Note on the Landweber-Stong
elliptic genus
by Don Zagier
University
of Maryland,
Max-Planck-Institut
College Park, MD 20742
fHr Mathematik,
5300 Bonn, FRG
In algebraic topology one studies genera, which are ring homomorphisms from the
SO
oriented bordism ring ~,
to a q-algebra R (commutative, with I). To a genus
SO
~: ~, ~ R one associates the following three power series with coefficients in R :
(i)
g(x), an odd power series with leading term
law of
x , the logarithm of the formal group
(this means that the formal ~rouD, whose definition we do not repeat here,
•
.
~
~.
.
~
2n x 2n+I
equals
g-1(g(x) +g(y)) ). It zs given expllcltly by g(x) = ~ n = 0 ~ ( ¢ ~
) ~
and
SO
hence, since the classes of the {p2n generate
~, ®~, determines ~ completely.
(ii)
~
P(u), an even power series with leading term
power series of
~.
This means that if
nential characteristic
oriented manifold
mental class of
(iii)
the stable
characteristic
H*(.;R)-valued
by
~(~)
~.
then the genus of an
M
on the homology
is obtained by evaluating
~(TM)
applying
the complexified
~(TM)
These three power series determine
u
g-1(u)
denotes
Recently,
of Landweber,
that
~4,
~(~)
in
= F(~ -[2])
KO(B~) ® R ,
on a certain KO,-fundamental
power
exponential
for
as one sees by
then the genus of an arbitrary
u/2
sinh u/2
Spin mani-
class of
M.
F ( e U + e -u- 2) ,
g .
Witten and others.
~(M)
vanishes
if
to an even-dimensional
and numerically
of degree
KO(.) ® R - v a l u e d
class of genera has come into prominence
Stong, Ochanine,
projective bundle associated
by
is nilpotent
the inverse power series of
logically by the property
a polynomial
the stable
one another by the formulas
P(u)
a particular
oriented manifold,
~ - [2]
Chern character),
fold is obtained by evaluating
-I
denotes
class on oriented bundles characterized
(this makes sense because
g
funda-
M.
as above
(I)
if
(regarded as a real 2-plane bundle),
This means that if ~
characteristic
expo-
= P(eI(~))
F(y), a power series with leading term I, the K0-theory characteristic
series of
where
denotes
class on oriented bundles characterized
is a complex line bundle
arbitrary
~
I, the Hirzebruch
by the property
i.e., that
through the work
These genera are characterized
M
topo-
is the total space of the complex
complex vector bundle over a closed
that the power series
g'(x) -2
is
217
x
(2)
g(x)
(The equivalence
elliptic
dt
j
0
of these two definitions
integral,
such
~
(a)
elliptic
R .
genera.
[5].)
Landweber
Since this is an
and Stong [4] dis-
genus with values in the power series ring
satisfying:
For
r ~ I the coefficient
of
(This, or rather the weaker statement
for
6, g E
is due to Ochanine
are called elliptic
covered that there is a particular
R = ~[[q]]
for some
~I - 26t 2 + gt 4
yr
in
~
conjectured
that condition
belongs
(a) characterizes
q by
of
y
r
is divisible by
property of the above-mentioned
which we do not formulate here.)
zation (i.e., up to replacing
q2r-1R
to
that the coefficient
r k 2, arises from a certain natural
istic class
F(y)
Based on numerical
q
r+1
KO-character-
computations,
they
the genus in question up to a reparametri-
aq +bq 2 + ...
with
a #0)
and that with a suitable
choice of parameter one has
(b)
F(y)
has coefficients
in
~[[q]] .
By what was said in (iii), this means that the genus takes on values
Spin manifolds.
These facts were proved by D. and G. Chudnovsky
show that with a suitable normalization
The leading term of the coefficient
(d)
The genus takes values
with rational
(We recall basic definitions
of equation
of
in the subring
y
r
about modular
(2) are certain modular forms
theories
to connections
in
F(y)
for
r ~ I is
M~(F0(2)) c ~[[q]]
algebra on
for all elliptic
forms below.)
