Post on 03-Feb-2022
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
Logarithmic conformal field theory has indecomposable structure
Introduction and Synopsis
Nils Carqueville
Logarithmic conformal field theory has indecomposable structure(and logarithms, too. . . )
Introduction and Synopsis
Nils Carqueville
Logarithmic conformal field theory has indecomposable structure(and logarithms, too. . . )
Study LCFT from an algebraic point of view:
Introduction and Synopsis
Nils Carqueville
Logarithmic conformal field theory has indecomposable structure(and logarithms, too. . . )
Study LCFT from an algebraic point of view:
vertex operator algebras, (generalized) modules and intertwining operators
Introduction and Synopsis
Nils Carqueville
Logarithmic conformal field theory has indecomposable structure(and logarithms, too. . . )
Study LCFT from an algebraic point of view:
vertex operator algebras, (generalized) modules and intertwining operators
nonmeromorphic OPE with P (z)-tensor product theory
Introduction and Synopsis
Nils Carqueville
Logarithmic conformal field theory has indecomposable structure(and logarithms, too. . . )
Study LCFT from an algebraic point of view:
vertex operator algebras, (generalized) modules and intertwining operators
nonmeromorphic OPE with P (z)-tensor product theory
results on triplet algebras
Introduction and Synopsis
Nils Carqueville
Logarithmic conformal field theory has indecomposable structure(and logarithms, too. . . )
Study LCFT from an algebraic point of view:
vertex operator algebras, (generalized) modules and intertwining operators
nonmeromorphic OPE with P (z)-tensor product theory
results on triplet algebras
C2-cofiniteness
Introduction and Synopsis
Nils Carqueville
Logarithmic conformal field theory has indecomposable structure(and logarithms, too. . . )
Study LCFT from an algebraic point of view:
vertex operator algebras, (generalized) modules and intertwining operators
nonmeromorphic OPE with P (z)-tensor product theory
results on triplet algebras
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
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 ∈ Ztogether with a linear vertex operator map
V −→ (EndV )[[x, x−1]] , v 7−→ Y (v, x) =∑
m∈Z vmx−m−1 .
1
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 ∈ Ztogether with a linear vertex operator map
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
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 ∈ Ztogether with a linear vertex operator map
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
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 ∈ Ztogether with a linear vertex operator map
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 :
(V1) the truncation condition umv = 0 for all m ≫ 0;
1
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 ∈ Ztogether with a linear vertex operator map
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 :
(V1) the truncation condition umv = 0 for all m ≫ 0;
(V2) the vacuum property Y (Ω, x) = 1V ;
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 ∈ Ztogether with a linear vertex operator map
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 :
(V1) the truncation condition umv = 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;
Vertex Operator Algebras
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
Vertex Operator Algebras
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) ;
(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;
Vertex Operator Algebras
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) ;
(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
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 ∈ Rtogether with a linear vertex operator map
V −→ (EndW )[[x, x−1]] , v 7−→ YW (v, x) =∑
m∈Z vWm x−m−1 .
1
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 ∈ Rtogether with a linear vertex operator map
V −→ (EndW )[[x, x−1]] , v 7−→ YW (v, x) =∑
m∈Z vWm x−m−1 .
The following axioms hold for all u, v ∈ V and w ∈ W :
1
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 ∈ Rtogether with a linear vertex operator map
V −→ (EndW )[[x, x−1]] , v 7−→ YW (v, x) =∑
m∈Z vWm x−m−1 .
The following axioms hold for all u, v ∈ V and w ∈ W :
(M1) the truncation condition uWm w = 0 for all m ≫ 0;
1
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 ∈ Rtogether with a linear vertex operator map
V −→ (EndW )[[x, x−1]] , v 7−→ YW (v, x) =∑
m∈Z vWm x−m−1 .
The following axioms hold for all u, v ∈ V and w ∈ W :
(M1) the truncation condition uWm w = 0 for all m ≫ 0;
(M2) the vacuum property YW (Ω, x) = 1W ;
Modules for Vertex Operator Algebras
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
Modules for Vertex Operator Algebras
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) ;
(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;
Modules for Vertex Operator Algebras
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) ;
(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 .
(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 .
