Vertex Operator Algebra Approach to Logarithmic Conformal Field Theory

Integrable Systems and Quantum Symmetries, Prague, June 2006

Vertex Operator Algebra Approach to

Logarithmic Conformal Field Theory

Nils Carqueville

Bonn University

Introduction and Synopsis

Nils Carqueville


Nils Carqueville

Logarithmic conformal field theory has indecomposable structure


Nils Carqueville

Logarithmic conformal field theory has indecomposable structure(and logarithms, too. . . )


Nils Carqueville


Study LCFT from an algebraic point of view:


Nils Carqueville



vertex operator algebras, (generalized) modules and intertwining operators


Nils Carqueville




nonmeromorphic OPE with P (z)-tensor product theory


Nils Carqueville





results on triplet algebras


Nils Carqueville






C2-cofiniteness


Nils Carqueville






C2-cofiniteness

logarithmic mode algebras

Vertex Operator Algebras

Nils CarquevilleFrenkel, Huang, Lepowsky 1989

Definition. A vertex operator algebra⋆ is a Z-graded C-vector space

V =∐

m∈ZV(m) with dimV(m) < ∞ for all m ∈ Z

Z

1




V =∐

m∈ZV(m) with dimV(m) < ∞ for all m ∈ Ztogether with a linear vertex operator map

V −→ (EndV )[[x, x−1]] , v 7−→ Y (v, x) =∑

m∈Z vmx−m−1 .

1




V =∐


V −→ (EndV )[[x, x−1]] , v 7−→ Y (v, x) =∑

m∈Z vmx−m−1 .

There are two special elements in V : the vacuum Ω ∈ V(0) and theconformal vector ω ∈ V(2).

1




V =∐


V −→ (EndV )[[x, x−1]] , v 7−→ Y (v, x) =∑

m∈Z vmx−m−1 .

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

1




V =∐


V −→ (EndV )[[x, x−1]] , v 7−→ Y (v, x) =∑

m∈Z vmx−m−1 .


(V1) the truncation condition umv = 0 for all m ≫ 0;

1




V =∐


V −→ (EndV )[[x, x−1]] , v 7−→ Y (v, x) =∑

m∈Z vmx−m−1 .



(V2) the vacuum property Y (Ω, x) = 1V ;




V =∐


V −→ (EndV )[[x, x−1]] , v 7−→ Y (v, x) =∑

m∈Z vmx−m−1 .



(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;


Nils Carqueville

(V4) the Jacobi identity

x−10 δ

(x1 − x2

x0

)Y (u, x1)Y (v, x2) − x−1

0 δ

(x2 − x1

−x0

)Y (v, x2)Y (u, x1)

= x−12 δ

(x1 − x0

x2

)Y (Y (u, x0)v, x2) ;

Z


Nils Carqueville


x−10 δ

(x1 − x2

x0

)Y (u, x1)Y (v, x2) − x−1

0 δ

(x2 − x1

−x0

)Y (v, x2)Y (u, x1)

= x−12 δ

(x1 − x0

x2

)Y (Y (u, x0)v, x2) ;

(V5) the modes Lm of the energy momentum operator Y (ω, x) =∑m∈Z Lmx−m−2 span a representation of the Virasoro algebra

[Lm, Ln] = (m − n)Lm+n +c

12(m3 − m)δm+n,0 ,

and the homogeneous subspaces V(m) are exactly the eigenspaces ofthe operator L0 with eigenvalues m;


Nils Carqueville


x−10 δ

(x1 − x2

x0

)Y (u, x1)Y (v, x2) − x−1

0 δ

(x2 − x1

−x0

)Y (v, x2)Y (u, x1)

= x−12 δ

(x1 − x0

x2

)Y (Y (u, x0)v, x2) ;

(V5) the modes Lm of the energy momentum operator Y (ω, x) =∑m∈Z Lmx−m−2 span a representation of the Virasoro algebra

[Lm, Ln] = (m − n)Lm+n +c

12(m3 − m)δm+n,0 ,

and the homogeneous subspaces V(m) are exactly the eigenspaces ofthe operator L0 with eigenvalues m;

(V6) the L−1-derivative property ddx

Y (v, x) = Y (L−1v, x).

