Rationality of Bershadsky-Polyakov vertex algebras

0.14 MB·English

by  Tomoyuki Arakawa

RATIONALITY OF W-ALGEBRAS, I: BERSHADSKY-POLYAKOV
VERTEX ALGEBRAS
0
1
TOMOYUKIARAKAWA
0
2

y
a
M
Abstract. WeprovetheconjectureofKac-Wakimotoontherationalityofex-
ceptionalW-algebrasforthefirstnon-trivialseries,namely,fortheBershadsky-
6 Polyakov vertex algebrasW(2) atlevelk=p/2−3withp=3,5,7,.... This
3
givesnewexamples ofrationalconformalfieldtheories.
]
A
Q
1. Introduction
.
h
Recently,aremarkablefamilyofW-algebrasassociatedwithsimpleLie algebras
t
a andtheirnon-principal nilpotentelements,calledexceptional W-algebras, hasbeen
m discoveredbyKacandWakimoto[KW08]. In[KW08]itwasconjecturedthatwith
[ an exceptional W-algebra one can associated a rational conformal field theory.

As a first step to resolve the Kac-Wakimoto conjecture we have proved in the
2
previous article [Ara10b] that (non-principal) exceptional W-algebras are lisse, or
v
5 equivalently [Ara10a], C2-cofinite. Therefore it remains [Zhu96, Hua08] to show
8 that exceptional W-algebras are rational, i.e., that the representations are com-
1 pletely reducible, in order to prove the conjecture. In this article we prove the
0
rationality of the first non-trivial series of exceptional W-algebras, that is, the
.
5 Bershadsky-Polyakov (vertex) algebras W(2) [Pol90, Ber91] at level k = p/2−3
0 3
with p = 3,5,7,.... The vertex algebra W(2) is the W-algebra associated with
0 3
1 g=sl3 and it minimal nilpotent element.
: Let us state our main result more precisely: Let W denote the unique simple
v k
i quotient of W(2) at level k 6=−3.
X 3
r Main Theorem (Conjectured by Kac and Wakimoto [KW08]). Let k = p/2−3
a with p = 3,5,7,.... Then the vertex algebra W is rational. The simple W -
k k
modules are parameterized by the set of integral dominant weights of sl of level
3
p−3. These simple modules can be obtained by the quantum BRST reduction from
irreducible admissible representations of sl of level k. b
3
For p = 3, W is one-dimensional. In the remaining cases W are con-
3/2−3 b p/2−3
formal with negative central charges. They have features in common with the
sl -integrableaffine vertex algebrasin the sense that their Zhu algebrasare closely
2
related with Smith’s algebra [Smi90] and that the following relations hold:
b
:G+(z)p−2 :=:G−(z)p−2 :=0.
This work is partially supported by the JSPS Grant-in-Aid for Scientific Research (B) No.
20340007.
1
2 TOMOYUKIARAKAWA
2. Bershadsky-Polyakov algebras at exceptional levels.
Let Wk denote the Bershadsky-Polyakov(vertex) algebra W(2) at level k 6=−3,
3
whichisthevertexalgebrafreelygeneratedbythefieldsJ(z),G±(z),T(z)withthe
following OPE’s:
2k+3
J(z)J(w)∼ , G±(z)G±(w)∼0,
3(z−2)2
1
J(z)G±(w)∼± G±(w),
z−w
(2k+3)(3k+1) 2 1
T(z)T(w)∼− + T(w)+ ∂T(w),
2(k+3)(z−w)4 (z−w)2 z−w
3 1
T(z)G±(w)∼ G±(w)+ ∂G±(w),
2(z−w)2 z−w
1 1
T(z)J(w)∼ J(w)+ ∂J(w),
(z−w)2 z−w
(k+1)(2k+3) 3(k+1)
G+(z)G−(w)∼ + J(w)
(z−w)3 (z−w)2
1 3(k+1)
+ 3:J(w)2 :+ ∂J(w)−(k+3)T(w) .
z−w 2
(cid:18) (cid:19)
As in introduction we denote by W the unique simple quotient of Wk.
k
Theorem 2.0.1 ([Ara10b]). Let k, p be as in Main Theorem. Then W is lisse.
k
Set
1
L(z)= L z−n−2 =T(z)+ ∂J(w).
