Mod 3 congruence families for twisted signaturesof string manifolds in dimensions up to twenty-eight

1. Introduction

The index of an elliptic operator is an integer expressed by a rational characteristic number. This relation underlies many divisibility and congruence theorems. The signature theorem of Hirzebruch [4] and the index theorem of Atiyah and Singer [2] are the classical examples. On spin manifolds, the Rokhlin–Ochanine divisibility theorem gives further restrictions on the signature; see [10, 11]. Elliptic genera add modular symmetry to this picture. Witten’s construction [13] and its mathematical developments by Landweber–Stong and Liu [5, 7, 8] show that a modular transformation can convert an equality of characteristic forms into a family of arithmetic identities between indices.

The present paper concerns the generalized anomaly-cancellation forms of Liu and Wang [9]. Their construction depends on two integers $(a,b)$ and an auxiliary even-rank Euclidean bundle $V$ . Chen and Han [3] applied the specialization

\dim M=24,\qquad V=TM,\qquad(a,b)=(0,1)

to obtain a mod $3$ congruence between a twisted signature and a twisted Dirac index. Together with the $(1,0)$ comparison and a second-coefficient identity for the Witten genus, they proved

\operatorname{Sig}(M,\Lambda^{2}T_{\mathbb{C}}M)\equiv\operatorname{Ind}(D\otimes T_{\mathbb{C}}M)\pmod{3}

for every $24$ -dimensional string manifold. In particular, the twisted signature on the left is divisible by $3$ because the Rarita–Schwinger index on the right is divisible by $3$ in this dimension [3, Theorem 0.1].

We retain both parameters and compute the relevant Fourier coefficients uniformly. Working with the reduced tangent bundle separates these coefficients from dimension-dependent trivial summands. A triangular expansion in the level-two modular generators then gives comparisons in dimensions $4,8,12,16,20,24,28$ . The parameter analysis below concerns specific cancellations of universal coefficients; it makes no claim that other parameter values cannot yield additional divisibility on particular manifolds.

Notation for the index pairings

Let $M^{4k}$ be closed and string. In particular, $M$ is spin, so its complex spinor bundle $\mathcal{S}(M)=\Delta(TM)$ is defined. For a virtual complex bundle $F$ and an integer $r\geq 0$ , set

(1.1)

\mathcal{I}_{r}(F):=\left\langle\widehat{A}(TM)\operatorname{ch}\!\left(\mathcal{S}(M)^{\otimes r}\right)\operatorname{ch}(F),[M]\right\rangle.

Thus $\mathcal{I}_{0}(F)$ is the index of the spin Dirac operator twisted by $F$ . The identity $\widehat{L}(TM)=\widehat{A}(TM)\operatorname{ch}(\mathcal{S}(M))$ gives

(1.2)

\mathcal{I}_{r}(F)=\operatorname{Sig}\!\left(M,\mathcal{S}(M)^{\otimes(r-1)}\otimes F\right),\qquad r\geq 1.

Both identities extend additively to virtual bundles. Complete definitions and normalization conventions are given in Section 2.

Put $E=\widetilde{T_{\mathbb{C}}M}=T_{\mathbb{C}}M-\mathbb{C}^{4k}$ . For $a,b\geq 0$ , define the following three virtual bundles:

(1.3)	$\displaystyle\mathcal{A}_{a,b}$	$\displaystyle=(1+a)E+b(\Lambda^{2}E-S^{2}E),$
(1.4)	$\displaystyle\mathcal{B}^{(1)}_{a,b}$	$\displaystyle=(b-a)E,$
(1.5)	$\displaystyle\mathcal{B}^{(2)}_{a,b}$	$\displaystyle=(1+b)E+(a+b)\Lambda^{2}E+\frac{(a-b)^{2}-(a+b)}{2}E\otimes E.$

The coefficient in the last line is integral because $(a-b)^{2}\equiv a+b\pmod{2}$ . These bundles arise as the $q$ and $q^{1/2},q$ coefficients of the two Liu–Wang theta bundles; see Proposition 3.2.

Our first main result is a uniform comparison across dimensions.

Theorem 1.1.

Let $M^{4k}$ be a closed string manifold for $1\leq k\leq 7$ , and let $a,b\in\mathbb{Z}_{\geq 0}$ . With the bundles defined in (1.3)–(1.5), we have the following congruences:

(1.6)	$\displaystyle k=1:$	$\displaystyle\qquad\mathcal{I}_{a}(\mathcal{A}_{a,b})\equiv 0\pmod{3};$
(1.7)	$\displaystyle k=2,3:$	$\displaystyle\qquad\mathcal{I}_{a}(\mathcal{A}_{a,b})+\mathcal{I}_{b}(\mathcal{B}^{(1)}_{a,b})\equiv 0\pmod{3};$
(1.8)	$\displaystyle k=4,5,6,7:$	$\displaystyle\qquad\mathcal{I}_{a}(\mathcal{A}_{a,b})-\mathcal{I}_{b}(\mathcal{B}^{(2)}_{a,b})\equiv 0\pmod{3}.$

Corollary 1.1 (Genuine-bundle form of Theorem 1.1).

Under the hypotheses of Theorem 1.1, put $d=4k$ and $T=T_{\mathbb{C}}M$ . Introduce the integer indices

J_{r}=\mathcal{I}_{r}(\mathbb{C}),\quad U_{r}=\mathcal{I}_{r}(T),\quad V_{r}=\mathcal{I}_{r}(\Lambda^{2}T),\quad W_{r}=\mathcal{I}_{r}(S^{2}T),\quad Z_{r}=\mathcal{I}_{r}(T\otimes T).

Every twisting bundle in these definitions is genuine; in particular, $J_{r}=\operatorname{Ind}(D\otimes\mathcal{S}(M)^{\otimes r})$ , and similarly for the other four indices. For $r\geq 1$ , these may equivalently be written as twisted signatures using (1.2). Set

	$\displaystyle c$	$\displaystyle=\frac{(a-b)^{2}-(a+b)}{2},$
	$\displaystyle\alpha$	$\displaystyle=1+b-d(a+b)-2dc,$
	$\displaystyle\beta$	$\displaystyle=-d(1+b)+(a+b)\binom{d+1}{2}+cd^{2}.$

Then the three congruences of that theorem become, respectively: for $k=1$ ,

(C1.1)

(1+a)U_{a}+bV_{a}+d(b-a-1)J_{a}\equiv bW_{a}\pmod{3};

for $k=2,3$ ,

(C1.2)

\begin{split}&(1+a)U_{a}+bV_{a}+d(b-a-1)J_{a}+(b-a)U_{b}\\ &\hskip 71.13188pt\equiv bW_{a}+d(b-a)J_{b}\pmod{3};\end{split}

for $k=4,5,6,7$ ,

(C1.3)

\begin{split}&(1+a)U_{a}+bV_{a}+d(b-a-1)J_{a}\\ &\hskip 34.1433pt\equiv bW_{a}+(a+b)V_{b}+cZ_{b}+\alpha U_{b}+\beta J_{b}\pmod{3}.\end{split}

Thus every index in these congruences is twisted by a genuine bundle.

Corollary 1.1 is the genuine-bundle reformulation of Theorem 1.1; it has the same hypotheses and modulus. The next theorem records the concise genuine-bundle congruences obtained from selected parameter choices, using the additional coefficient relations established in Sections 5 and 6.

Theorem 1.2.

Let $M^{d}$ be a closed string manifold, where $d=8,12,16,20,24$ , or $28$ . Use the genuine-bundle indices $J_{r},U_{r},V_{r},W_{r}$ of Corollary 1.1. The following two congruences hold in each indicated dimension. Every congruence in the table is modulo $3$ .

\begin{array}[]{c|c|c}d&\text{Dirac congruence}&\text{spinor-twisted congruence}\\ \hline\cr 8&U_{0}\equiv 2J_{0}&U_{2}+V_{0}+2J_{0}\equiv 2J_{2}+W_{0}\\ 12&U_{0}\equiv 0&U_{2}+V_{0}\equiv W_{0}\\ 16,28&W_{0}\equiv J_{0}&W_{2}+V_{0}+J_{0}\equiv U_{2}+U_{0}\\ 20&W_{0}\equiv U_{0}+J_{0}&W_{2}+V_{0}+J_{2}\equiv 2(U_{2}+U_{0})\\ 24&W_{0}+U_{0}\equiv 0&W_{2}+V_{0}\equiv 0\end{array}

In particular, the two twenty-four-dimensional formulas are

	$\displaystyle\operatorname{Ind}(D\otimes S^{2}T)+\operatorname{Ind}(D\otimes T)$	$\displaystyle\equiv 0\pmod{3},$
	$\displaystyle\operatorname{Sig}(M,\mathcal{S}(M)\otimes S^{2}T)+\operatorname{Ind}(D\otimes\Lambda^{2}T)$	$\displaystyle\equiv 0\pmod{3}.$

All twists appearing here are genuine complex vector bundles.

Theorem 1.2 gives selected mod $3$ consequences, whereas the following theorem strengthens its Dirac column in dimensions $16,20,24,28$ under the same string hypothesis. The extra input is the level-one Witten genus. No higher modulus is asserted for the spinor-twisted column or for the full $(a,b)$ -comparison in Theorem 1.1, and the displayed moduli are not claimed to be optimal. The proof is given in Section 6.8.

Theorem 1.3.

Let $M^{d}$ be a closed string manifold, where $d=16,20,24$ , or $28$ , and put $T=T_{\mathbb{C}}M$ . With the genuine-bundle indices

J_{0}=\operatorname{Ind}(D)=\widehat{A}(M),\qquad U_{0}=\operatorname{Ind}(D\otimes T),\qquad W_{0}=\operatorname{Ind}(D\otimes S^{2}T),

the following congruences hold:

(1.9)

\begin{array}[]{c|c|c}d&\text{congruence}&\text{modulus}\\ \hline\cr 16&W_{0}\equiv J_{0}&729\\ 20&W_{0}\equiv 0&27\\ 24&W_{0}+U_{0}\equiv 0&27\\ 28&W_{0}\equiv J_{0}&9\end{array}.

In dimension $20$ , one also has $U_{0}+J_{0}\equiv 0\pmod{27}$ , so that $W_{0}\equiv U_{0}+J_{0}\pmod{27}$ . In particular, all four Dirac congruences in dimensions $16,20,24,28$ in Theorem 1.2 remain valid with modulus $9$ in place of $3$ .

The two parameter choices used for the high-dimensional statements are $(2,0)$ and $(0,2)$ . They must be distinguished from the diagonal choice $(2,2)$ : for every $s\geq 0$ ,

\mathcal{B}^{(2)}_{s,s}=(1+s)E+2s\Lambda^{2}E-sE^{\otimes 2}=\mathcal{A}_{s,s}.

Consequently the high-dimensional comparison with $a=b=s$ is an identity and supplies no additional congruence by itself.

This paper is organized as follows. Section 2 introduces the geometric conventions and modular forms used throughout the paper. In Section 3, we establish the specialized modular transformation law and compute the Fourier coefficients in reduced $K$ -theory. These calculations lead to the proof of Theorem 1.1 in Section 4, where denominators are cleared before taking residues. Section 5 studies the three parameter families in dimension $24$ and recovers the Chen–Han comparisons. Together with the remaining dimension calculations in Section 6, this proves Theorem 1.2. The latter section also discusses the limitation in dimension $32$ and proves Theorem 1.3 using the level-one Witten genus. Finally, in Section 7, we present two non-string counterexamples illustrating the role of the string hypothesis.

2. Preliminaries

We recall the geometric and modular conventions used below.

2.1. Spin and string structures

All manifolds in this paper are smooth, closed, and oriented. Let $M$ be of real dimension $4k$ , with $TM$ equipped with a Riemannian metric and its Levi–Civita connection $\nabla^{TM}$ . The orientation then gives rise to an $\mathrm{SO}(4k)$ -principal bundle of oriented orthonormal frames.

Definition 2.1.

[6, Chapter II] and [12] A spin structure on $M$ is a lift of the oriented frame bundle from $SO(4k)$ to $\operatorname{Spin}(4k)$ . Such a lift exists precisely when $w_{2}(TM)=0$ . For a spin bundle, the class $p_{1}(TM)$ has a canonical half $\tfrac{1}{2}p_{1}(TM)\in H^{4}(M;\mathbb{Z})$ . A string structure is a spin structure together with a trivialization of this half Pontryagin class. In particular,

(2.1)

\frac{1}{2}p_{1}(TM)=0\quad\Longrightarrow\quad p_{1}(TM)=0.

Only the cohomological consequence in (2.1) enters the characteristic-number calculations. The stronger language of a string structure is retained because it is the natural hypothesis for the Witten genus; the underlying spin structure supplies the spinor bundle used below. In fact, the characteristic-number arguments only require that $M$ be spin and that $p_{1}(TM)$ vanish in rational cohomology. This weaker condition need not imply the existence of a string structure. Background on the obstruction-theoretic definition can be found in [6, Chapter II] and [12].

Let $\mathcal{S}(M)=\Delta(TM)$ be the complex spinor bundle associated with the complex spin representation. Since the dimension is even, it is $\mathbb{Z}_{2}$ -graded,

\mathcal{S}(M)=\mathcal{S}^{+}(M)\oplus\mathcal{S}^{-}(M).

We use $\mathcal{S}(M)^{\otimes r}$ for the ordinary $r$ -fold tensor power and put $\mathcal{S}(M)^{\otimes 0}=\mathbb{C}$ .