Modules for Vertex Operator Algebras

Nils Carqueville

Definition. A (generalized) V -module is an R-graded C-vector space

W =∐

h∈RW[h] with dimW[h] < ∞ for all h ∈ R

Z

1


Nils Carqueville


W =∐

h∈RW[h] with dimW[h] < ∞ for all h ∈ Rtogether with a linear vertex operator map

V −→ (EndW )[[x, x−1]] , v 7−→ YW (v, x) =∑

m∈Z vWm x−m−1 .

1


Nils Carqueville


W =∐


V −→ (EndW )[[x, x−1]] , v 7−→ YW (v, x) =∑


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

1


Nils Carqueville


W =∐


V −→ (EndW )[[x, x−1]] , v 7−→ YW (v, x) =∑



(M1) the truncation condition uWm w = 0 for all m ≫ 0;

1


Nils Carqueville


W =∐


V −→ (EndW )[[x, x−1]] , v 7−→ YW (v, x) =∑



(M1) the truncation condition uWm w = 0 for all m ≫ 0;

(M2) the vacuum property YW (Ω, x) = 1W ;


Nils Carqueville

(M3) the Jacobi identity

x−10 δ

(x1 − x2

x0

)YW (u, x1)YW (v, x2) − x−1

0 δ

(x2 − x1

−x0

)YW (v, x2)YW (u, x1)

= x−12 δ

(x1 − x0

x2

)YW (Y (u, x0)v, x2) ;

Z


Nils Carqueville


x−10 δ

(x1 − x2

x0

)YW (u, x1)YW (v, x2) − x−1

0 δ

(x2 − x1

−x0

)YW (v, x2)YW (u, x1)

= x−12 δ

(x1 − x0

x2

)YW (Y (u, x0)v, x2) ;

(M4) the modes LWm of the energy momentum operator

YW (ω, x) =∑

m∈ZLWm x−m−2

span a representation of the Virasoro algebra, and the homogeneoussubspaces W[h] are exactly the (generalized) eigenspaces of theoperator LW

0 with (generalized) eigenvalues h;


Nils Carqueville


x−10 δ

(x1 − x2

x0

)YW (u, x1)YW (v, x2) − x−1

0 δ

(x2 − x1

−x0

)YW (v, x2)YW (u, x1)

= x−12 δ

(x1 − x0

x2

)YW (Y (u, x0)v, x2) ;

(M4) the modes LWm of the energy momentum operator

YW (ω, x) =∑

m∈ZLWm x−m−2

span a representation of the Virasoro algebra, and the homogeneoussubspaces W[h] are exactly the (generalized) eigenspaces of theoperator LW

0 with (generalized) eigenvalues h;

(M5) the L−1-derivative property ddx

YW (v, x) = YW (L−1v, x).

(Logarithmic) Intertwining Operators

Nils Carqueville

Definition. Let (Wi, Yi), (Wj, Yj) and (Wk, Yk) be (generalized) V -modules.A (logarithmic) intertwining operator of type

(Wk

Wi Wj

)is a linear map

Wi −→ (Hom(Wj,Wk))[log x]x ,

w(i) 7−→ Ykij(w(i), x) =

∑

m∈C∑a∈N(w(i))Ym,ax

−m−1(log x)a .


Nils Carqueville


(Wk

Wi Wj

)is a linear map


w(i) 7−→ Ykij(w(i), x) =

∑


−m−1(log x)a .

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


Nils Carqueville


(Wk

Wi Wj

)is a linear map


w(i) 7−→ Ykij(w(i), x) =

∑


−m−1(log x)a .

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

(IO1) the truncation condition (w(i))Ym,aw(j) = 0 for all m with Rem ≫ 0,

independently of a;


Nils Carqueville


independently of a;


Nils Carqueville


independently of a;

(IO2) the Jacobi identity

x−10 δ

(x1 − x2

x0

)Yk(v, x1)Y

kij(w(i), x2)w(j)

− x−10 δ

(x2 − x1

−x0

)Yk

ij(w(i), x2)Yj(v, x1)w(j)

= x−12 δ

(x1 − x0

x2

)Yk

ij(Yi(u, x0)w(i), x2)w(j) ;


Nils Carqueville


independently of a;


x−10 δ

(x1 − x2

x0

)Yk(v, x1)Y

kij(w(i), x2)w(j)

− x−10 δ

(x2 − x1

−x0

)Yk


= x−12 δ

(x1 − x0

x2

)Yk


(IO3) the L−1-derivative property ddxYk

ij(w(i), x) = Ykij(L

Wi

−1w(i), x).


