${\mathcal{O}_{\mathcal{M}}}$${\mathcal{V}}$${\mathcal{W}}$${\mathcal{O}_{\mathcal{M}}}$${{\kappa_{j}}^{!}\widetilde{\mathcal{G}}}$${{\kappa_{j}}^{!}\mathcal{G}}$${\mathcal{O}_{\mathcal{M}}}$${\mathcal{A}_{{{\kappa_{j}}^{*}}\mathcal{d\\!e\\!t}}}$${\mathcal{T}_{\mathcal{M}}}$$\scriptstyle{\cdot
c_{j}}$$\scriptstyle{\varrho_{j}}$$\scriptstyle{b_{\varrho_{j}}}$$\scriptstyle{a_{\Lambda}}$$\scriptstyle{\operatorname{Id}}$$\scriptstyle{\kappa_{j}^{!}b_{\widetilde{L}}}$$\scriptstyle{\kappa_{j}^{!}a_{L}}$
(29)

######  Proposition A.21 ([Kon87]).

The pullback of the determinant line bundle from the Grassmannian along the
$j^{\textrm{th}}$ Krichever map is canonically isomorphic to $\lambda_{j}$.

${\kappa_{j}}^{*}{\mathcal{d\\!e\\!t}}_{\operatorname{Gr}(H)}\cong\lambda_{j}$

###### Proof.

We see $\kappa_{j}(X/S,P,z)=\pi_{*}\omega_{(X\setminus P)/S}^{j}$. Using the
A.12 of the determinant line bundle on the Grassmannian, we have

$\displaystyle\textstyle\det_{\operatorname{Gr}(H)}\Big{(}\pi_{*}\omega_{(X\setminus
P)/S}^{j}\Big{)}=\det R^{0}\pi_{*}\omega_{X/S}^{j}\Big{/}\det
R^{1}\pi_{*}\omega_{X/S}^{j}$

which can be seen using relative Čech cohomology. This is exactly the A.4 of
$\lambda_{j}$. ∎

######  Proposition A.22.

The Virasoro algebra acts by first order differential operators on
$\lambda_{j}$ with central charge $c_{j}$. Precisely, there is a Lie algebra
antihomomorphism
$\mathfrak{vir}\to\Gamma(\mathcal{M}_{g,1^{\infty}},\mathcal{A}_{\lambda_{j}})$
which is compatible with the action $\Lambda$ of $\mathfrak{witt}$ given in
A.2.

Using these standard properties of Atiyah algebras described in Section 2.2,
we describe the action of the Virasoro Lie algebroid on the line bundles of
the Mumford isomorphism.

Denote
$\mathcal{A}_{j}\colonequals\mathcal{A}_{\lambda_{j}\otimes\lambda_{1}^{-c_{j}}}$.
From A.22, $\mathcal{V}$ acts on $\mathcal{A}_{\lambda_{j/2}}$ with central
charge $c_{j}$, and further by 2.6 combined with 2.7, $\mathcal{V}$ acts on
$\mathcal{A}_{\lambda_{1/2}^{-c_{j}}}$ with central charge $-c_{j}$. Then the
action of $\mathcal{V}$ on $\mathcal{A}_{j}$ is defined via the action on
$\mathcal{A}_{\lambda_{j/2}}\times_{\mathcal{T}_{X}}-c_{j}\mathcal{A}_{\lambda_{1/2}}$.
Importantly, we see that $\mathcal{V}$ acts with central charge
$c_{j}-c_{j}=0$, as shown in the commutative diagram of Lie algebroids below.

${\mathcal{K}}$${0}$${\mathcal{O}_{\mathcal{M}}}$${\mathcal{V}}$${\mathcal{W}}$${0}$${0}$${\mathcal{O}_{\mathcal{M}}}$${\mathcal{A}_{j}}$${\mathcal{T}_{\mathcal{M}}}$${0}$$\scriptstyle{0}$$\scriptstyle{a_{\Lambda}}$$\scriptstyle{\alpha_{j}}$

######  Lemma A.23.

There exists a unique morphism of Lie algebroids
$\alpha_{j}\colon\thinspace\mathcal{W}\to\mathcal{A}_{j}$ which commutes with
the Virasoro action $\mathcal{V}\to\mathcal{A}_{j}$ and the projection
$\mathcal{V}\to\mathcal{W}$. In other words, the Virasoro action on
$\mathcal{A}_{j}$ descends to a Witt action.

###### Proof.

The proof is the same as for 5.12. ∎

######  Theorem A.24 ([Kon87]; [ADKP88]).

There exists a flat holomorphic connection on the line bundle
$\lambda_{1}^{-c_{j}}\otimes\lambda_{j}$.

###### Proof.

Define $\mathcal{K}$ to be the kernel of the Witt algebroid action
$a_{\Lambda}\colon\thinspace\mathcal{W}\to\mathcal{T}_{\mathcal{M}}$. Since
the action is surjective, by [Mac87, Theorem 1.4], $\mathcal{K}$ is a Lie
algebra bundle. In particular,
$\mathcal{K}\cong\pi_{*}\mathcal{T}_{(X\setminus P)/\mathcal{M}}$ is a bundle
with fiber the Lie algebra $\mathfrak{k}=\Gamma(C\setminus p,\mathcal{T}_{C})$
and an anchor map which is trivial. Therefore the bracket on $\mathcal{K}$ is
given by the pointwise bracket of its fibers. By A.1, we know every fiber is
equal to its commutant $\mathfrak{k}=[\mathfrak{k},\mathfrak{k}]$, therefore
$\mathcal{K}=[\mathcal{K},\mathcal{K}]$.

The rest of the proof is identical to the proof of 5.13. ∎

##  Appendix B Proof of 2.11 (a condition to be a pullback)

We label the morphisms in the diagram as follows.

${0}$${A}$${B}$${C}$${0}$${0}$${A^{\prime}}$${B^{\prime}}$${C^{\prime}}$${0}$$\scriptstyle{\iota}$$\scriptstyle{a}$$\scriptstyle{\wr}$$\scriptstyle{\pi}$$\scriptstyle{b}$$\scriptstyle{c}$$\scriptstyle{\iota}$$\scriptstyle{\pi}$

###### Proof.

Suppose we have $X$ such that the following square commutes.

${X}$${C}$${B^{\prime}}$${C^{\prime}}$$\scriptstyle{g}$$\scriptstyle{f}$$\scriptstyle{c}$$\scriptstyle{\pi}$

Let $x\in X$. Choose $y\in B$ such that $\pi(y)=g(x)$. Then $b(y)-f(x)\in
B^{\prime}$, and further $\pi^{\prime}(b(y)-f(x))=\pi^{\prime}\circ
b(y)-\pi^{\prime}\circ f(x)=c\circ\pi(y)-c\circ g(x)=c(\pi(y)-g(x))=0$.
Therefore we have $b(y)-f(x)\in A^{\prime}$. And using the isomorphism $a$ and
inclusion into $B$, we have $a^{-1}(b(y)-f(x))\in B$.

Define $\varphi\colon\thinspace X\to B$ as $\varphi(x)=y-a^{-1}(b(y)-f(x))$.
We first check this map commutes, then check it does not depend on the choice
of $y$.

$\displaystyle\pi\circ\varphi(x)=\pi(y-a^{-1}(b(y)-f(x)))=\pi(y)-\pi(a^{-1}(b(y)-f(x)))=\pi(y)=g(x)$
$\displaystyle
b\circ\varphi(x)=b(y-a^{-1}(b(y)-f(x)))=b(y)-b(a^{-1}(b(y)-f(x)))=b(y)-(b(y)-f(x))=f(x)$

Now consider two lifts $y,y^{\prime}\in B$ such that
$\pi(y)=\pi(y^{\prime})=g(x)$. Consider the subtraction of the resulting maps

$\displaystyle\varphi(x)-\varphi^{\prime}(x)$
$\displaystyle=[y-a^{-1}(b(y)-f(x))]-[y^{\prime}-a^{-1}(b(y^{\prime})-f(x))]=(y-y^{\prime})-a^{-1}(b(y-y^{\prime}))=0$

where the last equality results from noticing that $y-y^{\prime}\in A$ and
$b(y-y^{\prime})\in A^{\prime}$. ∎

##  Appendix C Proof of 3.5 (superconformal Noether normalization)

Before considering a family of open SRSs as in 3.5, we first provide the
required preliminaries from the setting of spin curves over a point, which is
the setting studied by [AP05].