2.2. Complex $K$ -theory and power operations

Let $K(M)$ denote complex topological $K$ -theory. An expression such as $F-G$ means the corresponding virtual bundle. Its Chern character is defined additively, $\operatorname{ch}(F-G)=\operatorname{ch}(F)-\operatorname{ch}(G)$ , and an elliptic index twisted by $F-G$ is the difference of the two ordinary twisted indices.

For a complex vector bundle $W$ , define the total exterior and symmetric powers by

(2.2)

\Lambda_{t}(W)=\sum_{j\geq 0}t^{j}\Lambda^{j}W,\qquad S_{t}(W)=\sum_{j\geq 0}t^{j}S^{j}W.

They extend multiplicatively to virtual bundles by

(2.3)

S_{t}(W)=\Lambda_{-t}(W)^{-1},\qquad\Lambda_{t}(W-Z)=\frac{\Lambda_{t}(W)}{\Lambda_{t}(Z)}.

These are identities in the completed ring $K(M)[[t]]$ ; see [1, Chapter III]. The first coefficients are

(2.4)		$\displaystyle\Lambda_{t}(W)$	$\displaystyle=1+tW+t^{2}\Lambda^{2}W+O(t^{3}),$
(2.5)		$\displaystyle S_{t}(W)$	$\displaystyle=1+tW+t^{2}S^{2}W+O(t^{3}).$

The decomposition of the tensor square gives

(2.6)

W\otimes W=S^{2}W+\Lambda^{2}W

in $K(M)$ . Since the power operations are defined on the whole $\lambda$ -ring $K(M)$ , all three expressions in (2.6) also make sense for a virtual class. The identity itself follows for virtual classes by comparing the coefficient of $t^{2}$ in $S_{t}(W)\Lambda_{-t}(W)=1$ .

For a rank- $r$ bundle $W$ , its reduced class is

\widetilde{W}=W-\mathbb{C}^{r}.

The central reduced class of this paper is

(2.7)

E=\widetilde{T_{\mathbb{C}}M}=T_{\mathbb{C}}M-\mathbb{C}^{4k}.

It has virtual rank zero. Reduction eliminates the constant Fourier contributions that would otherwise obscure the modular coefficients.

2.3. Characteristic forms

For a real Euclidean bundle $W$ with connection $\nabla^{W}$ and curvature $R^{W}=(\nabla^{W})^{2}$ , define the Hirzebruch forms

(2.8)		$\displaystyle\widehat{A}(W,\nabla^{W})$	$\displaystyle=\det{}^{1/2}\!\left(\frac{\frac{\sqrt{-1}}{4\pi}R^{W}}{\sinh(\frac{\sqrt{-1}}{4\pi}R^{W})}\right),$
(2.9)		$\displaystyle\widehat{L}(W,\nabla^{W})$	$\displaystyle=\det{}^{1/2}\!\left(\frac{\frac{\sqrt{-1}}{2\pi}R^{W}}{\tanh(\frac{\sqrt{-1}}{4\pi}R^{W})}\right).$

This is the normalization used in [9, 3]. For a Hermitian bundle $F$ with unitary connection and curvature $R^{F}$ , the Chern character form is

(2.10)

\operatorname{ch}(F,\nabla^{F})=\operatorname{tr}\exp\!\left(\frac{\sqrt{-1}}{2\pi}R^{F}\right).

When a characteristic form is paired with $[M]$ , we suppress the connections because the result depends only on the corresponding cohomology classes.

If the formal Chern roots of $T_{\mathbb{C}}M$ are $\{\pm x_{j}\}_{j=1}^{2k}$ , the splitting principle gives

(2.11)	$\displaystyle\widehat{A}(TM)$	$\displaystyle=\prod_{j=1}^{2k}\frac{x_{j}/2}{\sinh(x_{j}/2)},$
(2.12)	$\displaystyle\widehat{L}(TM)$	$\displaystyle=\prod_{j=1}^{2k}\frac{x_{j}}{\tanh(x_{j}/2)},$
(2.13)	$\displaystyle\operatorname{ch}(\mathcal{S}(M))$	$\displaystyle=\prod_{j=1}^{2k}\left(e^{x_{j}/2}+e^{-x_{j}/2}\right).$

Lemma 2.2.

On a spin manifold, with the normalization above,

(2.14)

\widehat{L}(TM)=\widehat{A}(TM)\operatorname{ch}(\mathcal{S}(M)).

Proof.

It suffices by the splitting principle to compare one factor for each $x_{j}$ . Multiplying the factors in (2.11) and (2.13) gives

\frac{x_{j}/2}{\sinh(x_{j}/2)}\left(e^{x_{j}/2}+e^{-x_{j}/2}\right)=\frac{x_{j}\cosh(x_{j}/2)}{\sinh(x_{j}/2)}=\frac{x_{j}}{\tanh(x_{j}/2)},

which is the factor in (2.12). Taking the product proves the identity. ∎

The normalization of $\widehat{L}$ must be distinguished from the usual total Hirzebruch class $L_{\mathrm{std}}(TM)=\prod_{j}x_{j}/\tanh x_{j}$ . In dimension $4k$ ,

\widehat{L}(TM)^{(4j)}=2^{2k-2j}L_{\mathrm{std}}(TM)^{(4j)}.

Their top-degree terms agree, but their lower-degree terms do not. Consequently $\widehat{L}$ cannot be replaced by $L_{\mathrm{std}}$ in a twisted index pairing. On a spin manifold, the Clifford-module identification of complex differential forms with $\mathcal{S}(M)\otimes\mathcal{S}(M)^{*}$ , with signature grading on the first factor, gives the index density $\widehat{A}\operatorname{ch}(\mathcal{S}(M))\operatorname{ch}(F)$ , as used below.

2.4. Twisted Dirac indices and twisted signatures

Let $F$ be a Hermitian bundle with connection. The spin Dirac operator twisted by $F$ has chiral part

D_{F}^{+}:\Gamma(\mathcal{S}^{+}(M)\otimes F)\longrightarrow\Gamma(\mathcal{S}^{-}(M)\otimes F).

The Atiyah–Singer index theorem [2] states that

(2.15)

\operatorname{Ind}(D_{F}^{+})=\langle\widehat{A}(TM)\operatorname{ch}(F),[M]\rangle.

The signature operator acts on complex differential forms and is graded by the usual chirality involution. Its twist by $F$ has index

(2.16)

\operatorname{Sig}(M,F)=\langle\widehat{L}(TM)\operatorname{ch}(F),[M]\rangle;

see [4, 6]. For $F=\mathbb{C}$ this is the ordinary signature of the cup-product form on middle-dimensional cohomology.

Definition 2.3.

For $r\geq 0$ and $F\in K(M)$ , define

(2.17)

\mathcal{I}_{r}(F)=\langle\widehat{A}(TM)\operatorname{ch}(\mathcal{S}(M)^{\otimes r})\operatorname{ch}(F),[M]\rangle.

The integrality of (2.17) follows from (2.15); tensoring the spinor bundle simply changes the twisting bundle. Lemma 2.2 and (2.16) imply, for $r\geq 1$ ,

(2.18)

\mathcal{I}_{r}(F)=\operatorname{Sig}\!\left(M,\mathcal{S}(M)^{\otimes(r-1)}\otimes F\right).

For $b=1$ , this is the ordinary $F$ -twisted signature; larger values introduce additional spinor tensor factors.

Remark 2.4.

If $F=F_{+}-F_{-}$ is virtual, then $\mathcal{I}_{r}(F)=\mathcal{I}_{r}(F_{+})-\mathcal{I}_{r}(F_{-})$ . Consequently every congruence below is an ordinary congruence of integers, even though reduced and virtual bundles are used to write it economically.

2.5. Theta functions

Let $\tau\in\mathbb{H}$ and $q=e^{2\pi\sqrt{-1}\tau}$ . We use the product normalizations

(2.19)	$\displaystyle\theta(v,\tau)$	$\displaystyle=2q^{1/8}\sin(\pi v)\prod_{n=1}^{\infty}(1-q^{n})(1-e^{2\pi\sqrt{-1}v}q^{n})(1-e^{-2\pi\sqrt{-1}v}q^{n}),$
(2.20)	$\displaystyle\theta_{1}(v,\tau)$	$\displaystyle=2q^{1/8}\cos(\pi v)\prod_{n=1}^{\infty}(1-q^{n})(1+e^{2\pi\sqrt{-1}v}q^{n})(1+e^{-2\pi\sqrt{-1}v}q^{n}),$
(2.21)	$\displaystyle\theta_{2}(v,\tau)$	$\displaystyle=\prod_{n=1}^{\infty}(1-q^{n})(1-e^{2\pi\sqrt{-1}v}q^{n-1/2})(1-e^{-2\pi\sqrt{-1}v}q^{n-1/2}),$
(2.22)	$\displaystyle\theta_{3}(v,\tau)$	$\displaystyle=\prod_{n=1}^{\infty}(1-q^{n})(1+e^{2\pi\sqrt{-1}v}q^{n-1/2})(1+e^{-2\pi\sqrt{-1}v}q^{n-1/2}).$

These conventions agree with [9, Section 2] and [3, Section 1]. Define

(2.23)		$\displaystyle\delta_{1}(\tau)$	$\displaystyle=\frac{1}{8}\bigl(\theta_{2}(0,\tau)^{4}+\theta_{3}(0,\tau)^{4}\bigr),$	$\displaystyle\varepsilon_{1}(\tau)$	$\displaystyle=\frac{1}{16}\theta_{2}(0,\tau)^{4}\theta_{3}(0,\tau)^{4},$
(2.24)		$\displaystyle\delta_{2}(\tau)$	$\displaystyle=-\frac{1}{8}\bigl(\theta_{1}(0,\tau)^{4}+\theta_{3}(0,\tau)^{4}\bigr),$	$\displaystyle\varepsilon_{2}(\tau)$	$\displaystyle=\frac{1}{16}\theta_{1}(0,\tau)^{4}\theta_{3}(0,\tau)^{4}.$

Direct multiplication of the products gives

(2.25)		$\displaystyle\delta_{1}$	$\displaystyle=\frac{1}{4}+6q+6q^{2}+O(q^{3}),$	$\displaystyle\varepsilon_{1}$	$\displaystyle=\frac{1}{16}-q+7q^{2}+O(q^{3}),$
(2.26)		$\displaystyle\delta_{2}$	$\displaystyle=-\frac{1}{8}-3q^{1/2}-3q-12q^{3/2}+O(q^{2}),$	$\displaystyle\varepsilon_{2}$	$\displaystyle=q^{1/2}+8q+28q^{3/2}+O(q^{2}).$

Let

	$\displaystyle\Gamma_{0}(2)$	$\displaystyle=\left\{\begin{pmatrix}a&b\\ c&d\end{pmatrix}\in SL_{2}(\mathbb{Z}):c\equiv 0\pmod{2}\right\},$
	$\displaystyle\Gamma^{0}(2)$	$\displaystyle=\left\{\begin{pmatrix}a&b\\ c&d\end{pmatrix}\in SL_{2}(\mathbb{Z}):b\equiv 0\pmod{2}\right\}.$

The following standard structure statement is used only to obtain the finite expansion in (4.1). It is proved, with the same normalizations, in [7, Lemma 2.2]; see also [3, Section 1.2].

Proposition 2.5 (Structure of the level-two rings).

The modular forms with real Fourier coefficients, trivial character, and holomorphic expansions at every cusp for $\Gamma_{0}(2)$ are generated by $\delta_{1}$ and $\varepsilon_{1}$ , of weights $2$ and $4$ , respectively. The real modular forms for $\Gamma^{0}(2)$ are generated by $\delta_{2}$ and $\varepsilon_{2}$ , with the same weights. Under $S:\tau\mapsto-1/\tau$ ,

(2.27)

\delta_{2}(-1/\tau)=\tau^{2}\delta_{1}(\tau),\qquad\varepsilon_{2}(-1/\tau)=\tau^{4}\varepsilon_{1}(\tau).

Proof.

The theta transformation laws interchange $\theta_{1}$ and $\theta_{2}$ under $S$ , leave $\theta_{3}$ fixed, and multiply each theta constant by $(-\sqrt{-1}\tau)^{1/2}$ . Substitution into (2.23)–(2.24) proves (2.27). Under $\tau\mapsto\tau+1$ , the constants $\theta_{2}$ and $\theta_{3}$ are interchanged. Combining these laws with the generators $T,ST^{2}S$ of $\Gamma_{0}(2)$ , up to the central element, gives the asserted weights and groups. The product expansions are holomorphic at infinity; the $S$ relations verify holomorphy at the other cusp. Conjugation by $S$ gives the corresponding statements for $\Gamma^{0}(2)$ .

