Available online at www.sciencedirect.com

Nuclear Physics B 871 [FS] (2013) 512–525

www.elsevier.com/locate/nuclphysb

Free field realization of the Dq algebra for the η–ξ

system, integrals of motion and characters

Alessandro Nigro

Dipartimento di Fisica and INFN – Sezione di Milano, Università degli Studi di Milano I, Via Celoria 16,I-20133 Milano, Italy

Received 13 November 2012; received in revised form 30 January 2013; accepted 19 February 2013

Available online 21 February 2013

Abstract

We introduce a free field realization of the central extension of the Lie algebra Dq of difference opera-tors on the circle in terms of the fermionic η–ξ system. This realization admits a nontrivial Jordan blockstructure. We also review the free field realization of W1+∞ algebra, and point out some relations betweenits generators of weight zero and the local integrals of motion of Bazhanov, Lukyanov and Zamolodchikov.Finally we compute the finitized characters, and the continuum characters of the Local Integrals of Motion,and find out and interesting analogy with the generating functions for the counting of branched covers ofelliptic curves.© 2013 Elsevier B.V. All rights reserved.

1. Introduction

In this paper we will study the 2-dimensional Conformal Field Theory corresponding to theη–ξ [2,3] ghost system. It is well known that this theory is a logarithmic CFT with central chargec = −2, this theory has a large number of nontrivial features. Firstly there is a U(1) symmetryassociated to the conservation of the ghost number. If one restricts himself to the “small” algebragenerated by ∂ξ , η this symmetry is enhanced to an SL(2) symmetry. This symmetry is onlyglobal and is related to the presence of a W3 algebra.

In this work we will introduce a two new q-deformation of the U(1) ghost current, thisq-deformation will give rise to a free field realization of the central extension Lie algebra ofdifference operators on the circle Dq [8]. We will analyze in detail the commutation relations of

E-mail address: Alessandro.Nigro@mi.infn.it.

0550-3213/$ – see front matter © 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.nuclphysb.2013.02.014

A. Nigro / Nuclear Physics B 871 [FS] (2013) 512–525 513

this algebra, and by introducing the expansion of the generators Λn(q) for q → 1 with q = eh andh → 0 one has that the order h0 reproduces the U(1) current, while the order h is proportionalto some Ln which form a Virasoro algebra with central charge c = 1. Moreover it is well known[8] that the algebras Dq admit a Virasoro subalgebra with continuously varying central charge.This is realized by adding to the stress energy tensor a total derivative of the U(1) current [5]. Asit is well known in other cases of deformed Algebras, the higher orders in h will generate higherspin operators which are related to the Integrals of Motion of Bazhanov Lukyanov and Zamolod-chikov [1]. We analyze in detail the relation between the higher orders in the h expansion and theBLZ integrals of Motion at c = −2 which have been obtained for Symplectic Fermions in [4].

We also give a brief review of the free field realization of the W1+∞ (also called D) algebrain terms of the ghost system [6,7], and show that the generators of weight zero are related to theintegrals of motion.

We finally analyze the characters of the deformed dilatation generator Λ0(q) by performingthe trace over all eigenstates of the deformed dilatation operator. These eigenstates are wellknown to be labeled by two-column diagrams Y and can be expressed in a simple way in termsof the modes of the ghost fields. We furthermore compute the finitized and continuum charactersof all the IOM by performing the trace over all eigenstates of the IOM, and notably among themthe character of the operator W

(0)0 which is the zero mode of the primary spin 3 field which

together with other 2 spin 3 fields forms the W algebra of the model. This character turns out tobe related to the character of the W3 algebra [9], and also related [10] to the generating functionof counting covers of an elliptic curve.

Moreover we compute the character of all the integrals of motion which also turn out to berelated to generating functions of counting covers of elliptic curves [11]. We also compute thecharacter of an infinite linear combination of integrals of motion, which depends on infinitemodular parameters and appeared before in [11,12], again as generating function of countingcovers of elliptic curves and as the character of the infinite wedge projective representation ofthe algebra of differential operators on the circle. These characters turn out to be quasi modularforms [10,12].

