03.06.24 · modern-geometry / characteristic-classes

Bott-Taubes rigidity theorem

shipped3 tiersLean: none

Anchor (Master): Bott-Taubes 1989 *J. Amer. Math. Soc.* 2, 137-186; Taubes 1989 *J. Differential Geom.* 29, 23-58; Witten 1988 *Commun. Math. Phys.* 109, 525-536; Witten 1988 *LNM* 1326, 161-181; Liu 1995 *J. Differential Geom.* 41, 343-396; Hirzebruch-Berger-Jung *Manifolds and Modular Forms* Lecture 9

Intuition Beginner

A closed shape with a rotational symmetry — a circle acting on it — looks like a single shape but really packages a whole family at once. Each angle of the circle gives a slightly different rearrangement of the shape's interior, and the family of rearrangements is the symmetry. A topological invariant of the shape can then be lifted into an equivariant invariant: a function on the circle that reads the symmetry into the answer. The lifted invariant is more refined, because it records how the original quantity sees each angle separately.

The Bott-Taubes rigidity theorem says that for one particular invariant, this refinement collapses. The Witten genus is a power series in a formal variable $q$ attached to closed spin shapes. When the shape carries a circle of symmetries, the equivariant Witten genus picks up a second variable that records the angle. Rigidity is the statement that the second variable does nothing: the equivariant Witten genus is the same function of $q$ as the ordinary Witten genus, for every angle.

Why does this matter? Because rigidity is the topological echo of a beautiful physical idea. Witten proposed that the genus is the index of a Dirac operator on the loop space of the shape, equivariant under loop rotation. A symmetry of the original shape lifts to a symmetry of the loop space, and the lifted symmetry should not change the index. Rigidity makes this rigorous and turns it into a counting theorem for spin manifolds with rotational symmetry.

Visual Beginner

A schematic of the upper half-plane crossed with the complex plane, with the modular parameter $q$ on the vertical axis and the rotation parameter $t$ on the horizontal axis. Horizontal lines mark constant values of the equivariant Witten genus: for each fixed $q$ , the genus as a function of $t$ is a flat horizontal line. The picture conveys that all the variation lives in the vertical $q$ -direction, and the $t$ -direction is rigid.

The picture captures the essential message: a two-variable function on the upper half-plane and the complex plane reduces to a one-variable function on the upper half-plane alone, because the equivariant variable contributes nothing.

Worked example Beginner

Take the simplest closed spin shape with a circle symmetry: the four-sphere $S^{4}$ . Give $S^{4}$ the standard $S^{1}$ -action that rotates a pair of coordinates and fixes the orthogonal pair, so the fixed-point set is a copy of the two-sphere $S^{2}$ sitting inside $S^{4}$ . The Witten genus of $S^{4}$ is the ordinary $\hat{A}$ -genus collapsed at the cusp, which is a single number: zero, because the dimension is four and there is no top-degree characteristic number to contribute.

Step 1. Compute the ordinary Witten genus of $S^{4}$ . The $\hat{A}$ -genus of any sphere of positive dimension is zero, so the constant term in $q$ vanishes. Higher coefficients in $q$ involve characteristic numbers of symmetric powers of the tangent bundle, and these also vanish for spheres because the tangent bundle is stably parallelisable. Result: $W (S^{4}) (q) = 0$ .

Step 2. Compute the equivariant Witten genus. The Atiyah-Bott formula localises the answer onto the fixed-point set, which is $S^{2}$ . Each summand depends on the rotation parameter $t$ through a Jacobi theta function, so individual summands are non-constant in $t$ .

Step 3. The rigidity theorem says these summands must add to zero or to a constant in $t$ . For $S^{4}$ the answer is exactly zero in both variables, because the ordinary genus is zero.

Step 4. The example shows the punchline at its cleanest: the ordinary answer is zero, the equivariant pieces are individually non-zero, and their sum cancels. Rigidity is the cancellation pattern.

What this tells us: the equivariant Witten genus is a sophisticated bookkeeping device. Each fixed-point contribution looks like a non-constant function of the rotation parameter, but the global sum is forced to be the ordinary genus by an arithmetic constraint on Jacobi forms. The cancellation is automatic and is the source of the topological rigidity.

Check your understanding Beginner

Exercise (easy, multiple choice).

The Bott-Taubes proof uses one major tool to convert fixed-point sums into rigidity. Which is it?

A. The Mayer-Vietoris sequence in singular cohomology.
B. The Atiyah-Bott Lefschetz fixed-point formula plus modular properties of Jacobi theta functions.
C. The Universal Coefficient Theorem.
D. The Künneth formula for products of spheres.

Hint

Two ingredients: equivariant index theory (Lefschetz) and arithmetic of theta functions.

Answer

B. The Atiyah-Bott Lefschetz fixed-point formula plus modular properties of Jacobi theta functions.

The Lefschetz formula expresses the equivariant index as a sum over fixed-point components. Each summand is a ratio of Jacobi theta functions whose transformation laws under $SL_{2} (Z) ⋉ Z^{2}$ force the sum to be doubly periodic and pole-free, hence constant in the rotation parameter.

Formal definition Intermediate+

Let $M$ be a closed oriented smooth manifold of even real dimension $2 n$ with a smooth $S^{1}$ -action. We use the convention $q = e^{2 π i τ}$ with $τ$ in the upper half-plane $H$ , and $t = e^{2 π i z}$ with $z \in C$ . The complexified tangent bundle is denoted $T M_{C}$ and dimension counts are taken in $C$ .

Definition (Witten genus, characteristic-class form). For $M$ a closed spin manifold of real dimension $4 k$ , the Witten genus is the multiplicative-sequence genus attached to the characteristic power series $$ Q_W(x, \tau) = \frac{x/2}{\sinh(x/2)} \cdot \prod_{n \geq 1} \frac{(1 - q^n)^2}{(1 - q^n e^x)(1 - q^n e^{-x})}. $$ Equivalently, in the $Θ$ -bundle formalism, $$ \varphi_W(M)(\tau) = \int_M \mathrm{ch}(\Theta(\tau, TM_\mathbb{C})) \cdot \hat A(TM), $$ where $$ \Theta(\tau, V) = \bigotimes_{n \geq 1} \mathrm{Sym}{q^n}(V\mathbb{C}) \otimes \bigotimes_{n \geq 1} \mathrm{Sym}{q^n}(\bar V\mathbb{C}) $$ is the theta line-bundle expansion (formal sum of symmetric powers weighted by $q^{n}$ ), and $\hat{A} (T M)$ is the $\hat{A}$ -genus density of the spin tangent bundle.

Definition (equivariant Witten genus). When $M$ carries an $S^{1}$ -action lifting to the spin structure, the $S^{1}$ -equivariant Witten genus is the equivariant index $$ \varphi_W^{S^1}(M)(\tau, z) = \mathrm{ind}{S^1}(\not D \otimes \Theta(\tau, TM\mathbb{C})) \in \mathbb{C}((q))[t, t^{-1}], $$ where $\neq D$ is the Dirac operator on $M$ twisted by the formal bundle $Θ (τ, T M_{C})$ , and the $S^{1}$ -equivariant index records the trace of the rotation-character action on the (virtual) kernel-minus-cokernel.

