- Email: [email protected]
Elliptic genera of N = 2 hermitian symmetric space models
12J~uaryl~5
' PHYSICS LETTERS B
PhysicsLettersB 342 (1995) 87-93
ELSEVIER
Elliptic genera of N = 2 hermitian symmetric space models Toshiya Kawai 1 National Laboratoryfor High Energy Physics (KEK), Tsukuba, lbaraki 305, Japan
Received 11 July 1994 Editor: M. Dine
Abstract
Expressions are given for the elliptic genera of the K~ama-Suzuki models associated with hermitian symmetric spaces when the problems of field identifications are absent. We use the models' known Coulomb gas descriptions.
Starting from the work of Witten [ 1 ] the past year has witnessed a resurgence of research activities in the subject of elliptic genus [2-4] in the particular context of N = 2 super(conformal) field theories [5-12]. Through this development the N = 2 elliptic genera have been computed for various models including the minimal models, Landau-Ginzburg models and their orbifolds, and sigma models. It is natural to ask if we can extend this list so as to cover another large class of N = 2 superconformal field theories, the so-called Kazama-Suzuki models [ 13]. Of course for some special cases of the models that admit Landau-Ginzburg formulations we already have one of the expressions at hand, however, for general Kazama-Suzuki models their elliptic genera have not been obtained. In this note we give one possible expression for the elliptic genera of the Kazama-Suzuki models associated with hermitian symmetric spaces (henceforth abbreviated as HSS models). The HSS models are known to admit the Coulomb gas descriptions, i.e. those in terms of a combined system of bosons and parafermions [ 14-17] and we will essentially make use of this fact for our purpose. In particular we will adhere to the point of view advocated in [ 17] and some of the technologies encountered there will be exploited here also. Our starting point is a slight extension of the branching relation considered in [18] as a Lie algebraic extension of Gepner's branching relation for the N = 2 minimal model [ 19]. Before writing this down we have to explain our notation. We fix, once and for all, a simple Lie algebra g of rank n. We use the convention in which the length of any long root is equal to x/~, thus 02 = 2 where 0 is the highest root of g. The simple roots and fundamental weights of g are denoted respectively by al ..... an and to1 ..... tOn. The Weyl vector is defined to n tOi where A + is the set of positive roots of g. Given a root a the corresponding be p = i1 )'-~a+ a = )--~i=l eoroot is written as a v = ~2 a . We frequently use the lattices P = )"~/~lZtOi (the weight lattice), Q = )-'~fl Za, (the root lattice) and QV = )'-~n=lZt~V(the coroot lattice) as well as the set p+t = {A E )"~1Z___o ~o/: A.0 < k}. l E-mail address: [email protected]. 0370-2693/95/$09.50 © 1995ElsevierScienceB.V. All rightsreserved. SSD10370-269 3 (94) 01334-9
88
T. Kawai/ Physics Letters B 342 (1995) 87-93
The dual Coxeter number of g is denoted by g. Let W be the Weyl group of g. We write the length of w E W as g(w). The notation e [x] = exp(27rix) will be used throughout. The theta function of level k is defined by
@V.k(r,u)=
E e[r~fl 2+ku.fl], flEQV+lzlk
/zEP/kQ
v,
(1)
and transforms as
O~,k(- , -~)=(-ir)n/2e[2r]
E B~,jz, (') O~,,k(r, u) , ~'Et'/kQV
(2)
where
B(k). = iP/kQVl-i/2e [ - - ~ ] #,Ix"
(3)
The Weyl-Kac character at level k of the affine Lie algebra ~ is x A ( r , U) =
)-~wEW( --1)g(W)Ow(A+p),k+g (I' U)
(4)
y~w~w(_l)e(W)Ow(o),g(r,u) ' where A E P+~ and transforms as
XA -7 7 =eL2rJ At~pk+ ~ a(k) "A'(r'u) '