2. The η–ξ system

The η–ξ system is defined by the following stress energy tensor:

T2(z) = :∂ξ(z)η(z): (2.1)

where the fields η–ξ are defined by

ξ(z) =∑n∈Z

ξn

zn(2.2)

η(z) =∑n∈Z

ηn

zn+1(2.3)

and the modes satisfy the following anticommutation relations:

{ξm,ηn} = δm+n,0 (2.4)

514 A. Nigro / Nuclear Physics B 871 [FS] (2013) 512–525

and : . . . : denotes fermionic normal ordering that means that the annihilation modes are moved tothe right with a minus sign whenever two fields are interchanged. There is however an ambiguityin the normal ordering :ξ0η0: which can be defined in general as:

:ξ0η0: = λη0ξ0 + (λ − 1)ξ0η0 (2.5)

the convention λ = 12 is employed in string theory, we shall however employ here the more

convenient convention for λ = 1.The fields ξ , η have the following correlation functions:

⟨η(z)ξ(w)

⟩ = ⟨ξ(z)η(w)

⟩ = 1

z − w(2.6)

and ⟨η(z)η(w)

⟩ = ⟨ξ(z)ξ(w)

⟩ = 0 (2.7)

The sl(2,C) invariant vacuum is characterized by ξnΩ = ηnΩ = 0 for n > 0.This theory admits an U(1) symmetry which is generated by the following current:

J (z) = T1(z) = :ξ(z)η(z): (2.8)

this current however does not define a well defined primary operator, as it can be seen by thefollowing operator product expansion:

T2(z)J (w) = −1

(z − w)3+ J (w)

(z − w)2+ ∂J (w)

(z − w)(2.9)

from the operator product expansion of the stress energy tensor with itself

T2(z)T2(w) = −1

(z − w)4+ 2T2(w)

(z − w)2+ ∂T2(w)

(z − w)(2.10)

one easily sees that the modes of the expansion

T2(z) =∑n∈Z

Ln

zn+2(2.11)

satisfy the following Virasoro Algebra with central charge c = −2:

[Ln,Lm] = (n − m)Ln+m − 1

6n(n2 − 1

)δn+m,0 (2.12)

The zero mode ξ0 can act on the vacuum, one has that there are 2 vacua Ω and ξ0Ω , and ingeneral for each state v there is always another state ξ0v.

It is also possible to define twisted boundary conditions for the fields

ξ(e2πiz

) = −ξ(z) (2.13)

η(e2πiz

) = −η(z) (2.14)

Where

ξ(z) =∑n∈Z

ξn− 1

2

zn− 12

(2.15)

η(z) =∑ η

n+ 12

zn+1+ 12

(2.16)

n∈Z

A. Nigro / Nuclear Physics B 871 [FS] (2013) 512–525 515

In this case there are no zero modes and the twisted ground state μ is nondegenerate.In [3] it was observed that in the untwisted case the η–ξ system contains a “small” algebra

generated by ∂ξ and η, this algebra does not contain the mode ξ0 and is characterized as thekernel of η0 on the space of states of the η–ξ system.

The states in the small algebra are of the following form

ξ−n1 . . . ξ−nlη−m1 . . . η−mr Ω (2.17)

and we will denote them as |Y〉 where Y is the two column diagram Y = (n1, . . . , nl,m1, . . . ,mr)

and the subscripts l, r stand for left and right.In the case of twisted boundary conditions one has the following generic state

ξ−n1 . . . ξ−nlη−m1 . . . η−mr μ (2.18)

which we will also label by a two column diagram Y and denote also as |Y〉 since there is basiclyno source of ambiguity since the 2 sectors always appear in separate contexts.

On the “small” algebra the original U(1) symmetry of the theory is enhanced to an sl(2)

symmetry.By defining

ψ+(z) = η(z) (2.19)

ψ−(z) = ∂ξ(z) (2.20)

the free fermion field �ψ transforms as a j = 12 representation of sl(2) [2]:[

J+, J−] = 2J 0 (2.21)[J±, J 0] = ±J 0 (2.22)[J 0,ψ±(z)

] = ±1

2ψ±(z) (2.23)[

J±,ψ±(z)] = 0 (2.24)[

J±,ψ∓(z)] = ψ±(z) (2.25)

J 0Ω = J±Ω = 0 (2.26)

by virtue of this global symmetry the highest weight states will always fall into irreducible rep-resentations of sl(2) carrying isospin j ∈ 1

