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Index theorems in N=2 superconformal theories
Volume 205, number 4
PHYSICS LETTERS B
5 May 1988
INDEX T H E O R E M S IN N - - 2 S U P E R C O N F O R M A L T H E O R I E S ~" W. L E R C H E t.2 and N.P. W A R N E R 3 CERN, CH-1211 Geneva 23, Switzerland Received 10 February 1988
Having emphasized the role of N=2 superconformal invariance in various string theories, we compute the "index" Tr( - 1)F for the N=2 supercharge on orbifold and manifold backgrounds. Resolving a quantization ambiguity in the appropriate way, we show that it generalizes the Dolbeault index ind 0". It has however, in general, no well-defined modular properties. The lack of modular invariance can be expressed in terms of the integral characteristic class ~cj, that is also related to the condition for spacetime supersymmetry.
1. Introduction There exists now a variety o f constructions o f lower-dimensional heterotic string theories. One type o f construction deals with compactification o f higher dimensions [ 1 ], another type works directly in lower dimensions [2,3 ]. The overlap between these two approaches appears to be essentially the symmetric orbifold construction. Many properties o f a compactified theory can be interpreted in terms o f topological properties o f the compact space. It is certainly important to investigate whether some o f these topological concepts can be translated or generalized to more general conformal field theories that cannot even be regarded as limiting cases o f manifold compactifications. We will consider here mainly a certain class o f (abelian) orbifolds and the theories described by the covariant lattice approach #t. The overlap between the latter theories and orbifolds is given by at least the set o f asymmetric orbifolds with inner automorphism twists and with rank 22 gauge groups. A particular property of all o f these theories is a global N = 2 right-moving superconformal symmetry, extending the local N = 1 world-sheet supersymmetry. In fact, most o f all known theories exhibit this symmetry. For example, it was shown [ 5-10 ] that N = 1 spacetime supersymmetry implies N = 2 sheet supersymmetry. We want to stress, however, that in the above-mentioned class o f models the occurrence o f N = 2 supersymmetry is independent o f any spacetime supersymmetry. It is thus interesting to study this N = 2 superconformal invariance in more detail. In particular, we are interested in the index Tr( - 1 ) r that one can define for the two supercharges. We will first compute this quantity for asymmetric orbifold and covariant lattice theories. In general such theories cannot be interpreted as compactifications o f higher-dimensional theories on manifolds, and the supercharges have no representation in terms o f differential operators. However, when such an interpretation is possible, we show that Tr( - l )F generalizes the Dolbeault index ind tg. We also find that even though T r ( , 1 )F can have this interpretation as an index, it is not a well-defined function in loop space in that it has no good modular properties unless a certain cohomological Work supported in part by NSF Grant # 84-07109. Supported by the Max-Planck-Gesellschaft, D-8000 Munich, Fed. Rep. Germany. 2 Address after 1 September 1988: California Institute of Technology, Pasadena, CA 91125, USA. 3 On leave of absence from: Department of Mathematics, Massachusetts Institute of Technology,Cambridge, MA 02139, USA. #l For a review see ref. [4]. 0 3 7 0 - 2 6 9 3 / 8 8 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
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condition is met. This condition is equivalent to the presence of spacetime supersymmetry, and if it is satisfied, Tr ( - 1 )F becomes identical to Tr F., the index of the Dirac-Ramond operator.
2. The N--- 2 superconformal algebra In conformal field theory, the N = 2 superconformal algebra [ 10,11 ] can be represented in the operator product form
T ( z ) . T ( w ) ~ (z_w)~ ½c + (z_w)~Z 2T(w) + OT(w) - z-w ' T(z)'J(w)~
J ( w ) + aJ(w) , (z-w) 2 z-w
G+(z)'G-(w)
~c (z-w) 3
J(z)'J(w)
+ J(w) ~( z - w +)
T(z).G+-(w)~ ~G+-(w) OG+-(w) ~( z - w ) ÷ - - z - w ' ~c (z-w) ~'
J ( z ) . C +-( w ) ~ + 6+-(w)
Z--W '
T(w)+½OJ(w) z-w , G+-(z)'G+-(w)~O.
(1)
Here, T(z) represents the stress-energy tensor, G -+(z) the supercharges and J(z) an U ( 1 ) Kac-Moody current; c is the central charge. All these operator products have well-known representations in terms of (anti-)commutation relations of their mode components. These components are defined by T(z)=Ez-Z-"L,,, G +-( Z ) ~- ~ Z -3/2-nq:aG++_ a and J(z) = Y~z- l-nJn, where n~Z and a is arbitrary. One such commutation relation is
{Gn++a, Gm_a } =Ln+m + ½( n - m + 2a)Jn+m + [-~C(n+ a) 2 - -~4C]~n+m •
(2)
This, together with the other corresponding (anti-)commutation relations, defines an N = 2 superconformal algebra, Aa, for any given value of a [ 8,12,3 ]. As the algebras Aa and Aa+ ~are isomorphic [ 8,12 ], the parameter, a, is restricted to the range 0 ~ annihilated by L,, G + , J, for n > 0, and have definite eigenvalues under a maximal set of commuting generators: Lo I h, q> = h Ih, q>, Jo Ih, q> = q l h, q>. The complete highest-weight representation is then obtained by applying the corresponding creation operators (n < 0). The NS sector is defined by a = ½, and the R sector by a = 0. In the R sector, the representation space splits further into two sectors P + ~ P - , characterized by the action of the two supercharges Gfflh, q ~ ½ > + = 0 ,
Ih, qT½>_+~P -+.
(3)
Furthermore, G~-: P + o P - , Gff = (G~-)*: P---*P+ and {( - 1 )J0, Gff }= [Lo, Gff ] =0. It follows that except for states annihilated by Gd- or Gff, all other states come in pairs with the same mass and with U( 1 ) charges differing by one unit. The states which are not paired are massless, i.e., have h = ~4c. Thus one is tempted to define as "index" for an arbitrary N = 2 superconformal theory ind Gd- = T r ( - 1 )Jo. This, however, can only represent a meaningful quantity if we can regulate the trace such that the regulator commutes with Jo and the mass operator. For closed string theories with a right-moving (2, 0) supersymmetry, the appropriate object to consider is ind GJ- = T r [ ( - 1 )Jo~[G~+(G~)*]2qL"o--c/24] = T r [ ( - 1 )Jo#Lo-c/24qLO-c/24] ,
(4)
where the trace runs over the R sector, Lo is the hamiltonian for the left-moving sector, and q=exp(2niz), Im r > 0. As only states with Lo = 2~c contribute, (4) is a function of q only. A priori, the value of the trace is not guaranteed to be integral as Jo need not take integer eigenvalues on all states. This, however, is only a sufficient, 472
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but not necessary condition: for example, in orbifold theories, integrality o f (4) need only occur after summing over all twisted sectors. I f ( - 1 )so = + 1, then one can indeed identify the right-hand side o f (4) with the usual notion o f index: ind Gd" = d i m kerG~ - d i m ker(Gff )*. We like to point out that one can also consider the one-parameter family o f indices ind G : = T r [ ( - 1 )so #Lo(a)--c/24qff.o--c/24], with L o ( a ) = L o + a J o + ~ca 2 and where the trace runs over states created from a shifted Dirac sea 112~) ("spectral flow" [8 ] ). This allows in particular to define also index theorems in the NS sector ( a = ½). Naively, as a moves over one period (e.g., from zero to one), one expects that ind Ga+ returns to its original value. It turns out, however, that this is in general not true; rather, there occurs a global anomaly if the first Chern class c~ (as defined below) does not vanish. Similarly, ind G ~/2 = ind Gd- if ½c~ = 0; this is relevant in supersymmetric theories. In the following, we will confine ourselves, however, to a = 0 and compute (4) for several classes o f string theories.