Nils Carqueville


independently of a;


x−10 δ

(x1 − x2

x0

)Yk(v, x1)Y

kij(w(i), x2)w(j)

− x−10 δ

(x2 − x1

−x0

)Yk


= x−12 δ

(x1 − x0

x2

)Yk


(IO3) the L−1-derivative property ddxYk

ij(w(i), x) = Ykij(L

Wi

−1w(i), x).

The dimensions of the spaces of all intertwining operators Ykij are called the

fusion rules Nkij.

Nonmeromorphic Operator Product Expansion

Nils CarquevilleHuang, Lepowsky 1995; Huang 1995, 2002; Buhl 2002; Huang, Lepowsky, Zhang 2003

Theorem. Let V be a vertex operator algebra. Given two logarithmicintertwining maps Y1 and Y2 of type

(W4

W1 M

)and

(M

W2 W3

), there exists a

logarithmic intertwining map Y of type(

W4

W1⊠P (z1−z2)W2 W3

)such that

〈w′4,Y1(w1, z1)Y2(w2, z2)w3〉 =

⟨w′

4,Y(w1 ⊠P (z1−z2) w2, z2

)w3

⟩,

R




(W4

W1 M

)and

(M

W2 W3

), there exists a


W4

W1⊠P (z1−z2)W2 W3

)such that

〈w′4,Y1(w1, z1)Y2(w2, z2)w3〉 =

⟨w′

4,Y(w1 ⊠P (z1−z2) w2, z2

)w3

⟩,

if the following conditions are satisfied for a full subcategory C of generalizedV -modules that is closed under the contragredient functor.

R




(W4

W1 M

)and

(M

W2 W3

), there exists a


W4

W1⊠P (z1−z2)W2 W3

)such that

〈w′4,Y1(w1, z1)Y2(w2, z2)w3〉 =

⟨w′

4,Y(w1 ⊠P (z1−z2) w2, z2

)w3

⟩,


(1) V is C2-cofinite, i.e.

dim (V/C2(V )) < ∞ with C2(V ) = spanu−2v

∣∣ u, v ∈ V.

R




(W4

W1 M

)and

(M

W2 W3

), there exists a


W4

W1⊠P (z1−z2)W2 W3

)such that

〈w′4,Y1(w1, z1)Y2(w2, z2)w3〉 =

⟨w′

4,Y(w1 ⊠P (z1−z2) w2, z2

)w3

⟩,




∣∣ u, v ∈ V.

(2) All generalized V -modules W in ob C are quasi-finite-dimensional, i.e. dim‘

m<RW[m] < ∞ for all R ∈ R.




(W4

W1 M

)and

(M

W2 W3

), there exists a


W4

W1⊠P (z1−z2)W2 W3

)such that

〈w′4,Y1(w1, z1)Y2(w2, z2)w3〉 =

⟨w′

4,Y(w1 ⊠P (z1−z2) w2, z2

)w3

⟩,




∣∣ u, v ∈ V.

(2) All generalized V -modules W in ob C are quasi-finite-dimensional, i.e. dim‘

m<RW[m] < ∞ for all R ∈ R.

(3) Every object which is a finitely generated lower-truncated generalized V -module, except that it may have

infinite-dimensional homogeneous subspaces, is an object in C.

Triplet Algebras

Nils CarquevilleBlumenhagen et al. 1991; Kausch 1991

An infinite family of logarithmic conformal field theories can be shownto satisfy the conditions: the triplet algebras W(2, (2p − 1)×3)p≥2

⋆ withcentral charge cp,1 = 1 − 6(p − 1)2/p.