Definition (Jacobi form). A holomorphic function $F : H \times C \to C$ is a weak Jacobi form of weight $k$ and index $m$ for the modular group $SL_{2} (Z) ⋉ Z^{2}$ if $$ F!\left(\frac{a \tau + b}{c \tau + d}, \frac{z}{c \tau + d}\right) = (c \tau + d)^k \exp!\left(\frac{2 \pi i m c z^2}{c \tau + d}\right) F(\tau, z) $$ for every $(a c b d) \in SL_{2} (Z)$ , and $$ F(\tau, z + \lambda \tau + \mu) = \exp(- 2 \pi i m (\lambda^2 \tau + 2 \lambda z)) F(\tau, z) $$ for every $(λ, μ) \in Z^{2}$ . The form is holomorphic (versus weak) if the $q$ -expansion at the cusp has non-negative powers of $q$ only.

Definition (rigidity). A power series $F (q, t) \in C ((q)) [t, t^{- 1}]$ is rigid if it is a Laurent polynomial in $t$ of degree zero — that is, if it is independent of $t$ as a formal expression. The Bott-Taubes theorem is the rigidity statement for the equivariant Witten genus.

Counterexamples to common slips

Rigidity holds for the Witten genus on spin manifolds — equivalently, the level-one Hirzebruch elliptic genus. The Ochanine elliptic genus is rigid for $Γ_{0} (2)$ on spin manifolds (Liu 1995); the Hirzebruch level- $N$ genera are rigid for the corresponding $Γ_{1} (N)$ . Without the spin hypothesis, the equivariant genus need not be rigid: the equivariant signature is rigid (Atiyah-Bott 1968), but the equivariant unsigned Pontryagin numbers are not.
Rigidity is the statement that the equivariant genus equals the ordinary genus, not that the equivariant genus vanishes. The vanishing of $\hat{A} (M)$ for spin $M$ with non-identity $S^{1}$ -action (Atiyah-Hirzebruch 1970) is the $q \to 0$ cuspidal limit of the Bott-Taubes statement, not the statement itself.
The string condition $p_{1} (M) /2 = 0$ is required for the Witten genus to land in modular forms of level one. Rigidity itself works for any spin $M$ with $S^{1}$ -action; the level-one modularity of the answer requires the string refinement. Without string structure, the answer is a modular form for a smaller congruence subgroup.
The equivariant parameter $t$ is the character of the rotation, not a generic complex number. The formal Laurent-polynomial structure $C ((q)) [t, t^{- 1}]$ captures this. Rigidity then says all Laurent coefficients except the constant term vanish.

Key theorem with proof Intermediate+

Theorem (Bott-Taubes rigidity; Bott-Taubes 1989). Let $M$ be a closed spin manifold of real dimension $4 k$ with a smooth $S^{1}$ -action lifting to the spin structure. The equivariant Witten genus $$ \varphi_W^{S^1}(M)(\tau, z) \in \mathbb{C}((q))[t, t^{-1}] $$ is independent of $t$ as a formal Laurent expression, and equals the ordinary Witten genus $φ_{W} (M) (τ)$ .

Proof (Liu's Jacobi-form route). The argument has three steps: identify the equivariant Witten genus as a Jacobi form, compute its index, and apply the structure theorem for Jacobi forms of index zero.

Step 1. The equivariant Witten genus is a weak Jacobi form of weight $2 k$ and index $0$ . By the Atiyah-Bott Lefschetz fixed-point formula applied to the Dirac operator twisted by $Θ (τ, T M_{C})$ , $$ \varphi_W^{S^1}(M)(\tau, z) = \sum_F \int_F \frac{\hat A(F) \cdot \mathrm{ch}(\Theta(\tau, TF_\mathbb{C})) \cdot \mathrm{ch}(\Theta(\tau, N_F^\mathbb{C}))}{e_{S^1}(N_F)(\tau, z)}, $$ where the sum runs over the connected components $F$ of the fixed-point set $M^{S^{1}}$ , $N_{F}$ is the normal bundle of $F$ in $M$ , and $e_{S^{1}} (N_{F}) (τ, z)$ is the equivariant theta-Euler class of $N_{F}$ , an explicit product of Jacobi theta functions in $τ$ and $z$ weighted by the rotation characters of $N_{F}$ .

Each component of this sum is a meromorphic function on $H \times C$ whose transformation law under $SL_{2} (Z) ⋉ Z^{2}$ follows from the standard transformation laws of the Jacobi theta function $θ_{1}$ . The total sum has weight $2 k$ (from the dimension count: each Euler-class denominator contributes weight $- 1$ per real codimension of $N_{F}$ in $M$ , and the numerator densities are weight zero). The index is computed from the equivariant-Euler-class quadratic form on the rotation characters; for the spin case, the index is zero (this is the place the spin hypothesis enters: the rotation characters of the normal bundle to a spin $S^{1}$ -action satisfy a parity constraint that forces the index to vanish; see Liu 1995 §3 for the explicit computation on each fixed-point component).

Step 2. No poles. The equivariant Witten genus is a priori meromorphic on $H \times C$ , with potential poles along the divisor where the equivariant Euler class vanishes — that is, along the lattice $τ Z + Z$ in $C$ . But $φ_{W}^{S^{1}} (M)$ is defined as an equivariant index, which is a formal power series in $q$ with Laurent-polynomial coefficients in $t$ ; in particular, it has no poles in $t$ along any divisor of finite order. So the meromorphic function is in fact holomorphic.

Step 3. Index-zero Jacobi forms are constant in $z$ . A weak holomorphic Jacobi form of index zero satisfies the periodicity $F (τ, z + λ τ + μ) = F (τ, z)$ for every $(λ, μ) \in Z^{2}$ . So $F$ is doubly periodic in $z$ on the elliptic curve $E_{τ} = C / (τ Z + Z)$ . A holomorphic doubly-periodic function on a compact Riemann surface is constant by Liouville's theorem. So $F (τ, z)$ is independent of $z$ , and equals its value at any single point — for instance $z = 0$ , where it specialises to the ordinary Witten genus $φ_{W} (M) (τ)$ .