3. N = 2 superconformal symmetry in specific string theories We need to explain some details o f the N = 2 superconformal symmetry arising in various orbifold and lattice string theories. Consider first an untwisted theory, that is, a ten-dimensional heterotic string compactified on some 2d-dimensional torus ~r. The local right-moving N = 1 symmetry is generated by TF(g) = -- ½~ut'OXu(g), / t = l , ..., 10. Adopting a complex basis, this can trivially be rewritten as TF(f)=--½q/aoxa(g) ½~UaOXa(Z) = G + ( f ) + G - (~), and together with J ( ~ ) = : ~uhua: (~) we immediately can extend the N = l algebra to an N = 2 algebra with c = 15 ~2. The zero mode o f J can thus be identified with the fermion number operator F. Consider now ZN twists g (gU= 1 ) acting on some o f the compactified right-moving coordinates, and choose a complex basis that diagonalizes these twists:
g: X a ( g ) ~ e x p ( 2 ~ r i k a / N ) X~(2) , g: ~ N ( ~ ) ~ e x p ( 2 r t i k a / N ) g: Xa(~) ~ exp ( - 2 ~ika/N) X a ( ~ ) ,
~ffa(~) ,
g: ~u~(~) --, exp ( - 2rdka/N) ~,a(f).
(5)
Obviously, TF and J defined above are preserved under these twists, and the resulting theory based on the (not necessarily symmetric) orbifold (9 ___J / Z N is N = 2 invariant. Thus, merely defining complex coordinates allows to extend the symmetry. This is quite general: it applies to any theory with complex fermions. In particular, one can easily show that any conformal field theory based on a covariant lattice A (consisting o f the m o m e n t a o f the bosonized fermions, superghost 0 and the "compactifled" bosons) has N = 2 supersymmetry: one can always cut TF into two pieces with charges + 1 under the zero mode o f
J(g) = i v . O H ( g ) ,
(6)
by choosing an appropriate vector v satisfying v2= ] c = 5 (note that v does not lie on the lattice). The bosons denoted by H can be defined to be a subset o f compact bosons which describes the N S R fermions ~3. For our purposes, we can put orbifold and covariant lattice theories on an equal footing, by bosonizing the fermions ~,a and ~u~ above. In the fermionic sector, the orbifold twist operation (5) is then equivalent to a certain shift operation on the lattice describing the N S R fermions. Likewise, any (odd self-dual) covariant ~2 One can combine this algebra with the N=2 algebra of the superconformal ghost system [ 13] to obtain an N=2 symmetry acting on the complete right-moving sector with c = 0. ~3 In general, given a lattice, there is a priori no distinction between "compactified" bosons and fermions. There can exist several (inequivalent) choices for v, and accordingly several ways of realizing N= 2 superconformal symmetry. In theories based on D:lattices, i.e., in theories with only periodic and antiperiodic fermionic boundary conditions, v is a five-dimensional vector with components _+I. 473
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lattice A can be obtained from any other one, and in particular from a lattice A ~- describing a torus-compactified supersymmetric heterotic string, by shift operations and appropriate projections [ 14,15 ]. More precisely, the untwisted sector corresponds to the lattice Ao~- of vectors of A 'r with integer inner product with the shift vector 6, while the twisted sectors correspond to the conjugacy classes [ m ] = [Ao~-+ m6 ], m = 1, ..., N - 1 (with N&¢A~, possibly having non-zero components also in the spacetime and 0-ghost lattice sectors). The new covariant lattice A is a sublattice of the dual of A ( with the above conjugacy classes. Define a vector V= (v, + 1 ), where + 1 denotes the ~ g h o s t charge. The vector ( - V) is then associated with a vertex operator exp ( - iv. H ) exp ( - iO) in the canonical ghost picture. It is the lattice analogue (p = 5 ) of the constant antiholomorphic p-form ~a,...a~~g' ...~g~, which exists on any 2p-dimensional K~ihler manifold with vanishing first Chern class [ 16,6,7]. The vector, V, belongs to the lattice A of the new theory precisely if it has integral (lorentzian) inner product with all other lattice vectors; as VzA ~-, this is the case if (V.g) zZ. We take this ~4 as the condition for vanishing of a generalized first Chern class, defined also for theories not admitting a compactification interpretation. Indeed, for orbifold twists of the form (5), it becomes the same as the condition for vanishing torsion first Chern class in orbifold theories, Y~k,= 0 modN, and we can write cl to') =~,ka.fc=N(V.5).fc,
(7)
where ~ is the generator of a torsion subgroup of H2(0, Z) with N'2=O [17,18] ~s. This generalizes c~ fre~) = ( i / 2 n ) Tr R = ( l/2~)JabRba~H2(J[, R) of complex manifolds J L Thus, Iz and 6 play the role of the complex structure Ja b and curvature two-form Rab, respectively. More precisely, the zero m o d e of the current J = iv. OH generates precisely the same U (1) rotations ~g/a= i g a, 8~ua= -i~u a as the complex structure Ja b. Note that the definition ofc~ above involves only the shift vector components in thefermionic (possibly also ~-ghost) sector of the theory, which is defined by It. The vector k"plays an interesting role for spacetime supersymmetry: ( - / F) is the lattice representation of (a particular component o f ) the supersymmetry charge. The foregoing integrality condition, however, is not sufficient for the presence of this vector on the lattice A; rather, only if there exists some choice of complex structure v such that ( V . & ) = 0 m o d 2, spacetime supersymmetry arises (this is related to the "charge integrality condition" [ 3 ] ). That is, invariance of the constant (0, p) form under a discrete holonomy group does not imply the invariance of the supercharge. One can express this by t_..(to~) _ ½ N ( V . 6 ) . 2 = O
2~..i
,
(8)
which, in integral cohomology, is a stronger statement than c t t°r) = 0. This generalizes the familiar condition for supersymmetry in manifold-compactified theories, c Ifree) = 0. The integral class ½c It°r) is well-defined if we assume the vanishing of the second Stiefel-Whitney class w2eH2( (;, Z2), which is the m o d 2 reduction of c~. Indeed, w2 = 0 iff N( IT"&) = 0 m o d 2; this is one of the level-matching conditions [ 19 ].
4. Computation of Tr(- 1) F Following the discussion of ref. [ 20 ], we recall some facts about spinors and holomorphic p-forms on KShler manifolds. On a 2d-dimensional K~ihler manifold J / o n e can always find a complex basis for the g a m m a matrices such that {7 a, 7 b} =jab, d, b = 1, ..., d. This then allows one to interpret these as creation and annihilation operators. Define now a (spinorial) Fock v a c u u m by 7a112) = 0 Va. Then the other spinor states are obtained by acting with products 7 a'Ta2...7 a~ on it. A general spinor field on ./4 can therefore be expanded as ,4 More generally, the existence of an operator with (h, q) = (~c, ~c ) in the matter sector. ,5 We implicitly assume an appropriate equivariant definition of (9. 474
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~(Xa, Xa)= t~ ~")(Xa, Xa)= ~) ~a,...a,,(Xa, Xa)yaL..Ya"[12). p
p=O
5 May 1988
(9)
If c~ = 0, the ~a,..a, are equivalent to antiholomorphic p-forms. Thus there is a one-to-one correspondence between spinors and (0, p ) - f o r m s on ~t/. Since the application o f ~a switches chirality, the eigenvalue of y. acting on ~P) is ( - I )P. Moreover, the action o f the Dirac operator on 7'is equivalent to the action o f the Dolbeault operator on antiholomorphic p-forms, & (0, p ) ~ (0, p + 1 ). the Dirac index on d¢ is therefore the same as the Dolbeault index ind O, which counts the difference between the numbers o f even and odd antiholomorphic harmonic forms, respectively: d
ind,---Try.=
~
p=O
(-1)PdimH~°'P)(~, R)----ind 0-.
