Modular Invariance and Anomaly Cancellation Formulas for Fiber Bundles

Abstract

By combining modular invariance of characteristic forms and the family index theory, we obtain some new anomaly cancellation formulas for any dimension under the not top degree component. For a fiber bundle of dimension $(4l-2)$, we obtain the anomaly cancellation formulas for the determinant line bundle. For the fiber bundle with a dimension of $(4l-3)$, we derive the anomaly cancellation formulas of the index gerbes. For the fiber bundle of dimension $(4l-1)$, we obtain some results of the eta invariants. Moreover, we give some anomaly cancellation formulas of the Chen–Connes character and the higher elliptic genera.

1. Introduction

The work of Alvarez-Gaumé and Witten in [1] established a key mathematical insight, often described as “miraculous cancellation”. They derived a diophantine formula that equates the top dimensional components of the Hirzebruch $\widehat{L}$-form to those of the $\widehat{A}$-form on a 12-dimensional smooth Riemannian manifold. This precise cancellation between these fundamental characteristic forms is not only structurally beautiful but also critically important, as it provides the essential mechanism for the cancellation of gravitational anomalies in physical theories, ensuring their mathematical consistency. In a significant advancement, Liu [2] pioneered the establishment of higher-dimensional “miraculous cancellation” formulas tailored for $(8k+4)$-dimensional Riemannian manifolds. This was achieved through the novel development of the modular invariance properties inherent to characteristic forms. A notable application of these foundational formulas lies in their utility for deriving certain divisibility results within this geometric context. Building upon this work, the research was further generalized by Han and Zhang in [3,4]. They succeeded in establishing a broader class of cancellation formulas that incorporate complex line bundles, and they explicitly demonstrated the applications of their more general framework. Subsequently, Chen and Han [5] pushed the boundaries further by deriving a family of twisted cancellation formulas, which apply to both $8k$ and $(8k+k)$-dimensional manifolds. They effectively leveraged these new formulas to investigate profound arithmetic properties, specifically divisibility on spin manifolds and congruence relations on ${\mathrm{spin}}^c$ manifolds. Building on the connection between modular forms and topology, Wang [6] conducted a study focused on the modular invariance of specific characteristic forms. This approach enabled the derivation of novel anomaly cancellation formulas. A key application of these formulas was to spin and spinc manifolds, leading to the derivation of non-trivial divisibility properties for certain characteristic numbers associated with these structures, thereby highlighting the utility of modular invariance in probing arithmetic aspects of manifold theory. A further generalization was achieved by Han, Liu, and Zhang [7], who obtained a broader cancellation formula through the use of Eisenstein series. Notably, in [8], they further showed that modular forms of weight 14 provide a unified derivation of both the $E_8\times E_8$ Green–Schwarz and the $E_8$ Horava–Witten anomaly factorization formulas, answering a question of J. H. Schwarz. They also established generalizations of these decomposition formulas and obtained a new Horava–Witten type decomposition formula on 12-dimensional manifolds. Expanding the toolkit available for such derivations, Han, Huang, Liu, and Zhang introduced in [9] new modular forms of weight 14 and weight 10 over $SL(2,\mathbf{Z})$. By utilizing these specific modular forms, they were able to derive a series of novel and interesting anomaly cancellation formulas, thereby enriching the catalog of topological constraints applicable to 12-dimensional manifolds. In a series of developments, the study of anomaly cancellation formulas has been advanced through the application of modular forms. Wang [10,11] derived new cancellation formulas by investigating specific $SL(2,\mathbf{Z})$ modular forms, leading to divisibility properties for the index of twisted Dirac operators. In a related direction, Liu [12] constructed a modular form of weight $2k$ associated to $2k$-dimensional spin manifolds. This line of research was further extended by Chen, Han, and Zhang [13], who introduced integral modular forms of weight $2k$ for both $4k$ and $(4k-1)$-dimensional $spi{n}^c$ manifolds, establishing an important extension of the modular formalism to broader geometric contexts.

On the other hand, anomaly quantifies the nontriviality of the determinant line bundle associated with a family of Dirac operators. The perturbative anomaly detects the real first Chern class of this bundle, whereas the global anomaly captures additional integral information beyond the real cohomology. For a family of Dirac operators parametrized over an even-dimensional closed manifold, the determinant line bundle is equipped with the Quillen metric and the compatible Bismut–Freed connection, whose curvature is given by the two-form component of the Atiyah–Singer family index density. Specifically, the integral along the fibers of the $\widehat{A}$-form of the vertical tangent bundle [14,15]. For a family of Dirac operators on an odd-dimensional manifold, Lott [16] constructed an Abelian gerbe-with-connection. This object is known as the index gerbe. It serves as a higher analogue of the determinant line bundle. Its curvature corresponds to the three-form component arising from the family index theorem. Building on these ideas, Han and Liu [17] combined modular forms with characteristic forms to derive anomaly cancellation formulas for both the determinant line bundle and the index gerbe, leading to new results on eta invariants.

A natural question arises: Can such cancellation formulas be extended to more general settings, particularly for fiber bundles? Furthermore, we observe that existing results are largely confined to the top degree component of modular forms, leaving the non-top degree components unexplored. Inspired by previous works [6,17], this paper moves beyond the study of top-degree components. We consider general-degree components of modular forms. This approach allows us to establish new anomaly cancellation formulas that are applicable to fiber bundles.

The structure of this paper is briefly described below: In Section 2, we have introduced some definitions and basic concepts that we will use in the paper. In Section 3, we prove anomaly cancellation formulas for determinant line bundles when the dimension of the fiber bundles is $(4l-2)$ and prove anomaly cancellation formulas for index gerbes when the dimension of the fiber bundles is $(4l-3)$ and obtain certain results about eta invariants when the dimension of the fiber is $(4l-1)$. Moreover, we give some anomaly cancellation formulas of the Chen–Connes character and the higher elliptic genera.

2. Characteristic Forms and Modular Forms

This section aims to review the theory of characteristic forms and modular forms that will be foundational to our derivation of anomaly cancellation formulas.

2.1. Characteristic Forms

Let $\widehat{A}(TM,{\nabla }^{TM})$ and $\widehat{L}(TM,{\nabla }^{TM})$ be the Hirzebruch characteristic forms defined, respectively, by (cf. [18])

$$\widehat{A}(TM,{\nabla }^{TM})={\det }^{{\displaystyle \frac{1}{2}}}\left({\displaystyle \frac{\frac{\sqrt{-1}}{4\pi }{R}^{TM}}{\sinh \left(\frac{\sqrt{-1}}{4\pi }{R}^{TM}\right)}}\right),$$

(1)

$$\widehat{L}(TM,{\nabla }^{TM})={\det }^{{\displaystyle \frac{1}{2}}}\left({\displaystyle \frac{\frac{\sqrt{-1}}{2\pi }{R}^{TM}}{\tanh \left(\frac{\sqrt{-1}}{4\pi }{R}^{TM}\right)}}\right).$$

(2)

Consider a smooth fiber bundle $\pi :M\to Y$ with compact fibers Z and a connected base Y. Let $TZ$ denote the vertical tangent bundle equipped with a metric $g^Z$. Suppose there exists a horizontal subbundle $T^{H}M$ such that $TM=T^{H}M\oplus TZ$, and let $T^{H}M$ carry the lift of a metric $g^Y$ from $TY$. Assuming orthogonality between $T^{H}M$ and $TZ$, the tangent bundle $TM$ is endowed with the metric $g^Y\oplus g^Z$. Let $P_Z$ be the orthogonal projection from $TM$ to $TZ$. Then, for $U\in TM$ and $V\in TZ$, the connection ${\nabla }^Z$ defined by ${\nabla }_U^{Z}V=P_Z{\nabla }_U^{L}V$ is metric-compatible with $g^Z$, and its curvature is given by $R^Z={\left({\nabla }^Z\right)}^2$. (cf. [19])

Given Hermitian vector bundles E and F over M with connections ${\nabla }^E,{\nabla }^F$ and curvatures $R^E$, $R^F$, we consider their formal difference $G=E-F$. This induces a natural connection ${\nabla }^G$, and we define the corresponding Chern character form as

$$\mathrm{ch}(G,{\nabla }^G)=\mathrm{Tr}\left[\exp \left({\displaystyle \frac{\sqrt{-1}}{2\pi }}{R}^E\right)\right]-\mathrm{Tr}\left[\exp \left({\displaystyle \frac{\sqrt{-1}}{2\pi }}{R}^F\right)\right].$$

(3)

For any complex number z, the total exterior and symmetric powers of E are defined as the formal sums

$${\wedge }_z\left(E\right)=\mathbf{C}{{|}_M+zE+z^2{\wedge }^2\left(E\right)+\cdots ,S_z\left(E\right)=\mathbf{C}|}_M+zE+z^2{S}^2\left(E\right)+\cdots ,$$

respectively, which are elements of $K\left(M\right)\left[\right[t\left]\right].$ These operations satisfy the following relations:

$$S_z\left(E\right)={\displaystyle \frac{1}{{\wedge }_{-z}\left(E\right)}},{\wedge }_z(E-F)={\displaystyle \frac{{\wedge }_z\left(E\right)}{{\wedge }_z\left(F\right)}}.$$

(4)

Moreover, let $\left\{{\sigma }_i\right\}$ and $\left\{{\sigma }_j^{\prime }\right\}$ denote the formal Chern roots of the Hermitian vector bundles E and F, respectively. Then

$$\mathrm{ch}\left({\wedge }_z\left(E\right)\right)=\prod _i(1+e^{{\sigma }_i}t).$$

(5)

These constructions give rise to the following Chern character formulas:

$$\mathrm{ch}\left(S_z\left(E\right)\right)={\displaystyle \frac{1}{{\prod }_i(1-e^{{\sigma }_i}t)}},\mathrm{ch}\left({\wedge }_z(E-F)\right)={\displaystyle \frac{{\prod }_i(1+e^{{\sigma }_i}t)}{{\prod }_j(1+e^{{\sigma }_j^{\prime }}t)}}.$$

(6)

Let W be a real Euclidean vector bundle over M equipped with a Euclidean connection ${\nabla }^W$. Its complexification $W_{\mathbf{C}}=W\otimes \mathbf{C}$ inherits a canonical Hermitian metric and a compatible Hermitian connection ${\nabla }^{W_{\mathbf{C}}}$ induced by those on W. Separately, for any vector bundle E (complex or real) over M, we define the formal difference $\tilde{E}=E-\dim E$ in $K\left(M\right)$ or $KO\left(M\right)$, respectively.

2.2. Some Properties About the Jacobi Theta Functions and Modular Forms

The modular group $SL(2,\mathbf{Z})$ consists of all integer matrices $\left(\begin{array}{cc}i& j\\ k& l\end{array}\right)$ with $il-jk=1$. It is generated by the elements

$$\mathcal{S}=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right),\mathcal{T}=\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$$

whose actions on the upper half-plane $\mathbf{H}$ are defined by $\mathcal{S}ϵ=-{\displaystyle \frac{1}{ϵ}}$ and $\mathcal{T}ϵ=ϵ+1$, with $ϵ\in \mathbf{H}$. Under these actions, the Jacobi theta functions transform as follows (cf. [20]):

$$\vartheta (u,ϵ+1)=exp\left({\displaystyle \frac{\pi \sqrt{-1}}{4}}\right)\vartheta (u,ϵ),\vartheta (u,-{\displaystyle \frac{1}{ϵ}})={\displaystyle \frac{1}{\sqrt{-1}}}{\left({\displaystyle \frac{ϵ}{\sqrt{-1}}}\right)}^{{\displaystyle \frac{1}{2}}}exp\left(\pi \sqrt{-1}ϵu^2\right)\vartheta (ϵu,ϵ);$$

(7)

$${\vartheta }_1(u,ϵ+1)=exp\left({\displaystyle \frac{\pi \sqrt{-1}}{4}}\right){\vartheta }_1(u,ϵ),{\vartheta }_1(u,-{\displaystyle \frac{1}{ϵ}})={\left({\displaystyle \frac{ϵ}{\sqrt{-1}}}\right)}^{{\displaystyle \frac{1}{2}}}exp\left(\pi \sqrt{-1}ϵu^2\right){\vartheta }_2(ϵu,ϵ);$$

(8)

$${\vartheta }_2(u,ϵ+1)={\vartheta }_3(u,ϵ),{\vartheta }_2(u,-{\displaystyle \frac{1}{ϵ}})={\left({\displaystyle \frac{ϵ}{\sqrt{-1}}}\right)}^{{\displaystyle \frac{1}{2}}}exp\left(\pi \sqrt{-1}ϵu^2\right){\vartheta }_1(ϵu,ϵ);$$

(9)

$${\vartheta }_3(u,ϵ+1)={\vartheta }_2(u,ϵ),{\vartheta }_3(u,-{\displaystyle \frac{1}{ϵ}})={\left({\displaystyle \frac{ϵ}{\sqrt{-1}}}\right)}^{{\displaystyle \frac{1}{2}}}exp\left(\pi \sqrt{-1}ϵu^2\right){\vartheta }_3(ϵu,ϵ);$$

(10)

$${\vartheta }^{\prime }(u,ϵ+1)=exp\left({\displaystyle \frac{\pi \sqrt{-1}}{4}}\right){\vartheta }^{\prime }(u,ϵ),{\vartheta }^{\prime }(0,-{\displaystyle \frac{1}{ϵ}})={\displaystyle \frac{1}{\sqrt{-1}}}{\left({\displaystyle \frac{ϵ}{\sqrt{-1}}}\right)}^{{\displaystyle \frac{1}{2}}}ϵ{\vartheta }^{\prime }(0,ϵ).$$

(11)

Definition 1.