Triplet Algebras




Definition. A W-algebra of type W(2, h1, . . . , hm) is a vertex operatoralgebra which has a minimal generating set consisting of the vacuum Ω, theconformal vector ω of weight 2 and m additional primary vectors W i ofweight hi, i ∈ 1, . . . ,m, with all singular vectors divided out.

Triplet Algebras





C2-cofiniteness is easily proven for the first triplet algebra with p = 2

Triplet Algebras

Nils CarquevilleBlumenhagen et al. 1991; Kausch 1991; Gaberdiel, Kausch 1996; Rohsiepe 1996




C2-cofiniteness is easily proven for the first triplet algebra with p = 2, as allrelevant commutators and singular vectors are explicitly known:⋆

Nab =W a−3W

b−3Ω − δab

(8

9L3

−2 +19

36L2

−3 +14

9L−4L−2 −

16

9L−6

)Ω

+ iεabc

(−2W c

−4L−2 +5

4W c

−6

)Ω .

Triplet Algebras

Nils Carqueville

Proposition. The vertex operator algebra W(2, 3×3) is C2-cofinite and thenonmeromorphic operator product expansion exists.

Z

Triplet Algebras

Nils Carqueville


For all other triplet algebras neither commutators nor singular vectors areexplicitly known.

Z

Triplet Algebras

Nils Carqueville



Problem: Prove C2-cofiniteness with very little information:

Z

Triplet Algebras

Nils Carqueville




1st step: prove existence of singular vectors

Z

Triplet Algebras

Nils CarquevilleFlohr 1995





⊲ use characters⋆ χV2p−1(q) = 1η(q)

∑n∈Z(2n + 1)q(2np+p−1)2/(4p)

Triplet Algebras







∑n∈Z(2n + 1)q(2np+p−1)2/(4p)

⊲ prove and use embedding of pure Virasoro modules into tripletalgebras, use Kac determinant

Triplet Algebras







∑n∈Z(2n + 1)q(2np+p−1)2/(4p)


2st step: analyze singular vectors

Triplet Algebras







∑n∈Z(2n + 1)q(2np+p−1)2/(4p)



⊲ use Nahm’s results on quasiprimary normal-ordered products

Triplet Algebras







∑n∈Z(2n + 1)q(2np+p−1)2/(4p)



⊲ use Nahm’s results on quasiprimary normal-ordered products⊲ do a lot of careful calculations!

Main result: C2-cofiniteness

Nils Carqueville

Theorem. For all p ∈ Z≥2, the nonmeromorphic operator product expansionexists and is associative for the vertex operator algebra W(2, (2p − 1)×3).Furthermore, all these vertex operator algebras are C2-cofinite.


Nils Carqueville


Why is the C2-cofiniteness property so interesting?


Nils Carqueville




∣∣ u, v ∈ V


Nils CarquevilleZhu 1996




∣∣ u, v ∈ V

crucial for convergence and modular covariance of characters⋆


Nils CarquevilleZhu 1996; Huang 2004




∣∣ u, v ∈ V


crucial for Huang’s proof of the Verlinde conjecture⋆


Nils CarquevilleZhu 1996; Huang 2004; Gaberdiel, Neitzke 2000




∣∣ u, v ∈ V



finite fusion rules⋆


Nils CarquevilleZhu 1996; Huang 2004; Gaberdiel, Neitzke 2000; Dong, Li, Mason 1998




∣∣ u, v ∈ V




finitely many inequivalent irreducible modules⋆


Nils CarquevilleZhu 1996; Huang 2004; Gaberdiel, Neitzke 2000; Dong, Li, Mason 1998; Miyamoto 2002




∣∣ u, v ∈ V





every weak module is a direct sum of generalized eigenspaces of L0⋆


Nils CarquevilleZhu 1996; Huang 2004; Gaberdiel, Neitzke 2000; Dong, Li, Mason 1998; Miyamoto 2002




∣∣ u, v ∈ V





every weak module is a direct sum of generalized eigenspaces of L0⋆

interesting relation to “rationality”. . .

Jordan Vertex Operator Algebras?

Nils Carqueville

All vertex algebraic approaches to LCFT so far place its characteristicfeatures on the level of modules.


