Relative Chern character number and super-connection

TL;DR

This paper derives formulas to compute the relative Chern character number for complex vector bundles with specific singularities and applies it to express the index of a twisted Dirac operator on spin manifolds with sphere bundle structures.

Contribution

It introduces a new formula for the relative Chern characteristic number in the presence of singularities on odd-dimensional spheres and relates it to the index of twisted Dirac operators on spin manifolds.

Findings

Formula for relative Chern characteristic number with singularities

Expression for the index of twisted Dirac operators on sphere bundles

Application to spin manifolds with specific bundle structures

Abstract

For two complex vector bundles admitting a homomorphism, whose singularity locates in the disjoint union of some odd--dimensional spheres, we give a formula to compute the relative Chern characteristic number of these two complex vector bundles. In particular, for a spin manifold admitting some sphere bundle structure, we give a formula to express the index of a special twisted Dirac operator.

Equations

$$K(M,M\setminus U)\stackrel{\text{mbox}ch}{⟶}{H}^{even}(M,M\setminus U),$$

$$0\to E_{+}\stackrel{v}{⟶}{E}_{-}\to 0,$$

$$v\big{|}_{\partial D(p)}:E_{+}\big{|}_{\partial D(p)}\to E_{-}\big{|}_{\partial D(p)}$$

$$v\big{|}_{\partial N(S^{2k_{i}-1})}:E_{+}\big{|}_{\partial N(S^{2k_{i}-1})}\to E_{-}\big{|}_{\partial N(S^{2k_{i}-1})}$$

$$⟨\text{mbox}ch(E_{+})-\text{mbox}ch(E_{-}),[M]⟩=(-1{)}^{n+1}\sum _{1\le i\le l}{\mathrm{deg}}^{\ast }(v_i)$$

$$Ind(D_{+}^{\otimes F})=(-1{)}^n\sum _{1\le i\le l}{\mathrm{deg}}^{\ast }(v_{x_i}).$$

$$\text{mbox}cs({\nabla }_0,{\nabla }_1)={\int }_0^1\text{mbox}Tr({\dot{\nabla }}_u{e}^{{\nabla }_u^2})du,$$

$$\tilde{\nabla }=du\frac{\partial }{\partial u}+{\nabla }_u.$$

$$\text{mbox}Ch(g)=\sum _{k=0}^{\infty }(-1{)}^k\frac{k!}{(2k+1)!}\text{mbox}Tr({\omega }^{2k+1}),$$

$$\text{mbox}Ch(\tilde{g})=\text{mbox}Ch(g_t)+dt\wedge \tilde{}\text{mbox}Ch(g_t).$$

$$\tilde{}\text{mbox}Ch(g_t)=\sum _{k=0}^{\infty }(-1{)}^k\frac{k!}{(2k)!}\text{mbox}Tr(g_t^{-1}{\dot{g}}_t\wedge {\omega }_t^{2k}),$$

$$\frac{\partial }{\partial t}\text{mbox}Ch(g_t)=d\tilde{}\text{mbox}Ch(g_t).$$

$$\frac{1}{(-2\pi \sqrt{-1}{)}^k}{\int }_{S^{2k-1}}\text{mbox}Ch(g)=-{\mathrm{deg}}^{top}(g),$$

$$S^{p+q-1}\to S^p\vee S^q\to S^p\times S^q\to S^{p+q}\to S^{p+1}\vee S^{q+1},$$

$$1\to {\pi }_{2n-1}(G{L}_N(\mathbb{C}))\to [S^{2n-2k}\times S^{2k-1},G{L}_N(\mathbb{C})]\to {\pi }_{2n-2k}(G{L}_N(\mathbb{C}))\oplus {\pi }_{2k-1}(G{L}_N(\mathbb{C}))\to 1.$$

$$g_{1}:S^{2n-2k}\times S^{2k-1}\to GL_{N}(\mathbb{C}),\mbox{ such that }g_{1}\Big{|}_{S^{2n-2k}\times q}=Id_{N},$$

$$\frac{1}{(-2\pi \sqrt{-1}{)}^n}{\int }_{S^{2n-2k}\times S^{2k-1}}\text{mbox}Ch(g)=\mathrm{deg}(h),$$