Let Γ be a subgroup of $SL(2,\mathbf{Z})$, A modular form of weight k for Γ with character χ is a holomorphic function $f:\mathit{H}\to \mathbf{C}$ on the upper half-plane $\mathit{H}$ satisfying the following transformation law for all $g=\left(\begin{array}{cc}{a}_1& b_1\\ z_1& d_1\end{array}\right)\in \mathrm{\Gamma }$:

$$f\left(gϵ\right):=\chi \left(g\right){(z_1ϵ+d_1)}^{k}f\left(ϵ\right),$$

(12)

where $gϵ={\displaystyle \frac{a_1ϵ+b_1}{z_1ϵ+d_1}}$ and $\chi :\mathrm{\Gamma }\to {\mathbf{C}}^{★}$ is a group character. The integer k is called the weight of the modular form.

• Let us define the following modular subgroups of $SL(2,\mathbf{Z})$:

$${\mathrm{\Gamma }}_0\left(2\right)=\left\{\left(\begin{array}{cc}{a}_2& b_2\\ z_2& d_2\end{array}\right)\in SL(2,\mathbf{Z})\mid z_2\equiv 0\left(\mathrm{mod}2\right)\right\},$$

$${\mathrm{\Gamma }}^0\left(2\right)=\left\{\left(\begin{array}{cc}{a}_2& b_2\\ z_2& d_2\end{array}\right)\in SL(2,\mathbf{Z})\mid b_2\equiv 0\left(\mathrm{mod}2\right)\right\}.$$

It is a standard result that ${\mathrm{\Gamma }}_0\left(2\right)$ is generated by T and $S{T}^{2}ST$, while ${\mathrm{\Gamma }}^0\left(2\right)$ is generated by $STS$ and $T^{2}STS$ (cf. [20]).

For a modular subgroup $\mathrm{\Gamma }$, denote by ${\mathcal{M}}_{\mathbf{R}}\left(\mathrm{\Gamma }\right)$ the ring of modular forms over $\mathrm{\Gamma }$ with real Fourier coefficients. Adopting the notation ${\vartheta }_j={\vartheta }_j(0,ϵ),$ $1\le j\le 3,$ we introduce four explicit modular forms (cf. [2]):

$${\delta }_1\left(ϵ\right)={\displaystyle \frac{1}{8}}({\vartheta }_2^4+{\vartheta }_3^4),{\epsilon }_1\left(ϵ\right)={\displaystyle \frac{1}{16}}{\vartheta }_2^4{\vartheta }_3^4,$$

$${\delta }_2\left(ϵ\right)=-{\displaystyle \frac{1}{8}}({\vartheta }_1^4+{\vartheta }_3^4),{\epsilon }_2\left(ϵ\right)={\displaystyle \frac{1}{16}}{\vartheta }_1^4{\vartheta }_3^4.$$

They have the following Fourier expansions in $q^{{\displaystyle \frac{1}{2}}}$:

$${\delta }_1\left(ϵ\right)={\displaystyle \frac{1}{4}}+6q+\cdots ,{\epsilon }_1\left(ϵ\right)={\displaystyle \frac{1}{16}}-q+\cdots ,$$

$${\delta }_2\left(ϵ\right)=-{\displaystyle \frac{1}{8}}-3q^{{\displaystyle \frac{1}{2}}}+\cdots ,{\epsilon }_2\left(ϵ\right)=q^{{\displaystyle \frac{1}{2}}}+\cdots ,$$

where the omitted terms (denoted by $“\cdots ”$) are of higher degree and feature integral coefficients. Furthermore, these functions adhere to the transformation laws:

$${\delta }_2(-{\displaystyle \frac{1}{ϵ}})={ϵ}^2{\delta }_1\left(ϵ\right),{\epsilon }_2(-{\displaystyle \frac{1}{ϵ}})={ϵ}^4{\epsilon }_1\left(ϵ\right).$$

(13)

Lemma 1

([2]). Let ${\delta }_1\left(ϵ\right)$ (resp. ${\epsilon }_1\left(ϵ\right)$) is a modular form of weight 2 (resp. 4) over ${\mathrm{\Gamma }}_0\left(2\right)$, ${\delta }_2\left(ϵ\right)$ (resp. ${\epsilon }_2\left(ϵ\right)$) is a modular form of weight 2 (resp. 4) over ${\mathrm{\Gamma }}^0\left(2\right)$, while ${\delta }_3\left(ϵ\right)$ (resp. ${\epsilon }_3\left(ϵ\right)$) is a modular form of weight 2 (resp. 4) over ${\mathrm{\Gamma }}_{\theta }\left(2\right)$ and moreover ${\mathcal{M}}_{\mathbf{R}}\left({\mathrm{\Gamma }}^0\left(2\right)\right)=\mathbf{R}[{\delta }_2\left(ϵ\right),{\epsilon }_2\left(ϵ\right)]$.

3. The Generalized Cancellation Formulas

Consider a $4k$-dimensional Riemannian manifold M and a real vector bundle V of rank $2m$ over M. Suppose the dimension of the fiber Z is $4l$ and $l<k.$ Let $T_{\mathbf{C}}Z$ be the complexification of $TZ$ and $a,b$ be two integers. Set

$$\begin{array}{cc}\hfill {\mathrm{\Theta }}_1(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)=& \underset{n=1}{\overset{\infty }{⨂}}{S}_{q^n}\left(\widetilde{T_{\mathbf{C}}Z}\right)\otimes {\left(\underset{n^{\prime }=1}{\overset{\infty }{⨂}}{\wedge }_{q^{n^{\prime }}}\left(\widetilde{V_{\mathbf{C}}}\right)\right)}^a\hfill \\ \hfill & \otimes {\left(\underset{n^{\prime \prime }=1}{\overset{\infty }{⨂}}{\wedge }_{q^{n^{\prime \prime }-{\displaystyle \frac{1}{2}}}}\left(\widetilde{V_{\mathbf{C}}}\right)\right)}^b\otimes {\left(\underset{n^{\prime \prime \prime }=1}{\overset{\infty }{⨂}}{\wedge }_{-q^{n^{\prime \prime \prime }-{\displaystyle \frac{1}{2}}}}\left(\widetilde{V_{\mathbf{C}}}\right)\right)}^b,\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill {\mathrm{\Theta }}_2(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)=& \underset{n=1}{\overset{\infty }{⨂}}{S}_{q^n}\left(\widetilde{T_{\mathbf{C}}Z}\right)\otimes {\left(\underset{n^{\prime }=1}{\overset{\infty }{⨂}}{\wedge }_{q^{n^{\prime }}}\left(\widetilde{V_{\mathbf{C}}}\right)\right)}^b\hfill \\ \hfill & \otimes {\left(\underset{n^{\prime \prime }=1}{\overset{\infty }{⨂}}{\wedge }_{q^{n^{\prime \prime }-{\displaystyle \frac{1}{2}}}}\left(\widetilde{V_{\mathbf{C}}}\right)\right)}^b\otimes {\left(\underset{n^{\prime \prime \prime }=1}{\overset{\infty }{⨂}}{\wedge }_{-q^{n^{\prime \prime \prime }-{\displaystyle \frac{1}{2}}}}\left(\widetilde{V_{\mathbf{C}}}\right)\right)}^a,\hfill \end{array}$$

(15)

where $\widetilde{T_{\mathbf{C}}Z}=T_{\mathbf{C}}Z-\dim Z$, $\widetilde{V_{\mathbf{C}}}=V_{\mathbf{C}}-\dim V$. Clearly, ${\mathrm{\Theta }}_1(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)$ and ${\mathrm{\Theta }}_2(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)$ admit formal Fourier expansion in $q^{{\displaystyle \frac{1}{2}}}$ as

$${\mathrm{\Theta }}_1(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)=A_0(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)+A_1(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)q+A_2(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)q^2+\cdots ,$$

(16)

$${\mathrm{\Theta }}_2(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)=B_0(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)+B_1(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)q^{{\displaystyle \frac{1}{2}}}+B_2(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)q+\cdots ,$$

(17)

where $A_j$ and $B_j$ are elements in the semi-group formally generated by Hermitian vector bundles over Z, each admitting a canonically induced Hermitian connection. Let $\{\pm 2\pi i{y}_v\}$ denote the formal Chern roots of $V_{\mathbf{C}}$. Furthermore, if V is a spin with an associated spinor bundle $△\left(V\right)$, the Chern character of $△\left(V\right)$ is given by

$$\mathrm{ch}(△\left(V\right))={\mathrm{\Pi }}_{v=1}^m(e^{\pi i{y}_v}+e^{-\pi i{y}_v}).$$

(18)

Without assuming V is spin, we adopt the formal notation $\mathrm{ch}\left({(△\left(V\right))}^a\right)$ as shorthand for the expression ${\left({\mathrm{\Pi }}_{v=1}^m(e^{\pi i{y}_v}+e^{-\pi i{y}_v})\right)}^a$, which is well-defined in the cohomology of M.

Let $p_1$ denote the first Pontryagin class. For any differential form $\omega$ on M, we write ${\omega }^{\left(4p\right)}$ for its component of degree. Note that this is a general term and may not represent the top degree component. We now state one of our main results.

Theorem 1.

If $(a+2b)p_1\left(V\right)=p_1\left(Z\right)$, then

$${\left\{\widehat{A}(TZ,{\nabla }^{TZ})\mathrm{ch}\left({(△\left(V\right))}^a\right)\right\}}^{\left(4p\right)}=2^{(a-b)m+p}\sum _{r=0}^{\left[{\displaystyle \frac{p}{2}}\right]}{2}^{-6r}{h}_r,$$

(19)

where each $h_r,1\le r\le \left[{\displaystyle \frac{p}{2}}\right],$ is a canonical integral linear combination of

$${\left\{\widehat{A}(TZ,{\nabla }^{TZ})\mathrm{ch}\left({(△\left(V\right))}^b\right)\mathrm{ch}\left(B_q(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)\right)\right\}}^{\left(4p\right)},0\le q\le r,$$

and $h_1,$ $h_2$ are given by (31) and (32).

Proof. 