For completeness, fix weight $2k$ and put $m=\lfloor k/2\rfloor$ . The $m+1$ forms

f_{r}=(8\delta_{2})^{k-2r}\varepsilon_{2}^{r},\qquad 0\leq r\leq m,

have leading terms $(-1)^{k}t^{r}$ , where $t=q^{1/2}$ is the local parameter at the infinity cusp of $\Gamma^{0}(2)$ . They are therefore linearly independent. The valence formula for this index-three group bounds the order at that cusp of a nonzero holomorphic weight- $2k$ form by $(2k)\cdot 3/12=k/2$ . Given any such form, successive subtraction of multiples of $f_{0},\ldots,f_{m}$ kills its coefficients through $t^{m}$ . The remainder has order at least $m+1>k/2$ and must vanish. Thus these forms span the weight- $2k$ space. The elimination uses real coefficients when the initial Fourier coefficients are real. Applying the $S$ relation proves the ring statement on the other group. ∎

3. Theta bundles, modularity, and Fourier coefficients

Let $V\to M$ be a real oriented Euclidean bundle of even rank $2l$ , and write $V_{\mathbb{C}}$ for its complexification. Liu and Wang introduced the following elements of $K(M)[[q^{1/2}]]$ in [9, equations (3.1)–(3.2)]:

	$\displaystyle\Theta_{1}(T_{\mathbb{C}}M,V_{\mathbb{C}},a,b):={}$	$\displaystyle\bigotimes_{n=1}^{\infty}S_{q^{n}}(\widetilde{T_{\mathbb{C}}M})\otimes\left(\bigotimes_{m=1}^{\infty}\Lambda_{q^{m}}(\widetilde{V_{\mathbb{C}}})\right)^{\!a}$
(3.1)			$\displaystyle\otimes\left(\bigotimes_{r=1}^{\infty}\Lambda_{q^{r-1/2}}(\widetilde{V_{\mathbb{C}}})\right)^{\!b}\otimes\left(\bigotimes_{s=1}^{\infty}\Lambda_{-q^{s-1/2}}(\widetilde{V_{\mathbb{C}}})\right)^{\!b},$
	$\displaystyle\Theta_{2}(T_{\mathbb{C}}M,V_{\mathbb{C}},a,b):={}$	$\displaystyle\bigotimes_{n=1}^{\infty}S_{q^{n}}(\widetilde{T_{\mathbb{C}}M})\otimes\left(\bigotimes_{m=1}^{\infty}\Lambda_{q^{m}}(\widetilde{V_{\mathbb{C}}})\right)^{\!b}$
(3.2)			$\displaystyle\otimes\left(\bigotimes_{r=1}^{\infty}\Lambda_{q^{r-1/2}}(\widetilde{V_{\mathbb{C}}})\right)^{\!b}\otimes\left(\bigotimes_{s=1}^{\infty}\Lambda_{-q^{s-1/2}}(\widetilde{V_{\mathbb{C}}})\right)^{\!a},$

where $a,b$ are integers in the formal power-series construction. We restrict to $a,b\geq 0$ so that the spinor powers entering the index pairings are actual tensor powers. The reduced bundles and Fourier coefficients themselves are integral virtual $K$ -classes.

Specializing to $V=TM$ gives $\widetilde{V_{\mathbb{C}}}=E$ and $l=2k$ . Define the scalar series

(3.3)		$\displaystyle R_{1}^{a,b}(\tau)$	$\displaystyle=\left\langle\widehat{A}(TM)\operatorname{ch}(\mathcal{S}(M)^{\otimes a})\operatorname{ch}(\Theta_{1}(T_{\mathbb{C}}M,T_{\mathbb{C}}M,a,b)),[M]\right\rangle,$
(3.4)		$\displaystyle R_{2}^{a,b}(\tau)$	$\displaystyle=\left\langle\widehat{A}(TM)\operatorname{ch}(\mathcal{S}(M)^{\otimes b})\operatorname{ch}(\Theta_{2}(T_{\mathbb{C}}M,T_{\mathbb{C}}M,a,b)),[M]\right\rangle.$

The modular input is the following specialization of the Liu–Wang construction. We include its theta-function proof to specify both the normalization and the cohomological use of the string condition.

Theorem 3.1.

Let $M^{4k}$ be a closed string manifold. For nonnegative $a,b$ , the series $R_{1}^{a,b}$ and $R_{2}^{a,b}$ are modular forms of weight $2k$ over $\Gamma_{0}(2)$ and $\Gamma^{0}(2)$ , respectively, and

(3.5)

R_{1}^{a,b}(-1/\tau)=2^{(a-b)2k}\tau^{2k}R_{2}^{a,b}(\tau).

Proof.

Let $v_{j}=x_{j}/(2\pi\sqrt{-1})$ and define

H(v,\tau)=v\frac{\theta^{\prime}(0,\tau)}{\theta(v,\tau)},\qquad J_{\nu}(v,\tau)=\frac{\theta_{\nu}(v,\tau)}{\theta_{\nu}(0,\tau)}.

The products defining the theta bundles give the total characteristic classes

	$\displaystyle\Phi_{1}(x,\tau)$	$\displaystyle=2^{2ka}\prod_{j=1}^{2k}H(v_{j},\tau)J_{1}(v_{j},\tau)^{a}J_{2}(v_{j},\tau)^{b}J_{3}(v_{j},\tau)^{b},$
	$\displaystyle\Phi_{2}(x,\tau)$	$\displaystyle=2^{2kb}\prod_{j=1}^{2k}H(v_{j},\tau)J_{1}(v_{j},\tau)^{b}J_{2}(v_{j},\tau)^{a}J_{3}(v_{j},\tau)^{b}.$

Here $R_{i}^{a,b}=\langle\Phi_{i}^{(4k)},[M]\rangle$ . Indeed, the constant term of $H$ is $(x/2)/\sinh(x/2)$ , and that of $J_{1}$ is $\cosh(x/2)$ ; the factors $2^{2ka}$ and $2^{2kb}$ account for the spinor characters.

The Jacobi transformation formulas imply

	$\displaystyle H(v,-1/\tau)$	$\displaystyle=e^{-\pi\sqrt{-1}\tau v^{2}}H(\tau v,\tau),$
	$\displaystyle J_{1}(v,-1/\tau)$	$\displaystyle=e^{\pi\sqrt{-1}\tau v^{2}}J_{2}(\tau v,\tau),$
	$\displaystyle J_{2}(v,-1/\tau)$	$\displaystyle=e^{\pi\sqrt{-1}\tau v^{2}}J_{1}(\tau v,\tau),$
	$\displaystyle J_{3}(v,-1/\tau)$	$\displaystyle=e^{\pi\sqrt{-1}\tau v^{2}}J_{3}(\tau v,\tau).$

Consequently

(3.6)

\Phi_{1}(x,-1/\tau)=2^{2k(a-b)}e^{\pi\sqrt{-1}\tau(a+2b-1)\sum_{j}v_{j}^{2}}\Phi_{2}(\tau x,\tau).

Since $\sum_{j}x_{j}^{2}=p_{1}(TM)=0$ in cohomology, the exponential factor equals $1$ in cohomology. This assertion does not require the Pontryagin differential form of the Levi–Civita connection to vanish pointwise. If that form is $d\eta$ , then for every closed form $\omega$ and every $j\geq 1$ ,

p_{1}^{j}\wedge\omega=d(\eta\wedge p_{1}^{j-1}\wedge\omega),\qquad\int_{M}p_{1}^{j}\wedge\omega=0.

Thus all nonconstant exponential terms disappear after integration. Replacing each degree-two root $x_{j}$ by $\tau x_{j}$ multiplies the degree- $4k$ term by $\tau^{2k}$ , proving (3.5). The inverse $S$ relation follows in the same way.

The normalized theta ratios show that $R_{1}^{a,b}(\tau+1)=R_{1}^{a,b}(\tau)$ and $R_{2}^{a,b}(\tau+2)=R_{2}^{a,b}(\tau)$ . Together with the two $S$ relations, these identities give the required transformation laws on $\Gamma_{0}(2)$ and $\Gamma^{0}(2)$ . The theta constants in the denominators have no zeros on the upper half-plane, and the factor $v$ removes the simple zero of $\theta(v,\tau)$ at $v=0$ . The Taylor coefficients in the roots are therefore holomorphic. The bundle products have no negative Fourier powers at infinity, and the $S$ relations give the same property at the other cusp. This proves holomorphy at all cusps.

Equivalently, in the general formula of [9, Theorem 3.1 and equation (3.15)], the anomaly class is $p_{1}(TM)-(a+2b)p_{1}(V)$ and the scalar is $2^{(a-b)l}$ . The specialization $V=TM$ , $l=2k$ gives exactly the formula above. ∎

We next perform the $K$ -theory calculation underlying all later results.

Write $t=q^{1/2}$ . For an integer $c\geq 0$ , the binomial expansion of (2.4) gives

(3.7)		$\displaystyle\Lambda_{t}(E)^{c}$	$\displaystyle=1+cEt+\left(c\Lambda^{2}E+\binom{c}{2}E\otimes E\right)t^{2}+O(t^{3}),$
(3.8)		$\displaystyle\Lambda_{-t}(E)^{c}$	$\displaystyle=1-cEt+\left(c\Lambda^{2}E+\binom{c}{2}E\otimes E\right)t^{2}+O(t^{3}).$

Indeed, a term of degree two is obtained either from the degree-two term in one factor, giving $c\Lambda^{2}E$ , or from the degree-one terms in two distinct factors, giving $\binom{c}{2}E\otimes E$ . Moreover, by (2.6), we get

	$\displaystyle\Lambda_{t}(E)^{b}\Lambda_{-t}(E)^{b}$	$\displaystyle=1+\Bigl(2b\Lambda^{2}E+2\binom{b}{2}E\otimes E-b^{2}E\otimes E\Bigr)t^{2}+O(t^{3})$
		$\displaystyle=1+b(2\Lambda^{2}E-E\otimes E)t^{2}+O(t^{3})$
(3.9)			$\displaystyle=1+b(\Lambda^{2}E-S^{2}E)t^{2}+O(t^{3}).$

These calculations give the following coefficient formulas.

Proposition 3.2.

For $V=TM$ and $E=\widetilde{T_{\mathbb{C}}M}$ , the theta bundles have expansions

(3.10)		$\displaystyle\Theta_{1}(T_{\mathbb{C}}M,T_{\mathbb{C}}M,a,b)$	$\displaystyle=1+\mathcal{A}_{a,b}q+O(q^{2}),$
(3.11)		$\displaystyle\Theta_{2}(T_{\mathbb{C}}M,T_{\mathbb{C}}M,a,b)$	$\displaystyle=1+\mathcal{B}^{(1)}_{a,b}q^{1/2}+\mathcal{B}^{(2)}_{a,b}q+O(q^{3/2}),$

where

(3.12)	$\displaystyle\mathcal{A}_{a,b}$	$\displaystyle=(1+a)E+b(\Lambda^{2}E-S^{2}E),$
(3.13)	$\displaystyle\mathcal{B}^{(1)}_{a,b}$	$\displaystyle=(b-a)E,$
(3.14)	$\displaystyle\mathcal{B}^{(2)}_{a,b}$	$\displaystyle=(1+b)E+(a+b)\Lambda^{2}E+\frac{(a-b)^{2}-(a+b)}{2}E\otimes E.$

Proof.

We calculate the two products separately. Up to order $q=t^{2}$ , only the $n=m=1$ integral factors and the $r=s=1$ half-integral factors can contribute.

For $\Theta_{1}$ , the integral factors are

	$\displaystyle\prod_{n\geq 1}S_{q^{n}}(E)$	$\displaystyle=1+Eq+O(q^{2}),$
	$\displaystyle\left(\prod_{m\geq 1}\Lambda_{q^{m}}(E)\right)^{a}$	$\displaystyle=1+aEq+O(q^{2}).$

The two half-integral products have equal exponent $b$ . Their linear terms cancel, and (3.9) gives

\left(\prod_{r\geq 1}\Lambda_{q^{r-1/2}}(E)\right)^{b}\left(\prod_{s\geq 1}\Lambda_{-q^{s-1/2}}(E)\right)^{b}=1+b(\Lambda^{2}E-S^{2}E)q+O(q^{2}).

Multiplying the three displayed expressions proves (3.12). The remainder is $O(q^{2})$ because each paired half-integral product is unchanged under $t\mapsto-t$ , and therefore contains only even powers of $t$ .

For $\Theta_{2}$ , the integral factors give

\left(1+Eq+O(q^{2})\right)\left(1+bEq+O(q^{2})\right)=1+(1+b)Eq+O(q^{2}).

The half-integral factors now have exponents $b$ and $a$ and opposite signs. Using (3.7)–(3.8), their product is

		$\displaystyle\left[1+bEt+\left(b\Lambda^{2}E+\binom{b}{2}E^{\otimes 2}\right)t^{2}\right]\left[1-aEt+\left(a\Lambda^{2}E+\binom{a}{2}E^{\otimes 2}\right)t^{2}\right]+O(t^{3})$
		$\displaystyle\quad=1+(b-a)Et$
		$\displaystyle\qquad+\left[(a+b)\Lambda^{2}E+\left(\binom{a}{2}+\binom{b}{2}-ab\right)E^{\otimes 2}\right]t^{2}+O(t^{3}).$

Finally,

\binom{a}{2}+\binom{b}{2}-ab=\frac{a^{2}-a+b^{2}-b-2ab}{2}=\frac{(a-b)^{2}-(a+b)}{2}.

Adding the integral contribution proves (3.13) and (3.14). ∎

Applying Definition 2.3 coefficientwise gives

(3.15)		$\displaystyle R_{1}^{a,b}(\tau)$	$\displaystyle=\mathcal{I}_{a}(1)+\mathcal{I}_{a}(\mathcal{A}_{a,b})q+O(q^{2}),$
(3.16)		$\displaystyle R_{2}^{a,b}(\tau)$	$\displaystyle=\mathcal{I}_{b}(1)+\mathcal{I}_{b}(\mathcal{B}^{(1)}_{a,b})q^{1/2}+\mathcal{I}_{b}(\mathcal{B}^{(2)}_{a,b})q+O(q^{3/2}).$

Put $T=T_{\mathbb{C}}M$ and $d=\dim M$ . From (2.3),

(3.17)	$\displaystyle\Lambda^{2}E$	$\displaystyle=\Lambda^{2}T-dT+\binom{d+1}{2}\mathbb{C},$
(3.18)	$\displaystyle S^{2}E$	$\displaystyle=S^{2}T-dT+\binom{d}{2}\mathbb{C},$
(3.19)	$\displaystyle E\otimes E$	$\displaystyle=T\otimes T-2dT+d^{2}\mathbb{C}.$

For example, subtracting (3.18) from (3.17) gives

(3.20)

\Lambda^{2}E-S^{2}E=\Lambda^{2}T-S^{2}T+d\mathbb{C}.

