Current Algebra of Super WZNW Models

13 Pages·0.1 MB·English

by  E. Abdalla

Current Algebra of Super WZNW Models
3
9 E. Abdalla1, M.C.B. Abdalla2, O.H.G. Branco3 and L.E. Saltini4 ∗
9
1
1,3,4 Instituto de F´ısica, Universidade de S˜ao Paulo,

n
Cx. Postal 20516, BR-01498-970 S˜ao Paulo SP / Brazil
a
J
6 2 Instituto de F´ısica Te´orica, UNESP,
2
Rua Pamplona 145, BR-01405-, S˜ao Paulo SP / Brazil

1
v
8
0
1 Abstract
1
0 We derive the current algebra of supersymmetric principal chiral
3
models with a Wess-Zumino term. At the critical point one obtains
9
/ two commuting super Kac-Moody algebra as expected, but in general
h
t there are intertwining fields connecting both right and left sectors,
-
p analogously to the bosonic case. Moreover, in the present supersym-
e metric extension we have a quadratic algebra, rather than an affine
h
Lie algebra, due to the mixing between bosonic and fermionic fields
:
v
since the purely fermionic sector displays a Lie algebra as well.
i
X
r
a
Universidade de S˜ao Paulo
IFUSP/P-1026
January 1993
∗Work supported in part by CNPq
1
Since the discovery of higher conservation laws for integrable models,
algebraic methods have been frequently advocated in order to display the
structure of the dynamics of field theory. Higher conserved charges imply
non trivial constraints for correlation functions, and one often finds such
a strong algebraic machinery that a calculable S-matrix turns out to be a
consequence. In the case of conformally invariant field theory, the Virasoro
(and Kac-Moody) constraints fix all correlation functions.
Current algebra for integrable non linear sigma models were, up to very
recently [1] largely unknown. Nevertheless, the Yang-Baxter relations are
the best tools to explore non perturbative properties of integrable models
[2]. The starting point of the formulation is very generally the consideration
of the Poisson brackets between the spatial part of the Lax pair, leading
to a Lie-Poisson algebra containing an antisymmetric numerical r-matrix
which obeys the Yang-Baxter relation. The Yang-Baxter relation leads to
an (almost) unique S-matrix of the theory [3, 4, 5, 15] . On the other hand,
an algebraic strategy has also been effectively and successfully used in the
case of 2-D conformally invariant theories, in order to compute correlation
functions [6], as well as in the case of two dimensional gravity in the light
cone gauge [7].
Recently, the non linear σ-model with a Wess-Zumino term has been
studied in this context [8]. The current, which corresponds to a piece of
the Lax pair of the model [9], is shown to fulfill a new affine algebra. Such
algebras have played an important role in several cases in field theory (see
refs. [10] and [11]). The hope is that after quantization we could be able to
address to the problem of computing exact Greens functions, by means of
higher conserved currents.
InthepresentpaperweconsiderthesupersymmetricWess-Zumino-Witten
model [12, 13], which is defined by the action
1
S (g,ψ) = d2xtr∂µg−1∂ g + (1)
susy λ2 Z µ
n 1
+ dr d2xtrg−1∂ gg−1∂ g g−1∂ g +
4π Z0 Z r r r µ r r ν r
1 1 nλ2 nλ2
+ tr d2x{ψ¯i∂/ψ − (1−( )2)(ψ¯ψ)2 + i g−1∂µgψ¯γ γ ψ}
4λ2 Z 4 4π 4π 5 µ
obtained from the superfield G(x,θ) = g(x) + θ¯ψ(x) + 1θ¯θF(x), satisfying
2
the constraint G†(x,θ)G(x,θ) = 1, after integrating over the Grassman vari-
2
able θ (see ref [12] for details, but noticing change in conventions see below
and ref [15]).