n
2
n∈Z
X
This defines a conformal vector of Wk with central charge
4(k+1)(2k+3) 4(p−4)(p−3)
c(k)=− =− ,
k+3 p
which gives J, G+, G− conformal weights 1, 1, and 2, respectively. Hence Wk
is Z -graded with respect to the Hamiltonian L . We expand the corresponding
≥0 0
fields accordingly:
J(z)= J z−n−1, G+(z)= G+z−n−1, G−(z)= G−z−n−2.
n n n
n∈Z n∈Z n∈Z
X X X
Then we have
2k+3
[J ,J ]= mδ , [J ,G ]=G , [J ,F ]=−F ,
m n m+n,0 m n m+n m n m+n
3
(2k+3)(m+1)m
[L ,J ]=−nJ − δ ,
m n m+n m+n,0
6
[L ,G+]=−nG+ , [L ,G−]=(m−n)G− ,
m n m+n m n m+n
[G+,G−]=3(J2) +(3(k+1)m−(2k+3)(m+n+1))J −(k+3)L
m n m+n m+n m+n
(k+1)(2k+3)m(m+1)
+ δ ,
m+n,0
2
where (J2) z−n−2 :=:J(z)2 :.
n∈Z n
P
RATIONALITY OF W-ALGEBRAS, I: BERSHADSKY-POLYAKOV VERTEX ALGEBRAS 3
For (ξ,χ)∈C2, let L(ξ,χ)be the irreducible representationof Wk generatedby
the vector |ξ,χi such that
J |ξ,χi=ξ|ξ,χi, J |ξ,χi=0 (n>0),
0 n
L |ξ,χi=χ|ξ,λi, L |ξ,χi=0 (n>0),
0 n
G−|ξ,χi=0 (n≥0), G+|ξ,χi=0 (n≥1).
n n
By Theorem 2.0.1, any simple W -module is of the form L(ξ,λ) with some ξ and
k
χ. (It is important that the lisse condition is defined independent of the grading.)
For a Wk-module M set M = {m ∈ M;J m = am, L m = dm}. It is
a,d 0 0
clear that L(ξ,χ) = L(ξ,χ) , dimL(ξ,χ) = 1. Let L(ξ,χ) = {v ∈
a,d ξ,χ top
(a,d)∈C2
d∈χL+Z≥0
L(ξ,χ);L v = χv} = L(ξ,χ) . By definition L(ξ,χ) is spanned by the
0 a a,χ top
vectors (G+)i|ξ,χi with i≥0.
0 L
Following [Smi90] set
g(ξ,χ)=−(3ξ2−(2k+3)ξ−(k+3)χ),
so that G−G+|ξ,χi=g(ξ,χ)|ξ,χi. We have
0 0
G−(G+)i|ξ,χi=ih (ξ,χ)(G+)i−1|ξ,χi,
0 0 i 0
where
1
h (ξ,χ)= (g(ξ,χ)+g(ξ+1,χ)+···+g(ξ+i−1,χ))
i
i
=−i2+ki−3ξi+3i−3ξ2−k+2kξ+6ξ+kχ+3χ−2.
Hence we have the following assertion.
Proposition 2.0.2. The space L(ξ,χ) is n-dimensional only if h (ξ,χ)=0.
top n
Define
∞
−J
∆(−J,z)=z−J0exp (−1)k+1 k ,
kzk
!
k=1
X
and set
ψ(a )z−n−1 =Y(∆(−J,z)a,z)
(n)
n∈Z
X
for a ∈Wk. Then for any Wk-module M, we can define on M a new WK-module
structure by twisting the action of Wk as a 7→ ψ(a ) ([Li97]). We denote by
(n) (n)
ψ(M) thus obtained Wk-module from M.
Proposition 2.0.3. Suppose that dimL(ξ,χ) =i. Then
top
2k+3 2k+3
ψ(L(ξ,χ))∼=L(ξ+i−1− ,χ−(ξ−i+1)+ ).
3 3
Proof. The assertion follows from the fact that
2k+3 2k+3
ψ(J )=J − δ , ψ(L )=L −J + ,
n n n,0 n n n
3 3
ψ(G+)=G+ , ψ(G−)=G− .
n n−1 n n+1
(cid:3)
4 TOMOYUKIARAKAWA
By solving the equation
2k+3 2k+3
h (ξ,χ)=h (ξ+i−1− ,χ−(ξ−i+1)+ )
i j
3 3
we obtain the following assertion.
Proposition 2.0.4. Suppose that dimL(ξ,χ) = i and dimψ(L(ξ,χ)) = j.
top top
Then
1
ξ =ξ := (−2i−j+2k+6),
i,j
3
i2+ji−ki−3i+j2−6j−2jk+3k+6
χ=χ := .
i,j
3(k+3)
Proposition 2.0.5. Let k, p be as in Main Theorem. Then (G+ )p−21 belongs to
−1
the maximal ideal of Wk.