######  Lemma C.1 ([AP05, Lemma 3]).

Let $A$ and $B$ be effective divisors on a compact Riemann surface $C$ such
that $[A]\cap[B]=\varnothing$ and $\deg(A-B)-\deg[A]-\deg[B]>2g-2$. Then

$\displaystyle H^{0}(C,\Omega^{1}_{C}(A-B))\to
H^{1}_{dR}\big{(}C\setminus[A]\big{)}$ $\displaystyle\sigma\to[\sigma]$

is onto.

######  Proposition C.2 ([AP05, Proposition 2]).

Let $C$ be a compact Riemann surface, let $\Omega_{C}^{1/2}$ be a spin
structure on $C$, let $p\in C$, and let $n\geq 0$. Consider the analytic map
below.

$\displaystyle\Upsilon_{C}\colon\thinspace
H^{0}\big{(}C,\Omega_{C}^{1/2}(np)\big{)}\to H^{1}_{dR}(C\setminus p)$
$\displaystyle\sigma\mapsto[\sigma^{2}]_{dR}$

If $n>6g-2$, then $\dim\Upsilon_{C}^{-1}(0)=n-2g$.

###### Proof.

The proof can be found in [AP05] and is very similar to the proof of C.3
below. In particular, C.1 is used in the proof. ∎

######  Proposition C.3.

Let $C$ be a compact Riemann surface, let $\Omega_{C}^{1/2}$ be a spin
structure on $C$, let $p,q\in C$ be distinct, and let $n\geq 0$. Consider the
analytic map below.

$\displaystyle\Upsilon_{q,C}\colon\thinspace
H^{0}\big{(}C,\Omega_{C}^{1/2}(np-q)\big{)}\to H^{1}_{dR}(C\setminus p)$
$\displaystyle\sigma\mapsto[\sigma^{2}]_{dR}$

If $n>6g$, then $\dim\Upsilon_{q,C}^{-1}(0)=n-2g-1$.

###### Proof.

Let $\sigma\in\Upsilon_{q,C}^{-1}(0)$ be nontrivial. Let the zero divisor of
$\sigma$ be $mp+rq+E$, where $n\geq m\geq 0$, $r\geq 1$, $E\geq 0$, $[E]\cap
p=\varnothing$, and $[E]\cap q=\varnothing$.

We use the technical lemma C.1 above. Set $A=(2n-m)p$ and $B=rq+E$.

${H^{0}\big{(}C,\Omega_{C}^{1/2}(np-q)\big{)}}$${H^{0}\big{(}C,\Omega^{1}_{C}((2n-m)p-2q-E)\big{)}}$${H^{1}_{dR}(C\setminus
p)}$${\upsilon}$${\sigma\upsilon}$${\left[\sigma\upsilon\right]_{dR}}$

The first map above is an isomorphism since it is multiplication by $\sigma$.
The second map above is surjective by C.1 when

  1. 1)

$m\leq 2n$ and

  2. 2)

$\deg(E)+\deg[E]\leq 2n-m-r-2g$.

We notice that the composite map is $\upsilon\mapsto[\sigma\upsilon]_{dR}$,
which is $d\Upsilon_{q,C,\sigma}$ up to a constant factor.

Let $t\in C$ be a generic point, distinct from $p$ and $q$. Let $s\geq 1$. Set

$\displaystyle\mathbb{P}_{\Upsilon^{-1}(0)}\colonequals\left\\{(\sigma)\in\mathbb{P}H^{0}(C,\Omega_{C}^{1/2}(np-q))\text{
such that }\sigma\in\Upsilon_{q,C}^{-1}(0)\right\\},$
$\displaystyle\mathbb{P}_{st}\colonequals\left\\{(\sigma)\in\mathbb{P}H^{0}(C,\Omega_{C}^{1/2}(np-q))\text{
such that }\operatorname{div}(\sigma)\geq st\right\\}.$

Notice that $\dim\mathbb{P}H^{0}(C,\Omega_{C}^{1/2}(np-q))=n-2$,
$\dim\mathbb{P}_{\Upsilon^{-1}(0)}\geq n-2g-2$, and
$\dim\mathbb{P}_{st}=n-s-2$. Therefore, if $n-2g-2\geq s$, then
$\mathbb{P}_{\Upsilon^{-1}(0)}\cap\mathbb{P}_{st}\neq\varnothing$.

By the last paragraph, we there exists $\sigma\in\Upsilon_{q,C}^{-1}(0)$
nontrivial with $mp+rq+E\geq(n-2g-2)t$. This implies that $n-2g-2\leq m+r+\deg
E\leq g-1+n$.

We see that $r+\deg E-(n-2g-2)\geq m$. Then since we have assumed $n>6g$, we
have that $2n>3g+1=(g-1+n)-(n-2g-2)\geq m$ which is condition 1) above.

Further, since we have assumed $n>6g$, we have that $n-3g+1\geq
3g+1\geq(r+m)-(n-2g-2)\geq\deg E\geq\deg[E]$. This implies that
$(g-1+n)+\deg[E]\leq 2n-2g$, which implies condition 2) above. ∎

We now return to the setting where $\pi\colon\thinspace X\to S$ is a family of
closed SRSs, $P$ is a relative divisor representing NS punctures, and
$(x|\xi)$ are relative superconformal coordinates on the trivial family
$\mathbb{A}^{1|1}\times S\to S$. Also assume that $U$ is a small enough open
supersubspace of $S$. Let $Y\to U$ be the restriction of the family of open
SRSs $X\setminus P\to S$ to the base $U$.

We next prove three claims before combining the results in the proof of 3.5 at
the end of this section.

######  Claim C.4.

Consider $\pi\colon\thinspace X\to S$ a family of closed SRSs with $P$ a
relative divisor representing NS punctures.

If $n>2g$, then for every $s\in S$ there exists a small enough open $U\ni s$
such that there exist nontrivial
$(f|\varphi)\in\Gamma(\pi^{-1}(U),\mathcal{O}_{X}(nP))$ such that $\delta
f=\varphi\,\delta\varphi$.

###### Proof.

Since we are interested in odd sections $\varphi$ of $\mathcal{O}_{X}(nP)$
which are relative de Rham exact, we consider
$H^{0}\big{(}\pi^{-1}(U),\Pi\mathcal{O}_{X}(nP)\big{)}$, where $\Pi$ is the
parity change operator, as a super vector bundle over $U$. For high enough
$n$, this super vector bundle is locally free. The relative de Rham cohomology
$H^{1}_{dR/U}(Y)$ results from the relative superconformal differential
$\delta$. Considered over $U$, this is a super vector bundle since the de Rham
cohomology is topological.

Consider the map between the relative affine superspaces over $U$

$\displaystyle\Upsilon\colon\thinspace\mathbb{H}^{0}\big{(}\pi^{-1}(U),\Pi\mathcal{O}_{X}(nP)\big{)}\to\mathbb{H}^{1}_{dR/U}(Y)$
$\displaystyle\sigma\mapsto[\sigma\,\delta\sigma]_{dR}.$

We wish to show that there exists an everywhere nonzero section $\varphi$ in
$\Upsilon^{-1}(0)$.

By super Riemann-Roch, we have that
$\dim_{U}\mathbb{H}^{0}\big{(}\pi^{-1}(U),\Pi\mathcal{O}_{X}(nP)\big{)}=n\,|\,n-g+1$
since $n>2g-1$. Further by de Rham theory,
$\dim_{U}\mathbb{H}^{1}_{dR/U}(Y)=2g\,|\,0$, since the cohomology classes are
the same as for the underlying Riemann surface. Therefore, $\Upsilon$ is a map
between superspaces of relative dimension $n\,|\,n+g-1$ and $2g\,|\,0$ over
$U$.

