#  ALE spaces and nodal curves

Nigel Hitchin

##  1 Introduction

In [12] [13] Kronheimer gave a construction of asymptotically locally
Euclidean (ALE) metrics in four dimensions by hyperkähler quotients. These
metrics are defined on the complex surfaces obtained by resolving the
singularity at the origin of $\mathbf{C}^{2}/\Gamma$ where $\Gamma\subset
SU(2)$ is a finite subgroup and the resolution is by a configuration of
rational curves which intersect according to a Dynkin diagram of type ${\bf
A,D,E}$. The construction gives a great deal of global information about the
metric – completeness, asymptotic behaviour, moduli. Nevertheless, some
aspects are impenetrable – for example the case of ${\bf E_{8}}$ entails
taking a vector space of dimension $4|\Gamma|=480$ and solving $357$ quadratic
equations even before taking a quotient to obtain a 4-manifold.

The aim of this paper is to give a different approach based on twistor theory,
using algebraic geometry to highlight different ingredients, providing a fresh
look at these spaces. Twistor theory was already used by the author 45 years
ago in the ${\bf A_{k}}$ and ${\bf D_{k}}$ cases [6],[3] (even though the
latter paper appeared much later) by using the algebraic equations defining
the deformations of $\mathbf{C}^{2}/\Gamma$ embedded in $\mathbf{C}^{3}$ as a
singular affine surface, or equivalently the other complex structures of the
hyperkähler family. This method produced some local formulae but proved
impossible for the consideration of the ${\bf E_{k}}$ series.

Here we replace the equations by a compactification of the twistor space. This
aspect appeared in a recent paper of the author [9] and in some respects this
one can be considered as a sequel. In both cases we consider the action of a
circle induced by the scalar action of $\mathbf{C}^{*}$ on $\mathbf{C}^{2}$
and consider the family of invariant metrics. The basis of twistor theory
consists of studying a distinguished family of rational curves – twistor lines
– in the twistor space. In [9] we identified the twistor lines which are
preserved by the $\mathbf{C}^{*}$-action in order to describe the induced
metric on the central 2-sphere of fixed points in the ALE space. They are
closures of $\mathbf{C}^{*}$-orbits. In this paper we identify the orbits of a
_general_ twistor line.

The twistor space is a complex 3-manifold which is a fibration
$Z\rightarrow{\rm P}^{1}$. The circle action fixes two distinguished points
$u=0,\infty$ on ${\rm P}^{1}$ and the fibre over these points is isomorphic to
the complex structure of the resolved singularity. We introduce a
compactification $\bar{Z}$ which extends the fibration and consider the fibre
over $u=1$. This is a projective rational surface obtained by a systematic
blowing up of points in the plane. An orbit of a twistor line is a surface in
$\bar{Z}$ which meets this fibre in a rational curve, and we identify in
Theorem 1 the linear system to which these curves belong. The curves are
however almost always singular, so earlier issues are transformed into the
serious problem of identifying nodal curves in a known linear system.
Nevertheless, we consider some examples – the ${\bf A_{2\ell-1}}$ case
involves hyperelliptic curves, and we also address the constraint on an
elliptic curve which appeared in the earlier treatment of the ${\bf D_{4}}$
case.

It is at this point that we make contact with the work of N.Honda [10],[11]
who adapted the original Penrose twistor approach by replacing smooth rational
curves with nodal ones. The relevant case here consists of the 3-parameter
family of nodal curves in the surface $u=1$ and this gives it an
interpretation as the minitwistor space for an Einstein-Weyl structure on the
3-dimensional quotient of the ALE space by the circle action.

In the case of ${\bf A_{1}}$ – the original Eguchi-Hanson metric which began
the study of the locally Euclidean property – the rational surface is a
quadric and the curves have no singularities – they are just plane sections.
As a consequence we deduce that the quotient of the ALE manifold can be
identified with hyperbolic 3-space, and we use this as a model for a brief
discussion of the orbifold Einstein-Weyl geometry in general. In particular,
the fixed 2-sphere in the ALE space becomes a 2-sphere at infinity in the
quotient. Infinity in the ALE space becomes a distinguished point with the
orbifold structure of $\mathbf{R}^{3}/\Gamma$. We do not develop further the
properties of these spaces but the interpretation of a minitwistor space as
the space of geodesics in an Einstein-Weyl manifold gives an explanation, in
terms of closed geodesics, of the existence of the singularities in the
rational curves.

This paper is based on a talk given in Oxford on July 25th 2023 at a
conference for Peter Kronheimer’s 60th birthday.

##  2 The twistor construction

The ALE metrics are hyperkähler, so have complex structures $I,J,K$ satisfying
the quaternionic identities and preserved by the Levi-Civita connection. They
define a 2-sphere of complex structures $aI+bJ+cK$ with $a^{2}+b^{2}+c^{2}=1$.
To obtain a 4-dimensional hyperkähler manifold via the twistor construction
one needs a complex 3-manifold $Z$ with a projection $\pi:Z\rightarrow{\rm
P}^{1}$ and a family of sections which we call _twistor lines_ with normal
bundle $N\cong\mathcal{O}(1)\oplus\mathcal{O}(1)$. Deformation theory gives a
complex 4-dimensional space of such sections where the tangent space at a
point is isomorphic to $H^{0}(P^{1},N)$. A conformal structure is defined by
declaring a null tangent vector to be a section of $N$ which vanishes at some
point. In addition one requires a symplectic form along the fibres – a section
of $\Lambda^{2}T^{*}_{Z/{\rm P}^{1}}(2)$ where $\mathcal{O}(k)$ is the pull
back by $\pi$ of the $\mathcal{O}(k)$ on ${\rm P}^{1}$. This fixes a complex
metric in the conformal class. Finally we require a real structure, an
antiholomorphic involution $\sigma:Z\rightarrow Z$ which covers the antipodal
map $u\mapsto-1/\bar{u}$ on ${\rm P}^{1}$, preserves the relative symplectic
form and induces a positive definite metric on the space $M$ of real sections.
Each fibre $Z_{u}$ meets a real section in a unique point and endows $M$ with
a complex structure and a complex symplectic form, with the fibre over
$u\in\mathbf{C}\cup\infty={\rm P}^{1}\cong S^{2}$ realizing the 2-sphere of
complex structures of the hyperkähler family.

Kronheimer’s construction identifies the twistor space, confirming the picture
suggested in [6]. A finite subgroup $\Gamma\subset SU(2)$ acts on
$\mathbf{C}^{2}$ and there is a basis $x,y,z$ of the ring of homogeneous
invariant polynomials in $(z_{1},z_{2})$ of degree $\ell,m,n$ respectively
satisfying an equation $f(x,y,z)=0$ which embeds the quotient
$\mathbf{C}^{2}/\Gamma$ as a singular affine surface in $\mathbf{C}^{3}$. A
versal deformation consists of adding lower degree polynomials to $f$, the
coefficients of which can be understood as Weyl-invariant polynomials on the
Cartan subalgebra $\mathfrak{h}$ of the Lie algebra of type $ADE$ associated
to $\Gamma$. Then $f(x,y,z,h_{1},\dots,h_{k})=0$ is a family of affine
surfaces, the generic one smooth. The twistor space for an $S^{1}$-invariant
metric is the hypersurface in the total space of
$\mathcal{O}(\ell)\oplus\mathcal{O}(m)\oplus\mathcal{O}(n)$ defined by
$f(x,y,z,a_{1}u,\dots,a_{k}u)=0$ with $a_{i}$ real, together with a resolution
of the singularities at $u=0,\infty$. The real structure is induced by the
involution $(z_{1},z_{2})\mapsto(\bar{z}_{2},-\bar{z}_{1})$ on the invariant
polynomials $x,y,z$ and $u=z_{1}/z_{2}$.