Nils Carqueville


But all known LCFTs have indecomposable structure in the vacuum sector,L0Ω = Ω.


Nils Carqueville



=⇒ notion of a Jordan vertex operator algebra?


Nils Carqueville




The “generalized Jacobi identity” is problematic.


Nils Carqueville





x−10 δ

(x1 − x2

x0

)Y (u, x1 )Y (v, x2 )

− x−10 δ

(x2 − x1

−x0

)Y (v, x2 )Y (u, x1 )

= x−12 δ

(x1 − x0

x2

)Y (Y (u, x0 )v, x2 )


Nils Carqueville





x−10 δ

(x1 − x2

x0

)Y (u, x1, log x1)Y (v, x2, log x2)

− x−10 δ

(x2 − x1

−x0

)Y (v, x2, log x2)Y (u, x1, log x1)

?= x−1

2 δ

(x1 − x0

x2

)Y (Y (u, x0, log x0)v, x2, log x2)


Nils Carqueville





x−10 δ

(x1 − x2

x0


− x−10 δ

(x2 − x1

−x0


?= x−1

2 δ

(x1 − x0

x2



Nils Carqueville





x−10 δ

(x1 − x2

x0

)(log x0)

−1δ

(log x1 − log x2

log x0


+ x−10 δ

(x2 − x1

−x0

)(log x0)

−1δ

(log x2 − log x1

− log x0


?= x−1

2 δ

(x1 − x0

x2

)(log x2)

−1δ

(log x1 − log x0

log x2



Nils Carqueville





x−10 δ

(x1 − x2

x0

)(log x0)

−1δ

(log x1 − log x2

log x0


+ x−10 δ

(x2 − x1

−x0

)(log x0)

−1δ

(log x2 − log x1

− log x0


?= x−1

2 δ

(x1 − x0

x2

)(log x2)

−1δ

(log x1 − log x0

log x2



Nils Carqueville





x−10 δ

(x1 − x2

x0


− x−10 δ

(x2 − x1

−x0


?= x−1

2 δ

(x1 − x0

x2

)Y (Y (u, x0, log(−x2 + x1))v, x2, log(x1 − x0))


Nils Carqueville





x−10 δ

(x1 − x2

x0


− x−10 δ

(x2 − x1

−x0


?= x−1

2 δ

(x1 − x0

x2

)Y (Y (u, x0, log(−x2 + x1))v, x2, log(x1 − x0))

Two steps back: Logarithmic Mode Algebras

Nils Carqueville

T (z)Ω(w) ∼1

(z − w)2+

1

(z − w)∂Ω(w)


Nils Carqueville

T (z)Ω(w) ∼1

(z − w)2+

1

(z − w)∂Ω(w)

[Lm, Ωn,b

]= (m + 1)δb,0δm+n,−1 − (m + n)Ωm+n,b + (b + 1)Ωm+n,b+1


Nils Carqueville

T (z)Ω(w) ∼1

(z − w)2+

1

(z − w)∂Ω(w)

[Lm, Ωn,b


Ω(z)Ω(w) ∼ − (log(z − w))2 − 2 log(z − w)Ω(w)


Nils Carqueville

T (z)Ω(w) ∼1

(z − w)2+

1

(z − w)∂Ω(w)

[Lm, Ωn,b


Ω(z)Ω(w) ∼ − (log(z − w))2 − 2 log(z − w)Ω(w)[Ωm,a, Ωn,b

]?= δa,0(1 − δm,0)

2

mΩm+n,b − δb,0(1 − δn,0)

2

nΩm+n,a

+ (δa,0δb,2 − δa,2δb,0) δm,0δn,0 − δa,1δm,02Ωn,b + δb,1δn,02Ωm,a

+ (δa,1δb,0 + δa,0δb,1) (1 − δm,0)δm+n,02

m

−

(m−1∑

i=1

1

i+

−m−1∑

i=1

1

i

)δa,0δb,0δm+n,0

2

m

Conclusion

Nils Carqueville

Vertex operator algebra approach to LCFT

Conclusion

Nils Carqueville


on the level of modules

Conclusion

Nils Carqueville



⊲ C2-cofiniteness and nonmeromorphic OPE for all triplet algebras

Conclusion

Nils Carqueville




⊲ C2-cofiniteness as an important finiteness property

Conclusion

Nils Carqueville





⊲ upper bounds on the dimensions of the Zhu algebras

Conclusion

Nils Carqueville






⊲ . . .

