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Admissible ŝl (2|1; C)k characters and parafermions
ELSEVIER
Nuclear Physics B 529 [PM] (1998) 588-610
Admissible s /(211; C) k characters and parafermions M. Hayes 1, A. Taormina 2 Department of Mathematical Sciences, University of Durham, Durham, DH1 3LE, England, UK Received 9 March 1998; revised 4 June 1998; accepted 2 July 1998
Abstract
The branching functions of a particular subclass of characters of the affine superalgebra s/(211; C)k into characters of the subalgebra s/(2;C)k are calculated for fractional levels k = 1 1, U E N. They involve rational torus A,~u-1) and Z,_I parafermion characters. @ 1998 u Elsevier Science B.V. PACS: 02.20.Sv; 11.25.Pm Keywords: Non-critical superstrings; Affine superalgebras; Characters
1. I n t r o d u c t i o n
It has long been stressed that non-critical strings might well be described in terms of a topological GIG Wess-Zumino-Novikov-Witten ( W Z N W ) model [ 1 - 3 ] , where G is a Lie supergroup or a Lie group, depending on whether the string theory considered is supersymmetric or not. Our original motivation is to re-analyse non-critical N = 2 strings in the light of this potentially new approach. Although non-critical N = 2 strings have been the subject of some investigation [ 4 - 6 ] , they remain largely understudied. They possess interesting features and technical challenges. In particular, as emphasized in [4], this string theory is not confined to the regime of weak gravity, i.e. the phase transition point between weak and strong gravity regimes is not of the same nature as in the N = 0, 1 cases. This absence of barrier in the central charge is a hope for new physics. i E-mail: [email protected] 2 E-mail: [email protected] 0550-3213/98/$ - see frontmatter @ 1998 Elsevier Science B.V. All rights reserved. PII S 0 5 5 0 - 3 2 1 3 ( 9 8 ) 0 0 5 2 2 - 7
M. Hayes, A. Taormina/Nuclear Physics B 529 [PM] (1998) 588-610
589
The exact correspondence between the traditional approach to non-critical strings and the latter is yet to be proven. However, a crucial ingredient in the description of the spectrum in the GIG picture is the representation theory of the corresponding affine Lie (super)algebra, ~, at fractional level k = p/u - h, p E Z \ {0},u C N, gcd(p,u) = 1, with h the dual Coxeter number of ~. For instance, the SL(211;R)/SL(211"R ) topological quantum field theory obtained by gauging the anomaly free diagonal subgroup S L ( 2 f l ' R ) of the global SL(211;R)L x SL(211;R)R symmetry of the WZNW model appears to be intimately related to the non-critical charged fermionic string, which is the prototype of N = 2 supergravity in two dimensions. A comparison of the ghost content of the two theories strongly suggests that the N = 2 non-critical string is equivalent to the tensor product of a twisted SL(211; R)/SL(211; R) WZNW model with the topological theory of a spin-½ system. It is however only when a one-to-one correspondence between the physical states and equivalence of the correlation functions of the two theories are established that one can view the twisted GIG model as the topological version of the corresponding non-critical string theory. For the bosonic string, the recent derivation of conformal blocks for admissible representations of s/(2; R) is a major step in this direction [7,8]. As far as the s/(211;C) and N = 2 algebras are concerned, a deeper investigation of their representations is a prerequisite to obtaining similar results for the corresponding conformal models. In this paper we therefore investigate further the characters of the complex affine superalgebra s/(211; C)k. When the matter, which is coupled to supergravity in the N = 2 non-critical string, is minimal, i.e. taken in an N = 2 super Coulomb gas representation with central charge, Cmatter = 3 ( l -- - ~ ) ,
p, uEN,
gcd(p,u) = 1,
(l.l)
the level of the 'matter' affine superalgebra s/(211;C ) appearing in the model of SL(211;R)/SL(2ll;R) is of the form
k= P- - l ,
(1.2)
b/
i.e., the level precisely takes values for which admissible representations of s/(211; C)~ do exist [9]. The character formulae obtained in [10] as functions of three complex variables T, cr and ~, are not directly suitable for the analysis of their behaviour under the modular group. In order to make this analysis straightforward, we provide here a decomposition of the s/(21 l ; C ) k characters into characters of the subalgebra s/(2: C)~.. It turns out that for a level of the form 1
k= --
1,
u=2,3 .....
(1.3)
bt
i.e. of the form (l.2) with p = 1, the branching functions involve characters of a rational torus model Au(u-t) as well as Z , - i parafermion characters. The identification of the branching functions relies on the residue analysis at the simple pole developed,
M. Hayes, A. Taormina/Nuclear Physics B 529 [PM] (1998) 588-610
590
in the limit where o- tends to zero, by a subset of s/(211; C ) / characters. One particular feature of ; / ( 2 l l ; C ) k is that the central charge of the associated Virasoro algebra is zero (a consequence of equal number of bosonic and fermionic generators in sl(2/1)), Csl(2ll ) = O,
(1.4)
so that the coset sl(211;C)l,/sl(2;C)k
has central charge Ccoset = Cs~(211) -cst(z) =
-3k/(k+2). Since we restrict ourselves here to levels of the form (1.3), the coset central charge is positive, 3 ( u - 1) - , u+l
Ccoset -- -
(1.5)
and its value is one when u = 2. The branching functions in this case are just the characters of the rational torus algebra A2, as already noticed in [ 10]. The paper is organised as follows. In Section 2 the sl(211;C)t characters given in [ 10] are rewritten in terms of infinite products and theta functions. The latter formulation leads in Section 3 to the decomposition of s / ( 2 l l ; C ) k characters into s/(2;C)k characters. The derivation of the crucial formula (3.5) is outlined in Appendix A. Section 4 uses the residues of singular ~l(211;C)k characters at the pole o-= 0 to rewrite the branching functions in such a way that the S transform of the ~/(211; C)k characters can be easily worked out. The decomposition formulae (4.4) and (4.5) are conjectured for higher values of the parameter u on the basis of a detailed analysis of the cases u = 3, 4 sketched in Appendix B. We offer some comments on the structure of the coset sl(211; C) ~/sl( 2; C) k in our conclusions.
2. N e w formulae for
s/(211; C)k admissible
characters
The algebra sl(211;C)k is the affinisation of the Lie superalgebra labelled A ( 1 , 0 ) in Kac's classification [ 11 ]. The latter has rank two. Let al and a2 be two fermionic simple roots. Those roots and their opposites, corresponding to the generators e±~,, e±,~2, have zero norm square. The even subalgebra of A ( 1 , 0 ) is the direct sum of an abelian algebra, with generator h+, and of the algebra Al with generators e+(al+a2), h_. The background material needed here, as well as the notations used, can be found in [12,10] and references therein. We however recall the defining commutation relations of sl(2ll;C)k. Introducing the following Laurent expansions for the currents, (z) : n
S(h+) (z) n
n
/-4-
J(e±~,)(z) =2..~jn z n
--n-- I
,
J(e±-2) ( z ) = ~ f " J +z-n-in , n
(2.1)
M. Hayes, A. Taormina/Nuclear Physics B 529 [PM] (1998) 588-610
the commutation relations of ~I(211;C) k in terms of the Laurent modes
591 are
3 { j + , j ; ] = 2J~,+n + kmtSm+n,o,
4- , [ J3,Jn~ ] = i J~,,+.
3 J~3 ] = -~m k 8 m+mO, [ Jm,
•-4r Ja,/oT = +Jm+,,,
-4- .~
.+'
[ J~,, , Yn ] = TYro+,,,
[2J3,, j; + ] = -}-Jm+n, "+
[2J~,,jf ] = +Jm+n, '+
[ 2Urn, j ' i ] = ijmi+,, [2Um,j~] = qZYm+n, "± k [u.,,G] = - ~ mSm+,,,o, 3 {L+,fo-}= ( Urn+,, -- Jm+n) .q-
._
{j,,,,a. }=
-
km~Sm+m0,
3
(U,,+,, + Jm+,,) + km~m+,,,o,
t
5: {j.,f, j#} = Jm+n"
(2.2)
Furthermore, the Sugawara energy-momentum tensor is given by T(Z)-
1
--{2(J3)2(Z)
2 ( k + 1)
-2uZ(z)
+j'+j'- (z) - j'-j'+(z)
+ J+J-(z)
- j+j-(z)
+ J-J+(z)
+ j - j + ( z ) },
(2.3)
and L0 is the zero-mode in its Laurent expansion. In [ 10], we derived formulae for the characters of irreducible representations of s/(211; C)k at fractional level k = pu - 1, where p and u are positive coprime integers. Those characters are obtained in a standard way from the knowledge of the quantum numbers and embedding diagrams of singular vectors within a given Verma module [ 12]. In the present paper, we are interested in sl(211;C) k characters belonging to classes IV and V in the notations introduced in [12,10]. Incidentally, the two classes share the same embedding diagram structure, as explicitly argued in [13]. The quantum numbers h_, h+ of the highest weight state in class V are obtained by requiring the following s/(2[I;C)~ Kac determinant [15] factors to vanish:
~ ° ) ( 6 q + &2,0, m ) = h _ + (k + 1 ) m - p , q~(')(-&,,0, 1 + m ' ) = - ½ h _ + ½h+ + ( k + 1)(1 + m ' ) ,
(2.4) (2.5)
for m,m' E Z+, 0 ~< m + m' ~< u - 2 and k + 1 = p/u, gcd(p,u) = 1. In class IV, these highest weight state quantum numbers are obtained by requiring ,~o~°) (,~, + ,~2,0, m) = 0 = ~ ( ' ) ( , ~ , , O , m ' ) ,
(2.6)
i.e.
h_+(k+l)m=O,
21 h _ - ~ h +1 + ( k + l ) m ' = O ,
(2.7)
592
M. Hayes, A. Taormina/Nuclear Physics B 529 [PM] (1998) 588-610
for m,m + E Z+, 0 ~< m' ~< m ~< u - 1 and k + 1 = p / u , gcd(p,u) = 1. It is still a conjecture that there exist no subsingular vectors in classes IV and V, and we stress that the corresponding characters have been constructed in [10] under this assumption. By definition, s/(211;C)~ characters are given by X/l(211;c)k (~-, o', v) ~a tr exp(27ri(~-Lo + o'J 3 + vUo)) h_, h t
(2.8)
where Jo3 and U0 are the zero-modes of the sl(211;C)~ Cartan generators, and the 1 isospin ½h_ and charge ~h+ are their eigenvalues when they act on the highest weight state Is2} of the representation,
S la> ='~h_ls2 ),
Uolg2)= lh+[~O).
(2.9)
The variables q, z and ( are defined by d
q= exp(2~ir),
TCC
d
z = exp(2~i~),
~ E C,
(~exp(2~iv),
v C C.
Im(r) > 0 =~
Iql
< 1,
(2.10)
The trace is over all states in the irreducible module with highest weight state Is2). For instance, the class IV Neveu-Schwarz characters corresponding to an irreducible highest weight module with highest weight state quantum numbers h~ s, hN+s read NS,lV,s/(2ll;C)k XhNS,h~S
" INS 1 NS £7", 0-, 9) = qhNSz ~h ~h+ FNS(7., O', 9) Z qa2pu+ap(l+m)zap aEZ 1 -- q2au+l+mz >( (1 -t- qau+m, + 51 z ~~ ( - 51) ( 1 + qaU-m'+m+½z½(½) ' +
(2.11)
wherem, m t E Z + a n d 0 ~ < m t ~ < m ~ < u - l , h N S = m ( k + l) + k , hN+s = (2m' - m ) ( k + 1),
(2.12)
and the function FNS(T, o-, v) is given by FNS(7", O', P) = H0(3( l + z ~ ( ~ q "1 - ~1 ) ( l +1z
1
( ~I q " - ~1) ( l + z l ( - ½ q ' - l ) ( l + z - ~ ( - ~ q n -1~ ) (1 - qn)2(1 - z q ' ) ( 1 - z - l q n - l )
1
1
n=l
(2.13)
On the other hand, the class V Neveu-Schwarz characters corresponding to an irreducible highest weight module with highest weight state quantum numbers hNs, h~ s read
M. Hayes, A. Taormina/Nuclear Physics B 529 [PM] (1998) 588-610 ."
1 NS
I NS
NS,V,~)(2]I;C), (75 O', P) = qhNSz ~h_ (sh+ FNS (75 O', U) ~ Xh~S'hr~ s
593
qa2pu+ap(M+M' +l ) .,~--ap
aCZ 1 -- q2au+M+M'+l Z--I X
( 1 + qau+M+½ Z _~(_~L
whereM, M ICZ+,0~
~) ( 1 + qau+M'+½Z. -'(~ -) '
(2.14)
t~
hN_s = - ( M + M ' + 2 ) ( k +
l) + k ,
hN+s = ( M - M t ) ( k + I ) .
(2.15)
In both classes, the conformal weight of the highest weight state is given by hNS - 4 ( k 1+ 1~ ((hN-S)2 -- (h~-S)2 + (2hN-S -- k)).
(2.16)
We recall here that the class IV (resp. class V) Ramond characters are easily obtained from class IV (resp. class V) Neveu-Schwarz characters by spectral flow, namely, R,s/(211;C)k
Xh~ ,I,~
(7, o-, v) = q
k/4 k/2
Z
NS,,~l(2[l;C)kr
XhN_S,h~S
T
t , - o - - 7, v),
(2.17)
where h R = hNS I
R
5h_ = - 1 R
_
I_hNS + 1
- 2--
~ hNS
~k,
1
+ ~k,
_lhNS
7h+ - 2,~+
(2.18)
.