* "A,A' -~
(5)
'
with
A(k) = ilZ~+lE A,A"
r wEW t
--
l~£(w) l~(k+g) *1 ""w(A+p) ,A'+p
(6)
The fundamental branching relation we consider is g
o
vEe/(k+g)Qv which reduces to the one for the N = 2 minimal models in the case g = su(2). Using the multiplication formula
~)A'm(1'u)~l~'n(1'v) =
(r u - v ~
E @)nA--mlz+mny'mn(m+n) \ 'm-t-n] )'EQV/(m+n)Q v
(r, mu+nv~ Oa+#+my'm+n m ~ n /
'
(8)
it is straightforward to find 7"
yEQ v
where c~a(r) is the string function defined through xA(r,u) = Y~aep/kQV cA(r)Oa,k(r'u)
kg
Z - - +-----~ k •
and (10)
89
T. Kawai / Physics Letters B 342 (1995) 8 7 - 9 3
As an immediate consequence of (7), we find that
xA.~,+~" O',v) = v+(k+g)Q v
xA'~'(r,V)
x A a ' ( r + 1, v ) = e [ h A'~
(11)
c w~' , , Aw.]jUa~ r ~ . 2,-, 4 V)
(12)
"
At
t
S(A,/z,v),(At,/~',v') , ¥ v ' ,/z ( ' r , O) ,
(13)
A'~P~ ~'eP/sQ" ~'eP/(k+g)Qv where t~'-Aa = A . (A + 2p) - / 2 2
~Z
+ 2g'
2(k+g)
kdimg cwzw = k+------g'
(14)
and " (A t , g ,' v' ) --- A (A,/.~,v),. " ' (a k, A) ' U /l:t(g) ~ , / z ' (\ ~Rv(,kv+' g ) ) ~ * "
(15)
It also easily follows from (9) that
X~.'t~(~',V+K~'+A)=e[A.(k-~gV--I~)]e[--f
.] A,Is,-gx . (u2~'+2x.vJJX~_g, ~ tT",v),
(16,
for K, A E P. We also need to introduce the alternating sum
:ZA(r,v)
=
Z (--1)e(w)xA'W(O)(r,v) ,
(17)
w~=W
which transforms as :zA (7, V + 1) --e [h A'p
CWZ-'--W 24 j1 ~Av , {'r , jv"~
Z~, ( _ 1 ) , ,v = e L~ v -Tj v2]F A,Z~p+, f ,,~pm,Z+,Q Aik,)A'(--i)ln+l,,,~,~,[D(k+g)' .~,*"rA'('r,o),]
(18)
(19)
where we used Z(
l"~(w)u(g) ~w(.),?
-",
= (-i)
IA+I
(20)
wEW
We will construct the elliptic genera of HSS models taking X~'t~(~", v) (or actually Z A(r, v) ) as basic building blocks. Before presenting the result, we need to make a digression on some mathematical technicalities relevant in our argument. Since almost all of these are gathered in appendices of [ 17], we shall be very brief. For a fuller explanation and notation the reader is advised to refer to that paper. Let J = {i E { 1 , 2 , . . . ,n} : ai = 1} where ai's are positive integers such that 0 = ~--]n] aioti and are tabulated for instance in [ 17]. The set J is non-empty iff g = A,, B,, C,, D,, E6 or E7 and we shall be exclusively eone.~,ned with these cases. Given a # E J then go is defined to be the semi-simple Lie algebra obtained by deleting the node # from the Dynldn diagram of g and we remark that a# is always a long root. For each # E J we obtain a hermitian symmetric space g/m# where m# is the reductive subalgebra such that m# _ g# ~ u ( 1 ) and we shall set D# = ½dimn(g/m#). Let W# be the subgroup of W corresponding to the Weyl group of go + C A + } where A# + is the subset of A + corresponding to the set of positive roots and W¢ = {w E W : w - 1 (A#)
T. Kawai / Physics Letters B 342 (1995) 87-93
90
of go. Kostant's lemma states that there is a unique decomposition of any w E W as w = w ' w " where w' E IV# and w" E I4,~. Hence 114"#[= [WI/IWo 1. For the elements of W~ we have g(~r) = ( p - t r ( p )
) .too,
(21)
Vtr E W °.
Denoting the translation by kto# as tk,o# and the longest element in W° as if0, we set (22)
~,l#,k = tkoj#l~ # .
Then the y#,k'S, as # runs through J, generate the group of diagram automorphism llk which is isomorphic to P/Q regardless of the value of the nonnegative integer k. It follows that Yo.8(P) = P or equivalently p - kO(p) = gtoo.
(23)
The following formula is worth noting: w(A+p)+j(k+g)too-w(w
(mod(k+g)QV),
#)-j(y#,~+g)j(A+p)
jEZ,
(24)
where A E p+k and w E W and especially (mod g Q V ) ,
w ( p ) + jgoJo - w ( ~ ° ) - J ( p )
j E Z.
(25)
Furthermore we have (26)
Do = g( ~#) = gw 2 = 2p " w#.
