A $\hbar$-deformed Virasoro Algebra as Hidden Symmetry of the Restricted sine-Gordon Model

9 Pages·0.13 MB·

by  Bo-yu Hou

h
A ¯-deformed Virasoro Algebra as Hidden Symmetry of the
Restricted sine-Gordon Model
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9
Bo-yu Hou† Wen-li Yang‡,†
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1

n
a ‡ CCAST (World Laboratory), P.O.Box 8730, Beijing 100080, China
J
† Institute of Modern Physics, Northwest University, Xian 710069, China ∗
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2
February 1, 2008

2
v
5
3
Abstract
2
2 As the Yangian double with center,which is deformed from affine algebra by the additive loop
1 parameter ¯h ,we get the commuting relation and the bosonization of quantum ¯h-deformed Virasoro
6 algebra. The corresponding Miura transformation , associated screening operators and the BRST
9 charge have been studied.Morever, we also constructe the bosonization for type I and type II inter-
/
h twinervertexoperators.Finally,we showthatthecommutingrelation ofthesevertexoperatorsinthe
t case of p′ = r p = r−1 and h¯ = π actually give the exact scattering matrix of the Restricted
-
p sine-Gordon model.
e Mathematics subject classification(1991): 81R10,81Txx,82B23, 58F07,17b37,16W30
h
:
v
1 Introduction
i
X
r
a Recently,more and more attention has been paid on the studies of q-deformation of infinite dimensional
algebra—q-deformedaffine algebra[1] ,q-deformedvirasoroalgebra andW-algebra[3,4] .These q-deformed
algebrawouldplaythesameroleinthestudiesofcompletelyintegrablemodelswhichareintegrablelyper-
turbed from the critical ones (conformal invariant models) as the non-deformed algebrain the studies of
conformalfieldtheories(CFT).Fromthebosonizationforq-deformedaffinealgebraU (sˆl )anditsvertex
q 2
operators,JimboetalsucceededingivingthecorrelationfunctionsofXXZmodelbothinthebulkcase[7]
and the boundary case[8] . Feigin and Frenkel obtained the q-deformed virasoro algebra,W-algebra,cor-
responding“screeningoperators”andquantumq-deformedMiuratransformationsfromthequantization
fortheq-deformationoftheclassicalvirasoroandW-algebra[4,5]. Ontheotherhand,Awataetalalsocon-
structed the quantum q-deformed Virasoroand W-algebra,associated “screeningoperators”and related
Miura transformation from the studies of Macdonald symmetric functions[3].The studies of bosonization
∗Mailingaddress
1
for q-deformed virasoro and W-algebra has been begined in Ref.[2] and Ref.[9]. Actually,Asai,Jimbo et
al has constructed the bosonization for vertex operators of q-deformed W-algebra (φ (z) in Ref.[9]).
µ
However,there exists another important deformation of infinite dimensional algebra which also plays
animportantroleinthe completelyintegrablefieldtheories(Inordertocomparationwithq-deformation
, we call it as ¯h -deformation ).This deformation was originated by Drinfeld in studies of Yangian[1].
Simirnov suggested that the Yangian double would be the dynamical non-abelian symmetry algebra
for SU(2)-invariant Thirring model[16]. Recently progresses has been achieved by Khoroshkin te al [11]
and Iohara et al[12] in the studies of Yangian double and its central extension .The bosonization for
Yangian double with center and its vertex operators make it possible to describe the structure of model
(local operators,the asymptotic states etc.)in terms of the representation theory of Yangian double with
center[11].It also ensure that the correlation functions of SU(2)-invariant Thirring model can be given
explicitly in the integral forms. In this paper,we will given a new deformation of Virasoro algebra—
¯h-deformed Virasoro algebra(HDVA) which has the similar relation as the Yangian algebra and the q-
deformed affine algebra.When the deformed parameter h¯ −→0, this HDVA will degenerate to the usual
Virasoro algebra. The HDVA is quantum version of some classical Poisson algebra which is constructed
byh¯-deformedSuggwareconstructionofYangiandoublewithcenteratthecriticalpoint[4,5](theclassical
versionofthisalgebrahasbeingstudiedbyourcolleagueandthepaperisinprepairation). Wealsoshow
that when the deformed parameter h¯ = π, this HDVA is the hidden non-abelian dynamical symmetry
algebra of the Restricted sin-Gordon model(This can be seen from the state space[14] and scattering
matrix[15,17]). Morever,weconstructethebosonizationforh¯-deformedvirasoroalgebraanditstypeIand
typeIIvertexoperators. Thecommutingrelationforthesevertexoperators(inthecaseofp′ =r p=r−1
and h¯ =π) actually give the exact scattering matrix of the Restricted sine-Gordon[13,14,15,17]. It is well-
known that the similar phenomena has occured in the studies of braid relations for the vertex operators
(primary fields) of Virasoro algebra in CFT[20].