(of weights
6 and
genera.
elliptic genus has been the object of considerable
cohomology
for all
-q
2r-I
, and
of modular forms on
Fourier coefficients.
known to be the free polynomial
Stong genus is universal
Z[[q]]
one also has
(c)
F0(2)
in
[2], whose formulas
(the "elliptic cohomology"
with index theory,
In particular,
2 and
4);
since
6 and
M~(F0(2))
g
is
g , it follows that the LandweberThis universal,
interest;
of Landweber,
string theory, etc.
modular form-valued
it gives rise to new
Stong, and Ravenel)
and
The purpose of this note is to
give elementary proofs of a variety of formulas for the power series
genus
the
g, P, and
F
associated
to the Landweber-Stong
(and in particular, easy proofs of the properties
(a)-(d)).
These proofs use ideas from the theory of elliptic functions and modular
forms but we will prove everything we need from scratch.
THEOREM.
Let
R=~[[q]].
Xhen the following five formulas define the same power series
P(u) E R[[u]] :
(3)
P(u)
=
I
!
k 0
21k
21k
Gk
t
~
2k-2~k-lj!
uk
218
(5)
P(u)
(6)
P(u)
(7)
F(u)
where
u
g-1(u)
u/2
sinh u/2
=
Gk' ~k
6, g and
.
Gk
(8)
=
2k-1 - 1
2k
Bk
~ q
m>_-1
=
)
n
q
'
~ ( ~ (-1)n/d d k-1 ) qn ,
+ n_->1 dln
-~
3~ 2
~ dk-1
dln
2#d
n=1
I Bk
2k
I
_ g(c
4*+7~4)
ar
~
-u
]
a r (eu +e
-2) r
r=1
r
~k (q)
= -3 G 2*
(-1) n
a E R are defined by
Gk(q )
~k
given by (2) ,
~ r
(1-qn) 2
]
£ I [(1-qneu)(1-qne-u)]
- -
=
g
u/2
[
sinh u/2 " I -
'
6
with
- 3
~
n>1
n
q ,
d
d n
2~d
d3)q n
X
n>-1
dln
2*n/d
(I +q2m)
_q2m)2r
=
~
n>1
~
dln
+
\
2r-I
q
/
-
2~d
(here
B 2 = ~, , B 4 = - ~ i
,
I denotes a sum over
dn
The genus with characteristic power series P(u) satis...
positive divisors of n).
fies properties
are Bernoulli numbers and
(a)-(d).
Each of the five formulas in the theorem describes some aspect of the genus with
characteristic power series
P(u) : (3) and (4) describe the genus in cohomolo~y and
make the modularity property (d) clear (since
Gk* and ~k
are the Fourier expansions
of well-known Eisenstein series, as recalled below), (5) shows that the genus is
elliptic, and (6) and (7) describe the genus in K-theory and (both) make the properties
(a)-(c) evident.
(To deduce (a) and (c) from (6) one has to split off the terms
and n = 2 from the infinite product.)
with a different proof.
n = I
Formula (6) was given by the Chudnovskys, but
It has been generalized by Witten [6] to get other genera
whose coefficients are modular forms, and in this form interpreted by him, using ideas
from quantum field theory, as the equivariant index formula (Atiyah-Bott-Singer fixed
point theorem) for a Dirac operator on the free loop space of a manifold.
We shall
return to these other genera at the end of the note.
Proof of the theorem.
~(u+2~i)
(9)
Consider meromorphic functions
= -~(u) ,
has poles only for
~(u+4~iT)
uCL,
= ~(u) ,
~(u)
~:C + ¢
satisfying
~(-u)
= - ~(u) ,
= ~ + O(I)
u
as
u÷O
,
219
where
r is in the complex upper half-plane and
L denotes the lattice
Z'4~i~ +7z'2~i.
Clearly there can be at most one such function, since the difference of any two would
be holomorphic and doubly periodic, hence constant, hence zero because odd.
give different constructions showing that
~ exists and equals
2ziT
by any of the five equations (3)-(7), with q = e
First, define
(I0)
~(u)
=
~P(u)
for
We will
P(u)
given
, hy the rapidly convergent series
~
m6Z
I
2wimT)
2 slnh(~-+
"
u
=
~
m62Z
I
u/2
qm e
- q
-m
Combining the terms
The properties (9) are immediately checked.
e
-u/2
m and
-m, we find
co
I
@(u)
or, setting
-
(eU/2
e_U/2)
eu/2 - e -u/2
~
qm(1 +q2m)
m =I
(l-q2meU) (]-q2me-U)
y = eu +e -u -2 ,
u/2
qm(1+q )y
2m ~ )
sinh u/2 ( I - m~= I ~
_2--~Tq~q
u~(u)
Expanding the geometric series in y , we find the function
P(u)
defined by (7), with
a
given by the first formula in (8). The second formula in (8) follows from the first
r
by applying the binomial theorem; either one makes properties (a)-(c) evident.