$$g=p{r}_2^{\ast }f\cdot {\varphi }^{\ast }h,$$

$$S^{2n-2k}\times S^{2k-1}\stackrel{\wedge }{\to }{S}^{2n-2k}\wedge S^{2k-1}=S^{2n-1}$$

$$S^{2n-2k}\times S^{2k-1}\to S^{2k-1}.$$

$$\left(\begin{array}[]{cc}pr^{*}_{2}f\cdot\phi^{*}h&\\ &Id_{N}\\ \end{array}\right)\in GL_{2N}(\mathbb{C}),$$

$$\left(\begin{array}[]{cc}pr^{*}_{2}f&\\ &\phi^{*}h\\ \end{array}\right).$$

$$[\text{mbox}Ch(g)]=[\text{mbox}Ch(p{r}_2^{\ast }f\cdot {\varphi }^{\ast }h)]=[p{r}_2^{\ast }\text{mbox}Ch(f)]+[{\varphi }^{\ast }\text{mbox}Ch(h)].$$

$$\frac{1}{(-2\pi \sqrt{-1}{)}^n}{\int }_{S^{2n-2k}\times S^{2k-1}}\text{mbox}Ch(g)$$

$$\phi (\alpha )=(2\pi \sqrt{-1}{)}^{-\frac{k}{2}}\alpha ,$$

$$A_T={\nabla }^E+TV.$$

$$\text{mbox}ch(E,A_T)=\phi (\text{mbox}Tr[e^{-A_T^2}]).$$

$$\frac{\partial \text{mbox}ch(E,A_T)}{\partial T}=-\frac{1}{\sqrt{2\pi \sqrt{-1}}}d\phi \text{mbox}Tr[V{e}^{-A_T^2}].$$

$$\gamma (T)=\frac{1}{\sqrt{2\pi \sqrt{-1}}}\phi {\int }_0^T\text{mbox}Tr(V{e}^{-A_t^2})dt.$$

$$\text{mbox}ch(E,A_0)-\text{mbox}ch(E,A_T)=d\gamma (T).$$

Full text

[TABLE]

**AMS Subject Classifications ** 53C23 55Q05 58A12

††footnotetext:

Manuscript last updated: .

†

Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan. E-mail: dexielin@ms.u-tokyo.ac.jp

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sectionIntroduction

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Let $M$ be a closed oriented $2n$–dimensional manifold. The aim of this paper is to localize relative Chern character in the following setting. The relative Chern character is formulated as

[TABLE]

where $U$ is a proper open submanifold of $M$. In this paper, we suppose all manifolds are smooth.

We fix an element $[E]$ of $K(M,M\setminus U)$ which is represented by the following data [1, Lemma 8.4]:

[TABLE]

where $E_{+}$ and $E_{-}$ are two complex vector bundles with the same rank, and $v$ is a complex linear homomorphism between the two vector bundles which is isomorphic on $M\setminus U$. We call $x\in M$ a singularity point of $v$, if $v_{x}$ is not isomorphic. The singularity of $v$ is the set of singular points, denoted by $Sing(v)$. We assume that $U$ is a tubular neighborhood of a closed submanifold $Y$. If we deform $v$ slightly, we can assume that $Sing(v)$ is a subset of $Y$. In this paper we assume $Sing(v)\subset Y$. Note that we do not assume the transversality of $v$.

The pairing between the relative Chern character class $\mbox{ch}(E_{+},E_{-},v)$ and the fundamental class $[M]$ is equal to $\langle\mbox{ch}(E_{+})-\mbox{ch}(E_{-}),[M]\rangle$, which we call relative Chern character number. The relative Chern character number depends only on the data $(E_{+},E_{-},v)$ restricted on the tubular neighborhood of $Y$. However, the explicit relationship between the relative Chern character number and the neighborhood is not clear, in particular when $v$ is not transverse to zero as a section of $\hom(E_{+},E_{-})$.

The notion of super–connection was introduced by Quillen [7], as a generalization of the notion of connection in the category of $\mathbb{Z}_{2}$–graded vector bundles. We can use the super–connection to localize the relative Chern character number. The following examples are typical localizations of the relative Chern character in different conditions on $Y$ by super–connection.