(10)
If, however, c~~ O, ~al..a,, (xa, :ca) is not equivalent to a (0, p ) - f o r m since it has wrong transformation behavior under the U ( 1 ) part o f the h o l o n o m y group, and the foregoing identifications, in particular the relation o f ( - 1 )P with chirality, do not hold. Analogously, in orbifold string theory, as soon as we can define a complex basis (i.e., an N = 2 supersymmetry), we can introduce creation and annihilation operators using {C/e_n, ~Ub } = ~ab. In the standard (untwisted) Fock vacuum, we can take ~ug, ~ue__,and ~uLn (n > 0) as creation and ~'8, ~g and ~u~ as annihilation operators. A spinor field in the untwisted sector can be expanded as in (9), with additional higher mode contributions. In the twisted sectors, creation and annihilation operators have to be appropriately redefined. It is easier to employ the bosonic formulation here; a generic spinor on (9 in the shifted sector [ m ] (m = 0, ..., N - 1 ) can then be expanded as (modulo derivatives) 2
--tml ) = ~ Cm,.~: e x p [ i ( 2 + m J ) - H ] : ( 0 ) I / 2 ) . 2
(11)
Here, the 2 are vectors with arbitrary integer components, and the exponentiated shift vector ~ represents the twist field in the fermionic sector o f the theory. We now compute ind G~- <~m) in the 2d-dimensional internal sector [we denote internal and spacetime sectors by (int) and (st), respectively; 6 has non-zero components only in (int) ]. The trace in (4) runs only over those states on which G ÷ (int) ( Z ) has an integral m o d e expansion. Since G + (z) = G + t~t) (z) + G + (int) ( Z ) is well-defined on all states, the states are precisely those on which also G - c~t) (z) has an integral mode expansion. Thus, ind G~ (int) = T r R . . [ ( -- 1 )FOnt) #Lo--c/24qff, O--C/24 ] •
(12)
R ~t) indicates summing only over states whose spacetime part is in the R a m o n d sector. To evaluate the trace, we need to know the action o f ( - 1 )F..o on the states ( 11 ). In principle, there occurs an ambiguity in defining the action o f ( - 1 )eo.,~ on the twisted orbifold ground states (this corresponds to the ambiguity in defining different regularization procedures in the a-model calculation below). For the index o f the N= 2 supercharge, however, the natural choice is to identify F ¢i~t)=j~int), and thus to determine the action o f ( - 1 )F..'} on a ground state by its U ( 1 ) charge ~6. Denoting by F . <~t) the chirality operator on the fermionic excitations o f the internal space, we have, therefore ( - 1 )FOre> I ~"/[m I ) = ( -- l )m(V'~)F.(int) [ ~'/[m] ) •
(13)
Thus, for arbitrary first Chern class, ( - 1 )eo,. ceases to have eigenvalues + 1 and to represent the chirality operator o f the internal sector. This is the p h e n o m e n o n o f charge fractionalization. It follows ~6lfc~ =0 mod 2, i.e., Wz=0, one can also choose ( - 1)r,~,, in such a way that it takes only eigenvalues + 1 on a//states, and thus can be identified with the chirality operator acting on spinors. With such a choice,the calculationwould lead to the index of the Dirac-Ramond operator. 475
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l N-- 1
ind G~-(inct) = ~m~__0 (-1)m
a ~ int)
5 May 1988
,
(14)
where indt,,jG~ ~"') is the index Tr f,.[int) of the Dirac-Ramond operator G~int) = G~-(int) ..~Go-(int) in the shift sector Ira]. For (V. ~ ) = 0 rood 2, i.e., lc~ = 0, the above-mentioned ambiguity disappears and ind G ff
(15)
It is interesting to note that expression (15) is precisely the chiral partition function [ 21 ] of the theory, as /"t.int) =/'~.st) (see, e.g. ref. [20] ); it is algebraically defined even for conformal field theories not admitting a compactification interpretation. Of course, for theories admitting such a interpretation, ind Go is the loop space generalization of the index of the Dirac operator ind I~ on J / [ 22 ]. In the same spirit we can regard ind G~- as an elliptic generalization of the Dolbeault index ind c~, which can be defined for a differential operator on some K~ihler manifold. It is obvious that ( 15 ) is a generalization of (10), which applies if c t f r e e ) = (i/21r) Tr R = 0 . However, (15) holds only for ~~.
,f
SE = -~n d2a gab (X) OXa OXb- vagab (X) [obo + F ba(X) OXc] ~a.
( 16 )
Separating out and rescaling the zero modes, the (in the small Im z limit) leading part reads SE----- ~
'I
(!7)
dZaOxa(oabOd-~ab)Xb--~ab!llao~b,
where X and ~t are now the quantum fluctuations around the classical zero modes x~, x~, q/~, ~,~, and __I a ~ c d is the curvature two-form. The index of the N = 2 ~ab-~Rabea(Xo, X0)~o~o supercharge G~- = f da~ ~u~[ - i D /Dxa + g~a( X)OXb / Oa~ ) is then obtained from
f
f
ind G~ = l mlim . dXdq/exp(-SE)= ~ 0 ,~
det' (06ab) dx~ dx~ d~tg d~t~ det'(08ab) det'(~tSab+~tab) '
(18)
.//
with periodic-periodic boundary conditions on the fermions. The computation of the remaining determinant is similar to the index calculation of the Dirac-Ramond operator [ 22 ], except that one has to regulate carefully using complex coordinates since ~ takes now values in the holonomy group U (d) instead of SO (2d). Let us first consider l°-{det' ~k d e t(0+ - ~ II)'\ ) = ~ ' l ° g ( mI +- - - ~ + n ) m,n
II k
= l i m ~ ~ ' (-1)k+L (mz+n) k+s s o O k= I m,n k
=lim 1 ~ (--1)k+l s~o 2 k=l k vk{1 +exp[irr(k+s)]}Gk+s (z) ,
(19)
where z is the modular parameter of a skewed torus, and where
m = l n=--oo
476
(mz+n)-S+ n= ~l n-S= (-27ri)~ ( ( -F~s)~i)s
n= ~t as_~ (n)q n) •
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Being careful about the contribution of the pole of the (-function for k= 1, (19) becomes i½nvlim s~0
[SGs+l('~)]--
p 2k
p2k
-~G2k(Z)=ircv-- k~. -~G2k('t') k= 1 = I ~
(20)
,
where the G2k(r) are the Eisenstein series (transforming with modular weight 2k for k > 1 ). Therefore, using the identities O~(vl O/vO~ (OIz) =exp [ - Y (v2k/2k)G2k(Z)], O~ (01 T) = 2q3(z) and denoting the skew eigenvalues of (i/2~) ~abby ~%, we get indG~-=
exp(lto,~)Ol(ioga/2nlz)]
dxgdxgd~vgd@,g ~=,I~l .17
I
=q-d/~2 dxg dxg d@'gd@'g a=l~I
(
½('Oa~ nf=il det[1 exp(l°ga) sinh-~-taa)]
__qn exp(i~?/2~z) ] - - 1
.