The ${\bf A}$ and ${\bf D}$ series are easy to write down because the
invariant polynomials are so well known, in topology for example as Chern
classes for ${\bf A_{k}}$ and Pontryagin and Euler classes for ${\bf D_{k}}$:

$\displaystyle{\bf A_{k}}$ $\displaystyle:$ $\displaystyle
xy=\prod_{i=1}^{k+1}(z-a_{i}u)$ (1) $\displaystyle{\bf D_{k}}$
$\displaystyle:$ $\displaystyle
x^{2}-zy^{2}=-\frac{1}{z}\left(\prod_{i=1}^{k}(z+a_{i}^{2}u^{2})-u^{2k}\prod_{i=1}^{k}a_{i}^{2}\right)+2yu^{k}\prod_{i=1}^{k}a_{i}.$
(2)

Then finding the twistor lines is the “Diophantine” problem of solving an
algebraic equation with polynomial coefficients in $u$ with polynomials
$x(u),y(u),z(u)$.

The first case is straightforward and was carried out in [6]:
$z(u)=cu^{2}+au-\bar{c}$ is a real quadratic and
$z-a_{i}u=c(u-\alpha_{i})(u-\beta_{i})$. Since the discriminant
$(a-a_{i})^{2}+4c\bar{c}$ is positive we can distinguish $\alpha_{i}$ and
$\beta_{i}$. Then one takes $x(u)=A(u-\alpha_{1})\cdots(u-\alpha_{k+1})$ and
$y(u)=B(u-\beta_{1})\cdots(u-\beta_{k+1})$ with $AB=c^{k+1}$ and $a,c$ and
$A/|A|$ give 4 real local coordinates for the metric.

The ${\bf D_{k}}$ case is more complicated – here $z(u)$ is a quartic
polynomial. Rewriting the equation as

$zx^{2}-(zy+\prod_{i=1}^{k}a_{i}u)^{2}=-\prod_{i=1}^{k}(z+a_{i}^{2}u^{2})$

we observe that a solution $x(u),y(u)$ means that

$f(u,w)=\frac{\sqrt{z(u)}x(u)-(z(u)y(u)+u^{k}\prod_{i=1}^{k}a_{i})}{\sqrt{z(u)}x(u)+(z(u)y(u)+u^{k}\prod_{i=1}^{k}a_{i})}$
(3)

is a rational function on the elliptic curve $w^{2}=z(u)$ with zeros $p_{i}$
and poles $q_{i}$ at $w=\pm ia_{j}u$ for some choice of signs. This is a
single constraint, implemented by an elliptic integral, on the divisor class
$D=\sum_{i}p_{i}-q_{i}$ which reduces the five coefficients of $z(u)$ to
produce 4 local coordinates on the ALE space.

  * Remark: Since this approach predates Kronheimer’s work it is perhaps useful to point out the context. Both cases produced formulae in terms of local coordinates, and in the ${\bf A_{k}}$ case one could relate this in [6] to a previously derived expression for the metric [5] by Gibbons and Hawking whose global interpretation was clear. In that sense the contribution of the twistor approach was to describe all the complex structures in the hyperkähler family. At the time, there was no alternative expression for the ${\bf D_{k}}$ case and the local coordinates used were not sufficient to give a global description of the space, but much later it made sense when moduli spaces of monopoles and their hyperkähler metrics were being analyzed. The constraint on the elliptic spectral curve of a charge 2 $SU(2)$ monopole with singularities was almost the same as the one above. The difference, which changes the asymptotic behaviour of the metric to ALF (asymptotically locally flat) is that the divisor class $D$, instead of being trivial and giving a rational function, is the restriction of a canonical element in $H^{1}(T{\rm P}^{1},\mathcal{O})$ to the curve $w^{2}=z(u)$ where $T{\rm P}^{1}=\mathcal{O}(2)$ is a non-compact surface, the tangent bundle of ${\rm P}^{1}$. The formulae in [3] require only slight modification to apply to the ALE case.

Proceeding to the ${\bf E_{k}}$ case following this approach proved difficult
and it becomes easier, but less explicit, to consider the algebraic geometry
of compactifications.

##  3 Compactifications

If we compactify $\mathbf{C}^{2}$ to ${\rm P}^{2}$ then
$\mathbf{C}^{2}/\Gamma$ can be compactified to ${\rm P}^{2}/\Gamma$ by adding
the orbifold ${\rm P}^{1}/\Gamma$ at infinity to the singular surface and its
resolution. For the ${\bf{D,E}}$ series there are three orbifold points which
give singularities of ${\rm P}^{1}/\Gamma$ corresponding to the stabilizers of
the vertices, midpoints of faces and edges of a dihedron, tetrahedron,
octahedron and icosahedron. These can be resolved [4] to give a configuration
of rational curves which support an anticanonical divisor: $-K=2C+E+F+G$. For
${\bf E_{k}}$ it is:

${E}$${F}$${G}$${G}$${3}-{k}$$-{2}$${C}$${-1}$${-3}$

and for ${\bf D_{k}}$ and ${\bf A_{2\ell-1}}$:

${E}$${F}$${G}$${G}$$-{(k-2)}$${-2}$${C}$${-1}$${-2}$${C}$${-\ell}$${-\ell}$${0}$${D_{1}}$${D_{2}}$

  * Remark: We consider ${\bf A_{k}}$ with $k$ odd for uniformity here. The point is that the ${\bf D_{k},E_{k}}$ Dynkin diagrams have a unique trivalent vertex which means the corresponding rational curve in the resolution of the singularity is pointwise fixed. This will play an important role later. In the ${\bf A_{k}}$ case there is a further circle action – a circle subgroup of $SU(2)$ commuting with the action of the cyclic group $\Gamma$ and some combination fixes any one of the chain of rational curves. It is convenient to take the central vertex as the fixed one when $k$ is odd. In addition $-1\in\Gamma$ for all cases except ${\bf A_{k}}$ for $k$ even.

Conversely a rational surface with a canonical divisor as above is the
compactification of one of the versal deformations of the singular surface.
This way the surface and its subvarieties replaces the explicit equation.
Following [9] consider the configuration in ${\rm P}^{2}$ in Figure 1: a conic
$E$ with 4 points $e_{1},e_{2},e_{3},e_{4}$ and a line through $x$ tangent to
the conic at $f$. Blowing up the six points gives a cubic surface (in fact the
natural projective compactification of the ${\bf D_{4}}$ equation (2)).
Because $E$ meets $G$ tangentially in the cubic we have three lines through a
point and blowing up that point to give a $-1$ curve $C$ gives three $-2$
curves and the compactification above for ${\bf D_{4}}$. Clearly blowing up
further points $e_{5},e_{6},\dots,e_{k}$ gives the ${\bf D_{k}}$
configuration.