When $Y$ is a set consisting of finitely many points, Feng, Li and Zhang [5] gave a Poincaré-Hopf type formula to calculate the relative Chern character number.

• 2)

When $M$ and $Y$ are closed complex manifolds, $Y\hookrightarrow M$ is a holomorphic embedding and $Sing(v)=Y$, Bismut(1990) gave several estimates of the convergence of the relative Chern character with respect to the deformation variable of the super–connection, under the ”quantization assumption” [3, Section 1 a. Assumptions and notaions]. One of the estimates implies the localization of the relative Chern character number, and the number turns out to be equal to some topological invariants on $Y$.

We suppose that $v$ satisfies the following assumption.

• •

for each $S^{2k_{i}-1}$, its normal bundle is trivial,

• •

the images of these embeddings are mutually disjoint, i.e. $\iota_{i}(S^{2k_{i}-1})\cap\iota_{j}(S^{2k_{j}-1})=\emptyset$, for $i\neq j$.

We suppose that $Sing(v)$ is a subset of the union of these images, i.e. $Sing(v)\subset\bigcup_{1\leq i\leq l}Im(\iota_{i})$.

For convenience, we also use $S^{2k_{i}-1}$ to denote $Im(\iota_{i})$. Let $N(S^{2k_{i}-1})$ denote the tubular neighborhood of $S^{2k_{i}-1}$. By the above assumption, we have a diffeomorphism $N(S^{2k_{i}-1})\cong S^{2k_{i}-1}\times D^{2n-2k_{i}+1}$, where $D^{2n-2k_{i}+1}$ denotes the standard unit disk of dimension $2n-2k_{i}+1$, and the boundary $\partial N(S^{2k_{i}-1})$ is diffeomorphic to $S^{2k_{i}-1}\times S^{2n-2k_{i}}$.

When $Sing(v)$ is just a set of finitely many points, Feng, Li and Zhang [5] gave an explicit formula of the relative Chern character number as a sum of contributions from each point $p\in Sing(v)$, which depends only on the restriction of $v$ on the boundary of a small disk $D(p)$ around $p$. Actually the homomorphism

[TABLE]

determines an element in $K^{1}(S^{2n-1})\cong\mathbb{Z}$. They used the notion of odd Chern character due to Getzler [6, Section 1,P. 492] to calculate the contribution.

In this paper, we follow the strategy of Feng-Li-Zhang’s work combined with a calculation of some homotopy set which will be stated in Lemma 3.1. Our method to localize the relative Chern character number is to use the odd Chern character on the product sphere, i.e. $\partial N(S^{2k_{i}-1})\cong S^{2k_{i}-1}\times S^{2n-2k_{i}}$. In the rest of this paper, we fix these diffeomorphisms. We will show that the homomorphism

[TABLE]

also determines an integer, which we will denote by $\deg^{*}(v_{i})$.

The main theorem of this paper is:

[TABLE]

A direct application of the above formula is the localization of the index of twisted Dirac operator for some spin manifold. To be more specific, let $M$ be a closed oriented $2n$–dimensional spin manifold, $\pi:M\to B$ be a bundle with sphere fiber of dimension $2k-1$, for some integer $2\leq k<n$ and $F$ be a complex vector bundle of rank $r$ over $M$. We consider the case that $E_{+}$ is a trivial rank $r$ bundle and $E_{-}$ is equal to $F$. Assume that $v$ is a homomorphism between $E_{+}$ and $E_{-}$ satisfying $Sing(v)\subset\coprod_{1\leq i\leq l}\pi^{-1}(x_{i})$, where $x_{1},\cdots,x_{l}$ are finitely many points in $B$, such that the homology class $[\pi^{-1}(x_{i})]$ belongs to the torsion part of $H_{2k-1}(M;\mathbb{Z})$ for each $x_{i}$. Let $D^{\otimes F}$ be the twisted Dirac operator, i.e. Dirac operator of the $F$–tensered spinor bundle. Then, the index can be localized as follows(see Proposition 5.3):

[TABLE]

The organization of the paper is as follows:

In Section Relative Chern character number and super–connection, we review some properties of odd Chern characteristic class. In Section Relative Chern character number and super–connection, we give the local calculation near the singularities contained in the product of two spheres. In Section Relative Chern character number and super–connection, we give the proof of our Main Theorem. In Section Relative Chern character number and super–connection, we show how to apply such a formula to localize the index of the twisted Dirac operator for the spin manifold mentioned above.