The above formulae are not suited to the discussion of the modular properties of characters. Our ultimate goal is to identify how the above sl(2[ I ; C ) k characters branch into s l ( 2 ; C ) k characters, in order to make their modular transforms straightforward to obtain. We restrict ourselves to levels of the form k = ,1- - 1, and we mainly concentrate on the Neveu-Schwarz sector of the theory. We first use standard techniques to produce expressions for the characters in terms of infinite products. Namely, starting with expression (2.11), we calculate the residue at the poles corresponding to 1 + qa',+,/+½ z ½( - " = 0 for some integer a ~ and to 1 + q'<"-m'+m+½zl(½ = 0 for some integer a ' . It is not too hard to show that the residues above coincide with those obtained from the following expression: NS,IV,sTI(2[ 1;C)k.
/~hNS hNs
~
~
[7, 0", P) = qIySz
I hNS
I hNS
~re
(~ + F " ° ( r , 0", v)
oo
x H
(1 - qU.)2(1 - zq "n-m) (1 - z - l q " ( ' - l ) + m )
n=l
x h~-j (m, m~; r, o-, v ) ,
where
(2.19)
M. Hayes, A. Taormina/Nuclear Physics B 529 [PM] (1998) 588-610
594 I
1
I
I
1
t
1
h n ( m , m ; r, o', v ) = ( 1 + z ½(3q "(n-1)+m-" +3 ) ( 1 + z 3 ( - ~ q u(n-1)+m +3 ) ×(l + z-½(½qU"-m'-½)(l
(2.20)
+ z-½f-½qU"+m'-m-½).
Although two functions having the same poles and residues are not necessarily equal, and in particular can differ by a function which is regular on the appropriate domain, we expanded (2.11 ) and (2.19) in a power series of the variable q for several representations and found the two expressions are identical, to a large finite order. An easy way to obtain the class V NS characters is to substitute M + M' for m, M for m' and z ~ z - 1 in the above expression, and multiply by an overall phase. Note that, in contrast to the case where the level k is integer, i.e. where u = 1 in the parametrisation (1.2), and for which the s/(2[l; C)~ characters are regular in the limit o- ---+ 0, the situation for a level of the form k = ~1 - 1 is quite different. Indeed, in this case, there exist u 2 NS characters in classes IV and V, u of which are regular at o- = 0. The others develop a simple pole at this value of o- and the residue at the pole is given by
.
~--+01imz~rto" Xh~S,hTS
.....
,
~w, o, t.,) --
o0,2 (7-, ½12) + < . 2 (7-, ½12) Ns /]3(7. )
.)(r,s'
! (7-, ~"2 ) ,
(2.21) where the N = 2 superconformal characters appear in the infinite product form first derived by Matsuo [16]. In the above, the N = 2 central charge is c = 3(1 - 2) and m=r÷sl,m'=r-½ in class IV, w h i l e M = r - ½ , M ' = s - ½ in classV. Similar formulae exist for singular Ramond characters,
.
lim zcrto- Xh" ,h:~
cr----~0
-
O', 12) =
(7-, ½12) + o_,.2 (7-,'212) /]3(7-)
R+ N=2~
X~,s'
t 7-, sr ½), (2.22)
withm=r+s,m'=rinclassIV, whileM=r-l,M'=s-1 in classV. Although interesting, the infinite product formula (2.19) has a denominator whose behaviour under the modular group is non-trivial. Using the Jacobi triple identity repetitively, as well as the standard properties of theta functions, a tedious calculation leads from (2.19) to the following elegant expression: NS,IV,s~/(2I I;C)k e 7-
X h~.~,h~s
~ , ~r, v ) =
O--u+2(m+l),2u (7-, ~) -- 19u+2(m+l),2u (7-, ~) /](r)_l/]3_2U(ur) O~,2(r, or) -- O_~,2(r, ~r)
u--I 1 12 0v X HZ~-~us+m--2m',u (7-,U)'l-~u(s+l)+m+l+2r, u ( 7 - , ~ ) . r= 1 s=0
(2.23)
The generalised theta functions appearing in the above expressions are defined by [ 17 ]
M. Hayes, A. Taormina/Nuclear Physics B 529 [PM] (1998) 588-610 Zgm,k(7", 0") d E
595 (2.24)
e2~rirk(a+~)2e27ricrk(a+~)'
aGZ where k is a positive integer, m E Z and Zgm,k(r, or) = Om+2kz,k(r, tr). The last obstacle to easy modular transformations is the presence of the function r/3 2"(ur). Our strategy to eliminate this function consists of two steps. The first is to rewrite the expression (2.23) in terms of s / ( 2 ; C ) k characters, as described in the next section. The branching coefficients are functions of v and r and still involve the function rl3-e"(ur). The second step, described in Section 4, eliminates this function from the branching coefficients by calculating the residue at the pole o- = 0 of each singular s~'l( 211 ; C ) k character when decomposed in sl ( 2; C ) k characters ( formulae ( 4.4 ), ( 4.5 ) ), and comparing the result obtained with the expressions (2.21) and 2.22).
3. Branching sl(2[1; C)k into sl(2; C)k The s/(2; C)k characters can be written as, see Ref. [9,18],
.,:,~2;c)~_ ~ ' t ,
Xn,n'
Ob+,o (r, ~) -- O~_,~, (r, ~) y = O1,2(r,o')-Oll,2(r,o')
(3.1)
'
where the level is parametrized as k=-,
t
H
gcd(t,u)=
with 0 ~< n ~ 2u + t -
1,
uEN,
(3.2)
tEZ,
2 and 0 ~< n' ~< u - l, and
b±du(+(n+l)-n'(k+2)),
aduZ(k+2).
(3.3)
In order to identify which s/(2; C)k characters enter in the decomposition of the class IV. NS s/(211;C) characters, we first rewrite (2.23) as
NS'IV"(I(2I I ;C)k (
Xh~,~,hT~
r, ~r, v ) =
O_u+2(m+l),2 u (T, -~) -- ~.~u+2(m+l),2u (T, ~) r / ( r ) - l r / 3 - 2 " (ur)
01,2(r,~r) - O-l,2(r,~r)
n=l u--n
0"
{pi}','~,."CS i=1
X H~qm+l+2pn--I ( ~O" "~ ...... , . + : , , . . j j=l
(3.4)
where the sum ~{piiTS,,cs is over the (u - 1 ) ! / ( u - n)!(n - 1)! possible subsets S(~) of (u - n) distinct integers Pi included in the set S = {1 . . . . . u - 1}. For each choice of subset S(,,), the variables pu-n+l pu-1 take the distinct values in S\S(,,). . . . . .
596
M. Hayes, A. Taormina/Nuclear Physics B 529 [PM] (1998) 588-610
The following expression is central in our discussion of branching functions. It is obtained from (3.4) along the lines described in Appendix A. It reads NS'IV';/(211;C)k(7",O" P)
Xh~,h'~
,,/=1
u--n {Pi}i= I CS
r=O
D(~I
' " " ]3,. . . .
l P; l " " ' Pn -- 2 ; r )
u--2
{Ou(u-n)(n-1)(1-20+2(n-l)(P,,-,,-uft~l-2(u-nl(P .........~-ur~tl,u(u-l)(u-nl(n-I)("I')
×~ ~=0 /,I
X
Z
Ou[(u-l)(4A+3)+2(u-n)(l-2t)-4r],2u(u-1)(u+l)
A=0
-~.
s/(2;C)
.
.
In the above formula, the quantum numbers of the representation considered are hNs_ = . l. ( u. . m
1),
h+Ns=l(2m,_m),
U
O<<.m'<~m<<.u--1.
U
(3.6) Given a set of n integers cri, i = 1 .... n, one also introduces, //
&i = Z ( n
(3.7)
+ 1 - j)olj
j=[ and the domain,
D(al ..... an;r) ={O/j
" O<~ cej <~ n + l - - j ,
j=l
. . . . . n :&l = k t ( n + l) + r , kl E N},
(3.8) with 0 ~ &l <~ I n ( n + 1 ) ( 2 n + 1).
(3.9)
In particular, one has D(/zl . . . . . / z . - n - l ; vl . . . . . Vn-2; r) = {(]~j,/,Jj,)
: 0 ~ 1.~.] ~ u - - n - - j ,
O<<.vj, ~ < n - - l - - j ' ,
j'=l
j = 1. . . . . u - - n - - 1;
..... n-2:/21+Vl=k'(u-1)+r,k'CN}, (3.10)
M. Hayes, A. Taormina/Nuclear Physics B 529 [PM] (1998) 588-610
597
with
O <<. [x~ <<. ~l ( u - n - l ) ( u - n ) ( 2 ( u - n - 1 ) + l ) ,
(3.~1)
0 < ~ 1 <~ ~ ( n - 2 ) ( n - 1 ) ( 2 ( n - 2 ) + l ) . The function G is given by the following product: u--n-I
~(Pl . . . . .
P u - l ; ]~; P ) --
II i=1
02P(O;u-n-i)-2ufii'( . . . . . i)(u-n-i+l)u(T)
11--2
(3.12)
X 1-I 0 2 e ( u - n ; n - I - j ) - 2 u / ' j , ( n - 1 - j ) ( n - j ) u ( g ) , j=l
where one defines
(3.13)
P ( a; fl) = fi,~,B - flP,~+B+l, with 17
/~./,n = ~
fio,n =- fin.
P./+k,
(3.14)
k=l
The function .U reads
.~ { f(n;~,-~) .=..Ul bt
s=0
{ t=0
z D(pl,...,p...... l;s)
{z{
~(o;#;,~)
D(cr]....,¢r,,_2;t)
u--2 ×Z
Ou[--(u--n)(n--l)(2A'+l)--2(n--1)s+2(u-n)t],u(u--1)(u--n)(n--I) (7")
A'=O
II}I (3.15)
Finally, the label [ne] in the s / ( 2 ; C ) ~ characters entering the formula (3.5) is the residue modulo 2(u + 1) of n~ defined by (e = 0, 1) n~=-l+(1-Ze)((u-l)(Z,~+l)+u-Zr+(u-n)(1-Zg)).
(3.16)
For each choice of variables ,~, r, n, g, either [no] or [n]] is in the set S = {1 . . . . . u 1 } , or else, [no] = [ n i l . In the latter case, there is no contribution proportional to X .C/(2;C) I,,,, I,,,-ill-J in (3.5), while in the former case, one gets a contribution X ,,:1(2;C) i,,, I , , - , , - 1 with • = 0 (resp. 1) according to whether [no] (resp. [hi ] ) is in the set S. Let us end this section by noting that the corresponding decomposition in s/(2;C)~. characters for NS class V s~/(2[1;C) characters is readily obtained by making the substitution M + M / for m, M for m/, z --~ z - I and multiply by an overall phase in (3.5).
M. Hayes,A. Taormina/NuclearPhysicsB 529 [PM](1998)588-610
598
4. Parafermionic characters as branching functions
As stressed in the introduction, the expression (3.5) neatly isolates the s/(2;C)k character dependence, but the branching functions are still written in a way which obscures their modular properties. However, it is easy to calculate the residue at the simple pole o- = 0 in the NS singular characters, both in class IV and class V, when they are decomposed in s/(2; C)k characters. Indeed, as discussed in [18], when the level k is of the form (1.3), the residue at the pole o-= 0 of singular s/(2; C)~ characters are unitary minimal Virasoro characters at level u, multiplied by r / - 2 ( r ) . On the other hand, we have the residue calculated as in (2.21), which can be rewritten using du(2) string functions at level (u - 2 ) [17,19], ,.,
.
NS,sl(21 l;C)k ,. -
lim zcrto-.ghNS h,y
or----+0
tr, O-, V)
1 = 1.90,2 (7-, 1/.-') -{- 'L92,2 (7-, 2P) g]3 (7-) = /.90,2 (7-, llt.,) _]_ /.92,2
NS N=2. ,¥r,s' t T'#½)
7- b' (7-,½12) Zu--2 C}:~2, (T)Otb'u--Bffu--2),u(u--2)('~, 2U)
r]3 ( r )
;qg=-u+3 u--2
1713(
T)
I
zU--2 C}:li72'(7-){r~=O(Z'~½(ii/u_fiffu_2))_(2r_s)(u_2),(u_ . )(u-2) (7-) 17d=-- u+3
" s=0
XL~½(ti,'u-~,(u--2),+u(2r--s),u(u--l)(7-'~) }}'
(4.1)
where g=m=r+s-
1,
Fn=2m'-m
(4.2)
in class IV. The above residues are thus expanded in a basis of theta functions at level u(u - 1) with arguments 7- and v/u, as are the residues calculated from (3.5) (see (3.15)). A comparison between these two methods of obtaining the residues of singular NS s/(211;C) characters provides enough information to express the branching functions in a form where the modular transformations can be carried out easily. We have worked out in detail the branching functions for the cases u = 3 and u = 4 (see Appendix B), and the truly remarkable result is that the branching functions involve a rational torus A,(,-1) [20] as well as Z,-1 parafermion characters [21,22]. With hindsight, this structure can be derived from the coset central charge value (1.5), since 3 ( u - 1) 2(u-2) -- 1 + - - , u+l u+l
Ccose t -- - -
(4.3)
where the first term is the torus central charge while the second term is precisely the central charge of the parafermionic algebra based on Z._l parafermions. When u = 2, there are no parafermions and the branching functions are just the Ae toms characters,
M. Hayes, A. Taormina/Nuclear Physics B 529 [PM] (1998) 588-610
599
as discovered in [ 10]. Based on the analysis for low values of the parameter u, we conjecture the following s/(2 I1;C)k character decomposition formulae. The class IV characters can be written u-1
NS'IV'sI(211;C)k/T
L
/I(hNzS,hNS
,
O" /,'~'
,
a~l(2;C)k .
I =ZXi,u_m_l~T,O" ) i=0 u-2
X
i,X(i,sl(7-)Lq,(u_ll(m_2m,+u)+uX(i,s),u(u_l) (T,. ~
, (4.4)
where hN_s and hN+s are given by (3.6), while the class V NS characters are decomposed as u-I NS,V,[/(211;C)k
/~ hr~_S,hNs
(7-,
,~':/(2;C) k
0", V) = Z Xi,M+M' +I ( T, 0") i=O u-2
s=0
(4.5) where hN_s = - - I ( M + M ' + I + u ) ,
h~ s = -I(M - M').
