M5 branes on $\mathbb{R}^{1,1}\times$Taub-NUT
Andreas Gustavsson

Physics Department, University of Seoul, Seoul 02504 KOREA

<EMAIL_ADDRESS>

###### Abstract

##  1 Introduction

Bosonization is the equivalence between a 2d theory of a Dirac fermion and a
theory of a scalar field whose non-Abelian generalization is a WZW theory [9].
The partition function factorizes, which can be better understood by coupling
the chiral part to a background gauge field, resulting in a chirally gauged
WZW theory [8]. An analogous situation happens in 6d for the selfdual three-
fom of the M5 brane. If we compute the partition function of a nonchiral
three-form then it factorizes. This factorization can be better understood if
one couples the chiral part to a background $C$-field that is a three-form
gauge potential in 11d supergravity [11]. A somewhat different kind of
chirally gauged WZW theory appears as the boundary theory of M2 branes [15].

If we put the M5 brane on $\mathbb{R}^{1,1}\times$Taub-NUT and view the Taub-
NUT space as a circle fibration over $\mathbb{R}^{3}$, then the radius of that
circle vanishes at the origin, where we have the submanifold
$\mathbb{R}^{1,1}$. This has a brane interpretation of a D4 brane intersecting
with a D6 brane. Near the intersection there are open strings stretching
between the two branes. More generally, on the intersection brane, which is
$\mathbb{R}^{1,1}$, we have chiral fermions in the bifundamental of
$U(N)\times U(Q)$ where in the brane picture there are $Q$ coincident D6
intersecting with $N$ coincident D4 branes. These chiral fermions have a gauge
anomaly that cancels the corresponding gauge anomaly of the 5d SYM that lives
on the D4 brane [10]. By bosonizing we get a chirally gauged WZW on
$\mathbb{R}^{1,1}$ and again that theory has a gauge anomaly that cancels the
corresponding gauge anomaly of the D4 brane [5]. The gauge anomaly of the D4
brane comes from a gravi-photon term in eq (LABEL:gravi). In this paper we
will extend this analyzis to the supersymmetric case. We find that the
combined system of 5d SYM on the D4 branes and supersymmetric chirally gauged
supersymmetric WZW theories plus additional mass terms for the five scalar
fields on $\mathbb{R}^{1,1}$ results in a supersymmetric and gauge invariant
theory.

##  2 The 5d super Yang-Mills

We put the Abelian M5 brane on a circle bundle with the metric

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle
G_{\mu\nu}dx^{\mu}dx^{\nu}+r^{2}\left(d\psi+\kappa_{\mu}dx^{\mu}\right)^{2}$

Here $x^{\mu}$ are coordinates on the five-dimensional base manifold,
$\psi\sim\psi+2\pi$ is the fiber coordinate parametrizing the circle fiber
with circumference $2\pi r$. The radius $r$ can depend on the base-manifold
coordinates. The gravi-photon is denoted $\kappa_{\mu}$ and its curvature is
denoted $w_{\mu\nu}=\partial_{\mu}\kappa_{\nu}-\partial_{\nu}\kappa_{\mu}$. We
perform dimensional reduction along the circle fiber. That results in an
Abelian 5d SYM. This has a non-Abelian generalization whose action is given by
[3]

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int
d^{5}x\sqrt{-G}\frac{1}{4\pi^{2}r}{\cal{L}}$

where the Lagrangian density is

$\displaystyle{\cal{L}}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}F_{\mu\nu}^{2}+\frac{r}{4}{\cal{\varepsilon}}^{\mu\nu\lambda\kappa\tau}\omega(A)_{\mu\nu\lambda}w_{\kappa\tau}$
$\displaystyle-\frac{1}{2}(D_{\mu}\phi^{A})^{2}-\frac{m^{2}}{2}(\phi^{A})^{2}$
$\displaystyle+\frac{i}{2}\bar{\psi}\Gamma^{\mu}D_{\mu}\psi-\frac{i}{4r}\bar{\psi}\Gamma^{\mu}\psi\partial_{\mu}r+\frac{ir}{16}\bar{\psi}\Gamma^{\mu\nu}\Gamma^{\widehat{\psi}}\psi
w_{\mu\nu}$
$\displaystyle-\frac{1}{2}\bar{\psi}\Gamma^{A}\Gamma^{\widehat{\psi}}[\psi,\phi^{A}]+\frac{1}{4}[\phi^{A},\phi^{B}]^{2}$

where a trace over the Lie algebra generators is understood and not written
out explicitly for notational simplicity. The field content is a gauge field
$A_{\mu}$, five scalar fields $\phi^{A}$ and a fermionic field $\psi$. We use
here an 11d notation, where the gamma matrices are 11d. Likewise the fermionic
field is an 11d Majorana spinor that is reduced to 6d where it is Weyl
projected, and subsequently dimensionally reduced to 5d. The details are
summarized in the Appendix D. The mass squared is given by the following
rather complicated expression

