Table Of ContentCOINVARIANTS OF NILPOTENT SUBALGEBRAS OF
THE VIRASORO ALGEBRA AND PARTITION
IDENTITIES
BORIS FEIGIN AND EDWARD FRENKEL
3
9 Dedicated to I.M. Gelfand on his 80th birthday
9
1
n
a 1. Introduction
J
3 Let Vm,n be the irreducible representation of the Virasoro algebra L
p,q
1 with central charge c = 1−6(p−q)2/pq and highest weight h =
p,q m,n
1 [(np−mq)2−(p−q)2]/4pq, where p,q > 1 are relatively prime integers,
v and m,n are integers, such that 0 < m < p,0 < n < q. For fixed p
9
and q the representations Vm,n form the (p,q) minimal model of the
3 p,q
0 Virasoro algebra [1].
1
For N > 0 let L be the Lie subalgebra of the Virasoro algebra,
0 N
3 generated by Li,i < −N. There is a map from the Virasoro algebra
9 to the Lie algebra of polynomial vector fields on C∗, which takes L to
i
/
h z−i+1 ∂ , where z is a coordinate. This map identifies L with the Lie
-t algebr∂az of vector fields on C, which vanish at the originNalong with the
p
firstN+1derivatives. TheLiealgebraL hasafamilyofdeformations
e 2N
h L(p ,...,p ), which consist of vector fields, vanishing at the points
1 N+1
v: p ∈ C along with the first derivative.
i
i
X For a Lie algebra g and a g−module M we denote by H(g,M) the
r spaceofcoinvariants (or0thhomology)ofg inM, which isthequotient
a
M/g·M, where g·M is the subspace of M, linearly spanned by vectors
a·x,a ∈ g,x ∈ M. We will prove the following result.
Theorem 1. For any irreducible representation V of the (2,2r + 1)
minimal model
dimH(L ,V) = dimH(L(p ,...,p ),V).
2N 1 N+1
This statement was proved in [2] for N = 2 and irreducible repre-
sentations of general minimal models.
As an application, we will give a new proof of the Gordon identities,
which relies on two different computations of the characters of the
irreducible representations of the (2,2r+1) minimal models.
Research of the second author was supported by a Junior Fellowship from the
Harvard Society of Fellows and by NSF grant DMS-9205303.
1
2 BORIS FEIGIN AND EDWARD FRENKEL
Letusnotethata prioridimH(L ,V) ≥ dimH(L(p ,... ,p ),V).
2N 1 N+1
Indeed, if we have a family of Lie algebras, then the dimension of ho-
mology is the same for generic points of the space of parameters of the
family, but it may increase at special points.
Each minimal model of the Virasoro algebra associates to a punc-
tured complex curve, with a representation inserted at each puncture,
a linear space, which is called thespace ofconformalblocks. This space
can be defined as the space of coinvariants of the Lie algebra of mero-
morphic vector fields on this curve, which are allowed to have poles
only at the punctures, in the tensor product of these representations
(cf. [1], mathematical aspects of this correspondence are treated in
detail in [2], [3]).
In this language, the space H(L(p ,... ,p ),Vm,n) is isomorphic
1 N+1 p,q
to the direct sum of the spaces of conformal blocks, associated to the
projective line with punctures p ,... ,p with all possible insertions
1 N+1
from the (p,q) minimal model, and ∞ with the insertion of Vm,n. The
p,q
dimension of this space can be calculated by a Verlinde type argument
by deforming our curve – the projective line with N +2 punctures – to
a joint union of N projective lines, each with 3 punctures. Under this
deformationthedimensionofthehomologydoesnotchange[2],[3],and
can therefore be reduced to the well-known result on the dimensions of
the spaces of conformal blocks, associated to the projective line with 3
insertions – the fusion coefficients.
ThisgivesanexplicitformulafordimH(L(p ,... ,p ),Vm,n),which
1 N+1 p,q
represents a lower bound for dimH(L ,Vm,n). We can also obtain an
2N p,q
upper bound by a different method.
