**Vertex algbra seminar**

Spring 2012. 4 ects.

Run by André Henriques and Johan van de Leur.

This seminar meets every Thursday at 13:15 in room 610.

*Schedule:*

**Johan** (Feb 9th): We will describe some of the most prominent examples of Lie algebras intimately related to VOAs: the Heisenberg algebra, the current algebra (central extension of the loop Lie algebra), and the Virasoro algebra.
*Hand in exercise: Show that the L*_{n}'s defined by L(z)=Σ_{n∈ℤ}L_{n}z^{-n-2}:=1/2 :α(z)α(z): satisfy the relation [L_{n},L_{m}] = (n-m)L_{n+m}+(n^{3}-n)/12 δ_{n,-m}.

**Simen** (Feb 16th): Formal distributions (=fields) will be introduced. These are formal
power series with values in End(V), where V is a vector space.
Locality: this is a weakening of the notion of commutativity, that
applies to formal distributions.
Finally, the operator product expansion and the normally ordered
product will be introduced.

*
Hand in exercise:
(alles verwijst naar H2 in het boek van Kac)
1) Gegeven formule (2.2.7), leidt formule (2.2.8) af (de a*_{(m,n)} zijn er net boven gedefinieerd).
2) Gegeven formules (2.2.10) en (2.2.11) (die op het college zijn bewezen/genoemd), leidt de uitdrukkingen voor c_{z}^{n}, c_{w}^{n} en ~c in opmerking 2.2a af.
3) Gegeven (i), (ii) en (iii) van stelling 2.3, leidt (iv), (v) en (vi) van dezelfde stelling af.
[Noz: chapter 1][FBZ: chapter 1][Kac: chapter 2]

**Julian** (Feb 23rd): Taylor's formula. How to write commutation relations in terms of
fields or terms of Fourier modes.
Examples: Heisenberg VOA, the affine VOA, and the Virasoro VOA.
The notion of conformal weight of a field.
[Noz: section on examples][Kac: section 2.4]

**Joost N.** (Mar 1st): Central extensions of Lie algebras. Lie algebra cohomology, with
special emphasis on H^2.
The computationof H^2(Witt Lie algebra) and H^2(loop Lie algebra) and
the corresponding central extensions:
Virasoro and affine lie algebra.
[paper by Goddard & Olive][...?...]

**André** (Mar 8th): Definition of VOA. The notation of "lambda-bracket". The Wightmann
axioms for quantum field theory.
*Hand in exercise: Prove the formal Cauchy formula:
if f(z,w) is in ℂ[[z,w]][z*^{-1}, w^{-1}, (z-w)^{-1}] then
Res_{z-w}(ι_{w,z-w}f(z,w))=
Res_{z}(ι_{z,w}f(z,w))-
Res_{z}(ι_{w,z}f(z,w))
[Kac: chapter 1][FBZ: section 1.3]

**Ralph** (Mar 15th): Tricks of the trade for proving the something is a VOA: Dong's
Lemma, the reconstruction theorem.
Proof that Heisenberg is a VOA.
*Hand in exercise: Show how a commutative (non-conformal) VOA gives rise to a Z*_{+}-graded commutative associative algebra V with unit 1_{V} and a derivation T of degree 1, and conversely that such an algebra V carries a canonical (non-conformal) VOA structure. - Show that L(z) = 1/2 : b(z) b(z) : gives rise to a Virasoro element for the Heisenberg VOA. Hint: note how omega can be recovered via L_{-2} |0⟩ = ω.
[Noz: section 3.5][Kac section 3.2][FBZ section 2.3]

**Shan** (Mar 29th): The Verma module for the loop Lie algebra and its irreducible quotient.
Proof that they are VOAs
*Hand in exercise: Laat zien: Zij I een ideaal in een VOA V. Dan I is gegradeed. D.w.z., I=⊕ I∩ V_n.* [FBZ: section 4.4]

**Jules** (Apr 5th):
Representation theory in finite dimensions: Cartan subalgebra, triangular degomposition of *g*, Chevalley generators of *g*, Serre relations, highest weight vectors, Verma modules
[Humphreys chapt 6] [EFK] [Fulton & Harris]

**Joost B.** (Apr 12th): The Sugawara construction, or, in other words, how to embed the Virasoro VOA
into (the completion of) the affine and Heisenberg VOAs.
*Hand in exercises: 1. Show that for S(z)= Σ*_{n∈ ℤ}S_{n} z^{-n-2} as given in the lecture we have S_{-1}J^{b}_{-1}v_{k}= (h^{v}+k)J^{b}_{-2}v_{k}.
2. Show using the OPE for S(z)J^{a}(w) that [S_{n}, J^{a}_{m}]= -(k+h^{v})mJ^{a}_{n+m}.
Recall that if a(z)b(w)~Σ_{j=0}^{N-1}c^{j}(w)/(z-w)^{j+1}, then [a_{(m)}, b__{(n)}]= Σ_{j}^{N-1}{m choose j} c^{j}_{(m+n-j)}.
[Kac 3.5][FBZ: sections 2.5.10 and 3.4.8]

**Reinier** (Apr 19th):
Representation theory of affine Lie algberas (& Virasoro?), Verma modules, Weyl modules, integrable representations.
[EFK 2.6 - 2.8] [ArXiv:0709.0105]
[Kass, Moody, Patera, Slansky: Affine Lie algebras, weight multiplicities, and branching rules. Vol. 1]

**Shan** (Apr 26th):
Lattice VOAs & isomorphism with affine VOA (and rep theory of lattice VOAs?)
[Kac 5.5, 5.6] [..?..]

**André** (May 3rd):
Bundles of VOAs [FBZ chap 6]

**Ralph** (May 10th):
Conformal blocks, proof of finite dimensionality [BK Section 7.3][ArXiv:0507086] [FBZ Chap. 9.1 - 9.3]
*Hand in exercise: Look up the precise definition of a quasi-conformal vertex algebra [FBZ, Definition 6.3.4] and show that any conformal vertex algebra which is ℤ-graded with its gradation bounded from below is quasi-conformal. Then discuss which of the vertex algebras known to you are quasi-conformal (i.e. Heisenberg, affine / Kac-Moody, Virasoro, commutative).*

**Joost B.** (May 24th):
The Knizhnik-Zamolodchikov connection on the bundle of conformal blocks: case of the Heisenberg VOA
[FBZ 13.1 - 13.2]

**Joost N.** (May 31st):
The Knizhnik-Zamolodchikov connection on the bundle of conformal blocks: case of the affine VOA [FBZ]

**References:**

[Noz]:Christophe NOZARADAN, "Introduction to Vertex Algebras".
Available at http://arxiv.org/pdf/0809.1380

[FBZ]: E. Frenkel, D. Ben-Zvi, "Vertex algebras and algebraic curves".
Second edition.

[Kac]: Victor Kac, "Vertex algebras for beginners".
Second edition. University Lecture Series, 10.
[EFK]: Etingof-Frenkel-Kirillov, "Lectures on Representation theory and Knizhnik-Zamolodchikov Equations".
[BK]: book by Bakalov and Kirillov