U
(4.6)
U
The function X(i, s) is given by X(i,s)=(u-1)i+2i
-
(4.7)
-2s,
and the symbol [ ~] is the integer part of ~. In the above expressions, one interprets ,q ~ 7-, ( u - l ) t )Ci,x(i,~)(7-) as the partition function for the Zu-1 parafermionic theory where the i lowest dimensional field is @x(/,s) in the notations of [23]. Recall that the s~u(2) string functions at level u - 1 have the following symmetries [17]: (u--l)
_ C(U--I)
c(U-- 1)
(u--l)
[
.
C~',m (~')-- ~,-m (r) = ~,m+2(,-l)Z(7-) = C,,-1-~,,-|-mtT-)' c (t,m U-l)
(T)=0for
g - - m 4~ 0 mod2.
(4.8)
The Au(~-l) torus characters are given by the theta functions at level u(u - 1) multiplied by r/-I (7-). The decomposition (4.4) encodes all information needed to obtain a parafermionic realisation of s/(211; C)k at fractional level [24].
600
M. Hayes, A. Taormina/Nuclear Physics B 529 [PM] (1998) 588-610
5. Conclusions Admissible representations of affine Lie algebras and superalgebras have been known to exist for fractional levels since the pioneering work of Kac and Wakimoto [9]. It is argued there that the corresponding characters can be expressed in terms of modular forms and provide a finite representation of the modular group. Over the last two years, we have calculated explicit character formulae for the affine superalgebra sl(211; C) k using the Kac-Kazhdan determinant formula [ 14,15 ] and the embedding diagrams for singular vectors one can derive from it [ 12,10]. A necessary condition for admissibility is that the level of the algebra sl(2]l;C)k is of the form (1.2). In this paper, we restrict ourselves to levels of the form k = ~ - 1, u E N, and to a certain class of highest weight representations labelled class IV and class V in [ 10], where the highest weight state quantum numbers are given by (3.6) and (4.6). The corresponding characters, first derived in [ 10], are decomposed here into characters of the bosonic subalgebra s/(2;C)k, which has been extensively studied. The branching functions involve characters of the rational torus Au(,-~) and partition functions of the parafermionic algebra Z,-1, given by s~u(2) string functions at level (u - 1). The decomposition (4.4) and (4.5) are conjectured for arbitrary value of u c N on the basis of a detailed analysis of the cases u = 2 [10] and u = 3,4 (see appendices). They reflect the structure of the central charge of the coset s/(2[1;C)k/s/(2; C)k, given by (4.3) as the sum of a toms and a Zu-1 parafermionic central charge. There are three immediate byproducts of these decomposition formulae. First of all, the S modular transform of the sl (2[ 1; C) ~ characters can easily be obtained from (4.4) and (4.5) since the modular transformations laws of theta functions, s~u(2) string functions and ~/(2; C)k characters are standard. We have explicitly checked that in the cases u = 2, 3, the irreducible sl(211;C) k characters provide a finite representation of the modular group, indicating the existence of an underlying rational theory. Second, as pointed out in the appendix, the more technical steps taken in obtaining the branching functions produce mathematical identities reminiscent of those discovered and discussed in [25,26], and may well put the latter in a new perspective. Third, the decomposition formulae encode all necessary information to obtain a new representation of the s/(211;C)k currents in terms of a primary parafermionic conformal field, as described in [24]. In particular, in the case u = 3 (resp. u = 4), the primary field corresponds to the Ising model (resp. 3-states Potts model) spin field o- of conformal weight 1/16 (resp. 1/15). The deep r61e of such a representation has yet to be understood, particularly in connection with our interest in the family of levels studied here, which stems from the non-trivial r61e s/(211; C)k plays in the description of non-critical unitary minimal N = 2 strings, when the matter central charge is given by Cmatter= 3 ( l - - 2). The most straightforward generalisation of this work would be to relax the condition p = 1 and investigate the theory of non-unitary minimal N = 2 strings.
M. Hayes, A. Taormina/Nuclear Physics B 529 [PM] (1998) 588-610
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Acknowledgements M. Hayes acknowledges the British EPSRC for a studentship. A.Taormina acknowledges the Leverhulme Trust for a fellowship and thanks the EC for support under a Training and Mobility of Researchers Grant No. FMRX-CT-96-0012.
Appendix A We outline here the derivation of expression (3.5) from (3.4). The use of the theta function identity, k+k' Om,k(r. o-)Om',k' (r, o') = ~ Om/e-m'k+2&k',kk'(/~+k')(r)Om+m'+Zgk,k+k'(r, or). g=l
(A.I) is crucial. In particular, it allows us to write N i= I N-I
N-I
=Z{ Z r=0
H
D(nl,..,nN-i;r
l~P(O;N--i)--2uhi'(N--i)(N--i+l)u('l')
OPu-2ur'Nu 7", -~
,
i=1
(A.2) where hi = ~",N-I ~.-,j=i (" N - J ) n l , D(nl
.....
={hi
nN--1;
r)
• O<.nj<~N-j,
j=l
..... N-1
"fix=k'N+r,k'
cN},
(A.3)
and (A.4) with N fi.j,N = Z k=l
P,j+k,
ff0,N ~-- PN.
(A.5)
Note that the invariance of the left-hand side of Eq. (A.2) under permutations of PN} provides identities between sums of products of theta functions at fixed values of r. This type of identity is used in deriving (3.5), and will be mentioned at the appropriate time below. We shall first discuss the occurrence of the function .Y'(n; r, ~,) in (3.5). It stems from the factors O,,,_2,,,,.(r, ~,, ) . - n and O,,,-2m,+...(r. 7,~ ) n - t in (3.4). To see that, first note {Pl .....
M. Hayes, A. Taormina/Nuclear Physics B 529 [PM] (1998) 588-610
602
that in the term w h e r e n = 1 (resp. n = u), one is only left with the first (resp. second) factor. A p p l y i n g f o r m u l a ( A . 2 ) with N = u - 1 and pi = m - 2 m N = u -
1, Pi
=
m - 2m ~ + u
~ Vi = ! . . . . .
u - 1 (resp.
Vi = 1 . . . . . u - 1) to this factor, one obtains St'( 1 ; r, ~)
( r e s p . . T ( n ; r, ~ ) , w h e r e one f o r m a l l y sets theta functions at level zero to 1, as well as sums and products r a n g i n g o v e r negative values of the index. In the generic case w h e r e n is neither 1 nor u, apply f o r m u l a ( A . 2 ) with N = u - n , p i to the first factor, and with N = n - 1, Pi
=
= m-2m
~ Vi = 1 .....
u-n
m - 2m' + u Vi = 1 . . . . . n - 1 to the second
factor. This leads to
p u--n
i.~)n--I
o ....
u--n-- 1 .-
×[
t=0
u--n-- 1 D(pl,..,p,
,,-l;S)
i=1
Z
D (o-i ,..,o-,,_ 2 ;t ) j = l
(A.6)
XO(n-l,(m--2m'+u)-2ut,(n-l)u ( 7 " , ~ ) ] • Now, use ( A . I ) to evaluate the product,
u-2
=Z
Lg--u[(u--n)(n--l)(2g'+l)+2((n--l)s--(u--n)t)l'u(u--l)(u-n)(n--l) ('I")
,V=0 P
XO(u_t)(m_2m,)_u[2(s+t)_(n_l)+2(u_n){,].u(u_l)
(T, ~)
(A.7)
,
where the s u m m a t i o n index g in ( A . 1 ) is rewritten as g = ( u - 1 ) p - ~ ' , p
= 1 .....
u; ~ ' =
0 . . . . . u - 2 and the f o l l o w i n g relation has been used:
Om,u(u_l)(u_n)(n_l)(7") = ~
02pu2(u_l)(u_n)(n_l)+um,u3(u_l)(u_n)(n_l)(7").
p=l
(A.8) F r o m here, it is straightforward to obtain expression ( 3 . 1 5 ) . Let us n o w rewrite the o--dependent factors in (3.4) in such a way that s l ( 2 ; C ) characters at level k = ±u - 1 appear. N o t e that the s l ( 2 ; C )
d e n o m i n a t o r O1,2(r, o-) -
O - 1 , 2 ( r , o-) is already explicitly written in ( 3 . 4 ) , but the n u m e r a t o r should be a differe n c e o f theta functions in the variables r and ~'~ at level a = u ( u +
1) see ( ( 3 . 1 ) ) . We
illustrate the derivation for the term n = 1 in ( 3 . 4 ) , and first use ( A . 2 ) to rewrite
M. Hayes,A. Taormina/NuclearPhysicsB 529 [PM](1998)588-610
603
u-I i=1 u-2
=Z r=0
u-2
~ 1-I ~'~--(u--l--i)(u--i)--2ufzi'(u--l--i)(u--i)u(T) D(p~l,...tx,,_2;r)i=1 O"
XO(m+l)(u--l)_2ur,u(u--l) (T, ~) .
(1.9)
In the above, we used that when n = 1, the possible values of Pi are all values in the set S = { 1,2 . . . . . u - 1 }. The next step is to evaluate
[IOm+l+u+2i,u (T,~)l~2(m+1, .... 2u
(T,~)--O2(m+l)+u,2u
(T,~)
.
(1.10)
i= I
To this effect, use ( A . I ) with g = (u + 1)p - g ' , p
= 1 . . . . . u , g ' = 0 . . . . . u as well as
the relation /,q~m,2u(u_l)(u+l)(T)
---- ~um+4u2p(u_l)(u+l),2u3(u
(A.11)
1)(u+l)(7"),
p=l and calculate, O"
O"
O"
O(m+l)(,,-l)-2ur,u(u-,) (T,U) [0"2(n,+I)--u,2u(T,;) --02(m+l)+u,2u(7",U) ] = ~-~{19u(u--1)(4g+l)--4ur,iu(u-1)(u+l)(T) t=0 O"
X~'qul(2~+l)(u--l)+l-2r]--(u--m--l)(u+l),u(u+l) (T, ~) -- Ou(u-- I ) (4g+ 1)+4ur,2u(u--1)(u+l ) ( T )
XIg_ul(2~+l)(u_l)+l+2rl_(u_m_l)(u+l),u(u+l) T, U
"
(A.12)
A l s o note that the above formula is obtained after defining g" = u + 1 - g' in the first sum, and using the fact that the term g" = u + 1 is identical to the term ~" = 0. N o w manipulate the above expression to make the s / ( 2 ; C )
character numerator
l~b, ,u(u+l) (T, ~ ) -- Ob_ ,u(u+l) (T, ~ ) appear with b± = ± u (n' + 1 ) - n " ( u + 1 ),
for s o m e n', n" /> 0.
(A. 13 t
One identifies n" with u - m - 1, n"=u-m-
1 ~>0,
n"=0 .....
u-
1
since
0~
1
in class IV.
(A.14) The identification o f n' is more involved. W h e n r = 0 in (A. 12), the s~l(2; C) character appearing is X ,,~l(2;C). , , , , tr, o-) (resp. - X , ,~"l(2;C) , , , , i~T , o - ) ) when 0 ~< /Tt ~ u - 1 and n' is the
M. Hayes,A. Taormina/Nuclear Physics B 529 [PM](1998)588-610
604
residue modulo 2 ( u + 1) of ( u - 1 ) ( 2 g + l ) (resp. - ( u l ) ( 2 g + 1) - 2 ) . I f n ' is the residue modulo 2 ( u + 1) of one of the two above expressions, but is greater than u - 1, then contributions cancel, and there is no s/(2; C) character contribution. Whenever r 4= 0 in ( A . 1 2 ) , the terms r and u - 1 - r must be combined in order to produce the s/(2; C) characters. The contribution to ( A . 1 0 ) from terms corresponding to r 4= 0 can be written,
u--2
u-2
E r=l
D(#l u
#, 2 r) i=1
X Z Ou(u-1)(4g+l)-4ur,2u(u-l)(u+l) (T) g=0
XOu,(2g+l)(u--l)--2r+l]--(u--m--l)(u+l).u(u+') (7", ~ ) u--2 r'=l
u--2 D(#b.../xu-2;r') i=l
X ~ ~u(u--l)(4g+l)--4ur',2u(u--l)(u+l)( T ) g=0
XL~_u[(2g+l)(u_l)_2r,+l]_(u_m_l)(u+l).u(u+l) "1",
(A.15)
,
where we have changed variable from r to r' = u - 1 - r in the second sum over r. Finally, use the following identity, t.t--2
Z
H O--(u--l--i)(u--i)--2U[Xi'(u--l--i)(u--i)u(T)
D (#l ,.../z,,_2;r) i=l u--2
(A.16) D(p.i,...#,-2;u--l--r) i=1 which follows from the invariance of ( A . 2 ) under the permutations of P l . . . . . p . - l , and write, for all r-terms, including r = 0,
i=1 u-2 u-2 r=0
D (/xl,.../x,_z;r) i=1
)< ~ Ou(u--l)(4e+l)--4ur,2u(u--l)(u+l) ( T ) f.=0
1
M. Hayes, A. Taormina/Nuclear
Physics B 529 [PM] (1998) 588-610
605
u
x
[
( > 77 ;
~~~~2P+l~~u-l~-2r+ll-_(u--n~-l)(u+I),~(u+l)
-I?_
u
(
.
73 ;
u~(2~+l)(u-l)-2r+l]-_(u--nt-l)(u+l),u(u+l)
(A.17)
)I (7, (T) (resp. -xL?ii?
(r, a) )
when 0 < n’ 6 u - 1 and n’ is the residue modulo 2(u + 1) of (U - 1)(2!+
1) - 2r
The si( 2; C) character
appearing
in the above is &!~?’