Combining the three steps: $φ_{W}^{S^{1}} (M) (τ, z) = φ_{W} (M) (τ)$ as functions on $H \times C$ , and equivalently as formal expressions in $C ((q)) [t, t^{- 1}]$ . $□$

Bridge. The proof above is the central insight of the Liu route: identifying the equivariant Witten genus with an index-zero Jacobi form turns rigidity into a one-line consequence of Liouville on the elliptic curve. The foundational reason it holds is that the spin hypothesis forces the index to vanish, and the index is exactly the bookkeeping quantity that measures how the equivariant variable can vary. Without the spin condition, the index can be non-zero, and the equivariant genus picks up genuine $z$ -dependence — this is the signature side, where the equivariant signature is rigid for a different reason (Atiyah-Bott 1968). Putting these together, rigidity builds toward the recognition that elliptic genera are Jacobi-form-valued invariants, and the equivariant refinement is the elliptic-curve direction of that bigrading. This is exactly the bridge to elliptic cohomology, where Hopkins's TMF programme realises the Witten genus as a ring-spectrum map and rigidity is the equivariant compatibility statement — putting these together, the bridge is the recognition that the Witten genus is the topological shadow of the moduli stack of elliptic curves, and the equivariant variable parametrises the elliptic-curve direction. The same pattern appears again in 03.06.23 (Modularity of the elliptic genus), where the modular transformation law on the $τ$ -variable plays the role that double-periodicity plays on the $z$ -variable here, and identifies the genus with a global section of an automorphic line bundle.

Exercises Intermediate+

Exercise 3 (medium, symbolic).

Show that an isolated fixed point of an $S^{1}$ -action on a closed spin $M^{4 k}$ contributes to $φ_{W}^{S^{1}} (M)$ a product of Jacobi theta function ratios $θ_{1}^{'} (0) / θ_{1} (m_{j} z, τ)$ where $m_{j}$ are the rotation weights at the fixed point.

Hint

Apply the Atiyah-Bott formula. At an isolated fixed point the normal bundle is $C^{2 k}$ with a diagonal $S^{1}$ -action of weights $m_{1}, \dots, m_{2 k}$ . The equivariant theta-Euler class is the product of $θ_{1} (m_{j} z, τ) / θ_{1}^{'} (0)$ .

Answer

At an isolated fixed point $p$ with normal-bundle weights $m_{1}, \dots, m_{2 k}$ , the Atiyah-Bott fixed-point contribution to the equivariant Witten genus is $$ \varphi_W^{S^1}(M)(\tau, z)\big|p = \prod{j=1}^{2k} \frac{\theta_1'(0, \tau)}{\theta_1(m_j z, \tau)} \cdot \prod_{j=1}^{2k} \frac{1}{(2 \pi i)} \cdot \mathrm{Td}\text{-style correction}, $$ where the leading factor comes from the equivariant theta-Euler class (the denominator is the product of Jacobi $θ_{1}$ 's evaluated at the rotation characters of the normal directions) and the numerator is the universal Witten-genus density at a point. The full equivariant genus is the sum of such contributions over all fixed points; for spin $M$ , the parity of the rotation weights ensures the sum is index-zero in the Jacobi-form sense. Rubric: full credit for the $θ_{1}$ -ratio with the rotation-weight identification.

Exercise 4 (medium, short-answer).

Explain why rigidity of the equivariant Witten genus implies the Atiyah-Hirzebruch vanishing theorem: $\hat{A} (M) = 0$ for $M$ closed spin with non-identity smooth $S^{1}$ -action.

Hint

Take the cuspidal limit $q \to 0$ of the Bott-Taubes equality. The equivariant Witten genus collapses to the equivariant $\hat{A}$ -genus, and constancy in $t$ forces a vanishing for non-fixed actions.

Answer

By Bott-Taubes, $φ_{W}^{S^{1}} (M) (τ, z) = φ_{W} (M) (τ)$ . Take the limit $q \to 0$ . The Witten genus collapses to the $\hat{A}$ -genus on the right, while the equivariant Witten genus collapses to the equivariant $\hat{A}$ -genus on the left, which by the Atiyah-Bott formula is a sum of contributions from the fixed-point set with poles at the rotation parameters of the normal bundle. For the limit to be constant in $t$ , the equivariant $\hat{A}$ -sum must reduce to its $z = 0$ value, which is just the integral over the fixed-point set of the ordinary $\hat{A}$ -density. When the $S^{1}$ -action is non-identity the fixed-point set has dimension strictly less than $dim M$ , and the integral of $\hat{A}$ over a lower-dimensional manifold is zero by degree counting. So $\hat{A} (M) = 0$ . Rubric: full credit for the cuspidal-limit reduction with the degree-counting argument.

Exercise 5 (medium, short-answer).

State the loop-space interpretation of the Witten genus due to Witten (1988). Why does the $S^{1}$ -action on $M$ lift to an action on the free loop space $L M$ that commutes with loop rotation?

Hint

A symmetry of $M$ acts pointwise on loops by post-composition. Loop rotation acts by reparametrising the source circle. The two commute because pointwise post-composition and source reparametrisation are independent operations on $Map (S^{1}, M)$ .

Answer

Witten's heuristic: the Witten genus $φ_{W} (M) (τ)$ is the index of a Dirac operator on the free loop space $L M = Map (S^{1}, M)$ , equivariant under loop rotation with character parameter $q$ . The $q$ -expansion records the loop-rotation eigenvalues. An $S^{1}$ -action on $M$ lifts to $L M$ by post-composition: if $ρ_{θ} : M \to M$ is the rotation by angle $θ$ , then $ρ_{θ} : L M \to L M$ sends $γ \mapsto ρ_{θ} \circ γ$ . This commutes with loop rotation $σ_{ϕ} : γ \mapsto γ (\cdot + ϕ)$ because $ρ_{θ} \circ γ (\cdot + ϕ) = (ρ_{θ} \circ γ) (\cdot + ϕ)$ — the operations act on different arguments of the map $γ : S^{1} \to M$ . So $L M$ carries a torus action by $S_{loop}^{1} \times S_{M}^{1}$ , and the Dirac index on $L M$ becomes a function of two equivariant parameters: $q$ for loop rotation, $t$ for the lifted $M$ -symmetry. Rigidity is the formal statement that the index is constant in $t$ . Rubric: full credit for the commutativity argument and the two-parameter index identification.

Exercise 7 (hard, short-answer).

Compute the Witten genus of the K3 surface as a closed spin manifold of real dimension four. Express the answer in terms of the Eisenstein series and the discriminant.

Hint

K3 has $\hat{A} (K 3) = 2$ , $sig (K 3) = - 16$ , $χ (K 3) = 24$ , $p_{1} (K 3) = - 48$ in $H^{4}$ . The Witten genus of a four-manifold is determined by $\hat{A}$ and the second symmetric-power correction in $q$ .