The existence of diagram automorphism is closely related to field identifications [20]. In fact from what we have explained we can easily deduce that s(k)
# (A,/x,u),(A',/zl,~ ' )
(27)
= S (k)
(A.u.v),(At,be',v ') ,
where y#(A, p., v) = (y#,k(A),/x + gto#, v + (k + g)to#),
(28)
u+(k+g)a~# [7", V) -- xA'*'(r,v) Xr,.k(a),j*+go•,,
(29)
and
general if the group of diagram automorphism acts non-freely it is difficult to construct modular invariant partition functions. In the following we simply assume that the group of diagram automorphism acts without fixed points thus evading subtle problems. Hence we may define In
1
-
-lelO.IA,Aep+ Z k veP/(I+g)Q Z v NiklZ, (,.o) (zj(,,o)) 1 IP/QI
~
l~e(w)M(k)'TA
y~(
_
-.
~
• • A,A ~w(£+O)
(30) (31)
(,7"0) ~-,
*
,
A,Xej,+~w~W where Nik,~'s are non-negative integers such that they define a modular invariant partition function of the WZW model at level k, i.e. it satisfies NA (k~- = 0 ,A
if
(£ +2p) A. (A + 2 p ) =0 2(k+g)
(modZ)
(32)
91
T. Kawai l Physics Letters B 342 (1995) 87-93
N(k) = A,A
,~A,A' ( k ) ( A ( k ) ~* lV ,,,(k) A' ,/~' \ L~'/
E
(33)
A' £ ' ¢pk+
N(k) (k) rfa).7(:,) = N a,a
'
"y E Ilk.
(34)
In the step going from (30) to (31) we used [18] ~'~A ( ' r , O ) = V ~ . ", ( - - 1 ) * ( w ) 8 x J
[P/(k+g)evl v,w(A+p) "
(35)
wEW
Using (18), (19), (32) and (33) we can show that
(a~+b
v
Z\c--~-d'c~--~)
=e.
I f cv2 1Z , c~-+ciJ (z,v)
b (ac d) ESL(2,Z).
(36)
Let us fix x E J and consider the HSS model associated with g/rex. The one third of its Virasoro central charge is
kD k+g
~ ,
:
(37)
where we have set D = D×. Now we come to the main assertion of this letter:
the elliptic genus of the HSS
model is given by ZO', z) = i T x i z(~-, oJ× z ) ,
(38)
so long as the diagram automorphism acts freely. To confirm this we check [6] the modular transformation property
(ar+b
z
Z\-~-~-~,cr+d
)[~
=e
cz2 ]Z(~.,z)
2cr+d
'
(a b ) ESL(2,Z )
(39)
and the double quasi-periodicity
Z(r,z+Ar+~)=(-1)e(A+~)e
-~(A~-+2Az) Z(r,z),
A,~EhZ,
(40)
where h is the least positive integer such that
hgtax E (k + g)Q v.
(41)
The modular transformation property (39) follows immediately from (36) since D -- gta2x. As for the double quasi-periodicity (40) it suffices to check ^
27wA(~,+,)(r, ~Ox(Z + A~"+ / X ) ) = (--1)e(~+')e [-2(A2~" + 2Az)]I/TA(h+a)(~', taxZ) ,
(42)
for A, b* E hZ. To prove this we first note that (--1) ~h = (--1) °h,
(43)
which follows from (41) since Z9
kh_~g+g tax • p = 2(khgD + g) "
(44)
T. Kawai/ PhysicsLettersB 342 (1995)87-93
92
We then apply (16) and use, besides (41) and (43), the properties presented earlier including (11 ), (25) and (26) while taking into account the fact that a× is a long root. In order to complete our identification we have to make sure that Z(~-, z) has the proper Xy genus (a.k.a the Poincar6 polynomial). The Xy genus is related to the elliptic genus by Z(0, z) = y-~/2Xy [6]. In the present case we have that 1
Xy = IP/QI IW×l "M(k)'AyW×.w((g/(k+g))(A+p)-p))+8/2 A wEWAEp~ " ~ 1l
=
~
~
^r(k) yQkCA,o')
l'aA
(45)
,
(46)
o'EW x AEp+~
where
g
Qk(A,o') = £(o') + ~-~gtOX • (tr(A + p) - p ) ,
(47)
which is in accordance with the results in [21-23,17]. Here we have used Kostant's lemma, (21) and the property that tax is stabilized by Wx. Note that with our definition of h we have hQk(A,o') E Z as must be so. The Witten index can be obtained either from Xy=1 or Z ( r , 0 ) using (35) as
Iw×l IP/QI AEP+~
AA •
(48)
A few remarks are in order. (i) It is obvious that if we take g = su(2) then our formula of the elliptic genus reduces to the one given in [6] for the N = 2 minimal model 2 . (Note that gx = 0, W = W x = { 1 , - 1 } and ]P/QI = 2 . ) (ii) It is straightforward to compute the elliptic genus of the theory orbifoldized by Zh. According to the formula presented in [6], it is given by h--I
Zorb(r, z) = -hl ~
( - 1 ) D ( a + f l + aBB ) [ [ (~',z),
(49)
a,B=O oe
where
IZZI
(~',z) = e
aft e
]
( o t 2 r + 2 a z ) Z(~,z + c e r + f l ) ,
(50)
O/
and we used (43). This can also be rewritten as 1
Zorb(r, z) =
1
Ie/at IW×l h
h-1
~
*
a,fl=OA,.~Epk+veP/(k+g)av 2 The elliptic genus of the diagonal N = 2 minimal model was first given by Witten [ 1 ]. In Ref. [5] the elliptic genera were erroneously written down for the remaining non-diagonal cases (i.e. D and E). The expressions there, contrary to the authors' claim, do not satisfy the correct modular transformation laws since they have included only the diagonal entries of the modular invariants.