Since the domain has higher even dimension than the even dimension of the
codomain, we see that there must exist a nontrivial section $\varphi\in
H^{0}\big{(}\pi^{-1}(U),\Pi\mathcal{O}_{X}(nP)\big{)}$ such that the de Rham
class $[\varphi\,\delta\varphi]$ is $0$. We have assumed that $U$ is small
enough, so we are allowed to shrink $U$ more if needed to make $\varphi$
everywhere nonzero over $U$. ∎

######  Claim C.5.

Consider $\pi\colon\thinspace X\to S$ a family of closed SRSs with $P$ a
relative divisor representing NS punctures.

If $n>6g-2$, then for every $s\in S$ there exists a small enough open $U\ni s$
such that the superspace of odd
$\varphi\in\Gamma(\pi^{-1}(U),\mathcal{O}_{X}(nP))$ such that there exists an
$f$ so that $\delta f=\varphi\,\delta\varphi$ has dimension $n-2g\,|\,n-g+1$.

###### Proof.

Consider the differential map between relative tangent bundles

${d\Upsilon_{\sigma}\colon\thinspace\Big{(}H^{0}\big{(}\pi^{-1}(U),\Pi\mathcal{O}_{X}(nP)\big{)}\Big{)}\Big{|}_{u}}$${\Big{(}H^{1}_{dR/U}(Y)\Big{)}\Big{|}_{u}}$${\upsilon}$${\left[\upsilon\,\delta\sigma+\sigma\,\delta\upsilon\right]_{dR}}$

where $\sigma\in\Upsilon^{-1}(0)$ is a point, with $\pi(\sigma)=u\in U$.

We wish to show that $d\Upsilon_{\sigma}$ is surjective for some nontrivial
point $\sigma\in\Upsilon^{-1}(0)$ in each fiber over $U$, and therefore by the
implicit function theorem, $\Upsilon^{-1}(0)$ has relative dimension over $U$
the difference $(n\,|\,n-g+1)-(2g\,|\,0)$.

We recall that a family of SRSs have as fiber over each point of the base a
split SRS, which is known to be exactly a spin curve. For $\Sigma$ a split
SRS,
$\Pi\mathcal{O}_{\Sigma}\cong\Pi\mathcal{J}_{\Sigma_{\operatorname{red}}}\oplus\Pi\mathcal{O}_{\Sigma_{\operatorname{red}}}$,
where $\Pi\mathcal{J}_{\Sigma_{\operatorname{red}}}$ is the spin structure on
$\Sigma_{\operatorname{red}}$. So the domain of $d\Upsilon_{\sigma}$ has
decomposition as a supervector space as
$H^{0}\big{(}\Sigma,\Pi\mathcal{J}_{\Sigma_{\operatorname{red}}}(np)\big{)}\oplus
H^{0}\big{(}\Sigma,\Pi\mathcal{O}_{\Sigma_{\operatorname{red}}}(np)\big{)}$.

Since the codomain of $d\Upsilon_{\sigma}$ is of relative dimension
$(2g\,|\,0)$, then the surjectivity of $d\Upsilon_{\sigma}$ just means the
surjectivity of the even-even block of
$H^{0}\big{(}\Sigma,\Pi\mathcal{J}_{\Sigma_{\operatorname{red}}}(np)\big{)}\xrightarrow{d\Upsilon_{\sigma}}H^{1}_{dR}\big{(}\Sigma\setminus
p\big{)}$.

The situation of a single spin curve was studied by [AP05]. We restate their
result in C.2. This proposition implies the even dimension of
$\Upsilon^{-1}(0)$ is $n-2g$. ∎

######  Claim C.6.

Consider $\pi\colon\thinspace X\to S$ a family of closed SRSs with $P$ a
relative divisor representing NS punctures. Let $R$ be the zero divisor of
$\delta\varphi$, for a $\varphi$ such that there exists $f$ where $\delta
f=\varphi\,\delta\varphi$. Let $Q$ be a relative irreducible divisor (of
degree $1$) contained in $R$.

If $n>6g$, then for every $s\in S$ there exists a small enough open $U\ni s$
such that the superspace of odd
$\varphi\in\Gamma(\pi^{-1}(U),\mathcal{O}_{X}(nP-Q))$ such that there exists
an $f$ so that $\delta f=\varphi\,\delta\varphi$ has dimension
$n-2g-1\,|\,n-g+1$.

###### Proof.

We assume that $Q$ and $P$ are disjoint without loss of generality.

Consider the slightly modified map

$\displaystyle\Upsilon_{Q}\colon\thinspace\mathbb{H}^{0}\big{(}\pi^{-1}(U),\Pi\mathcal{O}_{X}(nP-Q)\big{)}\to\mathbb{H}^{1}_{dR/U}(Y)$
$\displaystyle\sigma\mapsto[\sigma\,\delta\sigma]_{dR}.$

Consider the differential map between relative tangent bundles

${d\Upsilon_{Q,\sigma}\colon\thinspace\Big{(}H^{0}\big{(}\pi^{-1}(U),\Pi\mathcal{O}_{X}(nP-Q)\big{)}\Big{)}\Big{|}_{u}}$${\Big{(}H^{1}_{dR/U}(Y)\Big{)}\Big{|}_{u}}$${\upsilon}$${\left[\upsilon\,\delta\sigma+\sigma\,\delta\upsilon\right]_{dR}}$

where $\sigma\in\Upsilon^{-1}(0)$ is a point, with $\pi(\sigma)=u\in U$.

As in the proof of C.5, we wish to show that $d\Upsilon_{Q,\sigma}$ is
surjective for some nontrivial point $\sigma\in\Upsilon^{-1}(0)$ in each fiber
over $U$. As in the proof of C.5, the result follows from C.3 since we have
assumed $n>6g$. ∎

Lastly we combine the above three claims to show the below lemma which implies
3.5.

######  Lemma C.7.

Consider $\pi\colon\thinspace X\to S$ a family of closed SRSs with $P$ a
relative divisor representing NS punctures. Let $Y\to U$ be the restriction of
the family of open SRSs $X\setminus P\to S$ to the base $U$.

For every $s\in S$ there exists a small enough open $U\ni s$ such that there
exist global functions $f|\varphi\in\Gamma(Y,\mathcal{O}_{X\setminus P})$
where $f$ is even and $\varphi$ odd such that

$\displaystyle\delta f=\varphi\,\delta\varphi$ (30)

and the relative zero divisor of $\delta\varphi$ is finite over $U$, where
$\delta\colon\thinspace\mathcal{O}_{X}\to\omega_{X/S}$ is the relative
superconformal differential in 9.

Further, there exist global functions
$f^{\prime},\varphi^{\prime}\in\Gamma(Y,\mathcal{O}_{X\setminus P})$ as above,
with the relative zero divisor of $\varphi^{\prime}$ over $U$ disjoint from
the relative zero divisor of $\varphi$ over $U$.

###### Proof.

Assume $n>6g$.

C.4 implies the existence of $(f|\varphi)$ corresponding to a superconformal
Noether normalization. Since $\varphi$ was constructed on a compact SRS, then
$(f|\varphi)$ define a finite surjective morphism with finite relative zero
divisor.

C.5 and C.6 then show the existence of a $(f^{\prime}|\varphi^{\prime})$ with
disjoint divisor. In particular, C.6 shows that the space of
$\varphi^{\prime}\in\Gamma(\pi^{-1}(U),\mathcal{O}_{X}(nP))$ which have
relative zero divisors which intersect the relative zero divisor of
$\delta\varphi$ (denoted by $R$ previously) is of dimension $n-2g-1$.
Therefore, there exists an odd
$\varphi^{\prime}\in\Gamma(\pi^{-1}(U),\mathcal{O}_{X}(nP))$ with relative
zero divisor disjoint from $R$. ∎

###### Proof of 3.5.

We have reduced the proof to showing that C.7 implies the result.

Define the map $p$ to be the morphism over $U$ corresponding to the map of
functions $\mathbb{C}[x|\xi]\to\Gamma(Y,\mathcal{O}_{X\setminus P})$ given by
$x\mapsto f$ and $\xi\mapsto\varphi$.