Answer

For K3 the Witten genus is the Eisenstein series of weight two with the constant term adjusted by the $\hat{A}$ -genus. Explicitly, $φ_{W} (K 3) (τ) = 2 \cdot E_{4} (τ) /Δ (τ)^{1/6}$ in the leading approximation, and to leading order in $q$ , $$ \varphi_W(K3)(\tau) = 2 + 48 q + 600 q^2 + 5824 q^3 + O(q^4). $$ The coefficient pattern $2, 48, 600, 5824, \dots$ is the $q$ -expansion of $2 E_{4} / η^{16}$ in the conventions of Hirzebruch-Berger-Jung Lecture 5, which can be re-expressed in terms of $E_{4}$ and $Δ$ using $Δ = η^{24}$ . The 24-dimensional moonshine pattern of $j (τ)$ appears at the boundary where the dimension is $4 k = 24$ : there, $φ_{W} (M_{24}) = 24 \cdot E_{4}^{3} /Δ - 24 \cdot 720 = j (τ) - 744$ for the unique 24-dimensional string manifold contributing to monstrous moonshine. For K3 (dimension 4, $2 k = 2$ ), the Witten genus is a (quasi-)modular form of weight 2; the closed-form expression is read off from the $\hat{A}$ -density via the multiplicative-sequence machine. Rubric: full credit for the $q$ -expansion coefficients $2, 48, 600$ and the identification with $E_{4}$ -style Eisenstein data.

Exercise 8 (hard, short-answer).

Show that the equivariant signature is rigid: for a closed oriented $M^{4 k}$ with smooth $S^{1}$ -action, $sig^{S^{1}} (M) (t)$ is constant in $t$ . This is the Atiyah-Bott rigidity theorem of 1968. Compare with Bott-Taubes rigidity.

Hint

The equivariant signature is the equivariant index of the signature operator. Apply the Lefschetz formula; the contributions are rational functions of $t$ rather than Jacobi-form-valued.

Answer

By the Atiyah-Bott Lefschetz formula applied to the signature operator, $$ \mathrm{sig}^{S^1}(M)(t) = \sum_F \mathrm{sig}(F) \cdot \prod_j \frac{t^{m_j} + 1}{t^{m_j} - 1}, $$ where the sum runs over the connected fixed-point components $F$ , $m_{j}$ are the rotation weights of the normal bundle to $F$ , and the product is the equivariant signature contribution. Each summand is a rational function of $t$ with poles only at roots of unity. The sum, however, is a Laurent polynomial in $t$ (by the definition of the equivariant index as a virtual character); a rational function whose only finite poles are at roots of unity and which is a Laurent polynomial must be a constant Laurent polynomial — the poles cancel. The constant value at $t = 1$ recovers the ordinary signature. So $sig^{S^{1}} (M) (t) = sig (M)$ , constant in $t$ .

The comparison with Bott-Taubes: the equivariant signature is rigid by a pole-cancellation argument in the variable $t$ alone, whereas the equivariant Witten genus is rigid by an elliptic-curve double-periodicity argument in $z$ (with $q = e^{2 π i τ}$ fixed). Bott-Taubes upgrades Atiyah-Bott from a one-dimensional argument on the multiplicative group to a two-dimensional argument on the elliptic curve, and the spin hypothesis is what makes the upgrade possible. Rubric: full credit for the pole-cancellation argument and the elliptic-curve comparison.

Advanced results Master

Theorem (Liu's rigidity for general elliptic genera). Let $M$ be a closed spin manifold of real dimension $4 k$ with a smooth $S^{1}$ -action lifting to the spin structure. For each level- $N$ Hirzebruch elliptic genus $φ_{N}$ , the equivariant version $φ_{N}^{S^{1}} (M) (τ, z)$ is a weak Jacobi form of weight $2 k$ and index $0$ for the congruence subgroup $Γ_{1} (N) ⋉ Z^{2}$ , hence constant in $z$ .

This is Liu 1995 J. Diff. Geom. 41 §3-§4. The argument is the Bott-Taubes Jacobi-form route generalised: the relevant theta function is the level- $N$ theta function $θ_{N}$ with its own transformation law, and the index of the resulting Jacobi form is again zero by a spin-parity computation on the fixed-point set. Liu's framework reproduces Bott-Taubes ( $N = 1$ ), Hirzebruch level- $2$ rigidity (which is Ochanine rigidity for $Γ_{0} (2)$ ), and provides a uniform Jacobi-form proof for the whole family.

Theorem (cuspidal limits and rigidity of classical genera). The Bott-Taubes rigidity theorem implies, by cuspidal specialisation:

(i) Atiyah-Hirzebruch 1970: $\hat{A} (M) = 0$ for closed spin $M$ with non-identity smooth $S^{1}$ -action.

(ii) Atiyah-Bott 1968: $sig^{S^{1}} (M) (t) = sig (M)$ for closed oriented $M^{4 k}$ with smooth $S^{1}$ -action.

(iii) Witten's heuristic equality: $φ_{W} (M) (q) = ind_{S^{1}} (\neq D_{L M}) (q)$ interpreting the Witten genus as the loop-space Dirac index.

The first two statements are the $q \to 0$ and $q \to i \infty$ cuspidal limits of the Bott-Taubes rigidity equation; the third is the heuristic statement that gave the original physical motivation. Witten's claim is a heuristic identity at the level of formal power series in $q$ — making it rigorous requires the Bott-Taubes machinery (or the equivalent Hopkins-Miller TMF refinement).

Theorem (Hopkins-Miller TMF refinement; Hopkins 2002 ICM). The Witten genus $\varphi_W : \pi_ \mathrm{MString} \to \mathbb{Z}[[q]]$ refines to a ring-spectrum map* $$ \sigma : \mathrm{MString} \to \mathrm{TMF}, $$ where $TMF$ is the spectrum of topological modular forms. The equivariant Witten genus refines to an $S^{1}$ -equivariant ring-spectrum map $$ \sigma^{S^1} : \mathrm{MString}^{S^1} \to \mathrm{TMF}^{S^1}, $$ and the Bott-Taubes rigidity statement is the homotopical assertion that $σ^{S^{1}}$ is the pullback of $σ$ along the forgetful map $MString^{S^{1}} \to MString$ .

The TMF refinement realises elliptic cohomology as the universal home for Witten-genus-style invariants. Bott-Taubes rigidity is the elliptic-cohomology analogue of the well-known statement that ordinary cohomology with rational coefficients sees no rotational variation: the equivariant cohomology of a free $S^{1}$ -action collapses to the cohomology of the quotient. For Witten-genus-style invariants the analogous collapse happens at the level of $TMF$ , and rigidity is the precise version.

Theorem (failure of rigidity for non-spin elliptic genera). The Ochanine elliptic genus is rigid for $Γ_{0} (2) ⋉ Z^{2}$ on spin manifolds (Liu 1995). For oriented but non-spin manifolds the Ochanine genus is well-defined, but the equivariant version need not be rigid: examples are given by $CP^{2}$ with a non-identity $S^{1}$ -action, where $φ_{Och}^{S^{1}} (CP^{2}) (τ, z)$ has explicit $z$ -dependence.

This bounds the scope of rigidity: the spin hypothesis is essential for the index-zero Jacobi-form computation, and without it the equivariant Euler-class denominator carries a non-zero index that produces genuine $z$ -dependence. The boundary case $CP^{2}$ exhibits this dependence concretely; see Hirzebruch-Berger-Jung Lecture 8 for the computation.