Conclusion

Nils Carqueville






⊲ . . .

on the fundamental level

Conclusion

Nils Carqueville






⊲ . . .


⊲ Jordan vertex operator algebras?

Conclusion

Nils Carqueville






⊲ . . .



⊲ logarithmic mode algebras

Conclusion

Nils Carqueville






⊲ . . .



⊲ logarithmic mode algebras

References

Nils Carqueville

R. Blumenhagen, M. Flohr, A. Kliem, W. Nahm, A. Recknagel and R. Varnhagen, W-Algebras with

two and three Generators, Nucl. Phys. B 361 (1991), 255–289.

G. Buhl, A spanning set for VOA modules, J. Algebra 254 (2002), 125–151, [math.QA/0111296].

C. Dong, H. Li, and G. Mason, Modular invariance of trace functions in orbifold theory, Commun.Math. Phys. 214 (2001), 1–56, [q-alg/9703016].

N. Carqueville and M. Flohr, Nonmeromorphic operator product expansion and C2-cofiniteness for a

family of W-algebras, J. Phys. A: Math. Gen. 39 (2006), 951–966, [math-ph/0508015].

M. Flohr, On modular invariant partition functions of conformal field theories with logarithmic

operators, Int. J. Mod. Phys. A 11 (1996), 4147–4172, [hep-th/9509166].

I. B. Frenkel, Y.-Z. Huang and J. Lepowsky, On axiomatic approaches to vertex operator algebras and

modules, Memoirs Amer. Math. Soc. 104 (1989).

M. R. Gaberdiel and H. Kausch, A rational logarithmic conformal field theory, Phys. Lett. B 386(1996), 131–137, [hep-th/9606050].

M. R. Gaberdiel and A. Neitzke, Rationality, quasirationality and finite W -algebras, Commun. Math.Phys. 238 (2003), 305–331, [hep-th/0009235].

http://arxiv.org/abs/math.QA/0111296

http://arxiv.org/abs/q-alg/9703016

http://www.arxiv.org/abs/math-ph/0508015

http://www.arxiv.org/abs/hep-th/9509166



References

Nils Carqueville

Y.-Z. Huang and J. Lepowsky, A theory of tensor products for module categories for a vertex operator

algebra, I, Selecta Mathematica 1 (1995), 699–756, [hep-th/9309076].


algebra, II, Selecta Mathematica 1 (1995), 757–786, [hep-th/9309159].


algebra, III, J. Pure Appl. Algebra 100 (1995), 141–172, [q-alg/9505018].

Y.-Z. Huang, J. Lepowsky and L. Zhang, A logarithmic generalization of tensor product theory for

modules for a vertex operator algebra, [math.QA/0311235].

Y.-Z. Huang, A theory of tensor products for module categories for a vertex operator algebra, VI, J.Pure Appl. Algebra 100 (1995), 173–216, [q-alg/9505019].

Y.-Z. Huang, Differential equations and intertwining operators, Comm. Contemp. Math. 7 (2005),375–400, [math.QA/0206206].

Y.-Z. Huang, Vertex operator algebras and the Verlinde conjecture, [math.QA/0412261].

H. G. Kausch, Extended conformal algebras generated by a multiplet of primary fields, Phys. Lett. B259 (1991), 448–455.








References

Nils Carqueville

M. Miyamoto, Modular invariance of vertex operator algebras satisfying C2-cofiniteness,[math.QA/0209101].

F. Rohsiepe, On Reducible but Indecomposable Representations of the Virasoro Algebra,BONN-TH-96-17 (1996), [hep-th/9611160].

Y. Zhu, Modular invariance of characters of vertex operator algebras, J. Amer. Math. Soc. 9 (1996),237–302.

http://www.arxiv.org/abs/math.QA/0209101


Vertex Operator Algebra Approach to Logarithmic Conformal Field Theory

Documents