This method is based on the calculation of the annihilating ideal of a
minimal model [4]. Let us recall the definition of the annihilating ideal.
For c ∈ C let M be the vacuum Verma module with central charge
c
c. It is generated by a vector v , such that L v = 0,i ≥ −1. This
c i c
representation has the structure of a vertex operator algebra (VOA)
[5], which acts on any irreducible module M of the Virasoro algebra
with the same central charge. It means that each vector of M defines
c
a local current (or field), which is a formal power series in z and z−1,
whose Fourier coefficients are linear operators, acting on M. These
local currents can be constructed as follows. Consider the projective
line CP1 with three punctures: 0 and ∞ with the insertions of M,
and z with the insertion of M . The space of conformal blocks is one-
c
dimensional in this case. In other words, the space of coinvariants of
the Lie algebra A(0,z) of meromorphic vector fields on CP1, which
are allowed to have poles only at 0,∞ and z, in M ⊗ M ⊗ M , is
c
one-dimensional. It has a canonical generator – the projection of the
COINVARIANTS OF SUBALGEBRAS OF THE VIRASORO ALGEBRA 3
tensorproductofthehighest weight vectors. Thedualtothisgenerator
defines for eachvector of M alinear operatorfromM to M, depending
c
on z, which is our local current.
These local currents can be constructed explicitly. The monomials
{L ...L v |m ≥ m ≥ ··· ≥ m > 1}
−m1 −ml c 1 2 l
linearly span M . The corresponding local currents are equal to
c
1 1
... : ∂m1−2T(z)...∂ml−2T(z) :, (1)
(m −2)! (m −2)! z z
1 l
where T(z) = L z−i−2. The Fourier components of thus defined
i∈Z i
P
local currents form a Lie algebra U (L) , which is called the local
c loc
completion of the universal enveloping algebra of the Virasoro algebra
withcentralchargec[6],[4]. They actonanyrepresentationofL,which
has the same central charge.
The module M is irreducible, if and only if c is not equal to c .
c p,q
However, if c = c , then M contains a (unique) singular vector. The
p,q c
quotient of M by the submodule, generated by this singular vector,
cp,q
is isomorphic to V1,1. This representation, which is called the vacuum
p,q
representation ofthe(p,q)minimalmodel, defines another VOA,which
is the quotient of the VOA of M in the appropriate sense [7]. An ir-
cp,q
reducible representation M of the Virasoro algebra with central charge
c is a module over this VOA [7], if and only if the space of coinvari-
p,q
ants of the Lie algebra A(0,z) in M⊗M⊗V1,1 is one-dimensional. An
p,q
explicit computation of the dimension of this space, which was made in
[2], tells us that it is so, if and only if M is an irreducible representation
of the (p,q) minimal model.
The Fourier components of the local current, corresponding to the
singular vector of M , generate an ideal in U (L) , which is called
cp,q cp,q loc
the annihilating ideal of the (p,q) minimal model. Any element of this
ideal acts trivially on any representation of the (p,q) minimal model.
Imposing this condition with respect to the generators of the anni-
hilating ideal immediately leads to certain linear relations among the
monomial elements of Vm,n. This allows to estimate the dimension of
p,q
the coinvariants H (L ,Vm,n) from above.
0 2N p,q
The peculiarity of the (2,2r + 1) models is that this upper bound
coincides with the lower bound, and is therefore exact. This proves
Theorem 1 and leads to a nice combinatorial description of the minimal
representations of these models. Namely, as a Z−graded linear space,
such a representation is isomorphic to the quotient of the space of
polynomials in infinitely many variables by a certain monomial ideal.
Thus we obtain an expression for the character of this module, which
4 BORIS FEIGIN AND EDWARD FRENKEL
coincides with the right hand side of one of the Gordon identities. But
it is known that this character is equal to the left hand side of the
identity. Hence, we obtain a new proof of the Gordon identities.