${G}$${f}$${x}$${E}$${e}_{1}$${e}_{2}$${e}_{3}$${e}_{4}$ Figure 1: $D_{4}$
blowing up

The ${\bf E_{k}}$ series is similar but with an additional point $y$ on $G$
blown up.

The above describes a compactification of the fibres of the twistor space
$Z\rightarrow{\rm P}^{1}$. The ALE property of the metric means that the
4-manifold has a conformal orbifold compactification by a single point (as
used in [1]) and in consequence its twistor space has an orbifold
compactification by adding ${\rm P}^{1}/\Gamma$. After blowing up this gives a
compactification which extends the fibration so we obtain
$\bar{Z}\rightarrow{\rm P}^{1}$ where each fibre $\bar{Z}_{u}$ for $u\neq
0,\infty$ is holomorphically equivalent to one of the projective surfaces
described above.

##  4 The $\mathbf{C}^{*}$-action

The circle action on the ALE space induces a corresponding action on the
twistor space which extends to a $\mathbf{C}^{*}$-action, covering
$u\mapsto\lambda u$ under the projection $\bar{Z}\rightarrow{\rm P}^{1}$. It
preserves the fibres over $u=0,\infty$ which correspond to complex structures
$\pm I$, the resolution of the singularity and its complex conjugate. On the
resolution the action fixes pointwise the rational curve defined by the
trivalent vertex (we shall call this $C_{0}$) and the intersection points of
the other rational curves. We shall consider the orbit of a generic twistor
line $P$ under this action.

Firstly consider flat space $\mathbf{R}^{4}$. The twistor space $Z$ here is
the total space of the vector bundle
$\mathcal{O}(1)\oplus\mathcal{O}(1)\rightarrow{\rm P}^{1}$ and its
compactification by ${\rm P}^{1}$ is ${\rm P}^{3}$. Blowing up this line at
infinity gives $\bar{Z}\rightarrow{\rm P}^{1}$ with fibres ${\rm P}^{2}$. In
homogeneous coordinates we remove the line $z_{0}=z_{1}=0$ from ${\rm P}^{3}$
and the projection to ${\rm P}^{1}$ is $u=z_{1}/z_{0}$. The 4-parameter family
of twistor lines considered as sections of
$\mathcal{O}(1)\oplus\mathcal{O}(1)$ are
$z_{2}=az_{1}+bz_{0},z_{3}=cz_{1}+dz_{0}$. The real lines are given by
$d=\bar{a},c=-\bar{b}$. Fix $(a,b,c,d)$ to give a twistor line $P$, then
$\lambda\in\mathbf{C}^{*}$ transforms $P$ to

$z_{2}=\lambda az_{1}+\lambda^{-1}bz_{0},\qquad z_{3}=\lambda
cz_{1}+\lambda^{-1}dz_{0}.$

Eliminating $\lambda$ we see that the closure of the orbit of $P$ is the
quadric surface in ${\rm P}^{3}$ given by

$(cz_{2}-az_{3})(dz_{2}-bz_{3})+(ad-bc)z_{0}z_{1}=0.$

In particular the surface meets the line at infinity in two points
$[0,0,a,c],[0,0,b,d]$. The quadric meets the fibre $u=0$ where $z_{1}=0$ which
from the equation reads $(cz_{2}-az_{3})(dz_{2}-bz_{3})=0$ and consists of two
lines $L_{1},L_{2}$ meeting at the origin, the closure of two orbits in
$\mathbf{C}^{2}$ under the scalar $\mathbf{C}^{*}$-action. One line is given
by the orbit of the intersection of $P$ with the fibre $u=0$, the other is a
line in ${\rm P}^{3}$, the limiting transform as $\lambda\rightarrow\infty$.

In the ALE case we take a generic twistor line $P$ and consider the closure
$X$ of the orbit. The orbifold compactification of $Z$ clearly produces a
similar pair of lines $L_{1},L_{2}$ as the flat case. Now consider the
$\mathbf{C}^{*}$-action on the complement in $\bar{Z}$ of the fibre over
$u=\infty$, as in [9] Section 7. The open Bialynicki-Birula stratum for the
action consists of orbits which flow down to the fixed curve $C_{0}$ in the
fibre over $u=0$. Thus a generic point on $P$ has a limit on $C_{0}$ and so
this curve is another component of the intersection $X\cap\bar{Z}_{0}$. Note
that the intersection number $(L_{1}+L_{2}+C_{0}).C_{0}=1+1-2=0$. In fact the
cohomology class of $X$ in $H^{2}(\bar{Z},\mathbf{Z})$ must vanish on $C_{0}$
since the class $[C_{0}]\subset H_{2}(\bar{Z},\mathbf{Z})$ appears as a
holomorphic curve with conjugate complex structures in $u=0,\infty$.

The curve $C_{0}$ is the limit for the downward flow of a generic point in
$\bar{Z}_{0}$ and so the lines $L_{1},L_{2}$, orbit closures, only intersect
$C_{0}$ and not the other rational curves in the resolution. Then the
cohomology class of $X$ vanishes on all classes in $H_{2}(Z_{0},\mathbf{Z})$.

Now consider the intersection $X\cap\bar{Z}_{1}$, the fibre over $u=1$. Since
$H_{2}(Z_{1},\mathbf{Z})\cong H_{2}(Z_{0},\mathbf{Z})$, we learn from the
above discussion that the divisor class $Q$ of $X$ in $\bar{Z}_{1}$ has the
property that $Q$ is trivial on the affine part
$H_{2}(Z_{1},\mathbf{Z})\subset H_{2}(\bar{Z}_{1},\mathbf{Z})$ and also on
$E,F,G$ for the ${\bf D_{k},E_{k}}$ series and $D_{1},D_{2}$ for ${\bf
A_{2\ell-1}}$. This yields the following:

###### Theorem 1

The divisor class $Q$ of the image of a twistor line in the rational surface
$\bar{Z}_{1}$ is given by

$\displaystyle{\bf E_{6}}:Q$ $\displaystyle=12C+4E+6F+4G$ $\displaystyle{\bf
E_{7}}:Q$ $\displaystyle=24C+6E+12F+8G$ $\displaystyle{\bf E_{8}}:Q$
$\displaystyle=60C+12E+30F+20G$ (4) $\displaystyle{\bf A_{2\ell-1}}:Q$
$\displaystyle=\ell C+D_{1}+D_{2}$ $\displaystyle{\bf D_{k}}:Q$
$\displaystyle=(2k-4)C+2E+(k-2)F+(k-2)G.$

and in each case the intersection number $Q^{2}=|\Gamma|$ and $KQ=-4$ where
the ALE space is asymptotic to $\mathbf{C}^{2}/\Gamma$.