2N:

|j,m〉 = ψ(+−2j . . .ψ+

−j+mψ−−j+m+1 . . .ψ

−)−1Ω (2.27)

The fields with half integral isospin are fermionic whereas those with integral isospin are bosonic.The bosonic sector is generated by the isospin 1 fields:

W(+)(z) = :∂η(z)η(z): (2.28)

W(0)(z) = 1

2

(:∂η(z)∂ξ(z): + :∂2ξ(z)η(z):) (2.29)

W(−)(z) = :∂2ξ(z)∂ξ(z): (2.30)

The fields W(i)(z) are primary of spin 3:

T2(z)W(i)(w) = 3W(i)(w)

(z − w)2+ ∂W(i)(w)

(z − w)(2.31)

and they form a W3 algebra.

516 A. Nigro / Nuclear Physics B 871 [FS] (2013) 512–525

We notice in passing that the field W(0)(z) can be rewritten (for later convenience) as:

W(0)(z) = 1

2

(−2T3(z) + ∂T2(z))

(2.32)

where we have introduced:

T2n(z) = ∂nξ(z)∂n−1η(z) (2.33)

T2n−1(z) = ∂n−1ξ(z)∂n−1η(z) (2.34)

which will come in handy in the following sections.We notice in passing that the mode of weight zero of the field W(0)(z) is expressed in terms

of the modes of the Fermi fields as:

W(0)0 =

∞∑m=1

m2(N+m − N−

m

)(2.35)

where we have introduced the following number operators:

N+n = ξ−nηn (2.36)

N−n = η−nξn (2.37)

which for m > 0, k < 0 satisfy[N+

m , ξk

] = ξkδm+k,0 (2.38)[N−

m ,ηk

] = ηkδm+k,0 (2.39)[N+

m ,ηk

] = 0 (2.40)[N−

m , ξk

] = 0 (2.41)

we now introduce the following quantities

Ik =∞∑

m=1

mk(N+

m + (−1)k+1N−m

)(2.42)

which we shall call the Integrals of Motion, it is straightforward to see that their eigenvalues aregiven by

Ik|Y〉 =( ∑

m∈P1

mk + (−1)k+1∑

m∈P2

mk

)|Y〉 (2.43)

where we have used the notation Y = (P1|P2).It is straightforward to see that the I2k−1 have as eigenvalues the eigenvalues of the local

integrals of motion of [1], already obtained in [4].It is also straightforward to verify that they are in involution:

[Ik, Im] = 0 (2.44)

We shall have to comment on the integrals of motion in the next sections, especially on theirrelations to the W1+∞ and the Dq algebras.

For now it suffices to notice that:

W(0)0 = I2 (2.45)

A. Nigro / Nuclear Physics B 871 [FS] (2013) 512–525 517

3. The D and Dq algebras

The quantum W1+∞ [7,6], also denoted by D [8] is the quantum version of the w1+∞ algebragenerated by the polynomials in the variables z and D = z∂z. Given any regular function f wemay write its associated generator zmf (D).

The algebra of such generators is the classical w1+∞ algebra which is defined by the followingcommutation relations [8,7,6]:[

zrf (D), zsg(D)] = zr+s

(f (D + s)g(D) − f (D)g(D + r)

)(3.1)

The unique central extension of this algebra is the algebra D (where the hat is used to denote thecentral extension), we will denote the quantum generators of this algebra as W(zmf (D)), thealgebra associated to these generators is given by:[

W(zrf (D)

),W

(zsg(D)

)] = W(zr+sf (D + s)g(D)

) − W(zr+sf (D)g(D + r)

)+ C

r∑j=1

f (−j)g(r − j)δr+s,0 (3.2)

where C is the central charge, which is proportional to the central charge of the Virasoro algebra.We also introduce the following basis of generators V n

k = W(zkDn), which satisfy the fol-lowing algebra:

[V n

r ,V ms

] =n∑

k=0

(n

k

)sn−kV k+m

r+s −m∑

k=0

(m

k

)rm−kV k+n

r+s + Cδr+s

r∑j=1

(−j)n(r − j)m (3.3)

This algebra can be realized in terms of the η–ξ system as:

V nr =

∮dz

2πizr :Dnξ(z)η(z): (3.4)

The commutation relations can be proven explicitly by computing the Operator Product Ex-pansion of the fields by using the Wick’s rules and computing a double contour integral, thiscalculation turns out to be quite a challenge, for this reason we shall carry it out for the case of[V 2

r , V 2s ] to check that indeed we obtain the correct result.