Let $\{\pm 2\pi \sqrt{-1}{x}_j|1\le j\le l\}$ be the Chern roots of $T_{\mathbf{C}}Z.$ Set

$$\begin{array}{c}\hfill Q_1\left(ϵ\right)=\widehat{A}(TZ,{\nabla }^{TZ})\mathrm{ch}\left({(△\left(V\right))}^a\right)\mathrm{ch}\left({\mathrm{\Theta }}_1(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)\right),\end{array}$$

(20)

$$\begin{array}{c}\hfill Q_2\left(ϵ\right)=\widehat{A}(TZ,{\nabla }^{TZ})\mathrm{ch}\left({(△\left(V\right))}^b\right)\mathrm{ch}\left({\mathrm{\Theta }}_2(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)\right).\end{array}$$

(21)

Moreover, we can show by direct computations that

$$\widehat{A}(TZ,{\nabla }^{TZ})\mathrm{ch}\left(\underset{n=1}{\overset{\infty }{⨂}}{S}_{q^n}\left(\widetilde{T_{\mathbf{C}}Z}\right)\right)=\prod _{j=1}^{2k}{\displaystyle \frac{x_j{\vartheta }^{\prime }(0,ϵ)}{\vartheta (x_j,ϵ)}},$$

(22)

$$\mathrm{ch}\left(△\left(V\right)\otimes \underset{n^{\prime }=1}{\overset{\infty }{⨂}}{\wedge }_{q^{n^{\prime }}}\left(\widetilde{V_{\mathbf{C}}}\right)\right)=2^m\prod _{v=1}^m{\displaystyle \frac{{\vartheta }_1(y_v,ϵ)}{{\vartheta }_1(0,ϵ)}},$$

(23)

$$\mathrm{ch}\left(\underset{n^{\prime \prime }=1}{\overset{\infty }{⨂}}{\wedge }_{q^{n^{\prime \prime }-{\displaystyle \frac{1}{2}}}}\left(\widetilde{V_{\mathbf{C}}}\right)\right)=\prod _{v=1}^m{\displaystyle \frac{{\vartheta }_3(y_v,ϵ)}{{\vartheta }_3(0,ϵ)}},$$

(24)

$$\mathrm{ch}\left(\underset{n^{\prime \prime \prime }=1}{\overset{\infty }{⨂}}{\wedge }_{-q^{n^{\prime \prime \prime }-{\displaystyle \frac{1}{2}}}}\left(\widetilde{V_{\mathbf{C}}}\right)\right)=\prod _{v=1}^m{\displaystyle \frac{{\vartheta }_2(y_v,ϵ)}{{\vartheta }_2(0,ϵ)}}.$$

(25)

So we have

$$Q_1\left(ϵ\right)=2^{am}\left(\prod _{j=1}^{2k}{\displaystyle \frac{x_j{\vartheta }^{\prime }(0,ϵ)}{\vartheta (x_j,ϵ)}}\right)\left(\prod _{v=1}^m{\displaystyle \frac{{\vartheta }_1^a(y_v,ϵ){\vartheta }_2^b(y_v,ϵ){\vartheta }_3^b(y_v,ϵ)}{{\vartheta }_1^a(0,ϵ){\vartheta }_2^b(0,ϵ){\vartheta }_3^b(0,ϵ)}}\right).$$

(26)

Similarly, we have

$$Q_2\left(ϵ\right)=2^{bm}\left(\prod _{j=1}^{2k}{\displaystyle \frac{x_j{\theta }^{\prime }(0,ϵ)}{\theta (x_j,ϵ)}}\right)\left(\prod _{v=1}^m{\displaystyle \frac{{\theta }_2^a(y_v,ϵ){\theta }_1^b(y_v,ϵ){\theta }_3^b(y_v,ϵ)}{{\theta }_2^a(0,ϵ){\theta }_1^b(0,ϵ){\theta }_3^b(0,ϵ)}}\right).$$

(27)

Setting $P_1\left(ϵ\right)=Q_1{\left(ϵ\right)}^{4p}$ and $P_2\left(ϵ\right)=Q_2{\left(ϵ\right)}^{4p}$, and invoking by (7)–(11) together with the identity $(a+2b)p_1\left(V\right)=p_1\left(Z\right)$, we find that $P_1\left(ϵ\right)$ is a modular form of weight $2p$ over ${\mathrm{\Gamma }}^0\left(2\right)$, and $P_2\left(ϵ\right)$ is a modular form of weight $2p$ over ${\mathrm{\Gamma }}_0\left(2\right)$. Additionally, the following identity is satisfied:

$$P_1\left(-{\displaystyle \frac{1}{ϵ}}\right)=2^{(a-b)m}{ϵ}^{2p}{P}_2\left(ϵ\right).$$

(28)

At each point $x\in Z,$ modulate the volume form defined by the metric on $T_{x}Z,$ the functions $P_i\left(ϵ\right),i=1,2,$ admit expansions as power series in $q^{{\displaystyle \frac{1}{2}}}$ with real Fourier coefficients. It then follows from Lemma 1 that

$$P_2\left(ϵ\right)=h_0{\left(8{\delta }_2\right)}^p+h_1{\left(8{\delta }_2\right)}^{p-2}{\epsilon }_2+\cdots +h_{\left[{\displaystyle \frac{l}{2}}\right]}{\left(8{\delta }_2\right)}^{p-2\left[{\displaystyle \frac{p}{2}}\right]}{\epsilon }_2^{\left[{\displaystyle \frac{p}{2}}\right]},$$

(29)

where each $h_r,0\le r\le \left[{\displaystyle \frac{p}{2}}\right],$ is a real multiple of the volume form at $x.$ By (13), (28) and (29), we obtain

$$P_1\left(ϵ\right)=2^{(a-b)m}\left[h_0{\left(8{\delta }_1\right)}^p+h_1{\left(8{\delta }_1\right)}^{p-2}{\epsilon }_1+\cdots +h_{\left[{\displaystyle \frac{p}{2}}\right]}{\left(8{\delta }_1\right)}^{p-2\left[{\displaystyle \frac{p}{2}}\right]}{\epsilon }_1^{\left[{\displaystyle \frac{p}{2}}\right]}\right].$$

(30)

Comparing the constant terms in (30) yields (19). Then, by equating the coefficients of $q^{{\displaystyle \frac{j}{2}}}$, $j\ge 0$ on both sides of (29) and proceeding by induction, we find that each $h_r$ with $1\le r\le \left[{\displaystyle \frac{p}{2}}\right]$ can be expressed as an integral linear combination of the forms ${\left\{\widehat{A}(TZ,{\nabla }^{TZ})\mathrm{ch}\left({(△\left(V\right))}^b\right)\mathrm{ch}\left(B_q(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)\right)\right\}}^{\left(4p\right)}.$ Explicit expressions for $h_0$ and $h_1$ are given as follows:

$$h_0={(-1)}^p{\left[\widehat{A}(TZ,{\nabla }^{TZ})\mathrm{ch}\left({(△\left(V\right))}^b\right)\right]}^{\left(4p\right)},$$

(31)

$$h_1={(-1)}^p{\left\{\widehat{A}(TZ,{\nabla }^{TZ})\mathrm{ch}\left({(△\left(V\right))}^b\right)(-24p+\mathrm{ch}\left(B_1(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)\right))\right\}}^{\left(4p\right)},$$

(32)

where $B_1(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b))=-(a-b)\widetilde{V_{\mathbf{C}}}.$

The proof is completed. □

When $(a+2b)p_1\left(V\right)\ne p_1\left(M\right)$ and considering the top degree component, we can obtain Liu–Wang’s theorem.

Theorem 2

(Liu–Wang [21]).

$$\begin{gathered} {\left\{\widehat{A},(TM,{\nabla }^{TM})\mathrm{ch}\left({(△\left(V\right))}^a\right)\right\}}^{\left(4k\right)}-\sum _{r=0}^{\left[{\displaystyle \frac{k}{2}}\right]}{2}^{(a-b)m+k+6r} \\ ·{\left\{\widehat{A}(TM,{\nabla }^{TM})\mathrm{ch}\left({(△\left(V\right))}^b\right)\mathrm{ch}\left(b_r\right)\right\}}^{\left(4k\right)} \\ =[p_1\left(M\right)-(a+2b)p_1\left(V\right)]\mathcal{B}({\nabla }^{TM},{\nabla }^V,a,b), \end{gathered}$$

(33)

where

$$\begin{array}{cc}\hfill \mathcal{B}({\nabla }^{TM},{\nabla }^V,a,b)=& \sum _{r=0}^{\left[{\displaystyle \frac{k}{2}}\right]}{2}^{(a-b)m+k+6r}{\beta }_r-\{{\displaystyle \frac{e^{\frac{1}{24}[p_1\left(M\right)-(a+2b)p_1\left(V\right)]}-1}{[p_1\left(M\right)-(a+2b)p_1\left(V\right)]}}\hfill \\ \hfill & ·\widehat{A}(TM,{\nabla }^{TM})\mathrm{ch}\left({(△\left(V\right))}^a\right){\}}^{4k-4}.\hfill \end{array}$$

(34)

Example 1.

Let N be an 8-dimensional smooth closed oriented manifold. $\mathbf{H}{P}^2$ is a quaterionic projective plane and $dim\mathbf{H}{P}^2=8$. Let $M=\mathbf{H}{P}^2\times N,$ $V=TZ,$ $Z=\mathbf{H}{P}^2$ and Z be spin. We take $a=0,$ $b=1$ and $p=1$. By Theorem 1 and Formula (33), we consider the following anomaly cancellation formula.

$$\begin{gathered} {\left\{\widehat{A}(TZ,{\nabla }^{TZ})\mathrm{ch}\left({(△\left(V\right))}^o\right)\right\}}^{\left(4\right)}-\sum _{r=0}^{\left[{\displaystyle \frac{1}{2}}\right]}{2}^{(0-1)4+1-6r}{h}_r \\ =[p_1\left(Z\right)-(0+2)p_1\left(V\right)]\mathcal{B}({\nabla }^{TZ},{\nabla }^V,0,1). \end{gathered}$$

(35)

First, the total Pontryagin class of $\mathbf{H}{P}^2$

$$p\left(\mathbf{H}{P}^2\right)={(1+u)}^6{(1+4u)}^{-1},$$

where $u\in H^4\left(\mathbf{H}{P}^2\right)$ is the generator. So $p_1\left(\mathbf{H}{P}^2\right)=2u$ and $p_2\left(\mathbf{H}{P}^2\right)=7u^2$. Then, we have

$${\left\{\widehat{A}(TZ,{\nabla }^{TZ})\right\}}^{\left(4\right)}=-{\displaystyle \frac{1}{24}}{p}_1\left(TZ\right)=-{\displaystyle \frac{1}{24}}{p}_1\left(T\mathbf{H}{P}^2\right)=-{\displaystyle \frac{1}{12}}u,$$

$$\sum _{r=0}^{\left[{\displaystyle \frac{1}{2}}\right]}{2}^{(0-1)4+1-6r}{h}_r=2^{-3}{h}_0,$$

$$\begin{array}{cc}\hfill h_0=& (-1){\left\{\widehat{A}(TZ,{\nabla }^{TZ})\mathrm{ch}\left({(△\left(V\right))}^1\right)\right\}}^{\left(4\right)}\hfill \\ \hfill =& (-1)\left[{\left(\widehat{A}(TZ,{\nabla }^{TZ})\right)}^{\left(0\right)}{\left(\mathrm{ch}\left({(△\left(V\right))}^1\right)\right)}^{\left(4\right)}+{\left(\widehat{A}(TZ,{\nabla }^{TZ})\right)}^{\left(4\right)}{\left(\mathrm{ch}\left({(△\left(V\right))}^1\right)\right)}^{\left(0\right)}\right]\hfill \\ \hfill =& (-1)(1·4u+-{\displaystyle \frac{1}{12}}u·16)\hfill \\ \hfill =& -{\displaystyle \frac{1}{3}}u.\hfill \end{array}$$

Therefore, the left side of Equation (35) is equal to ${\displaystyle \frac{1}{4}}u$. For the right side of Equation (35),

$$\begin{array}{cc}\hfill [p_1\left(Z\right)-(0+2)p_1\left(V\right)]\mathcal{B}({\nabla }^{TZ},{\nabla }^V,0,1)& =-p_1\left(Z\right)\left(2^{-3}·(-1)·{\displaystyle \frac{1}{24}}·16-{\displaystyle \frac{1}{24}}\right)\hfill \\ \hfill & =-{\displaystyle \frac{1}{8}}(-p_1\left(Z\right))\hfill \\ \hfill & ={\displaystyle \frac{1}{4}}u.\hfill \end{array}$$

Therefore, Equation (35) holds.

