ELLIPTIC GENERA AND ELLIPTIC COHOMOLOGY 13
Similarly, letting δ =
1
8
, = 0, we recover the A-genus, which for a spin manifold
is the index of the Dirac operator:
log(x) =
x
0
dt
1 + (t/2)2
= 2
sinh−1(x/2);
Q(x) =
x/2
sinh(x/2)
.
The genera L and A are elliptic genera corresponding to singular elliptic curves.
This is explicitly seen in the fact that their logarithms invert to singly-periodic
functions as opposed to doubly-periodic functions.
The signature was long known to satisfy a stronger form of multiplicativity,
known as strict multiplicativity. If M is a fiber bundle over B with fiber F and
connected structure group, then L(M) = L(B)L(F ). The same statement holds
for the A-genus when F is a spin manifold. As more examples were discovered,
Ochanine introduced the notion of elliptic genera to explain the phenomenon and
classify strictly multiplicative genera.
Theorem 3.3 (Ochanine, Bott–Taubes). A genus ϕ satisfies the strict mul-
tiplicativity condition ϕ(M) = ϕ(B)ϕ(F ) for all bundles of spin manifolds with
connected structure group if and only if ϕ is an elliptic genus.
There is extra algebraic structure encoded within the values of the universal
elliptic genus. Using the Weierstrass function, to a lattice Λτ = + Z we can
canonically associate coefficients (τ) and δ(τ) satisfying the Jacobi quartic equa-
tion. The functions (τ) and δ(τ) are modular forms, of weight 2 and 4 respectively,
on the subgroup Γ0(2) := {A SL2(Z) | A = (

0
) mod 2} SL2(Z). Therefore,
the elliptic genus ϕJ associates to any compact oriented 4k-manifold a modular
form of weight 2k on the subgroup Γ0(2). Because modular forms are holomorphic
and invariant under the translation τ τ + 1, we can expand them in the variable
q =
e2πiτ
and consider ϕJ (M) Q q .
Using insights from quantum field theory, Witten gave an alternate interpreta-
tion of the elliptic genus which illuminated several of its properties. His definition
of ϕJ is as follows. Let M be a spin manifold of dimension n with complex spinor
bundle S(TM C). To a complex vector bundle V X, use the symmetric and
exterior powers to define the bundle operations
StV = 1 + tV +
t2V ⊗2
+ · · · K(X) t , ΛtV = 1 + tV +
t2Λ2V
+ · · · K(X) t .
Then, the power series in q defined by
A(TM)ch S(TM
C)

l=1
Sql (TM
C

Cn)


l=1
Λql (TM
C

Cn)
, [M
n]
Q q
is equal to the q-expansion of the elliptic genus ϕJ . When M is a spin manifold,
Witten formally defined the signature operator on the free loop space LM, and he
showed its S1-equivariant index equals ϕJ (up to a normalization factor involving
the Dedekind η function). The S1-action on LM = Map(S1,M) is induced by the
natural action of
S1
on itself.
Witten also defined the following genus, now known as the Witten genus:
ϕW (M) := A(TM)ch

l=1
Sql (TM
C

Cn))
, [M
n
] Q q .
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