Shrinking $U$ if needed, consider local relative superconformal coordinates
$(z|\zeta)$ on $Y$. In this chart we have that

$\displaystyle\varphi=\gamma+c\zeta$

for some even|odd functions respectively $c|\gamma$, functions of $z$ and on
$U$. Since 30 gives that $D_{\zeta}f=\varphi\,D_{\zeta}\varphi$ (see 11), then

$\displaystyle\frac{\partial}{\partial{\varphi}}+\varphi\frac{\partial}{\partial{f}}\cong\left(\frac{1}{c-\gamma^{\prime}\zeta}\right)\left(\frac{\partial}{\partial{\zeta}}+\zeta\frac{\partial}{\partial{z}}\right).$
(31)

We see that the vector field on $Y$ in 31 has poles exactly where
$D_{\zeta}\varphi=c-\gamma^{\prime}\zeta=0$, which is the local coordinate
expression for $\delta\varphi$ by 10.

Further, the Berezinian of the Jacobian matrix is $c-\gamma^{\prime}\zeta$.
And since $f=p^{*}(x)$ and $\varphi=p^{*}(\xi)$, the ramification of $p$ is
given by the zeros of $c-\gamma^{\prime}\zeta$.

This shows that
$p^{*}\left(\frac{\partial}{\partial{\xi}}+\xi\frac{\partial}{\partial{x}}\right)$,
which is given in local relative coordinates in 31, is a rational vector field
which generates $\mathcal{D}_{X/S}\big{|}_{Y}(R)$, where $R$ is the relative
divisor over $U$ given by $c-\gamma^{\prime}\zeta=0$. Thus, if the relative
zero divisor of $\delta\varphi$ is disjoint from the relative zero divisor of
$\delta\varphi^{\prime}$ over $U$, then the ramification of $p$ is also
disjoint from the ramification of $p^{\prime}$.

∎

##  Appendix D Proof of 4.3 (a condition to be a discrete submodule)

4.3 provides an alternative characterization of a discrete subspace. We first
provide a few lemmas.

######  Lemma D.1.

Consider a discrete $\mathcal{O}_{U}$-submodule $L$, such that
$L_{U}\oplus\widehat{K}_{U}\to(\widehat{H}_{j})_{U}$ is an isomorphism. For
every compact $K^{\prime}\supseteq K$, then
$L_{U}\cap\widehat{K}^{\prime}_{U}$ is locally free of finite type and
$(\widehat{H}_{j})_{U}/(L_{U}+\widehat{K}^{\prime}_{U})=0$.

###### Proof.

We have the exact sequence

$\displaystyle 0\to L_{U}\cap\widehat{K}_{U}\to
L_{U}\cap\widehat{K}^{\prime}_{U}\to\widehat{K}^{\prime}_{U}/\widehat{K}_{U}\to(\widehat{H}_{j})_{U}/(L_{U}+\widehat{K}_{U})\to(\widehat{H}_{j})_{U}/(L_{U}+\widehat{K}^{\prime}_{U})\to
0$

which simplifies to
$L_{U}\cap\widehat{K}^{\prime}_{U}\cong\widehat{K}^{\prime}_{U}/\widehat{K}_{U}$.
Further, we see that
$\widehat{K}^{\prime}_{U}/\widehat{K}_{U}\cong\widehat{(K^{\prime}/K)}_{U}$
which is locally free of rank $\dim(K^{\prime}/K)<\infty$. ∎

######  Lemma D.2.

Consider a discrete $\mathcal{O}_{S}$-submodule $L$. If
$(\widehat{H}_{j})_{S}/(L_{S}+\widehat{K}_{S})=0$, then
$L_{S}\cap\widehat{K}_{S}$ is locally free of finite type.

###### Proof.

By D.1, we see that locally we can find $K^{\prime}\supseteq K$, so that
$L_{U}\cap\widehat{K}^{\prime}_{U}$ is free of finite type and
$(\widehat{H}_{j})_{U}/(L_{U}+\widehat{K}^{\prime}_{U})=0$. Then we have the
exact sequence

$\displaystyle 0\to L_{U}\cap\widehat{K}_{U}\to
L_{U}\cap\widehat{K}^{\prime}_{U}\to\widehat{K}^{\prime}_{U}/\widehat{K}_{U}\to
0.$

Since $L_{U}\cap\widehat{K}_{U}$ is the kernel of a surjective map between
free modules of finite type, then the result follows. ∎

######  Lemma D.3.

Let $L$ be a super $\mathcal{O}_{S}$-submodule which for every $s\in S$ there
exists a neighborhood $U$ of $s$ and a compact $K$ such that
$L_{U}\cap\widehat{K}_{U}$ is free of finite type and
$(\widehat{H}_{j})_{U}/(L_{U}+\widehat{K}_{U})=0$.

Then for every compact $K^{\prime}$,
$(\widehat{H}_{j})_{S}/(L_{S}+\widehat{K}^{\prime}_{S})$ is locally of finite
presentation.

###### Proof.

Choose compact $K^{\prime\prime}\supseteq(K^{\prime}\cup K)$. Then by D.1,
$L_{U}\cap\widehat{K}^{\prime\prime}_{U}$ is free of finite type and
$(\widehat{H}_{j})_{U}/(L_{U}+\widehat{K}^{\prime\prime}_{U})=0$ for some
$U\ni s$. Then we have the exact sequence

$\displaystyle
L_{U}\cap\widehat{K}^{\prime\prime}_{U}\to(K^{\prime\prime}/K^{\prime})_{U}\to(\widehat{H}_{j})_{U}/(L_{U}+\widehat{K}^{\prime}_{U})\to
0$

from which the result follows. ∎

###### Proof of 4.3.

The definition of a discrete module clearly satisfies the condition of 4.3
automatically.

For the other direction, assume $L$ is a super $\mathcal{O}_{S}$-submodule
which for every $s\in S$ there exists a neighborhood $U$ of $s$ and a compact
$K$ such that $L_{U}\cap\widehat{K}_{U}$ is free of finite type and
$(\widehat{H}_{j})_{U}/(L_{U}+\widehat{K}_{U})=0$.

Then there exists $K^{\prime}\subseteq K$ such that
$L_{U}\cap\widehat{K}^{\prime}_{U}=0$. Then since
$\widehat{H}_{s}/(L_{s}+\widehat{K}^{\prime}_{s})$ is a finite dimensional
super vector space, we can find $K^{\prime\prime}\supseteq K^{\prime}$ such
that the natural map
$(K^{\prime\prime}/K^{\prime})_{s}\to(\widehat{H}_{j})_{s}/(L_{s}+\widehat{K}^{\prime}_{s})$
is an isomorphism.

Then since $(\widehat{H}_{j})_{S}/(L+\widehat{K}_{S})$ is locally of finite
presentation by D.3 and
$(\widehat{H}_{j})_{s}/(L_{s}+\widehat{K}^{\prime\prime}_{s})=0$, we may apply
the super Nakayama lemma [Var04, Lemma 4.7.1 ii] to conclude that there exists
a neighborhood $U^{\prime}\ni s$ such that
$(\widehat{H}_{j})_{U^{\prime}}/(L_{U^{\prime}}+\widehat{K}_{U^{\prime}})=0$.
Similarly, since $L\cap\widehat{K}^{\prime\prime}_{S}$ is locally free of
finite type by D.2 and $L_{s}\cap\widehat{K}^{\prime\prime}_{s}=0$, there
exists $U^{\prime\prime}\ni s$ such that
$L_{U^{\prime\prime}}\cap\widehat{K^{\prime\prime}}_{U^{\prime\prime}}=0$.

Therefore $U^{\prime}\cap U^{\prime\prime}\ni s$ and $K^{\prime\prime}$
exhibit $L$ as a discrete module. ∎

## Acknowledgments

This paper began as a collaboration with D. Diroff and I would like to thank
him for his contributions to the first results of this project. The advice and
encouragement from A. Voronov has been invaluable. I thank the anonymous
referee for suggestions on minor improvements throughout the paper, and
especially for suggesting major revision of 4.4 and the proof of 3.6.