Next, define
~(u)
as
u-Ip(u)
with
P(u)
given by the product formula (6).
Again it is easy to check that this function satisfies (9) and hence is the same as the
one just considered.
This proves (6) and gives a second proof of (a)-(c).
The Taylor expansions (3) and (4) of
from the above two constructions:
P(u)
and
log P(u)
are easily obtained
du
co
u/2
sinh u/2
P(u)
the first construction gives
( qmeU/2
~
2m u
m=1
I -q e
me-U/2 )
q
=
u/2
2m -u
sinh u/2
I -q e
u12
sinh ~
which (on substituting the Taylor series of
and
sinh
~ (e 2
m,d->1
d odd
-e
du
2 )q
md
) is seen to be
equivalent to (3), and the second gives
~
log e(u)
=
log ~
u/2
+
(-I) n
n=1
nd
(e du + e -du - 2) qd
'
d=1
which is similarly equivalent to (4) (the Taylor expansion of
u/2u/2
log sinh
can be
found by differentiation).
Finally, the "elliptic" property (5) also follows from the axiomatic characterization (9): the properties (9) imply that
~(u) 2
under translation by
u = 0 with leading terms
L and have poles at
and
~'(u) 2
are even and invariant
-2
-4
u
and u
as
their only singularities (mod L), so ~,2 must be a monic quadratic polynomial of
i.e.,
@,2
as
g-1(u)
of
6 and
of u 2 and
= ~4
-26
where
~2
+ S
for some
g(x) = x +...
g as functions of
u4
in P(u)
6, g ~ ~ , and then
I
~--~ = u +...
~2
can be written
is given by the elliptic integral (2). The expansions
2~ir
can be obtained by comparing the coefficients
q = e
obtained from formulas (3), (4), and (5).
It remains to check property (d), i.e., that the series
theorem has Taylor coefficients which are modular forms on
P(u)
Y0(2) .
defined in the
We recall that
220
F0(2)
for
is the subgroup of
F = F0(2)
or
SL2(~)
SL2(~)
even and nonnegative)
consisting of matrices
a modular form of weight
is a holomorphic
function
(2 ~)
k on
f:H+
F (k
with
c
even and that
an integer, necessarily
$ (l{=upper half-plane)
satisfy-
for all T6 H, (~ ~) 6r and having a Fourier expansion
ing f (aT+b)
cr+d = (cT+d)kf(r)
a(n) qn with a(n) of polynomial growth in n.
The C-vector space of such forms is
denoted by
Mk(£) , the Q-vector space (of the same dimension)
for all
is denoted by
SL2(Z)
n
and the graded ring
of forms with
@M~(£)
k
by
M~(F).
a(n) E
For
F=
this ring is ~[G4,G6], where
(11)
For
~(F),
Gk
k ~4
=
_
Gk(T)
the function
__
+
Bk
2k
Gk(~)
~ ( ~
n=; dln
m,n , not both zero).
(12)
G (aT+b.
-2 c--~ )
=
) qn
is modular of weight
the absolutely convergent Eisenstein
integers
d k-1
series
+
k even)
2(2%i)kp
~ - ~ k
k because
~'(mT+n) -k
The function
(cT+d) 2 G2(~)
( k > O,
is equal to
(summation over all pairs of
G 2 is "nearly" modular:
i
~-~ c(cr+d)
.
it satisfies
a
T 6 H , (c ~) ESL2(~) .
for
From these facts and the easily checked identities
(13)
G~(T)
=
it follows that
Gk(T) - 2k - 1 G k ( 2 T )
G k* and ~k
more precisely,
~k(T)
=
are modular forms on
Therefore each of the formulas
M~,(F0(2));
,
-Gk(T)
F0(2)
for all
(3), (4), or (5) shows that
the coefficient
of u k
+ 2Gk(2T)
= u
~
mEZ
~
u
n6 7Z
(-1)n
+4~im~ + 2"rrin
=
l
-
(including
I
sinh
x
u k
~ (4--~)
k>0
k even
with the inner sum clearly a modular form of weight
k
on
k=2)
has coefficients
is in KI<@(F°(2))
The modularity also follows from (10) because the expansion
u~(u)
k
P(u)
(k~2)
~,
m,nEZ
in
for each k>0.