Implementing this program for other minimal models would lead to
a result, analogous to Theorem 1, as well as to nice character formulas
for irreducible representations. However, for general minimal models,
for which the representations are “smaller” and the structure of the
annihilating ideal is more complicated, one should impose extra con-
ditions, corresponding to other elements of the annihilating ideal, to
obtain the exact upper bound.
2. Lower bound: genus 0 conformal blocks
InthissectionwewillcalculatethedimensionofH(L(p ,... ,p ),V)
1 N+1
for an irreducible representation V of a (p,q) minimal model.
Denote by A(p ,... ,p ) the Lie algebra of meromorphic vector
1 N+1
fields on CP1, which are allowed to have poles only at the distinct
points p ,... ,p ∈ C, and ∞. There is an embedding of this Lie
1 N+1
algebra into the direct sum of N+2 Virasoro algebras, and thus it acts
on tensor products V ⊗ ... ⊗ V of N + 2 reprersentations of the
1 N+2
Virasoro algebra with the same central charge.
Let us fix central charge c and an irreducible representation V =
p,q
V of the (p,q) minimal model. The following statement follows
N+2
from the general results on computation of conformal blocks, outlined
in Section 5 of [2] (cf. [3] for details).
Proposition 2. (1) dimH(L(p ,... ,p ),V) = dimH(A(p ,... ,p ),V ⊗
1 N+1 1 N+1 1
P
... ⊗ V ⊗ V), where the sum is taken over all (N + 1)−tuples
N+1
V ,... ,V of irreducible representations of the (p,q) minimal model.
1 N+1
(2) dimH(A(p ,... ,p ),V ⊗...⊗V ⊗V) = dimH(A(p ,... ,p ),V ⊗
1 N+1 1 N+1 1 N 1
P
...⊗V ⊗W)·dimH(A(0,1),W ⊗V ⊗V), where the sum is taken
N N+1
over all irreducible representations W of the (p,q) minimal model.
Using this result, we can calculate dimH(L(p ,... ,p ),V) by in-
1 N+1
duction.
LetV ,... ,V bethesetofallirreduciblerepresentations ofthe(p,q)
1 s
minimal model (s = (p − 1)(q − 1)/2), and let u be the s−vector,
N
whose components are ui = dimH(L(p ,... ,p ),V ),i = 1,... ,s.
N 1 N+1 i
Put c = dimH(A(0,1),V ⊗V ⊗V ). The numbers c are usually
ijk i j k ijk
called the fusion coefficients. They define the fusion algebra of this
minimalmodel, which hasgeneratorsg ,1 ≤ i ≤ s andrelationsg ·g =
i i j
c g .
k ijk k
P
COINVARIANTS OF SUBALGEBRAS OF THE VIRASORO ALGEBRA 5
Let us introduce the s × s matrix M = M , whose (i,j)th entry
p,q
is equal to c . This matrix can be interpreted as the matrix of
k ijk
P
action of the sum g on the fusion algebra.
k k
P
According to Proposition 2, ui = c uj . This gives us the
N Pj,k ijk N−1
recursion relation for u :
N
u = Mu .
N N−1
For any i, the representation V is generated from the highest weight
i
vector v by the action of the Lie algebra L , hence ui = 1. Therefore,
i 0 0
we obtain the formula
u = MNu ,
N 0
where ut = [1,1,... ,1].
0
This formula enables us to calculate the dimensions ui explicitly by
N
diagonalizing the matrix M.
Intherestofthepaperwewillfocusonthe(2,2r+1)models. Insuch
a model we have r irreducible representations V = V1,i ,i = 1,... ,r.
i 2,2r+1
The (i,j)th entry of the corresponding r × r matrix M is equal to
min{i,j}. This matrix is equal to the square of the matrix
0 0 ... 0 1
0 0 ... 1 1
M (x) = ................ (2)
r
0 1 ... 1 1
1 1 ... 1 1
This fact has the following interpretation: in the fusion algebra of
this model g is equal to the square of g , and the action of g is
k k r r
P
given by matrix M .
r
Let us introduce vectors x , whose ith component is equal to 1, and
i
all other component are equal to 0. Since u = M2x , we have ui =
0 r 1 N
xtM2N+2x . This gives us a lower bound for vi = dimH(L ,V ).
i r 1 N 2N i
Proposition 3. vi ≥ xtM2N+2x .