(resp. - (U - 1) (2! + 1) - 2 + 2r) ; n” is identified with u - m - 1. The discussion of the term II = u is completely similar to the above case. The generic case where II # 1 and II # u proceeds along the same lines, but is more involved since one has to apply (A.2) to the factors nyz<‘=T” 79,,L+~+u+2pi,u (7, %) and nyi’ 29~,+1+2,,,,+,,,, ,U(7, :). This derivation is very close in spirit to the technique used at the beginning of this appendix when we were discussing II -
the v-dependence
One obtains
n-l
II
I9 n?+l+u+2p,,u
n ,=I
of (3.4).
(7
5)
n
7%+1+2,,.-,~+,,u
(7
f)
j=l 11-n-l =
u-n-2
c
c
/L,=o
/L*=o
...
e
yy...
/.L,,-,!-I=ovl=o
G(p ,,...,
2
p,-l;&I)
v,,_*=o
v*=o
oX~(,-,)(nl+l+u)+21S,,_,,-2uP,,(u--n)u
(
7, ;
>
u
( >
X~(~-l)(nl+l)+2/i,,~,,:,,~,-2u~,,(n-l)u
7, I*
Now, the product of the last two theta functions
(A.18)
above can be written, using (A. 1) ,
u-2 c
~~1,‘1(11--1~(LI--n)~~-l)~~~~~U--n)ll(l--2~)+~U-l)~nt+l+u)-2u(~,+B,),u(u-l)
P=O (A.19) where a=U(L4-~)(n-1)(1-2C)+2(n-l)[p,_,,-u~,] -2( U - n) [&,;,_
of ,kt + Cl = k’( u - 1) + r, k’ E N, r = 0, .., u - 2 as &scribed
Note that the partitioning in (3.10)
(A.20)
I - u& 1.
is devised to simplify
the theta function
6 (LI- ~~)u~l-28)+~u-l)(n~+I+u)-2u(p,+P,),u(u-l) using the property u
292u(u-l),u(u-I)
Now, using
( > 7,;
= fiO,,(,-I)
(A. 1) , we calculate,
u
( > 7, ;
.
(A.21)
606
M. Hayes, A. Taormina/Nuclear Physics B 529 [PAl] (1998) 588-610 u-n
n-I
i=1
j=l
u--2 = ~
~
~(Pl .....
Pu-l;/2;
~)
(7",U)
,
r=O D(l~l,...,l~u_n_l;Vl,...,pn_2;r ) u-2 g=O
g~=0
--l-~b-c,2u(u+l)(u-l)(7")Od--e+f,u(u+l)
(A.22)
where
b = u ( u - 1) (4g' + 3), c=2u(u - n) (1 - 2g) - 4ur, 1 d=~c
e=u[(u-1)(2g' + l) +u], f=-(u-m-
(A.23)
1)(u+l).
One must now make the numerators of s/(2; C) characters appear in the sum over g~. Similar manipulations as in the case n = 1 described previously in this appendix allow one to rewrite the expression (A.22) as u--2
Z
G(pl . . . . . pu-1;/2; ~)
r=0 D (/Zl ,...,,u.. . . . . 1;Ul ,...,}',,- 2 ;r ) u--2
X ZOa,u(u_l)(u_n)(n_l)('l") g=0
Ob+c,2u(u+l)(u_l)(7") ,f' =0
(A.24)
_ ~1(2;C) " '~1(2;C) (7", 0-) (resp. -Xn',n" (7",0-)) Call n" = u - m - 1. The s/(2; C) character is Xn',,," when n t is the residue of ( u - 1) ( 2 g ' + 2 ) - 2 r + ( u - n ) ( l - 2 e ) (resp. - ( u - 1) ( 2 U + 2)+2r-(u-n)(1-2g)-2) in the r a n g e 0 ~ < n ' ~ < u 1.
M. Hayes,A. Taormina/NuclearPhysicsB 529 [PM](1998)588-610
607
Appendix B In this appendix, we look in some detail at the derivation of branching functions for the cases u = 3 and u = 4. Incidentally, the corresponding parafermionic theories are the Ising model and the 3-states Potts model, a fact which will be highlighted below by the natural occurrence of unitary Virasoro characters at level 3 and 5 in connection with s~u(2) string functions at level 2 and 3 respectively. As explained in Section 2, at fixed values of u, there exist u 2 characters corresponding to irreducible representations, u of which are regular when the variable o- tends to zero. We will view the latter as characters obtained by spectral flow from the other (Ramond or Neveu-Schwarz) sector, where they are singular. To fix the ideas, let us start in the NS sector of the u = 3 theory, where six characters are singular in the limit described above. We will study in detail the class IV character labelled by h Ns = - 2 / 3 , h Ns = 0. Using the decomposition formula (3.5), one gets
-¥-2/3,0IV'NS"~/(211;C)-2/3(7", O',/2)
(7",0") C1(7-)`00,6
0,2 X2,2
'
~ ) q-C2(7")O6,6
~)-}-(7-)"06,6
3)
3) (BI)
where
CI (7") = "//-3(37 ") [O15,36(T) -- 033,36(7")]O2,6(T ) -}- [`03,36(7") -- O21,36(T)]O4,6(T), C2 (7-) = 77-3 (37") [ O15,36 (7-) -- 033,36 (7-) ] 04,6 (7-) -}- [ 03,36 (T) -- O21,36 (7") ]/94,6 (T), C3(7") = 7/-3(37") [ OI,6(7-) q- 05,6(7") ] [ 06,36(7-) - "030,36(7-) ].
(B.2)
The residue of the above character when cr -+ 0 can be calculated in two different ways. From the decomposition (B.1), it is readily obtained by noticing that the three sl(2; C) characters have a simple pole singularity when cr --~ 0, and that their residue at the pole are Virasoro characters at level 3 multiplied by the function r / - 2 ( r ) [ 18]. On the other hand, formula (4.1) gives the residue in terms of the unique level one string function c~010)(r) = r / - l ( T ) (see e.g. [ 1 7 ] ) , i.e.
,.-.
.
l V , N o~, o ~,~'J+~, . ~ l 1 ;e~~) - - 2 / 3 /
lim Lcrtcr X-2/3,0
O--+0
,,
~7-, or, u)
{ "-~-O1,2(T)[03,6 (T, 3) -~-09,6 (7",3)]).
(B.3)
608
M. Hayes, A. Taormina/Nuclear Physics B 529 [PM] (1998) 588-610
Comparison of these two residue calculations yields the following expressions for the coefficients C i ( r ) ,
[xVir(3) (T)2 _ x~,i[(3)(r)2] CI (,r) = c0,0 A1) vir(3) [.¥1,1 ( r ) O 2 , 2 ( r ) - xvir(3) 1,1 2,1 (r)zgo,2(r)] ' Vir(3) (r)2 -- x~i[(3)(r)2]C2(T)=~O,O )(1,1 "(1) [ XlVI[(3)(r) z90,2(r) - XVi[(3)(r)z92,2(r)] , /. Vir xVir(3) 2,2( (r) [xVi[(3)(r) + , 2,1 (r)]C3(r)=C(ol)oZgL2(r)[X vir(3),,t(T) -I---)(2,1Vir(3)(T) ], (B.4) or again, using well-known identities relating the Ising model characters to square roots of theta functions at level 2 [27],
. Vir(3)/_s t',),
Cl(r) = x2,1
C2(r) = xvi[ (3)(r),
C3(r)
. Vir(3)/-~
= 2"2,2 t")-
(B.5)
It is also worthwhile noticing that the three unitary Virasoro characters at level 3 coincide with the three partition functions of the Z2 parafermionic theory whose lowest dimensional fields are ~P9 z, ~p2 and ~Pl [23]. The latter is actually a primary conformal field which is identified as the spin field of the Ising model. These partitions are given in terms of s~u(2) string functions at level 2. Namely,
..•Vir(3)
Vir(3).. )(2,1 I,T) -: T}(T)¢ 0(22) ( T),
)(Vir(3) 2,2 ( T ) = r / ( T ) C I ~ ? ( T ) '
(B.6)
and therefore, the above expressions can also be viewed as relating s~u(2) string functions at level one and two. The same analysis can be applied to the other five singular NS s/(2ll; C) characters at level k = - 2 / 3 , and to the three singular R characters which flow to the three regular NS characters at the same level. They all involve the three functions (B.5). One then obtains expressions which can be read off (4.4), (4.5) for the nine NS s/(2ll;C ) characters at this level. The modular S transform of these nine characters is encoded in a 9 x 9 matrix which is unitary and whose fourth power is the identity [ 13]. A similar analysis can be performed for the case u = 4. For instance, using again the decomposition formula (3.5), one obtains
IV,NS X_3/4,0 (7", o', p) f o ,.~1(2"C).. ='q-'(r)~X 3 ' t r , tr) [ D,(T)012,,2 (T, ~/] ) -~-D2(T){1-~4,12 (T, 4) -~-1920,12 (T, 4) /]
M. Hayes,A, Taormina/Nuclear Physics B 529 [PM](1998)588-610
609
_XI,3 ,;l(2;C)(r,~r)[D3(T)O0,12(T, ~/ p -~-D4(T){l~8,12 (T, 4) . s:I(2;C)(T,O.) ~-X2,3
D3(T)O,2,12 (T,-~) -}-D4(T){04,12 (T, 4)
q-O20,12 ( T ' 4 )
}] }'
(B.7)
where the four functions D i ( T ) , i = 1 . . . . . 4 are all sums of quintic expressions in theta functions (two at level 8, two at level 24 and one at level 120) times the function r/-5 (4r). The important remark is that all sixteen NS sl(211; C) characters at level k = - 3 / 4 have the same structure as the one above, with the same four D i functions. Here again, the residues can be calculated in two ways. The decomposition (B.7) yields Virasoro characters at level 4 multiplied by the function "q-Z(r), while formula (4.1) expresses the residue in terms of s~u(2) string functions at level two. By comparison of these two calculations, one can write, xVir(3) " 2,2 ( T ) D 1 (T) _-- C(2) 2,0 (T) [.)(~,~(4)(T)0_6,6(T) _ xV2(4)(T).00,6(T ) ] -~-C~2) (T)[xVi~(4)(T)00,6 (T)
--
.)(Vi~(4)(T)'066(T) ],
xVir(3) _ ¢(2) Vir(4) . Vir(4) l_~ 2,2 ( z ) D 2 ( r ) - 2,0 (r) [X3,3 ( r ) O 2 6 ( r ) - x3,2 t~jO4,6(r)] (Vir(3) Vir(4) (T)Z'90,6(T) 2.2 (T) D3(T)--_ C(2) 2,0 (T)[)(I,I
" xV'~(4)(T)t96, 6(T)]
--
,
--
avVir(3) Vir(4) t_x .q rT~) 2,2 (T)D4(7") = C(2) 2,0 ( r ) [Xl,t t~/u4,6t
-)(3,1
(r)O06(r) 1,
--xViI'(4)(T)O2,6(T)]
q_C(2) 2,2 (T) [xV'~ (4) (T)O2,6 (T)
--
Vir(4) (T)O4 6(T) ] , /1(3,2
(B.8)
where we have used the identity [25], xVir(4) Vir(4) . Vir(4) Vir(4) . Vir(3) 1,1 (T))(3,3 (T) -- A'3,1 (r)x3,2 ( r ) = d(2,2 (T).
(B.9)
It turns out that the functions Di(r) enter the partition function of the 3-states Ports model, and can therefore be interpreted as the partition functions for the parafermionic algebra Z3. One has . Vir(5) ~_ . Vir(5) D I ( T ) =X1,1 (T)--A'4,1 (T) . D o ( r )
=
)(Vir(5)
4,3
. Vir(5)
D 3 ( T ) ----X2,1
D4(T)
= ,~Vil'( 5 )
3,3
=T](T)C(3)(T),
n(T)C~31)(T), -I- X3,I (T) =n(r)c~3~o(r),
(T) = (T)
,
Vir(5)
(r) = "q(r)c(131)(r).
(B.10)
The relations (B.8) can therefore be viewed as relations between s~u(2) string functions at level two and level three. It should also be stressed that the relations (B.8), with the functions D i expressed in terms of Virasoro characters at level five as in (B.10), provide new identities very similar to the ones obtained in [25].
610
M. Hayes, A. Taormina/Nuclear Physics B 529 [PM] (1998) 588-610
References [1] G. Aharony, O. Ganor, N. Sochen, J. Sonnenschein and S. Yankielowicz, Nucl. Phys. B 399 (1993) 527. [2] Fan J-B. and M. Yu, Modules over affine Lie superalgebras, AS-ITP-93-14; G/G gauged supergroup valued WZWN field theory, hep-th 9304122. [3] H.L. Hu and M. Yu, Phys. Lett. B 289 (1992) 302. 14] J. Distler, Z. Hlousek and H. Kawai, Int. J. Mod. Phys. A 5 (1990) 391. [5] I. Antoniadis, C. Bachas and C. Kounnas, Phys. Lett. 242, 2 (1990) 185. [6] E. Abdalla, M.C.B. Abdalla and D. Dalmazi, Phys. Lett. 291 (1992) 32. [7] J.L. Petersen, J. Rasmussen and M. Yu, Nucl. Phys. B 457 (1995) 309. [8] P. Furlan, A.Ch. Ganchev and V.B. Petkova, Nucl. Phys. B 491 (1997) 635. [9] V.G. Kac and M. Wakimoto, Proc. Nat. Acad. Sci. 85 (1988) 4956. [10] P. Bowcock, M.R. Hayes and A. Taormina, Nucl. Phys. B 510 (1998) 739. [11] V.G. Kac, Adv. Math. 26 (1977) 8. [12] P. Bowcock and A. Taormina, Commun. Math. Phys. 185 (1997) 467. [13] M.R. Hayes, Admissible representations and characters of the affine superalgebras o~p(l/2) and ~7I(211;C), PhD Thesis (1998). 114] V.K. Dobrev, Multiplets of Verma Modules over the osp(2/2) tt~ super Kac-Moody algebra, in Topological and Geometrical Methods in Field Theory, Proc., ed. J. Hietarinta and J. Westerholm (Espoo 1986) (World Scientific, Singapore, 1986). [ 15 ] V.G. Kac, Highest weight representations of conformal current algebras, in Topological and Geometrical Methods in Field Theory, Proc., ed. J. Hietarinta and J. Westerholm (Espoo 1986) (World Scientific, Singapore, 1986). [16] Y. Matsuo, Progr. Theor. Phys. 77 (1987) 793; V.K. Dobrev, Phys. Lett. B 186 (1987) 43. [17] V.G. Kac and D.H. Peterson, Adv. in Math. 53 (1984) 125. [18] S. Mukhi and S. Panda, Nucl. Phys. B 338 (1990) 263. 119] F. Ravanini and S-K. Yang, Phys. Lett. B 195 (1988) 262. [20] R. Dijkgraaf, C. Vafa, E. Verlinde and H. Verlinde, Commun. Math. Phys. 123 (1989) 485. 121] V.A. Fateev and A.B. Zamolodchikov, Sov. Phys. JETP 62 (2) (1985) 215; Sov. Phys. JETP 63 (5) (1986) 913. [221 Z. Qiu, Phys. Lett. B 198 (1987) 497. [23] D. Gepner and Z. Qiu, Nucl. Phys. B 285 (1987) 423. 124] P. Bowcock, Hayes M.R. and A. Taormina, Parafermionic representation of the affine st(211; c)k algebra at fractional level; Durham preprint DTP 98/5, hep-th 980324. [25] A. Taormina, Commun. Math. Phys. 165 (1994) 69. 126] A. Taormina and S.J.M. Wilson, physics 9706004, to appear in Commun. Math. Phys.. 1271 P. Ginsparg, Applied conformal field theory, Les Houches lecture notes (1988).
