Table Of ContentFusion rules in N=1
superconformal minimal models
8
9
Pablo Minces , , M.A. Namazie and Carmen Nu´n˜ez
9 †∗ ‡ †
1
n Instituto de Astronom´ıa y F´ısica del Espacio (CONICET)
†
a
J C.C.67 - Suc.28, 1428 Buenos Aires, Argentina
8 Universidade de S˜ao Paulo, Instituto de F´ısica
∗
Caixa Postal 66.318 - CEP 05315-970 - S˜ao Paulo - Brasil
1
v ‡ namazie@pacific.net.sg
5
3
0
1
0 Abstract
8
9
The generalization to N=1 superconformal minimal models of the
/
h relation between the modular transformation matrix and the fusion
t
- rules in rational conformal field theories, the Verlinde theorem, is
p
shown to provide complete information about the fusion rules, in-
e
h cluding their fermionic parity. The results for the superconformalTri-
:
v critical Ising and Ashkin-Teller models agree with the known rational
i conformal formulation. The Coulomb gas description of correlation
X
functions in the Ramond sector of N=1 minimal models is also dis-
r
a cussed and a previous formulation is completed.
1 Introduction.
One of the most interesting results in 2D conformal field theory is the Ver-
linde theorem [1][2][3]. It gives the number of conformal blocks of a RCFT
on a punctured Riemann surface in terms of elements of the modular matrix.
The result arises as a consequence of the well established, but nevertheless
still surprising, fact that the modular matrix S implementing the modular
transformation τ 1/τ on the space of genus one conformal blocks, di-
→ −
agonalizes the fusion rules. The proof of the theorem relies on the technical
assumption that both left and right extended chiral algebras consist only of
generators with integral conformal weight, the so called rational conformal
fieldtheories(RCFT’s), andthusevidently excludes thesuperconformalcase.
A generalized Verlinde formula which describes the fusion rules in all
sectors of N = 1 superconformal theories was obtained in [4]: by consider-
ing some examples of correlation functions of primary fields and employing
certain bases of conformal blocks, it was argued that the Verlinde conjec-
ture extends to the N = 1 superconformal unitary series. In this letter we
show that the generalized Verlinde formula contains information about the
fermionic parity of the fusion rules.
InSection 2. we complete theformulationofthe Coulombgasdescription
[5] of correlation functions in the R sector performed in [6] and [4], by con-
sidering the parity of the R fields. In Section 3. we discuss the generalized
Verlinde formula and compute the fusion rules of the Tricritial Ising model
(TIM) and the critical Ashkin-Teller model (AT). These models are known
to have both RCFT as well as superconformal descriptions, so they can be
regarded as a check of our results. Since the superconformal Coulomb gas
method remains at the level of a prescription, the correlation functions have
to pass several checks such as null state decoupling and correct behavior un-
der degeneration of the torus to the plane as well as in the factorizationlimit.
Consistency with the conformal fusion rules can therefore be considered an-
other successful check on the superconformal blocks and on the extension of
the Verlinde formula.
1
2 Two-Point Conformal Blocks.
Contour integral representations of the conformal blocks can be computed
with the Dotsenko-Fateev Coulomb gas technique [5] by introducing Feigin-
Fuks screening operators to make correlation functions background charge
neutral. Correlation functions of N = 1 superconformal primary fields on
thetoruswereconsideredin[6][4]whereonecontourexamples, corresponding
to Ramond primary fields, and double contour integrals, necessary when
considering NS primary fields, were studied. In this section we rely heavily
on notation, results and the discussion contained in these references.
The N=1 superconformal minimal models are characterized by the fol-
lowing discrete series of central charges c and allowed conformal weights of
the primary fields ∆ [7].
r,s
3 12
c = (p = 3,4,...)