Consequently

(3.21)

\mathcal{A}_{a,b}=(1+a)T+b(\Lambda^{2}T-S^{2}T)+d(b-a-1)\mathbb{C}.

If

(3.22)

c_{a,b}:=\frac{(a-b)^{2}-(a+b)}{2},

then

	$\displaystyle\mathcal{B}^{(2)}_{a,b}={}$	$\displaystyle(a+b)\Lambda^{2}T+c_{a,b}T\otimes T$
		$\displaystyle+\bigl(1+b-d(a+b)-2dc_{a,b}\bigr)T$
(3.23)			$\displaystyle+\left[-d(1+b)+(a+b)\binom{d+1}{2}+c_{a,b}d^{2}\right]\mathbb{C}.$

The dimension-specific formulas below follow from (3.21) and (3.23).

4. The modular coefficient comparison

In this section we prove Theorem 1.1.

4.1. Triangular coefficients on the $\Gamma^{0}(2)$ side

Since $R_{2}^{a,b}$ has weight $2k$ , Proposition 2.5 gives numbers $h_{r}=h_{r}(M;a,b)$ such that

(4.1)

R_{2}^{a,b}(\tau)=\sum_{r=0}^{\lfloor k/2\rfloor}h_{r}(8\delta_{2})^{k-2r}\varepsilon_{2}^{r}.

By (3.5) and (2.27),

(4.2)

R_{1}^{a,b}(\tau)=2^{(a-b)2k}\sum_{r=0}^{\lfloor k/2\rfloor}h_{r}(8\delta_{1})^{k-2r}\varepsilon_{1}^{r}.

Set $t=q^{1/2}$ , from (2.26),

(4.3)

8\delta_{2}=-(1+24t+24t^{2})+O(t^{3}),\qquad\varepsilon_{2}=t+8t^{2}+O(t^{3}).

For an integer $n\geq 0$ , the binomial theorem gives

	$\displaystyle(8\delta_{2})^{n}$	$\displaystyle=(-1)^{n}\left[1+24nt+\left(24n+\binom{n}{2}24^{2}\right)t^{2}\right]+O(t^{3})$
(4.4)			$\displaystyle=(-1)^{n}\left[1+24nt+24n(12n-11)t^{2}\right]+O(t^{3}).$

Only $r=0,1,2$ can contribute through $t^{2}$ . For $r=0$ , the corresponding expansion follows by setting $n=k$ in the preceding formula. The $r=1$ term is present only if $k\geq 2$ ; in that case,

	$\displaystyle(8\delta_{2})^{k-2}\varepsilon_{2}$	$\displaystyle=(-1)^{k}\bigl[1+24(k-2)t+O(t^{2})\bigr]\bigl[t+8t^{2}+O(t^{3})\bigr]$
(4.5)			$\displaystyle=(-1)^{k}\bigl[t+(24k-40)t^{2}\bigr]+O(t^{3}),$

and the $r=2$ term, present only if $k\geq 4$ , satisfies

(4.6)

(8\delta_{2})^{k-4}\varepsilon_{2}^{2}=(-1)^{k}t^{2}+O(t^{3}).

Lemma 4.1.

For $1\leq k\leq 7$ , set $h_{r}=0$ whenever $r>\lfloor k/2\rfloor$ . The first three index coefficients of $R_{2}^{a,b}$ satisfy

(4.7)	$\displaystyle\mathcal{I}_{b}(1)$	$\displaystyle=(-1)^{k}h_{0},$
(4.8)	$\displaystyle\mathcal{I}_{b}(\mathcal{B}^{(1)}_{a,b})$	$\displaystyle=(-1)^{k}(24kh_{0}+h_{1}),$
(4.9)	$\displaystyle\mathcal{I}_{b}(\mathcal{B}^{(2)}_{a,b})$	$\displaystyle=(-1)^{k}\bigl(24k(12k-11)h_{0}+(24k-40)h_{1}+h_{2}\bigr).$

In particular, $h_{1}$ is omitted for $k=1$ , and $h_{2}$ is omitted for $k<4$ . Every $h_{r}$ in the expansion for $k\leq 7$ is an integer.

Proof.

Set $t=q^{1/2}$ , $m=\lfloor k/2\rfloor$ , and write

F_{r}(t):=(8\delta_{2})^{k-2r}\varepsilon_{2}^{r},\qquad R_{2}^{a,b}(\tau)=\sum_{r=0}^{m}h_{r}F_{r}(t).

Using

8\delta_{2}=-1-24t-24t^{2}+O(t^{3}),\qquad\varepsilon_{2}=t+8t^{2}+O(t^{3}),

we obtain

$\displaystyle F_{0}(t)$	$\displaystyle=(-1)^{k}\left[1+24kt+24k(12k-11)t^{2}\right]+O(t^{3}),$
$\displaystyle F_{1}(t)$	$\displaystyle=(-1)^{k}\left[t+(24k-40)t^{2}\right]+O(t^{3}),$	$\displaystyle k\geq 2,$
$\displaystyle F_{2}(t)$	$\displaystyle=(-1)^{k}t^{2}+O(t^{3}),$	$\displaystyle k\geq 4.$

Moreover, $F_{r}(t)=O(t^{3})$ for $3\leq r\leq m$ . Thus, with $h_{r}=0$ for $r>m$ ,

	$\displaystyle R_{2}^{a,b}(\tau)=(-1)^{k}\Bigl\{$	$\displaystyle h_{0}+(24kh_{0}+h_{1})t$
		$\displaystyle+\bigl[24k(12k-11)h_{0}+(24k-40)h_{1}+h_{2}\bigr]t^{2}\Bigr\}+O(t^{3}).$

Comparing this with

R_{2}^{a,b}(\tau)=\mathcal{I}_{b}(1)+\mathcal{I}_{b}(\mathcal{B}^{(1)}_{a,b})t+\mathcal{I}_{b}(\mathcal{B}^{(2)}_{a,b})t^{2}+O(t^{3})

gives the three asserted identities.

For integrality, put

N_{j}:=[t^{j}]R_{2}^{a,b}(\tau)\in\mathbb{Z},\qquad u_{r,j}:=[t^{j}]F_{r}(t),

where $N_{j}$ is an integer by its index interpretation. The expansions

	$\displaystyle 8\delta_{2}$	$\displaystyle=-1-24t-24t^{2}-96t^{3}+O(t^{4}),$
	$\displaystyle\varepsilon_{2}$	$\displaystyle=t+8t^{2}+28t^{3}+O(t^{4}),$

together with $m\leq 3$ , imply

u_{r,j}\in\mathbb{Z},\qquad u_{r,j}=0\quad(j<r),\qquad u_{r,r}=(-1)^{k},\qquad 0\leq r,j\leq m.

Comparing the coefficients of $t^{j}$ , we have

N_{j}=(-1)^{k}h_{j}+\sum_{r=0}^{j-1}h_{r}u_{r,j}.

Hence

h_{0}=(-1)^{k}N_{0}\in\mathbb{Z},\qquad h_{j}=(-1)^{k}\left(N_{j}-\sum_{r=0}^{j-1}h_{r}u_{r,j}\right),\quad 1\leq j\leq m.

Induction on $j$ therefore gives $h_{j}\in\mathbb{Z}$ for every $0\leq j\leq m$ . ∎

4.2. The first coefficient on the $\Gamma_{0}(2)$ side

From (2.25),

(4.10)

8\delta_{1}=2+48q+O(q^{2}),\qquad\varepsilon_{1}=\frac{1}{16}-q+O(q^{2}).

If $n=k-2r$ , then

	$\displaystyle(8\delta_{1})^{n}\varepsilon_{1}^{r}$	$\displaystyle=2^{n}(1+24nq+O(q^{2}))16^{-r}(1-16rq+O(q^{2}))$
(4.11)			$\displaystyle=2^{k-6r}\bigl[1+(24k-64r)q+O(q^{2})\bigr].$

Comparing the coefficient of $q$ in (3.15) and (4.2) proves the next lemma.

Lemma 4.2.

For every $k\geq 1$ ,

(4.12)

\mathcal{I}_{a}(\mathcal{A}_{a,b})=2^{(a-b)2k}\sum_{r=0}^{\lfloor k/2\rfloor}2^{k-6r}(24k-64r)h_{r}.

4.3. Clearing denominators before taking integer residues

The identity (4.12) has rational coefficients and an integer left-hand side. We clear its denominators before reducing modulo $3$ .

For fixed $a,b\in\mathbb{Z}_{\geq 0}$ , put

(4.13)

U_{k}=2^{2k\max(a-b,0)},\qquad V_{k}=2^{2k\max(b-a,0)}.

Both are positive integers, and $2^{2k(a-b)}=U_{k}/V_{k}$ as an exact rational equality. For $1\leq k\leq 7$ , choose

(4.14)

D_{k}=\begin{cases}1,&k=1,2,3,\\ 16,&k=4,5,\\ 256,&k=6,7.\end{cases}

The factors $D_{k}$ clear the denominators inside the sum in (4.12). They have been chosen as powers of $4$ so that no additional sign occurs modulo $3$ . Define the integers $J_{k}$ by the following table:

(4.15)

\begin{array}[]{c|l}k&J_{k}\\ \hline\cr 1&48h_{0}\\ 2&192h_{0}-h_{1}\\ 3&576h_{0}+h_{1}\\ 4&24576h_{0}+128h_{1}-2h_{2}\\ 5&61440h_{0}+448h_{1}-h_{2}\\ 6&2359296h_{0}+20480h_{1}+64h_{2}-3h_{3}\\ 7&5505024h_{0}+53248h_{1}+320h_{2}-3h_{3}.\end{array}

Every $h_{r}$ in this table is an integer by Lemma 4.1. Multiplying the exact coefficient sum in (4.12) by $D_{k}$ gives precisely $J_{k}$ . For example, at $k=4$ the coefficients $1536,8,-1/8$ become $24576,128,-2$ after multiplication by $16$ ; at $k=6$ the coefficients $9216,80,1/4,-3/256$ become $2359296,20480,64,-3$ after multiplication by $256$ .

Clearing also the outer denominator $V_{k}$ in (4.12) gives the integer identity

(4.16)

\boxed{D_{k}V_{k}\,\mathcal{I}_{a}(\mathcal{A}_{a,b})=U_{k}J_{k}.}

Since all exponents in (4.13) are nonnegative and even,

(4.17)

U_{k}\equiv V_{k}\equiv D_{k}\equiv 1\pmod{3}.

Reducing (4.16) therefore gives

(4.18)

\mathcal{I}_{a}(\mathcal{A}_{a,b})\equiv J_{k}\pmod{3}.

This also covers $a<b$ , since multiplication by $V_{k}$ clears the outer denominator before reduction.

4.4. Proof of the main comparison

Proof of Theorem 1.1.

We combine (4.18) with Lemma 4.1. All congruences in this proof are modulo $3$ .

If $k=1$ , (4.16) reads

V_{1}\mathcal{I}_{a}(\mathcal{A}_{a,b})=48U_{1}h_{0}.

Its right-hand side is divisible by $3$ , and $V_{1}\equiv 1$ . Consequently $\mathcal{I}_{a}(\mathcal{A}_{a,b})\equiv 0$ , proving (1.6).

If $k=2$ , then $J_{2}=192h_{0}-h_{1}\equiv-h_{1}$ , whereas (4.8) gives

\mathcal{I}_{b}(\mathcal{B}^{(1)}_{a,b})=48h_{0}+h_{1}\equiv h_{1}.

Together with (4.18), this shows that the sum of the two indices is congruent to zero modulo $3$ . For $k=3$ , the corresponding integer expressions are

J_{3}=576h_{0}+h_{1}\equiv h_{1},\qquad\mathcal{I}_{b}(\mathcal{B}^{(1)}_{a,b})=-72h_{0}-h_{1}\equiv-h_{1}.

This proves (1.7) in both cases.

For $k=4$ , the denominator-cleared expression is

J_{4}=24576h_{0}+128h_{1}-2h_{2}\equiv 2h_{1}+h_{2},

since $24576$ is divisible by $3$ , $128\equiv 2$ , and $-2\equiv 1$ . On the other side, (4.9) gives

\mathcal{I}_{b}(\mathcal{B}^{(2)}_{a,b})=3552h_{0}+56h_{1}+h_{2}\equiv 2h_{1}+h_{2}.

For $k=5$ , the two integer expressions give

	$\displaystyle J_{5}$	$\displaystyle=61440h_{0}+448h_{1}-h_{2}\equiv h_{1}+2h_{2},$
	$\displaystyle\mathcal{I}_{b}(\mathcal{B}^{(2)}_{a,b})$	$\displaystyle=-5880h_{0}-80h_{1}-h_{2}\equiv h_{1}+2h_{2}.$

The minus sign for odd $k$ is included in the second line.