Nuclear Physics B 529 [PM] (1998) 588-610
Admissible s /(211; C) k characters and parafermions M. Hayes 1, A. Taormina 2 Department of Mathematical Sciences, University of Durham, Durham, DH1 3LE, England, UK Received 9 March 1998; revised 4 June 1998; accepted 2 July 1998
Abstract
The branching functions of a particular subclass of characters of the affine superalgebra s/(211; C)k into characters of the subalgebra s/(2;C)k are calculated for fractional levels k = 1 1, U E N. They involve rational torus A,~u-1) and Z,_I parafermion characters. @ 1998 u Elsevier Science B.V. PACS: 02.20.Sv; 11.25.Pm Keywords: Non-critical superstrings; Affine superalgebras; Characters
1. I n t r o d u c t i o n
It has long been stressed that non-critical strings might well be described in terms of a topological GIG Wess-Zumino-Novikov-Witten ( W Z N W ) model [ 1 - 3 ] , where G is a Lie supergroup or a Lie group, depending on whether the string theory considered is supersymmetric or not. Our original motivation is to re-analyse non-critical N = 2 strings in the light of this potentially new approach. Although non-critical N = 2 strings have been the subject of some investigation [ 4 - 6 ] , they remain largely understudied. They possess interesting features and technical challenges. In particular, as emphasized in [4], this string theory is not confined to the regime of weak gravity, i.e. the phase transition point between weak and strong gravity regimes is not of the same nature as in the N = 0, 1 cases. This absence of barrier in the central charge is a hope for new physics. i E-mail: [email protected] 2 E-mail: [email protected] 0550-3213/98/$ - see frontmatter @ 1998 Elsevier Science B.V. All rights reserved. PII S 0 5 5 0 - 3 2 1 3 ( 9 8 ) 0 0 5 2 2 - 7
M. Hayes, A. Taormina/Nuclear Physics B 529 [PM] (1998) 588-610
589
The exact correspondence between the traditional approach to non-critical strings and the latter is yet to be proven. However, a crucial ingredient in the description of the spectrum in the GIG picture is the representation theory of the corresponding affine Lie (super)algebra, ~, at fractional level k = p/u - h, p E Z \ {0},u C N, gcd(p,u) = 1, with h the dual Coxeter number of ~. For instance, the SL(211;R)/SL(211"R ) topological quantum field theory obtained by gauging the anomaly free diagonal subgroup S L ( 2 f l ' R ) of the global SL(211;R)L x SL(211;R)R symmetry of the WZNW model appears to be intimately related to the non-critical charged fermionic string, which is the prototype of N = 2 supergravity in two dimensions. A comparison of the ghost content of the two theories strongly suggests that the N = 2 non-critical string is equivalent to the tensor product of a twisted SL(211; R)/SL(211; R) WZNW model with the topological theory of a spin-½ system. It is however only when a one-to-one correspondence between the physical states and equivalence of the correlation functions of the two theories are established that one can view the twisted GIG model as the topological version of the corresponding non-critical string theory. For the bosonic string, the recent derivation of conformal blocks for admissible representations of s/(2; R) is a major step in this direction [7,8]. As far as the s/(211;C) and N = 2 algebras are concerned, a deeper investigation of their representations is a prerequisite to obtaining similar results for the corresponding conformal models. In this paper we therefore investigate further the characters of the complex affine superalgebra s/(211; C)k. When the matter, which is coupled to supergravity in the N = 2 non-critical string, is minimal, i.e. taken in an N = 2 super Coulomb gas representation with central charge, Cmatter = 3 ( l -- - ~ ) ,
p, uEN,
gcd(p,u) = 1,
(l.l)
the level of the 'matter' affine superalgebra s/(211;C ) appearing in the model of SL(211;R)/SL(2ll;R) is of the form
k= P- - l ,
(1.2)
b/
i.e., the level precisely takes values for which admissible representations of s/(211; C)~ do exist [9]. The character formulae obtained in [10] as functions of three complex variables T, cr and ~, are not directly suitable for the analysis of their behaviour under the modular group. In order to make this analysis straightforward, we provide here a decomposition of the s/(21 l ; C ) k characters into characters of the subalgebra s/(2: C)~.. It turns out that for a level of the form 1
k= --
1,
u=2,3 .....
(1.3)
bt
i.e. of the form (l.2) with p = 1, the branching functions involve characters of a rational torus model Au(u-t) as well as Z , - i parafermion characters. The identification of the branching functions relies on the residue analysis at the simple pole developed,
M. Hayes, A. Taormina/Nuclear Physics B 529 [PM] (1998) 588-610
590
in the limit where o- tends to zero, by a subset of s/(211; C ) / characters. One particular feature of ; / ( 2 l l ; C ) k is that the central charge of the associated Virasoro algebra is zero (a consequence of equal number of bosonic and fermionic generators in sl(2/1)), Csl(2ll ) = O,
(1.4)
so that the coset sl(211;C)l,/sl(2;C)k
has central charge Ccoset = Cs~(211) -cst(z) =
-3k/(k+2). Since we restrict ourselves here to levels of the form (1.3), the coset central charge is positive, 3 ( u - 1) - , u+l
Ccoset -- -
(1.5)
and its value is one when u = 2. The branching functions in this case are just the characters of the rational torus algebra A2, as already noticed in [ 10]. The paper is organised as follows. In Section 2 the sl(211;C)t characters given in [ 10] are rewritten in terms of infinite products and theta functions. The latter formulation leads in Section 3 to the decomposition of s / ( 2 l l ; C ) k characters into s/(2;C)k characters. The derivation of the crucial formula (3.5) is outlined in Appendix A. Section 4 uses the residues of singular ~l(211;C)k characters at the pole o-= 0 to rewrite the branching functions in such a way that the S transform of the ~/(211; C)k characters can be easily worked out. The decomposition formulae (4.4) and (4.5) are conjectured for higher values of the parameter u on the basis of a detailed analysis of the cases u = 3, 4 sketched in Appendix B. We offer some comments on the structure of the coset sl(211; C) ~/sl( 2; C) k in our conclusions.
2. N e w formulae for
s/(211; C)k admissible
characters
The algebra sl(211;C)k is the affinisation of the Lie superalgebra labelled A ( 1 , 0 ) in Kac's classification [ 11 ]. The latter has rank two. Let al and a2 be two fermionic simple roots. Those roots and their opposites, corresponding to the generators e±~,, e±,~2, have zero norm square. The even subalgebra of A ( 1 , 0 ) is the direct sum of an abelian algebra, with generator h+, and of the algebra Al with generators e+(al+a2), h_. The background material needed here, as well as the notations used, can be found in [12,10] and references therein. We however recall the defining commutation relations of sl(2ll;C)k. Introducing the following Laurent expansions for the currents, (z) : n
S(h+) (z) n
n
/-4-
J(e±~,)(z) =2..~jn z n
--n-- I
,
J(e±-2) ( z ) = ~ f " J +z-n-in , n
(2.1)
M. Hayes, A. Taormina/Nuclear Physics B 529 [PM] (1998) 588-610
the commutation relations of ~I(211;C) k in terms of the Laurent modes
591 are
3 { j + , j ; ] = 2J~,+n + kmtSm+n,o,
4- , [ J3,Jn~ ] = i J~,,+.
3 J~3 ] = -~m k 8 m+mO, [ Jm,
•-4r Ja,/oT = +Jm+,,,
-4- .~
.+'
[ J~,, , Yn ] = TYro+,,,
[2J3,, j; + ] = -}-Jm+n, "+
[2J~,,jf ] = +Jm+n, '+
[ 2Urn, j ' i ] = ijmi+,, [2Um,j~] = qZYm+n, "± k [u.,,G] = - ~ mSm+,,,o, 3 {L+,fo-}= ( Urn+,, -- Jm+n) .q-
._
{j,,,,a. }=
-
km~Sm+m0,
3
(U,,+,, + Jm+,,) + km~m+,,,o,
t
5: {j.,f, j#} = Jm+n"
(2.2)
Furthermore, the Sugawara energy-momentum tensor is given by T(Z)-
1
--{2(J3)2(Z)
2 ( k + 1)
-2uZ(z)
+j'+j'- (z) - j'-j'+(z)
+ J+J-(z)
- j+j-(z)
+ J-J+(z)
+ j - j + ( z ) },
(2.3)
and L0 is the zero-mode in its Laurent expansion. In [ 10], we derived formulae for the characters of irreducible representations of s/(211; C)k at fractional level k = pu - 1, where p and u are positive coprime integers. Those characters are obtained in a standard way from the knowledge of the quantum numbers and embedding diagrams of singular vectors within a given Verma module [ 12]. In the present paper, we are interested in sl(211;C) k characters belonging to classes IV and V in the notations introduced in [12,10]. Incidentally, the two classes share the same embedding diagram structure, as explicitly argued in [13]. The quantum numbers h_, h+ of the highest weight state in class V are obtained by requiring the following s/(2[I;C)~ Kac determinant [15] factors to vanish:
~ ° ) ( 6 q + &2,0, m ) = h _ + (k + 1 ) m - p , q~(')(-&,,0, 1 + m ' ) = - ½ h _ + ½h+ + ( k + 1)(1 + m ' ) ,
(2.4) (2.5)
for m,m' E Z+, 0 ~< m + m' ~< u - 2 and k + 1 = p/u, gcd(p,u) = 1. In class IV, these highest weight state quantum numbers are obtained by requiring ,~o~°) (,~, + ,~2,0, m) = 0 = ~ ( ' ) ( , ~ , , O , m ' ) ,
(2.6)
i.e.
h_+(k+l)m=O,
21 h _ - ~ h +1 + ( k + l ) m ' = O ,
(2.7)
592
M. Hayes, A. Taormina/Nuclear Physics B 529 [PM] (1998) 588-610
for m,m + E Z+, 0 ~< m' ~< m ~< u - 1 and k + 1 = p / u , gcd(p,u) = 1. It is still a conjecture that there exist no subsingular vectors in classes IV and V, and we stress that the corresponding characters have been constructed in [10] under this assumption. By definition, s/(211;C)~ characters are given by X/l(211;c)k (~-, o', v) ~a tr exp(27ri(~-Lo + o'J 3 + vUo)) h_, h t
(2.8)
where Jo3 and U0 are the zero-modes of the sl(211;C)~ Cartan generators, and the 1 isospin ½h_ and charge ~h+ are their eigenvalues when they act on the highest weight state Is2} of the representation,
S la> ='~h_ls2 ),
Uolg2)= lh+[~O).
(2.9)
The variables q, z and ( are defined by d
q= exp(2~ir),
TCC
d
z = exp(2~i~),
~ E C,
(~exp(2~iv),
v C C.
Im(r) > 0 =~
Iql
< 1,
(2.10)
The trace is over all states in the irreducible module with highest weight state Is2). For instance, the class IV Neveu-Schwarz characters corresponding to an irreducible highest weight module with highest weight state quantum numbers h~ s, hN+s read NS,lV,s/(2ll;C)k XhNS,h~S
" INS 1 NS £7", 0-, 9) = qhNSz ~h ~h+ FNS(7., O', 9) Z qa2pu+ap(l+m)zap aEZ 1 -- q2au+l+mz >( (1 -t- qau+m, + 51 z ~~ ( - 51) ( 1 + qaU-m'+m+½z½(½) ' +
(2.11)
wherem, m t E Z + a n d 0 ~ < m t ~ < m ~ < u - l , h N S = m ( k + l) + k , hN+s = (2m' - m ) ( k + 1),
(2.12)
and the function FNS(T, o-, v) is given by FNS(7", O', P) = H0(3( l + z ~ ( ~ q "1 - ~1 ) ( l +1z
1
( ~I q " - ~1) ( l + z l ( - ½ q ' - l ) ( l + z - ~ ( - ~ q n -1~ ) (1 - qn)2(1 - z q ' ) ( 1 - z - l q n - l )
1
1
n=l
(2.13)
On the other hand, the class V Neveu-Schwarz characters corresponding to an irreducible highest weight module with highest weight state quantum numbers hNs, h~ s read
M. Hayes, A. Taormina/Nuclear Physics B 529 [PM] (1998) 588-610 ."
1 NS
I NS
NS,V,~)(2]I;C), (75 O', P) = qhNSz ~h_ (sh+ FNS (75 O', U) ~ Xh~S'hr~ s
593
qa2pu+ap(M+M' +l ) .,~--ap
aCZ 1 -- q2au+M+M'+l Z--I X
( 1 + qau+M+½ Z _~(_~L
whereM, M ICZ+,0~
~) ( 1 + qau+M'+½Z. -'(~ -) '
(2.14)
t~
hN_s = - ( M + M ' + 2 ) ( k +
l) + k ,
hN+s = ( M - M t ) ( k + I ) .