  * Proof: Since the curves $C,E,F,G$ and $D_{1},D_{2}$ are disjoint from the representative cycles in $H_{2}(Z_{0},\mathbf{Z})\cong H_{2}(Z_{1},\mathbf{Z})$ we can look for classes $Q=a_{0}C+a_{1}E+a_{2}F+a_{3}G$ etc. which satisfy $QE=QF=QG=0$ and also, since $L_{1},L_{2}$ intersect $C$ transversally, $QC=2$. Using the configurations in Section 3 this leads to the list. Then $Q^{2}$ and $KQ$ can be calculated directly using the adjunction formula for the rational curves $C,E,..$ etc. $\Box$

Since the image of a twistor line is a rational curve, the adjunction formula
$KQ+Q^{2}=2g-2$ says that these curves are singular and if they have simple
nodes then there must be $|\Gamma|/2-1$ singularities. Thus each twistor line
defines a nodal curve in the linear system $Q$. However, Honda and Nakata [10]
prove that if a smooth projective surface $S$ has a rational curve $C$ with
$\delta>0$ nodes as all its singularities and $C^{2}+1-2\delta>0$, then the
variety of such curves is non-singular and $C^{2}+1-2\delta$-dimensional. In
our case $Q^{2}+1-2\delta=3$ and so the images of the 4-parameter family of
twistor lines give an open set in the 3-parameter variety of nodal curves in
the linear system. The context of Honda’s work appears in Section 7. We shall
henceforth denote the surface $\bar{Z}_{1}$ by $S$.

  * Remark: Note that since $E$ came from a conic in the plane and $G$ a line under the blowing up from the plane, on blowing down a nodal curve in the linear system $Q$ to ${\rm P}^{2}$ the coefficients of $E,G$ in equations (4) show that the image is a plane curve in general of high degree and with singular behaviour at $f$, apart from the movable nodes.

##  5 Examples

###  5.1  ${\bf A_{2\ell-1}}$

The configuration of curves for ${\bf A_{2\ell-1}}$ can be achieved by taking
$S$ to be the projective bundle ${\rm
P}(\mathcal{O}\oplus\mathcal{O}(\ell))\rightarrow{\rm P}^{1}$ with $2\ell$
points $a_{1},\dots,a_{2\ell}$ on the zero section blown up. Then $C$ is a
fibre, $D_{1}$ the infinity section and $D_{2}$ the blown up zero section.

${C}$${-\ell}$${-\ell}$${0}$${D_{1}}$${D_{2}}$

From the equation (4) we have $Q=\ell C+D_{1}+D_{2}$ and $QC=2$ which means
that, if $x$ is an affine coordinate on ${\rm P}^{1}$, $Q$ has an equation of
the hyperelliptic form

$w^{2}+p(x)w+q(x)=0$

where $p(x)$ has degree $\ell$ and $q(x)$ degree $2\ell$ and vanishes at each
$x=a_{i}$, so $q(x)=c(x-a_{1})\dots(x-a_{2\ell})$. Here $\Gamma$ is the cyclic
group $\mathbf{Z}_{2\ell}$, so we require $|\Gamma|/2-1=\ell-1$ nodes. This
means $p(x)^{2}-4q(x)$ must have $\ell-1$ double zeros. This is $\ell-1$
constraints on the $\ell+1$ coefficients of $p(x)$ together with the
multiplier $c$ giving the expected $(\ell+2)-(\ell-1)=3$ parameters.

Note that for $\ell=1$ there are no singularities and the image of a twistor
line is a smooth rational curve of self-intersection $2$. In the model here we
can blow down the $-1$ curves $D_{1}$ and $D_{2}$ to obtain a quadric surface
and then the curves are plane sections. We shall meet this fact again in the
final section.

When $\ell=2$, by a projective transformation we can take three of the four
points to be $(0,1,\infty)$ and write $q(x)=\lambda^{2}\wp(z)$ using the
Weierstrass $\wp$-function. Then the general constraint is for
$\lambda\wp^{\prime}(z)+a_{0}+a_{1}\wp(z)+a_{2}\wp^{2}(z)$ to have a double
zero.

###  5.2  ${\bf D_{4}}$

In this case equation (4) gives $Q=4C+2E+2F+2G=-2K$. Now, as discussed in
Section 3, $S$ is a cubic surface $\Sigma\subset{\rm P}^{3}$ blown up at a
point $a$ giving the $-1$ curve $C$. Since the embedding of $\Sigma$ is the
anticanonical system, curves in the linear system $Q$ consist of intersections
with quadric surfaces in ${\rm P}^{3}$ meeting the cubic tangentially at that
point and three others. We shall see next in this approach the constrained
elliptic curve in the description of Section 2.

The cubic surface $\Sigma$ is the blow-up of the plane at the six points
$x,f,a_{1},a_{2},a_{3},a_{4}$ as in the diagram. We denote the images in
$\Sigma$ of the curves $E,F$ etc. in $S$ by $E_{*},F_{*}$ etc. and a generic
twistor line $P\cong{\rm P}^{1}$ maps to the nodal curve $Q_{*}\subset\Sigma$.
The lines $E_{*},F_{*},G_{*}$ meet at the point $a$. Blowing down further to
the plane we write the curves as $E_{**}$ etc.

${G_{**}}$${f}$${x}$${E_{**}}$${e}_{1}$${e}_{2}$${e}_{3}$${e}_{4}$ Figure 2:
the planar model

Consider the pencil of planes in ${\rm P}^{3}$ containing the line
$E_{*}\subset\Sigma$. This is the linear system $-K_{\Sigma}-E_{*}$, and

$Q_{*}(-K_{\Sigma}-E_{*})=(-2K)(-K+C-E)=4.$

A plane through $E_{*}$ meets the line $X_{*}$ in a point so $X_{*}$
parametrizes the pencil and $Q_{*}$ is a 4-fold cover of $X_{*}$.

In Figure 2, the lines through $x$ are the points of $X_{*}$, but also express
the conic $E_{**}$ as a double cover of $X_{*}$ branched over $f$ and the
point of contact of the other tangent from $x$. Let $t$ be an affine parameter
on $E_{*}$ where $t=0$ is the point $f$ and the covering involution is
$t\mapsto-t$. Then $t^{2}$ is an affine parameter on $X^{*}$ and $t^{2}=0$ is
the line $G_{**}$. The plane in the pencil defined by $t^{2}=0$ meets $Q_{*}$
in the image of the zeros of a quartic polynomial $z(u)$ on ${\rm P}^{1}$ and
then, if $t=a_{i}$ is the point $e_{i}$ on the conic $E_{**}$, $t=-a_{i}$ is
the other intersection of the line joining $x$ to $e_{i}$. The rational
function (3) on the elliptic curve $t^{2}=z(u)$ is then

$f(t)=\prod_{i=1}^{4}\frac{(t-a_{i})}{(t+a_{i})}.$

##  6 The central sphere

We take a small diversion here to see a rather different interpretation of the
surface $S=\bar{Z}_{1}$. The element $-1\in S^{1}$ defines a holomorphic
involution on the twistor space and composing with $\sigma$, a new real
structure $\tau$ which covers $u\mapsto 1/\bar{u}$ with fixed point set
$|u|=1$ – the reflective rather than antipodal real structure on ${\rm
P}^{1}$. So the fibre $S$ given by $u=1$ has a real structure. The holomorphic
curve $C_{0}\subset\bar{Z}_{0}$ is fixed by $-1$ and becomes a component of
the real points of $S$. From this point of view $S$ is the complexification of
this central 2-sphere and the paper [9] gave a description of the metric
induced from the hyperkähler metric on the ALE space.