The following axioms hold for all v ∈ V , w(i) ∈ Wi and w(j) ∈ Wj:
(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 .
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;
(Logarithmic) Intertwining Operators
Nils Carqueville
(IO1) the truncation condition (w(i))Ym,aw(j) = 0 for all m with Rem ≫ 0,
independently of a;
(Logarithmic) Intertwining Operators
Nils Carqueville
(IO1) the truncation condition (w(i))Ym,aw(j) = 0 for all m with Rem ≫ 0,
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) ;
(Logarithmic) Intertwining Operators
Nils Carqueville
(IO1) the truncation condition (w(i))Ym,aw(j) = 0 for all m with Rem ≫ 0,
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) ;
(IO3) the L−1-derivative property ddxYk
ij(w(i), x) = Ykij(L
Wi
−1w(i), x).
(Logarithmic) Intertwining Operators
Nils Carqueville
(IO1) the truncation condition (w(i))Ym,aw(j) = 0 for all m with Rem ≫ 0,
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) ;
(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
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
⟩,
if the following conditions are satisfied for a full subcategory C of generalizedV -modules that is closed under the contragredient functor.
R
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
⟩,
if the following conditions are satisfied for a full subcategory C of generalizedV -modules that is closed under the contragredient functor.
(1) V is C2-cofinite, i.e.
dim (V/C2(V )) < ∞ with C2(V ) = spanu−2v
∣∣ u, v ∈ V.
R
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
⟩,
if the following conditions are satisfied for a full subcategory C of generalizedV -modules that is closed under the contragredient functor.
(1) V is C2-cofinite, i.e.
dim (V/C2(V )) < ∞ with C2(V ) = spanu−2v
∣∣ 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.
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
⟩,
if the following conditions are satisfied for a full subcategory C of generalizedV -modules that is closed under the contragredient functor.
(1) V is C2-cofinite, i.e.
dim (V/C2(V )) < ∞ with C2(V ) = spanu−2v
∣∣ 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
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.
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
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.
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.
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
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.
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.
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
Proposition. The vertex operator algebra W(2, 3×3) is C2-cofinite and thenonmeromorphic operator product expansion exists.
For all other triplet algebras neither commutators nor singular vectors areexplicitly known.
Z
Triplet Algebras
Nils Carqueville
Proposition. The vertex operator algebra W(2, 3×3) is C2-cofinite and thenonmeromorphic operator product expansion exists.
For all other triplet algebras neither commutators nor singular vectors areexplicitly known.
Problem: Prove C2-cofiniteness with very little information:
Z
Triplet Algebras
Nils Carqueville
Proposition. The vertex operator algebra W(2, 3×3) is C2-cofinite and thenonmeromorphic operator product expansion exists.
For all other triplet algebras neither commutators nor singular vectors areexplicitly known.
Problem: Prove C2-cofiniteness with very little information:
1st step: prove existence of singular vectors
Z
Triplet Algebras
Nils CarquevilleFlohr 1995
Proposition. The vertex operator algebra W(2, 3×3) is C2-cofinite and thenonmeromorphic operator product expansion exists.
For all other triplet algebras neither commutators nor singular vectors areexplicitly known.
Problem: Prove C2-cofiniteness with very little information:
1st step: prove existence of singular vectors
⊲ use characters⋆ χV2p−1(q) = 1η(q)
∑n∈Z(2n + 1)q(2np+p−1)2/(4p)
Triplet Algebras
Nils CarquevilleFlohr 1995
Proposition. The vertex operator algebra W(2, 3×3) is C2-cofinite and thenonmeromorphic operator product expansion exists.
For all other triplet algebras neither commutators nor singular vectors areexplicitly known.
Problem: Prove C2-cofiniteness with very little information:
1st step: prove existence of singular vectors
⊲ use characters⋆ χV2p−1(q) = 1η(q)
∑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
Nils CarquevilleFlohr 1995
Proposition. The vertex operator algebra W(2, 3×3) is C2-cofinite and thenonmeromorphic operator product expansion exists.
For all other triplet algebras neither commutators nor singular vectors areexplicitly known.