(21)
.¢t'
Integrating over the fermionic zero modes results in substituting ~ b y products yields ind
G~ =q-all2 f
R =ReddXeA dx d, and rewriting the two
t d ( R ) Ch(q, R) [topform •
(22)
./t'
Here td (R) is the Todd genus, which is related as follows to the Dirac genus J (R) [ 23 ]: td(R)=
a=l
exp(½0)a)
(!o) 2 tma
~
1
sinh(½toa) - e x p ( : c l ) A ( R ) ,
(23)
with c I ~ C t free ). The series Ch (q, R ) = ~ T=o q"Tr 1, l exp (iR/2n) represents the Chern character for every string level. As the Todd genus is precisely the index density of the Dolbeault complex, the level n = 0 part of (22) is q-a/12 times ind d-=F td(R)[topform •
(24)
,h'
It follows that (22) is the string generalization of ind & For ct=0, it becomes equal to the index of the Dirac-Ramond operator ind Go=q-d/~2fA(R) Ch(q, R); this is the elliptic generalization of (10). The factor exP(½Cl ) above arises from the (-function regularization (20) of the determinants, and is in a sense an artefact of this regularization. However, as the same divergence as in (20) arises already in the field theory computation of ind O, we can appeal to the usual field theory reasoning [23 ] to resolve the regulator ambiguity as above. The index of Go has well-defined transformation properties under the modular group, as long as the first Pontryagin class p~ = - ( 1/82r 2 ) Tr R2e H 4(~/, R) vanishes [ 21,22 ] [ this is because Tr R 2 is multiplied by the anomalous modular function G2(z); cf. (20)]. Similarly, since there exists no compensating modular weight one function G, (z) to multiply Cl = 0 / 2 7 0 Tr R with in ( 2 0 ) - ( 2 2 ) , it follows that modular covariance of ind G~- is spoiled ifcl ~ 0. In other words, c tfree) # 0 andp tfre~)~ 0 are obstructions to generalizing the Dolbeault index to loop space. Both obstructions arise due to particular choices of regularization schemes that violate modular invariance: for the Dirac-Ramond operator, one chooses a regularization prescription which manifestly ensures holomorphicity; for the N = 2 supercharge one chooses a prescription which (in addition ) is adapted to complex coordinates. This feature is also reflected in the formula (14) for ind Gd- on orbifolds. For arbitrary first Chern class, the phases in the sum over twisted sectors destroy modular covariance. The phases disappear only for ~t, l~(tor) /') 1 ---~-V, This is similar to the role played by the integral class 2 (tor)= ~pltOr) ~ H 4 ( (9, Z ) for the Dirac-Ramond operator 477
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on orbifolds [ 19,24-26 ]. The relevance of ½cIt°r) instead of c ~tor) is already suggested by (23). More specifically, consider the character valued index associated with a group action g as in (5). According to the fixed point theorem, in the twisted sectors ind J = f td (R) is replaced by the Lefschetz number of the Dolbeault complex [23]:
E a=t exp(--ll0a) LD°'=Tr(gH~° ..... ))--Tr(gH~°'°dd))= f~x¢dEa=, 1--exp(--i0a) 2--2COS(0~) = nx¢O points points
2sin(½Oa)
with 0a= (2n/N)ka. The expression in the brackets is the index "density" for the twisted Dirac operator [ 23,27 ], as it appears in the expression for the Dirac-Ramond index ind Go on orbifolds [28 ]. The phase corresponds to that in (14), and vanishes if ~ ka = 0 mod 2N, i.e., ½c It°r) = 0. Acknowledgement We would like to thank C. Gomez, A. Lfitken and A. Schellekens for helpful discussions, and W.L. is grateful to the Theory Group of CERN for its hospitality. [ 1 ] P. Candelas, G. Horowitz, A. Strominger and E. Witten, Nucl. Phys. B 258 (1985) 46; L. Dixon, J. Harvey, C. Vafa and E. Witten, Nucl. Phys. B 261 ( 1985 ) 620; K. Narain, Phys. Lett. B 169 (1986) 41; M. Mueller and E. Witten, Phys. Lett. B 182 (1986) 28; L. Ib~ifiez,H. Nilles and F. Quevedo, Phys. Lett. B 187 (1987) 25. [2] H. Kawai, D. Lewellen and S. Tye, Nucl. Phys. B 288 (1987) 1; W. Lerche, D. Liist and A. Schellekens, Nucl. Phys. B 287 ( 1987 ) 477; I. Antoniadis, C. Bachas and C. Kounnas, Nucl. Phys. B 289 ( 1987 ) 87; K. Narain, M. Sarmadi and C. Vafa, Nucl. Phys. B 288 ( 1987 ) 551. [ 3 ] D. Gepner, Princeton preprint PUPT- 1056 (1987). [4] W. Lerche and A.N. Schellekens, preprint CERN-TH.4925/87. [5] C. Hull and E. Witten, Phys. Lett. B 160 (1985) 398. [6] W. Boucher, D. Friedan and A. Kent, Phys. Lett. B 172 (1986) 316. [7] A. Sen, Nucl. Phys. B 278 (1986) 289. [ 8 ] D. Friedan, A. Kent, S. Shenker and E. Winen, to be published. [9] J. Atick, L. Dixon and A. Sen, Nucl. Phys. B 292 (1987) 109. [ 10 ] T. Banks, L. Dixon, D. Friedan and E. Martinec, preprint SLAC-PUB-4377 ( 1987 ). [ 11 ] M. Ademollo et al., Phys. Lett. B 62 (1976) 105. [ 12] Z. Qiu, Princeton preprint (1986). [ 13 ] D. Friedan, E. Martinec and S. Shenker, Nucl. Phys. B 271 (1986) 33. [ 14 ] D. Ltist and S. Theisen, preprint MPI-PAE-PTh83/87 ( 1987 ). [ 15 ] A. Schellekens and N. Warner, preprint CERN-TH.4916/87. [ 16 ] A. Strominger and E. Witten, Commun. Math. Phys. 101 ( 1985 ) 341. [ 17] X. Wen and E. Witten, Nucl. Phys. B 261 (1985) 651. [ 18] D. Freed, Commun. Math. Phys. 107 (1986) 483. [ 19 ] D. Freed and C. Vafa, Commun. Math. Phys. 1 ! 0 ( 1987 ) 349. [20] M. Green, J. Schwarz and E. Witten, Superstring theory (Cambridge U.P., Cambridge, 1986). [21 ] A. Schellekens and N. Warner, Phys. Lett. B 177 (1986) 317; B 181 (1986) 339; Nucl. Phys. B 287 (1987) 87. [22] K. Pilch, A. Schellekens and N. Warner, Nucl. Phys. B 287 (1987) 362; E. Witten, Commun. Math. Phys. 109 (1987) 525; O. Alvarez, T. Killingback, M. Mangano and P. Windey, Berkeley preprint UCB-PTH-87/12 ( 1987 ). [ 23] See e.g.T. Eguchi, P. Gilkey and A. Hanson, Phys. Rep. 66 (1980) 213; L. Alvarez-Gaumr, Commun. Math. Phys. 90 (1983) 161. [24] E. Witten, Global anomalies in string theory, in: Anomalies, geometry and topology, eds. W. Bardeen and A. White (World Scientific, Singapore, 1985); Topological tools in ten dimensional physics, in: Unified string theories, eds. M. Green and D. Gross ( World Scientific, Singapore, 1986 ). [25 ] T. Killingback, Princeton preprint PUPT-1035 ( 1987 ). [26] K. Pilch and N. Warner, MIT preprint CTP# 1497 (1987). [27] M. Goodman, Commun. Math. Phys. 107 (1986) 391. [28] K. Li, MIT preprint CTP# 1460 (1987).