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sectionReview of odd Chern character

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Let $X$ be a closed manifold. We recall the notion of odd Chern character given by Getzler [6, P.490-495].

If $\nabla_{0}$ and $\nabla_{1}$ are two connections on a complex vector bundle $E$ over $X$, their Chern-Simons form is the differential form defined by

[TABLE]

where $\nabla_{u}=(1-u)\nabla_{0}+u\nabla_{1}$. By the projection $[0,1]\times X\to X$, we can define the connection on the product space $[0,1]\times X$,

[TABLE]

Let $N$ be a larger enough integer.

[TABLE]

where $\omega=g^{-1}dg$.

Let $g_{t}$, $t\in[0,1]$, be a family of differential maps from $X$ to $GL_{N}(\mathbb{C})$. It defines a differential mapping $\tilde{g}:[0,1]\times X\to GL_{N}(\mathbb{C})$. Hence, the odd Chern character $\mbox{Ch}(\tilde{g})$ can be decomposed into the following:

[TABLE]

[TABLE]

and satisfies the transgression formula

[TABLE]

The above proposition implies that the cohomology class $[\mbox{Ch}(g)]$ depends only on the homotopy class $[g]\in[X,GL_{N}(\mathbb{C})]$.

[TABLE]

where $\deg^{top}(g)$ denotes the topological degree of the homotopy class $[g]\in\pi_{2k-1}(GL_{N}(\mathbb{C}))$.

In this paper, we denote the integral $\frac{1}{(-2\pi\sqrt{-1})^{k}}\int_{S^{2k-1}}\mbox{Ch}(g)$ by $\deg(g)$.

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sectionLocal calculation on singularity

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Now we consider the local model of each $S^{2k_{i}-1}$. To simplify the notation, we drop the subscript $i$. We need the following technique in the homotopy theory to split the elements in $[S^{2n-2k}\times S^{2k-1},GL_{N}(\mathbb{C})]$.

Proof For any positive integers $p$ and $q$, we have the cofibration sequence,

[TABLE]

where the first map denotes the attaching and the forth map denotes the suspension, therefore null-homotopic. Taking $p=2n-2k$ and $q=2k-1$, we have the exact short sequence

[TABLE]

Since $\pi_{2n-2k}(GL_{N}(\mathbb{C}))=0$, we can find a homotopic equivalent map

[TABLE]

where $q$ is a fixed point in $S^{2k-1}$. Fixing a point $p$ in $S^{2n-2k}$, we set f=g_{1}\Big{|}_{p\times S^{2k-1}}. By the projection $p_{2}:S^{2n-2k}\times S^{2k-1}\to\{p\}\times S^{2k-1}$, we have that $(p^{*}_{2}f)^{-1}g_{1}$ is null-homotopy in the above exact sequence, hence it equals to a map $h:S^{2n-1}\to GL_{N}(\mathbb{C})$.

By Lemma 2.4 and Lemma 3.1, we can calculate the integral on the product sphere.

[TABLE]

where $h$ is a differential mapping from $S^{2n-1}$ to $GL_{N}(\mathbb{C})$ as in the above lemma.

Here, we denote the integral $\frac{1}{(-2\pi\sqrt{-1})^{n}}\int_{S^{2n-2k}\times S^{2k-1}}\mbox{Ch}(g)$ by $\deg^{*}(g)$.

Proof By Lemma 3.1, we can find two differential mappings $f:S^{2k-1}\to GL_{N}(\mathbb{C})$ and $h:S^{2n-1}\to GL_{N}(\mathbb{C})$, such that there exists a representation $g:S^{2n-2k}\times S^{2k-1}\to GL_{N}(\mathbb{C})$ of $[g]$ as the following

[TABLE]

where $\phi$ denotes the smash product

[TABLE]

and $pr_{2}$ denotes the canonical projection

[TABLE]

It suffices to show that the integral is independent on $f$. Note that we have a canonical embedding $GL_{N}(\mathbb{C})\hookrightarrow GL_{2N}(\mathbb{C})$, the image of $pr^{*}_{2}f\cdot\phi^{*}h$ under this embedding is

[TABLE]

which is homotopic to

[TABLE]

Hence, we have

[TABLE]

Moreover, we know that $2k-1<2n-1$ and the differential form $\mbox{Ch}(f)$ only depends on the component $S^{2k-1}$, therefore the integral $\int_{S^{2n-2k}\times S^{2k-1}}pr^{*}_{2}\mbox{Ch}(f)$ is zero.