We start by giving a name to the following quantity:

V (n)(z) = :Dnξ(z)η(z): (3.5)

the OPE of two such objects reads:

V (n)(z)V (m)(w) = Dnz

(1

z − w

):η(z)Dmξ(w): + Dm

w

(1

z − w

):Dnξ(z)η(w):

+ Dnz

(1

z − w

)Dm

w

(1

z − w

)(3.6)

where in general by expanding the propagator in a geometric series one obtains:

Dlz

(1

z − w

)= (−1)l

Li−l (wz)

w(3.7)

which can be simplified to:

518 A. Nigro / Nuclear Physics B 871 [FS] (2013) 512–525

Dlz

(1

z − w

)= (−1)l

w

(1 − w

z

)−l−1 l∑m=1

(w

z

)m m∑k=1

((−1)k+1

(l + 1

k − 1

)(m − k + 1)l

)

(3.8)

in particular this reduces for l = 2 to:

D2z

(1

z − w

)= z(w + z)

(z − w)3(3.9)

so that one has:[V 2

r , V 2s

] =∮

dz

2πi

∮dw

2πizrws

(z(w + z)

(z − w)3:η(z)D2ξ(w): + w(w + z)

(z − w)3:D2ξ(z)η(w):

+ zw(w + z)2

(z − w)6

)(3.10)

the double contour integral is evaluated in the following way: first one performs the z integrationon a circle enclosing w, and after one performs the w integration on a contour encircling theorigin. For this purpose are useful the following integrals:∮

dz

2πizr+1 (w + z)

(z − w)3η(z) = 1

2∂2z

(zr+1(w + z)η(z)

)z=w

(3.11)∮dz

2πizr w(w + z)

(z − w)3D2ξ(z) = 1

2∂2z

(zrw(w + z)D2ξ(z)

)z=w

(3.12)

∮dz

2πizr+1 w(w + z)2

(z − w)6= 1

5!∂5z

(zr+1w(w + z)2)

z=w(3.13)

one then computes the second derivatives and evaluates them at z = w and wants then to ex-press the result in terms of the V (n)(w) and their derivatives only, this is achieved by using thefollowing straightforward identities:

wD2ξ(w)∂η(w) = w∂V (2)(w) − V (3)(w) (3.14)

w2D2ξ(w)∂2η(w) = w2∂2V (2)(w) + V (3)(w) − 2w∂V (3)(w) + V (4)(w) (3.15)

w2∂2D2ξ(w)η(w) = V (4)(w) − V (3)(w) (3.16)

one then performs the w integration and when the smoke clears is left with:

[V 2

r , V 2s

] = (s2 − r2)V 2

r+s + 2(s − r)V 3r+s + 1

30r(r4 − 1

)δr+s,0 (3.17)

which is in perfect agreement with the general formula, although to prove the generic case in-volves lengthier calculations, this suffices to show how the calculation proceeds.

It is worth remarking that the modes of weight zero V n0 , by virtue of the identity:

Dk(z−n

) = (−1)knkz−n (3.18)

can be written as:

V n0 = In (3.19)

So that the subalgebra of modes of weight zero of W1+∞ in this free field realization is generatedby the integrals of motion, we shall comment on characters of this subalgebra in the last section.

A. Nigro / Nuclear Physics B 871 [FS] (2013) 512–525 519

We now turn our attention to the algebra Dq [8], which is defined as the Lie algebra of differ-ence operators on the circle. We first introduce the difference operator Dq :

Dqf (z) = f (qz) − f (z)

q − 1(3.20)

We now introduce the following generators:

Tm,n = q(m+1)n((q − 1)Dq + 1

)n(3.21)

It follows, that upon parametrizing q = eh that these generators satisfy the following algebra:

[Tm,n, Tm′,n′ ] = 2 sinh(h(m′n − mn′))Tm+m′,n+n′ (3.22)

The algebra Dq is defined as the central extension of this algebra, which is given by:

[Tm,n, Tm′,n′ ] = 2 sinh(h(m′n − mn′))Tm+m′,n+n′ + C

sinh(hm(n + n′))sinh(h(n + n′))

δm+m′,0 (3.23)

In the next section we shall introduce a free field realization of Dq which up to now did notappear in the literature.