7". Kawai / Physics Letters B 342 (1995) 87-93
93
In summary, we have presented a candidate formula of the elliptic genus of HSS model inspired by its Coulomb gas description and have checked that it has the pertinent properties. I am grateful to S.-K. Yang for discussion. References [ 1] E. Witten, On the Landau-Ginzburg Description of N = 2 Minimal Models, 1ASSNS-HEP-93/10 (hep-th.9304026). [2] A.N. Sclmllckens and N.E Warner, Phys. I.¢tt. B 177 (1986) 317; Nucl. Phys. B 287 (1987) 317; K. Pilch, A.N. Schellekans and N.E Warner, Nucl. Phys. B 287 (1987) 362. [3] E. Witten, Commun. Math. Phys. 109 (1987) 525; The Index of the Dirac Operator in Loop Space, in Elliptic Curves and Modular Forms in Algebraic Topology, ed. by ES. Landweber, Lecture Notes in Math. 1326 (Spdnger-Vedag, 1988). [4] O. Alvatez, T. Killingback, M. Mangano and E Windey, Commun. Math. Phys. I l l (1987) I; Nucl. Phys. B (Proc. Suppl.) A 1 (1987) 189. [5] E Di Francesco and S. Yankielowicz, Nucl. Phys. B 409 (1993) 186 (hep-th.9305037). [6] T. Kawai, Y. Yamada and S.-K. Yang, Nucl. Phys. B 414 (1994) 191 (hep-th.9306096). [7] P. Di Francesco, O. Aharony and S. Yankielowicz, Nucl. Phys. B 411 (1994) 584 (hep-th.9306157). [8] M. Henningson, Nucl. Phys. B 413 (1994) 73 (hep-th.9307040). [9] O. Aharony, S. Yankielowicz and A.N. Schellekans, Nucl. Phys. B 418 (1994) 157 (hep-th.9311128). [10] E Berglund and M. Henningson, Landau-Ginzburg orbifolds, mirror symmetry and the elliptic genus, IASSNS-HEP-93-92 (hepth.9401029). [ I l l T. Kawai and K. Mohri, Nucl. Phys. B 425 (1994) 191 (hep-th.9402148). [12] D. Nemcschansky and N.E Warner, Phys. Lett. B 329 (1994) 53 (hep-th.9403047). [13] Y. Kazama and H. Suzuki, Nucl. Phys. B 321 (1989) 232; Phys. I.~tt. B 216 (1989) ll2. [14] P. Fendley, W. Lerche, S.D. Mathur and N.P. Warner, Nucl. Phys. B 348 (1991) 66. [15] W. Lerche, Phys. Lett. B 252 (1990) 349. [16] T. Eguchi, S. Hosono and S.-K. Yang, Commun. Math. Phys. 140 (1991) 159. [17] T. Eguchi, T. Kawai, S. Mizoguchi and S.-K. Yang, Rev. Math. Phys. 4 (1992) 329. [18] T. Kawai, Phys. Lett. B 259 (1991) 460; B 261 (1991) 520 (E). [19] D. Gepner, Nucl. Phys. B 296 (1988) 757. [20] D. C~ptmr, Phys. Lett. B 222 (1989) 207. [21] W. Lerche, C. Vafa and N.E Warner, Nucl. Phys. B 324 (1989) 427. [22] S. Hosono and A. Tsuchiya, Commun. Math. Phys. 136 (1991) 451. [23] D. Gepner, Commun. Math. Phys. 142 (1991) 433.