2 − p(p+2)
[(p+2)r sp]2 4 1
∆ = − − + [1 ( 1)r s] (1)
r,s −
8p(p+2) 32 − −
where 1 s r p 1 with r s even in the NS sector and 1 s r 1
≤ ≤ ≤ − − ≤ ≤ −
for 1 r p 1 or 1 s r +1 for p+1 r p 1 with r s odd in
−
≤ ≤ 2 ≤ ≤ 2 ≤ ≤ − −
the R sector.h i h i
In order to compute correlation functions, the fields are represented by
vertex operators of the form [8]
NB(z) = eiαφ(z) NF(z) = ψ(z)eiαφ(z) R (z) = σ (z)eiαφ(z) (2)
α α α± ±
where φ and ψ are a free boson and a Majorana fermion, NB and NF are the
α α
bosonic and fermionic components of a NS field, σ are the two spin fields
±
with conformal weight 1 that are to be identified respectively with the spin
16
field and the disorder field of the Ising model, and R are the two R fields
α±
whose conformal weights correspond to the charge α (only the ground states
of conformal weight c are not degenerate). They obey the following OPE
24
1
ψ(z)σ (w) σ (w)+...
± ∓
∼ (z w)1
2
−
2
1
3
σ±(z)σ±(w) +(z w)8ψ(w)+...
∼ (z w)1 −
8
−
3
σ±(z)σ∓(w) (z w)8ψ(w)+... (3)
∼ −
Let us compute the conformal blocks corresponding to the correlator
φ φ using the Coulomb gas formalism and completing the results in
1,2 1,2
h i
[6] and [4] by taking into account the fermionic parities of the fields, i.e.,
considering the three possible two-point functions φ+ φ+ , φ φ and
1,2 1,2 −1,2 −1,2
φ+ φ . The conformal blocks are D E D E
1,2 −1,2
D E
Gi(sign1,sign2);ν = dz σsign1(z1)σsign2(z2)ψ(z) e−2iα−φ(z1)e−2iα−φ(z2)eiα−φ(z)
ν ν
CIi D E D E
(4)
where ν = 1,2,3,4 label the spin structure and
p+2 p
α = α = (5)
+
s 2p − −s2(p+2)
Theclosedcontourbasis C ,C hasbeenspecifiedin[6][4],andsign1,sign2
1 2
{ } ∈
+, .
{ −}
The correlation function of 2n spin fields and 2m disorder fields of the
Ising model on the torus is given by [9]
2
σ+(w ,w¯ )...σ+(w ,w¯ )σ (u ,u¯ )...σ (u ,u¯ ) =
1 1 2n 2n − 1 1 − 2m 2m
ν
D E
2
Θ εiwi+ ε′kuk εiεj
ν 2 Θ1(wi wj) 2
= εiεε+′kiX==±±11ε′k=0(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:18)PΘν(0)P (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Yk ε′kkYi<<jl(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Θ′1(−0) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:12) (cid:12)
P P ε′kε′l εiε′k
Θ (u u ) 2 Θ (w u ) 2
1 k l 1 i k
− − (6)
×(cid:12)(cid:12) Θ′1(0) (cid:12)(cid:12) Yi,k (cid:12)(cid:12) Θ′1(0) (cid:12)(cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
where Θ are t(cid:12)he usual Jac(cid:12)obi Θ-f(cid:12)unctions. (cid:12)
ν
3
We take the square root on both sides of this equality and keep the holo-
morphic part only (obtaining in this way the so called ‘holomorphic square
root’). We then specialize to n = 2, m = 0; n = 0, m = 2 and n = m = 1
and take the limits w w , u u and w u respectively. In all cases
3 4 3 4 2 3
→ → →
(because of (3)) the residue of the 3 order pole is kept. We thus obtain
8
σ+(z )σ+(z )ψ(z) = σ (z )σ (z )ψ(z)
1 2 − 1 − 2
ν ν
D E D 3 E 1
Θ1(z1 z2) 8 Θ1(z1 z) −2
= − −