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INDEX T H E O R E M S IN N - - 2 S U P E R C O N F O R M A L T H E O R I E S ~" W. L E R C H E t.2 and N.P. W A R N E R 3 CERN, CH-1211 Geneva 23, Switzerland Received 10 February 1988
Having emphasized the role of N=2 superconformal invariance in various string theories, we compute the "index" Tr( - 1)F for the N=2 supercharge on orbifold and manifold backgrounds. Resolving a quantization ambiguity in the appropriate way, we show that it generalizes the Dolbeault index ind 0". It has however, in general, no well-defined modular properties. The lack of modular invariance can be expressed in terms of the integral characteristic class ~cj, that is also related to the condition for spacetime supersymmetry.
1. Introduction There exists now a variety o f constructions o f lower-dimensional heterotic string theories. One type o f construction deals with compactification o f higher dimensions [ 1 ], another type works directly in lower dimensions [2,3 ]. The overlap between these two approaches appears to be essentially the symmetric orbifold construction. Many properties o f a compactified theory can be interpreted in terms o f topological properties o f the compact space. It is certainly important to investigate whether some o f these topological concepts can be translated or generalized to more general conformal field theories that cannot even be regarded as limiting cases o f manifold compactifications. We will consider here mainly a certain class o f (abelian) orbifolds and the theories described by the covariant lattice approach #t. The overlap between the latter theories and orbifolds is given by at least the set o f asymmetric orbifolds with inner automorphism twists and with rank 22 gauge groups. A particular property of all o f these theories is a global N = 2 right-moving superconformal symmetry, extending the local N = 1 world-sheet supersymmetry. In fact, most o f all known theories exhibit this symmetry. For example, it was shown [ 5-10 ] that N = 1 spacetime supersymmetry implies N = 2 sheet supersymmetry. We want to stress, however, that in the above-mentioned class o f models the occurrence o f N = 2 supersymmetry is independent o f any spacetime supersymmetry. It is thus interesting to study this N = 2 superconformal invariance in more detail. In particular, we are interested in the index Tr( - 1 ) r that one can define for the two supercharges. We will first compute this quantity for asymmetric orbifold and covariant lattice theories. In general such theories cannot be interpreted as compactifications o f higher-dimensional theories on manifolds, and the supercharges have no representation in terms o f differential operators. However, when such an interpretation is possible, we show that Tr( - l )F generalizes the Dolbeault index ind tg. We also find that even though T r ( , 1 )F can have this interpretation as an index, it is not a well-defined function in loop space in that it has no good modular properties unless a certain cohomological Work supported in part by NSF Grant # 84-07109. Supported by the Max-Planck-Gesellschaft, D-8000 Munich, Fed. Rep. Germany. 2 Address after 1 September 1988: California Institute of Technology, Pasadena, CA 91125, USA. 3 On leave of absence from: Department of Mathematics, Massachusetts Institute of Technology,Cambridge, MA 02139, USA. #l For a review see ref. [4]. 0 3 7 0 - 2 6 9 3 / 8 8 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
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condition is met. This condition is equivalent to the presence of spacetime supersymmetry, and if it is satisfied, Tr ( - 1 )F becomes identical to Tr F., the index of the Dirac-Ramond operator.
2. The N--- 2 superconformal algebra In conformal field theory, the N = 2 superconformal algebra [ 10,11 ] can be represented in the operator product form
T ( z ) . T ( w ) ~ (z_w)~ ½c + (z_w)~Z 2T(w) + OT(w) - z-w ' T(z)'J(w)~
J ( w ) + aJ(w) , (z-w) 2 z-w
G+(z)'G-(w)
~c (z-w) 3
J(z)'J(w)
+ J(w) ~( z - w +)
T(z).G+-(w)~ ~G+-(w) OG+-(w) ~( z - w ) ÷ - - z - w ' ~c (z-w) ~'
J ( z ) . C +-( w ) ~ + 6+-(w)
Z--W '
T(w)+½OJ(w) z-w , G+-(z)'G+-(w)~O.
(1)
Here, T(z) represents the stress-energy tensor, G -+(z) the supercharges and J(z) an U ( 1 ) Kac-Moody current; c is the central charge. All these operator products have well-known representations in terms of (anti-)commutation relations of their mode components. These components are defined by T(z)=Ez-Z-"L,,, G +-( Z ) ~- ~ Z -3/2-nq:aG++_ a and J(z) = Y~z- l-nJn, where n~Z and a is arbitrary. One such commutation relation is
{Gn++a, Gm_a } =Ln+m + ½( n - m + 2a)Jn+m + [-~C(n+ a) 2 - -~4C]~n+m •
(2)
This, together with the other corresponding (anti-)commutation relations, defines an N = 2 superconformal algebra, Aa, for any given value of a [ 8,12,3 ]. As the algebras Aa and Aa+ ~are isomorphic [ 8,12 ], the parameter, a, is restricted to the range 0 ~ annihilated by L,, G + , J, for n > 0, and have definite eigenvalues under a maximal set of commuting generators: Lo I h, q> = h Ih, q>, Jo Ih, q> = q l h, q>. The complete highest-weight representation is then obtained by applying the corresponding creation operators (n < 0). The NS sector is defined by a = ½, and the R sector by a = 0. In the R sector, the representation space splits further into two sectors P + ~ P - , characterized by the action of the two supercharges Gfflh, q ~ ½ > + = 0 ,
Ih, qT½>_+~P -+.
(3)
Furthermore, G~-: P + o P - , Gff = (G~-)*: P---*P+ and {( - 1 )J0, Gff }= [Lo, Gff ] =0. It follows that except for states annihilated by Gd- or Gff, all other states come in pairs with the same mass and with U( 1 ) charges differing by one unit. The states which are not paired are massless, i.e., have h = ~4c. Thus one is tempted to define as "index" for an arbitrary N = 2 superconformal theory ind Gd- = T r ( - 1 )Jo. This, however, can only represent a meaningful quantity if we can regulate the trace such that the regulator commutes with Jo and the mass operator. For closed string theories with a right-moving (2, 0) supersymmetry, the appropriate object to consider is ind GJ- = T r [ ( - 1 )Jo~[G~+(G~)*]2qL"o--c/24] = T r [ ( - 1 )Jo#Lo-c/24qLO-c/24] ,
(4)
where the trace runs over the R sector, Lo is the hamiltonian for the left-moving sector, and q=exp(2niz), Im r > 0. As only states with Lo = 2~c contribute, (4) is a function of q only. A priori, the value of the trace is not guaranteed to be integral as Jo need not take integer eigenvalues on all states. This, however, is only a sufficient, 472
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but not necessary condition: for example, in orbifold theories, integrality o f (4) need only occur after summing over all twisted sectors. I f ( - 1 )so = + 1, then one can indeed identify the right-hand side o f (4) with the usual notion o f index: ind Gd" = d i m kerG~ - d i m ker(Gff )*. We like to point out that one can also consider the one-parameter family o f indices ind G : = T r [ ( - 1 )so #Lo(a)--c/24qff.o--c/24], with L o ( a ) = L o + a J o + ~ca 2 and where the trace runs over states created from a shifted Dirac sea 112~) ("spectral flow" [8 ] ). This allows in particular to define also index theorems in the NS sector ( a = ½). Naively, as a moves over one period (e.g., from zero to one), one expects that ind Ga+ returns to its original value. It turns out, however, that this is in general not true; rather, there occurs a global anomaly if the first Chern class c~ (as defined below) does not vanish. Similarly, ind G ~/2 = ind Gd- if ½c~ = 0; this is relevant in supersymmetric theories. In the following, we will confine ourselves, however, to a = 0 and compute (4) for several classes o f string theories.