For $k=6$ , use

	$\displaystyle J_{6}$	$\displaystyle=2359296h_{0}+20480h_{1}+64h_{2}-3h_{3}\equiv 2h_{1}+h_{2},$
	$\displaystyle\mathcal{I}_{b}(\mathcal{B}^{(2)}_{a,b})$	$\displaystyle=8784h_{0}+104h_{1}+h_{2}\equiv 2h_{1}+h_{2}.$

In particular, $-3h_{3}$ disappears because $h_{3}\in\mathbb{Z}$ . For $k=7$ , the same integer comparison is

	$\displaystyle J_{7}$	$\displaystyle=5505024h_{0}+53248h_{1}+320h_{2}-3h_{3}\equiv h_{1}+2h_{2},$
	$\displaystyle\mathcal{I}_{b}(\mathcal{B}^{(2)}_{a,b})$	$\displaystyle=-12264h_{0}-128h_{1}-h_{2}\equiv h_{1}+2h_{2}.$

For each $k=4,5,6,7$ , equation (4.18) identifies the residue of the first index with that of $J_{k}$ , so the difference of the two indices is congruent to zero. This proves (1.8). ∎

Remark 4.3.

In dimension $24$ , the term $-3h_{3}$ in $J_{6}$ vanishes modulo $3$ , but integrality alone does not make it vanish modulo $9$ . A comparison modulo $9$ must retain the factors $D_{6},U_{6},V_{6}$ and account for all coefficients using additional divisibility or geometric relations. This limits the present argument, but does not rule out stronger geometric congruences.

5. The three parameter families in dimension twenty-four

We express all index formulas in this section and Section 6 using genuine complex vector bundles. For $d=\dim M$ and $T=T_{\mathbb{C}}M$ , put

(5.1)		$\displaystyle j_{r}$	$\displaystyle=\mathcal{I}_{r}(\mathbb{C}),$	$\displaystyle t_{r}$	$\displaystyle=\mathcal{I}_{r}(T),$	$\displaystyle\ell_{r}$	$\displaystyle=\mathcal{I}_{r}(\Lambda^{2}T),$
(5.1)		$\displaystyle s_{r}$	$\displaystyle=\mathcal{I}_{r}(S^{2}T),$	$\displaystyle z_{r}$	$\displaystyle=\mathcal{I}_{r}(T\otimes T)=s_{r}+\ell_{r}.$

Thus each symbol is the index of $D$ twisted by a genuine bundle, including the factor $\mathcal{S}(M)^{\otimes r}$ in $\mathcal{I}_{r}$ . These are the indices denoted by $J_{r},U_{r},V_{r},W_{r},Z_{r}$ in Corollary 1.1; the lower-case notation here distinguishes them from the clearing factors $U_{k},V_{k}$ and the scalar expression $J_{k}$ in Section 4. For $r\geq 1$ ,

\mathcal{I}_{r}(F)=\operatorname{Sig}(M,\mathcal{S}(M)^{\otimes(r-1)}\otimes F),\qquad F=\mathbb{C},T,\Lambda^{2}T,S^{2}T,T\otimes T.

To keep the exact coefficient comparisons concise, define the following integer combinations of genuine-bundle indices:

(5.2)	$\displaystyle P^{(d)}_{a,b}$	$\displaystyle=(a+1)t_{a}+b\ell_{a}-bs_{a}+d(b-a-1)j_{a},$
	$\displaystyle Q^{(d)}_{a,b}$	$\displaystyle=(b-a)(t_{b}-dj_{b}),$
	$\displaystyle H^{(d)}_{a,b}$	$\displaystyle=(a+b)\ell_{b}+c_{a,b}z_{b}+\alpha_{a,b}^{(d)}t_{b}+\beta_{a,b}^{(d)}j_{b},$

where

(5.3)	$\displaystyle c_{a,b}$	$\displaystyle=\binom{a}{2}+\binom{b}{2}-ab,$
	$\displaystyle\alpha_{a,b}^{(d)}$	$\displaystyle=1+b-d(a+b)-2dc_{a,b},$
	$\displaystyle\beta_{a,b}^{(d)}$	$\displaystyle=-d(1+b)+(a+b)\binom{d+1}{2}+d^{2}c_{a,b}.$

The exact substitutions in (3.17), (3.18), and (3.19) give

[q]R_{1}^{a,b}=P^{(d)}_{a,b},\qquad[q^{1/2}]R_{2}^{a,b}=Q^{(d)}_{a,b},\qquad[q]R_{2}^{a,b}=H^{(d)}_{a,b}.

The symbols $P,Q,H$ denote integer combinations of genuine-bundle indices.

Throughout the rest of this section $d=24$ . Since $24,\binom{25}{2},24^{2}$ are divisible by $3$ , Theorem 1.1 becomes

(5.4)

(a+1)t_{a}+b(\ell_{a}-s_{a})\equiv(b+1)t_{b}+(a+b)\ell_{b}+c_{a,b}z_{b}\pmod{3}.

We treat $(0,0)$ , $(a,0)$ , and $(0,b)$ separately.

For the parameter choices considered below, the corresponding coefficient classes in dimension $24$ are listed in the following table. These exact identities follow from (3.17)–(3.19), with $T=T_{\mathbb{C}}M$ and $E=T-\mathbb{C}^{24}$ .

(5.5)

\begin{array}[]{c|c|c|c}(a,b)&\mathcal{A}_{a,b}&\mathcal{B}^{(1)}_{a,b}&\mathcal{B}^{(2)}_{a,b}\\ \hline\cr(0,0)&T-24&0&T-24\\ (1,0)&2T-48&24-T&\Lambda^{2}T-23T+276\\ (0,1)&T+\Lambda^{2}T-S^{2}T&T-24&\Lambda^{2}T-22T+252\\ (2,0)&3T-72&48-2T&T^{\otimes 2}+2\Lambda^{2}T-95T+1152\\ (0,2)&T+2\Lambda^{2}T-2S^{2}T+24&2T-48&T^{\otimes 2}+2\Lambda^{2}T-93T+1104\end{array}

Here each integer denotes the corresponding multiple of the trivial complex line bundle, and all equalities are in $K(M)$ .

5.1. The weight-twelve coefficient calculation

Since $k=6$ , the modular expansion is

(5.6)

R_{2}^{a,b}=h_{0}(8\delta_{2})^{6}+h_{1}(8\delta_{2})^{4}\varepsilon_{2}+h_{2}(8\delta_{2})^{2}\varepsilon_{2}^{2}+h_{3}\varepsilon_{2}^{3}.

The last term starts at $q^{3/2}$ . Lemma 4.1 gives the exact integer relations

(5.7)	$\displaystyle j_{b}$	$\displaystyle=h_{0},$
(5.8)	$\displaystyle Q^{(24)}_{a,b}$	$\displaystyle=144h_{0}+h_{1},$
(5.9)	$\displaystyle H^{(24)}_{a,b}$	$\displaystyle=8784h_{0}+104h_{1}+h_{2}.$

The transformed expression is

(5.10)

\begin{split}R_{1}^{a,b}=2^{12(a-b)}\{&h_{0}(8\delta_{1})^{6}+h_{1}(8\delta_{1})^{4}\varepsilon_{1}\\ &+h_{2}(8\delta_{1})^{2}\varepsilon_{1}^{2}+h_{3}\varepsilon_{1}^{3}\}.\end{split}

Its $q$ coefficient is

(5.11)

P^{(24)}_{a,b}=2^{12(a-b)}\left(9216h_{0}+80h_{1}+\frac{1}{4}h_{2}-\frac{3}{256}h_{3}\right).

Multiplying by $256V_{6}$ gives an equality of integers:

(5.12)

256V_{6}P^{(24)}_{a,b}=U_{6}(2359296h_{0}+20480h_{1}+64h_{2}-3h_{3}).

Since $256,U_{6},V_{6}\equiv 1\pmod{3}$ ,

P^{(24)}_{a,b}\equiv 2h_{1}+h_{2}\equiv H^{(24)}_{a,b}\pmod{3}.

Expanding $P^{(24)}_{a,b},H^{(24)}_{a,b}$ by (5.2) proves (5.4).

5.2. The case $(a,b)=(0,0)$

For $(a,b)=(0,0)$ , the exact coefficient combinations are

P^{(24)}_{0,0}=H^{(24)}_{0,0}=t_{0}-24j_{0},\qquad Q^{(24)}_{0,0}=0.

Thus (5.4) is the tautology $t_{0}\equiv t_{0}$ . A nontrivial identity comes from the next coefficient of the Witten genus. The following is the identity of Chen and Han [3, Lemma 2.1].

Proposition 5.1 (Chen–Han).

For a closed $24$ -dimensional string manifold,

(5.13)

s_{0}+t_{0}=196884j_{0}.

Consequently,

(5.14)

\operatorname{Ind}(D\otimes S^{2}T)+\operatorname{Ind}(D\otimes T)\equiv 0\pmod{3}.

Proof.

Write the Witten bundle using the genuine bundle $T$ :

(5.15)

\Theta_{W}(T)=\left(\bigotimes_{n\geq 1}S_{q^{n}}(T)\right)\prod_{n\geq 1}(1-q^{n})^{24}.

Its low-order coefficients follow from

	$\displaystyle\bigotimes_{n\geq 1}S_{q^{n}}(T)$	$\displaystyle=1+Tq+(S^{2}T+T)q^{2}+O(q^{3}),$
	$\displaystyle\prod_{n\geq 1}(1-q^{n})^{24}$	$\displaystyle=1-24q+252q^{2}+O(q^{3}).$

Hence, with $\mathcal{W}_{M}=\langle\widehat{A}(TM)\operatorname{ch}(\Theta_{W}(T)),[M]\rangle$ ,

(5.16)

\mathcal{W}_{M}=j_{0}+(t_{0}-24j_{0})q+(s_{0}-23t_{0}+252j_{0})q^{2}+O(q^{3}).

The string condition makes this a weight-twelve modular form for $SL_{2}(\mathbb{Z})$ [13, 7]. Thus

(5.17)

\mathcal{W}_{M}=mE_{4}^{3}+n\Delta_{\mathrm{mod}}.

Using

(5.18)		$\displaystyle E_{4}^{3}$	$\displaystyle=1+720q+179280q^{2}+O(q^{3}),$
(5.19)		$\displaystyle\Delta_{\mathrm{mod}}$	$\displaystyle=q-24q^{2}+O(q^{3}),$

comparison gives

(5.20)	$\displaystyle j_{0}$	$\displaystyle=m,$
(5.21)	$\displaystyle t_{0}-24j_{0}$	$\displaystyle=720m+n,$
(5.22)	$\displaystyle s_{0}-23t_{0}+252j_{0}$	$\displaystyle=179280m-24n.$

In particular $m,n\in\mathbb{Z}$ and $n=t_{0}-744m$ . Substitution yields

s_{0}-23t_{0}+252m=197136m-24t_{0},\qquad s_{0}+t_{0}=196884m.

Since $3\mid 196884$ , (5.14) follows. ∎

5.3. The family $(a,0)$

Set $b=0$ and $C_{a}=\binom{a}{2}$ . The exact genuine-index combinations are

(5.23)	$\displaystyle P^{(24)}_{a,0}$	$\displaystyle=(a+1)(t_{a}-24j_{a}),$
(5.24)	$\displaystyle Q^{(24)}_{a,0}$	$\displaystyle=-a(t_{0}-24j_{0}),$
(5.25)	$\displaystyle H^{(24)}_{a,0}$	$\displaystyle=a\ell_{0}+C_{a}z_{0}+(1-24a-48C_{a})t_{0}+(-24+300a+576C_{a})j_{0}.$

Corollary 5.2.

For every $a\geq 0$ and every closed $24$ -dimensional string manifold,

(5.26)

(a+1)t_{a}\equiv t_{0}+a\ell_{0}+\binom{a}{2}z_{0}\pmod{3}.

Proof.

Substitute $b=0$ in (5.4). Every omitted rank-correction coefficient in (5.23)–(5.25) is divisible by $3$ . ∎

The coefficient $a+1$ vanishes modulo $3$ precisely when $a\equiv 2\pmod{3}$ ; $t_{a}$ itself may have additional divisibility for other parameters.

Corollary 5.3.

For every closed $24$ -dimensional string manifold,

(5.27)		$\displaystyle t_{0}+2\ell_{0}+z_{0}$	$\displaystyle\equiv 0\pmod{3},$
(5.28)		$\displaystyle s_{0}+t_{0}$	$\displaystyle\equiv 0\pmod{3}.$

Proof.

Setting $a=2$ in (5.26) gives

0\equiv 3t_{2}\equiv t_{0}+2\ell_{0}+z_{0}=t_{0}+s_{0}+3\ell_{0}\pmod{3}.

Both conclusions follow because $z_{0}=s_{0}+\ell_{0}$ and $\ell_{0}\in\mathbb{Z}$ . ∎

The level-two comparison therefore recovers the mod $3$ consequence of Proposition 5.1. More generally, $a\equiv 2\pmod{3}$ implies $\binom{a}{2}\equiv 1\pmod{3}$ and produces the same right-hand residue $t_{0}+s_{0}$ .

5.4. The family $(0,b)$

Set $a=0$ and $C_{b}=\binom{b}{2}$ . Then

(5.29)	$\displaystyle P^{(24)}_{0,b}$	$\displaystyle=t_{0}+b(\ell_{0}-s_{0})+24(b-1)j_{0},$
(5.30)	$\displaystyle Q^{(24)}_{0,b}$	$\displaystyle=b(t_{b}-24j_{b}),$
(5.31)	$\displaystyle H^{(24)}_{0,b}$	$\displaystyle=b\ell_{b}+C_{b}z_{b}+(1-23b-48C_{b})t_{b}+(-24+276b+576C_{b})j_{b}.$

Corollary 5.4.

