In conformal field theory and representation theory, a W-algebra is an associative algebra that generalizes the Virasoro algebra. W-algebras were introduced by Alexander Zamolodchikov, and the name "W-algebra" comes from the fact that Zamolodchikov used the letter W for one of the elements of one of his examples.
A W-algebra is an associative algebra that is generated by the modes of a finite number of meromorphic fields
W(h)(z)
T(z)=W(2)(z)
h ≠ 2
W(h)(z)
h\in | 12N |
* |
(h) | |
(W | |
n) |
n\inZ
W(h)(z)=\sumn\inZ
(h) | |
W | |
n |
z-n-h
(2) | |
L | |
n |
c\inC
[Lm,
(h) | |
W | |
n] |
=
(h) | |
((h-1)m-n)W | |
m+n |
W(h)(z)
h
Given a finite set of conformal dimensions
H
(W(h))h\in
c\inC
A W-algebra is called freely generated if its generators obey no other relations than the commutation relations. Most commonly studied W-algebras are freely generated, including the W(N) algebras. In this article, the sections on representation theory and correlation functions apply to freely generated W-algebras.
While it is possible to construct W-algebras by assuming the existence of a number of meromorphic fields
W(h)(z)
From a finite-dimensional Lie algebra
ak{g}
ak{sl}2\hookrightarrowak{g}
\hat{ak{g}}
Given a finite-dimensional Lie algebra
ak{g}
ak{h}\hookrightarrowak{g}
W(\hatak{g}/\hatak{h})
\hatak{h}\hookrightarrow\hatak{g}
W(\hatak{g}/\hatak{h})
\hatak{g}
\hatak{h}
W(\hatak{g}/\hatak{h})
\hatak{g}
\hatak{h}
Given a holomorphic field
\phi(z)
Rn
n
a1,...,an\inRn
\phi
\oint
(ai,\phi(z)) | |
e |
dz
ai
ak{g}
ak{g}
For any integer
N\geq2
N-1
2,3,...,N
\widehat{ak{sl}}N
The embeddings
ak{sl}2\hookrightarrowak{sl}N
N
F
ak{sl}N
ak{sl}2
H
F ⊗ F=R1 ⊕ oplush\inR2h-1
Rd
d
ak{sl}2
The trivial partition
N=N
N=1+1+...+1
\widehat{ak{sl}}N
N=3
3=2+1
2, |
| |||
The central charge of the W(N) algebra is given in terms of the level
k
cW(N)=(N-1)\left(1-N(N+1)\left(
1 | |
k+N |
+k+N-2\right)\right)
c\widehat{ak{sl
It is possible to choose a basis such that the commutation relations are invariant under
W(h)\to(-1)hW(h)
While the Virasoro algebra is a subalgebra of the universal enveloping algebra of
\widehat{ak{sl}}2
N\geq3
\widehat{ak{sl}}N
(Ln)n\inZ
(Wn)n\inZ
(3) | |
=(W | |
n) |
n\inZ
[Lm,Ln]=(m-n)Lm+n+
c | |
12 |
2-1)\delta | |
m(m | |
m+n,0 |
[Lm,Wn]=(2m-n)Wm+n
[Wm,Wn]=
c | |
360 |
m(m2-1)(m2-4)\deltam+n,0+
16 | |
22+5c |
Λm+n+
(m-n)(2m2-mn+2n2-8) | |
30 |
Lm+n
c\inC
Λn=
-2 | |
\sum | |
m=-infty |
LmLn-m
infty | |
+\sum | |
m=-1 |
Ln-mLm-
3 | |
10 |
(n+2)(n+3)Ln
Λ(z)=\sumn\inZΛnz-n-4
Λ=(TT)-
3 | |
10 |
T''
A highest weight representation of a W-algebra is a representation that is generated by a primary state: a vector
v
(h) | |
W | |
n>0 |
v=0 ,
(h) | |
W | |
0v |
=q(h)v
q(h)
q(2)=\Delta
Given a set
\vec{q}=(q(h))h\in
\left\{\prodh\in
(h) | |
W | |
-\vec{N |
h}v\right\}\vec{Nh\inl{V}}
l{V}
\vec{N}=(n1,n2,...,np)
0<n1\leqn2\leq...\leqnp
W-\vec{N
v
For generic values of the charges, the Verma module is the only highest weight representation. For special values of the charges that depend on the algebra's central charge, there exist other highest weight representations, called degenerate representations. Degenerate representations exist if the Verma module is reducible, andthey are quotients of the Verma module by its nontrivial submodules.
If a Verma module is reducible, any indecomposible submodule is itself a highest weight representation, and is generated by a state that is both descendant and primary, called a null state or null vector. A degenerate representation is obtained by setting one or more null vectors to zero. Setting all the null vectors to zero leads to an irreducible representation.
The structures and characters of irreducible representations can be deduced by Drinfeld-Sokolov reduction from representations of affine Lie algebras.
The existence of null vectors is possible only under
c
\vec{q}
For example, in the case of the algebra W(3), the Verma module with vanishing charges
q(2)=q(3)=0
L-1v,W-1v,W-2v
An alternative characterization of a fully degenerate representation is that its fusion product with any Verma module is a sum of finitely many indecomposable representations.