=
~
(-1)n gives
n~Tzx+~in
(_l)n
(m,r+n/2)~
F0(2) (here some care is
needed with the conditionally convergent double series), or from the axiomatic characterization
(9) by noting that for
~(u) = (cT+d) ~((cT+d) u)
(a b ) 6 F0(2)
satisfies
and
~ satisfying
This completes the proof of the theorem.
~ = aT+b
cT+d "
From a purely modular point of view,
there are two surprising aspects of the formulas it contains.
of modular forms
Mk(F0(2))
divisors of n with congruence conditions)
a(n) =O(n k/2) ).
and
The former is spanned by
2 for k > 2
First of all, the space
breaks up in a natural way as the direct sum of a space
of Eisenstein series (forms whose Fourier coefficients
k=2
(9) the function
(9) with respect to
G k• and ~k
(the dimension of the full space
of both
a(n)
are sums of powers of
and a space of "cusp forms" (forms satisfying
is k/2).
log P(u)
I for
It is quite
that the coefficients
snbspaceo
The other surprising fact is that, although the Eisenstein series
linear combination of
and
Mk
remarkable
have rational Fourier coefficients
P(u)
and hence has dimension
belong to this tiny
G~, G~,...
and non-zero constant terms, there is a rational
I, G 2 , .o. , G2r , namely
ar,
which vanishes to order
2r-I
(i.e. twice ~as far as one has any right to expect) and is monic with integral coeffi-
221
cients.
This, and also the fact that the
that the
a r have much smaller coefficients
(i.e.,
Gk* satisfy congruences to high moduli), are illustrated by the first values:
q2 +
4q3 + q4 +
6q5 +
4q6 +
8q7 + q8 +
13q9 + ...
7 + q + q2 +
G4, = -240
28q3 + q4 +
126q5 +
28q6 +
344q7 + q8 +
757q 9 + ...
G2*
_
I
24
G6,
+
q
+
31 + q + q2 + 244q3 + q4 + 3126q5 + 244q6 + 16808q7 + q8 + 59293q 9 + ...
504
al
q + q2 +
4q3 + q4 +
a2
q
3
+
6q5 +
4q6 +
q6
5q 5 +
q5
a3
8q7 + q8 +
+
14q
+
7
7q 7
We now turn to the other genera introduced by Witten.
13q9 + ...
+ ...
+
31q 9
+
27q 9 + ...
The same formal power
series calculation as that which showed the equivalence of (4) and (6) gives
u/2
sinh u/2
(14)
where
~W
(W
Gk
~
=
(1-qn) 2
(I -qneU)(1 -qne-U)
is defined by (11).
=
exp < ! 0 ~-,-G k u k )
k
Call this power series
21k
Pw(U)
and the associated genus
for "Witten" or "Weierstrass"; terminology due to Peter Landweber).
Let us
compare this genus and power series with those of Landweber-Stong:
- The left side of (14) shows that the KO-theory characteristic power series of
~W
can be written in the form
~ [
In=1
where
br
properties
n
]-I
q
(1-q n)2 y
belongs to
Z[[q]]
=
~
r
~ bry
r=O
(y =
and has leading coefficient
(a)-(c) hold for the Witten genus, and ~W(M)
a Spin manifold.
eu
_2+e
_
u )
r
q . Thus the analogues of
belongs to
~[[q]]
The product in (14) is simpler than that in (6).
if M
is
It can be neatly
expressed by saying that the associated KO-valued characteristic class ~J~W($) is a
certain tensor product of sums of symmetric powers of
~, and in this form has a
natural interpretation as an index formula for a Dirac-like operator on the free loop
space of
M; the corresponding expression for the Landweber-Stong genus also has an
interpretation as an index formula for an operator, but this time one which has no
finite-dimensional version.
(For all this, see Witten's paper [6].)
- The right side of (14) shows t h a t ~W(M)
is a modular form on
first Pontryagin class of M vanishes rationally, since then
characteristic class ~w(TM) .
In this case
~w(M)
G2
SL2(Z)
if the
drops out of the
is simpler than ~Ls(M)
because it
is a modular form on the full modular group rather than a congruence subgroup.
other hand, if P1(M)
does not vanish then ~w(M)
On the
is not a modular form at all, but
belongs instead to the larger ring
A
M,
M, = M~(SL2 (Z)) = ~[G4 , G6] .