For every $b\geq 0$ and every closed $24$ -dimensional string manifold,

(5.32)

t_{0}+b(\ell_{0}-s_{0})\equiv(b+1)t_{b}+b\ell_{b}+\binom{b}{2}z_{b}\pmod{3}.

Proof.

This is (5.4) with $a=0$ ; all five types of indices in (5.1) are integers. ∎

If $b=3m+2$ , then

\binom{b}{2}-1=\frac{9m(m+1)}{2}\in 3\mathbb{Z},\qquad b+1\equiv 0,\quad b\equiv-1\pmod{3}.

It follows that

(5.33)

H^{(24)}_{0,b}\equiv-\ell_{b}+z_{b}=s_{b}\pmod{3}.

The least such parameter is $b=2$ . For comparison, $b=1$ gives an ordinary twisted signature, since $\mathcal{I}_{1}(F)=\operatorname{Sig}(M,F)$ ; $b=2$ gives a signature with one additional spinor factor.

Theorem 5.5.

For every closed $24$ -dimensional string manifold,

(5.34)

s_{2}+\ell_{0}\equiv 0\pmod{3}.

Equivalently, with genuine twisting bundles,

(5.35)

\operatorname{Sig}(M,\mathcal{S}(M)\otimes S^{2}T)+\operatorname{Ind}(D\otimes\Lambda^{2}T)\equiv 0\pmod{3}.

Proof.

At $b=2$ , Corollary 5.4 and $z_{2}=s_{2}+\ell_{2}$ give

t_{0}+2\ell_{0}-2s_{0}\equiv 3t_{2}+2\ell_{2}+z_{2}=s_{2}+3(t_{2}+\ell_{2})\equiv s_{2}\pmod{3}.

Corollary 5.3 gives $t_{0}+s_{0}\equiv 0$ , so

t_{0}+2\ell_{0}-2s_{0}\equiv t_{0}-\ell_{0}+s_{0}\equiv-\ell_{0}\pmod{3}.

Thus $s_{2}+\ell_{0}\equiv 0$ . The signature formulation follows from (2.18). ∎

For a fixed index subscript $r$ , the coefficient reductions above depend only on the parameters modulo $3$ . This does not imply periodicity of the full pairings $\mathcal{I}_{a}$ and $\mathcal{I}_{b}$ , since their twisting bundles contain the actual tensor powers $\mathcal{S}(M)^{\otimes a}$ and $\mathcal{S}(M)^{\otimes b}$ .

5.5. Recovery of the Chen–Han comparison

The choices $(a,b)=(0,1)$ and $(1,0)$ in (5.4) give

(5.36)		$\displaystyle t_{0}+\ell_{0}-s_{0}$	$\displaystyle\equiv 2t_{1}+\ell_{1}\equiv\ell_{1}-t_{1}\pmod{3},$
(5.37)		$\displaystyle t_{1}$	$\displaystyle\equiv-t_{0}-\ell_{0}\pmod{3}.$

Together with $s_{0}+t_{0}\equiv 0$ , these imply

\ell_{1}\equiv t_{1}+t_{0}+\ell_{0}-s_{0}\equiv-s_{0}\equiv t_{0}\pmod{3},

or

\operatorname{Sig}(M,\Lambda^{2}T)\equiv\operatorname{Ind}(D\otimes T)\pmod{3}.

This recovers [3, Theorem 0.1]. The additional divisibility of the Rarita–Schwinger index by $3$ is a known geometric result; it is not inferred solely from this two-index comparison.

For completeness, the same coefficients recover the exact identity of [3, Lemma 2.3]. At $(a,b)=(1,0)$ , put $m=j_{0}$ ; then

h_{0}=m,\qquad h_{1}=-t_{0}-120m,\qquad h_{2}=\ell_{0}+81t_{0}+3972m.

The constant and first coefficients of $R_{1}^{1,0}$ give

	$\displaystyle j_{1}$	$\displaystyle=2^{12}\left(64h_{0}+h_{1}+\frac{h_{2}}{64}+\frac{h_{3}}{4096}\right),$
	$\displaystyle 2t_{1}-48j_{1}$	$\displaystyle=2^{12}\left(9216h_{0}+80h_{1}+\frac{h_{2}}{4}-\frac{3h_{3}}{256}\right).$

Eliminating $h_{3}$ yields

2t_{1}=2^{12}(12288h_{0}+128h_{1}+h_{2}).

Substitution therefore gives the genuine-index identity

(5.38)

t_{1}=2^{11}(\ell_{0}-47t_{0}+900j_{0}).

Indeed, $12288-15360+3972=900$ and $-128+81=-47$ . For this parameter pair, $1-a-2b=0$ , so this last exact identity also holds for closed $24$ -dimensional spin manifolds without the string hypothesis.

6. The remaining dimensions

We now use the genuine-bundle indices (5.1) and the exact integer combinations (5.2) with $d=\dim M$ . The clearing factors $U_{k},V_{k}$ retain their definitions from Section 4. All congruences in the first seven subsections are modulo $3$ . Together with Section 5, these computations prove Theorem 1.2.

6.1. Dimension four

Here $d=4$ and $k=1$ . The modular expansions are

	$\displaystyle R_{2}^{a,b}$	$\displaystyle=-h_{0}-24h_{0}q^{1/2}-24h_{0}q+O(q^{3/2}),$
	$\displaystyle R_{1}^{a,b}$	$\displaystyle=2^{2(a-b)}h_{0}(2+48q+O(q^{2})).$

Consequently,

V_{1}P^{(4)}_{a,b}=48U_{1}h_{0},

and $V_{1}\equiv 1\pmod{3}$ gives

(6.1)

(a+1)t_{a}+b(\ell_{a}-s_{a})+4(b-a-1)j_{a}\equiv 0\pmod{3}.

In fact the degree-four component of each relevant index density is a multiple of $p_{1}(TM)$ . The string condition therefore gives $j_{r}=t_{r}=\ell_{r}=s_{r}=z_{r}=0$ for every $r\geq 0$ . In particular the three families satisfy

(6.2)	$\displaystyle(0,0):$	$\displaystyle\quad t_{0}=0,$
(6.3)	$\displaystyle(a,0):$	$\displaystyle\quad(a+1)(t_{a}-4j_{a})=0,$
(6.4)	$\displaystyle(0,b):$	$\displaystyle\quad t_{0}+b(\ell_{0}-s_{0})+4(b-1)j_{0}=0.$

These equalities provide the dimension-four specializations without using a virtual twisting bundle in an index.

6.2. Dimension eight

Here $d=8$ in (5.2).

Now $k=2$ , and

(6.5)

R_{2}^{a,b}=h_{0}(8\delta_{2})^{2}+h_{1}\varepsilon_{2}.

Using (4.3),

	$\displaystyle(8\delta_{2})^{2}$	$\displaystyle=1+48q^{1/2}+624q+O(q^{3/2}),$
	$\displaystyle\varepsilon_{2}$	$\displaystyle=q^{1/2}+8q+O(q^{3/2}).$

Therefore

(6.6)	$\displaystyle j_{b}$	$\displaystyle=h_{0},$
(6.7)	$\displaystyle Q^{(8)}_{a,b}$	$\displaystyle=48h_{0}+h_{1},$
(6.8)	$\displaystyle H^{(8)}_{a,b}$	$\displaystyle=624h_{0}+8h_{1}.$

On the other side,

R_{1}^{a,b}=2^{4(a-b)}\left[h_{0}(8\delta_{1})^{2}+h_{1}\varepsilon_{1}\right].

After clearing the outer denominator, the $q$ coefficient identity is

(6.9)

V_{2}P^{(8)}_{a,b}=U_{2}(192h_{0}-h_{1}).

As $U_{2}\equiv V_{2}\equiv 1\pmod{3}$ , reduction of this integer equality gives $P^{(8)}_{a,b}\equiv-h_{1}\pmod{3}$ . Meanwhile (6.7) is congruent to $h_{1}$ . This reproduces, with all scalar coefficients visible,

(6.10)

P^{(8)}_{a,b}+Q^{(8)}_{a,b}\equiv 0\pmod{3}.

The genuine-bundle parameter families are

(6.11)

t_{0}\equiv 2j_{0}\pmod{3}.

(6.12)

(a+1)(t_{a}-8j_{a})-a(t_{0}-8j_{0})\equiv 0\pmod{3}.

(6.13)

t_{0}+b(\ell_{0}-s_{0})+8(b-1)j_{0}+b(t_{b}-8j_{b})\equiv 0\pmod{3}.

At $a=2$ , the first term of the $(a,0)$ relation is divisible by 3 and hence $t_{0}-dj_{0}\equiv 0$ . At $b=2$ , the $(0,b)$ relation is

t_{0}+2\ell_{0}-2s_{0}+dj_{0}+2(t_{2}-dj_{2})\equiv 0.

Subtracting $t_{0}-dj_{0}\equiv 0$ and multiplying by $2$ gives

t_{2}-dj_{2}+\ell_{0}-s_{0}+dj_{0}\equiv 0.

Thus

(6.14)

t_{2}+\ell_{0}+2j_{0}\equiv 2j_{2}+s_{0}\pmod{3}.

Here $t_{2}=\operatorname{Sig}(M,\mathcal{S}(M)\otimes T)$ and $j_{2}=\operatorname{Sig}(M,\mathcal{S}(M))$ .

6.3. Dimension twelve

Here $d=12$ in (5.2).

For $k=3$ the modular basis is

(6.15)

R_{2}^{a,b}=h_{0}(8\delta_{2})^{3}+h_{1}(8\delta_{2})\varepsilon_{2}.

Equations (4.4) and (4.5) give

(6.16)	$\displaystyle j_{b}$	$\displaystyle=-h_{0},$
(6.17)	$\displaystyle Q^{(12)}_{a,b}$	$\displaystyle=-(72h_{0}+h_{1}),$
(6.18)	$\displaystyle H^{(12)}_{a,b}$	$\displaystyle=-(1800h_{0}+32h_{1}).$

Indeed, $24k(12k-11)=72\cdot 25=1800$ and $24k-40=32$ .

The transformed expression is

R_{1}^{a,b}=2^{6(a-b)}\left[h_{0}(8\delta_{1})^{3}+h_{1}(8\delta_{1})\varepsilon_{1}\right].

Clearing the outer denominator in its $q$ coefficient gives

(6.19)

V_{3}P^{(12)}_{a,b}=U_{3}(576h_{0}+h_{1}).

All coefficients and factors are integers, so $U_{3}\equiv V_{3}\equiv 1\pmod{3}$ implies $P^{(12)}_{a,b}\equiv h_{1}\pmod{3}$ . Since (6.17) is $-h_{1}$ modulo $3$ , we obtain

(6.20)

P^{(12)}_{a,b}+Q^{(12)}_{a,b}\equiv 0\pmod{3}.

The genuine-bundle parameter families are

(6.21)

t_{0}\equiv 0\pmod{3}.

(6.22)

(a+1)(t_{a}-12j_{a})-a(t_{0}-12j_{0})\equiv 0\pmod{3}.

(6.23)

t_{0}+b(\ell_{0}-s_{0})+12(b-1)j_{0}+b(t_{b}-12j_{b})\equiv 0\pmod{3}.

At $a=2$ , the first term of the $(a,0)$ relation is divisible by 3 and hence $t_{0}-dj_{0}\equiv 0$ . At $b=2$ , the $(0,b)$ relation is

t_{0}+2\ell_{0}-2s_{0}+dj_{0}+2(t_{2}-dj_{2})\equiv 0.

Subtracting $t_{0}-dj_{0}\equiv 0$ and multiplying by $2$ gives

t_{2}-dj_{2}+\ell_{0}-s_{0}+dj_{0}\equiv 0.

Thus

(6.24)

t_{2}+\ell_{0}\equiv s_{0}\pmod{3}.

Here $t_{2}=\operatorname{Sig}(M,\mathcal{S}(M)\otimes T)$ and $j_{2}=\operatorname{Sig}(M,\mathcal{S}(M))$ .

6.4. Dimension sixteen

Here $d=16$ in (5.2).

When $k=4$ , the basis acquires a third term:

(6.25)

R_{2}^{a,b}=h_{0}(8\delta_{2})^{4}+h_{1}(8\delta_{2})^{2}\varepsilon_{2}+h_{2}\varepsilon_{2}^{2}.

The first three coefficients are

(6.26)	$\displaystyle j_{b}$	$\displaystyle=h_{0},$
(6.27)	$\displaystyle Q^{(16)}_{a,b}$	$\displaystyle=96h_{0}+h_{1},$
(6.28)	$\displaystyle H^{(16)}_{a,b}$	$\displaystyle=3552h_{0}+56h_{1}+h_{2}.$

Here $24\cdot 4(48-11)=3552$ . On the $R_{1}$ side, multiply the exact coefficient identity by $16V_{4}$ to obtain

(6.29)

16V_{4}P^{(16)}_{a,b}=U_{4}(24576h_{0}+128h_{1}-2h_{2}).

Reducing only this integer equality, using $16\equiv U_{4}\equiv V_{4}\equiv 1\pmod{3}$ , gives

P^{(16)}_{a,b}\equiv 2h_{1}+h_{2}\pmod{3}.

Equation (6.28) has the same reduction, and hence

(6.30)

P^{(16)}_{a,b}-H^{(16)}_{a,b}\equiv 0\pmod{3}.