Theorem (refinement to families). Let $π : E \to B$ be a smooth fibre bundle of closed spin manifolds with $S^{1}$ -action along the fibres. The family-equivariant Witten genus $\varphi_W^{S^1}(E/B)(\tau, z) \in H^(B; \mathbb{C})((q))[t, t^{-1}] $i sr i g i d in$ z $: e q u a l s$ \varphi_W(E/B)(\tau)$ as a cohomology-valued formal expression. The proof applies the family-Lefschetz formula fibrewise.*

This is the family refinement of Bott-Taubes, conjectured by Liu and proved in Liu 1995 §4. It plays the same role in fibre bundles that the original rigidity plays for a single manifold, and is the relevant statement for elliptic-cohomology computations on bundles such as $MString$ over $B String$ .

Synthesis. The Bott-Taubes rigidity theorem is the foundational reason that the Witten genus is the right elliptic-cohomology refinement of the $\hat{A}$ -genus. The central insight is exactly the identification of the equivariant Witten genus with an index-zero Jacobi form on $H \times C$ : this places elliptic genera inside the bigraded ring of Jacobi forms, and the spin hypothesis forces the index to vanish so the elliptic-direction collapses. Putting these together, rigidity generalises the Atiyah-Bott rigidity of the signature and the Atiyah-Hirzebruch vanishing of the $\hat{A}$ -genus to a single statement at the level of Witten-genus-valued invariants, and identifies the equivariant Witten genus with the ordinary Witten genus. This pattern appears again in 03.06.23 (Modularity of the elliptic genus), where the modular transformation in $τ$ controls the level structure of the genus, and the bridge is that the two transformations together — modular in $τ$ and double-periodic in $z$ — are exactly the two-variable symmetry of a Jacobi form. Rigidity identifies the equivariant Witten genus with the ordinary Witten genus, modularity identifies the ordinary Witten genus with a modular form, and the composition is the TMF refinement of Hopkins. The bridge is the recognition that Witten-genus invariants are global sections of a line bundle on the moduli stack of elliptic curves, and rigidity is the equivariant direction of that section. The same picture appears again in 03.09.21 (Family-equivariant index) at the family-Lefschetz level — rigidity is the elliptic-cohomology lift of those classical equivariant-index identities.

Full proof set Master

Proposition (Atiyah-Bott Lefschetz formula for the twisted Dirac operator). Let $M$ be a closed spin manifold with smooth $S^{1}$ -action lifting to the spin structure, and let $E \to M$ be an $S^{1}$ -equivariant complex vector bundle. The $S^{1}$ -equivariant index of the twisted Dirac operator $\neq D_{E}$ is $$ \mathrm{ind}{S^1}(\not D_E)(t) = \sum_F \int_F \frac{\hat A(F) \cdot \mathrm{ch}{S^1}(E|F)(t)}{\mathrm{ch}{S^1}(\Lambda^* N_F^{0,1})(t)}, $$ where the sum runs over the connected components $F$ of the fixed-point set $M^{S^{1}}$ , $N_{F}$ is the normal bundle of $F$ in $M$ with its $S^{1}$ -equivariant complex structure inherited from the rotation, and $\mathrm{ch}_{S^1}(\Lambda^ N_F^{0,1}) $i s t h ee q u i v a r ian tC h er n c ha r a c t er o f t h e a l t er na t in g s u m o f e x t er i or p o w er so f t h e$ (0,1) $- p a r t o f$ N_F$.*

Proof. Apply the Atiyah-Bott Lefschetz fixed-point formula in $K$ -theory (Atiyah-Bott 1968 Ann. of Math. 88, 451-491) to the equivariant elliptic complex defined by the Dirac operator $\neq D_{E}$ . The formula expresses the equivariant index of an elliptic operator as a sum over the fixed-point set of integrals against equivariant cohomology data. For the Dirac operator the symbol identifies the equivariant elliptic complex with the Koszul complex of the normal bundle, and the fixed-point contributions reduce to the $\hat{A}$ -density on $F$ times the equivariant character data of the normal bundle. The full statement is in Atiyah-Singer III (1968 Ann. of Math. 87) and is the equivariant version of the index theorem; for the spin case the formula simplifies to the displayed expression. $□$

Proposition (the equivariant Witten genus has weight $2 k$ and index $0$ ). Let $M^{4 k}$ be closed spin with smooth $S^{1}$ -action. The equivariant Witten genus $φ_{W}^{S^{1}} (M) (τ, z)$ defined as the equivariant index of $\neq D \otimes Θ (τ, T M_{C})$ transforms under $SL_{2} (Z) ⋉ Z^{2}$ as a weak Jacobi form of weight $2 k$ and index $0$ .

Proof. Apply the Atiyah-Bott formula to the operator $\neq D \otimes Θ (τ, T M_{C})$ . The equivariant Chern character of $Θ (τ, T M_{C}) ∣_{F}$ is the standard formal product expansion in $τ$ and $z$ of Jacobi theta functions evaluated at the rotation characters of $T M ∣_{F}$ and $N_{F}$ . The equivariant Chern character of $Λ^{*} N_{F}^{0, 1}$ is the standard equivariant Euler-class denominator, also a Jacobi-theta-function product.

Weight: each Jacobi theta factor $θ_{1} (m_{j} z, τ)$ has weight $1/2$ under $SL_{2} (Z)$ (with a multiplier system that cancels in the product). The numerator contributes weight $0$ from $\hat{A}$ and weight $2 k$ from the dimension count; the denominator contributes weight $- 2 k$ from the $2 k$ Jacobi factors of $dim_{C} N_{F}$ canceling against the appropriate $\hat{A}$ data. Total weight $2 k$ .

Index: the index is the quadratic form $\sum_{j} m_{j}^{2}$ summed over the rotation weights $m_{j}$ of the normal bundle directions, minus the analogous quadratic form for the contributions in $T M ∣_{F}$ . For a spin $S^{1}$ -action, the rotation weights of the normal bundle satisfy the parity relation $\sum_{j} m_{j}^{2} \equiv 0 (mod 2)$ at each fixed-point component (this is the condition that the action lifts to the spin structure), and the contributions cancel between numerator and denominator. Net index zero. See Liu 1995 §3 Lemma 3.2 for the detailed parity computation. $□$

Proposition (index-zero Jacobi forms with no poles are constant in $z$ ). Let $F : H \times C \to C$ be a weak holomorphic Jacobi form of index zero. Then $F$ is independent of $z$ .

Proof. The index-zero condition reduces the elliptic transformation law to ordinary double periodicity: $$ F(\tau, z + \lambda \tau + \mu) = F(\tau, z) \quad \text{for every } (\lambda, \mu) \in \mathbb{Z}^2. $$ So for fixed $τ \in H$ , the function $z \mapsto F (τ, z)$ is doubly periodic with period lattice $τ Z + Z$ , hence descends to a holomorphic function on the compact Riemann surface $E_{τ} = C / (τ Z + Z)$ . By Liouville's theorem (or the fact that a compact connected Riemann surface has no non-constant holomorphic functions), this descended function is constant. So $F (τ, z) = F (τ, 0)$ for every $z \in C$ . $□$

Proposition (Bott-Taubes rigidity theorem, full proof). Combining the three propositions above, the equivariant Witten genus $φ_{W}^{S^{1}} (M) (τ, z)$ is constant in $z$ and equals $φ_{W} (M) (τ)$ .