$\displaystyle m^{2}$ $\displaystyle=$
$\displaystyle\frac{R}{5}-\frac{r^{2}}{20}w_{\mu\nu}^{2}+\frac{3}{5}\frac{\nabla^{2}r}{r}-\left(\frac{\nabla_{\mu}r}{r}\right)^{2}$

and

$\displaystyle\omega(A)_{\mu\nu\lambda}$ $\displaystyle=$ $\displaystyle
A_{\mu}\partial_{\nu}A_{\lambda}-\frac{2i}{3}A_{\mu}A_{\nu}A_{\lambda}$

is the Chern-Simons three-form, where it is understood that the indices
$\mu,\nu,\lambda$ shall be antisymmetrized. We notice that the graviphoton
term

$\displaystyle\int
d^{5}x\sqrt{-G}\frac{1}{4\pi^{2}r}\frac{r}{4}{\cal{\varepsilon}}^{\mu\nu\lambda\kappa\tau}\omega(A)_{\mu\nu\lambda}w_{\kappa\tau}$
$\displaystyle=$ $\displaystyle\frac{1}{16\pi^{2}}\int dx^{\mu}\wedge
dx^{\nu}\wedge dx^{\lambda}\wedge dx^{\kappa}\wedge
dx^{\tau}\omega(A)_{\mu\nu\lambda}w_{\kappa\tau}$ (2.1)

makes no reference to the 5d metric. The supersymmetry variations are

$\displaystyle\delta\phi^{A}$ $\displaystyle=$ $\displaystyle
i\bar{\cal{\varepsilon}}\Gamma^{A}\psi$ $\displaystyle\delta A_{\mu}$
$\displaystyle=$ $\displaystyle
i\bar{\cal{\varepsilon}}\Gamma_{\mu}\Gamma^{\widehat{\psi}}\psi$
$\displaystyle\delta\psi$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Gamma^{\mu\nu}\Gamma^{\widehat{\psi}}{\cal{\varepsilon}}F_{\mu\nu}$
$\displaystyle+\Gamma^{\mu}\Gamma^{A}{\cal{\varepsilon}}\left(D_{\mu}\phi^{A}+\frac{\partial_{\mu}r}{r}\phi^{A}\right)$
$\displaystyle+\frac{r}{2}\Gamma^{A}\Gamma^{\mu\nu}\Gamma^{\widehat{\psi}}{\cal{\varepsilon}}w_{\mu\nu}\phi^{A}$
$\displaystyle-\frac{i}{2}\Gamma^{AB}\Gamma^{\widehat{\psi}}{\cal{\varepsilon}}[\phi^{A},\phi^{B}]$

where the supersymmetry parameter satisfies the following Killing spinor
equation

$\displaystyle\nabla_{\mu}{\cal{\varepsilon}}$ $\displaystyle=$ $\displaystyle
M_{\mu}{\cal{\varepsilon}}$ (2.3) $\displaystyle M_{\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{2r}\Gamma_{\mu}\Gamma^{\nu}\partial_{\nu}r-\frac{r}{8}\Gamma_{\mu}\Gamma^{\rho\sigma}\Gamma^{\widehat{\psi}}w_{\rho\sigma}-\frac{r}{4}w_{\mu\nu}\Gamma^{\nu}\Gamma^{\widehat{\psi}}$
(2.4)

We may introduce a Weyl covariant derivative

$\displaystyle{\cal{D}}_{\mu}\phi$ $\displaystyle=$ $\displaystyle
D_{\mu}\phi^{A}+\frac{\partial_{\mu}r}{r}\phi^{A}$
$\displaystyle{\cal{D}}_{\mu}\psi$ $\displaystyle=$ $\displaystyle
D_{\mu}\psi+\frac{3}{2}\psi\frac{\partial_{\mu}r}{r}-\frac{1}{2}\Gamma_{\mu}{}^{\nu}\psi\frac{\partial_{\nu}r}{r}$