(2.15)
In both classes, the conformal weight of the highest weight state is given by hNS - 4 ( k 1+ 1~ ((hN-S)2 -- (h~-S)2 + (2hN-S -- k)).
(2.16)
We recall here that the class IV (resp. class V) Ramond characters are easily obtained from class IV (resp. class V) Neveu-Schwarz characters by spectral flow, namely, R,s/(211;C)k
Xh~ ,I,~
(7, o-, v) = q
k/4 k/2
Z
NS,,~l(2[l;C)kr
XhN_S,h~S
T
t , - o - - 7, v),
(2.17)
where h R = hNS I
R
5h_ = - 1 R
_
I_hNS + 1
- 2--
~ hNS
~k,
1
+ ~k,
_lhNS
7h+ - 2,~+
(2.18)
.
The above formulae are not suited to the discussion of the modular properties of characters. Our ultimate goal is to identify how the above sl(2[ I ; C ) k characters branch into s l ( 2 ; C ) k characters, in order to make their modular transforms straightforward to obtain. We restrict ourselves to levels of the form k = ,1- - 1, and we mainly concentrate on the Neveu-Schwarz sector of the theory. We first use standard techniques to produce expressions for the characters in terms of infinite products. Namely, starting with expression (2.11), we calculate the residue at the poles corresponding to 1 + qa',+,/+½ z ½( - " = 0 for some integer a ~ and to 1 + q'<"-m'+m+½zl(½ = 0 for some integer a ' . It is not too hard to show that the residues above coincide with those obtained from the following expression: NS,IV,sTI(2[ 1;C)k.
/~hNS hNs
~
~
[7, 0", P) = qIySz
I hNS
I hNS
~re
(~ + F " ° ( r , 0", v)
oo
x H
(1 - qU.)2(1 - zq "n-m) (1 - z - l q " ( ' - l ) + m )
n=l
x h~-j (m, m~; r, o-, v ) ,
where
(2.19)
M. Hayes, A. Taormina/Nuclear Physics B 529 [PM] (1998) 588-610
594 I
1
I
I
1
t
1
h n ( m , m ; r, o', v ) = ( 1 + z ½(3q "(n-1)+m-" +3 ) ( 1 + z 3 ( - ~ q u(n-1)+m +3 ) ×(l + z-½(½qU"-m'-½)(l
(2.20)
+ z-½f-½qU"+m'-m-½).
Although two functions having the same poles and residues are not necessarily equal, and in particular can differ by a function which is regular on the appropriate domain, we expanded (2.11 ) and (2.19) in a power series of the variable q for several representations and found the two expressions are identical, to a large finite order. An easy way to obtain the class V NS characters is to substitute M + M' for m, M for m' and z ~ z - 1 in the above expression, and multiply by an overall phase. Note that, in contrast to the case where the level k is integer, i.e. where u = 1 in the parametrisation (1.2), and for which the s/(2[l; C)~ characters are regular in the limit o- ---+ 0, the situation for a level of the form k = ~1 - 1 is quite different. Indeed, in this case, there exist u 2 NS characters in classes IV and V, u of which are regular at o- = 0. The others develop a simple pole at this value of o- and the residue at the pole is given by
.
~--+01imz~rto" Xh~S,hTS
.....
,
~w, o, t.,) --
o0,2 (7-, ½12) + < . 2 (7-, ½12) Ns /]3(7. )
.)(r,s'
! (7-, ~"2 ) ,
(2.21) where the N = 2 superconformal characters appear in the infinite product form first derived by Matsuo [16]. In the above, the N = 2 central charge is c = 3(1 - 2) and m=r÷sl,m'=r-½ in class IV, w h i l e M = r - ½ , M ' = s - ½ in classV. Similar formulae exist for singular Ramond characters,
.
lim zcrto- Xh" ,h:~
cr----~0
-
O', 12) =
(7-, ½12) + o_,.2 (7-,'212) /]3(7-)
R+ N=2~
X~,s'
t 7-, sr ½), (2.22)
withm=r+s,m'=rinclassIV, whileM=r-l,M'=s-1 in classV. Although interesting, the infinite product formula (2.19) has a denominator whose behaviour under the modular group is non-trivial. Using the Jacobi triple identity repetitively, as well as the standard properties of theta functions, a tedious calculation leads from (2.19) to the following elegant expression: NS,IV,s~/(2I I;C)k e 7-
X h~.~,h~s
~ , ~r, v ) =
O--u+2(m+l),2u (7-, ~) -- 19u+2(m+l),2u (7-, ~) /](r)_l/]3_2U(ur) O~,2(r, or) -- O_~,2(r, ~r)
u--I 1 12 0v X HZ~-~us+m--2m',u (7-,U)'l-~u(s+l)+m+l+2r, u ( 7 - , ~ ) . r= 1 s=0
(2.23)
The generalised theta functions appearing in the above expressions are defined by [ 17 ]
M. Hayes, A. Taormina/Nuclear Physics B 529 [PM] (1998) 588-610 Zgm,k(7", 0") d E
595 (2.24)
e2~rirk(a+~)2e27ricrk(a+~)'
aGZ where k is a positive integer, m E Z and Zgm,k(r, or) = Om+2kz,k(r, tr). The last obstacle to easy modular transformations is the presence of the function r/3 2"(ur). Our strategy to eliminate this function consists of two steps. The first is to rewrite the expression (2.23) in terms of s / ( 2 ; C ) k characters, as described in the next section. The branching coefficients are functions of v and r and still involve the function rl3-e"(ur). The second step, described in Section 4, eliminates this function from the branching coefficients by calculating the residue at the pole o- = 0 of each singular s~'l( 211 ; C ) k character when decomposed in sl ( 2; C ) k characters ( formulae ( 4.4 ), ( 4.5 ) ), and comparing the result obtained with the expressions (2.21) and 2.22).
3. Branching sl(2[1; C)k into sl(2; C)k The s/(2; C)k characters can be written as, see Ref. [9,18],
.,:,~2;c)~_ ~ ' t ,
Xn,n'
Ob+,o (r, ~) -- O~_,~, (r, ~) y = O1,2(r,o')-Oll,2(r,o')
(3.1)
'
where the level is parametrized as k=-,
t
H
gcd(t,u)=
with 0 ~< n ~ 2u + t -
1,
uEN,
(3.2)
tEZ,
2 and 0 ~< n' ~< u - l, and
b±du(+(n+l)-n'(k+2)),
aduZ(k+2).
(3.3)
In order to identify which s/(2; C)k characters enter in the decomposition of the class IV. NS s/(211;C) characters, we first rewrite (2.23) as
NS'IV"(I(2I I ;C)k (
Xh~,~,hT~
r, ~r, v ) =
O_u+2(m+l),2 u (T, -~) -- ~.~u+2(m+l),2u (T, ~) r / ( r ) - l r / 3 - 2 " (ur)
01,2(r,~r) - O-l,2(r,~r)
n=l u--n
0"
{pi}','~,."CS i=1
X H~qm+l+2pn--I ( ~O" "~ ...... , . + : , , . . j j=l
(3.4)
where the sum ~{piiTS,,cs is over the (u - 1 ) ! / ( u - n)!(n - 1)! possible subsets S(~) of (u - n) distinct integers Pi included in the set S = {1 . . . . . u - 1}. For each choice of subset S(,,), the variables pu-n+l pu-1 take the distinct values in S\S(,,). . . . . .
596
M. Hayes, A. Taormina/Nuclear Physics B 529 [PM] (1998) 588-610
The following expression is central in our discussion of branching functions. It is obtained from (3.4) along the lines described in Appendix A. It reads NS'IV';/(211;C)k(7",O" P)
Xh~,h'~
,,/=1
u--n {Pi}i= I CS
r=O
D(~I
' " " ]3,. . . .
l P; l " " ' Pn -- 2 ; r )
u--2
{Ou(u-n)(n-1)(1-20+2(n-l)(P,,-,,-uft~l-2(u-nl(P .........~-ur~tl,u(u-l)(u-nl(n-I)("I')
×~ ~=0 /,I
X
Z
Ou[(u-l)(4A+3)+2(u-n)(l-2t)-4r],2u(u-1)(u+l)
A=0
-~.
s/(2;C)
.
.
In the above formula, the quantum numbers of the representation considered are hNs_ = . l. ( u. . m
1),
h+Ns=l(2m,_m),
U
O<<.m'<~m<<.u--1.
U
(3.6) Given a set of n integers cri, i = 1 .... n, one also introduces, //
&i = Z ( n
(3.7)
+ 1 - j)olj
j=[ and the domain,
D(al ..... an;r) ={O/j
" O<~ cej <~ n + l - - j ,
j=l
. . . . . n :&l = k t ( n + l) + r , kl E N},
(3.8) with 0 ~ &l <~ I n ( n + 1 ) ( 2 n + 1).
(3.9)
In particular, one has D(/zl . . . . . / z . - n - l ; vl . . . . . Vn-2; r) = {(]~j,/,Jj,)
: 0 ~ 1.~.] ~ u - - n - - j ,
O<<.vj, ~ < n - - l - - j ' ,
j'=l
j = 1. . . . . u - - n - - 1;
..... n-2:/21+Vl=k'(u-1)+r,k'CN}, (3.10)
M. Hayes, A. Taormina/Nuclear Physics B 529 [PM] (1998) 588-610
597
with
O <<. [x~ <<. ~l ( u - n - l ) ( u - n ) ( 2 ( u - n - 1 ) + l ) ,
(3.~1)
0 < ~ 1 <~ ~ ( n - 2 ) ( n - 1 ) ( 2 ( n - 2 ) + l ) . The function G is given by the following product: u--n-I
~(Pl . . . . .
P u - l ; ]~; P ) --
II i=1
02P(O;u-n-i)-2ufii'( . . . . . i)(u-n-i+l)u(T)
11--2
(3.12)
X 1-I 0 2 e ( u - n ; n - I - j ) - 2 u / ' j , ( n - 1 - j ) ( n - j ) u ( g ) , j=l
where one defines
(3.13)
P ( a; fl) = fi,~,B - flP,~+B+l, with 17
/~./,n = ~
fio,n =- fin.
P./+k,
(3.14)
k=l
The function .U reads
.~ { f(n;~,-~) .=..Ul bt
s=0
{ t=0
z D(pl,...,p...... l;s)
{z{
~(o;#;,~)
D(cr]....,¢r,,_2;t)
u--2 ×Z
Ou[--(u--n)(n--l)(2A'+l)--2(n--1)s+2(u-n)t],u(u--1)(u--n)(n--I) (7")
A'=O
II}I (3.15)
Finally, the label [ne] in the s / ( 2 ; C ) ~ characters entering the formula (3.5) is the residue modulo 2(u + 1) of n~ defined by (e = 0, 1) n~=-l+(1-Ze)((u-l)(Z,~+l)+u-Zr+(u-n)(1-Zg)).
(3.16)
For each choice of variables ,~, r, n, g, either [no] or [n]] is in the set S = {1 . . . . . u 1 } , or else, [no] = [ n i l . In the latter case, there is no contribution proportional to X .C/(2;C) I,,,, I,,,-ill-J in (3.5), while in the former case, one gets a contribution X ,,:1(2;C) i,,, I , , - , , - 1 with • = 0 (resp. 1) according to whether [no] (resp. [hi ] ) is in the set S. Let us end this section by noting that the corresponding decomposition in s/(2;C)~. characters for NS class V s~/(2[1;C) characters is readily obtained by making the substitution M + M / for m, M for m/, z --~ z - I and multiply by an overall phase in (3.5).
M. Hayes,A. Taormina/NuclearPhysicsB 529 [PM](1998)588-610
598
4. Parafermionic characters as branching functions
As stressed in the introduction, the expression (3.5) neatly isolates the s/(2;C)k character dependence, but the branching functions are still written in a way which obscures their modular properties. However, it is easy to calculate the residue at the simple pole o- = 0 in the NS singular characters, both in class IV and class V, when they are decomposed in s/(2; C)k characters. Indeed, as discussed in [18], when the level k is of the form (1.3), the residue at the pole o-= 0 of singular s/(2; C)~ characters are unitary minimal Virasoro characters at level u, multiplied by r / - 2 ( r ) . On the other hand, we have the residue calculated as in (2.21), which can be rewritten using du(2) string functions at level (u - 2 ) [17,19], ,.,
.