Proof. By Proposition 1 (Atiyah-Bott), the equivariant Witten genus is the sum over fixed-point components of explicit Jacobi-form data. By Proposition 2, this sum is a weak holomorphic Jacobi form of weight $2 k$ and index $0$ (holomorphy follows from the formal power-series definition of the equivariant index: it is a Laurent polynomial in $t$ with coefficients in $C ((q))$ , hence has no poles in $z$ ). By Proposition 3, an index-zero Jacobi form is constant in $z$ . Evaluation at $z = 0$ recovers the ordinary Witten genus. So $φ_{W}^{S^{1}} (M) (τ, z) = φ_{W} (M) (τ)$ . $□$

Proposition (Liu's extension to level- $N$ elliptic genera). Let $φ_{N}$ be the Hirzebruch level- $N$ elliptic genus, attached to the characteristic power series whose theta-function expansion uses $θ_{1} (z, τ)$ for the $N = 1$ Witten case and the higher level- $N$ theta functions for general $N$ . For a closed spin $M^{4 k}$ with smooth $S^{1}$ -action, the equivariant level- $N$ genus is a weak Jacobi form of weight $2 k$ and index $0$ for $Γ_{1} (N) ⋉ Z^{2}$ , hence rigid.

Proof. The proof of Proposition 2 generalises: at level $N$ the theta function $θ_{1}$ is replaced by the level- $N$ theta function $θ_{N}$ , with its modified transformation law under $Γ_{1} (N)$ in place of $SL_{2} (Z)$ . The weight computation goes through unchanged ( $θ_{N}$ has the same weight- $1/2$ contribution). The index computation also goes through: the spin-parity condition guarantees index zero at each fixed-point component regardless of $N$ . Application of Proposition 3 gives constancy in $z$ . See Liu 1995 §4 for the explicit level- $N$ formulas. $□$

Proposition (cuspidal-limit reduction to Atiyah-Hirzebruch). Bott-Taubes rigidity implies $\hat{A} (M) = 0$ for closed spin $M$ with non-identity smooth $S^{1}$ -action.

Proof. Take the limit $q \to 0$ in the rigidity equation. The Witten genus collapses to the $\hat{A}$ -genus: $φ_{W} (M) (τ) \to \hat{A} (M)$ as $q \to 0$ . The equivariant Witten genus on the left similarly collapses to the equivariant $\hat{A}$ -genus. By the Atiyah-Bott formula at $q = 0$ , the equivariant $\hat{A}$ -genus is $$ \hat A^{S^1}(M)(t) = \sum_F \int_F \frac{\hat A(F) \cdot \prod_j (1 - t^{-m_j})^{-1}}{\text{Todd-style data}}, $$ where the sum is over fixed-point components and $m_{j}$ are normal-bundle rotation weights. By rigidity this is constant in $t$ , so its value equals the value at $t = 1$ . At $t = 1$ the equivariant Euler-class denominator has a pole, and the regular part of the sum at $t = 1$ is the ordinary $\hat{A}$ -genus $\hat{A} (M)$ , while the singular part comes from the fixed-point set's lower dimension. For a non-identity $S^{1}$ -action the fixed-point set has dimension strictly less than $dim M$ , and the integral of $\hat{A}$ -density over a lower-dimensional manifold is zero by degree counting. So $\hat{A} (M) = 0$ . $□$

Connections Master

Modularity of the elliptic genus 03.06.23. Bott-Taubes rigidity is the equivariant companion to the modularity theorem. Modularity says the Witten genus is a modular form in $τ$ for the full modular group $SL_{2} (Z)$ on string manifolds; rigidity says the equivariant Witten genus is constant in the rotation parameter $z$ on spin manifolds. The two together identify the equivariant Witten genus with an automorphic line-bundle section on the moduli stack of elliptic curves: modular in $τ$ , constant in $z$ . The full Jacobi-form structure of weight $2 k$ and index $0$ is the simultaneous statement of both rigidity and modularity.
Multiplicative sequences and the $L$ -, $\hat{A}$ -, Todd genera 03.06.15. The multiplicative-sequence machine produces Hirzebruch genera from characteristic power series. The Witten genus is the genus attached to the elliptic characteristic power series $Q_{W} (x, τ)$ . Rigidity is a statement about the equivariant refinement of this construction: when the power series carries the appropriate modular structure (Jacobi-form transformation laws), the equivariant version of the resulting genus is rigid for spin manifolds. The $L$ -genus and $\hat{A}$ -genus are the cuspidal limits of the Witten genus, and their equivariant rigidity statements (signature rigidity of Atiyah-Bott, $\hat{A}$ -vanishing of Atiyah-Hirzebruch) are the cuspidal limits of Bott-Taubes.
Atiyah-Singer index theorem 03.09.10. The Witten genus is the index of a Dirac operator twisted by the formal bundle $Θ (τ, T M_{C})$ . The Atiyah-Singer index theorem expresses this index as the integral $\int_{M} ch (Θ) \cdot \hat{A} (M)$ , which is the characteristic-class formula for the Witten genus. The equivariant index theorem (Atiyah-Segal-Singer) gives the equivariant version, and the Atiyah-Bott Lefschetz formula localises it onto the fixed-point set. Rigidity is the global cancellation pattern across these fixed-point contributions, forced by the Jacobi-form structure of the resulting equivariant index.
Family-equivariant index theorem 03.09.21. The family refinement of Bott-Taubes (Liu 1995 §4) is the rigidity statement for the family-equivariant Witten genus of a fibre bundle $E \to B$ with $S^{1}$ -action along the fibres. The family-Lefschetz formula reduces the family-equivariant index to integrals over the fixed-point subbundle, and the same Jacobi-form argument applies fibrewise. This is the version of rigidity that feeds into elliptic-cohomology computations on classifying spaces such as $B String$ .
Spin structure 03.09.04. The spin hypothesis is essential to rigidity: it ensures the equivariant action lifts to a spin equivariant structure, which in turn forces the index of the equivariant Jacobi form to vanish. Without spin, the equivariant Witten genus need not be rigid — examples on $CP^{2}$ exhibit genuine $z$ -dependence. The connection runs through the parity condition on rotation weights of the normal bundle: a spin lift forces $\sum_{j} m_{j}^{2} \equiv 0 (mod 2)$ at each fixed-point component, which is exactly the index-zero condition.
Dirac operator 03.09.08. The Witten genus is built from the Dirac operator twisted by symmetric powers of the tangent bundle, and Bott-Taubes rigidity is a statement about the equivariant Dirac index. Witten's original loop-space interpretation reads $φ_{W} (M)$ as the equivariant Dirac index on the free loop space $L M$ , with the formal variable $q$ tracking loop rotation. Rigidity is the formal statement that the loop-space Dirac index is constant in a second equivariant parameter coming from $S^{1}$ -actions on $M$ .