A
This may not be all bad: the ring M, is also studied in the theory of modular forms
A
and is in many respects nearly as good as M, (the elements of M, are "almost modular"
= ~[G2. G4, G 6] m
222
A
by (12), and the
p ).
rood p
In one respect
I d~d
D = q dqd _ 2~i
reductions
of
M , N Zg[[q]] and
it is even better:
(specifically ,
- 12 G2G 6 , and more generally
it is closed under the differentiation
DG 2 = ~5G 4 - 2 G 22 , DG 4 = 7 G
Df + 2 k G 2 f E M k + 2
- There is no analogue of the additive
property
(5), so k0W
in cohomology.
factor
f CM k) .
(3) and (7) or of the elliptic
genus and does not have a simple description
the axiomatic
characterization
2G2
minus the Weierstrass £ - f u n c t i o n
This is closely related to the non-modularity
as functions
of
~ (cf. comments
Witten discusses
Let ~(~)
6 - 8 G 2 G 4 , DG 6 = ~400
- - G 42
(9) no longer applies:
is nearly, but not quite, doubly periodic.
(It changes by a
d2
u-1
under u ÷ u + 2 ~ i T . The function d~-~log
Pw(U)
is periodic,
-ql/2eU
series
for
formulae
operator
u-IPw(U)
and in fact equals
2g'2~i .)
is not an elliptic
This is because
the function
M , N 2g[[q]] agree for all primes
u u
tanh
the one associated
(n >--I ) be the characteristic
and
I -q~ee
+ qne2U
I
2~'
of
Pw(u)
where
q
to the signature operator.
classes with characteristic
is a parameter.
Tile G-signature
Atiyah and Singer says that for a smooth finite-dimensional
manifold
the equivariant
ring
signature,
Z.2~i~+
at the end of the note).
one other genus,
and ~n(~)
for the lattice
of the coefficients
an element of the representation
power
theorem of
X with
S l-action,
R(S |) ®£--~C[q,q -I] ,
is given by
co
(15)
<~(TM)
where
M
M = X $I
in X
on which
because
of
X
S I acts via
This equivariant
it is an element of
idea was to apply
manifold
M;
then
M
(16)
signature
to
T M ® C ),
u/2,
Modifying
neither
A
_
X
q=0
and
q=oo.
is the free loop space of a smooth
~
is finite-dimensional (and in
n
sense even though X itself is
(i) nor (ii) applies and we get a
by dividing
its defining characteristic
u =0
(i.e., looking at the associated stable class), and
T 1_--7~
1+qn ~
we find that this power series is (
\_p ~ T ~,
/ d i m M q0S (H) , where
is the ~[[q]]-valued genus with characteristic
Ps(u)
(cf. [I])
since S I is connected, or
-I
] which is regular at both
so that the formula makes
q.
either
S I on the middle cohomology
¢[q,q
On the other hand,
power series in
u by
is constant,
(15) to the case when
power series by its value at
replacing
the subbundle of the normal bundle of
is the fixed point set and each
infinite-dimensional.
q0S
n
it is defined in terms of the action of
Witten's
fact isomorphic
v
[ c o s n 6 sinnO]
(We are being very brief and a little
- s i n n O cos nO ~ "
and this action is trivial
(ii) because
non-trivial
,
is the fixed point set and
imprecise here.)
(i)
TT ~ n ( V n ) , [M] >
n=1
u/2
n~=1( I + qneU
tanh u/2 =
I - qneU
power series
I + qne-U~/(1+qn)
I
q--h--~e-u/I\ I -qn
For this genus, analogues of all five formulas
2
in the theorem about
P(u)
hold.
analogue of (6) is (16) itself, and the analogues of (3), (4), (5), and (7) are:
The
223
1-
es(U)
(17)
~
2~'k
(18)
es(U)
(19)
es(U)
=
gs1 (u)
u /
es(U)
,
'
where
gS(x)
I
x
i
= ;0 (I -26st2 +gst4) -g dt
n
I
4 + 6n~1 (d~n d ) q
tanh u/2
I - 2
,
-2) r
[
!1 (1- qm) 2r
~
m=
.r+d-1. ] qn
(21)
G~( -I )
2T
so (writing
P(T;u)
dependence on
(22)
=
instead of
T , and similarly for
r m[[q]]
.
(12) for
~ k ( ~-I
)
'
differentiation
for the Landweber-Stong
PS ) equation
(13) and the
k=2)
2 Tk G~(T)
=
(like the
it follows that
'
genus to emphasize the
(18) says
ps(~;u ) = p ( -21~ ; 7u ) "
In other words,
oo of
H/F0(2)
P and
to an
PS
are the same function, but expanded at the two cusps
(which are interchanged by
(5) into (22) and using
-I
Ps(U)
from which
T ÷ -I/2~).