' PHYSICS LETTERS B
PhysicsLettersB 342 (1995) 87-93
ELSEVIER
Elliptic genera of N = 2 hermitian symmetric space models Toshiya Kawai 1 National Laboratoryfor High Energy Physics (KEK), Tsukuba, lbaraki 305, Japan
Received 11 July 1994 Editor: M. Dine
Abstract
Expressions are given for the elliptic genera of the K~ama-Suzuki models associated with hermitian symmetric spaces when the problems of field identifications are absent. We use the models' known Coulomb gas descriptions.
Starting from the work of Witten [ 1 ] the past year has witnessed a resurgence of research activities in the subject of elliptic genus [2-4] in the particular context of N = 2 super(conformal) field theories [5-12]. Through this development the N = 2 elliptic genera have been computed for various models including the minimal models, Landau-Ginzburg models and their orbifolds, and sigma models. It is natural to ask if we can extend this list so as to cover another large class of N = 2 superconformal field theories, the so-called Kazama-Suzuki models [ 13]. Of course for some special cases of the models that admit Landau-Ginzburg formulations we already have one of the expressions at hand, however, for general Kazama-Suzuki models their elliptic genera have not been obtained. In this note we give one possible expression for the elliptic genera of the Kazama-Suzuki models associated with hermitian symmetric spaces (henceforth abbreviated as HSS models). The HSS models are known to admit the Coulomb gas descriptions, i.e. those in terms of a combined system of bosons and parafermions [ 14-17] and we will essentially make use of this fact for our purpose. In particular we will adhere to the point of view advocated in [ 17] and some of the technologies encountered there will be exploited here also. Our starting point is a slight extension of the branching relation considered in [18] as a Lie algebraic extension of Gepner's branching relation for the N = 2 minimal model [ 19]. Before writing this down we have to explain our notation. We fix, once and for all, a simple Lie algebra g of rank n. We use the convention in which the length of any long root is equal to x/~, thus 02 = 2 where 0 is the highest root of g. The simple roots and fundamental weights of g are denoted respectively by al ..... an and to1 ..... tOn. The Weyl vector is defined to n tOi where A + is the set of positive roots of g. Given a root a the corresponding be p = i1 )'-~a+ a = )--~i=l eoroot is written as a v = ~2 a . We frequently use the lattices P = )"~/~lZtOi (the weight lattice), Q = )-'~fl Za, (the root lattice) and QV = )'-~n=lZt~V(the coroot lattice) as well as the set p+t = {A E )"~1Z___o ~o/: A.0 < k}. l E-mail address: [email protected]. 0370-2693/95/$09.50 © 1995ElsevierScienceB.V. All rightsreserved. SSD10370-269 3 (94) 01334-9
88
T. Kawai/ Physics Letters B 342 (1995) 87-93
The dual Coxeter number of g is denoted by g. Let W be the Weyl group of g. We write the length of w E W as g(w). The notation e [x] = exp(27rix) will be used throughout. The theta function of level k is defined by
@V.k(r,u)=
E e[r~fl 2+ku.fl], flEQV+lzlk
/zEP/kQ
v,
(1)
and transforms as
O~,k(- , -~)=(-ir)n/2e[2r]
E B~,jz, (') O~,,k(r, u) , ~'Et'/kQV
(2)
where
B(k). = iP/kQVl-i/2e [ - - ~ ] #,Ix"
(3)
The Weyl-Kac character at level k of the affine Lie algebra ~ is x A ( r , U) =
)-~wEW( --1)g(W)Ow(A+p),k+g (I' U)
(4)
y~w~w(_l)e(W)Ow(o),g(r,u) ' where A E P+~ and transforms as
XA -7 7 =eL2rJ At~pk+ ~ a(k) "A'(r'u) '
* "A,A' -~
(5)
'
with
A(k) = ilZ~+lE A,A"