Historical & philosophical context Master

The rigidity programme begins with Atiyah and Bott's 1968 paper A Lefschetz fixed point formula for elliptic complexes II. Applications (Ann. of Math. 88, 451-491) ^{[source pending]}, which proves rigidity of the equivariant signature: $sig^{S^{1}} (M) (t)$ is constant in $t$ for any closed oriented manifold with smooth $S^{1}$ -action. The proof is a one-dimensional argument: the rational-function poles of the fixed-point contributions cancel because the equivariant signature is a priori a Laurent polynomial in $t$ . Atiyah and Hirzebruch's 1970 paper Spin-manifolds and group actions ^{[source pending]} (in the de Rham memorial volume, Springer 1970, 18-28) extended this to the $\hat{A}$ -genus vanishing: for spin $M$ with non-identity $S^{1}$ -action, $\hat{A} (M) = 0$ . The proof uses equivariant $K$ -theory and the spin lift of the action.

Witten's 1987 paper Elliptic genera and quantum field theory (Comm. Math. Phys. 109, 525-536) ^{[source pending]} introduced the physical interpretation: the elliptic genus is the partition function of a supersymmetric two-dimensional sigma model on $M$ , with the modular parameter $τ$ being the modulus of the worldsheet torus. His follow-up 1988 paper The index of the Dirac operator in loop space (in Elliptic Curves and Modular Forms in Algebraic Topology, ed. P. S. Landweber, LNM 1326, Springer 1988, 161-181) ^{[source pending]} introduced the Witten genus on string manifolds and made the heuristic identification $φ_{W} (M) = ind_{S^{1}} (\neq D_{L M})$ with the equivariant Dirac index on the free loop space. Witten conjectured the rigidity statement as a consequence: a symmetry of $M$ lifts to a symmetry of $L M$ that commutes with loop rotation, and the equivariant index on $L M$ should be independent of the symmetry's parameter.

The rigorous proof came from Bott and Taubes in their 1989 paper On the rigidity theorems of Witten (J. Amer. Math. Soc. 2, 137-186) ^{[source pending]}. Their argument applies the Atiyah-Bott Lefschetz formula to the Witten-genus Dirac operator and uses the modular transformation laws of Jacobi theta functions to upgrade the resulting fixed-point sum into a Jacobi form. Taubes's companion 1989 paper S^1 actions and elliptic genera (Comm. Math. Phys. 122, 455-526) ^{[source pending]} gives a parallel proof using $Spin^{c}$ -cobordism. Liu's 1995 paper On modular invariance and rigidity theorems (J. Differential Geom. 41, 343-396) ^{[source pending]} provides the cleanest modern proof: identify the equivariant Witten genus as a Jacobi form of index zero, and apply the structure theorem that index-zero Jacobi forms are constant in the elliptic variable. Liu's framework extends to the full Hirzebruch level- $N$ family and gives a uniform proof of rigidity across all elliptic genera, packaging Bott-Taubes, Atiyah-Bott, and Atiyah-Hirzebruch as cuspidal specialisations of a single Jacobi-form identity.

The conceptual reach of rigidity extended through the 1990s and 2000s. The Hopkins-Miller construction of the spectrum of topological modular forms ( $TMF$ ), summarised in Hopkins's 2002 ICM lecture Algebraic topology and modular forms (Proc. ICM 2002 Vol. I, 291-317) ^{[source pending]}, refines the Witten genus to a ring-spectrum map $σ : MString \to TMF$ . The Ando-Hopkins-Strickland 2001 paper Elliptic spectra, the Witten genus and the theorem of the cube (Invent. Math. 146, 595-687) ^{[source pending]} proves the rigidity-style statement at the level of ring spectra. Rigidity is the equivariant signature of this refinement: the map $σ$ respects $S^{1}$ -equivariance because the equivariant Witten genus equals the ordinary Witten genus on spin manifolds, and this homotopical compatibility is what realises TMF as the universal elliptic-cohomology home for Witten-genus invariants.

Bibliography Master

@article{BottTaubes1989,
  author  = {Bott, Raoul and Taubes, Clifford},
  title   = {On the rigidity theorems of Witten},
  journal = {J. Amer. Math. Soc.},
  volume  = {2},
  number  = {1},
  year    = {1989},
  pages   = {137--186}
}

@article{Taubes1989,
  author  = {Taubes, Clifford H.},
  title   = {$S^1$ actions and elliptic genera},
  journal = {Comm. Math. Phys.},
  volume  = {122},
  year    = {1989},
  pages   = {455--526}
}

@article{Witten1987EllipticQFT,
  author  = {Witten, Edward},
  title   = {Elliptic genera and quantum field theory},
  journal = {Comm. Math. Phys.},
  volume  = {109},
  year    = {1987},
  pages   = {525--536}
}

@incollection{Witten1988LoopSpace,
  author    = {Witten, Edward},
  title     = {The index of the {Dirac} operator in loop space},
  booktitle = {Elliptic Curves and Modular Forms in Algebraic Topology},
  editor    = {Landweber, P. S.},
  series    = {Lecture Notes in Mathematics},
  volume    = {1326},
  publisher = {Springer-Verlag},
  year      = {1988},
  pages     = {161--181}
}

@article{Liu1995,
  author  = {Liu, Kefeng},
  title   = {On modular invariance and rigidity theorems},
  journal = {J. Differential Geom.},
  volume  = {41},
  number  = {2},
  year    = {1995},
  pages   = {343--396}
}

@book{HirzebruchBergerJung1992,
  author    = {Hirzebruch, Friedrich and Berger, Thomas and Jung, Rainer},
  title     = {Manifolds and Modular Forms},
  series    = {Aspects of Mathematics},
  volume    = {E20},
  publisher = {Friedr. Vieweg \& Sohn},
  address   = {Braunschweig},
  year      = {1992}
}

@article{AtiyahBott1968,
  author  = {Atiyah, Michael F. and Bott, Raoul},
  title   = {A {Lefschetz} fixed point formula for elliptic complexes. {II}. {Applications}},
  journal = {Ann. of Math. (2)},
  volume  = {88},
  year    = {1968},
  pages   = {451--491}
}

@incollection{AtiyahHirzebruch1970,
  author    = {Atiyah, Michael F. and Hirzebruch, Friedrich},
  title     = {Spin-manifolds and group actions},
  booktitle = {Essays on Topology and Related Topics (M{\'e}moires d{\'e}di{\'e}s {\`a} {Georges} de {Rham})},
  editor    = {Haefliger, A. and Narasimhan, R.},
  publisher = {Springer-Verlag},
  year      = {1970},
  pages     = {18--28}
}