In terms of this derivative the Lagrangian becomes

$\displaystyle{\cal{L}}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}F_{\mu\nu}^{2}+\frac{r}{4}{\cal{\varepsilon}}^{\mu\nu\lambda\kappa\tau}\omega(A)_{\mu\nu\lambda}w_{\kappa\tau}$
$\displaystyle-\frac{1}{2}({\cal{D}}_{\mu}\phi^{A})^{2}+\frac{i}{2}\bar{\psi}\Gamma^{\mu}{\cal{D}}_{\mu}\psi$
$\displaystyle+\frac{ir}{16}\bar{\psi}\Gamma^{\mu\nu}\Gamma^{\widehat{\psi}}\psi
w_{\mu\nu}$
$\displaystyle-\frac{1}{2}\bar{\psi}\Gamma^{A}\Gamma^{\widehat{\psi}}[\psi,\phi^{A}]+\frac{1}{4}[\phi^{A},\phi^{B}]^{2}$

We now notice that the complicated mass term for the scalars has got
completely absorbed into the Weyl covariant derivative. The Weyl
transformations act as

$\displaystyle G_{\mu\nu}$ $\displaystyle\rightarrow$ $\displaystyle
e^{2\Omega}G_{\mu\nu}$ $\displaystyle r$ $\displaystyle\rightarrow$
$\displaystyle e^{\Omega}r$ $\displaystyle\phi^{A}$ $\displaystyle\rightarrow$
$\displaystyle e^{-\Omega}\phi^{A}$ $\displaystyle\psi$
$\displaystyle\rightarrow$ $\displaystyle e^{-\frac{3}{2}\Omega}\psi$
$\displaystyle A_{\mu}$ $\displaystyle\rightarrow$ $\displaystyle A_{\mu}$

Under these transformations the Weyl covariant derivative transforms Weyl
covariantly as

$\displaystyle{\cal{D}}_{\mu}\phi^{A}$ $\displaystyle\rightarrow$
$\displaystyle e^{-\Omega}{\cal{D}}_{\mu}\phi^{A}$
$\displaystyle{\cal{D}}_{\mu}\psi$ $\displaystyle\rightarrow$ $\displaystyle
e^{-\frac{3}{2}\Omega}{\cal{D}}_{\mu}\psi$

Using this, it can be easily seen that the action is Weyl invariant. In
addition to this, one may also show that if we transform the supersymmetry
parameter as

$\displaystyle{\cal{\varepsilon}}$ $\displaystyle\rightarrow$ $\displaystyle
e^{\frac{1}{2}\Omega}{\cal{\varepsilon}}$

then the supersymmetry variations are also Weyl invariant.

##  3 Conditions on the geometry

The Killing spinor equation (2.4) is derived from the 6d conformal Killing
spinor equation of the M5 brane by imposing the condition
$\partial_{\psi}{\cal{\varepsilon}}=0$. Solutions to the 6d conformal Killing
spinor equation have been summarized in [2], but the condition
$\partial_{\psi}{\cal{\varepsilon}}=0$ was not considered there.

The most general condition on the geometry from the Killing spinor equation is
obtained by analyzing $[\nabla_{\mu},\nabla_{\nu}]{\cal{\varepsilon}}$. But
not only will this lead to fairly complicated computations, but also we will
not need this strong condition for our purposes here. For the purpose of
checking supersymmetry of the action, we will only need the weaker conditions
that arises from

$\displaystyle\Gamma^{\mu\nu}\nabla_{\mu}\nabla_{\nu}{\cal{\varepsilon}}$
$\displaystyle=$ $\displaystyle-\frac{R}{4}{\cal{\varepsilon}}$ (3.1)

This equation limis our search for possible geometries, but since (3.1) gives
a weaker condition on the geometry than the Killing spinor equation itself, we
will still need to show the existence of solutions to the Killing spinor
equation. From (3.1) we obtain the following conditions

$\displaystyle
20\left(\frac{\nabla_{\mu}r}{r}\right)^{2}-8\frac{\nabla^{2}r}{r}+\frac{r^{2}}{4}w_{\mu\nu}^{2}$
$\displaystyle=$ $\displaystyle R$ (3.2)
$\displaystyle\nabla^{\nu}\left(\frac{1}{r}w_{\nu\mu}\right)+\frac{1}{4}{\cal{E}}_{\mu}{}^{\nu\lambda\rho\sigma}w_{\nu\lambda}w_{\rho\sigma}$
$\displaystyle=$ $\displaystyle 0$ (3.3)