3. N = 2 superconformal symmetry in specific string theories We need to explain some details o f the N = 2 superconformal symmetry arising in various orbifold and lattice string theories. Consider first an untwisted theory, that is, a ten-dimensional heterotic string compactified on some 2d-dimensional torus ~r. The local right-moving N = 1 symmetry is generated by TF(g) = -- ½~ut'OXu(g), / t = l , ..., 10. Adopting a complex basis, this can trivially be rewritten as TF(f)=--½q/aoxa(g) ½~UaOXa(Z) = G + ( f ) + G - (~), and together with J ( ~ ) = : ~uhua: (~) we immediately can extend the N = l algebra to an N = 2 algebra with c = 15 ~2. The zero mode o f J can thus be identified with the fermion number operator F. Consider now ZN twists g (gU= 1 ) acting on some o f the compactified right-moving coordinates, and choose a complex basis that diagonalizes these twists:
g: X a ( g ) ~ e x p ( 2 ~ r i k a / N ) X~(2) , g: ~ N ( ~ ) ~ e x p ( 2 r t i k a / N ) g: Xa(~) ~ exp ( - 2 ~ika/N) X a ( ~ ) ,
~ffa(~) ,
g: ~u~(~) --, exp ( - 2rdka/N) ~,a(f).
(5)
Obviously, TF and J defined above are preserved under these twists, and the resulting theory based on the (not necessarily symmetric) orbifold (9 ___J / Z N is N = 2 invariant. Thus, merely defining complex coordinates allows to extend the symmetry. This is quite general: it applies to any theory with complex fermions. In particular, one can easily show that any conformal field theory based on a covariant lattice A (consisting o f the m o m e n t a o f the bosonized fermions, superghost 0 and the "compactifled" bosons) has N = 2 supersymmetry: one can always cut TF into two pieces with charges + 1 under the zero mode o f
J(g) = i v . O H ( g ) ,
(6)
by choosing an appropriate vector v satisfying v2= ] c = 5 (note that v does not lie on the lattice). The bosons denoted by H can be defined to be a subset o f compact bosons which describes the N S R fermions ~3. For our purposes, we can put orbifold and covariant lattice theories on an equal footing, by bosonizing the fermions ~,a and ~u~ above. In the fermionic sector, the orbifold twist operation (5) is then equivalent to a certain shift operation on the lattice describing the N S R fermions. Likewise, any (odd self-dual) covariant ~2 One can combine this algebra with the N=2 algebra of the superconformal ghost system [ 13] to obtain an N=2 symmetry acting on the complete right-moving sector with c = 0. ~3 In general, given a lattice, there is a priori no distinction between "compactified" bosons and fermions. There can exist several (inequivalent) choices for v, and accordingly several ways of realizing N= 2 superconformal symmetry. In theories based on D:lattices, i.e., in theories with only periodic and antiperiodic fermionic boundary conditions, v is a five-dimensional vector with components _+I. 473
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lattice A can be obtained from any other one, and in particular from a lattice A ~- describing a torus-compactified supersymmetric heterotic string, by shift operations and appropriate projections [ 14,15 ]. More precisely, the untwisted sector corresponds to the lattice Ao~- of vectors of A 'r with integer inner product with the shift vector 6, while the twisted sectors correspond to the conjugacy classes [ m ] = [Ao~-+ m6 ], m = 1, ..., N - 1 (with N&¢A~, possibly having non-zero components also in the spacetime and 0-ghost lattice sectors). The new covariant lattice A is a sublattice of the dual of A ( with the above conjugacy classes. Define a vector V= (v, + 1 ), where + 1 denotes the ~ g h o s t charge. The vector ( - V) is then associated with a vertex operator exp ( - iv. H ) exp ( - iO) in the canonical ghost picture. It is the lattice analogue (p = 5 ) of the constant antiholomorphic p-form ~a,...a~~g' ...~g~, which exists on any 2p-dimensional K~ihler manifold with vanishing first Chern class [ 16,6,7]. The vector, V, belongs to the lattice A of the new theory precisely if it has integral (lorentzian) inner product with all other lattice vectors; as VzA ~-, this is the case if (V.g) zZ. We take this ~4 as the condition for vanishing of a generalized first Chern class, defined also for theories not admitting a compactification interpretation. Indeed, for orbifold twists of the form (5), it becomes the same as the condition for vanishing torsion first Chern class in orbifold theories, Y~k,= 0 modN, and we can write cl to') =~,ka.fc=N(V.5).fc,
(7)
where ~ is the generator of a torsion subgroup of H2(0, Z) with N'2=O [17,18] ~s. This generalizes c~ fre~) = ( i / 2 n ) Tr R = ( l/2~)JabRba~H2(J[, R) of complex manifolds J L Thus, Iz and 6 play the role of the complex structure Ja b and curvature two-form Rab, respectively. More precisely, the zero m o d e of the current J = iv. OH generates precisely the same U (1) rotations ~g/a= i g a, 8~ua= -i~u a as the complex structure Ja b. Note that the definition ofc~ above involves only the shift vector components in thefermionic (possibly also ~-ghost) sector of the theory, which is defined by It. The vector k"plays an interesting role for spacetime supersymmetry: ( - / F) is the lattice representation of (a particular component o f ) the supersymmetry charge. The foregoing integrality condition, however, is not sufficient for the presence of this vector on the lattice A; rather, only if there exists some choice of complex structure v such that ( V . & ) = 0 m o d 2, spacetime supersymmetry arises (this is related to the "charge integrality condition" [ 3 ] ). That is, invariance of the constant (0, p) form under a discrete holonomy group does not imply the invariance of the supercharge. One can express this by t_..(to~) _ ½ N ( V . 6 ) . 2 = O
2~..i
,
(8)
which, in integral cohomology, is a stronger statement than c t t°r) = 0. This generalizes the familiar condition for supersymmetry in manifold-compactified theories, c Ifree) = 0. The integral class ½c It°r) is well-defined if we assume the vanishing of the second Stiefel-Whitney class w2eH2( (;, Z2), which is the m o d 2 reduction of c~. Indeed, w2 = 0 iff N( IT"&) = 0 m o d 2; this is one of the level-matching conditions [ 19 ].
4. Computation of Tr(- 1) F Following the discussion of ref. [ 20 ], we recall some facts about spinors and holomorphic p-forms on KShler manifolds. On a 2d-dimensional K~ihler manifold J / o n e can always find a complex basis for the g a m m a matrices such that {7 a, 7 b} =jab, d, b = 1, ..., d. This then allows one to interpret these as creation and annihilation operators. Define now a (spinorial) Fock v a c u u m by 7a112) = 0 Va. Then the other spinor states are obtained by acting with products 7 a'Ta2...7 a~ on it. A general spinor field on ./4 can therefore be expanded as ,4 More generally, the existence of an operator with (h, q) = (~c, ~c ) in the matter sector. ,5 We implicitly assume an appropriate equivariant definition of (9. 474
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~(Xa, Xa)= t~ ~")(Xa, Xa)= ~) ~a,...a,,(Xa, Xa)yaL..Ya"[12). p
p=O
5 May 1988
(9)
If c~ = 0, the ~a,..a, are equivalent to antiholomorphic p-forms. Thus there is a one-to-one correspondence between spinors and (0, p ) - f o r m s on ~t/. Since the application o f ~a switches chirality, the eigenvalue of y. acting on ~P) is ( - I )P. Moreover, the action o f the Dirac operator on 7'is equivalent to the action o f the Dolbeault operator on antiholomorphic p-forms, & (0, p ) ~ (0, p + 1 ). the Dirac index on d¢ is therefore the same as the Dolbeault index ind O, which counts the difference between the numbers o f even and odd antiholomorphic harmonic forms, respectively: d
ind,---Try.=
~
p=O
(-1)PdimH~°'P)(~, R)----ind 0-.