The second term in the right hand side of equation (1) is the so called
Wess-Zumino term and only depends linearly on the time derivative of the
field g. Hence it proves useful to rewrite it in terms of A(g), introduced
in order to permit the canonical quantization of the theory [14, 15], been
defined in such a way that
1
dr d2xεµν tr g−1∂ g g−1∂ g g−1∂ g ≡ d2x tr A(g)∂ g (2)
Z0 Z r r r r µ r r ν r Z 0
and
∂A ∂A
ij − kl = ∂ g−1g−1 −g−1∂ g−1 . (3)
∂g ∂g 1 il kj il 1 kj
lk ji
We shall use the following notation: α = nλ2, ε01 = 1, γ0 = 0 −i ,
4π (cid:18)i 0 (cid:19)
0 i
γ1 = , and γ5 = γ0γ1. When α = ±1 we have a super conformally
(cid:18)i 0(cid:19)
invariant theory. Canonical quantization is straightforward, on account of
ref [8] and [14, 15]); we have the following canonically conjugated momenta:
∂L 1 n iα
Π¯g = = ∂0g−1 − A (g) + ψ¯ γ1ψ g−1 (4)
ij ∂(∂ g ) λ2 ji 4π ji 4λ2 jk km mi
0 ij
and
∂L i i
Πψ = = ψ¯ γ = ψ† , (5)
ij ∂(∂ ψ ) 4λ2 ji 0 4λ2 ji
0 ij
were the first term in the right side at the equation (4), is the momentum
canonically conjugated to the field g in the principal chiral model only:
1
Πg = ∂0g−1 (6)
ij λ2 ji
with the following Poisson algebra:
{g (x),g (y)} = 0 , (7)
ij kl
{Π¯g (x),Π¯g (y)} = 0 ,
ij kl
{g (x),Π¯g (y)} = δ δ δ(x − y ) ,
ij kl ik jl 1 1
3
{ψ (x),ψ (y)} = 0 ,
ij kl +
{Πψ(x),Πψ(y)} = 0 ,
ij kl +
{ψ (x),Πψ(y)} = δ δ δ(x −y ) .
ij kl + ik jl 1 1
However, as we have already mentioned this model has been obtained via
a superfield formulation where we have imposed the constraint
G†(x,θ)G(x,θ) = 1 (8)
which leads, for the fermion component, to the relation
ψ† = −g−1ψ g−1 . (9)
ij ik kl lj
In the phase space we have the constraint
i
Ω = Πψ + g−1ψ g−1 (10)
il li 4λ2 im mk kl
which must be implemented using the Dirac method [16]. The basic element
of the method is the dirac matrix
i
Q = {Ω (x),Ω (y)} = g† g† δ(x − y ) (11)
ij kl 2λ2 il kj 1 1
satisfying
dz{Ω (x),Ω (z)}{Ω (z),Ω (y)}−1 = δ δ δ(x − y ) . (12)
ij mn mn kl ik jl 1 1
Z
From Q we can obtain the Dirac brackets (notice that as we have
{g (x),Ω (y)} = 0 the Dirac bracket of g with any other field is the
ij kl ij
Poisson bracket itself). Writing the equation (4) for the field Πg instead the
ij
field Π¯g , they read:
ij
{g (x),g (y)} = 0 , (13)
ij kl D
4
{g (x),ψ (y)} = 0 ,
ij kl D
{g (x),Πψ(y)} = 0 ,
ij kl D
{g (x),Πg (y)} = δ δ δ(x −y ) ,
ij kl D ik jl 1 1
1 α
g g 2 † ψ † † ψ †
{Π (x),Π (y)} = { (1−α )[g Π ψ g −g Π ψ g ]+ F }δ(x −y ) ,
ij kl D 2 jk ml mn ni li mj mn nk λ2 ji,lk 1 1
1
g † † † †
{Π (x),ψ (y)} = − {δ [g ψ −αg γ ψ ]+δ [ψ g +αψ γ g ]}δ(x −y ) ,
ij kl D 2 ik jm ml jm 5 ml jl km mi km 5 mi 1 1
1
{Πg (x),Πψ(y)} = {g†Πψ +Πψg† −α[Πψγ g† −g†γ Πψ ]}δ(x −y ) ,
ij kl D 2 li kj il jk il 5 jk li 5 kj 1 1
2λ2
{ψ (x),ψ (y)} = − g g δ(x −y ) ,
ij kl D il kj 1 1
i
1
{ψ (x),Πψ(y)} = δ δ δ(x −y ) ,
ij kl D 2 ik jl 1 1
−i
{Πψ(x),Πψ(y)} = g†g† δ(x −y ) ,
ij kl D 8λ2 li jk 1 1
where
F = (g†)′(x)g† (x)−g†(x)(g† )′(x) .
ij,kl il kj il kj
We are now ready to study the relevant algebraic properties of the super-
symetric WZW theory. In analogy with the bosonic case [8], we consider the
conserved Noether currents, which in the supersymetric case are given by:
1 i
JR (x) = − tr{(1∓α)[g((g˙†)±(g†)′)+ ψ(1±γ )ψ†]t }(x) (14)
±,a λ2 4 5 a
5
and
1 i
JL (x) = − tr{(1±α)[(−(g˙†)∓(g†)′)g + ψ†(1±γ )ψ]t }(x) . (15)
±,a λ2 4 5 a
Here we introduced the notation with the indices of a basis t of the
a
Lie algebra G were the fields g are defined, with the structure constants fc
ab
defined as:
[t ,t ] = fc t
a b ab c
and any field JRorL is defined as:
±,a
JRorL = (J ,tRorL) = −tr(JRorLt ) .