4. The free field realization of the algebra Dq

We start this section by introducing a new q-deformed current, this current is a deformationof the natural U(1) current, but as we shall see it gives rise to a free field realization of thealgebra Dq , namely the central extension of the Lie Algebra of finite difference operators on thecircle.

We thus introduce the following spin-1 current:

Λ(z, q) = :ξ(q−1z

)η(qz): (4.1)

it can be seen by working out the calculation with Wick’s rules that this current has the followingOPE with itself:

Λ(z, q)Λ(w, t) = −qΛ(qw,qt)

(z − qtw)+ Λ(q−1w,qt)

q(z − (tq)−1w)+ 1

(z − qtw)(z − (tq)−1w)(4.2)

now by introducing the following mode expansion

Λ(z, q) =∑l∈Z

Λl(q)

zl+1(4.3)

one defines the commutator of two modes by the following contour integral:

[Λn(q),Λm(t)

] =∮

dw

2πi

∮Cw

dz

2πiznwmΛ(z, q)Λ(w, t) (4.4)

where the contour Cw encircles all poles of the OPE.It is easy to see that the modes of the deformed current satisfy the following commutation

relations

[Λn(q),Λm(t)

] = (qmt−n − q−mtn

)Λm+n(qt) +

((qt)n − (tq)−n

−1

)δn+m,0 (4.5)

qt − (tq)

520 A. Nigro / Nuclear Physics B 871 [FS] (2013) 512–525

We notice that for q = eγ h, t = eδh this algebra reduces to the well known [8] central extensionof the algebra Dq of difference operators on the circle:[

Λn

(eγ h

),Λm

(eδh

)] = 2 sinh(h(γm − δn)

)Λm+n

(e(γ+δ)h

)+ sinh(h(n(γ + δ)))

sinh(h(γ + δ))δn+m,0 (4.6)

where however γ , δ are not restricted to be integers, so this algebra is somewhat more general.If we now consider the expansion for q → 1 with q = eh one has:

Λ(z, q) ∼ T1(z) − 2z

(T2(z) − 1

2∂T1(z)

)h + · · · (4.7)

It is straightforward to see that this expansion can be expressed in closed form in terms of theTn(z) (2.33) fields introduced in Section 2. We may view these fields as integrals of motion, andsuitable linear combinations of the Tn(z) with higher derivatives of the lower spin fields will giverise to the local integrals of motion of [1] as we shall see.

Moreover we can introduce the analogous expansion for the modes of the q-deformed current:

Λn(q) =∞∑

m=0

hmL(m)n (4.8)

in particular one has that

L(1)n = −2Ln (4.9)

where Ln satisfy the Virasoro algebra with central charge c = 1.Moreover the combination

Ln(α) = Ln + (n + 1)α

2Jn (4.10)

(where the Ln form a Virasoro algebra with central charge c = −2 and the Jn are the modes ofthe U(1) current) forms a Virasoro Algebra

[Ln(α),Lm(α)

] = (n − m)Ln+m(α) + c(α)

12n(n2 − 1

)δn+m,0 (4.11)

with central charge

c(α) = −3α2 + 6α − 2 (4.12)

It is worth pointing out, that for c = 1 which corresponds to α = 2 the U(1) current T1(z) be-comes a primary field of spin 1 with respect to this Virasoro algebra.

The fact that the Virasoro algebra appears together with the U(1) current (when it is presentin the theory) in the h expansion, is in principle expected when dealing with deformations ofthe Virasoro algebra, so that this observation tells us that we are dealing with a well poseddeformation of the Virasoro algebra.

We now turn our attention to the eigenvalues of Λ0(q), in terms of the number operators wecan write Λ0(q) as:

Λ0(q) = q−1ξ0η0 + q−1∞∑(

q−2nN+n − q2nN−

n

)(4.13)

n=1

A. Nigro / Nuclear Physics B 871 [FS] (2013) 512–525 521

from which it is straightforward to obtain the eigenvalues:

Λ0(q)|Y〉 = q−1( ∑

n∈P1

q−2n −∑n∈P2

q2n

)|Y〉 (4.14)

where again we have used the notation Y = (P1|P2).We now consider the expansion

Λ0(q) =∞∑

k=0

hkL(k)0 (4.15)

where

Λ(k)0 = (−1)k

k!