N i r 1
In the next Section we will show that this bound is exact.
3. Upper bound: the annihilating ideal
The annihilating ideal of the (2,2r+1) minimal model is the ideal of
the Lie algebra U (L) , generated by the Fourier components of
c2,2r+1 loc
the local current, corresponding to the singular vector of the vacuum
Verma module M . All elements of this ideal act by 0 on any irre-
c2,2r+1
ducible representation of the (2,2r+1) minimal model. In particular,
6 BORIS FEIGIN AND EDWARD FRENKEL
the generators of the ideal act by 0. This leads to certain relations be-
tween vectors in the irreducible minimal representations, as explained
in [4].
Introduce a filtration on the irreducible representation V of the
i
(2,2r + 1) models as follows: 0 ⊂ V0 ⊂ V1 ⊂ ... ⊂ V∞ = V , where
i i i i
Vk is linearly spanned by the monomials L ...L v , such that
i −m1 −ml i
m ≥ m ≥ ··· ≥ m ≥ 1, and l ≤ k. Let Ω be the adjoint graded
1 2 l i
space:
Ω = ⊕ Ωk, Ωk = Vk/Vk−1.
i k≥0 i i i i
The space Ω is a polynomial algebra in variables a ,j > 0, where a is
i j j
the image of L v .
−j i
The symbol of the singular vector of M is equal to Lr v . Ac-
c2,2r+1 −2 0
cording to formula (1), the symbol of the corresponding local current
is equal to : T(z)r :. Hence the symbols of the Fourier components of
this current are given by the formula
: L ...L : .
X −j1 −jr
j1+...+jr=n
Their action on Ω is given by the multiplication by
i
S = a ...a , (3)
n X j1 jr
jk>0;j1+...+jr=n
for n ≥ r, and 0 for n < r.
It is known that V is the quotient of the Verma module with high-
i
est weight h by the maximal submodule, generated by two singular
1,i
vectors. The symbol of one of them is equal to Li v . This gives
−1 i
us a surjective map from the quotient Ω′ of C[a ] by the ideal I
i j j>0 i
generated by S ,n > r, and ai, to Ω .
n 1 i
Let Ωmon be the quotient of C[a ] by the ideal Imon, generated by
i j j>0 i
the monomials auav ,0 ≤ u < r,u+v = r,j > 0, and ai.
j j+1 1
We can introduce a Z−grading on the module V by putting degv =
i i
0,degL = j. The space Ω inherits this grading. We can also
−j i
introduce a compatible Z−grading on the space C[a ] by putting
j j>0
dega = j. The spaces Ω′ and Ωmon inherit this grading.
−j i i
ForanyZ−gradedlinearspaceV = ⊕ V(n),suchthatdimV(n) ≤
n≥0
∞, let chV = dimV(n)qn be its character. We will write chV ≤
Pn≥0
chV′, if dimV(n) ≤ dimV′(n) for any n ≥ 0.
We have
chV = chΩ ≤ chΩ′ ≤ chΩmon.
i i i i
The last inequality follows from the fact that each of the generators
S of the ideal I has as a summand one and only one generator of the
n i
ideal Imon, namely, auav is a summand of S .
i j j+1 rj+v
COINVARIANTS OF SUBALGEBRAS OF THE VIRASORO ALGEBRA 7
Now, by the construction of our filtration, the character of coinvari-
ants H(L ,V ) is equal to the character of the quotient Ω of the
2N i i,N
space Ω by the ideal, generated by a with j > 2N. Again, we have:
i j
chH(L ,V ) = chΩ ≤ chΩmon, (4)
2N i i,N i,N
where Ωmon is the quotient of Ωmon by the ideal, generated by a ,j >
i,N i j
2N. Therefore,
vi = dimH(L ,V ) ≤ ωi = dimΩmon.