±,a ± a ± a
In addition, it is important in the case of theories containing fermions to
introduce the fermionic currents:
i
iR (x) = − tr[ψ(1∓γ )ψ†t ](x) (16)
±,a 4λ4 5 a
and
i
iL (x) = − tr[ψ†(1∓γ )ψt ](x) . (17)
±,a 4λ4 5 a
Since we know, from the case of the study of supersymetric non local
charges, that the above purely fermionic objects do also show up indepen-
dently [15, 17, 18] of the currents [10]. Moreover, there are also intertwining
operators already in the bosonic case [8, 1], which are described by the fields
1
j (x) = tr[g†t gt ](x) . (18)
ab λ2 a b
In the case of supersymetric theories we have also the fermionic partner
1
jψR (x) = tr[g† iR t gt ](x) (19)
±,ab λ2 ± a b
and
1
jψL (x) = tr[g†t giL t ](x) . (20)
±,ab λ2 a ± b
We shall see that after coupling bosons with fermions, an infinite number
of fields will be present, as a consequence of a quadratic algebra, which is
6
absent in the purely bosonic, as well as in the purely fermionic models. We
introduce
1
KR (x) = − tr[g†t iRt gt ](x) , (21)
±,abc λ2 a ± b c
1
KL (x) = − tr[g†t gt iLt ](x) , (22)
±,abc λ2 a b ± c
1
YR (x) = tr[g†t gt g†iRt gt ,](x) (23)
±,abcd λ2 a b ± c d
1
YL (x) = tr[g†t gt g†t giLt ](x) . (24)
±,abcd λ2 a b c ± d
We are now in position to write down the full algebra. The purely right
sector is very simple. For the purely left sector one substitutes (R → L) and
α → −α. We have:
1
{JR (x),JR (y)} = − (1∓α)fc {(3±α)JR −(1∓α)JR + (25)
±,a ±,b 2 ab ±,c ∓,c
λ2
+ (1∓α)[(3±2α−α2)iR −(5∓2α+α2)iR ]}δ(x −y )±2(1∓α)2η δ′(x −y ) ,
2 ±,c ∓,c 1 1 ab 1 1
1
{JR (x),JR (y)} = − fc {(1−α)2JR +(1+α)2JR + (26)
±,a ∓,b 2 ab −,c +,c
λ2
−(1−α2)2 [iR +iR ]}δ(x −y ) ,
2 −,c +,c 1 1
1
{iR (x),JR (y)} = − (1∓α)2fc iR δ(x −y ) , (27)
±,a ±,b 2 ab ±,c 1 1
1
{iR (x),JR (y)} = (1±α)2fc iR δ(x −y ) , (28)
±,a ∓,b 2 ab ±,c 1 1
1
{iR (x),iR (y)} = fc iR δ(x −y ) , (29)
±,a ±,b λ2 ab ±,c 1 1
{iR (x),iR (y)} = 0 , (30)
±,a ∓,b
7
where
1
η = (t ,t ) = − tr(t t )
ab a b λ2 a b
Notice that the intertwining operators did not appear at all up to this
point, The mixed sector is more involved. We have:
{JR (x),JL (y)} = ±(1−α2)[(1±α)j (x)+(1∓α)j (y)]δ′(x −y )+ (31)
±,a ±,b ab ab 1 1
λ2
−(1−α2) [(1−α2)(jψR +jψR −jψL −jψL )+8(jψR −jψL )]δ(x −y )
4 −ab +ab −ab +ab ∓ab ∓ab 1 1
{JR (x),JL (y)} = (1−α2)[∓(1∓α)j′ (x)+ (32)
±,a ∓,b ab
λ2
− (1∓α)(3∓α)[jψR +jψR −jψL −jψL ]]δ(x −y )
4 −ab +ab −ab +ab 1 1
1
{iR (x),JL (y)} = (1−α2)[jψL −jψR ]δ(x −y ) , (33)
±,a ±,b 2 ±ab ±ab 1 1
1
{iR (x),JL (y)} = (1∓α)(3∓α)[jψL −jψR ]δ(x −y ) , (34)
±,a ∓,b 2 ±ab ±ab 1 1
1
{iL (x),JR (y)} = − (1−α2)[jψL −jψR ]δ(x −y ) , (35)
±,a ±,b 2 ±ba ±ba 1 1
1
{iL (x),JR (y)} = − (1±α)(3±α)[jψL −jψR ]δ(x −y ) , (36)
±,a ∓,b 2 ±ba ±ba 1 1
1
{iR (x),iL (y)} = [jψL −jψR ]δ(x −y ) , (37)
±,a ±,b λ2 ±ab ±ab 1 1
{iR (x),iL (y)} = 0 . (38)
±,a ∓,b
We see now in equations (31, 32) the explicit appearance of the bosonic
intertwiners as well the fermionic one in equations (31, 32, 33, 34, 35, 36,
37). Notice also the simplicity of the purely fermionic components brackets.