The 2-sphere lies in an affine surface in $\mathbf{R}^{3}$ and inherits the
natural volume form. Its conformal structure is defined by two complex
conjugate foliations by null curves of the compexification. In [9] these are
shown to be given by two conjugate pencils of rational curves of zero self-
intersection. Moreover these curves are characterized by intersecting $C_{0}$
and $C$ in one point and having zero intersection with the cycles coming from
the other rational curves in the resolution. Given our derivation of $Q$ in
the previous section, it is clear that $Q=D+\tau(D)$ where $D$ is the divisor
class for one pencil.

  * Example: For ${\bf A_{1}}$, $S$ blows down to a quadric surface ${\rm P}^{1}\times{\rm P}^{1}$ and the null curves are the two families of lines $D\cong\mathcal{O}(1,0),\tau(D)\cong\mathcal{O}(0,1)$. The projected twistor lines are plane sections, in the divisor class $\mathcal{O}(1,1)$. The Eguchi-Hanson metric on the cotangent bundle of the 2-sphere is rotation-invariant and therefore a standard metric on the real quadric $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=1$ in $\mathbf{R}^{3}$.

  * Remark: An expression for $D$ is given in [9] in terms of the exceptional curves arising from blowing up the points $e_{i},f,g,x,y$ on ${\rm P}^{2}$ but there is an unfortunate choice of notation with $\bar{E}_{i}$ denoting the $-1$ curve coming from the line in the plane joining $x$ to $e_{i}$. If $E_{i}$ is the blow up of the point $e_{i}$ then in the ${\bf D_{k}}$ case $\bar{E}_{i}$ is indeed its conjugate but this does not necessarily hold for the ${\bf E_{k}}$ series.

One way to determine the action of the real structure $\tau$ on the cohomology
of $S$ is to note that, because it changes orientation on the holomorphic
2-spheres in the resolution, the action is $-1$ on the cohomology of the
affine part $Z_{1}$. The circle action acts trivially on the $E,F,G$
configuration at infinity and so $\tau$ coincides with the antipodal real
form. Then the total class $E+F+G$ in $H^{2}(S,\mathbf{Z})$ is acted on by
$+1$ but in the case of ${\bf E_{6}}$ consideration of the real structure
$(z_{1},z_{2})\mapsto(\bar{z}_{1},-\bar{z}_{1})$ acting on the invariant
polynomials

$z_{1}^{4}+2i\sqrt{3}z_{1}^{2}z_{2}^{2}+z_{2}^{4},\quad
z_{1}^{4}-2i\sqrt{3}z_{1}^{2}z_{2}^{2}+z_{2}^{4},\quad
z_{1}z_{2}(z_{1}^{4}-z_{2}^{4})$

shows that $\tau$ interchanges the cohomology classes $E$ and $G$.

##  7 Einstein-Weyl manifolds

We can now place the theorem we used of Honda and Nakata in the context they
were studying, which is to regard the complex surface $S$ as a minitwistor
space for a 3-dimensional Einstein-Weyl manifold $M$. This is a manifold $M$
with a conformal structure and a torsion-free connection preserving it such
that the symmetrized Ricci tensor $R_{(i,j)}$ is a scalar. The skew-symmetric
component $R_{[i,j]}$ is a 2-form which is essentially the curvature of the
line bundle of volume forms. This was the original idea in 4 dimensions of
incorporating the electromagnetic field into general relativity. In practical
terms one can choose a metric $g$ in the conformal equivalence class and then
the Weyl connection satisfies $\nabla g=2\omega\otimes g$ for a 1-form
$\omega$.

The differential-geometric link with the ALE spaces is the fact that the
quotient of a hyperkähler 4-manifold by an isometric circle action is an
Einstein-Weyl 3-manifold. In local coordinates a representative metric can [2]
be put in the form

$g=e^{u}(dx^{2}+dy^{2})+dt^{2},\qquad\omega=-u_{t}dt$ (5)

where $u$ satisfies the $SU(\infty)$ Toda equation
$u_{xx}+u_{yy}+(e^{u})_{tt}=0$.

We can describe this more globally, following [8]. The circle action on the
ALE space preserves the Kähler form $\omega_{1}$ for this complex structure.
Then $t$ in the equation is, up to a constant, the moment map for this action.
The quotient by the circle action of a level set $t=c$ is a Kähler quotient,
with an induced holomorphic structure and $u_{t}e^{u}(dx^{2}+dy^{2})$ at $t=c$
is this metric. The time-dependent metric $h=e^{u}(dx^{2}+dy^{2})$ on this
surface satisfies the second order equation $h_{tt}=\kappa h$ where $\kappa$
is the Gaussian curvature.

On the hyperkähler manifold the circle action generates a vector field $X$ and
the vector field $IX$ is horizontal and is the gradient of the moment map $t$.
Thus the $dt^{2}$ term in formula (5) shows that this choice of metric in the
conformal class is determined by rescaling the quotient metric with the norm
square of $X$ to make $dt$ have length $1$.

The _twistor theory_ for Einstein-Weyl 3-manifolds was established in [7] and
based on the minitwistor quotient of a twistor space: if there is a good
quotient of the complex 3-manifold $Z$ by a $\mathbf{C}$-action, then the
normal bundle of a twistor line is an extension
$\mathcal{O}\rightarrow\mathcal{O}(1)\oplus\mathcal{O}(1)\rightarrow\mathcal{O}(2)$
where the trivial subbundle is tangent to an orbit of the action. Then the
quotient is a surface $S$ and the twistor lines project to rational curves of
self-intersection $2$. The two models for this are $S=\mathcal{O}(2)$ the
tangent bundle of ${\rm P}^{1}$, and $S={\rm P}^{1}\times{\rm P}^{1}$ with the
corresponding Einstein-Weyl structures being $\mathbf{R}^{3}$ with the flat
metric or hyperbolic metric respectively. Furthermore, the only compact
complex surfaces which admit such curves give locally the same geometry.

This is where the work of [10] comes to the rescue, considering families of
nodal curves. In that paper a local tubular neighbourhood of a rational nodal
curve is constructed which can be “normalized” to the neighbourhood of a
smooth curve of self-intersection $2$. Our surface $S$ is a global substitute
for the quotient of the ALE twistor space – locally around a twistor line we
can find a quotient, and this will constitute an example of this local
normalization. The authors of [10], [11] apply this to introduce many examples
of Einstein-Weyl manifolds from the geometry of compact rational surfaces. In
fact, as the first author of [10] has pointed out, Section 5.2 of that paper
contains a family which includes our ${\bf A_{2\ell-1}}$ case. Note that there
are now two real structures $\sigma$, $\tau$ on the surface $S$. The antipodal
structure $\sigma$ gives an Einstein-Weyl conformal structure which is
positive-definite whereas $\tau$, whose fixed points give the real surface
$C_{0}\subset S$ for the ALE metric, gives the quotient Lorentzian signature.