For the two parameter families, put $C_{a}=\binom{a}{2}$ and $C_{b}=\binom{b}{2}$ . Then

(6.31)

\begin{split}(a+1)(t_{a}-16j_{a})&\equiv a\ell_{0}+C_{a}z_{0}+(1-16a-32C_{a})t_{0}\\ &\quad+(-16+136a+256C_{a})j_{0}\pmod{3}.\end{split}

(6.32)

\begin{split}t_{0}+b(\ell_{0}-s_{0})+16(b-1)j_{0}&\equiv b\ell_{b}+C_{b}z_{b}+(1-15b-32C_{b})t_{b}\\ &\quad+(-16+120b+256C_{b})j_{b}\pmod{3}.\end{split}

Taking $(a,b)=(2,0)$ and using $z_{0}=s_{0}+\ell_{0}$ gives

0\equiv H^{(d)}_{2,0}\equiv s_{0}+(1-d)t_{0}+\frac{d(d-3)}{2}j_{0}\pmod{3}.

(6.33)

s_{0}\equiv j_{0}\pmod{3}.

For $(a,b)=(0,2)$ , reduction of the exact combinations gives

	$\displaystyle P^{(d)}_{0,2}$	$\displaystyle\equiv t_{0}-\ell_{0}+s_{0}+dj_{0},$
	$\displaystyle H^{(d)}_{0,2}$	$\displaystyle\equiv s_{2}-dt_{2}+\binom{d}{2}j_{2}.$

Using the preceding Dirac congruence and $P^{(d)}_{0,2}\equiv H^{(d)}_{0,2}$ ,

s_{2}-dt_{2}+\binom{d}{2}j_{2}+\ell_{0}-dt_{0}+\binom{d+1}{2}j_{0}\equiv 0\pmod{3}.

Substituting the dimension and reducing modulo $3$ gives

(6.34)

s_{2}+\ell_{0}+j_{0}\equiv t_{2}+t_{0}\pmod{3}.

Here $s_{2}=\operatorname{Sig}(M,\mathcal{S}(M)\otimes S^{2}T)$ and $t_{2}=\operatorname{Sig}(M,\mathcal{S}(M)\otimes T)$ .

6.5. Dimension twenty

Here $d=20$ in (5.2).

For $k=5$ the modular basis is

(6.35)

R_{2}^{a,b}=h_{0}(8\delta_{2})^{5}+h_{1}(8\delta_{2})^{3}\varepsilon_{2}+h_{2}(8\delta_{2})\varepsilon_{2}^{2}.

The triangular relations are

(6.36)	$\displaystyle j_{b}$	$\displaystyle=-h_{0},$
(6.37)	$\displaystyle Q^{(20)}_{a,b}$	$\displaystyle=-(120h_{0}+h_{1}),$
(6.38)	$\displaystyle H^{(20)}_{a,b}$	$\displaystyle=-(5880h_{0}+80h_{1}+h_{2}),$

because $24\cdot 5(60-11)=5880$ . On the transformed side, clearing all denominators gives

(6.39)

16V_{5}P^{(20)}_{a,b}=U_{5}(61440h_{0}+448h_{1}-h_{2}).

Since $16\equiv U_{5}\equiv V_{5}\equiv 1\pmod{3}$ , this integer identity implies

P^{(20)}_{a,b}\equiv h_{1}+2h_{2}\pmod{3}.

The reduction of (6.38) is also $h_{1}+2h_{2}$ , since the leading minus sign changes $80h_{1}+h_{2}$ into $h_{1}+2h_{2}$ modulo $3$ . Therefore

(6.40)

P^{(20)}_{a,b}-H^{(20)}_{a,b}\equiv 0\pmod{3}.

For the two parameter families, put $C_{a}=\binom{a}{2}$ and $C_{b}=\binom{b}{2}$ . Then

(6.41)

\begin{split}(a+1)(t_{a}-20j_{a})&\equiv a\ell_{0}+C_{a}z_{0}+(1-20a-40C_{a})t_{0}\\ &\quad+(-20+210a+400C_{a})j_{0}\pmod{3}.\end{split}

(6.42)

\begin{split}t_{0}+b(\ell_{0}-s_{0})+20(b-1)j_{0}&\equiv b\ell_{b}+C_{b}z_{b}+(1-19b-40C_{b})t_{b}\\ &\quad+(-20+190b+400C_{b})j_{b}\pmod{3}.\end{split}

Taking $(a,b)=(2,0)$ and using $z_{0}=s_{0}+\ell_{0}$ gives

0\equiv H^{(d)}_{2,0}\equiv s_{0}+(1-d)t_{0}+\frac{d(d-3)}{2}j_{0}\pmod{3}.

(6.43)

s_{0}\equiv t_{0}+j_{0}\pmod{3}.

For $(a,b)=(0,2)$ , reduction of the exact combinations gives

	$\displaystyle P^{(d)}_{0,2}$	$\displaystyle\equiv t_{0}-\ell_{0}+s_{0}+dj_{0},$
	$\displaystyle H^{(d)}_{0,2}$	$\displaystyle\equiv s_{2}-dt_{2}+\binom{d}{2}j_{2}.$

Using the preceding Dirac congruence and $P^{(d)}_{0,2}\equiv H^{(d)}_{0,2}$ ,

s_{2}-dt_{2}+\binom{d}{2}j_{2}+\ell_{0}-dt_{0}+\binom{d+1}{2}j_{0}\equiv 0\pmod{3}.

Substituting the dimension and reducing modulo $3$ gives

(6.44)

s_{2}+\ell_{0}+j_{2}\equiv 2(t_{2}+t_{0})\pmod{3}.

Here $s_{2}=\operatorname{Sig}(M,\mathcal{S}(M)\otimes S^{2}T)$ and $t_{2}=\operatorname{Sig}(M,\mathcal{S}(M)\otimes T)$ .

6.6. Dimension twenty-eight

Here $d=28$ in (5.2).

Here $k=7$ and

(6.45)

\begin{split}R_{2}^{a,b}={}&h_{0}(8\delta_{2})^{7}+h_{1}(8\delta_{2})^{5}\varepsilon_{2}\\ &+h_{2}(8\delta_{2})^{3}\varepsilon_{2}^{2}+h_{3}(8\delta_{2})\varepsilon_{2}^{3}.\end{split}

The coefficient identities are

(6.46)	$\displaystyle j_{b}$	$\displaystyle=-h_{0},$
(6.47)	$\displaystyle Q^{(28)}_{a,b}$	$\displaystyle=-168h_{0}-h_{1},$
(6.48)	$\displaystyle H^{(28)}_{a,b}$	$\displaystyle=-12264h_{0}-128h_{1}-h_{2}.$

For the transformed $q$ coefficient, multiply by $256V_{7}$ :

(6.49)

256V_{7}P^{(28)}_{a,b}=U_{7}(5505024h_{0}+53248h_{1}+320h_{2}-3h_{3}).

This is an integer identity, and $256,U_{7},V_{7}$ are all congruent to $1$ modulo $3$ . It follows that both $P^{(28)}_{a,b}$ and $H^{(28)}_{a,b}$ have residue $h_{1}+2h_{2}$ modulo $3$ .

Taking $(a,b)=(2,0)$ and using $z_{0}=s_{0}+\ell_{0}$ gives

0\equiv H^{(d)}_{2,0}\equiv s_{0}+(1-d)t_{0}+\frac{d(d-3)}{2}j_{0}\pmod{3}.

(6.50)

s_{0}\equiv j_{0}\pmod{3}.

For $(a,b)=(0,2)$ , reduction of the exact combinations gives

	$\displaystyle P^{(d)}_{0,2}$	$\displaystyle\equiv t_{0}-\ell_{0}+s_{0}+dj_{0},$
	$\displaystyle H^{(d)}_{0,2}$	$\displaystyle\equiv s_{2}-dt_{2}+\binom{d}{2}j_{2}.$

Using the preceding Dirac congruence and $P^{(d)}_{0,2}\equiv H^{(d)}_{0,2}$ ,

s_{2}-dt_{2}+\binom{d}{2}j_{2}+\ell_{0}-dt_{0}+\binom{d+1}{2}j_{0}\equiv 0\pmod{3}.

Substituting the dimension and reducing modulo $3$ gives

(6.51)

s_{2}+\ell_{0}+j_{0}\equiv t_{2}+t_{0}\pmod{3}.

Here $s_{2}=\operatorname{Sig}(M,\mathcal{S}(M)\otimes S^{2}T)$ and $t_{2}=\operatorname{Sig}(M,\mathcal{S}(M)\otimes T)$ .

Together with the dimension- $24$ formulas $s_{0}+t_{0}\equiv 0$ and $s_{2}+\ell_{0}\equiv 0$ , these complete the proof of Theorem 1.2.

6.7. The next coefficient in dimension thirty-two

The exact formula of Lemma 4.2 still applies when $k=8$ , but involves $h_{4}$ . Put $D_{8}=1024$ and define the integer expression

J_{8}=50331648h_{0}+524288h_{1}+4096h_{2}-h_{4}.

Clearing the denominators in that formula gives

1024V_{8}P^{(32)}_{a,b}=U_{8}J_{8}.

To justify integrality of $h_{4}$ , the triangular induction needs one more Fourier order. The theta products give

	$\displaystyle 8\delta_{2}$	$\displaystyle=-1-24t-24t^{2}-96t^{3}-24t^{4}+O(t^{5}),$
	$\displaystyle\varepsilon_{2}$	$\displaystyle=t+8t^{2}+28t^{3}+64t^{4}+O(t^{5}).$

Thus all coefficients needed in this additional step are integral, and $h_{4}\in\mathbb{Z}$ follows as before. Reducing the cleared integer identity, since $1024\equiv U_{8}\equiv V_{8}\equiv 1\pmod{3}$ , gives

P^{(32)}_{a,b}\equiv 2h_{1}+h_{2}+2h_{4}\pmod{3}.

On the other side the integer coefficient formula is

H^{(32)}_{a,b}=16320h_{0}+152h_{1}+h_{2}\equiv 2h_{1}+h_{2}\pmod{3}.

The difference therefore has residue $2h_{4}$ . The same comparison does not automatically extend to dimension $32$ : it requires divisibility of $h_{4}$ by $3$ or another relation. This observation does not assert that such a condition fails for any particular string manifold.

6.8. Concise congruences modulo higher powers of three

Proof of Theorem 1.3.

Let $m=J_{0}=j_{0}\in\mathbb{Z}$ , so $U_{0}=t_{0}$ and $W_{0}=s_{0}$ . In dimension $d$ , the genuine-bundle product expression for the Witten bundle is

\Theta_{W}(T)=\left(\bigotimes_{n\geq 1}S_{q^{n}}(T)\right)\prod_{n\geq 1}(1-q^{n})^{d}.

The two factors have expansions

	$\displaystyle\bigotimes_{n\geq 1}S_{q^{n}}(T)$	$\displaystyle=1+Tq+(S^{2}T+T)q^{2}+O(q^{3}),$
	$\displaystyle\prod_{n\geq 1}(1-q^{n})^{d}$	$\displaystyle=1-dq+\frac{d(d-3)}{2}q^{2}+O(q^{3}).$

Their product and the index theorem give

(6.52)

\begin{split}\mathcal{W}_{M}(\tau)&:={\big\langle}\widehat{A}(TM)\operatorname{ch}(\Theta_{W}(T)),[M]{\big\rangle}\\ &=m+(U_{0}-dm)q+\left(W_{0}+(1-d)U_{0}+\frac{d(d-3)}{2}m\right)q^{2}+O(q^{3}).\end{split}

Under the string hypothesis this is a holomorphic modular form of weight $d/2$ for $SL_{2}(\mathbb{Z})$ . The classical structure theorem $M_{*}(SL_{2}(\mathbb{Z});\mathbb{C})=\mathbb{C}[E_{4},E_{6}]$ (see also [3, Theorem 1.1]) gives

M_{8}=\mathbb{C}E_{4}^{2},\qquad M_{10}=\mathbb{C}E_{4}E_{6},\qquad M_{14}=\mathbb{C}E_{4}^{2}E_{6}.

Here the subscript denotes modular weight. Each indicated generator has constant coefficient one. Consequently, in dimensions $16,20,28$ , respectively, $\mathcal{W}_{M}=mE_{4}^{2}$ , $mE_{4}E_{6}$ , and $mE_{4}^{2}E_{6}$ . The standard expansions

E_{4}=1+240q+2160q^{2}+O(q^{3}),\qquad E_{6}=1-504q-16632q^{2}+O(q^{3})

yield

(6.53)

\begin{array}[]{c|r|r}f&[q]f&[q^{2}]f\\ \hline\cr E_{4}^{2}&480&61920\\ E_{4}E_{6}&-264&-135432\\ E_{4}^{2}E_{6}&-24&-196632\end{array}.

If $A=[q]f$ and $B=[q^{2}]f$ , comparison with (6.52) gives the exact identities

U_{0}=(d+A)m,\qquad W_{0}=\left(B+(d-1)(d+A)-\frac{d(d-3)}{2}\right)m.

Thus, before taking any residues,

(6.54)

\begin{array}[]{c|r|r}d&U_{0}&W_{0}\\ \hline\cr 16&496m&69256m\\ 20&-244m&-140238m\\ 28&4m&-196874m\end{array}.