N 2N i N i,2N
We can easily calculate ωi by induction [8].
N
Let Cr = {(m ,...,m )|m ≥ ··· ≥ m ≥ 1,m ≥ m +
i 1 l 1 l i i+r−1
2,m > 1}. The monomials
l−i+1
{a ...a |(m ,...,m ) ∈ Cr}
m1 ml 1 l i
constitute a linear basis in Ωmon.
i
For every integer N > 0, introduce the subspaces Wi ,1 ≤ k ≤ r,
k,N
of Ωmon, which are linearly spanned by the monomials
i
{a ...a |(m ,...,m ) ∈ Cr,m ≤ N,m ≤ N −1}.
m1 ml 1 l i 1 k
Clearly, Wi is isomorphic to Ωmon .
1,N i,N−1
The dimension of the space Wi , which is spanned by our monomi-
k,N
als, in which a is allowed in the power less than k, is equal to the sum
N
of dimensions of the spaces Wi with l = 1,... ,r −k +1, because
l,N−1
we have the relations au av = 0 for u+v = r. Let us introduce the
N−1 N
r−vector wi , whose components are wi = dimWi ,k = 1,... ,r.
N k,N k,N
We obtain the formula [8]:
wi = M wi ,
N r N−1
where M is given by (2), which shows that wi = M2Nwi.
r 2N+1 r 1
Remark 1. As shown in [8], there is a q−deformation of this formula,
which gives an expression for the characters of Wi .
k,N
By definition, wi = min{i,k}. One can check that wi = M2x ,
k,1 1 r i
and so wi = M2N+2x . Therefore, ωi = wi = xtM2N+2x ,
2N+1 r i N 1,2N+1 1 r i
and this gives us an upper bound for vi .
N
Proposition 4.
vi ≤ xtM2N+2x .
N 1 r i
ButsinceMt = M ,thisupperboundcoincideswiththelowerbound
r r
from Proposition 3. Therefore, we have the equality
ui = vi = ωi = xtM2N+2x .
N N N 1 r i
This completes the proof of Theorem 1.
NotethatwehavealsoprovedthatΩ isisomorphictoΩ′ = C[a ] /I .
i i j j>0 i
8 BORIS FEIGIN AND EDWARD FRENKEL
Remark 2. For general minimal models the graded space Ω of an ir-
reducible representation is also isomorphic to the quotient of C[a ]
j j>0
by a certain ideal I. This ideal contains the operators S , given by
n
formula (3) with r = (p−1)(q −1)/2. They correspond to the action
of the symbols of generators of the annihilating ideal of the minimal
model. In general, however, these elements do not generate the ideal I.
There are other generators, corresponding to the action of the symbols
of other elements of the annihilating ideal on Ω. It is an interesting
problem to find explicit formulas for them. This will hopefully lead to
a nice combinatorial description of general irreducible representations.
4. Application: Gordon identities
By formula (4), dimH(L ,V )(n) ≤ dimΩmon(n) for any n and N.
2N i i,2N
In the previous Section we proved that dimH(L ,V ) = dimΩmon,
2N i i,2N
therefore dimH(L ,V )(n) = dimΩmon(n) for any n and N. But,
2N i i,2N
clearly,dimH(L ,V )(n) = dimV (n)anddimΩmon(n) = dimΩmon(n)
2N i i i,2N i
for N large enough. Hence dimV (n) = dimΩmon(n) for any n, and
i i
chV = chΩmon. This can be interpreted as follows.
i i
Proposition 5. The monomials
{L ...L v |(m ,...,m ) ∈ Cr}
−m1 −ml i 1 l i
constitute a linearbasisin the irreduciblerepresentationV of the (2,2r+
i
1) minimal model.