Now this geometry is best understood not as a five-manifold, but as the six-
manifold that is a circle bundle over the base five-manifold. We may express
the first integrability condition in terms of the curvature scalar of the six-
manifold if we note the relation

$\displaystyle R_{6d}$ $\displaystyle=$ $\displaystyle
R-\frac{r^{2}}{4}w_{\mu\nu}^{2}-2\frac{\nabla^{2}r}{r}$

that relates the curvature scalars $R_{6d}$ of the six-manifold with the
curvature scalar $R$ of the base five-dimensional base-manifold. By using this
relation, the integrability condition (3.2) can be expressed as

$\displaystyle R_{6d}$ $\displaystyle=$ $\displaystyle
20\left(\frac{\nabla_{\mu}r}{r}\right)^{2}-10\frac{\nabla^{2}r}{r}$ (3.4)

###  3.1 Conformally flat spacetimes

The constraint (3.4) is satisfied for a conformally flat metric, where in
order to preserve supersymmetry under dimensional reduction, we need to
restrict such a metric to be on the form

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle
r^{2}\eta_{\mu\nu}dx^{\mu}dx^{\nu}+r^{2}d\psi^{2}$

where $r=r(x^{\mu})$ does not depend on the fiber direction parametrized by
$\psi\sim\psi+2\pi$. To show this, we use the a standard formula for how the
curvature scalar transforms under a Weyl rescaling $G_{\mu\nu}\rightarrow
r^{2}G_{\mu\nu}$, [6]

$\displaystyle R$ $\displaystyle\rightarrow$
$\displaystyle\frac{1}{r^{2}}R-2(D-1)\frac{1}{r^{3}}G^{\mu\nu}\nabla_{\mu}\nabla_{\nu}r-(D-1)(D-4)\frac{1}{r^{4}}G^{\mu\nu}\nabla_{\mu}r\nabla_{\nu}r$

for a $D$-dimensional manifold. Here we take $D=6$ and
$G_{\mu\nu}=\eta_{\mu\nu}$. This formula then gives

$\displaystyle R$ $\displaystyle\rightarrow$
$\displaystyle-10\frac{1}{r^{3}}\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}r-10\frac{1}{r^{4}}\eta^{\mu\nu}\partial_{\mu}r\partial_{\nu}r$

In order to see that this corresponds to the constraint (3.4), we need to
express the expressions on the right-hand side that are to be computed with
respect to the metric $G_{\mu\nu}=r^{2}\eta_{\mu\nu}$ on the 5d base manifold,
in terms of the flat matric $\eta_{\mu\nu}$. We have

$\displaystyle(\nabla_{\mu}r)^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{r^{2}}\eta^{\mu\nu}\partial_{\mu}r\partial_{\nu}r$
$\displaystyle\nabla^{2}r$ $\displaystyle=$
$\displaystyle\frac{1}{r^{2}}\eta^{\mu\nu}\partial_{\mu}r\partial_{\nu}r+\frac{3}{r^{3}}\eta^{\mu\nu}\partial_{\mu}r\partial_{\nu}r$

Using these results we find the result

$\displaystyle R$ $\displaystyle\rightarrow$ $\displaystyle
20\left(\frac{\nabla_{\mu}r}{r}\right)^{2}-10\frac{\nabla^{2}r}{r}$

for the curvature scalar of the Weyl transformed metric, in agreement with the
integrability constraint (3.4).

This, however, is not sufficient to prove the existence of a Killing spinor,
since we only study a weaker version of the integrability condition. But it is
not difficult to construct Killing spinor solutions to (2.4) explicitly for
conformally flat metrics. The Killing spinor equation is

$\displaystyle\partial_{\mu}{\cal{\varepsilon}}$ $\displaystyle=$
$\displaystyle\frac{1}{2r}{\cal{\varepsilon}}\partial_{\mu}r$

It has the general solution

$\displaystyle{\cal{\varepsilon}}$ $\displaystyle=$ $\displaystyle\sqrt{r}\xi$

for a constant spinor $\xi$. This solution can also be obtained by starting
from the flat metric $ds^{2}=\eta_{\mu\nu}dx^{\mu}dx^{\nu}+d\psi^{2}$ and the
Killing spinor $\xi$ and then making a Weyl rescaling
$\xi\rightarrow\sqrt{r}\xi$.