By comparing the coefficients of $q,q^2$ in (30), we have

Theorem 3.

If $(a+2b)p_1\left(V\right)=p_1\left(Z\right)$, then

$$\begin{array}{cc}\hfill \{\widehat{A}& (TZ,{\nabla }^{TZ})\mathrm{ch}\left({(△\left(V\right))}^a\right)\mathrm{ch}(A_1(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)-24p){\}}^{\left(4p\right)}\hfill \\ \hfill =& -2^{(a-b)m+p+6}\sum _{r=0}^{\left[{\displaystyle \frac{p}{2}}\right]}r{2}^{-6r}{h}_r,\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill \{\widehat{A}& (TZ,{\nabla }^{TZ})\mathrm{ch}\left({(△\left(V\right))}^a\right)\mathrm{ch}(A_2(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)\hfill \\ \hfill & -(24p+6)(A_1(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)-24p)-(288p^2-264p)){\}}^{\left(4p\right)}\hfill \\ \hfill =& 2^{(a-b)m+p+11}\sum _{r=0}^{\left[{\displaystyle \frac{p}{2}}\right]}{r}^2{2}^{-6r}{h}_r.\hfill \end{array}$$

(37)

where

$$A_1(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)=\widetilde{T_{\mathbf{C}}Z}+a\widetilde{V_{\mathbf{C}}}+b(2{\wedge }^2\widetilde{V_{\mathbf{C}}}-\widetilde{V_{\mathbf{C}}}\otimes \widetilde{V_{\mathbf{C}}}),$$

(38)

$$\begin{array}{cc}\hfill A_2& (T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)\hfill \\ \hfill =& S^2\widetilde{T_{\mathbf{C}}Z}+\widetilde{T_{\mathbf{C}}Z}+a(\widetilde{T_{\mathbf{C}}Z}\otimes \widetilde{V_{\mathbf{C}}})+{\displaystyle \frac{a(a-1)}{2}}\widetilde{V_{\mathbf{C}}}\otimes \widetilde{V_{\mathbf{C}}}\hfill \\ \hfill & +a({\wedge }^2\widetilde{V_{\mathbf{C}}}+\widetilde{V_{\mathbf{C}}})+b(2{\wedge }^2\widetilde{V_{\mathbf{C}}}-\widetilde{V_{\mathbf{C}}}\otimes \widetilde{V_{\mathbf{C}}})\otimes (\widetilde{T_{\mathbf{C}}Z}+a\widetilde{V_{\mathbf{C}}})\hfill \\ \hfill & +{\displaystyle \frac{b(b-1)}{2}}(2{\wedge }^2\widetilde{V_{\mathbf{C}}}-\widetilde{V_{\mathbf{C}}}\otimes \widetilde{V_{\mathbf{C}}})\otimes (2\wedge \widetilde{V_{\mathbf{C}}}-\widetilde{V_{\mathbf{C}}}\otimes \widetilde{V_{\mathbf{C}}})\hfill \\ \hfill & +b(2(2{\wedge }^2\widetilde{V_{\mathbf{C}}}+\widetilde{V_{\mathbf{C}}}\otimes \widetilde{V_{\mathbf{C}}})-2({\wedge }^3\widetilde{V_{\mathbf{C}}}+\widetilde{V_{\mathbf{C}}})\otimes \widetilde{V_{\mathbf{C}}}\hfill \\ \hfill & +{\wedge }^2\widetilde{V_{\mathbf{C}}}\otimes {\wedge }^2\widetilde{V_{\mathbf{C}}}),\hfill \end{array}$$

(39)

and each $h_r,1\le r\le \left[{\displaystyle \frac{p}{2}}\right],$ is a canonical integral linear combination of

$${\left\{\widehat{A}(TZ,{\nabla }^{TZ})\mathrm{ch}\left({(△\left(V\right))}^b\right)\mathrm{ch}\left(B_q(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)\right)\right\}}^{\left(4p\right)},0\le q\le r.$$

We define virtual complex vector bundles ${\mathcal{B}}_r(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)$ on M, parameterized by integers $0\le r\le \left[{\displaystyle \frac{l}{2}}\right]$, for the cases where the fiber Z has dimension $4l-1,$$4l-2$ or $4l-3,$ via the equality:

$${\mathrm{\Theta }}_2(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)=\sum _{r=0}^{\left[{\displaystyle \frac{l}{2}}\right]}{\mathcal{B}}_r(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b){\left(8{\delta }_2\right)}^{l-2r}{\epsilon }_2^r\mathrm{mod}{q}^{(\left[{\displaystyle \frac{l}{2}}\right]+1)/2}·K\left(M\right)\left[q^{{\displaystyle \frac{1}{2}}}\right].$$

(40)

It is straightforward to show that each ${\mathcal{B}}_r(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)$, for $0\le r\le \left[{\displaystyle \frac{l}{2}}\right]$, can be expressed as a canonical linear combination of the terms $B_q(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)$ with $0\le q\le r.$ Moreover, these bundles carry canonically induced metrics and connections. We then obtain the following:

$$h_r={\left\{\widehat{A}(TZ,{\nabla }^{TZ})\mathrm{ch}\left({(△\left(V\right))}^b\right)\mathrm{ch}\left({\mathcal{B}}_r(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)\right)\right\}}^{\left(4l\right)},0\le r\le \left[{\displaystyle \frac{l}{2}}\right].$$

Let us recall the definition of Chern–Connes character represented by [22] the cocycle in the entire cyclic cohomology, which plays an important role in noncommutative geometry.

Definition 2

(cf. [22,23,24]). The Chern character of θ-summable even spectral triple ${\mathrm{ch}}_{\ast }(A,H,D,\gamma )=\left\{{\mathrm{ch}}_j\left(D\right)\right|j\ge 0andeven\}$ in the entire cyclic cohomology is defined by

$$\begin{array}{cc}\hfill {\mathrm{ch}}_j& \left(D\right)(f^0,\cdots ,f^j)\hfill \\ \hfill & ={\int }_{{▵}_j}\mathrm{str}(f^0{e}^{-s_1{D}^2}[D,f^1]e^{-(s_2-s_1)D^2[D,f^2]}\cdots e^{-(s_j-s_{j-1})D^2}[D,f^j]e^{-(1-s_j)D^2})ds,\hfill \end{array}$$

(41)

where $f^i\in A$ and ${▵}_j=\left\{(s_1,\cdots ,s_j)\right|0\le s_1\le \cdots \le s_j\le 1\}.$ For $t>0,$ considering the deformed Chern–Connes character ${\mathrm{ch}}_{\ast }\left(\sqrt{t}D\right)=\left\{{\mathrm{ch}}_j\left(\sqrt{t}D\right)\right|j\ge 0andeven\}$ is expressed by

$$\begin{array}{cc}\hfill {\mathrm{ch}}_j& \left(\sqrt{t}D\right)(f^0,\cdots ,f^j)\hfill \\ \hfill & =t^{{\displaystyle \frac{j}{2}}}{\int }_{{▵}_j}\mathrm{str}(f^0{e}^{-s_{1}t{D}^2}[D,f^1]e^{-(s_2-s_1)t{D}^2[D,f^2]}\cdots e^{-(s_j-s_{j-1})t{D}^2}[D,f^j]e^{-(1-s_j)t{D}^2})ds.\hfill \end{array}$$

(42)

Now let $\omega$ be a differential form over M with degree $2n$ and $4l+2n=4k$. If the fiber Z is spin and V is a complex bundle, we can construct the twisted Dirac triple $(C^{\infty }\left(Z\right),L^2(Z,S\left(TZ\right)\otimes V),D\otimes E,{\mathrm{\Gamma }}_1)$ where $S\left(TZ\right)$ denotes the spinors bundle and $D\otimes V$ is the twisted Dirac operator. Using the Getzler symbol calculus, we can prove

$$\begin{array}{cc}\hfill {\widehat{\mathrm{ch}}}_{2n}(D\otimes V,f^0,\cdots ,f^{2n}):=& \underset{t\to 0}{lim}{\mathrm{ch}}_{2n}\left(\sqrt{t}D\right)(f^0,\cdots ,f^j)\hfill \\ \hfill =& {\int }_Z\widehat{A}(TZ,{\nabla }^{TZ})\mathrm{ch}\left(V\right)f^{0}d{f}^1\wedge \cdots \wedge d{f}^{2n},\hfill \end{array}$$

(43)

where $\omega =f^{0}d{f}^1\wedge \cdots \wedge d{f}^{2n}$. We call ${\widehat{\mathrm{ch}}}_{2n}(D\otimes V,f^0,\cdots ,f^{2n})$ the Chern–Connes numbers. So by Theorems 1 and 2, we have

Corollary 1.

The following identity holds

$${\widehat{\mathrm{ch}}}_{2n}(D\otimes {(▵\left(V\right))}^a,f^0,\cdots ,f^{2n})=2^{(a-b)m+l}\sum _{r=0}^{\left[{\displaystyle \frac{l}{2}}\right]}{2}^{-6r}{h}_r,$$

(44)

$$\begin{array}{cc}\hfill {\widehat{\mathrm{ch}}}_{2n}(D\otimes {(△\left(V\right))}^a\otimes (A_1(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)& -24l),f^0,\cdots ,f^{2n})\hfill \\ \hfill =& -2^{(a-b)m+l+6}\sum _{r=0}^{\left[{\displaystyle \frac{l}{2}}\right]}r{2}^{-6r}{h}_r,\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill {\widehat{\mathrm{ch}}}_{2n}{(D\otimes (△\left(V\right))}^a\otimes (A_2(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)-(24l& +6)(A_1(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)-24l)\hfill \\ & -(288l^2-264l)),f^0,\cdots ,f^{2n})\hfill \\ \hfill =& 2^{(a-b)m+l+11}\sum _{r=0}^{\left[{\displaystyle \frac{l}{2}}\right]}{r}^2{2}^{-6r}{h}_r,\hfill \end{array}$$

(46)

where each $h_r,$ $0\le r\le \left[{\displaystyle \frac{l}{2}}\right],$ is a canonical integral linear combination of ${\widehat{\mathrm{ch}}}_{2n}(D\otimes {(▵\left(V\right))}^b\otimes B_q(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b),f^0,\cdots ,f^{2n})$, $0\le q\le r.$

Below, we consider anomaly cancellation formulas of determinant line bundles.

Let O be the principal $SO\left(4l\right)$ bundle of oriented orthonormal frames associated to $TZ$. Suppose $TZ$ is spin, then the bundle $O\underset{\varrho }{⟶}M$ admits a lift to a principal $SO\left(4l\right)$ bundle

$$O^{\prime }\underset{\sigma }{⟶}O\underset{\varrho }{⟶}M$$

such that the map $\sigma$ restricts fiberwise to the standard covering projection $\mathrm{Spin}\left(4l\right)\to SO\left(4l\right)$. We define the Hermitian spinor bundles

$$F=O^{\prime }{\times }_{\mathrm{Spin}\left(4l\right)}{S}_{4l},F_{\pm }=O^{\prime }{\times }_{\mathrm{Spin}\left(4l\right)}{S}_{\pm ,4l},$$

where $S_{4l}=S_{+,4l}\oplus S_{-,4l}$ denotes the complex spinor representation. The connection ${\nabla }^Z$ on O naturally lifts to a connection on $O^{\prime }$, which, in turn, induces a unitary connection on each of the spinor bundles F and $F_{\pm }$. For simplicity, we denote all these induced connections by ∇.

Define $H^{\infty }$ and $H_{\pm }^{\infty }$ as the spaces of $C^{\infty }$ sections of $F\otimes V$ and $F_{\pm }\otimes V$ over the total space M. These spaces may also be interpreted as comprising $C^{\infty }$ sections over the base manifold Y of corresponding infinite-dimensional bundles (retaining the notation $H^{\infty }$ and $H_{\pm }^{\infty }$). At a point $y\in Y$, the fibers $H^{\infty }$ and $H_{\pm }^{\infty }$ are given by the $C^{\infty }$ sections of $F\otimes V$ and $F_{\pm }\otimes V$ restricted to the fiber $Z_y$.