Problem: Prove C2-cofiniteness with very little information:
1st step: prove existence of singular vectors
⊲ use characters⋆ χV2p−1(q) = 1η(q)
∑n∈Z(2n + 1)q(2np+p−1)2/(4p)
⊲ prove and use embedding of pure Virasoro modules into tripletalgebras, use Kac determinant
2st step: analyze singular vectors
Triplet Algebras
Nils CarquevilleFlohr 1995
Proposition. The vertex operator algebra W(2, 3×3) is C2-cofinite and thenonmeromorphic operator product expansion exists.
For all other triplet algebras neither commutators nor singular vectors areexplicitly known.
Problem: Prove C2-cofiniteness with very little information:
1st step: prove existence of singular vectors
⊲ use characters⋆ χV2p−1(q) = 1η(q)
∑n∈Z(2n + 1)q(2np+p−1)2/(4p)
⊲ prove and use embedding of pure Virasoro modules into tripletalgebras, use Kac determinant
2st step: analyze singular vectors
⊲ use Nahm’s results on quasiprimary normal-ordered products
Triplet Algebras
Nils CarquevilleFlohr 1995
Proposition. The vertex operator algebra W(2, 3×3) is C2-cofinite and thenonmeromorphic operator product expansion exists.
For all other triplet algebras neither commutators nor singular vectors areexplicitly known.
Problem: Prove C2-cofiniteness with very little information:
1st step: prove existence of singular vectors
⊲ use characters⋆ χV2p−1(q) = 1η(q)
∑n∈Z(2n + 1)q(2np+p−1)2/(4p)
⊲ prove and use embedding of pure Virasoro modules into tripletalgebras, use Kac determinant
2st step: analyze singular vectors
⊲ 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.
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.
Why is the C2-cofiniteness property so interesting?
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.
Why is the C2-cofiniteness property so interesting?
dim (V/C2(V )) < ∞ with C2(V ) = spanu−2v
∣∣ u, v ∈ V
Main result: C2-cofiniteness
Nils CarquevilleZhu 1996
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.
Why is the C2-cofiniteness property so interesting?
dim (V/C2(V )) < ∞ with C2(V ) = spanu−2v
∣∣ u, v ∈ V
crucial for convergence and modular covariance of characters⋆
Main result: C2-cofiniteness
Nils CarquevilleZhu 1996; Huang 2004
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.
Why is the C2-cofiniteness property so interesting?
dim (V/C2(V )) < ∞ with C2(V ) = spanu−2v
∣∣ u, v ∈ V
crucial for convergence and modular covariance of characters⋆
crucial for Huang’s proof of the Verlinde conjecture⋆
Main result: C2-cofiniteness
Nils CarquevilleZhu 1996; Huang 2004; Gaberdiel, Neitzke 2000
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.
Why is the C2-cofiniteness property so interesting?
dim (V/C2(V )) < ∞ with C2(V ) = spanu−2v
∣∣ u, v ∈ V
crucial for convergence and modular covariance of characters⋆
crucial for Huang’s proof of the Verlinde conjecture⋆
finite fusion rules⋆
Main result: C2-cofiniteness
Nils CarquevilleZhu 1996; Huang 2004; Gaberdiel, Neitzke 2000; Dong, Li, Mason 1998
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.
Why is the C2-cofiniteness property so interesting?
dim (V/C2(V )) < ∞ with C2(V ) = spanu−2v
∣∣ u, v ∈ V
crucial for convergence and modular covariance of characters⋆
crucial for Huang’s proof of the Verlinde conjecture⋆
finite fusion rules⋆
finitely many inequivalent irreducible modules⋆
Main result: C2-cofiniteness
Nils CarquevilleZhu 1996; Huang 2004; Gaberdiel, Neitzke 2000; Dong, Li, Mason 1998; Miyamoto 2002
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.
Why is the C2-cofiniteness property so interesting?
dim (V/C2(V )) < ∞ with C2(V ) = spanu−2v
∣∣ u, v ∈ V
crucial for convergence and modular covariance of characters⋆
crucial for Huang’s proof of the Verlinde conjecture⋆
finite fusion rules⋆
finitely many inequivalent irreducible modules⋆
every weak module is a direct sum of generalized eigenspaces of L0⋆
Main result: C2-cofiniteness
Nils CarquevilleZhu 1996; Huang 2004; Gaberdiel, Neitzke 2000; Dong, Li, Mason 1998; Miyamoto 2002
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.