(ξ0η0 +

k∑s=0

(k

s

)2sIs

)(4.16)

where once more the Is are the integrals of motion.We now turn our attention to the possibility of forming a representation of Dq which has a

nontrivial logarithmic structure, it turns out that this is possible. We start by observing that thenaive stress energy tensor T2(z) doesn’t have any zero modes of the fields appearing in its explicitexpression, therefore it cannot give rise to an indecomposable structure.

However if we add a total derivative of the η field to the stress energy tensor:

T2(z) = T2(z) + δ∂η(z) (4.17)

The central charge remains the same, in this case, however the new generator of dilatations hasnontrivial Jordan block structure:

L0Ω = 0 (4.18)

L0ω = Ω (4.19)

from this observation one notices that if one defines a new deformed current:

D(q, z) = Λ(q, z) − 2η(qz) (4.20)

now by using the following additional OPEs together with the Λ–Λ OPE:

Λ(q, z)η(tw) = − q

z − qtwη(q2tw

)(4.21)

η(qz)Λ(t,w) = q−1

z − (qt)−1wη(tw) (4.22)

it is straightforward to check that the D current satisfies the same commutation relations of theold one, however the D0(q) generator has the nontrivial Jordan block structure:

D0(q)Ω = 0 (4.23)

D0ω = −2q−1Ω (4.24)

522 A. Nigro / Nuclear Physics B 871 [FS] (2013) 512–525

5. Characters

We start this section by computing the following character:

χ(q0, q1) = Tr(q

I11 q

I00

)(5.1)

where the trace is over all the eigenstates of the IOM in the vacuum sector, this can be expressedas the following sum over two column diagrams:

χ(q0, q1) =∑Y

qI1(Y)1 q

I0(Y)0 (5.2)

if we consider this sum over all two column diagrams of maximal height L, that is the length ofeach partition P1, P2 cannot exceed the value L, and the heights inside the partitions are alsoless than L, we have the following finitized character:

χ(L)(q0, q1) =L∏

n=1

(1 + qn

1 q0)(

1 + qn1 q−1

0

)(5.3)

in the limit L → ∞ we obtain the well known sl(2) ⊗ Vir character (in the Noveau–Schwartzsector) [3]:

χ(q,w) =∞∏

n=1

(1 + qnw

)(1 + qnw−1) (5.4)

The calculation can be repeated in the twisted sector (Ramond) and one obtains the result:

χ(q,w) =∞∏

n=1

(1 + qn− 1

2 w)(

1 + qn− 12 w−1) (5.5)

We now turn our attention to of I2, which is also W(0)0 , the generator of the W3 algebra, if we

consider the following character:

χ(q0, q1, q2) =N (q0, q2)Tr(q

I22 q

I11 q

I00

)(5.6)

we have the following finitization in the Noveau–Schwartz (NS) sector:

χ(L)(q0, q1, q2) =N (q0, q2)

∞∏n=1

(1 + qn2

2 qn1 q0

)(1 + q−n2

2 qn1 q−1

0

)(5.7)

whereas in the Ramond (R) sector:

χ(L)(q0, q1, q2) =N (q0, q2)

∞∏n=1

(1 + q

(n− 12 )2

2 qn− 1

21 q0

)(1 + q

−(n− 12 )2

2 qn− 1

21 q−1

0

)(5.8)

where we have inserted a normalization factor N (q0, q2) to make the product convergent in theL → ∞ limit, this normalization looks like in the NS sector:

N (q0, q2) = q

∑Ln=1 n0

0 q

∑Ln=1 n2

2 (5.9)

whereas in the R sector:

N (q0, q2) = q

∑Ln=1(n− 1

2 )0

q

∑Ln=1(n− 1

2 )2

(5.10)

0 2

A. Nigro / Nuclear Physics B 871 [FS] (2013) 512–525 523

If we choose to use ζ function regularization for the normalization factors and use the fact thatthe ζ function vanishes at negative even integers, we can replace the normalization factors by 1in the L = ∞ limit so that in this limit we obtain the W3 character of [9], which has been provenin [10] to be related to the generating function counting maps of curves of genus g > 1 to a curveof genus 1.