Clifford multiplication gives an action of $TZ$ on $F\otimes V$. Given a local orthogonal frame $\{e_1,e_2,\cdots ,e_{4l}\}$ for $TZ$, the Dirac family twisted by V is defined by $D^Z\otimes V={\sum }_{i=1}^{4l}{e}_i{\nabla }_{e_i}.$ We write ${(D^Z\otimes V)}_{\pm }$ for the restriction of this operator to $H_{\pm }^{\infty }$. For every $y\in Y$,

$${(D^Z\otimes V)}_y=\left[\begin{array}{cc}0& {(D^Z\otimes V)}_{y,-}\\ {(D^Z\otimes V)}_{y,+}& 0\end{array}\right]\in {\mathrm{End}}^{odd}(H_{y,+}^{\infty }\oplus H_{y,-}^{\infty })$$

is the twisted Dirac operator on the fiber $Z_y$.

Let ${\mathcal{L}}_{D^Z\otimes V}=det{\left(\mathrm{Ker}{(D^Z\otimes V)}_{+}\right)}^{\ast }\otimes det\left(\mathrm{Ker}{(D^Z\otimes V)}_{-}\right)$ be the determinant line bundle of the family operator $D^Z\otimes V$ over Y. The determinant line bundle ${\mathcal{L}}_{D^Z\otimes V}$ is equipped with the Quillen metric $g^{{\mathcal{L}}_{D^Z\otimes V}}$ and the compatible Bismut–Freed connection ${\nabla }^{{\mathcal{L}}_{D^Z\otimes V}}$. According to the Atiyah–Singer family index theorem [14,15], the curvature ${\mathbf{R}}^{{\mathcal{L}}_{D^Z\otimes V}}$ of this connection equals the two-form component of the fiberwise integral of $\widehat{A}(TZ,{\nabla }^{TZ})\mathrm{ch}(V,{\nabla }^V)$. Consequently, the form $\sqrt{-1}/\left(2\pi \right){\mathbf{R}}^{{\mathcal{L}}_{D^Z\otimes V}}$ represents the local anomaly.

Theorem 4

(Bismut and Freed [15]).

$${\mathbf{R}}^{{\mathcal{L}}_{D^Z\otimes V}}=2\pi \sqrt{-1}{\left\{{\int }_Z\widehat{A}(TZ,{\nabla }^Z)\mathrm{ch}(V,{\nabla }^V)\right\}}^{\left(2\right)}.$$

(47)

By Theorem 1, we have

Corollary 2.

If $(a+2b)p_1\left(V\right)=p_1\left(Z\right)$, ${\mathbf{R}}^{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^a}}$ is a modular form of weight $2p$ over ${\mathrm{\Gamma }}^0\left(2\right)$, while ${\mathbf{R}}^{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^b\otimes {\mathcal{B}}_r(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)}}$ is a modular form of weight $2p$ over ${\mathrm{\Gamma }}_0\left(2\right)$.

In the context of the global anomaly, Bismut and Freed [15] established a heat equation-based proof of the holonomy theorem, building on the formulation suggested by Witten [25]. In a related approach, Freed [26] employed $\mathbb{Z}/k$ manifolds to detect the integral first Chern class of the determinant line bundle ${\mathcal{L}}_{D^Z\otimes V}$.

Theorem 5

(Freed [26]). If $(\Sigma ,S)$ is a $\mathbb{Z}/k$ surface and $f:\overline{\Sigma }\to Y$ is a map, then

$$\begin{array}{cc}\hfill c_1\left(f^{\ast }\left({\mathcal{L}}_{D^Z\otimes V}\right)\right)\left[\overline{\Sigma }\right]={\displaystyle \frac{1}{k}}{\displaystyle \frac{\sqrt{-1}}{2\pi }}{\int }_{\Sigma }{f}^{\ast }& \left({\int }_Z\widehat{A}(TZ,{\nabla }^Z)\mathrm{ch}(V,{\nabla }^V)\right)\hfill \\ \hfill & +{\displaystyle \frac{\sqrt{-1}}{2\pi }}{\mathrm{lnhol}}_{{\mathcal{L}}_{D^Z\otimes V}}\left(S\right)\mathrm{mod}1,\hfill \end{array}$$

(48)

where we view $\mathbb{Z}/k\cong \mathbb{Z}[1/k]/\mathbb{Z}\subset \mathbb{Q}/\mathbb{Z}$.

For local anomalies, in [17], Han–Liu derived the local anomalies cancellation formula related to the curvature of the determinant line bundle in the dimension of $(8m+2)$ and $(8m-2)$. Let the dimension of $TZ$ be $4l-2$. In the case of considering not the top degree component, we have the following cancellation formulas.

Theorem 6.

When the fiber Z is of dimension $4l-2$, the local anomaly cancellation formula takes the following form:

$${\mathbf{R}}^{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^a}}-\sum _{r=0}^{\left[{\displaystyle \frac{l}{2}}\right]}{2}^{(a-b)m+l-6r}{\mathbf{R}}^{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^b\otimes {\mathcal{B}}_r(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)}}=0.$$

(49)

Proof. 

If $TZ$ is $(4l-2)$-dimensional, integrating both sides of (19) along the fiber Z, we have

$$\begin{array}{ll}\hfill \{& {\int }_Z\widehat{A}(TZ,{\nabla }^{TZ})\mathrm{ch}\left({(△\left(V\right))}^a\right){\}}^{\left(2\right)}-{\displaystyle \sum _{r=0}^{\left[{\displaystyle \frac{l}{2}}\right]}}{2}^{(a-b)m+l-6r}\{{\int }_Z\widehat{A}(TZ,{\nabla }^{TZ})\\ \hfill & ·\mathrm{ch}\left({(△\left(V\right))}^b\right)\mathrm{ch}\left({\mathcal{B}}_r(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)\right){\}}^{\left(2\right)}=0.\end{array}$$

(50)

By Theorem 4, we have

$$\begin{array}{c}{\mathbf{R}}^{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^a}}-\sum _{r=0}^{\left[{\displaystyle \frac{l}{2}}\right]}{2}^{(a-b)m+l-6r}{\mathbf{R}}^{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^b\otimes {\mathcal{B}}_r(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)}}\hfill \\ =2\pi \sqrt{-1}{\left\{{\int }_Z\widehat{A}(TZ,{\nabla }^{TZ})\mathrm{ch}\left({(△\left(V\right))}^a\right)\right\}}^{\left(2\right)}-\sum _{r=0}^{\left[{\displaystyle \frac{l}{2}}\right]}{2}^{(a-b)m+l-6r}2\pi \sqrt{-1}\hfill \\ ·{\left\{{\int }_Z\widehat{A}(TZ,{\nabla }^{TZ})\mathrm{ch}\left({(△\left(V\right))}^b\right)\mathrm{ch}\left({\mathcal{B}}_r(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)\right)\right\}}^{\left(2\right)}\hfill \\ =0.\hfill \end{array}$$

(51)

□

Remark 1.

Our cancellation formula is different from that of Han–Liu because the bundle and module forms are different, and the resulting coefficient relationships are also different.

As a consequence of Theorem 3, we obtain the following result.

Theorem 7.

When the fiber Z is of dimension $4l-2$, the local anomaly cancellation formula takes the following form:

$$\begin{array}{c}{\mathbf{R}}^{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^a\otimes A_1(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)}}-24l{\mathbf{R}}^{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^a}}\hfill \\ +\sum _{r=0}^{\left[{\displaystyle \frac{l}{2}}\right]}{2}^{(a-b)m+l-6r+6}r{\mathbf{R}}^{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^b\otimes {\mathcal{B}}_r(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)}}=0,\hfill \end{array}$$

(52)

and

$$\begin{array}{c}{\mathbf{R}}^{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^a\otimes (A_2(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)-(24l+6)(A_1(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)-24l)-(288l^2-264l))}}\hfill \\ -\sum _{r=0}^{\left[{\displaystyle \frac{l}{2}}\right]}{2}^{(a-b)m+l-6r+11}{r}^2{\mathbf{R}}^{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^b\otimes {\mathcal{B}}_r(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)}}=0.\hfill \end{array}$$

(53)

Proof. 

Similar to Theorem 6. □

For global anomalies, in the case of $(8m+2)$ dimension and $(8m-2)$ dimension, Han–Liu obtained the cancellation formula for the holonomies of Bismut–Freed connections on the determinant line bundles [17]. Let the dimension of $TZ$ be $4l-2$. We have the following theorem.

Theorem 8.

If the fiber Z is $(4l-2)$-dimensional, $(\Sigma ,S)$ is a $\mathbb{Z}/k$ surface and $f:\overline{\Sigma }\to Y$ is a map, then

$$\begin{array}{l}{\displaystyle \frac{\sqrt{-1}}{2\pi }}{\mathrm{lnhol}}_{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^a}}\left(S\right)-{\displaystyle \sum _{r=0}^{\left[{\displaystyle \frac{l}{2}}\right]}}{2}^{(a-b)m+l-6r}{\displaystyle \frac{\sqrt{-1}}{2\pi }}{\mathrm{lnhol}}_{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^b\otimes {\mathcal{B}}_r(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)}}\left(S\right)\\ \equiv c_1\left(f^{\ast }\left({\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^a}\right)\right)\left[\overline{\Sigma }\right]-{\displaystyle \sum _{r=0}^{\left[{\displaystyle \frac{l}{2}}\right]}}{2}^{(a-b)m+l-6r}{c}_1\left(f^{\ast }\left({\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^b\otimes {\mathcal{B}}_r(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)}\right)\right)\left[\overline{\Sigma }\right]\mathrm{mod}1,\end{array}$$

(54)

where we view $\mathbb{Z}/k\cong \mathbb{Z}[1/k]/\mathbb{Z}\subset \mathbb{Q}/\mathbb{Z}$.

Proof. 

Freed’s Theorem 5 and (50) give us

$$\begin{array}{cc}\hfill & c_1\left(f^{\ast }\left({\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^a}\right)\right)\left[\overline{\Sigma }\right]-\sum _{r=0}^{\left[{\displaystyle \frac{l}{2}}\right]}{2}^{(a-b)m+l-6r}{c}_1\left(f^{\ast }\left({\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^b\otimes {\mathcal{B}}_r(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)}\right)\right)\left[\overline{\Sigma }\right]\hfill \\ \hfill & -\left({\displaystyle \frac{\sqrt{-1}}{2\pi }}{\mathrm{lnhol}}_{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^a}}\left(S\right)-\sum _{r=0}^{\left[{\displaystyle \frac{l}{2}}\right]}{2}^{(a-b)m+l-6r}{\displaystyle \frac{\sqrt{-1}}{2\pi }}{\mathrm{lnhol}}_{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^b\otimes {\mathcal{B}}_r(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)}}\left(S\right)\right)\hfill \\ \hfill & \equiv 0\mathrm{mod}1,\hfill \end{array}$$

(55)

so (54) follows. □

As a consequence of Theorem 3, we also obtain the following result.

Theorem 9.