Part of this work was completed while at the Max Planck Institute for
Mathematics in Bonn, and the author acknowledges their funding.

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††institutetext: aSchool of Science, Xihua University, Chengdu 610039, China

# Report on chaos bound outside Taub-NUT black holes

Yucheng He 111E-mail<EMAIL_ADDRESS>a Zeqiang Wang 222E-mail:
<EMAIL_ADDRESS>a Deyou Chen 333E-mail<EMAIL_ADDRESS>
###### Abstract

Positions of a charged particle’s equilibrium orbits and spatial regions where
the chaos bound is violated are found through circular motions of the particle
around charged Taub-NUT black holes. Lyapunov exponent is gotten by
calculating eigenvalues of a Jacobian matrix in a phase space ($r,\pi_{r}$).
When the particle’s charge is fixed, the positions of the equilibrium orbits
gradually move away from the event horizons with the increase of the angular
momentum.The result shows that the bound is violated in the near-horizon
regions and at a certain distance from the horizons when the charge and NUT
parameter are fixed. The spatial regions increase with the increase of the NUT
parameter’s value.

###### Keywords:

Chaos bound, spatial regions, Taub-NUT black holes

##  1 Introduction

Motions of particles near black holes convey important information on
background spacetimes. For example, null geodesics effectively explore
quasinormal modes for test fields, and these modes are related to both the
internal information and surface quantization of black holes KZS1 ; KZS2 ;
HOD1 ; OD1 ; GK1 . Tunneling behaviors of particles near the event horizons
can reflect the temperature of black holes PW1 . The formation of black holes’
shadows can be understood by studying motions of photons around black holes
EHT1 ; EHT2 ; EHT3 . Research on these motions can bridge black holes physics
and information theory.

Due to the nonlinearity of the Einstein’s field equation, motions of particles
may cause chaos. Chaos is a nonlinear phenomenon that is very sensitive to
initial conditions. This sensitivity is represented by a Lyapunov exponent. A
lot of work has been done on the chaos and exponent Pappalardi:2022zbn ;
CamposDelgado:2022cwu ; Caputa:2022eye ; Balasubramanian:2022tpr ;
Baskan:2022dys ; Giataganas:2021ghs ; Rosenhaus:2021xhm ; Gong:2022nwp ;
Blake:2021hjj ; Qu:2021ius ; Pappalardi:2021ahe ; Susskind:2021esx ;
Kundu:2021mex ; Craps:2021bmz ; Morita:2021syq ; Chan:2020ujs ; Nosaka:2020nuk
; Lei:2020clg ; Liu:2020yaf ; Bianchi:2020des . Focusing on the near-horizon
region of the black hole, Hashimoto and Tanahashi studied the chaos generated
by the radial motion of the particle HT . The particle is subjected to an
strong external force so that it can come very close to the hole without
falling into it. They found the value of the exponent is independent on the
strength and species of the external force and determined by the surface
gravity of the hole,

$\lambda\leq\kappa,$ (1)

where $\lambda$ is the exponent and $\kappa$ is the surface gravity. This
result favorably supported the conjecture proposed recently by Maldacena,
Shenker and Stanford MSS . In this seminal conjecture, they proposed that
there is a general upper bound for the Lyapunov exponent of chaos in quantum
field theory with a large number of degrees of freedom. The exponent satisfies

$\lambda\leq\frac{2\pi T}{\hbar},$ (2)

where $T$ is the system’s temperature originally derived by thought
experiments of shock waves near black holes’ horizons JM . For a black hole,
its temperature is determined by the surface gravity. Therefore, Eqs. (1) and
(2) are equivalent. This bound has been extensively studied and confirmed by a
large amount of work. It was proved to be saturated in Sachdev-Ye-Kitaev model
and speculated that this saturation is dual to an Einstein gravity AK1 ;
Kitaev_2018 ; IH1 .

In the recent work, Zhao et al considered the particles’ equilibrium in the
near-horizon regions, and expanded the exponent at the event horizons ZLL .
When the contribution of sub-leading terms was considered, they found that the
upper bound of the exponent (chaos bound) is violated. Cases of the chaos
bound for the violations have also been found in KG1 ; KG2 . In KG1 ; KG2 ,
Kan et al found that the bound is violated when the influence of the
particle’s angular momentum on the exponent was taken into account. In their
work, the particle’s mass and charge are fixed at certain values and the
exponent was gotten by the effective potential method. Also considering the
contribution of the angular momentum, Lei and Ge et al obtained the expression
of the exponent by the matrix method, and studied the bound through the
circular motion of the particle around the black hole and the exponent’s
expansion at the event horizon LG2 . They found that the bound was violated
both in the near-horizon regions and at a certain distance from the horizons
of Reissner-Nordström and Reissner-Nordström anti-de Sitter black holes.

In this paper, we investigate the influence of the angular momentum of a
charged particle around the charged Taub-NUT black holes on the Lyapunov
exponent, and find spatial regions where the chaos bound is violated. Although
the exponent is also obtained by the matrix method, we always fix the
particle’s charge as a constant in our calculation. This is different from the
work of Ge et al., in which the charge is not fixed. Therefore, our
calculation is a special case of their work. The NUT charge has both rotation-
like and electromagnetic charge-like characteristics. Therefore, it is
interesting to investigate its influence on the exponent and bound.

The rest is organized as follows. In the next section, we review
thermodynamics of the charged Taub-NUT black holes and deriving the exponent
by calculating the eigenvalue of the Jacobian matrix in the phase space
($r,\pi_{r}$). In Section 3, we investigate the influence of the particle’s
angular momentum and NUT charge on the exponent, and find the spatial regions
where the bound is violated. The last section is devoted to our conclusions.

##  2 Circular motion of particles around Taub-NUT black holes

In this section we investigate the circular motion of a charged particle
around the charged Taub-NUT black hole to derive the Lyapunov exponent. As an
anisotropic cosmological model, the Taub-NUT solution is characterized by a
Misner string similar to a singularity on the axis. This black hole is a
solution of the Einstein-Maxwell theory. Its Lagrangian is

$\displaystyle S=\frac{1}{16}\int d^{4}x\sqrt{-g}(R-2\Lambda-
F_{\mu\nu}F^{\mu\nu}).$ (2.1)

From the Lagrangian, the solution of the black hole is given by JGM

$\displaystyle ds^{2}=-\frac{f(r)}{r^{2}+n^{2}}(dt+2ncos\theta
d\phi)^{2}+\frac{r^{2}+n^{2}}{f(r)}dr^{2}+(r^{2}+n^{2})(d\theta^{2}+sin^{2}\theta
d\phi^{2}),$ (2.2)

with the electromagnetic potential

$A_{\mu}=\frac{-Q_{0}r}{r^{2}+n^{2}}\left(dt+2ncos\theta d\phi\right),$ (2.3)

where $f(r)=r^{2}-2Mr-n^{2}+Q_{0}^{2}$, $M,n$, $Q_{0}$ are the mass, NUT
parameter, and electric parameter of the black hole, respectively. This black
hole has two Misner string singularities located at $\theta=0$ and
$\theta=\pi$. There are two roots from $f(r)=0$, which yields event and inner
horizons,

$r_{\pm}=M\pm\sqrt{M^{2}+n^{2}-Q_{0}^{2}}.$ (2.4)

The surface gravity is

$\kappa=\frac{r_{+}-M}{r_{+}^{2}+n^{2}}.$ (2.5)

The Hawking temperature and the entropy are Bordo:2019slw

$T=\frac{1}{4\pi r_{+}}\left(1-\frac{Q_{0}^{2}}{r_{+}^{2}+n^{2}}\right),\quad
S=\pi\left(r_{+}^{2}+n^{2}\right).$ (2.6)

The electric charge $Q$ is expressed by the electric parameter $Q_{0}$, which
is

$Q=\frac{Q_{0}(r_{+}^{2}-n^{2})}{r_{+}^{2}+n^{2}}.$ (2.7)

Its electric potential is $\Phi=\frac{Q_{0}r_{+}}{r_{+}^{2}+n^{2}}$. $N$ is
the Misner charge associated with the NUT parameter and $\Psi$ is its
conjugate quantity