2 h¯-deformed virasoro algebra and its bosonization
Firstly,we give some review of q-deformed virasoro algebra[4].This deformed algebra depends on two
parameters p and q,and is generated by current T (z) with the following relations
q
w z (1−q)(1−p/q) w wp
f ( )T (z)T (w)−f ( )T (w)T (z)= (δ( )−δ( )) (1)
q q q q q q
z w 1−p zp z
1 (xq;p2)(xpq−1;p2) ∞
f (x)= , (z;p)= (1−zpn)
q 1−x(xpq;p2)(xp2q−1;p2)
n=0
Y
2
It can be shown that the h¯-deformation of affine algebra(or Yangian double) can be considered as the
operators scaling limit of q-deformation of affine algebra[2,19] (but not the scaling limit of bosonic field
realization ) .Thus, we give the HDVA as the operator scaling limit of q-deformation of virasoro algebra
in Eq.(1).The scaling limit is taken as following way
z =p−h¯iβ q =p−ξ , f(β)= limfq(pih¯β)
p→1
−iβ
T(β)= limTq(p h¯ )
p→1
From direct calculation,we obtain the HDVA which is generated by the current T(β) as the following
relation
f(β −β )T(β )T(β )−f(β −β )T(β )T(β )=ξ(ξ+1)(δ(β −β +i¯h)−δ(β −β −i¯h))(2)
2 1 1 2 1 2 2 1 1 2 1 2
2Γ(iβ + 1 − ξ)Γ(iβ + ξ +1)
f(β)= 2h¯ 2 2 2h¯ 2
βΓ(iβ − ξ)Γ(iβ + 1 + ξ)
2h¯ 2 2h¯ 2 2
In order to study the bosonization for HDVA ,we will start from the the quantum Miura transformation
associated with HDVA in the same way as that of the studies for q-deformed W-algebra given by Feigin
and Frenkel[4].Let us consider free bosons λ(t) with continuous parameter t (∈R−0) which satisfy
4shh¯tshh¯ξtshh¯(ξ+1)t
[λ(t),λ(t′)]= 2 2 2 δ(t+t′) (3)
tsh¯ht
We also need to define the “Zero mode” operators P and Q ,which commute with λ(t) and satisfy the
relation
[P,Q]=−i (4)
We considerthe Fock spaceF ofHeisenberg algebradefined by Eq.(3)andEq.(4)whichis generatedby
p
the heighest weight vector v . The heighest weight vector v satisfies
p p
λ(t)v =0 , t>0 and Pv =pv (5)
p p p
Now we introduce the field Λ(β)
∞
Λ(β)=:exp{− λ(t)eitβdt}: (6)
−∞
Z
andgivetheassociatedquantumMiuratransformation(i.ethetransformationfromΛ(β)toh¯-deformation
virasoroalgebra T(β) )
i¯h i¯h
T(β)=Λ(β+ )+Λ−1(β− ) (7)
2 2
3
In orderto get the commuting relationfor bosonic field,we must give a comment:When one compute the
exchange relation of operators,one often encounter an integral as follows
∞
F(t)dt
Z0
which is divergent at t=0 .Here we adopt the regularizationgiven by Jimbo et al[10].Namely, the above
integral should be understood as the contour integral
log(−t)
F(t) dt
2iπ
Zc
where the contour C is chosen as in the Ref.[10]. After straightforward calculation,we show that the
bosonicrepresentationfor h¯-deformedvirasorocurrentT(β) inEq.(7)satisfies the definationrelationfor
HDVA as in Eq.(2).Thus,we constructe the bosonization for HDVA.