Thus, weobtainthefollowingformulaforthecharacterofthemodule
V :
i
chV = |Cr(n)|qn, (5)
i X i
n≥0
whereCr(n)isthesubsetofCr,whichconsistsoftheelements(m ,... ,m ) ∈
i i 1 l
Cr, for which m = n. The right hand side of the formula (5) is
i j
P
known to be equal to
qN12+...+Nr2−1+Ni+...+Nr−1
,
X (q) ...(q)
n1,...,nr−1≥0 n1 nr−1
where N = n +...+n , and (q) = l=n(1−ql) (cf. [9]).
j j r−1 n Ql=1
On the other hand, it is known [10, 11] that
chV = (1−qn)−1.
i Y
n>0,n6=0,±imod(2r+1)
This formula follows from the Weyl-Kac type character formula for
irreducible minimal representations of the Virasoro algebra.
COINVARIANTS OF SUBALGEBRAS OF THE VIRASORO ALGEBRA 9
Thus, we have obtained a new proof of the Gordon identities:
qN12+...+Nr2−1+Ni+...+Nr−1
(1−qn)−1 = .
Y X (q) ...(q)
n>0,n6=0,±imod(2r+1) n1,...,nr−1≥0 n1 nr−1
For r = 2 these are the Rogers-Ramanujan identities:
qn(n+1) qn2
(1−qn)−1 = and (1−qn)−1 = .
Y X (q) Y X (q)
n n
n>0,n6=0,±1mod5 n≥0 n>0,n6=0,±2mod5 n≥0
Theycorrespondtotwoirreduciblerepresentationsofthe(2,5)minimal
model.
Acknowledgements The main part of this work was done while B.F.
was visiting the Isaac Newton Institute of the University ofCambridge.
He would like to the thank the Institute for hospitality. E.F. thanks
A.Szenes for valuable discussions.
References
1. A.Belavin,A.Polyakov,A.Zamolodchikov,Infiniteconformal symmetry in two-
dimensional quantum field theory, Nucl. Phys. B241 (1984) 333-380.
2. B.L.Feigin, D.B.Fuchs, Cohomology of some nilpotent subalgebras of the Vi-
rasoro and Kac-Moody algebras, in Geometry and Physics, Essays in honor
of I.M.Gelfand on the occasion of his 75th birthday, eds. S.Gindikin and
I.M.Singer, 209-235.North-Holland, 1991.
3. A.Beilinson,B.Feigin,B.Mazur,Introductiontoalgebraicfieldtheoryoncurves,
to appear.
4. B.Feigin, T.Nakanishi, H.Ooguri, The annihilating ideals of minimal models,
Int. J. Mod. Phys. Suppl. 1A (1992) 217-238.
5. I.Frenkel,Y.Zhu,Vertexoperatoralgebrasassociatedtorepresentationsofaffine
and Virasoro algebras, Duke Math. Journal 66 (1992) 123-168.
6. B.Feigin,E.Frenkel,AffineKac-MoodyalgebrasatthecriticallevelandGelfand-
Dikii algebras, Int. J. Mod. Phys. Suppl. 1A (1992) 197-215.
7. I.Frenkel,Y.-Z.Huang,J.LepowskyOnaxiomatic approaches to vertexoperator
algebras and modules, Preprint, 1990.
8. E.Frenkel,A.Szenes,Dilogarithmidentities,q−differenceequations,andtheVi-
rasoro algebra, Preprinthep-th/9212094,to appear in Int. Math. Res. Notices,
Duke Math. Journal.
9. G.E.Andrews, The Theory of Partitions, Addison-Wesley, Reading, 1976.
10. V.Kac,M.Wakimoto,Modularinvariantrepresentationsofinfinite-dimensional
Liealgebrasandsuperalgebras,Proc.Natl.Acad.Sci.USA,85(1988)4956-4960
11. V.Kac, Modular invariance in mathematics and physics, Address at the Cen-
tennial of the American Mathematical Society, Preprint, 1988.
10 BORIS FEIGIN AND EDWARD FRENKEL
Landau Institute for Theoretical Physics, Moscow 117334, Russia
Department of Mathematics, Harvard University, Cambridge, MA
02138, USA