It is convenient to parametrize highest-weight representations not by the set of charges
\vec{q}=(q(2),...,q(N))
P
ak{sl}N
Let
e1,...,eN-1
ak{sl}N
Kij=(ei,ej)
ak{sl}N
Kii=2,Ki,i+1=Ki,i-1=-1
12 | |
N(N-1) |
\rho=
12 | |
\sum |
e>0e
(\rho,\rho)= | 1 |
12 |
N(N2-1)
\omega1,...,\omegaN-1
(\omegai,ej)=\deltaij
P=
N-1 | |
\sum | |
i=1 |
Pi\omegai i.e. (ei,P)=Pi
q(h)
q(h)
h
* | |
e | |
i |
=eN-i
q(h)(P*)=(-1)hq(h)(P)
q(2)=
c+1-N | |
24 |
-(P,P)
b
c=(N-1)(1+N(N+1)\left(b+b-1\right)2)
e>0
r,s\inN*
(e,P)=rb+sb-1
P
rs
P-rbe
P-sb-1e
(e,P)\inN*b+N*b-1
The maximal number of null vectors is the number of positive roots
12N(N-1) | |
P=(b+b-1)\rho+b\Omega++b-1\Omega-
\Omega+,\Omega-
N-1 | |
\sum | |
i=1 |
N\omegai
ak{sl}N
l{R} | |
\Omega+,\Omega- |
The irreducible finite-dimensional representation
R\Omega
ak{sl}N
\Omega
Λ\Omega
|Λ\Omega|=\dim(R\Omega)
Vp
p\in
N-1 | |
\sum | |
i=1 |
R\omegai
R\Omega ⊗ Vp=
oplus | |
λ\inΛ\Omega |
Vp+λ
l{R} | |
\Omega+,\Omega- |
l{V}P
P
l{R} | |
\Omega+,\Omega- |
x l{V}P=
\sum | |||||||||||
|
\sum | |||||||||||
|
l{V} | ||||||||||
|
To a primary state of charge
\vec{q}=(q(h))h\in
V\vec{q
W(h)(z)
W(h)(y)V\vec{q
V(z)
L-1
L-1V(z)=
\partial | |
\partialz |
V(z)
On the Riemann sphere, if there is no field at infinity, we have
W(h)(y)\underset{y\toinfty}{=}O\left(y-2h\right)
n=0,1,...,2h-2
\ointinftydy ynW(h)(y)=0
W(h)(y)
2h-1
Local Ward identities are obtained by inserting
\ointinftydy \varphi(y)W(h)(y)=0
\varphi(y)
\varphi(y)\underset{y\toinfty}{=}O\left(y2h-2\right)
(h) | |
W | |
-n |
n\geqh
(h) | |
W | |
-n |
n\leqh-1
For example, in the case of a three-point function on the sphere
3 | |
\left\langle\prod | |
i=1 |
V\vec{qi}(zi)\right\rangle
L-1,W-1,W-2
\left\langleV\vec{q1}(z1)V\vec{q2}(z2)W
k | |
-1 |
V\vec{q3}(z3)\right\rangle
k\inN
In the W(3) algebra, as in generic W-algebras, correlation functions of descendant fields can therefore not be deduced from correlation functions of primary fields using Ward identities, as was the case for the Virasoro algebra. A W(3)-Verma module appears in the fusion product of two other W(3)-Verma modules with a multiplicity that is in general infinite.
A correlation function may obey a differential equation that generalizes the BPZ equations if the fields have sufficiently many vanishing null vectors.
A four-point function of W(N)-primary fields on the sphere with one fully degenerate field obeys a differential equation if
N=2
N\geq3
P1=(b+b-1)\rho+b\omega1
P2=(b+b-1)\rho+x\omegaN-1
x\inC
{}NFN-1
W-minimal models are generalizations of Virasoro minimal models based on a W-algebra. Their spaces of states are made of finitely many fully degenerate representations. They exist for certain rational values of the central charge: in the case of the W(N) algebra, values of the type
(N) | |
c | |
p,q |
=N-1-
| ||||
N(N |
with p,q\inN*
ck+N,k+N+1
SU(N)k x SU(N)1 | |
SU(N)k+1 |
For example, the two-dimensional critical three-state Potts model has central charge
(2) | |
c | |
5,6 |
(3) | ||
=c | = | |
4,5 |
45 | |
(p,q)=(5,6)
(p,q)=(4,5)
Conformal Toda theory is a generalization of Liouville theory that is based on a W-algebra. Given a simple Lie algebra
ak{g}
\phi
ak{g}
L[\phi]=
1 | |
2\pi |
(\partial\phi,\bar\partial\phi)+\mu\sume\in\}}\exp\left(b(e,\phi)\right)
\mu
b
ak{g}
\phi
The methods that lead to the solution of Liouville theory may be applied to W(N)-conformal Toda theory, but they only lead to the analytic determination of a particular class of three-point structure constants, and W(N)-conformal Toda theory with
N\geq3
At central charge
(2) | |
c=c | |
1,q |
2q-1
H=\{2,2q-1,2q-1,2q-1\}
q=2
c=-2
Finite W-algebras are certain associative algebras associated to nilpotent elements of semisimple Lie algebras.
The original definition, provided by Alexander Premet, starts with a pair
(ak{g},e)
ak{g}
Z
ak{g}
ak{g}=oplusak{g}(i).
\chi
ak{g}
\chi(x)=\kappa(e,x)
\kappa
\omega\chi(x,y)=\chi([x,y]).
l
ak{m}=l+oplusiak{g}(i).
I
U(ak{g})
\{x-\chi(x):x\inak{m}\}
U(ak{g})/I
(ak{m})
U(ak{g})
(U(ak{g})/I)ad(ak{m)}
(ak{g},e)
U(ak{g},e)