In what follows we will be interested in a particular representions of HDVA in Fock space which
correspondstominimalmodelsofCFT[13].Thisrepresentationsareparameterizedbytwopostivecoprime
integer numbers p′ , p (p′ > p) and two type boson a(t) ,a′(t) which are associated with α series and
+
α− series respectively
p
ξ = , α = ξ(1+ξ) (8)
p′−p 0
p
p′ (ξ+1) λ(t)
α =− =− , a(t)= (9)
+ sp s ξ 2shth¯ξ
2
p ξ λ(t)
α− = p′ =s(1+ξ) , a(t)= 2shth¯(ξ+1) (10)
r 2
We consider the action of HDVA on the Fock space F
l,k
′
PFl,k =αl,kFl,k , and , αl,k =α+l+α−k , 0≤l ≤p ,0≤k≤p (11)
as bosonic operator T(β) .This action is highly reducible and we have to throw out some states from
Fock module to obtain the irreducible component. The procedure depends on BRST charge Q± which
are defined as follows
Q+ = dβS+(β)f(β,α0P) , Q− = dβS−(β)f′(β,α0P) (12)
IC+ IC−
∞
screening current S+(β)=e−2iα+Q :exp{− a(t)(et2h¯ +e−t2h¯)eitβdt}: (13)
−∞
Z
∞
screening current S−(β)=e−2iα−Q :exp{− a′(t)(et2h¯ +e−t2h¯)eitβdt}: (14)
−∞
Z
sinπ(iβ − 1 − α0P)
h¯ξ 2ξ ξ
f(β,α P)= (15)
0 sinπ(iβ + 1 )
h¯ξ 2ξ
4
sinπ( iβ + 1 + α0P)
′ h¯(ξ+1) 2(1+ξ) 1+ξ
f (β,α P)= (16)
0 sinπ( iβ − 1 )
h¯(1+ξ) 2(1+ξ)
wheretheintegrationcontoursarechosenasfollows: thecontourC enclosethepolesβ = ih¯ −i¯hξn(0≤
+ 2
n) ,the contour C− enclose the poles β =−ih¯ +i¯h(ξ+1)n(0≤n) .An important properties of screening
2
chargesQ± is that they commute with the HDVA currentT(β)andhaveBRST propertiesacting onthe
Fock space F
l,k
Qp| =0 , Qp′| =0 (17)
+ Fl,k − Fl,k
As a result, we could have the following complexes
...X−−2→=Ql+ F2p−l,k X−1−=→Qp+−l Fl,k X−0=→Ql+ F−l,k X1−=→Qp−l ... (18)
Y−2=Qk− Y−1=Q−p′−k Y0=Qk− Y1=Q−p′−k
... −→ Fl,2p′−k −→ Fl,k −→ Fl,−k −→ ... (19)
we assume the following cohomologicalproperties
KerXj/ImXj−1 =0 , if j 6=0
KerYj/ImYj−1 =0 , if j 6=0
KerX0 =KerY0 , ImX−1 =ImY−1
Ll,k =KerX0/ImX−1 =KerY0/ImY−1 (20)
where L is an irreducible representation of HDVA.
l,k
InordertorelatewiththeRestrictedsin-Gordonmodel,weshouldtakeh¯ =π.Inthefollowingpartof
thispaperwerestrictusto the caseof¯h=π ,butthe vertexoperators((21)—(24))andtheir commuting
relations ( (25)—-(32)) are easy to generalize to the generic ¯h. If p=r−1 and p′ =r , the above space
L will be the space of the Restricted sine-Gordon model [14].
l,k
3 Vertex operators and commuting relations
It is well-known that the vertex operators of Virasoro algebra ( or of q-deformed affine algebra) is of
great importance in the CFT[13] (or in the solvable lattice model[7]). Hence, we construct the simplest
two type vertex operators: Z′(β) for type I and Z (β) for type II
a a
−∞
V (β)≡Z (β)=eiα+Q :exp{ a(t)eitβdt}: (21)
(1,0) +
∞
Z
−∞
V (β)≡Z′ (β)=eiα−Q :exp{− a′(t)eitβdt}: (22)
(0,1) +
∞
Z
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V((11,,00))(β)≡Z−(β)= dηZ+(β)S+(η)f(η−β,α0P) (23)
IC1
(0,1) ′ − ′
V (β)≡Z (β)= dηZ (β)S (η)f (η−β,α P) (24)
(0,1) − + 0
IC2
where the integration contour are chosen as follows: the contour C enclose the poles η = β + iπ −
1 2
iπξn (0 ≤ n), the contour C enclose the poles η = β + iπ +iπ(1+ξ)n (0 ≤ n). Because the other
2 2
vertex operators Vm,n(β) (corresponds to Vm,n(z) in the Ref.[11]) can be constructed by the symmetric
l,k l,k
fusion of the simplest two type vertex operators Z (β) and Z′(β), here we only consider these two type
a a
basical vertex operators.