" Θ′1(0) # " Θ′1(0) #
1
Θ1(z2 −z) −21 Θν 12(z1 −z2) −2
×" Θ′1(0) # (cid:16)Θν(0) (cid:17)
Θ 1(z +z 2z)
ν 2 1 2 − (7)
× (cid:16) Θ (0) (cid:17)
ν
1 1 1
σ+(z )σ (z )ψ(z) = Θ1(z1 −z2) −8 Θ1(z1 −z) −2 Θ1(z2 −z) −2
1 − 2
ν " Θ′1(0) # " Θ′1(0) # " Θ′1(0) #
D E
[Θ′1(z1 −z)Θ1(z2 −z)Θν 21(z1 −z2)
× Θ (0) Θ (0) (cid:16)Θ (0) (cid:17)
′1 ′1 ν
Θ1(z1 −z)Θ′1(z2 −z)Θν 21(z1 −z2)
− Θ (0) Θ (0) (cid:16)Θ (0) (cid:17)
′1 ′1 ν
2Θ1(z1 −z)Θ1(z2 −z)Θ′ν 12(z1 −z2) ]12 (8)
− Θ (0) Θ (0) (cid:16)Θ (0) (cid:17)
′1 ′1 ν
By calculating the lattice contribution in (4) we get
G(sign1,sign2);ν(r,s) = dz σsign1(z )σsign2(z )ψ(z)
i 1 2
ν
CIi D E
1α2 α2
Θ1(z1 z2) 2 − Θ1(z1 z)Θ1(z2 z) − −
− − −
×" Θ′1(0) # " Θ′1(0)2 #
Γν(λ) ( 1)rδ1ν+r(s+1)δ4νΓν λ˜ (9)
× − −
h (cid:16) (cid:17)i
4
where
Γν(λ) = Θν(0)12eiπ 2λN2−112 δ4ν ( 1)np(δ1ν+δ4ν)
η3 (cid:16) (cid:17) −
2 n Z
X∈
q(λ+4nNN)2eiπλ√+nNNα−(2z−z1−z2) (10)
×
˜
λ = r(p+2) sp λ = r(p+2) sp N = 2p(p+2) (11)
− − −
Notice that only the fermionic correlator (7) was considered in [6][4].
(+,+);ν ( , );ν
Clearly from eq. (7) Gi = Gi−− . Nevertheless, it is possible to
see that both σ+(z )σ+(z )ψ(z) and σ+(z )σ (z )ψ(z) have the same
h 1 2 iν h 1 − 2 iν
monodromyandmodularproperties aswell asdegenerationandfactorization
limits. Indeed, in the degeneration limit,q 0, the four-point functions on
→
the sphere are recovered, namely
V (0)Rsign1(x )Rsign2(x )V ( ) (12)
α 1,2 1 1,2 2 2α0−α ∞
D E
where the conjugate vertices V and V are either NB, NF or R and
α 2α0 α ∆ ∆ ∆±
−
the contours C and C on the torus degenerate to Pochhammer contours on
1 2
the extended complex plane.
Thepropertiesoftheconformalblocksundermonodromytransformations
are not altered. Recall that φ(a) and φ(b) are the Verlinde operators which
implement these transformations as the point z (or z ) is transported once
1 2
around either an a or b cycle on the torus. These operators are not modified
when the correlation functions contain spin fields of different parity.
Another check on the conformal blocks is provided by the factorization
limit z z , where the intermediate states are precisely those dictated
1 2
→
by the fusion rules. The problematic feature appearing for even p however,
remains. In fact, the N = 1 superconformal partition function can always
be obtained by factorizing the modular and monodromy invariant two point
correlation function on the identity intermediate state. However in the p = 4
case or AT model at criticality, the C contour in the factorization limit
1
reproduces all the terms of the corresponding superconformal partition func-
tion [10], except for the contribution Tr (-1)F of the ν = 1 sector. This can
[R]
5
be understood since the intermediate state in the ν = 1 sector is a fermion,
so we do not expect to obtain the partition function but the fermion expec-
tation value. Actually when ν = 1 the residue is zero, reflecting the fact that
the fermion is a null state.