r wEW t
--
l~£(w) l~(k+g) *1 ""w(A+p) ,A'+p
(6)
The fundamental branching relation we consider is g
o
vEe/(k+g)Qv which reduces to the one for the N = 2 minimal models in the case g = su(2). Using the multiplication formula
~)A'm(1'u)~l~'n(1'v) =
(r u - v ~
E @)nA--mlz+mny'mn(m+n) \ 'm-t-n] )'EQV/(m+n)Q v
(r, mu+nv~ Oa+#+my'm+n m ~ n /
'
(8)
it is straightforward to find 7"
yEQ v
where c~a(r) is the string function defined through xA(r,u) = Y~aep/kQV cA(r)Oa,k(r'u)
kg
Z - - +-----~ k •
and (10)
89
T. Kawai / Physics Letters B 342 (1995) 8 7 - 9 3
As an immediate consequence of (7), we find that
xA.~,+~" O',v) = v+(k+g)Q v
xA'~'(r,V)
x A a ' ( r + 1, v ) = e [ h A'~
(11)
c w~' , , Aw.]jUa~ r ~ . 2,-, 4 V)
(12)
"
At
t
S(A,/z,v),(At,/~',v') , ¥ v ' ,/z ( ' r , O) ,
(13)
A'~P~ ~'eP/sQ" ~'eP/(k+g)Qv where t~'-Aa = A . (A + 2p) - / 2 2
~Z
+ 2g'
2(k+g)
kdimg cwzw = k+------g'
(14)
and " (A t , g ,' v' ) --- A (A,/.~,v),. " ' (a k, A) ' U /l:t(g) ~ , / z ' (\ ~Rv(,kv+' g ) ) ~ * "
(15)
It also easily follows from (9) that
X~.'t~(~',V+K~'+A)=e[A.(k-~gV--I~)]e[--f
.] A,Is,-gx . (u2~'+2x.vJJX~_g, ~ tT",v),
(16,
for K, A E P. We also need to introduce the alternating sum
:ZA(r,v)
=
Z (--1)e(w)xA'W(O)(r,v) ,
(17)
w~=W
which transforms as :zA (7, V + 1) --e [h A'p
CWZ-'--W 24 j1 ~Av , {'r , jv"~
Z~, ( _ 1 ) , ,v = e L~ v -Tj v2]F A,Z~p+, f ,,~pm,Z+,Q Aik,)A'(--i)ln+l,,,~,~,[D(k+g)' .~,*"rA'('r,o),]
(18)
(19)
where we used Z(
l"~(w)u(g) ~w(.),?
-",
= (-i)
IA+I
(20)
wEW
We will construct the elliptic genera of HSS models taking X~'t~(~", v) (or actually Z A(r, v) ) as basic building blocks. Before presenting the result, we need to make a digression on some mathematical technicalities relevant in our argument. Since almost all of these are gathered in appendices of [ 17], we shall be very brief. For a fuller explanation and notation the reader is advised to refer to that paper. Let J = {i E { 1 , 2 , . . . ,n} : ai = 1} where ai's are positive integers such that 0 = ~--]n] aioti and are tabulated for instance in [ 17]. The set J is non-empty iff g = A,, B,, C,, D,, E6 or E7 and we shall be exclusively eone.~,ned with these cases. Given a # E J then go is defined to be the semi-simple Lie algebra obtained by deleting the node # from the Dynldn diagram of g and we remark that a# is always a long root. For each # E J we obtain a hermitian symmetric space g/m# where m# is the reductive subalgebra such that m# _ g# ~ u ( 1 ) and we shall set D# = ½dimn(g/m#). Let W# be the subgroup of W corresponding to the Weyl group of go + C A + } where A# + is the subset of A + corresponding to the set of positive roots and W¢ = {w E W : w - 1 (A#)
T. Kawai / Physics Letters B 342 (1995) 87-93
90
of go. Kostant's lemma states that there is a unique decomposition of any w E W as w = w ' w " where w' E IV# and w" E I4,~. Hence 114"#[= [WI/IWo 1. For the elements of W~ we have g(~r) = ( p - t r ( p )
) .too,
(21)
Vtr E W °.
Denoting the translation by kto# as tk,o# and the longest element in W° as if0, we set (22)
~,l#,k = tkoj#l~ # .
Then the y#,k'S, as # runs through J, generate the group of diagram automorphism llk which is isomorphic to P/Q regardless of the value of the nonnegative integer k. It follows that Yo.8(P) = P or equivalently p - kO(p) = gtoo.
(23)
The following formula is worth noting: w(A+p)+j(k+g)too-w(w
(mod(k+g)QV),
#)-j(y#,~+g)j(A+p)
jEZ,
(24)
where A E p+k and w E W and especially (mod g Q V ) ,
w ( p ) + jgoJo - w ( ~ ° ) - J ( p )
j E Z.
(25)
Furthermore we have (26)
Do = g( ~#) = gw 2 = 2p " w#.