We now make a leap forward and define in complete generality:

χ(L)(q0, q1, q2, . . .) =N (q0, q2, . . .)Tr

( ∞∏i=0

qIi

i

)(5.11)

which can be evaluated in the NS sector as:

χ(L)(q0, q1, q2, . . .) =N (q0, q2, . . .)

L∏n=1

(1 +

∞∏i=0

qni

i

)(1 +

∞∏i=0

q(−1)i+1ni

i

)(5.12)

whereas in the R sector one has

χ(L)(q0, q1, q2, . . .) =N (q0, q2, . . .)

L∏n=1

(1 +

∞∏i=0

q(n− 1

2 )i

i

)(1 +

∞∏i=0

q(−1)i+1(n− 1

2 )i

i

)

(5.13)

where the normalization in the NS sector is

N (q0, q2, . . .) = q

∑Ln=1 n0

0 q

∑Ln=1 n2

2 q

∑Ln=1 n4

4 . . . (5.14)

whereas in the R sector

N (q0, q2, . . .) = q

∑Ln=1(n− 1

2 )0

0 q

∑Ln=1(n− 1

2 )2

2 q

∑Ln=1(n− 1

2 )4

4 . . . (5.15)

By using again the ζ function regularization we can replace these factors by 1 in the L = ∞limit.

These characters are almost with the characters projective representation of the algebra ofdifferential operators on the circle [12] which is related to the counting function of the m-simplebranched covers with a fixed genus of an elliptic curve [11]. This character is proven in [12] tobe quasi modular of weight 0. In order to make our characters coincide with those of [12], andhence be quasi modular, we need to multiply them by the normalization factor:

N (q1, q3, q5, . . .) = qζ(−1,1− δ

2 )

1 qζ(−3,1− δ

2 )

3 qζ(−5,1− δ

2 )

5 . . . (5.16)

where δ = 0 in the NS sector whereas δ = 1 in the R sector. Multiplying our character by thisfactor can be understood as shifting the I2n−1 by a constant and thus defining:

I2n−1 = I2n−1 + ζ

(1 − 2n,1 − δ

2

)(5.17)

This is very natural since the I2n−1 coincide with the local integrals of motion of [1] only uponsubtracting the vacuum eigenvalue, which we have now restored, the I2n−1 are now really thelocal integrals of motion and their character:

χ(q0, q1, q2, . . .) = Tr(q

I00 q

I11 q

I22 q

I33 . . .

)(5.18)

coincides with the character of [12] and is now quasi modular of weight zero.

524 A. Nigro / Nuclear Physics B 871 [FS] (2013) 512–525

We now want to compute the finitized character of the Λ0(t) generator of the algebra Dt

which is obviously defined as:

χ(L)(Λ0(t), q

) =∑Y

qΛ0(t,Y) (5.19)

this can be evaluated to:

χ(L)(Λ0(t), q

) =L∏

n=1

q−t2n−1−δ (1 + qt−2n−1+δ )(

1 + qt2n−1−δ )(5.20)

which in the L → ∞ limit becomes:

χ(Λ0(t), q

) = qt1−δ

t2−1

∞∏n=1

(1 + qt−2n−1+δ )(

1 + qt2n−1−δ )(5.21)

where δ = 0 in the NS sector and δ = 1 in the R sector.We now recall that

L(k)0 (Y) = (−1)k

k!k∑

s=0

(k

s

)2sIs(Y) (5.22)

from this it is clear that we can compute the character of the L(k)0 in terms of χ(q0, q1, q2, . . .)

by setting:

qs = q(−1)k2s

k! (ks) (5.23)

6. Conclusion

In this paper we have introduced a new free field realization of the central extension of theLie algebra of difference operators on the circle in terms of the η–ξ ghost system. We haveshown that this free field realization admits a nontrivial Jordan block structure for the deformeddilatation generator. We have also elucidated the relation of this algebra and of the W1+∞ algebrato the Local Integrals of Motion of [1]. Furthermore we have computed the characters of all theintegrals of motion and shown that they are related to the generating functions for the countingof the m-simple branched covers with a fixed genus of an elliptic curve. We have also computedthe character of the deformed dilatation generator.

Acknowledgements

The author acknowledges financial support from Fondo Sociale Europeo (Regione Lombar-dia), through the grant “Dote ricerca”.

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