In the critical case, α → +1 we see that a whole sector decouples completely,
8
the same being true for α → −1. We have in that case a super Kac-Moody
algebra, as expected, and the model is completely soluble. In fact, the con-
served charges can be used in order to provide a complete solution of the
Green functions, the only missing ingredient with respect to our computa-
tion being the super Virasoro generators. We finally write down the part of
the algebra involving the intertwiners. We have:
{JL (x),j (y)} = −(1±α)fdj δ(x −y ) , (39)
±,a bc ac bd 1 1
{JR (x),j (y)} = −(1∓α)fdj δ(x −y ) , (40)
±,a bc ab dc 1 1
{j (x),j (y)} = 0 , (41)
ab cd
{j (x),iL (y)} = 0 , (42)
ab ±,c
{j (x),iR (y)} = 0 , (43)
ab ±,c
{j (x),jψR (y)} = 0 , (44)
ab ±,cd
1
{jψR (x),JL (y)} = − (1±α)[2jψR −(1±α)jψR −(1∓α)KL ]δ(x −y ) ,
±,ab ±,c 2 ±ac,b ±ab,c ±,abc 1 1
(45)
1
{jψR (x),JL (y)} = − (1∓α)[2jψR +(1∓α)jψR −(3∓α)KL ]δ(x −y ) ,
±,ab ∓,c 2 ±ac,b ±ab,c ±,abc 1 1
(46)
1
{jψR (x),JR (y)} = (1∓α)[2jψR −(1∓α)jψR −(1±α)KR ]δ(x −y ) ,
±,ab ±,c 2 ±ac,b ±ca,b ±cab 1 1
(47)
1
{jψR (x),JR (y)} = (1±α)[2jψR +(1±α)jψR −(3±α)KR ]δ(x −y ) ,
±,ab ∓,c 2 ±ac,b ±ca,b ±cab 1 1
(48)
9
1
{jψR (x),iL (y)} = − [jψR −KL ]δ(x −y ) , (49)
±,ab ±,c λ2 ±ab,c ±abc 1 1
{jψR (x),iL (y)} = 0 , (50)
±,ab ∓,c
1
{jψR (x),iR (y)} = [jψR −KR ]δ(x −y ) , (51)
±,ab ±,c λ2 ±ca,b ±cab 1 1
{jψR (x),iR (y)} = 0 , (52)
±,ab ∓,c
1
{jψR (x),jψL (y)} = [KL −KR ]δ(x −y ) , (53)
±,ab ±,cd λ2 ±cab,d ±cab,d 1 1
{jψR (x),jψL (y)} = 0 , (54)
±,ab ∓,cd
1
{jψR (x),jψR (y)} = [YR −YR ]δ(x −y ) , (55)
±,ab ±,cd λ2 ±abcd ±cdab 1 1
{jψR (x),jψR (y)} = 0 (56)
±,ab ∓,cd
where
1
(jψR ) = (g†iRt gt )
±ab ij λ2 ± a b ij
1
(jψR ) = (g†iRt t g)
±ab ij λ2 ± a b ij
1
(KR ) = − (g†t iRt gt )
±abc ij λ2 a ± b c ij
1
(KR ) = − (g†t t iRt g)
±abc ij λ2 a b ± c ij
1
(jψL ) = (g†t giLt )
±ab ij λ2 a ± b ij
10