In the case of ${\bf A_{2\ell-1}}$ the surface $S$ and its nodal curves have
already been described here in Section 5.1. There is another action – a
triholomorphic action – of the circle on the 4-manifold, and the quotient
Einstein-Weyl structure is here just the Levi-Civita connection of the flat
metric on $\mathbf{R}^{3}$. This is essentially the Gibbons-Hawking Ansatz
[5]. We can see this from the twistor point of view: the
$\mathbf{C}^{*}$-action $(x,y,z)\mapsto(\lambda x,\lambda^{-1}y,z)$ on
$\mathcal{O}(k+1)\oplus\mathcal{O}(k+1)\oplus\mathcal{O}(2)$ preserves $z$,
and twistor lines project to sections $z(u)$ of $\mathcal{O}(2)\rightarrow{\rm
P}^{1}$, the minitwistor space for $\mathbf{R}^{3}$.

For the action we are considering in this paper an explicit formula for the
${\bf A_{k}}$ Einstein-Weyl structure can be found in [2], based on earlier
work of Ward [14]. For $k=1$, we already observed that $S$ is a quadric and so
the Einstein-Weyl quotient is hyperbolic 3-space, but it is useful to see this
concretely. One standard form of this ${\bf A_{1}}$ metric is the following,
using left-invariant 1-forms $\sigma_{i}$ on $SU(2)$:

$g=\frac{dr^{2}}{(1-(1/r)^{4})}+\frac{r^{2}}{4}(\sigma_{1}^{2}+\sigma_{2}^{2})+\frac{r^{2}}{4}(1-(1/r)^{4})\sigma_{3}^{2}$

which clearly shows that as $r\rightarrow\infty$ it becomes asymptotically a
quotient of Euclidean space and as $r\rightarrow 1$ we get the 2-sphere
$C_{0}$ with metric $(\sigma_{1}^{2}+\sigma_{2}^{2})/4$. The vector field $X$
given by the circle action is dual to $\sigma_{3}$.

If we take the Kähler form to be
$2rdr\wedge\sigma_{3}+{r^{2}}\sigma_{1}\wedge\sigma_{2}$ then the moment map
is $t=r^{2}$ and $(X,X)=(t-t^{-1})/4$. This gives the data for the Toda form
for the metric (5) and one finds $\omega=-d\log(t^{2}-1)$ and so $d\omega=0$.
This means the volume bundle for the conformal structure is flat and we can
rescale the metric by a power of $(t^{2}-1)$ to obtain a Levi-Civita
connection for the hyperbolic 3-space:

$\frac{\rho^{2}}{(1-\rho^{4})^{2}}\left(d\rho^{2}+\frac{1}{4}\rho^{2}(\sigma_{1}^{2}+\sigma_{2}^{2})\right).$

Here $2r^{2}=\rho^{2}+\rho^{-2}$ and so $\rho\rightarrow 1$ corresponds to
$r\rightarrow 1$ – the central 2-sphere becomes the 2-sphere at infinity in
hyperbolic space. As $r\rightarrow\infty$, $\rho\rightarrow\infty$ and
infinity in the ALE space defines a distinguished origin in the hyperbolic
space.

These two features hold in the general ALE case: the Einstein-Weyl quotient
has an asymptotic 2-sphere and an orbifold central point modelled on
$\mathbf{R}^{3}/\Gamma$ where $\Gamma$ is acting via its image $\Gamma/\pm 1$
in $SO(3)$. There will be other orbifold singularities pertaining to isolated
fixed points of the circle action. This is not the place to go into further
details but it allows for the opportunity to give a geometrical meaning to the
appearance of nodes in the twistor lines.

In the original situation of smooth rational curves of self-intersection $2$,
the space $S$ is interpreted as the space of oriented geodesics for the Weyl
connection, the antipodal real structure being the change of orientation. Then
two real twistor lines intersect in two points because two points in the
3-manifold have a unique geodesic connecting them (with two orientations). In
our case the self-intersection $Q^{2}=|\Gamma|$ so we expect $|\Gamma|/2$
geodesics between them. In a neighbourhood of the orbifold point this is
clear: take a fundamental domain in the $\Gamma/\pm 1$ \- covering and two
points $p,q$. Then we shall find geodesics joining $p$ to each of the
$\Gamma$-translates of $q$.

The same principal applies to a single point $p$. In a local normalized
neighbourhood the smooth rational curve represents the geodesics corresponding
to the tangent directions at $p$ along each of which passes a geodesic. But,
as with the geodesics on a cone, globally we obtain geodesics which return
with a different direction, giving an identification of pairs of points on the
curve, and hence a node on its image in the space of geodesics. We lose one of
the geodesics when we had two points to give $|\Gamma|/2-1$ singularities.

## References

  * [1] M.F.Atiyah, Green’s functions for self-dual four-manifolds, Adv. Math. Suppl. Stud. 7 (1981) 129 –158.
  * [2] D.Calderbank, The geometry of the Toda equation, J. Geometry and Physics 36 (2000) 152–162.
  * [3] S.Cherkis & N.Hitchin, Gravitational instantons of type $D_{k}$, Comm.Math.Phys. 260 (2005) 299–317.
  * [4] M.Demazure, H.Pinkham & B.Teissier (eds.), “Séminaire sur les singularités des surfaces”, Lecture Notes in Mathematics 777, Springer Verlag Heidelberg (1980).
  * [5] G.Gibbons & S.Hawking, Gravitational multi-instantons, Phys.Lett. B 78 (1978) 430–432.
  * [6] N.J.Hitchin, Polygons and gravitons, Math.Proc.Camb.Phil.Soc. 85 (1979) 465–476.
  * [7] N.J.Hitchin, Complex manifolds and Einstein’s equations, in “Twistor geometry and nonlinear systems” , H.Doebner et al (Eds), Lecture Notes in Mathematics 970, 73-99, Springer, Berlin (1982).
  * [8] N.J.Hitchin, Higgs bundles and diffeomorphism groups, in “Surveys in Differential Geometry Vol 21” H-D Cao and S-T Yau, (eds.), International Press, Cambridge, Mass. (2016), 139–163
  * [9] N.J.Hitchin, The central sphere of an ALE space, Quarterly Journal of Mathematics 72 (2021) 277–337.
  * [10] N.Honda and F.Nakata, Minitwistor spaces, Severi varieties, and Einstein–Weyl structure, Ann. Glob. Anal. Geom. 39 (2011) 293–323.
  * [11] N.Honda, Segre quartic surfaces and minitwistor spaces, New York J. Math. 28 (2022) 672–704.
  * [12] P.B.Kronheimer, The construction of ALE spaces as hyperkähler quotients, J.Differential Geometry 29 (1989) 665–683.
  * [13] P.B.Kronheimer, A Torelli-type theorem for gravitational instantons, J.Differential Geom. 29 (1989) 685–697.
  * [14] R.S.Ward, Einstein-Weyl spaces and $SU(\infty)$ Toda fields, Class. Quantum Grav. 7 (1990) 95–98.

