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 Ln's defined by L(z)=Σn∈ℤLnz-n-2:=1/2 :α(z)α(z): satisfy the relation [Ln,Lm] = (n-m)Ln+m+(n3-n)/12 δn,-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 czn, cwn 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 Resz-ww,z-wf(z,w))= Reszz,wf(z,w))- Reszw,zf(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 1V 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)= Σn∈ ℤSn z-n-2 as given in the lecture we have S-1Jb-1vk= (hv+k)Jb-2vk. 2. Show using the OPE for S(z)Ja(w) that [Sn, Jam]= -(k+hv)mJan+m. Recall that if a(z)b(w)~Σj=0N-1cj(w)/(z-w)j+1, then [a(m), b_(n)]= ΣjN-1{m choose j} cj(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