NS,sl(21 l;C)k ,. -
lim zcrto-.ghNS h,y
or----+0
tr, O-, V)
1 = 1.90,2 (7-, 1/.-') -{- 'L92,2 (7-, 2P) g]3 (7-) = /.90,2 (7-, llt.,) _]_ /.92,2
NS N=2. ,¥r,s' t T'#½)
7- b' (7-,½12) Zu--2 C}:~2, (T)Otb'u--Bffu--2),u(u--2)('~, 2U)
r]3 ( r )
;qg=-u+3 u--2
1713(
T)
I
zU--2 C}:li72'(7-){r~=O(Z'~½(ii/u_fiffu_2))_(2r_s)(u_2),(u_ . )(u-2) (7-) 17d=-- u+3
" s=0
XL~½(ti,'u-~,(u--2),+u(2r--s),u(u--l)(7-'~) }}'
(4.1)
where g=m=r+s-
1,
Fn=2m'-m
(4.2)
in class IV. The above residues are thus expanded in a basis of theta functions at level u(u - 1) with arguments 7- and v/u, as are the residues calculated from (3.5) (see (3.15)). A comparison between these two methods of obtaining the residues of singular NS s/(211;C) characters provides enough information to express the branching functions in a form where the modular transformations can be carried out easily. We have worked out in detail the branching functions for the cases u = 3 and u = 4 (see Appendix B), and the truly remarkable result is that the branching functions involve a rational torus A,(,-1) [20] as well as Z,-1 parafermion characters [21,22]. With hindsight, this structure can be derived from the coset central charge value (1.5), since 3 ( u - 1) 2(u-2) -- 1 + - - , u+l u+l
Ccose t -- - -
(4.3)
where the first term is the torus central charge while the second term is precisely the central charge of the parafermionic algebra based on Z._l parafermions. When u = 2, there are no parafermions and the branching functions are just the Ae toms characters,
M. Hayes, A. Taormina/Nuclear Physics B 529 [PM] (1998) 588-610
599
as discovered in [ 10]. Based on the analysis for low values of the parameter u, we conjecture the following s/(2 I1;C)k character decomposition formulae. The class IV characters can be written u-1
NS'IV'sI(211;C)k/T
L
/I(hNzS,hNS
,
O" /,'~'
,
a~l(2;C)k .
I =ZXi,u_m_l~T,O" ) i=0 u-2
X
i,X(i,sl(7-)Lq,(u_ll(m_2m,+u)+uX(i,s),u(u_l) (T,. ~
, (4.4)
where hN_s and hN+s are given by (3.6), while the class V NS characters are decomposed as u-I NS,V,[/(211;C)k
/~ hr~_S,hNs
(7-,
,~':/(2;C) k
0", V) = Z Xi,M+M' +I ( T, 0") i=O u-2
s=0
(4.5) where hN_s = - - I ( M + M ' + I + u ) ,
h~ s = -I(M - M').
U
(4.6)
U
The function X(i, s) is given by X(i,s)=(u-1)i+2i
-
(4.7)
-2s,
and the symbol [ ~] is the integer part of ~. In the above expressions, one interprets ,q ~ 7-, ( u - l ) t )Ci,x(i,~)(7-) as the partition function for the Zu-1 parafermionic theory where the i lowest dimensional field is @x(/,s) in the notations of [23]. Recall that the s~u(2) string functions at level u - 1 have the following symmetries [17]: (u--l)
_ C(U--I)
c(U-- 1)
(u--l)
[
.
C~',m (~')-- ~,-m (r) = ~,m+2(,-l)Z(7-) = C,,-1-~,,-|-mtT-)' c (t,m U-l)
(T)=0for
g - - m 4~ 0 mod2.
(4.8)
The Au(~-l) torus characters are given by the theta functions at level u(u - 1) multiplied by r/-I (7-). The decomposition (4.4) encodes all information needed to obtain a parafermionic realisation of s/(211; C)k at fractional level [24].
600
M. Hayes, A. Taormina/Nuclear Physics B 529 [PM] (1998) 588-610
5. Conclusions Admissible representations of affine Lie algebras and superalgebras have been known to exist for fractional levels since the pioneering work of Kac and Wakimoto [9]. It is argued there that the corresponding characters can be expressed in terms of modular forms and provide a finite representation of the modular group. Over the last two years, we have calculated explicit character formulae for the affine superalgebra sl(211; C) k using the Kac-Kazhdan determinant formula [ 14,15 ] and the embedding diagrams for singular vectors one can derive from it [ 12,10]. A necessary condition for admissibility is that the level of the algebra sl(2]l;C)k is of the form (1.2). In this paper, we restrict ourselves to levels of the form k = ~ - 1, u E N, and to a certain class of highest weight representations labelled class IV and class V in [ 10], where the highest weight state quantum numbers are given by (3.6) and (4.6). The corresponding characters, first derived in [ 10], are decomposed here into characters of the bosonic subalgebra s/(2;C)k, which has been extensively studied. The branching functions involve characters of the rational torus Au(,-~) and partition functions of the parafermionic algebra Z,-1, given by s~u(2) string functions at level (u - 1). The decomposition (4.4) and (4.5) are conjectured for arbitrary value of u c N on the basis of a detailed analysis of the cases u = 2 [10] and u = 3,4 (see appendices). They reflect the structure of the central charge of the coset s/(2[1;C)k/s/(2; C)k, given by (4.3) as the sum of a toms and a Zu-1 parafermionic central charge. There are three immediate byproducts of these decomposition formulae. First of all, the S modular transform of the sl (2[ 1; C) ~ characters can easily be obtained from (4.4) and (4.5) since the modular transformations laws of theta functions, s~u(2) string functions and ~/(2; C)k characters are standard. We have explicitly checked that in the cases u = 2, 3, the irreducible sl(211;C) k characters provide a finite representation of the modular group, indicating the existence of an underlying rational theory. Second, as pointed out in the appendix, the more technical steps taken in obtaining the branching functions produce mathematical identities reminiscent of those discovered and discussed in [25,26], and may well put the latter in a new perspective. Third, the decomposition formulae encode all necessary information to obtain a new representation of the s/(211;C)k currents in terms of a primary parafermionic conformal field, as described in [24]. In particular, in the case u = 3 (resp. u = 4), the primary field corresponds to the Ising model (resp. 3-states Potts model) spin field o- of conformal weight 1/16 (resp. 1/15). The deep r61e of such a representation has yet to be understood, particularly in connection with our interest in the family of levels studied here, which stems from the non-trivial r61e s/(211; C)k plays in the description of non-critical unitary minimal N = 2 strings, when the matter central charge is given by Cmatter= 3 ( l - - 2). The most straightforward generalisation of this work would be to relax the condition p = 1 and investigate the theory of non-unitary minimal N = 2 strings.
M. Hayes, A. Taormina/Nuclear Physics B 529 [PM] (1998) 588-610
601
Acknowledgements M. Hayes acknowledges the British EPSRC for a studentship. A.Taormina acknowledges the Leverhulme Trust for a fellowship and thanks the EC for support under a Training and Mobility of Researchers Grant No. FMRX-CT-96-0012.
Appendix A We outline here the derivation of expression (3.5) from (3.4). The use of the theta function identity, k+k' Om,k(r. o-)Om',k' (r, o') = ~ Om/e-m'k+2&k',kk'(/~+k')(r)Om+m'+Zgk,k+k'(r, or). g=l
(A.I) is crucial. In particular, it allows us to write N i= I N-I
N-I
=Z{ Z r=0
H
D(nl,..,nN-i;r
l~P(O;N--i)--2uhi'(N--i)(N--i+l)u('l')
OPu-2ur'Nu 7", -~
,
i=1
(A.2) where hi = ~",N-I ~.-,j=i (" N - J ) n l , D(nl
.....
={hi
nN--1;
r)
• O<.nj<~N-j,
j=l
..... N-1
"fix=k'N+r,k'
cN},
(A.3)
and (A.4) with N fi.j,N = Z k=l
P,j+k,
ff0,N ~-- PN.
(A.5)
Note that the invariance of the left-hand side of Eq. (A.2) under permutations of PN} provides identities between sums of products of theta functions at fixed values of r. This type of identity is used in deriving (3.5), and will be mentioned at the appropriate time below. We shall first discuss the occurrence of the function .Y'(n; r, ~,) in (3.5). It stems from the factors O,,,_2,,,,.(r, ~,, ) . - n and O,,,-2m,+...(r. 7,~ ) n - t in (3.4). To see that, first note {Pl .....
M. Hayes, A. Taormina/Nuclear Physics B 529 [PM] (1998) 588-610
602
that in the term w h e r e n = 1 (resp. n = u), one is only left with the first (resp. second) factor. A p p l y i n g f o r m u l a ( A . 2 ) with N = u - 1 and pi = m - 2 m N = u -
1, Pi
=
m - 2m ~ + u
~ Vi = ! . . . . .
u - 1 (resp.
Vi = 1 . . . . . u - 1) to this factor, one obtains St'( 1 ; r, ~)
( r e s p . . T ( n ; r, ~ ) , w h e r e one f o r m a l l y sets theta functions at level zero to 1, as well as sums and products r a n g i n g o v e r negative values of the index. In the generic case w h e r e n is neither 1 nor u, apply f o r m u l a ( A . 2 ) with N = u - n , p i to the first factor, and with N = n - 1, Pi
=
= m-2m
~ Vi = 1 .....
u-n
m - 2m' + u Vi = 1 . . . . . n - 1 to the second
factor. This leads to
p u--n
i.~)n--I
o ....
u--n-- 1 .-
×[
t=0
u--n-- 1 D(pl,..,p,
,,-l;S)
i=1
Z
D (o-i ,..,o-,,_ 2 ;t ) j = l
(A.6)
XO(n-l,(m--2m'+u)-2ut,(n-l)u ( 7 " , ~ ) ] • Now, use ( A . I ) to evaluate the product,
u-2
=Z
Lg--u[(u--n)(n--l)(2g'+l)+2((n--l)s--(u--n)t)l'u(u--l)(u-n)(n--l) ('I")
,V=0 P
XO(u_t)(m_2m,)_u[2(s+t)_(n_l)+2(u_n){,].u(u_l)
(T, ~)
(A.7)
,
where the s u m m a t i o n index g in ( A . 1 ) is rewritten as g = ( u - 1 ) p - ~ ' , p
= 1 .....
u; ~ ' =
0 . . . . . u - 2 and the f o l l o w i n g relation has been used:
Om,u(u_l)(u_n)(n_l)(7") = ~
02pu2(u_l)(u_n)(n_l)+um,u3(u_l)(u_n)(n_l)(7").
p=l
(A.8) F r o m here, it is straightforward to obtain expression ( 3 . 1 5 ) . Let us n o w rewrite the o--dependent factors in (3.4) in such a way that s l ( 2 ; C ) characters at level k = ±u - 1 appear. N o t e that the s l ( 2 ; C )
d e n o m i n a t o r O1,2(r, o-) -
O - 1 , 2 ( r , o-) is already explicitly written in ( 3 . 4 ) , but the n u m e r a t o r should be a differe n c e o f theta functions in the variables r and ~'~ at level a = u ( u +
1) see ( ( 3 . 1 ) ) . We
illustrate the derivation for the term n = 1 in ( 3 . 4 ) , and first use ( A . 2 ) to rewrite
M. Hayes,A. Taormina/NuclearPhysicsB 529 [PM](1998)588-610
603
u-I i=1 u-2
=Z r=0
u-2
~ 1-I ~'~--(u--l--i)(u--i)--2ufzi'(u--l--i)(u--i)u(T) D(p~l,...tx,,_2;r)i=1 O"
XO(m+l)(u--l)_2ur,u(u--l) (T, ~) .
(1.9)
In the above, we used that when n = 1, the possible values of Pi are all values in the set S = { 1,2 . . . . . u - 1 }. The next step is to evaluate
[IOm+l+u+2i,u (T,~)l~2(m+1, .... 2u
(T,~)--O2(m+l)+u,2u
(T,~)
.
(1.10)
i= I
To this effect, use ( A . I ) with g = (u + 1)p - g ' , p
= 1 . . . . . u , g ' = 0 . . . . . u as well as
the relation /,q~m,2u(u_l)(u+l)(T)
---- ~um+4u2p(u_l)(u+l),2u3(u
(A.11)
1)(u+l)(7"),
p=l and calculate, O"
O"
O"
O(m+l)(,,-l)-2ur,u(u-,) (T,U) [0"2(n,+I)--u,2u(T,;) --02(m+l)+u,2u(7",U) ] = ~-~{19u(u--1)(4g+l)--4ur,iu(u-1)(u+l)(T) t=0 O"
X~'qul(2~+l)(u--l)+l-2r]--(u--m--l)(u+l),u(u+l) (T, ~) -- Ou(u-- I ) (4g+ 1)+4ur,2u(u--1)(u+l ) ( T )
XIg_ul(2~+l)(u_l)+l+2rl_(u_m_l)(u+l),u(u+l) T, U
"
(A.12)
A l s o note that the above formula is obtained after defining g" = u + 1 - g' in the first sum, and using the fact that the term g" = u + 1 is identical to the term ~" = 0. N o w manipulate the above expression to make the s / ( 2 ; C )
character numerator
l~b, ,u(u+l) (T, ~ ) -- Ob_ ,u(u+l) (T, ~ ) appear with b± = ± u (n' + 1 ) - n " ( u + 1 ),
for s o m e n', n" /> 0.
(A. 13 t
One identifies n" with u - m - 1, n"=u-m-
1 ~>0,
n"=0 .....
u-
1
since
0~
1
in class IV.
(A.14) The identification o f n' is more involved. W h e n r = 0 in (A. 12), the s~l(2; C) character appearing is X ,,~l(2;C). , , , , tr, o-) (resp. - X , ,~"l(2;C) , , , , i~T , o - ) ) when 0 ~< /Tt ~ u - 1 and n' is the
M. Hayes,A. Taormina/Nuclear Physics B 529 [PM](1998)588-610
604
residue modulo 2 ( u + 1) of ( u - 1 ) ( 2 g + l ) (resp. - ( u l ) ( 2 g + 1) - 2 ) . I f n ' is the residue modulo 2 ( u + 1) of one of the two above expressions, but is greater than u - 1, then contributions cancel, and there is no s/(2; C) character contribution. Whenever r 4= 0 in ( A . 1 2 ) , the terms r and u - 1 - r must be combined in order to produce the s/(2; C) characters. The contribution to ( A . 1 0 ) from terms corresponding to r 4= 0 can be written,
u--2
u-2
E r=l
D(#l u
#, 2 r) i=1
X Z Ou(u-1)(4g+l)-4ur,2u(u-l)(u+l) (T) g=0
XOu,(2g+l)(u--l)--2r+l]--(u--m--l)(u+l).u(u+') (7", ~ ) u--2 r'=l
u--2 D(#b.../xu-2;r') i=l
X ~ ~u(u--l)(4g+l)--4ur',2u(u--l)(u+l)( T ) g=0
XL~_u[(2g+l)(u_l)_2r,+l]_(u_m_l)(u+l).u(u+l) "1",
(A.15)
,
where we have changed variable from r to r' = u - 1 - r in the second sum over r. Finally, use the following identity, t.t--2
Z
H O--(u--l--i)(u--i)--2U[Xi'(u--l--i)(u--i)u(T)
D (#l ,.../z,,_2;r) i=l u--2
(A.16) D(p.i,...#,-2;u--l--r) i=1 which follows from the invariance of ( A . 2 ) under the permutations of P l . . . . . p . - l , and write, for all r-terms, including r = 0,
i=1 u-2 u-2 r=0
D (/xl,.../x,_z;r) i=1
)< ~ Ou(u--l)(4e+l)--4ur,2u(u--l)(u+l) ( T ) f.=0
1
M. Hayes, A. Taormina/Nuclear
Physics B 529 [PM] (1998) 588-610
605
u
x
[
( > 77 ;
~~~~2P+l~~u-l~-2r+ll-_(u--n~-l)(u+I),~(u+l)
-I?_
u
(
.