Mathematical Institute, Woodstock Road, Oxford OX2 6GG, UK

<EMAIL_ADDRESS>


# Quench Detection in a Superconducting Radio Frequency Cavity with Combined
Temperature and Magnetic Field Mapping††thanks: The work is partially
supported by the U.S.Department of Energy, Office of Science, Office of High
Energy Physics under Awards No. DE-SC 0009960, NSF Grant 100614-010, and by
the U.S. Department of Energy, Office of Science, Office of Nuclear Physics
under contract DE-AC05-06OR23177.

B. D. Khanal1<EMAIL_ADDRESS>P. Dhakal2 and G. Ciovati1,2
1Center for Accelerator Science  Department of Physics
Old Dominion University  Norfolk  VA 23529  USA
2Thomas Jefferson National Accelerator Facility  Newport News  VA 23606  USA

###### Abstract

Local dissipation of RF power in superconducting radio frequency cavities
create so called "hot-spots", primary precursors of cavity quench driven by
either thermal or magnetic instability. These hot-spots are detected by a
temperature mapping system, and a large increase in temperature on the outer
surface is detected during cavity quench events. Here, we have used combined
magnetic and temperature mapping systems using anisotropic magneto-resistance
(AMR) sensors and carbon resisters to locate the hot spots and areas with high
trapped flux on a 3.0 GHz single-cell Nb cavity during the RF tests at 2.0 K.
The quench location and hot spots were detected near the equator when the
residual magnetic field in the Dewar is kept < 1 mG. The hot spots and quench
locations moved when the magnetic field is trapped locally, as detected by
T-mapping system. No significant dynamics of trapped flux is detected by AMR
sensors, however change in magnetic flux during cavity quench is detected by a
flux gate magnetometer, close to the quench location. The result provide the
direct evidence of hot spots and quench events due to localized trapped
vortices.

##  1 Introduction

Superconducting radio frequency (SRF) cavities made from elemental niobium are
the building blocks of modern particle accelerators, superconducting
electronics and quantum computers because of their formability in complex
structures and lithographic thin films. Recent advances in surface engineering
by doping and annealing led to an unprecedented quality factor [1, 2, 3],
however the process is vulnerable to residual flux trapping when the cavity
transitions to superconducting state during cooldown [4]. The trapped vortices
within the RF penetration depth oscillates under the RF field and lowers the
quality factor with additional RF dissipation.

The temperature mapping technique is able to detect the regions of large RF
power dissipation, referred as "hot-spots" [5]. The source of hot-spots could
be due to the normal conducting inclusion or segregation of impurities in
dislocations sites or grain boundaries. If the origin of hot spot is due to
trapped vortices, those could be moved or change their intensity an
application of a thermal gradient [6, 7]. The combined temperature and
magnetic field mapping system used in 1.3 GHz cavity showed an enhancement of
magnetic field trapping due to quench events either by trapping of thermo
current induced flux or trapping to additional residual flux [8, 9] . Here, we
present the results of RF measurements done on a 3.0 GHz single cell cavity
with combined temperature and magnetic field mapping where the hot-spots are
created on the cavity surface by trapping magnetic field locally.

##  2 Experimental setup

The 3 GHz single cell cavity used in this study was made from high purity
large grain Nb and scaled down from the TESLA end-cell cavity shape [10, 11].
After previous measurements reported in Ref.[12] the cavity was subjected to
$\sim$ 25 $\mu$m electropolishing followed by high pressure rinse with
deionized water. The cavity was assembled with input and pick up antennas. A
cavity vacuum of < 10-8 mbar was maintained with active pumping during the
cooldown and RF tests.

Figure 1: (a) and (b) AMR sensors and (c) temperature sensors board, (d)
assembled on the surface of cavity [13].

The combined temperature and magnetic field mapping system relies on 100
$\Omega$ Allen-Bradely carbon resistors to measure the temperature and AMR
sensors to measure the magnetic field. The details of set up, sensors
calibration, sensitivity and measurement can be found in Refs. [14, 15, 13].
The AMR sensors boards and temperature sensors boards are place alternatively
on every 22.5∘ around the cavity. There are 10 AMR sensors on each board
measure the radial and tangential component of the magnetic field in the
vicinity of cavity’s outer surface. 11 temperature sensors are mounted on each
board which are kept in contact with the cavity surface with spring-loaded
Pogo sticks. Apiezon-N cryogenic vacuum grease was applied on temperature
sensors to make good thermal contact with the cavity surface. Magnetic sensor
#3 and temperature sensors # 6 lies on the equator of the cavity. Three flux
gate magnetometers (FGM) and 4 Cernox sensors were attached on the cavity
surface to measure the residual magnetic field and temperature of outer cavity
surface and helium bath. One of FGM was place away from the cavity surface and
parallel to the cavity axis to measure the residual magnetic field on the
Dewar. Two FGM were placed along the axis of small coils and perpendicular to
the cavity axis as shown in Fig. 1. The Cernox sensor were place on the two
beam tube, closer to the cavity irises. The carbon resistors were calibrated
against the Cernox sensors for each measurements.

The cryogenic Dewar is equipped with compensation coils capable of maintaining
the residual magnetic field on cavity under test < 1 mG. In our current set
up, two small coils of radius $\sim$ 10 mm with 10 turns made from 38 AWG
enameled copper wire were mounted on the outer surface of the cavity in
equatorial region. The coils were mounted in such a way that temperature and
magnetic field sensors are placed on the axis of the coil. The magnetic field
produced by the coil will be trapped locally when the cavity undergoes
superconducting transitions.

The measurement procedure is as follow: (i) The magnetic field in Dewar was
set below 1 mG and offset voltage of AMR sensors was recorded at 10 K. (ii)
The cavity was cooled to 4.2 K and filled with liquid helium by maintaining
temperature difference of > 2 K between the cavity irises. (iii) $Q_{0}$(T) at
low rf field (peak surface rf magnetic field $B_{p}\sim$ 15 mT) from 4.2-2.0 K
was measured while calibrating the carbon resistors against the calibrated
Cernox sensors. (iv) $Q_{0}$ vs $E_{acc}$ at 2.0 K was measured along with the
temperature and magnetic field maps. (v) The cavity is warmed-up above $T_{c}$
($\sim$ 9.2 K) and 200 mA current on the coil that was mounted on the cavity
surface, producing $\sim 100$ $\mu$T magnetic field perpendicular to the
cavity surface as measured by FGM. (vi) The cavity was cooled down with less
than 0.1 K temperature difference between the cavity irises in order to
maximize the flux trapping. When the cavity reached superconducting state
($\sim$ 4.2 K), the current to the coiled is switched off. The magnetic field
reading after the current being turned off is the amount of magnetic field
trapped on the cavity surface. It was found that about 40 % of the magnetic
flux is trapped on the cavity surface. (vii) Steps (iii) and (iv) were
repeated by placing the coil at different locations on cavity surface.

