AW-algebra is an extension of the Virasoro algebra, with additional conserved currents Wpsiqpzqwhich are the quasi-primaries of conformal dimensions si. It includesWp2qpzq “ Tpzqand the OPE ofW-currents has be closed, which means that every terms should be written in terms ofWpsiqpzqand their derivatives. One can make a formal definition of the W-algebras using the notion of meromorphic conformal field theory [94].
The simplest example of the W-algebra called W3 is first written by Zamolodchikov [101]. It consists of two currentsTpzqandWpzqwhich are quasi-primaries of dimension 2 and 3. The OPE of dimension 3 currents are
WpzqWpwq „ c{3
pz´wq6 ` 2Tpwq
pz´wq4 ` BTpwq pz´wq3
` 1
pz´wq2 ˆ
2βΛpwq ` 3
10B2Tpwq
˙
(4.42)
` 1
pz´wq2 ˆ
βΛpwq ` 1
15B2Tpwq
˙ ,
where
Λpwq “ pTTqpwq ´ 3
10B2Tpwq, β“ 16
22`5c. (4.43)
The form of OPE is fixed by requiring the crossing symmetry of the correlation function.
In terms of Laurent modes
Tpzq “ ÿ
nPZ
Ln
zn`2, Wpzq “ ÿ
nPZ
Wn
zn`3, (4.44)
the commutation relations are given by
rLm,Wns “ p2m´nqWm`n (4.45) and
rWm,Wns “ c
360mpm2´1qpm2´4qδm`n,0
` pm´nq
„1
15pm`n`3qpm`n`2q ´1
6pm`2qpn`2q
Lm`n(4.46)
`βpm´nqΛm`n
where
Λm “ ÿ
nPZ
pLm´nLnq ´ 3
10pm`3qpm`2qLm. (4.47)
One can use these commutation relations and construct the Verma module corresponding to the primary states with respect to bothTpzqandWpzq. One can start with a state with
L0|h,wy “h|h,wy, W0|h,wy “w|h,wy, Ln|h,wy “Wn|h,wy “0, pną0q (4.48) and then form a W-descendeant states by applyingW´k orL´k1successively
W´k1¨ ¨ ¨L´k1
1¨ ¨ ¨ |h,wy, ki,k1j PN. (4.49) These states form the highest weight module of theW3-algebra.
Even though it is possible to directly construct ofW-algebra by starting with currents with higher conformal dimension and try to make the algebra close, it is very hard to construct the W-algebra with many high-dimensional currents. The direct construction has been done only up to the case with 3 generators [102, 103], but not beyond.
There is no complete classification allWalgebras, but many examples are known and have been studied. One particular family of examples are the the so-called Casimir alge- bras, which are based on simply laced Lie algebras. Its generators are constructed from theg-invariant contractions of the current field Jpzqof the affine Lie algebrag. The series of WN-algebras, for instance, is related to the AN Lie algebras. In [91] WN-blocks have
been related to the instanton partition functions corresponding toUpNq gauge theories.
It is natural to expect that also the other Casimir algebras appear as dual descriptions of instanton counting.
There is a more systematic method of obtaining the W-algberas called the quantum Drinfeld-Sokolov (DS) reduction [104, 105, 106]. It starts from an affine Lie algebra ˆgand impose some constraints on the generators by the BRST procedure. The reduced algebra Wpgqis given by the BRST cohomology.
4.2.1 Chiral Blocks and Twisted Representations
Since the spectrum of the CFT decomposes into representations of theW algebra, we can use generalized Ward identities to relate correlation functions of (W-)descendant fields to correlation functions of (W-)primary fields. In the case of the Virasoro algebra, we can always reduce them to functions of primary fields only. For generalW-algebras this is only possible if one restricts to primary fields on which theW-fields satisfy additional null relations.
We can make use of this property by computing chiral blocks. For a given configuration of a punctured Riemann surface, we define the chiral block by picking a representationφ for every tube, inserting the projector on the representationPHφ at that point in the corre- lator, and dividing by the product of all three point functions of the primary fields. By the above remarks the result is then independent of the three-point functions of the theory, i.e., it only depends on the kinematics of the theory.
In the simplest configuration, the sphere with four punctures, the chiral block is thus given by
F “ xV1p8qV2p1qPHφV3pqqV4p0qy
xV1p8qV2p1q|φyxφ|V3p1qV4p0qy . (4.50) On the gauge theory side it corresponds to the partition function including the perturba- tive contribution. Also, let us take the convention in what follows that whenever we write a correlator, we assume that it is divided by the appropriate primary three point functions.
The projector is usually written as PHφ “ÿ
I,J
|φIyxφJ|pK´1qI,J (4.51)
where I “ pi1,i2, . . .q denotes the W descendants, such thatφI “ W´i1W´i2¨ ¨ ¨φ is a W descendant.Kis the inner product matrix and the sum runs over allW descendants ofφ.
A representationφofWis called untwisted if it is local with respect toW, so that one can freely moveW-fields around it. TheW-fields then have integer mode representations around that representation.
More generally, theW-fields can pick up phases when circling aroundφ, so that the correlation function has a branch cut extending fromφ. Suchφare called twisted repre- sentations. Because of the phaseαpicked up by the W-fields, their modes are no longer integer, but given byrPZ`α.
A particular case of twisted representations can appear when theW has an outer auto- morphism such as aZN-symmetry. Let us say that by circling around a twisted represen- tation the algebraW gets mapped to an image underZN, such as
WkÞÑWk`1, k“1, . . .N. (4.52)
By choosing linear combinations Wpkq of the modes Wk that are eigenvectors under the automorphisms, theWpkqindeed pick up phases 2πik{N. ForN “2, the case that we are interested in below, theW-algebra thus decomposes into generatorsW`,W´ of integer and half-integer modes, respectively.
Let us finally note that in the case of Liouville theory the W-algebra is simply the Virasoro algebra. Example of conformal field theories with biggerW-algebras are Toda theories.