Why is the C2-cofiniteness property so interesting?
dim (V/C2(V )) < ∞ with C2(V ) = spanu−2v
∣∣ u, v ∈ V
crucial for convergence and modular covariance of characters⋆
crucial for Huang’s proof of the Verlinde conjecture⋆
finite fusion rules⋆
finitely many inequivalent irreducible modules⋆
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.
Jordan Vertex Operator Algebras?
Nils Carqueville
All vertex algebraic approaches to LCFT so far place its characteristicfeatures on the level of modules.
But all known LCFTs have indecomposable structure in the vacuum sector,L0Ω = Ω.
Jordan Vertex Operator Algebras?
Nils Carqueville
All vertex algebraic approaches to LCFT so far place its characteristicfeatures on the level of modules.
But all known LCFTs have indecomposable structure in the vacuum sector,L0Ω = Ω.
=⇒ notion of a Jordan vertex operator algebra?
Jordan Vertex Operator Algebras?
Nils Carqueville
All vertex algebraic approaches to LCFT so far place its characteristicfeatures on the level of modules.
But all known LCFTs have indecomposable structure in the vacuum sector,L0Ω = Ω.
=⇒ notion of a Jordan vertex operator algebra?
The “generalized Jacobi identity” is problematic.
Jordan Vertex Operator Algebras?
Nils Carqueville
All vertex algebraic approaches to LCFT so far place its characteristicfeatures on the level of modules.
But all known LCFTs have indecomposable structure in the vacuum sector,L0Ω = Ω.
=⇒ notion of a Jordan vertex operator algebra?
The “generalized Jacobi identity” is problematic.
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 )
Jordan Vertex Operator Algebras?
Nils Carqueville
All vertex algebraic approaches to LCFT so far place its characteristicfeatures on the level of modules.
But all known LCFTs have indecomposable structure in the vacuum sector,L0Ω = Ω.
=⇒ notion of a Jordan vertex operator algebra?
The “generalized Jacobi identity” is problematic.
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)
Jordan Vertex Operator Algebras?
Nils Carqueville
All vertex algebraic approaches to LCFT so far place its characteristicfeatures on the level of modules.
But all known LCFTs have indecomposable structure in the vacuum sector,L0Ω = Ω.
=⇒ notion of a Jordan vertex operator algebra?
The “generalized Jacobi identity” is problematic.
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)
Jordan Vertex Operator Algebras?
Nils Carqueville
All vertex algebraic approaches to LCFT so far place its characteristicfeatures on the level of modules.
But all known LCFTs have indecomposable structure in the vacuum sector,L0Ω = Ω.
=⇒ notion of a Jordan vertex operator algebra?
The “generalized Jacobi identity” is problematic.
x−10 δ
(x1 − x2
x0
)(log x0)
−1δ
(log x1 − log x2
log x0
)Y (u, x1, log x1)Y (v, x2, log x2)
+ x−10 δ
(x2 − x1
−x0
)(log x0)
−1δ
(log x2 − log x1
− log x0
)Y (v, x2, log x2)Y (u, x1, log x2)
?= x−1
2 δ
(x1 − x0
x2
)(log x2)
−1δ
(log x1 − log x0
log x2
)Y (Y (u, x0, log x0)v, x2, log x2)
Jordan Vertex Operator Algebras?
Nils Carqueville
All vertex algebraic approaches to LCFT so far place its characteristicfeatures on the level of modules.
But all known LCFTs have indecomposable structure in the vacuum sector,L0Ω = Ω.
=⇒ notion of a Jordan vertex operator algebra?
The “generalized Jacobi identity” is problematic.
x−10 δ
(x1 − x2
x0
)(log x0)
−1δ
(log x1 − log x2
log x0
)Y (u, x1, log x1)Y (v, x2, log x2)
+ x−10 δ
(x2 − x1
−x0
)(log x0)
−1δ
(log x2 − log x1
− log x0
)Y (v, x2, log x2)Y (u, x1, log x2)
?= x−1
2 δ
(x1 − x0
x2
)(log x2)
−1δ
(log x1 − log x0
log x2
)Y (Y (u, x0, log x0)v, x2, log x2)
Jordan Vertex Operator Algebras?