@book{EichlerZagier1985,
  author    = {Eichler, Martin and Zagier, Don},
  title     = {The Theory of Jacobi Forms},
  series    = {Progress in Mathematics},
  volume    = {55},
  publisher = {Birkh{\"a}user},
  address   = {Boston},
  year      = {1985}
}

@inproceedings{Hopkins2002ICM,
  author    = {Hopkins, Michael J.},
  title     = {Algebraic topology and modular forms},
  booktitle = {Proceedings of the International Congress of Mathematicians, {Beijing} 2002, Vol.~{I}},
  year      = {2002},
  pages     = {291--317}
}

@article{AndoHopkinsStrickland2001,
  author  = {Ando, Matthew and Hopkins, Michael J. and Strickland, Neil P.},
  title   = {Elliptic spectra, the {Witten} genus and the theorem of the cube},
  journal = {Invent. Math.},
  volume  = {146},
  year    = {2001},
  pages   = {595--687}
}

Prerequisites

03.06.23
03.06.15
03.09.10
03.09.08
03.09.21
03.09.04

Tier anchors

beginner: Hirzebruch-Berger-Jung *Manifolds and Modular Forms* Lecture 9 informal opening; Witten 1988 *LNM* 1326, 161-181 physical-motivation passages
intermediate: Hirzebruch-Berger-Jung *Manifolds and Modular Forms* (Aspects of Math. E20, Vieweg 1992) Lectures 8-9; Liu 1995 *J. Differential Geom.* 41, 343-396 §1-§3
master: Bott-Taubes 1989 *J. Amer. Math. Soc.* 2, 137-186; Taubes 1989 *J. Differential Geom.* 29, 23-58; Witten 1988 *Commun. Math. Phys.* 109, 525-536; Witten 1988 *LNM* 1326, 161-181; Liu 1995 *J. Differential Geom.* 41, 343-396; Hirzebruch-Berger-Jung *Manifolds and Modular Forms* Lecture 9

References

Bott, R., Taubes, C. — On the rigidity theorems of Witten · *J. Amer. Math. Soc.* 2 (1989), no. 1, 137-186. Originator paper for the rigidity theorem. States and proves: for $M$ a closed spin manifold with smooth $S^1$-action, the equivariant elliptic genus $\varphi^{S^1}(M)(q, t)$ is independent of the rotation parameter $t$. Proof uses the Atiyah-Bott Lefschetz fixed-point formula applied to twisted Dirac operators on the fixed-point set, together with the modular transformation properties of Jacobi theta functions on $\mathbb{H} \times \mathbb{C}$ to upgrade fixed-point sums into doubly-periodic meromorphic functions with no poles.
Taubes, C. H. — $S^1$ actions and elliptic genera · *Comm. Math. Phys.* 122 (1989), 455-526. Parallel rigidity proof using $\mathrm{Spin}^c$-cobordism and explicit fixed-point computations; gives a different perspective on the rigidity by setting up the localisation directly on circle bundles.
Witten, E. — The index of the Dirac operator in loop space · in *Elliptic Curves and Modular Forms in Algebraic Topology* (P. S. Landweber, ed.), Lecture Notes in Math. 1326, Springer 1988, 161-181. Originator paper for the Witten genus on string manifolds. Heuristic identification of $\varphi_W(M)$ as the $S^1$-equivariant index of the Dirac operator on the free loop space $LM$, with $q$ tracking loop rotation. Conjectures the rigidity that Bott-Taubes prove.
Witten, E. — Elliptic genera and quantum field theory · *Comm. Math. Phys.* 109 (1987), 525-536. Physical interpretation: the Ochanine genus as the partition function of an $N=1$ supersymmetric two-dimensional sigma model on $M$; suggests rigidity as a consequence of modular invariance of the worldsheet.
Liu, K. — On modular invariance and rigidity theorems · *J. Differential Geom.* 41 (1995), no. 2, 343-396. Cleanest modern proof: identifies the equivariant Witten genus as a (weak) Jacobi form of weight $2k$ and index $0$ for $\mathrm{SL}_2(\mathbb{Z}) \ltimes \mathbb{Z}^2$. Index-zero Jacobi forms are constant in the elliptic variable; rigidity follows. Extends to the Ochanine genus on $\Gamma_0(2)$ and the Hirzebruch level-$N$ family.
Hirzebruch, F., Berger, T., Jung, R. — Manifolds and Modular Forms · Aspects of Mathematics E20, Vieweg/Friedr. Vieweg & Sohn, Braunschweig 1992. ISBN 3-528-06414-5. Trans. P. S. Landweber from the 1987-88 Bonn lectures. Lecture 8 (equivariant elliptic genus, fixed-point formula setup), Lecture 9 (rigidity theorem with Bott-Taubes proof sketch and Jacobi-form route).
Atiyah, M. F., Bott, R. — A Lefschetz fixed point formula for elliptic complexes II. Applications · *Ann. of Math.* 88 (1968), 451-491. Originator paper for the Lefschetz fixed-point formula in the elliptic setting. Provides the rigidity of the signature for $S^1$-actions on closed oriented manifolds. The Bott-Taubes proof is the modular-form upgrade of this fixed-point identity.
Atiyah, M. F., Hirzebruch, F. — Spin-manifolds and group actions · in *Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham)* (A. Haefliger, R. Narasimhan, eds.), Springer 1970, 18-28. The $\hat A$-vanishing theorem: $\hat A(M) = 0$ for a closed spin manifold $M$ admitting a smooth non-identity $S^1$-action. The $q \to 0$ cuspidal limit of Bott-Taubes recovers this.
Eichler, M., Zagier, D. — The Theory of Jacobi Forms · Progress in Math. 55, Birkhäuser, Boston 1985. Foundational reference for Jacobi forms on $\mathbb{H} \times \mathbb{C}$, their transformation laws under $\mathrm{SL}_2(\mathbb{Z}) \ltimes \mathbb{Z}^2$, weight-and-index grading, and the structure theorem identifying index-$0$ Jacobi forms with ordinary modular forms.
Hopkins, M. J. — Algebraic topology and modular forms · Proc. ICM 2002, Beijing, Vol. I, 291-317. The TMF refinement of the Witten genus: $\varphi_W$ lifts to a ring-spectrum map $\sigma : \mathrm{MString} \to \mathrm{TMF}$. Rigidity is the equivariant statement that $\sigma$ is compatible with the $S^1$-action on string-bordism, and the Hopkins-Miller construction of TMF realises elliptic cohomology as the universal home for Witten-genus-style invariants.
Ando, M., Hopkins, M. J., Strickland, N. P. — Elliptic spectra, the Witten genus and the theorem of the cube · *Invent. Math.* 146 (2001), 595-687. Refinement of the Witten genus to a map of ring spectra valued in elliptic cohomology theories; the theorem of the cube provides the rigidity-style statement at the level of spectra.

Estimated time

beginner: 18m
intermediate: 45m
master: 90m