Max Planck Institute for Mathematics, Bonn, February 2006

Vertex Algebras, Operator Product Expansion, and C2-cofiniteness⋆

Nils Carqueville Uni Bonn

N. Carqueville, M. Flohr, J. Phys. A: Math. Gen. 39 (2006), 951–966, [math-ph/0508015]

Synopsis 

conformal field theory



vertex operator algebras. . .



. . . and related structures



meromorphic operator product expansion



nonmeromorphic operator product expansion



results on triplet W-algebras



C2 -cofiniteness and rationality

Nils Carqueville

Conformal Field Theory 

A two-dimensional conformal field theory⋆ is a projective functor T : C → V (motivated by string theory).



Constructing a general CFT is very difficult.



For g = 0, the problem has been completely solved, building on the notion of vertex operator algebra.



Historically, vertex operator algebras arose in the study of the Monster finite group and infinite-dimensional Lie algebras.

Segal 1987

Nils Carqueville

Vertex Operator Algebras Definition. A vertex operator algebra⋆ is a Z-graded C-vector space a V = V(m) with dimV(m) < ∞ for all m ∈ Z

Z

m∈

together with a linear vertex operator map −1

V −→ (EndV )[[x, x ]] ,

v 7−→ Y (v, x) =

X

Z

vm x−m−1 .

m∈

There are two special elements in V : the vacuum Ω ∈ V(0) and the conformal vector ω ∈ V(2) . The following axioms hold for all u, v ∈ V : (V1)

the truncation condition um v = 0 for all m ≫ 0;

(V2)

the vacuum property Y (Ω, x) = 1V ;

(V3)

the creation property Y (v, x)Ω ∈ V [[x]] and Y (v, x)Ω x=0 = v;

Frenkel, Huang, Lepowsky 1989

Nils Carqueville

Vertex Operator Algebras the Jacobi identity     x1 − x2 x2 − x1 −1 −1 x0 δ Y (u, x1 )Y (v, x2 ) − x0 δ Y (v, x2 )Y (u, x1 ) x0 −x0   x1 − x0 −1 Y (Y (u, x0 )v, x2 ) ; = x2 δ x2

(V4)

(V5)

the modes Lm of the energy momentum operator Y (ω, x) = P −m−2 L x span a representation of the Virasoro algebra m m∈Z c [Lm , Ln ] = (m − n)Lm+n + (m3 − m)δm+n,0 , 12 and the homogeneous subspaces V(m) are exactly the eigenspaces of the operator L0 with eigenvalues m;

(V6)

the L−1 -derivative property

d Y dx

(v, x) = Y (L−1 v, x). Nils Carqueville

Modules for Vertex Operator Algebras Definition. A (generalized) V -module is an R-graded C-vector space a W = W[h] with dimW[h] < ∞ for all h ∈ R

R

h∈

together with a linear vertex operator map −1

V −→ (EndW )[[x, x ]] ,

v 7−→ YW (v, x) =

X

Z

W −m−1 x vm .

m∈

The following axioms hold for all u, v ∈ V and w ∈ W : (M1)

the truncation condition uW m w = 0 for all m ≫ 0;

(M2)

the vacuum property YW (Ω, x) = 1W ;

Nils Carqueville

Modules for Vertex Operator Algebras the Jacobi identity     x1 − x2 x2 − x1 −1 −1 x0 δ YW (u, x1 )YW (v, x2 ) − x0 δ YW (v, x2 )YW (u, x1 ) x0 −x0   x1 − x0 −1 YW (Y (u, x0 )v, x2 ) ; = x2 δ x2

(M3)

(M4)

the modes LW m of the energy momentum operator X −m−2 x YW (ω, x) = LW m

Z

m∈

span a representation of the Virasoro algebra, and the homogeneous subspaces W[h] are exactly the (generalized) eigenspaces of the operator LW 0 with (generalized) eigenvalues h; (M5)

the L−1 -derivative property

d Y (v, x) dx W

= YW (L−1 v, x). Nils Carqueville

Modules for Vertex Operator Algebras ′ ′ Important example: For a V -module W , the structure (W , Y ) defined by ` ∗ with the vertex operator W ′ = h∈R W[h]

V −→ (EndW ′ )[[x, x−1 ]] , X ′ ′ v 7−→ Y (v, x) = x−m−1 , vm

Z

m∈

given by the relation  E D E D 
L Y ′ (v, x)w′ , w = w′ , Y exL1 −x−2 0 v, x−1 w

is the contragredient module, where h · , · i is the natural pairing between W and W ′ . P ′ ′ ′ =⇒ hψm w , wi = hw , ψ−m wi for primary fields m∈Z ψm x−m−wtψ Nils Carqueville