Therefore taking into account the parities of the R fields does not modify
the checks on the conformal blocks that have been performed previously in
[6][4].
Let us now consider the modular transformation matrix S on the basis of
conformalblocksG(sign1,sign2);ν(r,s). Itispossibletoseethat σ+(z )σ+(z )ψ(z)
i=1 h 1 2 iν
and σ+(z )σ (z )ψ(z) have the same S transformation properties, so the
h 1 − 2 iν
form of the S matrix does not depend on the parity of the fields. Notice that
the S 1 matrices are just those given in [4]. We list them here (up to
−ν =1,2,3,4
′
phase factors) for completeness.
S 1 r′,s′ = 4 sinπ rr α2 rs′ sinπ ssα2 sr′ (13)
−ν′=3,4 r,s − p(p+2) ′ + − 2 ! ′ − − 2 !
(cid:16) (cid:17)
q
S 1 r′,s′ = 4γr′,s′ sinπ rr α2 sr′ sinπ ssα2 rs′ (14)
−ν′=2 r,s − p(p+2) ′ + − 2 ! ′ − − 2 !
(cid:16) (cid:17)
q
S 1 r′,s′ = 4 sinπ rr α2 sr′ + r′ cosπ ssα2 rs′ r′
−ν′=1 r,s − p(p+2) ′ + − 2 2! ′ − − 2 − 2!
(cid:16) (cid:17)
q (15)
with
1
γr′,s′ = 1− 2δp,2Zδr′,p2δs′,p+22 (16)
Recall that the physical origin of the factor γ is the double degeneracy of
r,s
′ ′
the R states, apart from the vacuum (r ,s) = (p, p+2).
′ ′ 2 2
These equations are essentially the modular transformations of supercon-
formal characters [11] except for SR˜ R˜ or S 1 which is not defined except
→ −ν =1
′
for the (r,s) = p, p+2 state where it is the identity.
2 2
(cid:16) (cid:17)
6
3 The Verlinde theorem in N=1 superconfor-
mal models.
Even though the proof of the Verlinde theorem [1],[2], [3] requires that left
and right extended chiral algebras consist only of generators with integral
conformal weight, it was shown in [4] that it is possible to construct a com-
plete Verlinde basis in N=1 superconformal minimal models, namely
G = (E E O O ) (17)
+ +
− −
with
E±(r,s) = 12[Gν1=3(r,s)±e−iπ(2λN2−112)Gν1=4(r,s)],
1
O (r,s) = [Gν=2(r,s) ei4πGν=1(r,s)] (18)
± 2 1 ± 1
which isaneigenstate of φ(a) andwithrespect towhich φ(b) yields thefusion
rule coefficients. In this basis the descendants in a q expansion are always at
integer level spacing above the highest weight state (regarding superdescen-
dants as Virasoro primaries). Moreover, the proof of the Verlinde theorem
relies only on conformal and duality properties under certain manipulations
of conformal blocks in their degeneration and factorization limits. Since we
have checked that these limits are indeed consistent with the fusion rules,
this suggests that by working in the appropriate basis the proof can always
be carried through. Even though we have no general proof, we will show
in this section that the generalized Verlinde formula gives the correct fusion
coefficients in the TIM and AT models, including the fermionic parity of the
fields.