Fromthedirectcaculation,weobtainthecommutingrelationsofvertexoperators(orthebraidmatrix
for the primary fields in CFT[13,20])
a+b=c+d
l+a+b l+d
Z (β )Z (β )| = Z (β )Z (β )U |β −β | (25)
a 1 b 2 Fl,k c 2 d 1 l+b l 1 2 Fl,k
c,d (cid:18) (cid:19)
X
a+b=c+d
′ ′ ′ ′ ′ k+a+b k+d
Z (β )Z (β )| = Z (β )Z (β )U |β −β | (26)
a 1 b 2 Fl,k c 2 d 1 k+b k 1 2 Fl,k
c,d (cid:18) (cid:19)
X
i(β −β ) π
′ 1 2 ′
Z (β )Z (β )| =abctg( + )Z (β )Z (β )| (27)
a 1 b 2 Fl,k 2 4 b 2 a 1 Fl,k
l±2 l±1 ′ k±2 k±1 ′
U |β =κ(β) , U |β =κ(β) (28)
l±1 l k±1 k
(cid:18) (cid:19) (cid:18) (cid:19)
l l±1 sinπξsinπ(πiβξ ∓ ξl)
U |β =±κ(β) (29)
l±1 l sinlπsinπ(iβ + 1)
(cid:18) (cid:19) ξ πξ ξ
l l∓1 sinπ(1ξ±l)sin(iξβ)
U |β =∓κ(β) (30)
l±1 l sinlπsinπ(iβ + 1)
(cid:18) (cid:19) ξ πξ ξ
′ k k±1 ′ sin1+πξsinπ(π(1iβ+ξ) ± 1+kξ)
U |β =±κ(β) (31)
k±1 k sin kπ sinπ( iβ − 1 )
(cid:18) (cid:19) 1+ξ π(1+ξ) 1+ξ
′ k k∓1 ′ sin1i+βξsinπ(1±ξk)
U |β =∓κ(β) (32)
k±1 k sin kπ sinπ( iβ − 1 )
(cid:18) (cid:19) 1+ξ π(1+ξ) 1+ξ
∞ shtπ(1+ξ)shitβdt
κ(β)=exp{− 2 }
shπξchtπ t
Z0 2 2
∞ shtπξshitβ dt
κ′(β)=exp{−Z0 shπ(122+ξ)cht2π t }=−κ(−β)|ξ−→1+ξ
we also find that
′
[Za(β),Q−]=0 , [Za(β),Q+]=0 (33)
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Due to the cohomological properties Eq.(20), the periodic properties of matrices U and U′ with regard
to l and k ,and the Eq.(33), we obtain
a+b=c+d
l+a+b l+d
Za(β1)Zb(β2)|Ll,k = Zc(β2)Zd(β1)U l+b l |β1−β2 |Ll,k (34)
c,d (cid:18) (cid:19)
X
a+b=c+d
′ ′ ′ ′ ′ k+a+b k+d
Za(β1)Zb(β2)|Ll,k = Zc(β2)Zd(β1)U k+b k |β1−β2 |Ll,k (35)
c,d (cid:18) (cid:19)
X
i(β −β ) π
′ 1 2 ′
Za(β1)Zb(β2)|Ll,k =ctg( 2 + 4)Zb(β2)Za(β1)|Ll,k (36)
Itis well-knownthat the Restrictedsine-Gordonmodel is a massiveintegrablemodel which is perturbed
from the minimal conformal field model with center C = 1− 6 (4 ≤ r)[14] .We find that when
r(r−1)
p′ = r and p = r − 1 (i.e ξ = r −1 ) ,besides the space of states are the same , the braid matrix
l+a+b l+d
U |β istheexactscatteringmatrixoftheRestrictedsine-Gordonmodel[14,15,17]. So,
l+b l
(cid:18) (cid:19)
we strongly suggest that the h¯-deformed Virasoro algebra defined in Eq.(2) is the just hidden symmetry
in the Restricted sine-Gordon model.
4 Conclusion
In this paper,we obtain the HDVA, its bosonic realization in space L and show that this deformed
l,k
algebra with the deformed parameter h¯ = π is the hidden symmetry of the Restricted sine-Gordon
model. The bosonization will make it possible to describe the structure of the Restricted sine-Gordon
model in terms of the representation theory of the HDVA and to caculate the correlation functions of
the Restricted sine-Gordon model. The correlation functions would be the scaling limit (q = p−ξ, ζ =
iβ
pπ , r−1=ξ,p−→1) ofLukyanovand Pugain’s inthe Ref.[2].We will presentthis resultin the other
paper.Of course,the studies of the Restricted sine-Gordon model with integrable boundary condition
would be important and the bosonization of HDVA make it accessible .Some results abount boundary
q-deformed virasoro algebra has been obtained[8,18].
The ¯h-deformationW-algebra, its bosonic representationand its vertex operatorsare also worthyto
be investigated.Some results have been obtained by us which is in prepairation. We expect that these
deformed W-algebra plays the role of symmetry algebra for some integrable field model.
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