Now, we obtain

[TABLE]

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sectionProof of Theorem 1.2

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Without loss of generality, we suppose that the rank of $E_{\pm}$ is large enough.

Let $E=E_{+}\oplus E_{-}$ be the $\mathbb{Z}_{2}$–graded complex vector bundle over $M$. Choose two unitary connections $\nabla^{E_{+}}$ and $\nabla^{E_{-}}$ on $E_{+}$ and $E_{-}$ respectively, such that near each small tubular neighborhood of $S^{2k_{i}-1}$, the connections are both trivial. Let $\nabla^{E}=\nabla^{E_{+}}\oplus\nabla^{E_{-}}$ be the $\mathbb{Z}_{2}$–graded connection on $E$, $v:E_{+}\to E_{-}$ extend to an endomorphism of $E$ by acting as zero on $E_{-}$ and $v^{*}$ be the adjoint of $v$ with respect to the hermitian metrics on $E_{\pm}$ respectively. Set $V=v+v^{*}$, then $V$ is an odd endomorphism of $E$ and $V^{2}$ is fiberwise positive on $M_{1}=M\setminus\bigcup\limits_{i}N(S^{2k_{i}-1})$.

For the convenience of computation, we define a function $\varphi:\Omega^{*}(M)\to\Omega^{*}(M)$, by

[TABLE]

where $\alpha$ is a differential form of degree $k$.

For any $T\in\mathbb{R}^{\geq 0}$, let $A_{T}$ be the super–connection on $E$ in the sense of Quillen(cf. [7, Section 2]), defined by

[TABLE]

Let $\mbox{ch}(E,A_{T})$ be the associated Chern character form defined by

[TABLE]

By straightforward calculation, one can derive the following lemmas.

We set

[TABLE]

Then, we get the transgression formula

[TABLE]

[TABLE]

[TABLE]

Now, we can give the proof of Theorem 1.2.

Proof When $v$ is restricted on $\partial N(S^{2k_{i}-1})$, we get that $v^{*}=v^{-1}$ and $V^{2}$ is the identity map acting on $E\mid_{\partial N(S^{2k_{i}-1})}$. Since $\nabla^{E}$ is trivial over $N(S^{2k_{i}-1})$ which is denoted by $d$, then on $\partial N(S^{2k_{i}-1})$ we have the following two identities

[TABLE]

Then, we get

[TABLE]

Since e^{-tdV}=e^{-t\left(\begin{array}[]{cc}&dv^{*}\\ dv&\\ \end{array}\right)}, after taking the super–trace only the odd degree terms of \sum\limits_{n}\left(\begin{array}[]{cc}&dv^{*}\\ dv&\\ \end{array}\right)^{n} remain. By the identity

[TABLE]

we have

[TABLE]

Therefore, we get

[TABLE]

The last equality follows from the formula (3.1). After taking the sum of $i\in\{1,\cdots,l\}$, we obtain the formula (1.2).

At the end of this section, we want to point out that the above argument can also be applied to the case, when $Sing(v)$ is a set consisting of finite points. In other words, Theorem 1.2 is a generalization of Feng-Li-Zhang’s result.

[TABLE]

where $\deg(v_{p})$ denotes the degree of the map v\big{|}_{\partial D(p)}:E_{+}\big{|}_{\partial D(p)}\to E_{-}\big{|}_{\partial D(p)}.

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sectionAn application to the index of twisted Dirac operator

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In this section, we show that the index of some twisted Dirac operator can be localized to the relative Chern character number. We assume that $M$ is a closed oriented $2n$–dimensional spin manifold, and satisfies the following assumption.