$N=-\frac{4\pi
n^{3}}{r_{+}}\left(1-\frac{Q_{0}^{2}(n^{2}+3r_{+}^{2})}{(r_{+}^{2}+n^{2})^{2}}\right),\quad\Psi=\frac{1}{8\pi
n}.$ (2.8)

The above thermodynamic quantities obey the first law of thermodynamics

$dM=TdS+\Phi dQ+\Psi dN.$ (2.9)

When a particle with mass $m$ and charge $q$ moves in a circular motion in the
equatorial plane of the charged black hole, its Lagrangian is

$\mathcal{L}=\frac{1}{2}\left[-f\dot{t}^{2}+\frac{\dot{r}^{2}}{f}+(r^{2}+n^{2})\dot{\phi}^{2}\right]-qA_{t}\dot{t},$
(2.10)

where $f=\frac{f(r)}{r^{2}+n^{2}}$, $\dot{x}^{\mu}=\frac{dx^{\mu}}{d\tau}$ and
$\tau$ is proper time. Using the above equation and generalized momenta
$\pi_{\mu}=\frac{\partial\mathcal{L}}{\partial\dot{x}}$, we get

$\pi_{t}=-f\dot{t}-qA_{t}=-E,\quad\pi_{r}=\frac{\dot{r}}{f},\quad\pi_{\phi}=(r^{2}+n^{2})\dot{\phi}=L.$
(2.11)

In the above equation, $E$ and $L$ represent the energy and angular momentum
of the particle, respectively. The Hamiltonian is

$H=\frac{-(\pi_{t}+qA_{t})^{2}+\pi_{r}^{2}f^{2}+\pi_{\phi}^{2}(r^{2}+n^{2})^{-1}f}{2f}.$
(2.12)

From the Hamiltonian, the equation of motion of the particle can be obtained,
which is

$\displaystyle\dot{t}=\frac{\partial
H}{\partial\pi_{t}}=-\frac{\pi_{t}+qA_{t}}{f},\quad\dot{\pi_{t}}=-\frac{\partial
H}{\partial t}=0,\quad\dot{r}=\frac{\partial H}{\partial\pi_{r}}=\pi_{r}f,$
(2.13) $\displaystyle\dot{\pi_{r}}=-\frac{\partial H}{\partial
r}=-\frac{1}{2}\left[\pi_{r}^{2}f^{{}^{\prime}}-\frac{2qA_{t}^{{}^{\prime}}(\pi_{t}+qA_{t})}{f}+\frac{(\pi_{t}+qA_{t})^{2}f^{{}^{\prime}}}{f^{2}}-\pi_{\phi}^{2}((r^{2}+n^{2})^{-1})^{{}^{\prime}}\right],$
$\displaystyle\dot{\phi}=\frac{\partial
H}{\partial\pi_{\phi}}=\frac{\pi_{\phi}}{(r^{2}+n^{2})},\quad\dot{\pi_{\phi}}=-\frac{\partial
H}{\partial\phi}=0,$

where $"^{\prime}"$ denotes a derivative with respect to $r$. In this paper,
we use the matrix method to derive the exponent in the phase space
$(r,\pi_{r})$. Therefore, we need to get the radial coordinate and momentum at
a coordinate time $t$ from the equations of motion Eq.(2.13),

$\displaystyle\frac{\mathrm{d}r}{\mathrm{d}t}=\frac{\dot{r}}{\dot{t}}=-\frac{\pi_{r}f^{2}}{\pi_{t}+qA_{t}}$
(2.14)
$\displaystyle\frac{\mathrm{d}\pi_{r}}{\mathrm{d}t}=\frac{\dot{\pi_{r}}}{\dot{t}}=-qA_{t}^{{}^{\prime}}+\frac{1}{2}\left[\frac{\pi_{r}f^{2}f^{{}^{\prime}}}{\pi_{t}+qA_{t}}+\frac{\pi_{t}+qA_{t}f^{{}^{\prime}}}{f}-\frac{\pi_{\phi}^{2}((r^{2}+n^{2})^{-1})^{{}^{\prime}}f}{\pi_{t}+qA_{t}}\right].$

The four-velocity of a particle obeys a normalization condition
$g_{\mu\nu}\dot{x^{\mu}}\dot{x^{\nu}}=\eta$, where $\eta=0$ describes a case
of a massless particle and $\eta=-1$ denotes a case of a massive particle. The
particle is charged is in this paper and the normalization condition yields
$\pi_{t}+qA_{t}=-\sqrt{f\left[1+\pi_{r}^{2}f+\pi_{\phi}^{2}(r^{2}+n^{2})^{-1}\right]}$.
We define $F_{1}=\frac{\mathrm{d}r}{\mathrm{d}t}$ and
$F_{2}=\frac{\mathrm{d}\pi_{r}}{\mathrm{d}t}$. Then Eq.(2.14) is rewritten as

$\displaystyle
F_{1}=\frac{\pi_{r}f^{2}}{\sqrt{f\left(1+\pi_{r}^{2}f+\pi_{\phi}^{2}(r^{2}+n^{2})^{-1}\right)}},$
(2.15) $\displaystyle
F_{2}=-qA_{t}^{{}^{\prime}}+\frac{(2\pi_{r}^{2}f+1)f^{{}^{\prime}}}{2\sqrt{f\left(1+\pi_{r}^{2}f+\pi_{\phi}^{2}(r^{2}+n^{2})^{-1}\right)}}-\frac{\pi_{\phi}^{2}((r^{2}+n^{2})f)^{{}^{\prime}}}{2\sqrt{f\left(1+\pi_{r}^{2}f+\pi_{\phi}^{2}(r^{2}+n^{2})^{-1}\right)}}.$

In the phase space, the elements of the matrix is defined by

$K_{11}=\frac{\partial F_{1}}{\partial r},\quad K_{12}=\frac{\partial
F_{1}}{\partial\pi_{r}},\quad K_{21}=\frac{\partial F_{2}}{\partial r},\quad
K_{22}=\frac{\partial F_{2}}{\partial\pi_{r}}.$ (2.16)

For a particle whose equilibrium orbit is a circular should satisfies
$\pi_{r}=\frac{d\pi_{r}}{dt}=0$. The location of the equilibrium orbit is
calculated from this constraint and Eq.(2.15). By calculating the eigenvalues
of the matrix, the exponent of the chaos of the charged particle in the orbit
is obtained as follows

$\lambda^{2}=\frac{1}{4}\Big{[}\frac{f^{{}^{\prime}}+\pi_{\phi}^{2}((r^{2}+n^{2})^{-1}f)^{{}^{\prime}}}{1+\pi_{\phi}^{2}(r^{2}+n^{2})^{-1}}\Big{]}^{2}-\frac{1}{2}f\frac{f^{{}^{\prime\prime}}+\pi_{\phi}^{2}((r^{2}+n^{2})^{-1}f)^{{}^{\prime\prime}}}{1+\pi_{\phi}^{2}(r^{2}+n^{2})^{-1}}-\frac{qA_{t}^{{}^{\prime\prime}}f^{2}}{\sqrt{f(1+\pi_{\phi}^{2}(r^{2}+n^{2})^{-1})}}.$
(2.17)

When the Lyapunov exponent $\lambda>0$, the system of charged particle is
unstable and chaos system. When the charge of the particle is fixed, we will
calculate the Lyapunov exponent by different parameters of the black hole and
particle. Clearly, the charge and angular momentum of particle affect the
Lyapunov exponent. When the angular momentum is neglected, we get

$\lambda^{2}=\frac{1}{4}(f^{{}^{\prime}})^{2}-\frac{1}{2}ff^{{}^{\prime\prime}}-qA_{t}^{{}^{\prime\prime}}f^{\frac{3}{2}}.$
(2.18)

At the event horizon, $f(r)=0$, and (2.17) is reduced to

$\lambda^{2}=\frac{1}{4}(f^{{}^{\prime}})^{2}=\kappa^{2}.$ (2.19)

Clearly, the exponent is saturated at the horizon. This result is consistent
with that obtained in HT .