If the fiber Z is $(4l-2)$-dimensional, $(\Sigma ,S)$ is a $\mathbb{Z}/k$ surface and $f:\overline{\Sigma }\to Y$ is a map, then

$$\begin{array}{cc}\hfill {\displaystyle \frac{\sqrt{-1}}{2\pi }}& {\mathrm{lnhol}}_{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^a\otimes (A_1(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)-24l)}}\left(S\right)\hfill \\ \hfill & +\sum _{r=0}^{\left[{\displaystyle \frac{l}{2}}\right]}{2}^{(a-b)m+l-6r+6}r{\displaystyle \frac{\sqrt{-1}}{2\pi }}{\mathrm{lnhol}}_{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^b\otimes {\mathcal{B}}_r(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)}}\left(S\right)\hfill \\ \hfill \equiv & c_1\left(f^{\ast }\left({\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^a\otimes (A_1(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)-24l)}\right)\right)\left[\overline{\Sigma }\right]\hfill \\ \hfill & +\sum _{r=0}^{\left[{\displaystyle \frac{l}{2}}\right]}{2}^{(a-b)m+l-6r+6}r{c}_1\left(f^{\ast }\left({\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^b\otimes {\mathcal{B}}_r(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)}\right)\right)\left[\overline{\Sigma }\right]\mathrm{mod}1,\hfill \end{array}$$

(56)

and

$$\begin{array}{cc}\hfill {\displaystyle \frac{\sqrt{-1}}{2\pi }}& {\mathrm{lnhol}}_{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^a\otimes (A_2(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)-(24l+6)(A_1(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)-24l)-(288l^2-264l))}}\left(S\right)\hfill \\ \hfill & -\sum _{r=0}^{\left[{\displaystyle \frac{l}{2}}\right]}{2}^{(a-b)m+l-6r+11}{r}^2{\displaystyle \frac{\sqrt{-1}}{2\pi }}{\mathrm{lnhol}}_{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^b\otimes {\mathcal{B}}_r(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)}}\left(S\right)\hfill \\ \hfill \equiv & c_1\left(f^{\ast }\left({\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^a\otimes (A_2(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)-(24l+6)(A_1(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)-24l)-(288l^2-264l))}\right)\right)\left[\overline{\Sigma }\right]\hfill \\ \hfill & -\sum _{r=0}^{\left[{\displaystyle \frac{l}{2}}\right]}{2}^{(a-b)m+l-6r+11}{r}^2{c}_1\left(f^{\ast }\left({\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^b\otimes {\mathcal{B}}_r(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)}\right)\right)\left[\overline{\Sigma }\right]\mathrm{mod}1,\hfill \end{array}$$

(57)

where we view $\mathbb{Z}/k\cong \mathbb{Z}[1/k]/\mathbb{Z}\subset \mathbb{Q}/\mathbb{Z}$.

Putting $l=1$ and $l=2$ in the above theorems, we have the following.

Corollary 3.

In the case of a 2-dimensional fiber Z, we have the following local anomaly cancellation formula:

$${\mathbf{R}}^{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^a}}-2^{(a-b)m+1}{\mathbf{R}}^{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^b}}=0,$$

(58)

$$\begin{array}{c}\hfill {\mathbf{R}}^{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^a\otimes A_1(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)}}-24{\mathbf{R}}^{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^a}}=0,\end{array}$$

(59)

$${\mathbf{R}}^{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^a\otimes A_2(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)}}-24{\mathbf{R}}^{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^a}}=0.$$

(60)

If $(\Sigma ,S)$ is a $\mathbb{Z}/k$ surface and $f:\overline{\Sigma }\to Y$ is a map, then

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\sqrt{-1}}{2\pi }}{\mathrm{lnhol}}_{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^a}}\left(S\right)-2^{(a-b)m}{\displaystyle \frac{\sqrt{-1}}{\pi }}{\mathrm{lnhol}}_{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^b}}\left(S\right)\hfill \\ \hfill & \equiv c_1\left(f^{\ast }\left({\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^a}\right)\right)\left[\overline{\Sigma }\right]-2^{(a-b)m+1}{c}_1\left(f^{\ast }\left({\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^b}\right)\right)\left[\overline{\Sigma }\right]\mathrm{mod}1,\hfill \end{array}$$

(61)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\sqrt{-1}}{2\pi }}& {\mathrm{lnhol}}_{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^a\otimes \left(A_1(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)\right)}}\left(S\right)-12{\displaystyle \frac{\sqrt{-1}}{\pi }}{\mathrm{lnhol}}_{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^a}}\hfill \\ \hfill \equiv & c_1\left(f^{\ast }\left({\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^a\otimes \left(A_1(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)\right)}\right)\right)\left[\overline{\Sigma }\right]-24c_1\left(f^{\ast }\left({\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^a}\right)\right)\left[\overline{\Sigma }\right]\mathrm{mod}1,\hfill \end{array}$$

(62)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\sqrt{-1}}{2\pi }}& {\mathrm{lnhol}}_{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^a\otimes \left(A_2(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)\right)}}\left(S\right)-12{\displaystyle \frac{\sqrt{-1}}{\pi }}{\mathrm{lnhol}}_{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^a}}\hfill \\ \hfill \equiv & c_1\left(f^{\ast }\left({\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^a\otimes \left(A_1(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)\right)}\right)\right)\left[\overline{\Sigma }\right]-24c_1\left(f^{\ast }\left({\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^a}\right)\right)\left[\overline{\Sigma }\right]\mathrm{mod}1.\hfill \end{array}$$

(63)

Corollary 4.

In the case of a 6-dimensional fiber Z, we have the following local anomaly cancellation formula:

$${\mathbf{R}}^{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^a}}-2^{(a-b)m+2}{\mathbf{R}}^{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^b}}+(a-b)2^{(a-b)m-4}{\mathbf{R}}^{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^b\otimes \widetilde{V_{\mathbf{C}}}}}=0,$$

(64)

$$\begin{array}{c}{\mathbf{R}}^{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^a\otimes A_1(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)}}-48{\mathbf{R}}^{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^a}}\hfill \\ -6·2^{(a-b)m+5}{\mathbf{R}}^{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^b}}-(a-b)2^{(a-b)m+2}{\mathbf{R}}^{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^b\otimes \widetilde{V_{\mathbf{C}}}}}=0,\hfill \end{array}$$

(65)

$$\begin{array}{c}{\mathbf{R}}^{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^a\otimes A_2(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)}}-54{\mathbf{R}}^{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^a\otimes A_1(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)}}+1968{\mathbf{R}}^{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^a}}\hfill \\ +6·2^{(a-b)m+10}{\mathbf{R}}^{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^b}}+(a-b)2^{(a-b)m+5}{\mathbf{R}}^{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^b\otimes \widetilde{V_{\mathbf{C}}}}}=0.\hfill \end{array}$$

(66)

If $(\Sigma ,S)$ is a $\mathbb{Z}/k$ surface and $f:\overline{\Sigma }\to Y$ is a map, then

$$\begin{array}{cc}\hfill {\displaystyle \frac{\sqrt{-1}}{2\pi }}& {\mathrm{lnhol}}_{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^a}}\left(S\right)-2^{(a-b)m+1}{\displaystyle \frac{\sqrt{-1}}{\pi }}{\mathrm{lnhol}}_{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^b}}\left(S\right)\hfill \\ \hfill & +(a-b)2^{(a-b)m-4}{\displaystyle \frac{\sqrt{-1}}{2\pi }}{\mathrm{lnhol}}_{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^b\otimes \widetilde{V_{\mathbf{C}}}}}\left(S\right)\hfill \\ \hfill \equiv & c_1\left(f^{\ast }\left({\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^a}\right)\right)\left[\overline{\Sigma }\right]-2^{(a-b)m+2}{c}_1\left(f^{\ast }\left({\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^b}\right)\right)\left[\overline{\Sigma }\right]\hfill \\ \hfill & +(a-b)2^{(a-b)m-4}{c}_1\left(f^{\ast }\left({\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^b\otimes \widetilde{V_{\mathbf{C}}}}\right)\right)\left[\overline{\Sigma }\right]\mathrm{mod}1,\hfill \end{array}$$

(67)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\sqrt{-1}}{2\pi }}& {\mathrm{lnhol}}_{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^a\otimes \left(A_1(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)\right)}}\left(S\right)-24{\displaystyle \frac{\sqrt{-1}}{\pi }}{\mathrm{lnhol}}_{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^a}}\hfill \\ \hfill & -3·2^{(a-b)m+5}{\displaystyle \frac{\sqrt{-1}}{\pi }}{\mathrm{lnhol}}_{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^b}}\left(S\right)\hfill \\ \hfill & -(a-b)2^{(a-b)m+1}{\displaystyle \frac{\sqrt{-1}}{\pi }}{\mathrm{lnhol}}_{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^b\otimes \widetilde{V_{\mathbf{C}}}}}\left(S\right)\hfill \\ \hfill \equiv & c_1\left(f^{\ast }\left({\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^a\otimes \left(A_1(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)\right)}\right)\right)\left[\overline{\Sigma }\right]-48c_1\left(f^{\ast }\left({\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^a}\right)\right)\left[\overline{\Sigma }\right]\hfill \\ \hfill & -6·2^{(a-b)m+5}{c}_1\left(f^{\ast }\left({\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^b}\right)\right)\hfill \\ \hfill & -(a-b)2^{(a-b)m+2}{c}_1\left(f^{\ast }\left({\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^b\otimes \widetilde{V_{\mathbf{C}}}}\right)\right)\left[\overline{\Sigma }\right]\mathrm{mod}1,\hfill \end{array}$$

(68)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\sqrt{-1}}{2\pi }}& {\mathrm{lnhol}}_{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^a\otimes \left(A_2(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)\right)}}\left(S\right)-27{\displaystyle \frac{\sqrt{-1}}{\pi }}{\mathrm{lnhol}}_{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^a\otimes \left(A_1(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)\right)}}\left(S\right)\hfill \\ \hfill & +984{\displaystyle \frac{\sqrt{-1}}{\pi }}{\mathrm{lnhol}}_{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^a}}+3·2^{(a-b)m+10}{\displaystyle \frac{\sqrt{-1}}{\pi }}{\mathrm{lnhol}}_{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^b}}\left(S\right)\hfill \\ \hfill & +(a-b)2^{(a-b)m+4}{\displaystyle \frac{\sqrt{-1}}{\pi }}{\mathrm{lnhol}}_{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^b\otimes \widetilde{V_{\mathbf{C}}}}}\left(S\right)\hfill \\ \hfill \equiv & c_1\left(f^{\ast }\left({\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^a\otimes \left(A_2(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)\right)}\right)\right)\left[\overline{\Sigma }\right]-54c_1\left(f^{\ast }\left({\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^a\otimes \left(A_1(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)\right)}\right)\right)\left[\overline{\Sigma }\right]\hfill \\ \hfill & +1968c_1\left(f^{\ast }\left({\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^a}\right)\right)\left[\overline{\Sigma }\right]+6·2^{(a-b)m+10}{c}_1\left(f^{\ast }\left({\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^b}\right)\right)\hfill \\ \hfill & +(a-b)2^{(a-b)m+5}{c}_1\left(f^{\ast }\left({\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^b\otimes \widetilde{V_{\mathbf{C}}}}\right)\right)\left[\overline{\Sigma }\right]\mathrm{mod}1.\hfill \end{array}$$

(69)

Below, we consider anomaly cancellation formulas of index gerbes.

We now maintain the assumption that $TZ$ is oriented, but consider the case where the fiber dimension is $4l-3$, i.e., the fibers are odd-dimensional. Furthermore, assume that $TZ$ is spin. In this setting, one can still define the family of Dirac operators $D^Z\otimes V$ analogously to the even-dimensional fiber case. A key difference, however, is that the spinor bundle $F^{\prime }$ associated with $TZ$ is no longer ${\mathbb{Z}}_2$-graded. Let $H^{\infty }$ denote the space of $C^{\infty }$ sections of $F^{\prime }\otimes V$ over the total space M. This space can also be viewed as the space of $C^{\infty }$ sections over the base Y of certain infinite-dimensional bundles, which, by abuse of notation, we also denote by $H^{\infty }$. Specifically, for a point $y\in Y$, the fiber $H_y^{\infty }$ consists of the $C^{\infty }$ sections of $F^{\prime }\otimes V$ restricted to the fiber $Z_y$. For each $y\in Y$,

$${(D^Z\otimes V)}_y\in \mathrm{End}\left(H_y^{\infty }\right)$$

is the twisted Dirac operator on the fiber $Z_y$.

The index gerbe ${\mathcal{G}}_{D^Z\otimes V}$ with connection on Y, a higher analogue of the determinant line bundle for the family $D^Z\otimes V$, was constructed by Lott [16]. Its curvature $R^{{\mathcal{G}}_{D^Z\otimes V}}$ is a closed 3-form on Y equal to the three-form component of the fiber integral $\widehat{A}(TZ,{\nabla }^Z)\mathrm{ch}(V,{\nabla }^V)$ from the Atiyah–Singer families index theorem.