In particular,

	$\displaystyle d=16:$	$\displaystyle\quad W_{0}-J_{0}=69255m=729\cdot 95m,$
	$\displaystyle d=20:$	$\displaystyle\quad W_{0}=-140238m=-27\cdot 5194m,\qquad U_{0}+J_{0}=-243m,$
	$\displaystyle d=28:$	$\displaystyle\quad W_{0}-J_{0}=-196875m=-9\cdot 21875m.$

These integer identities prove three rows of (1.9) and the additional dimension- $20$ assertion.

For $d=24$ , the space has two generators, so the one-dimensional argument must be replaced by the coefficient elimination already given in Proposition 5.1. Its exact Chen–Han identity (5.13) says

W_{0}+U_{0}=196884m=27\cdot 7292m.

This proves the remaining row as an arithmetic consequence of the Chen–Han identity. ∎

Remark 6.1.

The refinements in (1.9) use additional relations among the selected Dirac indices, so they are compatible with the limitation in Remark 4.3. They do not strengthen the spinor-twisted column or the general parameter comparison, and the moduli are not claimed to be optimal on string bordism classes. The corresponding non-string examples in Example 7.2 also violate these refinements, since they already fail modulo $3$ .

7. Examples

This section presents counterexamples showing that the congruences in Theorems 1.1 and 1.2 do not extend to all closed spin manifolds without the string hypothesis. The first example tests the uniform comparison in dimensions $8$ and $16$ , while the second treats both selected congruences in every dimension of Theorem 1.2. The corresponding Dirac counterexamples also rule out such an extension of the stronger congruences in Theorem 1.3.

Example 7.1.

As in [3, Section 3], let $H=\mathbb{HP}^{2}$ with its standard orientation. It is spin since $H^{2}(H;\mathbb{Z}/2)=0$ , and

H^{*}(H;\mathbb{Z})=\mathbb{Z}[u]/(u^{3}),\qquad\langle u^{2},[H]\rangle=1,\qquad p_{1}(TH)=2u,\quad p_{2}(TH)=7u^{2}.

For an eight-dimensional manifold, the characteristic expansions needed here are

	$\displaystyle\widehat{A}$	$\displaystyle=1-\frac{p_{1}}{24}+\frac{7p_{1}^{2}-4p_{2}}{5760},$
	$\displaystyle\operatorname{ch}(T)$	$\displaystyle=8+p_{1}+\frac{p_{1}^{2}-2p_{2}}{12},$
	$\displaystyle\operatorname{ch}(\Lambda^{2}T)$	$\displaystyle=28+6p_{1}+\frac{p_{1}^{2}}{2},$
	$\displaystyle\operatorname{ch}(S^{2}T)$	$\displaystyle=36+10p_{1}+\frac{11p_{1}^{2}}{6}-\frac{8p_{2}}{3},$
	$\displaystyle\widehat{A}\operatorname{ch}(\mathcal{S})^{2}$	$\displaystyle=256\left(1+\frac{5p_{1}}{24}+\frac{67p_{1}^{2}+116p_{2}}{5760}\right),$

where only terms of degree at most eight are displayed. The square-power identities used here are $\operatorname{ch}(S^{2}T)=(\operatorname{ch}(T)^{2}+\operatorname{ch}(\psi^{2}T))/2$ and $\operatorname{ch}(\Lambda^{2}T)=(\operatorname{ch}(T)^{2}-\operatorname{ch}(\psi^{2}T))/2$ , where $\psi^{2}$ doubles each Chern root. Integration gives

\begin{array}[]{c|rrrr}&J_{r}(H)&U_{r}(H)&V_{r}(H)&W_{r}(H)\\ \hline\cr r=0&0&-1&1&-13\\ r=2&48&384&3136&960\end{array}.

For $a=b=0$ , (C1.2) would assert $U_{0}-8J_{0}\equiv 0\pmod{3}$ , whereas on $H$ this integer is $-1$ . This disproves an extension of that congruence to all spin manifolds.

For the higher-dimensional comparison, take $M=H\times H$ and $(a,b)=(2,0)$ . Multiplicativity of the index and the decomposition of squares of a direct sum give

(J_{0},U_{0},V_{0},W_{0})=(0,0,1,1),\qquad(J_{2},U_{2})=(2304,36864),\qquad Z_{0}=V_{0}+W_{0}=2.

Here $d=16$ , $c=1$ , $\alpha=-63$ , and $\beta=512$ . The left side of (C1.3) is $3U_{2}-48J_{2}=0$ , while its right side is $2V_{0}+Z_{0}-63U_{0}+512J_{0}=4\not\equiv 0\pmod{3}$ . Moreover, $p_{1}(TM)=2u_{1}+2u_{2}\neq 0$ . These examples show that the $p_{1}$ -vanishing hypothesis cannot simply be dropped from the uniform conclusion. They do not assert failure for every non-string manifold or every parameter choice.

Example 7.2.

We follow the product-manifold method of [3, Section 3]. Retain $H=\mathbb{HP}^{2}$ and the index values from Example 7.1. Let $K$ be a K3 surface with its complex orientation. Thus $K$ is spin, $\langle p_{1}(TK),[K]\rangle=-48$ , and $\widehat{A}(K)=2$ . Let $B$ be the eight-dimensional spin Bott manifold used in [3, Section 3], with $\widehat{A}(B)=1$ and $\operatorname{Sig}(B)=0$ . Consider the following closed spin manifolds:

\begin{array}[]{c|l}d&M_{d}\\ \hline\cr 8&H\\ 12&K\times H\\ 16&H\times H\\ 20&K\times H\times H\\ 24&B\times H\times H\\ 28&K\times B\times H\times H\end{array}.

Each has $p_{1}(TM_{d})\neq 0$ : restriction to any displayed $H$ factor gives $2u\neq 0$ . In dimension $24$ this is exactly the product used by Chen and Han to show the necessity of the string hypothesis for their congruence; here we test the different congruences in Theorem 1.2.

For clarity, define the integer residuals corresponding to the two congruences before coefficient reduction:

R_{D}(M)=\begin{cases}U_{0}-dJ_{0},&d=8,12,\\ W_{0}+(1-d)U_{0}+\dfrac{d(d-3)}{2}J_{0},&d=16,20,24,28,\end{cases}

R_{S}(M)=\begin{cases}U_{2}-dJ_{2}+V_{0}-W_{0}+dJ_{0},&d=8,12,\\ W_{2}-dU_{2}+\dbinom{d}{2}J_{2}+V_{0}-dU_{0}+\dbinom{d+1}{2}J_{0},&d=16,20,24,28.\end{cases}

Both would have to vanish modulo $3$ if the corresponding row of (1.2) held without the $p_{1}$ condition. Instead, direct calculation gives

\begin{array}[]{c|rr}d&R_{D}(M_{d})\bmod 3&R_{S}(M_{d})\bmod 3\\ \hline\cr 8&2&2\\ 12&1&1\\ 16&1&1\\ 20&2&2\\ 24&1&1\\ 28&2&2\end{array}.

We give the computations to verify both columns. The full spinor bundle and the $\widehat{A}$ -class are multiplicative on these even-dimensional products. For two factors $X,Y$ , with pulled-back bundles understood,

	$\displaystyle J_{r}(X\times Y)$	$\displaystyle=J_{r}(X)J_{r}(Y),$
	$\displaystyle U_{r}(X\times Y)$	$\displaystyle=U_{r}(X)J_{r}(Y)+J_{r}(X)U_{r}(Y),$
	$\displaystyle V_{r}(X\times Y)$	$\displaystyle=V_{r}(X)J_{r}(Y)+J_{r}(X)V_{r}(Y)+U_{r}(X)U_{r}(Y),$
	$\displaystyle W_{r}(X\times Y)$	$\displaystyle=W_{r}(X)J_{r}(Y)+J_{r}(X)W_{r}(Y)+U_{r}(X)U_{r}(Y).$

These are the same direct-sum and multiplicativity identities as those used in [3, Lemma 3.1]. For $H$ the values already computed yield

R_{D}(H)=-1,\qquad R_{S}(H)=384-8\cdot 48+1-(-13)=14.

For $K$ , the degree-four expansions

	$\displaystyle\widehat{A}$	$\displaystyle=1-\frac{p_{1}}{24},$	$\displaystyle\operatorname{ch}(\mathcal{S})$	$\displaystyle=4\left(1+\frac{p_{1}}{8}\right),$	$\displaystyle\operatorname{ch}(T)$	$\displaystyle=4+p_{1},$
	$\displaystyle\operatorname{ch}(\Lambda^{2}T)$	$\displaystyle=6+2p_{1},$	$\displaystyle\operatorname{ch}(S^{2}T)$	$\displaystyle=10+6p_{1}$

give

(J_{0},U_{0},V_{0},W_{0})(K)=(2,-40,-84,-268),\qquad(J_{2},U_{2})(K)=(-160,-1408).

Consequently, on $K\times H$ ,

(J_{0},U_{0},V_{0},W_{0})=(0,-2,42,14),\qquad(J_{2},U_{2})=(-7680,-129024),

and hence

R_{D}(K\times H)=-2,\qquad R_{S}(K\times H)=-129024+12\cdot 7680+42-14=-36836.

Both integers are $1$ modulo $3$ .

For the four higher dimensions, write $M_{d}=H\times H\times X$ , where $X$ is respectively a point, $K$ , $B$ , or $K\times B$ . The product formulas give

J_{0}(M_{d})=U_{0}(M_{d})=0,\qquad V_{0}(M_{d})=W_{0}(M_{d})=\widehat{A}(X).

Moreover, $J_{2}(H)=48$ , $U_{2}(H)=384$ , and $W_{2}(H)=960$ are all divisible by $3$ . The product formulas therefore imply

J_{2}(M_{d})\equiv U_{2}(M_{d})\equiv W_{2}(M_{d})\equiv 0\pmod{3}.

It follows that

R_{D}(M_{d})\equiv R_{S}(M_{d})\equiv\widehat{A}(X)\pmod{3},\qquad\widehat{A}(X)=1,2,1,2,

which proves the remaining four rows. In particular, on the reference product $B\times H\times H$ ,

\operatorname{Ind}(D\otimes S^{2}T)+\operatorname{Ind}(D\otimes T)=1,

\operatorname{Sig}(M,\mathcal{S}(M)\otimes S^{2}T)+\operatorname{Ind}(D\otimes\Lambda^{2}T)\equiv 1\pmod{3}.

Thus both concise formulas fail in each listed dimension after the $p_{1}$ -vanishing hypothesis is removed. The conclusion is the failure of a general extension to spin manifolds, not a claim that nonzero $p_{1}$ forces failure on every individual manifold.

Acknowledgements

The authors were supported by Science and Technology Development Plan Project of Jilin Province, China: No.20260102245JC, NSFC. No.12401059, the Fundamental Research Funds for the Central Universities N2405015 and Liaoning Province Science and Technology Plan Joint Project 2023-BSBA-118. The authors thank the referee for his (or her) careful reading and helpful comments.

Data availability

No data were collected for this article.

Conflict of interest

The authors have no relevant financial or non-financial interests to disclose.

Mod 33 congruence families for twisted signatures of string manifolds in dimensions up to twenty-eight

Abstract.

Key words and phrases:

2010 Mathematics Subject Classification

1. Introduction

Notation for the index pairings

Theorem 1.1.

Corollary 1.1 (Genuine-bundle form of Theorem 1.1).

Theorem 1.2.

Theorem 1.3.

2. Preliminaries

2.1. Spin and string structures

Definition 2.1.

2.2. Complex KK-theory and power operations

2.3. Characteristic forms

Lemma 2.2.

Proof.

2.4. Twisted Dirac indices and twisted signatures

Definition 2.3.

Remark 2.4.

2.5. Theta functions

Proposition 2.5 (Structure of the level-two rings).

Proof.

3. Theta bundles, modularity, and Fourier coefficients

Theorem 3.1.

Proof.

Proposition 3.2.

Proof.

4. The modular coefficient comparison

4.1. Triangular coefficients on the Γ0​(2)\Gamma^{0}(2) side

Lemma 4.1.

Proof.

4.2. The first coefficient on the Γ0​(2)\Gamma_{0}(2) side

Lemma 4.2.

4.3. Clearing denominators before taking integer residues

4.4. Proof of the main comparison

Proof of Theorem 1.1.

Remark 4.3.

5. The three parameter families in dimension twenty-four

5.1. The weight-twelve coefficient calculation

5.2. The case (a,b)=(0,0)(a,b)=(0,0)

Proposition 5.1 (Chen–Han).

Proof.

5.3. The family (a,0)(a,0)

Corollary 5.2.

Proof.

Corollary 5.3.

Proof.

5.4. The family (0,b)(0,b)

Corollary 5.4.

Proof.

Theorem 5.5.

Proof.

5.5. Recovery of the Chen–Han comparison

6. The remaining dimensions

6.1. Dimension four

6.2. Dimension eight

6.3. Dimension twelve

6.4. Dimension sixteen

6.5. Dimension twenty

6.6. Dimension twenty-eight

6.7. The next coefficient in dimension thirty-two

6.8. Concise congruences modulo higher powers of three

Proof of Theorem 1.3.

Remark 6.1.

7. Examples

Example 7.1.

Example 7.2.

Acknowledgements

Data availability

Conflict of interest

References

Mod $3$ congruence families for twisted signatures
of string manifolds in dimensions up to twenty-eight

2.2. Complex $K$ -theory and power operations

4.1. Triangular coefficients on the $\Gamma^{0}(2)$ side

4.2. The first coefficient on the $\Gamma_{0}(2)$ side

5.2. The case $(a,b)=(0,0)$

5.3. The family $(a,0)$

5.4. The family $(0,b)$