Proof. Because ξ = χ = 0, the corresponding 1 7→ |ξ ,χ i gives
1,p−2 1,p−2 1,p−2 1,p−2
anisomorphismW ∼=L(ξ ,χ ). Becauseh (ξ −(2k+3)/2,χ +
k 1,p−2 1,p−2 p−2 1,p−2 1,p−2
(2k+3)/3)= 0, from Proposition 2.0.3 it follows that ψ(W ) is at most p−2-
k top
dimensional. Hence (G+ )p−21=0. (cid:3)
−1
Remark 2.0.6. One can show that in fact (G+ )p−2 generatesthe maximalideal of
−1
Wk. However we do not need this fact.
Proposition2.0.7. Let k, p beas in Main Theorem. Then any simple W -module
k
isisomorphic toL(ξ ,χ )for some(i,j)suchthat1≤i≤p−2,1≤j ≤p−i−1.
i,j i,j
Proof. LetL(ξ,χ)beasimpleW -module. Because:G+(z)p−2 :=0byProposition
k
2.0.5,L(ξ,χ) isatmost(p−2)-dimensional. Sinceψ(L(ξ,χ))isalsoaW -module
top k
we have (ξ,χ)=(ξ ,χ ) for some 1≤i,j ≤p−2. Because ψ(ψ(L(ξ ,χ ))) is
i,j i,j i.j i,j
alsoa W -moduleit followsthatξ +i−1−2k+3 = i−j ≤ −2j−1+2k+6 = p−2j−1.
k i,j 3 3 3 3
Hence j ≤p−i−1. (cid:3)
Remark 2.0.8. The simple Wk-modules L(ξ ,χ ) with 1 ≤ i ≤ p−2, 1 ≤ j ≤
i,j i,j
p−i−1, are mutually non-isomorphic because their highest weights are distinct.
3. Proof of Main Theorem
Let k, p be as in Main Theorem.
Let g = sl as in introduction, h ⊂ g be the Cartan subalgebra of g consisting
3
of diagonal matrixes. Set h = E −E , h = h +h , e = e = E ,
i i,i i+1,i+1 θ 1 2 i αi i,i+1
f = f = E for i = 1,2, e = E , f = E , where E is the matrix
i αi i+1,i θ 1,3 θ 3,1 i,j
element. We equip g the invariant form (x|y) = tr(xy). Set Λ¯ = (2h +h )/3,
1 1 2
Λ¯ =(h +2h )/3, so that (Λ¯ |h )=δ .
1 1 2 i j i,j
Let g = g[t,t−1]⊕CK⊕CD be the (non-twisted) affine Kac-Moody algebra
associated with g, where K is the central element and D is the degree operator.
Let h =bh⊕CK⊕CD ⊂ g the standard Cartan subalgebra, h∗ = h∗⊕CΛ0⊕Cδ
the dual of h, where Λ and δ are elements dual to K and D, respectively.
0
Lebtus consider the minbimalnilpotent elementfθ andthe cobrrespondingDynkin
gradingofgb: g ={u∈g;[h ,u]=2ju}. Denote by H∞2+0(?) the BRST cohomol-
j θ fθ
ogy ofthe generalizedquantized Drinfeld-Sokolovreduction associatedwith (g,f )
θ
RATIONALITY OF W-ALGEBRAS, I: BERSHADSKY-POLYAKOV VERTEX ALGEBRAS 5
andthe correspondingDyndin grading. Then we have[KRW03, KW04] the vertex
algebra isomorphism
Wk →∼ H∞2+0(Vk(g)),
fθ
which is given by the following assignment:
J(z)7→J−Λ¯1+Λ¯2(z)−:Φ (z)Φ (z):,
1 2
G+(z)7→Jf1(z)−:Jh1(z)Φ (z):+:Φ (z)Φ (z)2 :−(k+1)∂Φ (z),
2 1 2 2
G+(z)7→−Jf2(z)−:Jh2(z)Φ (z):−:Φ (z)2Φ (z):−(k+1)∂Φ (z),
1 1 2 1
Here Ju(z) = u(z) − cfuγ,fβ : ψβ∗(z)ψγ(z) : for u ∈ g, cuu31,u2 is the
β,γ∈{α1,α2,θ}
structureconstant,ψ (z),ψ∗P(z)withα∈{α ,α ,θ}arefermionicghostssatisfying
α α 1 2
δ
(1) ψ (z)ψ∗(w)∼ α,β , ψ (z)ψ (w)∼ψ∗(z)ψ (∗w)∼0,
α β z−w α β α β
Φ (z), Φ (z) are bosonic ghosts satisfying
1 2
1
Φ (z)Φ (w)∼ , Φ (z)Φ (w)∼0,
1 2 i i
z−w
and the BRST differential is the zero mode of the field
Q(z)= e (z)ψ∗(w)−:ψ∗ (z)ψ∗ (z)ψ (z):
α α α1 α2 θ
α∈{αX1,α2,θ}
+Φ (z)ψ∗ (z)+Φ (z)ψ (z)+ψ (z),
1 α1 2 α2 θ
Let O be the category O of g at level k, L the irreducible representation of g
k λ
with highest weight λ. Let Wk-Mod be the category of Wk-modules.