The existence of diagram automorphism is closely related to field identifications [20]. In fact from what we have explained we can easily deduce that s(k)
# (A,/x,u),(A',/zl,~ ' )
(27)
= S (k)
(A.u.v),(At,be',v ') ,
where y#(A, p., v) = (y#,k(A),/x + gto#, v + (k + g)to#),
(28)
u+(k+g)a~# [7", V) -- xA'*'(r,v) Xr,.k(a),j*+go•,,
(29)
and
general if the group of diagram automorphism acts non-freely it is difficult to construct modular invariant partition functions. In the following we simply assume that the group of diagram automorphism acts without fixed points thus evading subtle problems. Hence we may define In
1
-
-lelO.IA,Aep+ Z k veP/(I+g)Q Z v NiklZ, (,.o) (zj(,,o)) 1 IP/QI
~
l~e(w)M(k)'TA
y~(
_
-.
~
• • A,A ~w(£+O)
(30) (31)
(,7"0) ~-,
*
,
A,Xej,+~w~W where Nik,~'s are non-negative integers such that they define a modular invariant partition function of the WZW model at level k, i.e. it satisfies NA (k~- = 0 ,A
if
(£ +2p) A. (A + 2 p ) =0 2(k+g)
(modZ)
(32)
91
T. Kawai l Physics Letters B 342 (1995) 87-93
N(k) = A,A
,~A,A' ( k ) ( A ( k ) ~* lV ,,,(k) A' ,/~' \ L~'/
E
(33)
A' £ ' ¢pk+
N(k) (k) rfa).7(:,) = N a,a
'
"y E Ilk.
(34)
In the step going from (30) to (31) we used [18] ~'~A ( ' r , O ) = V ~ . ", ( - - 1 ) * ( w ) 8 x J
[P/(k+g)evl v,w(A+p) "
(35)
wEW
Using (18), (19), (32) and (33) we can show that
(a~+b
v
Z\c--~-d'c~--~)
=e.
I f cv2 1Z , c~-+ciJ (z,v)
b (ac d) ESL(2,Z).
(36)
Let us fix x E J and consider the HSS model associated with g/rex. The one third of its Virasoro central charge is
kD k+g
~ ,
:
(37)
where we have set D = D×. Now we come to the main assertion of this letter:
the elliptic genus of the HSS
model is given by ZO', z) = i T x i z(~-, oJ× z ) ,
(38)
so long as the diagram automorphism acts freely. To confirm this we check [6] the modular transformation property
(ar+b
z
Z\-~-~-~,cr+d
)[~
=e
cz2 ]Z(~.,z)
2cr+d
'
(a b ) ESL(2,Z )
(39)
and the double quasi-periodicity
Z(r,z+Ar+~)=(-1)e(A+~)e
-~(A~-+2Az) Z(r,z),
A,~EhZ,
(40)
where h is the least positive integer such that
hgtax E (k + g)Q v.
(41)
The modular transformation property (39) follows immediately from (36) since D -- gta2x. As for the double quasi-periodicity (40) it suffices to check ^
27wA(~,+,)(r, ~Ox(Z + A~"+ / X ) ) = (--1)e(~+')e [-2(A2~" + 2Az)]I/TA(h+a)(~', taxZ) ,
(42)
for A, b* E hZ. To prove this we first note that (--1) ~h = (--1) °h,
(43)
which follows from (41) since Z9
kh_~g+g tax • p = 2(khgD + g) "
(44)
T. Kawai/ PhysicsLettersB 342 (1995)87-93
92
We then apply (16) and use, besides (41) and (43), the properties presented earlier including (11 ), (25) and (26) while taking into account the fact that a× is a long root. In order to complete our identification we have to make sure that Z(~-, z) has the proper Xy genus (a.k.a the Poincar6 polynomial). The Xy genus is related to the elliptic genus by Z(0, z) = y-~/2Xy [6]. In the present case we have that 1
Xy = IP/QI IW×l "M(k)'AyW×.w((g/(k+g))(A+p)-p))+8/2 A wEWAEp~ " ~ 1l
=
~
~
^r(k) yQkCA,o')
l'aA
(45)
,
(46)
o'EW x AEp+~
where
g
Qk(A,o') = £(o') + ~-~gtOX • (tr(A + p) - p ) ,
(47)
which is in accordance with the results in [21-23,17]. Here we have used Kostant's lemma, (21) and the property that tax is stabilized by Wx. Note that with our definition of h we have hQk(A,o') E Z as must be so. The Witten index can be obtained either from Xy=1 or Z ( r , 0 ) using (35) as
Iw×l IP/QI AEP+~
AA •
(48)
A few remarks are in order. (i) It is obvious that if we take g = su(2) then our formula of the elliptic genus reduces to the one given in [6] for the N = 2 minimal model 2 . (Note that gx = 0, W = W x = { 1 , - 1 } and ]P/QI = 2 . ) (ii) It is straightforward to compute the elliptic genus of the theory orbifoldized by Zh. According to the formula presented in [6], it is given by h--I
Zorb(r, z) = -hl ~
( - 1 ) D ( a + f l + aBB ) [ [ (~',z),
(49)
a,B=O oe
where
IZZI
(~',z) = e
aft e
]
( o t 2 r + 2 a z ) Z(~,z + c e r + f l ) ,
(50)
O/
and we used (43). This can also be rewritten as 1
Zorb(r, z) =
1
Ie/at IW×l h
h-1
~
*
a,fl=OA,.~Epk+veP/(k+g)av 2 The elliptic genus of the diagonal N = 2 minimal model was first given by Witten [ 1 ]. In Ref. [5] the elliptic genera were erroneously written down for the remaining non-diagonal cases (i.e. D and E). The expressions there, contrary to the authors' claim, do not satisfy the correct modular transformation laws since they have included only the diagonal entries of the modular invariants.