##  3 Bound on Lyapunov exponent

###  3.1 Lyapunov exponent in non-extremal charged Taub-NUT black holes

When a charged particle moves around the charged Taub-NUT black hole, we can
make the position of the particle’s equilibrium orbit near the horizon by
adjusting the charge-to-mass ratio and angular momentum of the particle.
Different values of the Lyapunov exponent represent different motions.
$\lambda^{2}>0$, $\lambda^{2}=0$ or $\lambda^{2}<0$ denote unstable, critical
or stable motions, respectively. When $\lambda^{2}>\kappa^{2}$, the bound of
the exponent is violated. We investigate the influence of the angular momentum
of the particle on the exponent and find the angular momentum’s range and
spatial region where the exponent is violated. We set $M=1,$ and $q=15$. The
position $r_{0}$ of the equilibrium orbit is listed in the following tables.

$r_{0}$ | L | 0 | 1 | 3 | 5 | 7 | 10 | 15
---|---|---|---|---|---|---|---|---
$Q_{0}$=0.50 | 2.27589 | 2.28942 | 2.37311 | 2.47695 | 2.57133 | 2.68608 | 2.82149
$Q_{0}$=0.80 | 2.03071 | 2.03672 | 2.07901 | 2.14326 | 2.21223 | 2.30866 | 2.43907
$Q_{0}$=0.95 | 1.87711 | 1.88132 | 1.91193 | 1.96138 | 2.01774 | 2.10121 | 2.22118
$Q_{0}$=1.28 | 1.04148 | 1.04239 | 1.0513 | 1.07936 | 1.13341 | 1.22565 | 1.35941

  * •

Table 1. Positions of equilibrium orbits of the charged particle around the
charged Taub-NUT black hole. For $n=0.80$, the event horizon is located at
$r_{+}=2.1789$ when $Q_{0}=0.50$, at $r_{+}=2.0000$ when $Q_{0}=0.80$, at
$r_{+}=1.8587$ when $Q_{0}=0.95$, and at $r_{+}=1.0400$ when $Q_{0}=1.28$.

$r_{0}$ | L | 0 | 1 | 3 | 5 | 7 | 10 | 15
---|---|---|---|---|---|---|---|---
$Q_{0}$=0.70 | 1.8343 | 1.83861 | 1.87001 | 1.92062 | 1.97786 | 2.06121 | 2.17736
$Q_{0}$=0.90 | 1.59727 | 1.59933 | 1.61509 | 1.64327 | 1.67912 | 1.73848 | 1.83387
$Q_{0}$=1.00 | 1.40263 | 1.40386 | 1.41348 | 1.43164 | 1.45648 | 1.50154 | 1.58273
$Q_{0}$=1.05 | 1.24104 | 1.24178 | 1.24767 | 1.25935 | 1.27653 | 1.31119 | 1.38354

  * •

Table 2. Positions of equilibrium orbits of the charged particles around the
charged Taub-NUT black hole. For $n=0.40$, the event horizon is located at
$r_{+}=1.81854$ when $Q_{0}=0.70$,at $r_{+}=1.59161$ when $Q_{0}=0.90$, at
$r_{+}=1.4000$ when $Q_{0}=1.00$, at $r_{+}=1.23976$ when $Q_{0}=1.05$,.

$r_{0}$ | L | 0 | 1 | 3 | 5 | 7 | 10 | 15
---|---|---|---|---|---|---|---|---
n=0.30 | 1.67851 | 1.68111 | 1.70073 | 1.73484 | 1.77677 | 1.84338 | 1.94522
n=0.60 | 1.86561 | 1.86989 | 1.90101 | 1.95129 | 2.0084 | 2.09226 | 2.21081
n=0.80 | 2.03071 | 2.03672 | 2.07901 | 2.14326 | 2.21223 | 2.30866 | 2.43907
n=0.90 | 2.12257 | 2.12958 | 2.17805 | 2.24951 | 2.32427 | 2.42668 | 2.56276

  * •

Table 3. Positions of equilibrium orbits of the charged particles around the
charged Taub-NUT black hole. For $Q_{0}=0.80$, the event horizon is located at
$r_{+}=1.6708$ when $n=0.30$, at $r_{+}=1.8485$ when $n=0.60$, at
$r_{+}=2.0000$ when $n=0.80$,and at $r_{+}=2.0816$ when $n=0.90$.

$r_{0}$ | L | 0 | 1 | 3 | 5 | 7 | 10 | 15
---|---|---|---|---|---|---|---|---
n=0.20 | 1.20074 | 1.20124 | 1.20525 | 1.21332 | 1.22553 | 1.25147 | 1.31129
n=0.40 | 1.40263 | 1.40386 | 1.41348 | 1.43164 | 1.45648 | 1.50154 | 1.58273
n=0.60 | 1.60689 | 1.60919 | 1.62669 | 1.65752 | 1.69612 | 1.75899 | 1.85861
n=0.80 | 1.81537 | 1.81913 | 1.84669 | 1.89195 | 1.94446 | 2.02361 | 2.13964

  * •

Table 4. Positions of equilibrium orbits of the charged particles around the
charged Taub-NUT black hole. For $Q_{0}=1.00$, the event horizon is located at
$r_{+}=1.2000$ when $n=0.20$, at $r_{+}=1.4000$ when $n=0.40$, at
$r_{+}=1.6000$ when $n=0.60$, and at $r_{+}=1.8000$ when $n=0.80$.

In the above tables, when we fixed the NUT parameter and increased the angular
momentum of the particle, the positions of the equilibrium orbits move away
the horizon, and finally tend to a certain position. When the angular momentum
is zero or small enough, the positions of the equilibrium orbits are closed to
the horizon. In Table 1 and 2, when the angular momentum and NUT parameter
were fixed, the positions of the equilibrium orbits moves closed to the
horizon with the increase of the value of the electric parameter.

Using Eq. (2.17), we numerically calculate the values of the Lyapunov exponent
at the equilibrium orbits and plot them in Figure 1 \- Figure 4. In Figure 1,
when the parameter $Q_{0}$ is small, the value of the exponent increases to a
maximum value with the increase of the angular momentum, and then decreases to
a constant value. However, for any values of the angular momentum, the chaos
bound is violated when the parameter $Q_{0}$ is large enough. In this case,
the exponent does not have a maximum value and tends to a certain value with
the increase of the angular momentum. The range of the angular momentum where
the bound is violated are increased with the increase of the parameter
$Q_{0}$. For different values of $Q_{0}$, the locations of equilibrium orbits
and range of the angular momentum for the violation are different. The
relative size of the spatial region of the particle’s motion when the bound is
violated is $1.22368>\frac{r_{0}}{r_{+}}>1.06782$ when $Q_{0}=0.80$, is
$1.250729>\frac{r_{0}}{r_{+}}>1.0230483$ when $Q_{0}=0.95$, and is
$2.1763>\frac{r_{0}}{r_{+}}>1.0014$ when $Q_{0}=1.28$. Therefore, the relative
size increases with the increase of the electric parameter’s value.

Figure 1: The Lyapunov exponent of chaos of the charged particle outside the
charged Taub-NUT black hole when $n=0.80$. The chaos bound is violated when
$Q_{0}=0.80$ and $11.5558>L>4.8284$ (the corresponding spatial region is
$2.4473>r_{0}>2.1356$), when $Q_{0}=0.95$ and $20.5396>L>2.4772$ (
$2.3247>r_{0}>1.9015$) and when $Q_{0}=1.28$ and $L>0$
($2.2634>r_{0}>1.0414$).

When the NUT parameter is fixed at $n=0.40$, we calculate values of the
exponent at the equilibrium orbits with different values of the electric
parameter. From Figure 2, we find that the violation still occurs when the
electric parameter is relative large. The range of the angular momentum and
spatial region for the violation increase with the increase of the electric
parameter. The relative size of the spatial region is given by
$1.1626>\frac{r_{0}}{r_{+}}>1.0256$ when $Q_{0}=0.90$, by
$1.5994>\frac{r_{0}}{r_{+}}>1.0316$ when $Q_{0}=1.00$ and by
$1.75156>\frac{r_{0}}{r_{+}}>1.2449$ when $Q_{0}=1.05$.