##  3 Experimental Results

###  3.1 rf results

Figure 2 shows the $Q_{0}(B_{p})$ measured for 4 different flux trapping
condition. The test labeled test 1 is after the first cooldown with < 1 mG
residual flux during the cooldown. The low field $Q_{0}$ is $\sim$ 4$\times
10^{9}$ and increases with the increase in RF field. The Q-rise is similar to
the doped niobium cavities [3], however similar results were also reported in
electropolished cavities resonating at higher frequencies [16]. The cavity
showed high field Q-slope starting at $B_{p}\sim$ 110 mT before quenching at
$B_{p}\sim$ 139 mT.

Figure 2: $Q_{0}(B_{p})$ measured at 2 K for different condition of trapped
magnetic field. The experimental uncertainties on $Q_{0}$ and $B_{p}$ are
$<10\%$, and $<5\%$ respectively. All rf tests were limited by quench.

The second RF test, test 2, was done with residual flux being trapped on two
different locations. Location 1 corresponds to the 0∘ (reference point for B-T
mapping system), whereas location 2 corresponds to 202.5∘. The quality factor
was lowered by $\sim$25 % and $Q_{0}(B_{p})$ doesn’t show any Q-rise
phenomenon. The cavity encounter a sudden quench at $B_{p}\sim$ 116 mT and
quality factor jumped to higher value and cavity finally quenched at
$B_{p}\sim$ 127 mT. The measurement was repeated by moving the coil to
location 3 (67. 5∘) and location 4 (202.5∘). In both cases, the cavity
quenched at $B_{p}\sim$ 135 mT with hysteresis in $Q_{0}(B_{p})$, due to the
release of trapped flux during quench. The flux gate magnetometer reads a drop
in trapped flux by $\sim$ 50 % after the quench. The drop in flux is due to
redistribution of vortices at quench location.

Figure 3: Change in temperature on cavity surface during RF test at 2.0 K
measured by the temperature sensors on hot-spots for (a) test 1, (b) test 2,
(c) test 3 and (d) test 4. The arrow in (c) represents the temperature during
RF power increasing (solid symbols) and decreasing (open symbols). After the
quench the hot-spot strength reduces and hysteresis in $\Delta T$ vs $B_{p}$
is observed.

###  3.2 B and T mapping

The first RF test was done with <1 mG residual magnetic field in the Dewar,
measured by flux gate magnetometer and also verified by AMR sensors at 10 K.
When the Dewar is filled with liquid helium, a small change in residual field
was measured by flux gate and AMR sensors. This change in field doesn’t affect
the RF performance, since the cavity is already in Meissner state. During the
RF test #1, as shown in Fig. 2, the temperature sensors didn’t show any
significant change as RF field is increased up to $B_{p}\sim$ 120 mT. On
further increase in RF field, the temperature on sensors # 5, 6, and 7 at
157.5∘ showed sudden increase in temperature and the cavity quenched at
$B_{p}\sim$ 139 mT as shown in Fig.3 (a). The quench location was identified
to be at 157.5∘, sensor #6 as shown in Fig. 4(a). The precursor heating was
also measured by the adjacent sensors # 5 and #7 as shown in Fig. 3(a). The
sensor #6 is located at the equator of the cavity whereas, senors #5 and #7
are located at $\sim$ 5 mm away from the equator weld. The sudden increase in
temperature $B_{p}\sim 120\leavevmode\nobreak\ mT$ on hot-spot locations
correspond to the onset of Q-slope in $Q_{0}(B_{p})$ during rf test #1 as
shown in Fig. 2.

During test #2, 200 mA current was applied to the coil encircling AMR sensors
#3 at 0∘ and temperature sensors # 5, 6, and 7 at 202.5∘. The total magnetic
field $\sim$ 300 mG was measured during the RF test, which corresponds to the
trapped magnetic field on AMR sensor #3 at location 0∘. On increasing RF power
during test #2, no significant change in magnetic field was detected by the
AMR sensors at 0∘, however, temperature sensors at 202.5∘ detected an increase
in temperature starting $B_{p}\sim$ 50 mT. On further increase in RF field,
the temperature increased until the cavity quench first at $B_{p}\sim$ 117 mT
and repetitive quench at $B_{p}\sim$ 127 mT. Based on the T-mapping data, the
quench location is identified to be sensor #6 at 202.5∘ as shown in Fig. 4,
shifted from sensor #6 at 157.5∘ as indicated by arrow. It is unknown that if
there is any quench events occurred at 0∘ due to no change in magnetic field
measured by AMR sensors. This result showed the clear demonstration of hot-
spot and cavity quench due to locally trapped magnetic field.

To further investigate the effect of local trapped flux on the hot-spots and
quench, we placed the coils at azimuthal angles 67.5∘ and 292.5∘ encircling
temperature sensors # 5, 6, and 7. Again, the rise in temperature was measured
at the particular location corresponding to the magnetic field trapped as
shown in Fig. 3. Interestingly, when the cavity quenched, some trapped flux is
released or migrated away from the sensors, but not far enough to be detected
by AMR sensors on boards neighboring the T-sensors board.

Figure 4: Unfolded temperature and magnetic field map during the cavity quench
for (a) test #1, and (b) test #2.

##  4 Discussion and Summary

Figure 5 shows the normalized surface resistance as a function of the peak RF
field. The surface resistance decrease with the increase in RF field during
test 1, typically observed in impurity doped cavities, even though no doping
treatment was applied to the cavity. However with the trapped flux, the
surface resistance increases with the increase in RF field. The field at which
the Q-slope starts corresponds to the onset of temperature increase as shown
in Figs. 2 and 3(a), which demonstrate that the medium field Q-slope is a
result of addition RF loss due to trapped vortices. The RF loss in hot-spots
can be characterized as $\Delta T\sim B_{p}^{n}$. The $\Delta T$ vs. $B_{p}$
was fitted with least square fits. The exponent is found to be n > 4 in the
case of local trapped flux suggesting the RF loss due to vortices is stronger
than the ohmic-type (n =2) loss.

Figure 5: Normalized surface resistance as a function of peak RF magnetic
field.

The combined temperature and magnetic field mapping system allows us to
identify the hot-spots and quench location on 3.0 GHz cavity. Even though AMR
sensors didn’t show any significant change in magnetic flux distribution, the
flux gate magnetometer mounted near the trapped flux location showed the re-
distribution or release of vortices as a result of cavity quench. After the
cavity quenched first time at the trapped flux location, the strength of the
hot-spots decreases as seen from the drop in temperature of sensors as shown
in Fig. 3 (c). Further studies are planned to investigate the vortex induced
hot-spots and quench locations in 1.3 GHz and 3.0 GHz cavities subjected to
different surface preparations.

##  5 ACKNOWLEDGEMENTS

We would like to acknowledge Peter Owen for RF and Justin Kent for cryogenic
support as well as technical staff members for the cavity fabrication,
processing and assembly.

## References

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