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 = Zτ + 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 .