Figure 2: The Lyapunov exponent of chaos of the charged particle outside the
charged Taub-NUT black hole when $n=0.40$. The chaos bound is violated when
$Q_{0}=0.90$ and $16.3314>L>4.6857$ (the corresponding spatial region is
$1.8569>r_{0}>1.6382$), when $Q_{0}=1$ and $339.7641>L>3.1137,$
($2.2391>r_{0}>1.4143$) and when $Q_{0}=1.05$ and $L>2.30113$
($2.17152>r_{0}>1.24494$).

When the electric parameter is fixed at $Q_{0}=0.80$, we calculate values of
the exponent at the equilibrium orbits and plot them in Figure 3. In the
figure, we find that a smaller value of the NUT parameter does not cause the
violation for the bound, and a larger value of the parameter causes the
violation When the angular momentum is within a certain range. When values of
the NUT parameter are different, the equilibrium orbital positions and the
range of the angular momentum are different. The relative spatial region is
$1.1095>\frac{r_{0}}{r_{+}}>1.0429$ when $n=0.60$, is $1.17062>r>1.04843$ when
$n=0.80$, and is $1.0995>\frac{r_{0}}{r_{+}}>1.0278$ when $n=0.90$. From the
figure, we find that the angular momentum’s range and spatial region are
increase with the increase of the NUT parameter’s value. As this parameter
increases, the black hole gradually approaches an extremal black hole. We
infer that the extremal black hole is more likely to violate the bound. This
will be discussed in the next section.

Figure 3: The Lyapunov exponent of chaos of the charged particle outside the
charged Taub-NUT black hole when when $Q_{0}=0.80$. The chaos bound is
violated when $n=0.60$, at $9.1781>L>4.7954,$ (the corresponding spatial
region is $2.0700>r_{0}>1.9456$), when $n=0.80$, at $11.1126>L>3.5954,$
($2.3412>r_{0}>2.0968$), when $n=0.90$, at $11.6743>L>3.2788,$
($2.4769>r_{0}>2.1816$).

When the electric parameter is fixed at $Q_{0}=1.00$, we calculate values of
the exponent at the equilibrium orbits and plot them in Figure 4. When the
bound is violated, the minimum value and region of the angular momentum
decreases as the NUT parameter’s value increases. When the NUT parameter’s
values are different, the angular momentum range and relative spatial region
are different. The relative spatial region is given by
$1.7401>\frac{r_{0}}{r_{+}}>1.0065$ when $n=0.20$, by
$1.5994>\frac{r_{0}}{r_{+}}>1.0316$ when $n=0.40$, by
$1.3298>\frac{r_{0}}{r_{+}}>1.0134$ when $n=0.60$, and is by
$1.2904>\frac{r_{0}}{r_{+}}>1.0180$ when $n=0.80$. Clearly, the relative
spatial region decreases with the increase of the NUT parameter’s value.

Figure 4: The Lyapunov exponent of chaos of the charged particle outside the
charged Taub-NUT black hole when $Q_{0}=1.00$. The chaos bound is violated in
the range $L>3.96016$ ( $2.0895>r_{0}>1.2086$) when $n=0.20$, in the range
$339.7641>L>3.1137$ ($2.2391>r_{0}>1.4143$) when $n=0.40$, in the range
$36.3545>L>2.5581,$ ($2.1277>r_{0}>1.6215$) when $n=0.60$, and in the range
$26.2532>L>2.1749$ ($2.3228>r_{0}>1.8325$) when $n=0.80$.

For the non-extremal charged Taub-NUT black hole, the violation for the chaos
bound are affected by the electric parameter, NUT parameter and the particle’s
angular momentum.

###  3.2 Lyapunov exponent in extremal charged Taub-NUT black holes

For an extremal charged Taub-NUT black hole, the inner and event horizons
coincides with each other and the surface gravity is zero. There is
$M^{2}+n^{2}=Q_{0}^{2}$ and $r_{+}=1$. Using (2.13), we get positions of
equilibrium orbits and list them in Table 5. The relationship between the
Lyapunov exponent and angular momentum is plotted in Figure 5.

$r_{0}$ | L | 0 | 10 | 20 | 30 | 40 | 50
---|---|---|---|---|---|---|---
n=0.2,$Q_{0}=\sqrt{1.04}$ | * | * | 1.16866 | 1.35506 | 1.47461 | 1.55774
n=0.4,$Q_{0}=\sqrt{1.16}$ | * | * | 1.23725 | 1.41428 | 1.53009 | 1.61158
n=0.6,$Q_{0}=\sqrt{1.36}$ | * | 1.06827 | 1.33819 | 1.50427 | 1.6155 | 1.69495
n=0.8,$Q_{0}=\sqrt{1.64}$ | * | 1.21692 | 1.46144 | 1.61723 | 1.72397 | 1.80138

  * •

Table5: Position of equilibrium orbits of a charged particle around the
charged extremal Taub-NUT black hole. An asterisk indicates that a equilibrium
orbit does not exist.

Figure 5: The Lyapunov exponent of chaos of the charged particle outside the
extremal Taub-NUT black hole when when. The chaos bound is violated in the
range $L>14.0836$ ($2.0197>r_{0}>1.0000$) when $n=0.20$ and
$Q_{0}=\sqrt{1.04}$, in the range $L>11.6518$ ($2.0771>r_{0}>1.0000$) when
$n=0.40$ and $Q_{0}=\sqrt{1.16}$, in the range $L>8.1515$
($2.16616>r_{0}>1.0000$) when $n=0.60$ and $Q_{0}=\sqrt{1.36}$, and in the
range $L>4.0176$ ($2.28059>r_{0}>1.0000$) when $n=0.80$ and
$Q_{0}=\sqrt{1.64}$.

For the extremal case, when the particle’s angular momentum is small, the
positions of the equilibrium orbits does not exist and the particle drops into
the black hole. A significant difference between it and non-extremal black
hole is that the equilibrium orbit here can approach the event horizon
infinitely without falling into it, which does not occur in the non-extremal
black hole. The exponent is always positive at equilibrium orbits in the
figure, which shows there always exists the violation for the chaos in the
near-horizon region and at a certain distance from the horizon of the black
hole. The range of the angular momentum and spatial region increase with the
increase of the NUT parameter’s value. Therefore, the extremal black hole is
more likely to violate the chaos bound than the non-extremal black hole.

##  4 Conclusions

In this paper, we investigated the influences of the particle’s angular
momentum and NUT parameter on the Lyapunov exponent and found the spatial
regions where the chaos bound is violated by changing the angular momentum and
fixing other parameters. The Lyapunov exponent was gotten by calculating the
eigenvalues of the Jacobian matrix. For the non-extremal black hole, when the
particle’s angular momentum is fixed and the electric parameter increases, the
equilibrium orbits are closer to the event horizon. With the increase of the
NUT parameter, the spatial region for the violation is closer to the event
horizon. Therefore, the relatively large electric parameter more likely
violate the bound in the near-horizon region. For the extremal case, when the
angular momentum is small, the positions of the equilibrium orbits do not
exist and there always exists the violation. The extremal black hole is more
likely to violate the chaos bound than the non-extremal black hole.

There are two explanations for the violation of the chaos bound. In LG2 , Ge
et al believed that this violation was related to the stability of the black
holes, and it is possible to solve this violation phenomenon through research
on stability. In LWT1 ; LWT2 , the authors found the violation by exploring
the influence of the minimum length effects on the chaotic motion. They
believed that this result is not a violation for the conjecture proposed in
MSS , and the bound can be corrected by the minimum length in the bulk. The
weak gravity conjecture reveals that the charge-to-mass ratio of a particle
can be greater than 1. In this paper, we set the particle charge to 15,
therefore, this conjecture is an implicit condition in our investigation. If
this conjecture was not considered, the result maybe changed. In addition, the
backreaction of the particle on the background spacetime has not been taken
into account LG2 . It will be meaningful to study of the chaos bound by
considering this effect.

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