(10)
If, however, c~~ O, ~al..a,, (xa, :ca) is not equivalent to a (0, p ) - f o r m since it has wrong transformation behavior under the U ( 1 ) part o f the h o l o n o m y group, and the foregoing identifications, in particular the relation o f ( - 1 )P with chirality, do not hold. Analogously, in orbifold string theory, as soon as we can define a complex basis (i.e., an N = 2 supersymmetry), we can introduce creation and annihilation operators using {C/e_n, ~Ub } = ~ab. In the standard (untwisted) Fock vacuum, we can take ~ug, ~ue__,and ~uLn (n > 0) as creation and ~'8, ~g and ~u~ as annihilation operators. A spinor field in the untwisted sector can be expanded as in (9), with additional higher mode contributions. In the twisted sectors, creation and annihilation operators have to be appropriately redefined. It is easier to employ the bosonic formulation here; a generic spinor on (9 in the shifted sector [ m ] (m = 0, ..., N - 1 ) can then be expanded as (modulo derivatives) 2
--tml ) = ~ Cm,.~: e x p [ i ( 2 + m J ) - H ] : ( 0 ) I / 2 ) . 2
(11)
Here, the 2 are vectors with arbitrary integer components, and the exponentiated shift vector ~ represents the twist field in the fermionic sector o f the theory. We now compute ind G~- <~m) in the 2d-dimensional internal sector [we denote internal and spacetime sectors by (int) and (st), respectively; 6 has non-zero components only in (int) ]. The trace in (4) runs only over those states on which G ÷ (int) ( Z ) has an integral m o d e expansion. Since G + (z) = G + t~t) (z) + G + (int) ( Z ) is well-defined on all states, the states are precisely those on which also G - c~t) (z) has an integral mode expansion. Thus, ind G~ (int) = T r R . . [ ( -- 1 )FOnt) #Lo--c/24qff, O--C/24 ] •
(12)
R ~t) indicates summing only over states whose spacetime part is in the R a m o n d sector. To evaluate the trace, we need to know the action o f ( - 1 )F..o on the states ( 11 ). In principle, there occurs an ambiguity in defining the action o f ( - 1 )eo.,~ on the twisted orbifold ground states (this corresponds to the ambiguity in defining different regularization procedures in the a-model calculation below). For the index o f the N= 2 supercharge, however, the natural choice is to identify F ¢i~t)=j~int), and thus to determine the action o f ( - 1 )F..'} on a ground state by its U ( 1 ) charge ~6. Denoting by F . <~t) the chirality operator on the fermionic excitations o f the internal space, we have, therefore ( - 1 )FOre> I ~"/[m I ) = ( -- l )m(V'~)F.(int) [ ~'/[m] ) •
(13)
Thus, for arbitrary first Chern class, ( - 1 )eo,. ceases to have eigenvalues + 1 and to represent the chirality operator o f the internal sector. This is the p h e n o m e n o n o f charge fractionalization. It follows ~6lfc~ =0 mod 2, i.e., Wz=0, one can also choose ( - 1)r,~,, in such a way that it takes only eigenvalues + 1 on a//states, and thus can be identified with the chirality operator acting on spinors. With such a choice,the calculationwould lead to the index of the Dirac-Ramond operator. 475
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l N-- 1
ind G~-(inct) = ~m~__0 (-1)m
a ~ int)
5 May 1988
,
(14)
where indt,,jG~ ~"') is the index Tr f,.[int) of the Dirac-Ramond operator G~int) = G~-(int) ..~Go-(int) in the shift sector Ira]. For (V. ~ ) = 0 rood 2, i.e., lc~ = 0, the above-mentioned ambiguity disappears and ind G ff
(15)
It is interesting to note that expression (15) is precisely the chiral partition function [ 21 ] of the theory, as /"t.int) =/'~.st) (see, e.g. ref. [20] ); it is algebraically defined even for conformal field theories not admitting a compactification interpretation. Of course, for theories admitting such a interpretation, ind Go is the loop space generalization of the index of the Dirac operator ind I~ on J / [ 22 ]. In the same spirit we can regard ind G~- as an elliptic generalization of the Dolbeault index ind c~, which can be defined for a differential operator on some K~ihler manifold. It is obvious that ( 15 ) is a generalization of (10), which applies if c t f r e e ) = (i/21r) Tr R = 0 . However, (15) holds only for ~~.
,f
SE = -~n d2a gab (X) OXa OXb- vagab (X) [obo + F ba(X) OXc] ~a.
( 16 )
Separating out and rescaling the zero modes, the (in the small Im z limit) leading part reads SE----- ~
'I
(!7)
dZaOxa(oabOd-~ab)Xb--~ab!llao~b,
where X and ~t are now the quantum fluctuations around the classical zero modes x~, x~, q/~, ~,~, and __I a ~ c d is the curvature two-form. The index of the N = 2 ~ab-~Rabea(Xo, X0)~o~o supercharge G~- = f da~ ~u~[ - i D /Dxa + g~a( X)OXb / Oa~ ) is then obtained from
f
f
ind G~ = l mlim . dXdq/exp(-SE)= ~ 0 ,~
det' (06ab) dx~ dx~ d~tg d~t~ det'(08ab) det'(~tSab+~tab) '
(18)
.//
with periodic-periodic boundary conditions on the fermions. The computation of the remaining determinant is similar to the index calculation of the Dirac-Ramond operator [ 22 ], except that one has to regulate carefully using complex coordinates since ~ takes now values in the holonomy group U (d) instead of SO (2d). Let us first consider l°-{det' ~k d e t(0+ - ~ II)'\ ) = ~ ' l ° g ( mI +- - - ~ + n ) m,n
II k
= l i m ~ ~ ' (-1)k+L (mz+n) k+s s o O k= I m,n k
=lim 1 ~ (--1)k+l s~o 2 k=l k vk{1 +exp[irr(k+s)]}Gk+s (z) ,
(19)
where z is the modular parameter of a skewed torus, and where
m = l n=--oo
476
(mz+n)-S+ n= ~l n-S= (-27ri)~ ( ( -F~s)~i)s
n= ~t as_~ (n)q n) •
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Being careful about the contribution of the pole of the (-function for k= 1, (19) becomes i½nvlim s~0
[SGs+l('~)]--
p 2k
p2k
-~G2k(Z)=ircv-- k~. -~G2k('t') k= 1 = I ~
(20)
,
where the G2k(r) are the Eisenstein series (transforming with modular weight 2k for k > 1 ). Therefore, using the identities O~(vl O/vO~ (OIz) =exp [ - Y (v2k/2k)G2k(Z)], O~ (01 T) = 2q3(z) and denoting the skew eigenvalues of (i/2~) ~abby ~%, we get indG~-=
exp(lto,~)Ol(ioga/2nlz)]
dxgdxgd~vgd@,g ~=,I~l .17
I
=q-d/~2 dxg dxg d@'gd@'g a=l~I
(
½('Oa~ nf=il det[1 exp(l°ga) sinh-~-taa)]
__qn exp(i~?/2~z) ] - - 1
.