[TABLE]

where $B$ and $S$ are closed manifolds, such that the following conditions hold:

$M$* is locally split, i.e. each $x\in B$ has a small neighborhood $U(x)$ in $B$, such that $\pi^{-1}(U(x))$ is diffeomorphic to $U(x)\times S$,*

• 2)

$S$* is diffeomorphic to an odd dimensional sphere $S^{2k-1}$, for some integer $2\leq k<n$.*

For technical reason, we need the following lemma.

Proof We can choose a tubular neighborhood of $Y$ by equipping a metric on $X$, and by the pull-back of the projection $\mathcal{N}\to Y$, we need only to show that $\omega\mid_{Y}$ is $d$-exact. By the Poincaré duality, it suffices to show that the following formula holds

[TABLE]

for any element $[Z]\in H_{*}(Y;\mathbb{Z})$. Since $Y$ is a rational homology sphere and $[Y]\in Tor(H_{*}(X;\mathbb{Z}))$, the above formula holds.

[TABLE]

Proof Since $S$ can be viewed as the foliation of the manifold $M$, and it admits a positive scalar curvature. By Connes Vanishing Theorem [4, Theorem $02$] we know $\langle\hat{A}(TM),[M]\rangle=0$. We omit the proof here and refer to a geometric proof [8], which is given by Zhang. Actually, in our setting we can use adiabatic limits method of Dirac operator as in Bismut’s work [2, Proposition $5.2$] to show that $\hat{A}(TM)$ vanishes.

Let $E:=\underline{\mathbb{C}}^{r}\oplus F$ be the $\mathbb{Z}_{2}$–graded vector bundle. We need to show that

[TABLE]

Since we can write

[TABLE]

where $R^{TM}$ denotes the curvature of $TM$ and $\beta$ denotes the form with high degree, it suffices to show a more general formula:

[TABLE]

where $\omega^{0}$ denotes the [math]-degree part of $\omega$.

Let $N_{\epsilon}(\pi^{-1}(x_{i}))$ be the tubular neighborhood of $\pi^{-1}(x_{i})$ of radius $\epsilon$ in normal direction and $\mathcal{U}_{\epsilon}=\coprod_{1\leq i\leq l}N_{\epsilon}(\pi^{-1}(x_{i}))$. Find a cut-off function whose support locates in $\mathcal{U}_{2\epsilon}$ and satisfying $\rho\mid_{\mathcal{U}_{\epsilon}}\equiv 1$. By Lemma 5.2, there is a form $\alpha$ on $\mathcal{U}_{2\epsilon}$ such that $\omega^{*}=d\alpha$ in $\mathcal{U}_{\epsilon}$, where $\omega^{*}$ denotes $\omega-\omega^{0}$. Hence, we get

[TABLE]

By the formula (4.6) on $M\setminus\mathcal{U}_{\epsilon}$ and $\rho\omega^{*}=d\alpha$ on $\mathcal{U}_{\epsilon}$, we say that the above formula equals zero. Therefore, we get

[TABLE]

[TABLE]

Proof The global section $s$ can be viewed as the complex homomorphism between the trivial line bundle and $L$, i.e. we have a homomorphism

[TABLE]

By the similar arguments in the proof of Proposition 5.3, we have

[TABLE]

Up to here we have considered the cases for odd-dimensional singularity. However, we also have the next corollary for a special case in even–dimensional singularity.

[TABLE]

Proof We know that the normal bundle over $\pi^{-1}(x_{i})$ is trivial and we can choose a connection of $L$, such that near $\pi^{-1}(x_{i})$ it is trivial, because any complex line bundle over $\pi^{-1}(x_{i})\cong S^{2n-2}$ is trivial for $n\geq 3$. The tubular neighborhood of $\pi^{-1}(x_{i})$ is diffeomorphic to $D^{2}\times S^{2n-2}$, whose boundary is diffeomorphic to $S^{1}\times S^{2n-2}$. Thus, the argument in the proof of Proposition 5.3 also works in this setting and we obtain

[TABLE]

Acknowledgement: First, the author wants to show the best gratitude to Professor Huitao Feng, who brought this problem to the focus. The author also would like to express the special thanks to Professor Mikio Furuta, for the long time discussion and helping, and comments that greatly improved the manuscript. This research is supported by the Todai Fellowship.