## March 14, 2006

### 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…

Posted at March 14, 2006 7:47 PM UTC

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### 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 $F$ 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.

Posted by: Scott on March 15, 2006 11:04 PM | Permalink | Reply to this

### Re: Vertex Algebras et al. - Some Literature

James Lepowksy, Haisheng Li
Introduction to Vertex Operator Algebras and Their Representations
Birkhauser 2003

I don’t know the subject well enough to say how well it covers the material, but the exposition was enjoyable and clear.

Posted by: Greg on March 16, 2006 3:09 AM | Permalink | Reply to this

### Re: Vertex Algebras et al. - Some Literature

Scott and Greg,

many thanks for this information. I’ll take a look at this stuff as soon as possible. (Right now I am preparing for a vacation, though…)

Posted by: urs on March 16, 2006 4:00 PM | Permalink | Reply to this
Read the post CFT in Oberwolfach
Weblog: The n-Category Café
Excerpt: An Oberwolfach meeting on conformal field theory.
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