

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)

$\to$ .

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

$\to$

or to mirror symmetry in

R. Borcherds
*Vertex algebras, Kac-Moody algebras and the monster*

Proc. Nat. Acad. Sci. USA **83** (1986) 3068-3071

$\to$,

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)

$\to$.

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

correctdefinition of vertex algebra.You can find the definition and further discussion in Chapter 3, Section 5 of

Chiral Algebras, or in the last chapter ofVertex Algebras and Algebraic Curves.A factorication algebra is simply a sheaf $F$ of vector spaces on the space of finite subsets of a curve, together with natural isomorphisms ${F}_{S}\otimes {F}_{T}\to {F}_{S\cup T}$ for disjoint $S$ and $T$. 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.