Theorem 3.0.9 ([Ara05]). b(i) The functor H∞2+0(?) : O → Wk-Mod ibs
fθ k
exact.
(ii) For L ∈O we have H∞2+0(L )=0 if and only if λ(α∨)∈{0,1,2,...}.
λ k fθ λ 0
Otherwise H∞2+0(L ) is irreducible.
fθ λ
Let Admk be the set of admissible weights [KW89] of g of level k, and put
Admk ={λ∈Admk;λ¯ is an integral dominant weight of g},
+
b
where h∗ ∋λ7→λ¯ ∈h∗ is the restriction. Then
Admk ={µ¯+kΛ ;µ∈Pp−3},
b + 0 ++
where Pp−3 is the set of integral dominant weights of g of level p−3. Explicitly,
++ b
we have
b Admk ={λ ;1≤i≤p−2, 1≤j ≤bp−i−1},
+ i,j
where
λ =(i−1)Λ¯ +(p−i−j−1)Λ¯ +kΛ .
i,j 1 2 0
Note that
(λ |λ +2ρ¯)
(2) ξ =(λ |−Λ¯ +Λ¯ ), χ = i,j i,j −(λ |Λ¯ ),
i,j i,j 1 2 i,j i,j 2
2(k+3)
where ρ¯=Λ¯ +Λ¯ .
1 2
Recall the following result of Malikov and Frenkel [MF99].
6 TOMOYUKIARAKAWA
Theorem 3.0.10 ([MF99]). For λ∈Admk, L is a module over L .
+ λ kΛ0
Proposition 3.0.11. For λ ∈Admk, H∞2+0(L ) is a simple W -module iso-
i,j + fθ λi,j k
morphic to L(ξ ,χ ).
i,j i,j
Proof. By Theorem 3.0.9 we have W ∼=H∞2+0(L ). Hence by the functoriality
k fθ kΛ0
ofH∞2+0(?),Theorem3.0.10immediatelygivesthatH∞2+0(L )isaW -module.
fθ fθ λi,j k
This module is (nonzero and) irreducible by Theorem 3.0.9. Let v be the image
∞+0
of the highest weight vector of L in H 2 (L ). By (2) and the fact [KW04]
λi,j fθ λi,j
that the image of L(z) in Wk is cohomologous to
Lg(z)+Lch(z)+LΦ(z)+∂JΛ¯2(z),
where L (z) is the Sugawara operator of g, L (z)=− : φ (z)∂φ∗(z),
g ch α=α1,α2,θ α α
L (z)= 1(:Φ (z)∂Φ (z):−:∂Φ (z)Φ (z)),itisstraightforwardtocheckthatthe
Φ 2 2 1 1 2 P
assignment|ξ ,χ i7→vgivesaWk-modulehomomorphism. Bytheirreducibility,
i,j i,j
this must be an isomorphism. (cid:3)
By Propositions 2.0.7 and 3.0.11, {H∞2+0(L );λ ∈ Admk} gives the complete
set of isomorphism classes of simple Wf-θmodulλes. Therefor+e Main Theorem now
k
followsimmediatelyfromthefollowingimportantresultofGorelikandKac[GK09].
Theorem 3.0.12 ([GK09, Corollary 8.8.9]). For any λ,µ∈Admk, we have
Ext1Wk-Mod(Hf∞2θ+0(Lλ),Hf∞2θ+0(Lµ))=0.
.
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Departmentof Mathematics,Nara Women’s University,Nara 630-8506,JAPAN
E-mail address: [email protected]