7". Kawai / Physics Letters B 342 (1995) 87-93
93
In summary, we have presented a candidate formula of the elliptic genus of HSS model inspired by its Coulomb gas description and have checked that it has the pertinent properties. I am grateful to S.-K. Yang for discussion. References [ 1] E. Witten, On the Landau-Ginzburg Description of N = 2 Minimal Models, 1ASSNS-HEP-93/10 (hep-th.9304026). [2] A.N. Sclmllckens and N.E Warner, Phys. I.¢tt. B 177 (1986) 317; Nucl. Phys. B 287 (1987) 317; K. Pilch, A.N. Schellekans and N.E Warner, Nucl. Phys. B 287 (1987) 362. [3] E. Witten, Commun. Math. Phys. 109 (1987) 525; The Index of the Dirac Operator in Loop Space, in Elliptic Curves and Modular Forms in Algebraic Topology, ed. by ES. Landweber, Lecture Notes in Math. 1326 (Spdnger-Vedag, 1988). [4] O. Alvatez, T. Killingback, M. Mangano and E Windey, Commun. Math. Phys. I l l (1987) I; Nucl. Phys. B (Proc. Suppl.) A 1 (1987) 189. [5] E Di Francesco and S. Yankielowicz, Nucl. Phys. B 409 (1993) 186 (hep-th.9305037). [6] T. Kawai, Y. Yamada and S.-K. Yang, Nucl. Phys. B 414 (1994) 191 (hep-th.9306096). [7] P. Di Francesco, O. Aharony and S. Yankielowicz, Nucl. Phys. B 411 (1994) 584 (hep-th.9306157). [8] M. Henningson, Nucl. Phys. B 413 (1994) 73 (hep-th.9307040). [9] O. Aharony, S. Yankielowicz and A.N. Schellekans, Nucl. Phys. B 418 (1994) 157 (hep-th.9311128). [10] E Berglund and M. Henningson, Landau-Ginzburg orbifolds, mirror symmetry and the elliptic genus, IASSNS-HEP-93-92 (hepth.9401029). [ I l l T. Kawai and K. Mohri, Nucl. Phys. B 425 (1994) 191 (hep-th.9402148). [12] D. Nemcschansky and N.E Warner, Phys. Lett. B 329 (1994) 53 (hep-th.9403047). [13] Y. Kazama and H. Suzuki, Nucl. Phys. B 321 (1989) 232; Phys. I.~tt. B 216 (1989) ll2. [14] P. Fendley, W. Lerche, S.D. Mathur and N.P. Warner, Nucl. Phys. B 348 (1991) 66. [15] W. Lerche, Phys. Lett. B 252 (1990) 349. [16] T. Eguchi, S. Hosono and S.-K. Yang, Commun. Math. Phys. 140 (1991) 159. [17] T. Eguchi, T. Kawai, S. Mizoguchi and S.-K. Yang, Rev. Math. Phys. 4 (1992) 329. [18] T. Kawai, Phys. Lett. B 259 (1991) 460; B 261 (1991) 520 (E). [19] D. Gepner, Nucl. Phys. B 296 (1988) 757. [20] D. C~ptmr, Phys. Lett. B 222 (1989) 207. [21] W. Lerche, C. Vafa and N.E Warner, Nucl. Phys. B 324 (1989) 427. [22] S. Hosono and A. Tsuchiya, Commun. Math. Phys. 136 (1991) 451. [23] D. Gepner, Commun. Math. Phys. 142 (1991) 433.