Nils Carqueville
All vertex algebraic approaches to LCFT so far place its characteristicfeatures on the level of modules.
But all known LCFTs have indecomposable structure in the vacuum sector,L0Ω = Ω.
=⇒ notion of a Jordan vertex operator algebra?
The “generalized Jacobi identity” is problematic.
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 x2)
?= x−1
2 δ
(x1 − x0
x2
)Y (Y (u, x0, log(−x2 + x1))v, x2, log(x1 − x0))
Jordan Vertex Operator Algebras?
Nils Carqueville
All vertex algebraic approaches to LCFT so far place its characteristicfeatures on the level of modules.
But all known LCFTs have indecomposable structure in the vacuum sector,L0Ω = Ω.
=⇒ notion of a Jordan vertex operator algebra?
The “generalized Jacobi identity” is problematic.
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 x2)
?= 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)
[Lm, Ωn,b
]= (m + 1)δb,0δm+n,−1 − (m + n)Ωm+n,b + (b + 1)Ωm+n,b+1
Two steps back: Logarithmic Mode Algebras
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
Ω(z)Ω(w) ∼ − (log(z − w))2 − 2 log(z − w)Ω(w)
Two steps back: Logarithmic Mode Algebras
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
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
on the level of modules
⊲ C2-cofiniteness and nonmeromorphic OPE for all triplet algebras
Conclusion
Nils Carqueville
Vertex operator algebra approach to LCFT
on the level of modules
⊲ C2-cofiniteness and nonmeromorphic OPE for all triplet algebras
⊲ C2-cofiniteness as an important finiteness property
Conclusion
Nils Carqueville
Vertex operator algebra approach to LCFT
on the level of modules
⊲ C2-cofiniteness and nonmeromorphic OPE for all triplet algebras
⊲ C2-cofiniteness as an important finiteness property
⊲ upper bounds on the dimensions of the Zhu algebras
Conclusion
Nils Carqueville
Vertex operator algebra approach to LCFT
on the level of modules
⊲ C2-cofiniteness and nonmeromorphic OPE for all triplet algebras
⊲ C2-cofiniteness as an important finiteness property
⊲ upper bounds on the dimensions of the Zhu algebras
⊲ . . .
Conclusion
Nils Carqueville
Vertex operator algebra approach to LCFT
on the level of modules
⊲ C2-cofiniteness and nonmeromorphic OPE for all triplet algebras
⊲ C2-cofiniteness as an important finiteness property
⊲ upper bounds on the dimensions of the Zhu algebras
⊲ . . .
on the fundamental level
Conclusion
Nils Carqueville
Vertex operator algebra approach to LCFT
on the level of modules
⊲ C2-cofiniteness and nonmeromorphic OPE for all triplet algebras
⊲ C2-cofiniteness as an important finiteness property
⊲ upper bounds on the dimensions of the Zhu algebras
⊲ . . .
on the fundamental level
⊲ Jordan vertex operator algebras?
Conclusion
Nils Carqueville
Vertex operator algebra approach to LCFT
on the level of modules
⊲ C2-cofiniteness and nonmeromorphic OPE for all triplet algebras
⊲ C2-cofiniteness as an important finiteness property
⊲ upper bounds on the dimensions of the Zhu algebras
⊲ . . .
on the fundamental level
⊲ Jordan vertex operator algebras?
⊲ logarithmic mode algebras
Conclusion
Nils Carqueville
Vertex operator algebra approach to LCFT
on the level of modules
⊲ C2-cofiniteness and nonmeromorphic OPE for all triplet algebras
⊲ C2-cofiniteness as an important finiteness property
⊲ upper bounds on the dimensions of the Zhu algebras
⊲ . . .
on the fundamental level
⊲ Jordan vertex operator algebras?
⊲ logarithmic mode algebras
References
Nils Carqueville
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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].
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algebra, II, Selecta Mathematica 1 (1995), 757–786, [hep-th/9309159].
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algebra, III, J. Pure Appl. Algebra 100 (1995), 141–172, [q-alg/9505018].
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modules for a vertex operator algebra, [math.QA/0311235].
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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.