(21)
.¢t'
Integrating over the fermionic zero modes results in substituting ~ b y products yields ind
G~ =q-all2 f
R =ReddXeA dx d, and rewriting the two
t d ( R ) Ch(q, R) [topform •
(22)
./t'
Here td (R) is the Todd genus, which is related as follows to the Dirac genus J (R) [ 23 ]: td(R)=
a=l
exp(½0)a)
(!o) 2 tma
~
1
sinh(½toa) - e x p ( : c l ) A ( R ) ,
(23)
with c I ~ C t free ). The series Ch (q, R ) = ~ T=o q"Tr 1, l exp (iR/2n) represents the Chern character for every string level. As the Todd genus is precisely the index density of the Dolbeault complex, the level n = 0 part of (22) is q-a/12 times ind d-=F td(R)[topform •
(24)
,h'
It follows that (22) is the string generalization of ind & For ct=0, it becomes equal to the index of the Dirac-Ramond operator ind Go=q-d/~2fA(R) Ch(q, R); this is the elliptic generalization of (10). The factor exP(½Cl ) above arises from the (-function regularization (20) of the determinants, and is in a sense an artefact of this regularization. However, as the same divergence as in (20) arises already in the field theory computation of ind O, we can appeal to the usual field theory reasoning [23 ] to resolve the regulator ambiguity as above. The index of Go has well-defined transformation properties under the modular group, as long as the first Pontryagin class p~ = - ( 1/82r 2 ) Tr R2e H 4(~/, R) vanishes [ 21,22 ] [ this is because Tr R 2 is multiplied by the anomalous modular function G2(z); cf. (20)]. Similarly, since there exists no compensating modular weight one function G, (z) to multiply Cl = 0 / 2 7 0 Tr R with in ( 2 0 ) - ( 2 2 ) , it follows that modular covariance of ind G~- is spoiled ifcl ~ 0. In other words, c tfree) # 0 andp tfre~)~ 0 are obstructions to generalizing the Dolbeault index to loop space. Both obstructions arise due to particular choices of regularization schemes that violate modular invariance: for the Dirac-Ramond operator, one chooses a regularization prescription which manifestly ensures holomorphicity; for the N = 2 supercharge one chooses a prescription which (in addition ) is adapted to complex coordinates. This feature is also reflected in the formula (14) for ind Gd- on orbifolds. For arbitrary first Chern class, the phases in the sum over twisted sectors destroy modular covariance. The phases disappear only for ~t, l~(tor) /') 1 ---~-V, This is similar to the role played by the integral class 2 (tor)= ~pltOr) ~ H 4 ( (9, Z ) for the Dirac-Ramond operator 477
Volume 205, number 4
PHYSICS LETTERS B
5 May 1988
on orbifolds [ 19,24-26 ]. The relevance of ½cIt°r) instead of c ~tor) is already suggested by (23). More specifically, consider the character valued index associated with a group action g as in (5). According to the fixed point theorem, in the twisted sectors ind J = f td (R) is replaced by the Lefschetz number of the Dolbeault complex [23]:
E a=t exp(--ll0a) LD°'=Tr(gH~° ..... ))--Tr(gH~°'°dd))= f~x¢dEa=, 1--exp(--i0a) 2--2COS(0~) = nx¢O points points
2sin(½Oa)
with 0a= (2n/N)ka. The expression in the brackets is the index "density" for the twisted Dirac operator [ 23,27 ], as it appears in the expression for the Dirac-Ramond index ind Go on orbifolds [28 ]. The phase corresponds to that in (14), and vanishes if ~ ka = 0 mod 2N, i.e., ½c It°r) = 0. Acknowledgement We would like to thank C. Gomez, A. Lfitken and A. Schellekens for helpful discussions, and W.L. is grateful to the Theory Group of CERN for its hospitality. [ 1 ] P. Candelas, G. Horowitz, A. Strominger and E. Witten, Nucl. Phys. B 258 (1985) 46; L. Dixon, J. Harvey, C. Vafa and E. Witten, Nucl. Phys. B 261 ( 1985 ) 620; K. Narain, Phys. Lett. B 169 (1986) 41; M. Mueller and E. Witten, Phys. Lett. B 182 (1986) 28; L. Ib~ifiez,H. Nilles and F. Quevedo, Phys. Lett. B 187 (1987) 25. [2] H. Kawai, D. Lewellen and S. Tye, Nucl. Phys. B 288 (1987) 1; W. Lerche, D. Liist and A. Schellekens, Nucl. Phys. B 287 ( 1987 ) 477; I. Antoniadis, C. Bachas and C. Kounnas, Nucl. Phys. B 289 ( 1987 ) 87; K. Narain, M. Sarmadi and C. Vafa, Nucl. Phys. B 288 ( 1987 ) 551. [ 3 ] D. Gepner, Princeton preprint PUPT- 1056 (1987). [4] W. Lerche and A.N. Schellekens, preprint CERN-TH.4925/87. [5] C. Hull and E. Witten, Phys. Lett. B 160 (1985) 398. [6] W. Boucher, D. Friedan and A. Kent, Phys. Lett. B 172 (1986) 316. [7] A. Sen, Nucl. Phys. B 278 (1986) 289. [ 8 ] D. Friedan, A. Kent, S. Shenker and E. Winen, to be published. [9] J. Atick, L. Dixon and A. Sen, Nucl. Phys. B 292 (1987) 109. [ 10 ] T. Banks, L. Dixon, D. Friedan and E. Martinec, preprint SLAC-PUB-4377 ( 1987 ). [ 11 ] M. Ademollo et al., Phys. Lett. B 62 (1976) 105. [ 12] Z. Qiu, Princeton preprint (1986). [ 13 ] D. Friedan, E. Martinec and S. Shenker, Nucl. Phys. B 271 (1986) 33. [ 14 ] D. Ltist and S. Theisen, preprint MPI-PAE-PTh83/87 ( 1987 ). [ 15 ] A. Schellekens and N. Warner, preprint CERN-TH.4916/87. [ 16 ] A. Strominger and E. Witten, Commun. Math. Phys. 101 ( 1985 ) 341. [ 17] X. Wen and E. Witten, Nucl. Phys. B 261 (1985) 651. [ 18] D. Freed, Commun. Math. Phys. 107 (1986) 483. [ 19 ] D. Freed and C. Vafa, Commun. Math. Phys. 1 ! 0 ( 1987 ) 349. [20] M. Green, J. Schwarz and E. Witten, Superstring theory (Cambridge U.P., Cambridge, 1986). [21 ] A. Schellekens and N. Warner, Phys. Lett. B 177 (1986) 317; B 181 (1986) 339; Nucl. Phys. B 287 (1987) 87. [22] K. Pilch, A. Schellekens and N. Warner, Nucl. Phys. B 287 (1987) 362; E. Witten, Commun. Math. Phys. 109 (1987) 525; O. Alvarez, T. Killingback, M. Mangano and P. Windey, Berkeley preprint UCB-PTH-87/12 ( 1987 ). [ 23] See e.g.T. Eguchi, P. Gilkey and A. Hanson, Phys. Rep. 66 (1980) 213; L. Alvarez-Gaumr, Commun. Math. Phys. 90 (1983) 161. [24] E. Witten, Global anomalies in string theory, in: Anomalies, geometry and topology, eds. W. Bardeen and A. White (World Scientific, Singapore, 1985); Topological tools in ten dimensional physics, in: Unified string theories, eds. M. Green and D. Gross ( World Scientific, Singapore, 1986 ). [25 ] T. Killingback, Princeton preprint PUPT-1035 ( 1987 ). [26] K. Pilch and N. Warner, MIT preprint CTP# 1497 (1987). [27] M. Goodman, Commun. Math. Phys. 107 (1986) 391. [28] K. Li, MIT preprint CTP# 1460 (1987).
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