(21) again gives
analogue of the property
u
(17) and (19).
Finally,
Equation
I/2
tanh u/2
(20) expresses
(I) in (iii) of the introduction,
(the one associated
formulas
equation
(22) leads
(-I) m (eU-e -u)
~ qm+q -m_eU_e -u
m->1
Ps(U)
as
like the one given for
but using a different K0,-Fundamental
to the signature operator).
We make a final remark.
Throughout
this note there has been an interplay between
their elliptic properties with respect to the variable
having this dual modular/elliptic
(a b)
in
SL2(~)
=
u.
m
is a function
%: H×C -+ $
(cT+d) k e 2~imcz2/(cx+d)
or a congruence
subgroup and
T and
Functions of two variables
nature are called Jacobi forms.
k and index
% ( c~+d
aT+b ' c~$d
z )
for
'
u/2
u -u
tanh u/2G(e +e -2)
the modularity properties of the various functions with respect to the variable
Jacobi form of weight
0 and
(3) and
(9) and hence to an analogue of (10), namely
(-I) m
~
2 tanh(u/2 + ~im~)
m E2~
(20) easily follows.
Substituting
G(y) E Z[[q, qy]] ; such a formula has an interpretation
equation
class
P(u)
q
But from equation
(respectively
2 k-1 Tk ~k(T)
) qn ,
din
proof of (14) or of the equality of (4) and (6)).
Gk(-~1) = TkGk(T)
c
~ (-1)n/d [ 2r-1 )
n>1
with
with
(18) is obtained directly from (16) by logarithmic
modularity property
with
)d d 3
~S = "~-+ n~l (d~n (-1
~ c r (eU+e
r>_1
(-1)mq mr
cr =
Indeed,
,
4 GI~ uk
exp ( k>~O -~--.v )
21k
6S
(20)
k
(k-l) ! u
k>0
21k
%(T,z)
More precisely,
satisfying
a
224
0( T , z+ ~r + ~)
for all
(~ ,p)
in
~2
was developed in [3].)
e -2~im(~2T +2Xz)
~(r,z)
or a sublattice of finite index.
(The theory of such forms
The most important examples of Jacobi forms are theta series.
Using the Jaeobi triple product identity, we find that
u-IPw(u)
can be expressed as
a quotient of theta-series,
(here
(~)
u-1Pw(U)
=
equals
0 for
(n!0 ( ~ ) n
qn2/8)/( n~
(~n4) qn2/8enU/2 )
n even, ( I) ½(n-j) for n odd), and is therefore a Jacobi
u
I
form with respect to T and z = ~
of weight -I and index -~.
(It is because
2~i ~
the index is non-zero that u-IPw(u)
is not quite elliptic in u and that its Taylor
coefficients are not quite modular forms in
T .)
series we have been considering are related to
P(u)
so
u-qP(u)
and
=
~2
PW(2~;u~ /Pw(T;u) ,
u-IPs(u)
The other two characteristic power
PW by
Ps(U)
PW(r;u)2/Pw(2T;2u) ,
are also Jacobi forms of weight
-I, but of index
0 .
It
is interesting to note that in all of the recent occurrences of modular forms in
algebraic topology, string theory, and the theory of Kac-Moody algebras, it is in
fact Jacobi forms which are entering.
REFERENCES
I.
M.F. Atiyah and F. Hirzebruch:
Spin-manifolds and group actions.
In: Essays on
Topology and Related Topics (M6moires d6di6s ~ Georges de Rham), pp. 18-26.
Springer, New York 1970.
2.
D.V. Chudnovsky and G.V. Chudnovsky:
genera.
To appear in Topology.
3.
M. Eichler and D. Zagier: The Theory of Jacobi Forms.
Birkh~user, Boston-Basel-Stuttgart 1985.
4.
P. Landweber and R.E. Stong: Circle actions on Spin manifolds and characteristic
numbers.
To appear in Topology.
5.
S. 0chanine:
Sur los genres multiplicatifs d&finis par des int&grales elliptiques.
To appear in Topology 26 (1987) 143-~5]
6.
E. Witten:
vo iume.
Elliptic modular functions and elliptic
Progress in Math. 55,
The index of the Dirac operator in loop space.
To appear in this