(Logarithmic) Intertwining Operators Definition. Let (Wi , Yi ), (Wj , Yj ) and (Wk , Yk ) be (generalized) V -modules.
k is a linear map A (logarithmic) intertwining operator of type WW i Wj Wi −→ (Hom(Wj , Wk ))[log x]{x} , XX k (w(i) )Ym,a x−m−1 (log x)a . w(i) 7−→ Yij (w(i) , x) =

C a∈N

m∈

The following axioms hold for all v ∈ V , w(i) ∈ Wi and w(j) ∈ Wj : (IO1)

the truncation condition (w(i) )Ym,a w(j) = 0 for all m with Rem ≫ 0, independently of a;

Nils Carqueville

(Logarithmic) Intertwining Operators (IO1)

the truncation condition (w(i) )Ym,a w(j) = 0 for all m with Rem ≫ 0, independently of a;

(IO2)

the Jacobi identity   x1 − x2 −1 x0 δ Yk (v, x1 )Yijk (w(i) , x2 )w(j) x0   x2 − x1 −1 Yijk (w(i) , x2 )Yj (v, x1 )w(j) − x0 δ −x0   x1 − x0 −1 = x2 δ Yijk (Yi (u, x0 )w(i) , x2 )w(j) ; x2

(IO3)

the L−1 -derivative property

d k (w(i) , x) Y dx ij

i = Yijk (LW −1 w(i) , x).

The dimensions of the spaces of all intertwining operators Yijk are called the fusion rules Nijk . Nils Carqueville

Visualization W b b b

Wk Wi

vm

b bW (i+2) bW (i+1) bW (i) b b b b

Yijk YW

b b b

vm

b bV (i+2) bV (i+1) bV (i) b

Wj vertex operator algebra V

b b b

module for V

Ω,ω

vertex operator on W

V

intertwining operator

Nils Carqueville

Meromorphic Operator Product Expansion An essential notion in field theories are correlation functions like hw′ , Y (v1 , x1 )Y (v2 , x2 ) . . . Y (vn , xn )wi . They can be used to compute physical observables. For large n such computations are quite involved. Operator product expansion expresses the product of two fields as the sum of single fields. More precisely, the axioms yield
′ ′ −1 hw , ι−1 hw , Y (v , x )Y (v , x )wi = ι Y (Y (v , x )v , x )wi 1 1 2 2 1 0 2 2 12 20

x0 =x1 −x2

or expanded and shortened

X Y (v1 , x1 )Y (v2 , x2 ) ∼ (x1 − x2 )−i−1 Y ((v1 )i v2 , x2 ) . i≥0

Nils Carqueville

Nonmeromorphic Operator Product Expansion Correlation functions for intertwining operators Yijk are particularly important. Do they also satisfy an operator product expansion of the from
′ ′ −1 hw , Y (Y (u, x )v, x )wi hw , Y (u, x )Y (v, x )wi = ι ι−1 ? 0 2 1 2 20 12 x0 =x1 −x2

Problem: the existence of the product Y1 (w1 , x1 )Y2 (w2 , x2 ) does not guarantee the existence of the iterate Y1 (Y2 (w1 , x1 − x2 )w2 , x2 ).

This problem is solved by P (z)-tensor product theory,⋆ replacing the ordinary vector space tensor product ⊗ by the more complicated operation ⊠P (z) . Motivation: Let W1 , W2 be V -modules, then W1 ⊗ W2 is a module for V ⊗ V , but not for V . W1 ⊠P (z) W2 is a module for V .

Huang, Lepowsky 1995; Huang 1995; Huang 2002; Huang, Lepowsky, Zhang 2003

Nils Carqueville

Nonmeromorphic Operator Product Expansion Theorem. Given two logarithmic intertwining maps Y1 and Y2 of type

W4 M , there exists a logarithmic intertwining map Y of type and W2 W
3 W1 M W4 such that W1 ⊠P (z1 −z2 ) W2 W3

′
 ′ hw4 , Y1 (w1 , z1 )Y2 (w2 , z2 )w3 i = w4 , Y w1 ⊠P (z1 −z2 ) w2 , z2 w3 ,

if the following conditions are satisfied for a full subcategory C of generalized V -modules that is closed under the contragredient functor. (1)

(2) (3)

All generalized V -modules W in ob C are C1 -cofinite, i.e. dim(W/C1 (W )) < ∞ with ` C1 (W ) = span{u−1 w | u ∈ m>0 V(m) , w ∈ W }. All generalized V -modules W in ob C are quasi-finite-dimensional, i.e. dim

‘

m