Theorem 10

(Lott [16]).

$${\mathbf{R}}^{{\mathcal{G}}_{D^Z\otimes V}}={\left\{{\int }_Z\widehat{A}(TZ,{\nabla }^Z)\mathrm{ch}(V,{\nabla }^V)\right\}}^{\left(3\right)}.$$

(70)

By Theorem 1, we have

Corollary 5.

If $(a+2b)p_1\left(V\right)=p_1\left(Z\right)$, ${\mathbf{R}}^{{\mathcal{G}}_{D^Z\otimes {(△\left(V\right))}^a}}$ is a modular form of weight $2p$ over ${\mathrm{\Gamma }}^0\left(2\right)$, while ${\mathbf{R}}^{{\mathcal{G}}_{D^Z\otimes {(△\left(V\right))}^b\otimes {\mathcal{B}}_r(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)}}$ is a modular form of weight $2p$ over ${\mathrm{\Gamma }}_0\left(2\right)$.

We have the following anomaly cancellation formulas for index gerbes.

Theorem 11.

When the fiber Z is $(4l-3)$-dimensional, the anomaly cancellation formula is given by the following:

$${\mathbf{R}}^{{\mathcal{G}}_{D^Z\otimes {(△\left(V\right))}^a}}-\sum _{r=0}^{\left[{\displaystyle \frac{l}{2}}\right]}{2}^{(a-b)m+l-6r}{\mathbf{R}}^{{\mathcal{G}}_{D^Z\otimes {(△\left(V\right))}^b\otimes {\mathcal{B}}_r(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)}}=0.$$

(71)

Proof. 

If $TZ$ is $(4l-3)$-dimensional, integrating both side of (19) along the fiber Z, we have

$$\begin{array}{cc}\hfill & {\left\{{\int }_Z\widehat{A}(TZ,{\nabla }^{TZ})\mathrm{ch}\left({(△\left(V\right))}^a\right)\right\}}^{\left(3\right)}\hfill \\ \hfill & -\sum _{r=0}^{\left[{\displaystyle \frac{l}{2}}\right]}{2}^{(a-b)m+l-6r}{\left\{{\int }_Z\widehat{A}(TZ,{\nabla }^{TZ})\mathrm{ch}\left({(△\left(V\right))}^b\right)\mathrm{ch}\left({\mathcal{B}}_r(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)\right)\right\}}^{\left(3\right)}=0.\hfill \end{array}$$

(72)

By Theorem 10, we have

$$\begin{array}{c}{\mathbf{R}}^{{\mathcal{G}}_{D^Z\otimes {(△\left(V\right))}^a}}-\sum _{r=0}^{\left[{\displaystyle \frac{l}{2}}\right]}{2}^{(a-b)m+l-6r}{\mathbf{R}}^{{\mathcal{G}}_{D^Z\otimes {(△\left(V\right))}^b\otimes {\mathcal{B}}_r(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)}}\hfill \\ ={\left\{{\int }_Z\widehat{A}(TZ,{\nabla }^{TZ})\mathrm{ch}\left({(△\left(V\right))}^a\right)\right\}}^{\left(3\right)}-\sum _{r=0}^{\left[{\displaystyle \frac{l}{2}}\right]}{2}^{(a-b)m+l-6r}\{{\int }_Z\widehat{A}(TZ,{\nabla }^{TZ})\hfill \\ ·\mathrm{ch}\left({(△\left(V\right))}^b\right)\mathrm{ch}\left({\mathcal{B}}_r(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)\right){\}}^{\left(3\right)}\hfill \\ =0.\hfill \end{array}$$

(73)

□

As a consequence of Theorem 3, we also obtain the following result.

Theorem 12.

When the fiber Z is $(4l-3)$-dimensional, the anomaly cancellation formula is given by the following:

$$\begin{array}{c}{\mathbf{R}}^{{\mathcal{G}}_{D^Z\otimes {(△\left(V\right))}^a\otimes A_1(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)}}-24l{\mathbf{R}}^{{\mathcal{G}}_{D^Z\otimes {(△\left(V\right))}^a}}\hfill \\ +\sum _{r=0}^{\left[{\displaystyle \frac{l}{2}}\right]}{2}^{(a-b)m+l-6r+6}r{\mathbf{R}}^{{\mathcal{G}}_{D^Z\otimes {(△\left(V\right))}^b\otimes {\mathcal{B}}_r(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)}}=0,\hfill \end{array}$$

(74)

and

$$\begin{array}{c}{\mathbf{R}}^{{\mathcal{G}}_{D^Z\otimes {(△\left(V\right))}^a\otimes (A_2(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)-(24l+6)(A_1(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)-24l)-(288l^2-264l))}}\hfill \\ -\sum _{r=0}^{\left[{\displaystyle \frac{l}{2}}\right]}{2}^{(a-b)m+l-6r+11}{r}^2{\mathbf{R}}^{{\mathcal{G}}_{D^Z\otimes {(△\left(V\right))}^b\otimes {\mathcal{B}}_r(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)}}=0.\hfill \end{array}$$

(75)

Putting $l=1$ and $l=2$ in the above theorems, we have the following.

Corollary 6.

In the case of a 1-dimensional fiber Z, we have the following anomaly cancellation formula:

$${\mathbf{R}}^{{\mathcal{G}}_{D^Z\otimes {(△\left(V\right))}^a}}-2^{(a-b)m+1}{\mathbf{R}}^{{\mathcal{G}}_{D^Z\otimes {(△\left(V\right))}^b}}=0,$$

(76)

$$\begin{array}{c}\hfill {\mathbf{R}}^{{\mathcal{G}}_{D^Z\otimes {(△\left(V\right))}^a\otimes A_1(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)}}-24{\mathbf{R}}^{{\mathcal{G}}_{D^Z\otimes {(△\left(V\right))}^a}}=0,\end{array}$$

(77)

$${\mathbf{R}}^{{\mathcal{G}}_{D^Z\otimes {(△\left(V\right))}^a\otimes A_2(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)}}-24{\mathbf{R}}^{{\mathcal{G}}_{D^Z\otimes {(△\left(V\right))}^a}}=0.$$

(78)

Corollary 7.

In the case of a 5-dimensional fiber Z, we have the following anomaly cancellation formula:

$${\mathbf{R}}^{{\mathcal{G}}_{D^Z\otimes {(△\left(V\right))}^a}}-2^{(a-b)m+2}{\mathbf{R}}^{{\mathcal{G}}_{D^Z\otimes {(△\left(V\right))}^b}}+(a-b)2^{(a-b)m-4}{\mathbf{R}}^{{\mathcal{G}}_{D^Z\otimes {(△\left(V\right))}^b\otimes \widetilde{V_{\mathbf{C}}}}}=0,$$

(79)

$$\begin{array}{c}{\mathbf{R}}^{{\mathcal{G}}_{D^Z\otimes {(△\left(V\right))}^a\otimes A_1(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)}}-48{\mathbf{R}}^{{\mathcal{G}}_{D^Z\otimes {(△\left(V\right))}^a}}\hfill \\ -6·2^{(a-b)m+5}{\mathbf{R}}^{{\mathcal{G}}_{D^Z\otimes {(△\left(V\right))}^b}}-(a-b)2^{(a-b)m+2}{\mathbf{R}}^{{\mathcal{G}}_{D^Z\otimes {(△\left(V\right))}^b\otimes \widetilde{V_{\mathbf{C}}}}}=0,\hfill \end{array}$$

(80)

$$\begin{array}{c}{\mathbf{R}}^{{\mathcal{G}}_{D^Z\otimes {(△\left(V\right))}^a\otimes A_2(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)}}-54{\mathbf{R}}^{{\mathcal{G}}_{D^Z\otimes {(△\left(V\right))}^a\otimes A_1(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)}}+1968{\mathbf{R}}^{{\mathcal{G}}_{D^Z\otimes {(△\left(V\right))}^a}}\hfill \\ +6·2^{(a-b)m+10}{\mathbf{R}}^{{\mathcal{G}}_{D^Z\otimes {(△\left(V\right))}^b}}+(a-b)2^{(a-b)m+5}{\mathbf{R}}^{{\mathcal{L}}_{D^Z\otimes {(△\left(V\right))}^b\otimes \widetilde{V_{\mathbf{C}}}}}=0.\hfill \end{array}$$

(81)

Below, we consider $\eta$-invariants.

For $y\in Y$, let ${\eta }_y(D^Z\otimes V)\left(s\right)$ be the eta function associated with ${(D^Z\otimes V)}_y$. Define as Atiyah, Patodi and Singer in [27]

$${\overline{\eta }}_y(D^Z\otimes V)\left(s\right)={\displaystyle \frac{{\eta }_y(D^Z\otimes V)\left(s\right)+ker{(D^Z\otimes V)}_y}{2}}.$$

Denote ${\overline{\eta }}_y(D^Z\otimes V)\left(0\right)$ (a function on Y) by $\overline{\eta }(D^Z\otimes V)$.

We continue to work within the framework of family twisted Dirac operators on odd-dimensional manifolds, as it pertains to index gerbes. For odd-dimensional fibers, an analogous Bismut–Freed theorem holds for the reduced $\eta$-invariants, as stated below.

Theorem 13

(Bismut and Freed [15]).

$$d\left\{\overline{\eta }(D^Z\otimes V)\right\}={\left\{{\int }_Z\widehat{A}(TZ,{\nabla }^Z)\mathrm{ch}(V,{\nabla }^V)\right\}}^{\left(1\right)}.$$

(82)

The following theorems establish properties of the reduced $\eta$-invariants.

Theorem 14.

If the fiber Z is $(4l-1)$-dimensional, then

$$\begin{array}{c}\hfill \exp \{2\pi \sqrt{-1}\left(\overline{\eta }(D^Z\otimes {(△\left(V\right))}^a)\right)-\sum _{r=0}^{\left[{\displaystyle \frac{l}{2}}\right]}{2}^{(a-b)m+l-6r}\left(\overline{\eta }\right(D^Z\otimes {(△\left(V\right))}^b\\ \hfill \otimes {\mathcal{B}}_r(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)\left)\right)\}\end{array}$$

(83)

is a constant function on Y.

Proof. 

If $TZ$ is $(4l-1)$-dimensional, integrating both sides of (19) along the fiber Z, we obtain

$$\begin{array}{cc}\hfill \{& {\int }_Z\widehat{A}(TZ,{\nabla }^{TZ})\mathrm{ch}\left({(△\left(V\right))}^a\right){\}}^{\left(1\right)}-\sum _{r=0}^{\left[{\displaystyle \frac{l}{2}}\right]}{2}^{(a-b)m+l-6r}\{{\int }_Z\widehat{A}(TZ,{\nabla }^{TZ})\hfill \\ \hfill & ·\mathrm{ch}\left({(△\left(V\right))}^b\right)\mathrm{ch}\left({\mathcal{B}}_r(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)\right){\}}^{\left(1\right)}=0.\hfill \end{array}$$

(84)

By Theorem 13, we have

$$\begin{array}{cc}\hfill d\{\overline{\eta }& (D^Z\otimes {(△\left(V\right))}^a)\}-\sum _{r=0}^{\left[{\displaystyle \frac{l}{2}}\right]}{2}^{(a-b)m+l-6r}d\left\{\overline{\eta }(D^Z\otimes {(△\left(V\right))}^b\otimes {\mathcal{B}}_r(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b))\right\}\hfill \\ \hfill =& {\left\{{\int }_Z\widehat{A}(TZ,{\nabla }^{TZ})\mathrm{ch}\left({(△\left(V\right))}^a\right)\right\}}^{\left(1\right)}\hfill \\ \hfill & -\sum _{r=0}^{\left[{\displaystyle \frac{l}{2}}\right]}{2}^{(a-b)m+l-6r}{\left\{{\int }_Z\widehat{A}(TZ,{\nabla }^{TZ})\mathrm{ch}\left({(△\left(V\right))}^b\right)\mathrm{ch}\left({\mathcal{B}}_r(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)\right)\right\}}^{\left(1\right)}\hfill \\ \hfill =& 0.\hfill \end{array}$$

(85)

therefore, we have

$$d(\left\{\overline{\eta }(D^Z\otimes {(△\left(V\right))}^a)\right\}-\sum _{r=0}^{\left[{\displaystyle \frac{l}{2}}\right]}{2}^{(a-b)m+l-6r}\left\{\overline{\eta }(D^Z\otimes {(△\left(V\right))}^b\otimes {\mathcal{B}}_r(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b))\right\})=0.$$

(86)

Since Y is connected,

$$\left\{\overline{\eta }(D^Z\otimes {(△\left(V\right))}^a)\right\}-\sum _{r=0}^{\left[{\displaystyle \frac{l}{2}}\right]}{2}^{(a-b)m+l-6r}\left\{\overline{\eta }(D^Z\otimes {(△\left(V\right))}^b\otimes {\mathcal{B}}_r(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b))\right\}$$

(87)

must be a constant function on Y. Therefore, it is not hard to see that Theorem 14 follows. □

As a consequence of Theorem 3, we obtain the following result.