73 ;
u~(2~+l)(u-l)-2r+l]-_(u--nt-l)(u+l),u(u+l)
(A.17)
)I (7, (T) (resp. -xL?ii?
(r, a) )
when 0 < n’ 6 u - 1 and n’ is the residue modulo 2(u + 1) of (U - 1)(2!+
1) - 2r
The si( 2; C) character
appearing
in the above is &!~?’
(resp. - (U - 1) (2! + 1) - 2 + 2r) ; n” is identified with u - m - 1. The discussion of the term II = u is completely similar to the above case. The generic case where II # 1 and II # u proceeds along the same lines, but is more involved since one has to apply (A.2) to the factors nyz<‘=T” 79,,L+~+u+2pi,u (7, %) and nyi’ 29~,+1+2,,,,+,,,, ,U(7, :). This derivation is very close in spirit to the technique used at the beginning of this appendix when we were discussing II -
the v-dependence
One obtains
n-l
II
I9 n?+l+u+2p,,u
n ,=I
of (3.4).
(7
5)
n
7%+1+2,,.-,~+,,u
(7
f)
j=l 11-n-l =
u-n-2
c
c
/L,=o
/L*=o
...
e
yy...
/.L,,-,!-I=ovl=o
G(p ,,...,
2
p,-l;&I)
v,,_*=o
v*=o
oX~(,-,)(nl+l+u)+21S,,_,,-2uP,,(u--n)u
(
7, ;
>
u
( >
X~(~-l)(nl+l)+2/i,,~,,:,,~,-2u~,,(n-l)u
7, I*
Now, the product of the last two theta functions
(A.18)
above can be written, using (A. 1) ,
u-2 c
~~1,‘1(11--1~(LI--n)~~-l)~~~~~U--n)ll(l--2~)+~U-l)~nt+l+u)-2u(~,+B,),u(u-l)
P=O (A.19) where a=U(L4-~)(n-1)(1-2C)+2(n-l)[p,_,,-u~,] -2( U - n) [&,;,_
of ,kt + Cl = k’( u - 1) + r, k’ E N, r = 0, .., u - 2 as &scribed
Note that the partitioning in (3.10)
(A.20)
I - u& 1.
is devised to simplify
the theta function
6 (LI- ~~)u~l-28)+~u-l)(n~+I+u)-2u(p,+P,),u(u-l) using the property u
292u(u-l),u(u-I)
Now, using
( > 7,;
= fiO,,(,-I)
(A. 1) , we calculate,
u
( > 7, ;
.
(A.21)
606
M. Hayes, A. Taormina/Nuclear Physics B 529 [PAl] (1998) 588-610 u-n
n-I
i=1
j=l
u--2 = ~
~
~(Pl .....
Pu-l;/2;
~)
(7",U)
,
r=O D(l~l,...,l~u_n_l;Vl,...,pn_2;r ) u-2 g=O
g~=0
--l-~b-c,2u(u+l)(u-l)(7")Od--e+f,u(u+l)
(A.22)
where
b = u ( u - 1) (4g' + 3), c=2u(u - n) (1 - 2g) - 4ur, 1 d=~c
e=u[(u-1)(2g' + l) +u], f=-(u-m-
(A.23)
1)(u+l).
One must now make the numerators of s/(2; C) characters appear in the sum over g~. Similar manipulations as in the case n = 1 described previously in this appendix allow one to rewrite the expression (A.22) as u--2
Z
G(pl . . . . . pu-1;/2; ~)
r=0 D (/Zl ,...,,u.. . . . . 1;Ul ,...,}',,- 2 ;r ) u--2
X ZOa,u(u_l)(u_n)(n_l)('l") g=0
Ob+c,2u(u+l)(u_l)(7") ,f' =0
(A.24)
_ ~1(2;C) " '~1(2;C) (7", 0-) (resp. -Xn',n" (7",0-)) Call n" = u - m - 1. The s/(2; C) character is Xn',,," when n t is the residue of ( u - 1) ( 2 g ' + 2 ) - 2 r + ( u - n ) ( l - 2 e ) (resp. - ( u - 1) ( 2 U + 2)+2r-(u-n)(1-2g)-2) in the r a n g e 0 ~ < n ' ~ < u 1.
M. Hayes,A. Taormina/NuclearPhysicsB 529 [PM](1998)588-610
607
Appendix B In this appendix, we look in some detail at the derivation of branching functions for the cases u = 3 and u = 4. Incidentally, the corresponding parafermionic theories are the Ising model and the 3-states Potts model, a fact which will be highlighted below by the natural occurrence of unitary Virasoro characters at level 3 and 5 in connection with s~u(2) string functions at level 2 and 3 respectively. As explained in Section 2, at fixed values of u, there exist u 2 characters corresponding to irreducible representations, u of which are regular when the variable o- tends to zero. We will view the latter as characters obtained by spectral flow from the other (Ramond or Neveu-Schwarz) sector, where they are singular. To fix the ideas, let us start in the NS sector of the u = 3 theory, where six characters are singular in the limit described above. We will study in detail the class IV character labelled by h Ns = - 2 / 3 , h Ns = 0. Using the decomposition formula (3.5), one gets
-¥-2/3,0IV'NS"~/(211;C)-2/3(7", O',/2)
(7",0") C1(7-)`00,6
0,2 X2,2
'
~ ) q-C2(7")O6,6
~)-}-(7-)"06,6
3)
3) (BI)
where
CI (7") = "//-3(37 ") [O15,36(T) -- 033,36(7")]O2,6(T ) -}- [`03,36(7") -- O21,36(T)]O4,6(T), C2 (7-) = 77-3 (37") [ O15,36 (7-) -- 033,36 (7-) ] 04,6 (7-) -}- [ 03,36 (T) -- O21,36 (7") ]/94,6 (T), C3(7") = 7/-3(37") [ OI,6(7-) q- 05,6(7") ] [ 06,36(7-) - "030,36(7-) ].
(B.2)
The residue of the above character when cr -+ 0 can be calculated in two different ways. From the decomposition (B.1), it is readily obtained by noticing that the three sl(2; C) characters have a simple pole singularity when cr --~ 0, and that their residue at the pole are Virasoro characters at level 3 multiplied by the function r / - 2 ( r ) [ 18]. On the other hand, formula (4.1) gives the residue in terms of the unique level one string function c~010)(r) = r / - l ( T ) (see e.g. [ 1 7 ] ) , i.e.
,.-.
.
l V , N o~, o ~,~'J+~, . ~ l 1 ;e~~) - - 2 / 3 /
lim Lcrtcr X-2/3,0
O--+0
,,
~7-, or, u)
{ "-~-O1,2(T)[03,6 (T, 3) -~-09,6 (7",3)]).
(B.3)
608
M. Hayes, A. Taormina/Nuclear Physics B 529 [PM] (1998) 588-610
Comparison of these two residue calculations yields the following expressions for the coefficients C i ( r ) ,
[xVir(3) (T)2 _ x~,i[(3)(r)2] CI (,r) = c0,0 A1) vir(3) [.¥1,1 ( r ) O 2 , 2 ( r ) - xvir(3) 1,1 2,1 (r)zgo,2(r)] ' Vir(3) (r)2 -- x~i[(3)(r)2]C2(T)=~O,O )(1,1 "(1) [ XlVI[(3)(r) z90,2(r) - XVi[(3)(r)z92,2(r)] , /. Vir xVir(3) 2,2( (r) [xVi[(3)(r) + , 2,1 (r)]C3(r)=C(ol)oZgL2(r)[X vir(3),,t(T) -I---)(2,1Vir(3)(T) ], (B.4) or again, using well-known identities relating the Ising model characters to square roots of theta functions at level 2 [27],
. Vir(3)/_s t',),
Cl(r) = x2,1
C2(r) = xvi[ (3)(r),
C3(r)
. Vir(3)/-~
= 2"2,2 t")-
(B.5)
It is also worthwhile noticing that the three unitary Virasoro characters at level 3 coincide with the three partition functions of the Z2 parafermionic theory whose lowest dimensional fields are ~P9 z, ~p2 and ~Pl [23]. The latter is actually a primary conformal field which is identified as the spin field of the Ising model. These partitions are given in terms of s~u(2) string functions at level 2. Namely,
..•Vir(3)
Vir(3).. )(2,1 I,T) -: T}(T)¢ 0(22) ( T),
)(Vir(3) 2,2 ( T ) = r / ( T ) C I ~ ? ( T ) '
(B.6)
and therefore, the above expressions can also be viewed as relating s~u(2) string functions at level one and two. The same analysis can be applied to the other five singular NS s/(2ll; C) characters at level k = - 2 / 3 , and to the three singular R characters which flow to the three regular NS characters at the same level. They all involve the three functions (B.5). One then obtains expressions which can be read off (4.4), (4.5) for the nine NS s/(2ll;C ) characters at this level. The modular S transform of these nine characters is encoded in a 9 x 9 matrix which is unitary and whose fourth power is the identity [ 13]. A similar analysis can be performed for the case u = 4. For instance, using again the decomposition formula (3.5), one obtains
IV,NS X_3/4,0 (7", o', p) f o ,.~1(2"C).. ='q-'(r)~X 3 ' t r , tr) [ D,(T)012,,2 (T, ~/] ) -~-D2(T){1-~4,12 (T, 4) -~-1920,12 (T, 4) /]
M. Hayes,A, Taormina/Nuclear Physics B 529 [PM](1998)588-610
609
_XI,3 ,;l(2;C)(r,~r)[D3(T)O0,12(T, ~/ p -~-D4(T){l~8,12 (T, 4) . s:I(2;C)(T,O.) ~-X2,3
D3(T)O,2,12 (T,-~) -}-D4(T){04,12 (T, 4)
q-O20,12 ( T ' 4 )
}] }'
(B.7)
where the four functions D i ( T ) , i = 1 . . . . . 4 are all sums of quintic expressions in theta functions (two at level 8, two at level 24 and one at level 120) times the function r/-5 (4r). The important remark is that all sixteen NS sl(211; C) characters at level k = - 3 / 4 have the same structure as the one above, with the same four D i functions. Here again, the residues can be calculated in two ways. The decomposition (B.7) yields Virasoro characters at level 4 multiplied by the function "q-Z(r), while formula (4.1) expresses the residue in terms of s~u(2) string functions at level two. By comparison of these two calculations, one can write, xVir(3) " 2,2 ( T ) D 1 (T) _-- C(2) 2,0 (T) [.)(~,~(4)(T)0_6,6(T) _ xV2(4)(T).00,6(T ) ] -~-C~2) (T)[xVi~(4)(T)00,6 (T)
--
.)(Vi~(4)(T)'066(T) ],
xVir(3) _ ¢(2) Vir(4) . Vir(4) l_~ 2,2 ( z ) D 2 ( r ) - 2,0 (r) [X3,3 ( r ) O 2 6 ( r ) - x3,2 t~jO4,6(r)] (Vir(3) Vir(4) (T)Z'90,6(T) 2.2 (T) D3(T)--_ C(2) 2,0 (T)[)(I,I
" xV'~(4)(T)t96, 6(T)]
--
,
--
avVir(3) Vir(4) t_x .q rT~) 2,2 (T)D4(7") = C(2) 2,0 ( r ) [Xl,t t~/u4,6t
-)(3,1
(r)O06(r) 1,
--xViI'(4)(T)O2,6(T)]
q_C(2) 2,2 (T) [xV'~ (4) (T)O2,6 (T)
--
Vir(4) (T)O4 6(T) ] , /1(3,2
(B.8)
where we have used the identity [25], xVir(4) Vir(4) . Vir(4) Vir(4) . Vir(3) 1,1 (T))(3,3 (T) -- A'3,1 (r)x3,2 ( r ) = d(2,2 (T).
(B.9)
It turns out that the functions Di(r) enter the partition function of the 3-states Ports model, and can therefore be interpreted as the partition functions for the parafermionic algebra Z3. One has . Vir(5) ~_ . Vir(5) D I ( T ) =X1,1 (T)--A'4,1 (T) . D o ( r )
=
)(Vir(5)
4,3
. Vir(5)
D 3 ( T ) ----X2,1
D4(T)
= ,~Vil'( 5 )
3,3
=T](T)C(3)(T),
n(T)C~31)(T), -I- X3,I (T) =n(r)c~3~o(r),
(T) = (T)
,
Vir(5)
(r) = "q(r)c(131)(r).
(B.10)
The relations (B.8) can therefore be viewed as relations between s~u(2) string functions at level two and level three. It should also be stressed that the relations (B.8), with the functions D i expressed in terms of Virasoro characters at level five as in (B.10), provide new identities very similar to the ones obtained in [25].
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M. Hayes, A. Taormina/Nuclear Physics B 529 [PM] (1998) 588-610
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