Let us summarize the arguments leading to the generalization of the Ver-
linde formula. The fusion coefficients NK are defined as
IJ
ϕ ϕ = NK ϕ (19)
I × J IJ K
where ϕ is one of the operators NB, NF or R , and the upper case indices
I α α α±
denote both r,s and spin structure sector ν (or, rather, the appropriate com-
binations of spin structures discussed above). As shown in [4], the b cycle
monodromy operator in the t channel, φt(b), yields the expected superconfor-
mal fusion rules. Since we are interested in the action of φ (b) on characters
I
7
rather than two-point blocks, we may factorize the t channel blocks on the
identity intermediate state. In general one has an equation of the form
φ (b)χ = NKχ (20)
I J IJ K
K
X
where χ denotes a particular character (or combination of characters) and
I
the normalization condition is NK = δK, where J = 0 denotes the identity
I0 I
or NS vacuum character. Furthermore, with respect to the same basis of
conformal blocks, one has under φt(a)
(J)
φ (a)χ = λ χ (21)
I J I J
One may now proceed as in [1], using the conjugation relation between a
and b cycle monodromy, φt(a) = S 1φt(b)S, to express the fusion coefficients
−
in terms of the modular matrices and the eigenvalue of φ (a):
I
NK = SLλ(L)(S 1)K (22)
IJ J I − L
L
X
The modular matrix K acts on the Verlinde basis (17) as
SI
1
G(r,s|τ) = (S)rr,′s,s′G(r′,s′|− τ) (23)
r,s
X′ ′
and is given by
S 1 S 1 S 1 S 1
−ν =3 −ν =3 −ν =2 −ν =2
S ′1 S ′1 S′ 1 S′ 1
= −ν′=3 −ν′=3 − −ν′=2 − −ν′=2 (24)
S SS−ν−ν′′11==44 −−SS−ν−ν′′11==44 −SS−ν−ν′1=′1=11 −SS−ν−ν′1=′1=11
Using the normalization condition allows the eigenvalue to be expressed as
SL
λ(L) = I (25)
I SL
I=0
whence
SLSL(S 1)K
NK = I J − L (26)
IJ S
L I=0,L
X
8
Since 2 = 1, multiplying by 2 on the right yields the result for the number
S S
of couplings between three operators labeled by I,J,K:
S S S
I,L J,L L,K
N = (27)
IJK
S
L I=0,L
X
Notice that the matrix (24) is symmetric and unitary for p odd. This is in
S
accord with the well known result that for p = 3 in particular, there are two
equivalent representations of the TIM- either as a p = 4 minimal conformal
model or as the N = 1 superconformal model that we are discussing. In fact,
for p = 3 the combinations of blocks (17) in the factorization limit are the
Virasoro characters
E (1,1) χVir E (1,1) χVir
+ ∼ 0 − ∼ 32
E (1,3) χVir E (1,3) χVir
+ 1 3
∼ 10 − ∼ 5
O (2,3) χVir O (2,1) χVir (28)
3 7
± ∼ 80 ± ∼ 16
However for p even the factor γ in the matrix S 1 , associated with
r,s −ν =2
′
the Ramond vacuum state, and the vanishing of S 1 for (r,s) = p, p+2 ,
−ν′=1 2 2
appear to prevent from being either symmetric or unitary, unlik(cid:16)e in th(cid:17)e
S
conformal case.
Let us compute the fusion coefficients in the first two cases of the N = 1
superconformal minimal series, p = 3 and 4. Notice that in equation (27)
I = 0 corresponds to E (1,1) and L = E (r,s),E (r,s),O (r,s),O (r,s).
+ + +
− −
We have to consider the three possibilities NS NS NS, R R NS
× ∼ × ∼
and NS R R. In the first case we find, for p = 3, i.e. the TIM,
× ∼
N = N = 1
(1,3, )(1,3, )(1,1,+) (1,3, )(1,3, )(1,3, )
± ± ± ± −
N = N = 1
(1,3,+)(1,3, )(1,1, ) (1,3,+)(1,3, )(1,3,+)
− − −
N = N = 1 (29)
(1,1, )(1,1, )(1,1,+) (1,3, )(1,1, )(1,3, )
− − ± − ∓
where we have written only the non-trivial nonvanishing coefficients. They
correspond respectively to the conformal fusion rules [12, 13]
1 1 3 3 3 3 1 3 3 1
= 0+ , = 0+ , = +
10 × 10 5 5 × 5 5 10 × 5 2 10
1 3 3 3 3 1 3 3
= , = , = 0 (30)
10 × 2 5 5 × 2 10 2 × 2
9