Theorem 15.

If the fiber Z is $(4l-1)$-dimensional, then

$$\begin{array}{cc}\hfill \exp \{2\pi \sqrt{-1}(\overline{\eta }& (D^Z\otimes {(△\left(V\right))}^a\otimes A_1(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b))-24l\overline{\eta }(D^Z\otimes {(△\left(V\right))}^a)\hfill \\ \hfill & +\sum _{r=0}^{\left[{\displaystyle \frac{l}{2}}\right]}{2}^{(a-b)m+l-6r+6}r\overline{\eta }(D^Z\otimes {(△\left(V\right))}^b\otimes {\mathcal{B}}_r(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b))\left)\right\}\hfill \end{array}$$

(88)

and

$$\begin{array}{cc}\hfill \exp \{2& \pi \sqrt{-1}(\overline{\eta }(D^Z\otimes {(△\left(V\right))}^a\otimes (A_2(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)\hfill \\ \hfill & -(24l+6)(A_1(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)-24l)-(288l^2-264l)\left)\right)\hfill \\ \hfill & -\sum _{r=0}^{\left[{\displaystyle \frac{l}{2}}\right]}{2}^{(a-b)m+l-6r+11}{r}^2\overline{\eta }(D^Z\otimes {(△\left(V\right))}^b\otimes {\mathcal{B}}_r(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b))\left)\right\}\hfill \end{array}$$

(89)

are a constant function on Y.

Setting $l=1$ and $l=2$ in the preceding theorems leads to the following corollaries.

Corollary 8.

If the fiber Z is 3-dimensional, then

$$\exp \left\{2\pi \sqrt{-1}(\overline{\eta }(D^Z\otimes {(△\left(V\right))}^a)-2^{(a-b)m+1}\overline{\eta }(D^Z\otimes {(△\left(V\right))}^b))\right\},$$

(90)

$$\begin{array}{c}\hfill \exp \left\{2\pi \sqrt{-1}(\overline{\eta }(D^Z\otimes {(△\left(V\right))}^a\otimes A_1(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b))-24\overline{\eta }(D^Z\otimes {(△\left(V\right))}^a))\right\},\end{array}$$

(91)

and

$$\exp \left\{2\pi \sqrt{-1}(\overline{\eta }(D^Z\otimes {(△\left(V\right))}^a\otimes A_2(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b))-24\overline{\eta }(D^Z\otimes {(△\left(V\right))}^a))\right\}$$

(92)

are a constant function on Y.

Corollary 9.

If the fiber Z is 7-dimensional, then

$$\begin{array}{cc}\hfill \exp \{2\pi \sqrt{-1}(\overline{\eta }& (D^Z\otimes {(△\left(V\right))}^a)-2^{(a-b)m+2}\overline{\eta }(D^Z\otimes {(△\left(V\right))}^b)\hfill \\ \hfill & +(a-b)2^{(a-b)m-4}\overline{\eta }(D^Z\otimes {(△\left(V\right))}^b\otimes \widetilde{V_{\mathbf{C}}})\left)\right\},\hfill \end{array}$$

(93)

$$\begin{array}{cc}\hfill \exp & \{2\pi \sqrt{-1}(\overline{\eta }(D^Z\otimes {(△\left(V\right))}^a\otimes A_1(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b))-48\overline{\eta }(D^Z\otimes {(△\left(V\right))}^a)\hfill \\ \hfill & -6·2^{(a-b)m+5}\overline{\eta }(D^Z\otimes {(△\left(V\right))}^b)-(a-b)2^{(a-b)m+2}\overline{\eta }(D^Z\otimes {(△\left(V\right))}^b\otimes \widetilde{V_{\mathbf{C}}})\left)\right\},\hfill \end{array}$$

(94)

and

$$\begin{array}{cc}\hfill \exp & \{2\pi \sqrt{-1}(\overline{\eta }(D^Z\otimes {(△\left(V\right))}^a\otimes A_2(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b))\hfill \\ \hfill & -54\overline{\eta }(D^Z\otimes {(△\left(V\right))}^a\otimes A_1(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b))+1968\overline{\eta }(D^Z\otimes {(△\left(V\right))}^a)\hfill \\ \hfill & +6·2^{(a-b)m+10}\overline{\eta }(D^Z\otimes {(△\left(V\right))}^b)+(a-b)2^{(a-b)m+5}\overline{\eta }(D^Z\otimes {(△\left(V\right))}^b\otimes \widetilde{V_{\mathbf{C}}})\left)\right\}\hfill \end{array}$$

(95)

are a constant function on Y.

Below, let us recall the definition of the higher elliptic genus (cf. [28] also see [29,30]). Let X be a compact complex manifold. The elliptic genus evaluated on the fundamental class $\left[X\right]$ can be expressed in terms of the Chern roots $x_i$ of the tangent bundle $TX$ as $\mathcal{ELL}\left(X\right)\left[X\right]$ where

$$\mathcal{ELL}\left(X\right)=\prod _i{x}_i{\displaystyle \frac{\theta (\frac{x_i}{2\pi i}-z,ϵ)}{\theta (\frac{x_i}{2\pi i},ϵ)}}.$$

(96)

Consider a manifold X as described above, and denote by $\pi$ its fundamental group. Let $f:X\to B\pi$ be the corresponding classifying map. Given a cohomology class $\alpha \in H^k(B\pi \left(X\right),\mathbf{Q})$.

Definition 3.

The higher elliptic genus is

$$El{l}_{\alpha }\left(X\right)=(\mathcal{ELL}\left(X\right)\cup f^{\ast }\left(\alpha \right))\left[X\right],$$

(97)

where the elliptic class $\mathcal{ELL}\left(X\right)\in H^{\ast }(X,\mathbf{Q})$ is given by (96).

So by Theorem 1, we have

Definition 4.

Let Z be spin and π be the fundamental group of Z. Let $B\pi \left(Z\right)$ be the classifying space of $\pi \left(Z\right)$ and $f:Z\to B\pi \left(Z\right)$ be the classifying map. Let $\alpha \in H^{2n}(B\pi \left(Z\right),\mathbf{Q})$ and $2n=4k-4l$. Then we call

$${\int }_Z\widehat{A}(TZ,{\nabla }^{TZ})\mathrm{ch}\left({(△\left(V\right))}^a\right)\mathrm{ch}\left({\mathrm{\Theta }}_1(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)\right)f^{\ast }\left(\alpha \right),$$

(98)

$${\int }_Z\widehat{A}(TZ,{\nabla }^{TZ})\mathrm{ch}\left({(△\left(V\right))}^b\right)\mathrm{ch}\left({\mathrm{\Theta }}_2(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)\right)f^{\ast }\left(\alpha \right),$$

(99)

the higher elliptic genus.

Corollary 10.

The following anomaly cancellation formula of higher genus holds:

(1) 

$${\int }_Z\widehat{A}(TZ,{\nabla }^{TZ})\mathrm{ch}\left({(▵\left(V\right))}^a\right)f^{\ast }\left(\alpha \right)=2^{(a-b)m+l}\sum _{r=0}^{\left[{\displaystyle \frac{l}{2}}\right]}{2}^{-6r}\widetilde{h_r},$$

(100)

where each $\widetilde{h_r},$ $0\le r\le \left[{\displaystyle \frac{l}{2}}\right],$ is a higher genus integral linear combination of

$${\int }_Z\widehat{A}(TZ,{\nabla }^{TZ})\mathrm{ch}\left({(▵\left(V\right))}^b\right)\mathrm{ch}\left(B_q(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)\right)f^{\ast }\left(\alpha \right),0\le q\le r.$$

(2) 

$$\begin{array}{cc}\hfill {\int }_Z\widehat{A}& (TZ,{\nabla }^{TZ})\mathrm{ch}\left({(△\left(V\right))}^a\right)\mathrm{ch}(A_1(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)-24l)f^{\ast }\left(\alpha \right)\hfill \\ \hfill =& -2^{(a-b)m+l+6}\sum _{r=0}^{\left[{\displaystyle \frac{l}{2}}\right]}r{2}^{-6r}\widetilde{h_r},\hfill \end{array}$$

(101)

$$\begin{array}{cc}\hfill {\int }_Z\widehat{A}& (TZ,{\nabla }^{TZ})\mathrm{ch}\left({(△\left(V\right))}^a\right)\mathrm{ch}(A_2(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)\hfill \\ \hfill & -(24l+6)(A_1(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)-24l)-(288l^2-264l))f^{\ast }\left(\alpha \right)\hfill \\ \hfill =& 2^{(a-b)m+l+11}\sum _{r=0}^{\left[{\displaystyle \frac{p}{2}}\right]}{r}^2{2}^{-6r}\widetilde{h_r},\hfill \end{array}$$

(102)

where each $\widetilde{h_r},$ $0\le r\le \left[{\displaystyle \frac{l}{2}}\right],$ is a higher genus integral linear combination of

$${\int }_Z\widehat{A}(TZ,{\nabla }^{TZ})\mathrm{ch}\left({(▵\left(V\right))}^b\right)\mathrm{ch}\left(B_q(T_{\mathbf{C}}Z,V_{\mathbf{C}},a,b)\right)f^{\ast }\left(\alpha \right),0\le q\le r.$$

4. Conclusions

In the process of quantizing classical systems, physicists discovered quantum anomalous effects, which are manifested as the symmetry being disrupted by the radiation correction. The violation of symmetry implies the inconsistency of quantum theory. To ensure the consistency and self-consistency of quantum theory, a method must be explored to “cancel” these anomalies. Elliptic genera not only provide a framework for constructing theoretical models capable of canceling these anomalies, but also reveal the root cause of quantum anomalies. Therefore, by constructing anomaly cancellation formulas, the ellipse genera can help scientists establish a more precise and complete picture of quantum theory, thereby promoting a deeper understanding of the structure of physics.

In this paper, we establish cancellation formulas for anomalies associated with families of tangent-twisted Dirac operators across multiple geometric settings. For $(4l-2)$-dimensional manifolds, we obtain curvature cancellation formulas for determinant line bundles in the local anomaly case, and holonomy cancellation formulas for Bismut–Freed connections over torsion loops arising from $\mathbf{Z}/\mathbf{k}$ surfaces in the global anomaly case. Furthermore, for $(4l-3)$-dimensional manifolds, we derive curvature cancellation formulas for index gerbes, representing a higher analogue of the determinant line bundle framework. These results are systematically derived through an investigation of modular invariance properties of characteristic forms. Moreover, we derive some results for the reduced eta invariants of family tangent twisted Dirac operators on $(4l-1)$-dimensional manifolds, which are interesting in spectral geometry. Furthermore, we give some anomaly cancellation formulas of the Chen–Connes character and the higher elliptic genera. These anomaly cancellation formulas can tell us how many fermions of different types should be coupled to the gravity to make the theory anomaly-free. These anomaly cancellation formulas can be applied to the Standard Model of particle physics, telling us how many different types of fermions need to be coupled with gravity to keep the theory free from anomalies. It can be applied to condensed matter physics to understand the body-edge relationship of topological states of matter and provide the self-consistency of low-energy effective theories. It is important to note that these cancellation formulas are direct consequences of the modular properties of characteristic forms, which are themselves deeply rooted in the theory of elliptic genera. Therefore, for future work, we can focus our attention on the study of Toeplitz operators in odd dimensions, and we hope that the results of our anomaly cancellation formulas can find more specific applications and deeper connections in physics.