###  3.2 $\mathbb{R}^{1,1}\times$ multi-Taub-NUT

Another class of six-manifolds that satisfy the constraints (3.3) and (3.4)
are of the form $\mathbb{R}^{1,1}\times X$ where we take $X$ to be a hyper-
Kahler manifold with the metric

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle Ud\vec{x}\cdot
d\vec{x}+\frac{1}{U}\left(d\psi+\vec{\kappa}\cdot d\vec{x}\right)^{2}$ (3.5)

Here $\vec{x}$ parametrizes $\mathbb{R}^{3}$ and $U$ is a function on
$\mathbb{R}^{3}$. Since $X$ is Ricci flat we have that $R_{6d}=0$ and (3.4)
reduces to

$\displaystyle\frac{2}{r^{2}}G^{\mu\nu}\nabla_{\mu}r\nabla_{\nu}r$
$\displaystyle=$ $\displaystyle\frac{1}{r}G^{\mu\nu}\nabla_{\mu}\nabla_{\nu}r$
(3.6)

where the radius is $r=\frac{1}{\sqrt{U}}$. The left-hand side of (3.6) is

$\displaystyle\frac{1}{2U^{3}}\vec{\nabla}U\cdot\vec{\nabla}U$

and the right-hand side is

$\displaystyle\frac{1}{2U^{3}}\vec{\nabla}U\cdot\vec{\nabla}U-\frac{1}{2U^{2}}{\nabla}^{2}U$

We see that (3.6) is satisfied if $U$ is harmonic everywhere on
$\mathbb{R}^{3}$

$\displaystyle{\nabla}^{2}U$ $\displaystyle=$ $\displaystyle 0$

except for points where $\frac{1}{U^{2}}$ vanishes. Let us now look at the
constraint (3.3). With the metric (3.5) this constraint becomes

$\displaystyle\partial_{i}(Uw_{ij})$ $\displaystyle=$ $\displaystyle 0$ (3.7)

in Cartesian coordinates on $\mathbb{R}^{3}$. In addition, it is necessary for
$w_{ij}$ to be closed everywhere outside the singular points. This leads to
the solution

$\displaystyle w_{ij}$ $\displaystyle=$
$\displaystyle\epsilon_{ijk}\partial_{k}U$

which is automatically closed

$\displaystyle\epsilon_{ijl}\partial_{l}w_{ij}=\partial_{k}\partial_{k}U=0$

outside singular points, and moreover it satisfies (3.7),

$\displaystyle\partial_{i}(Uw_{ij})=\epsilon_{ijk}\partial_{i}U\partial_{k}U+U\epsilon_{ijk}\partial_{i}\partial_{k}U=0$

A general harmonic function on $\mathbb{R}^{3}$ has the form

$\displaystyle U$ $\displaystyle=$
$\displaystyle\frac{1}{R^{2}}+\frac{1}{2}\sum_{i=1}^{N}\frac{1}{|\vec{x}-\vec{x}_{i}|}$

which leads to the multi Taub-NUT metric [1]. There are singularities at
$\vec{x}=\vec{x}_{i}$ for $i=1,...,N$. One may notice that such singularities
are fine since close to a singularity we have $\frac{1}{U^{2}}\nabla^{2}U\sim
r^{2}\nabla^{2}\frac{1}{r}=0$. Let us now take a closer look at a singularity,
starting with $N=1$ for which we have the Taub-NUT metric

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle
Udx^{i}dx^{i}+\frac{1}{U}\left(d\psi+\kappa_{i}dx^{i}\right)^{2}$

where

$\displaystyle U$ $\displaystyle=$
$\displaystyle\frac{1}{R^{2}}+\frac{1}{2|\vec{x}|}$

We can view the Taub-NUT space as a circle bundle over a base-manifold that is
a rescaled version of $\mathbb{R}^{3}$ with the metric $G_{ij}=U\delta_{ij}$.
If we use polar coordinates on $\mathbb{R}^{3}$ then the Taub-NUT metric is

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle
U\left(dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}\right)\right)+\frac{1}{U}\left(d\psi+\frac{1}{2}\cos\theta
d\varphi\right)^{2}$

and the radius of the circle fiber is $1/\sqrt{U}$ where

$\displaystyle U$ $\displaystyle=$ $\displaystyle\frac{1}{R^{2}}+\frac{1}{2r}$

The radius of the circle fibration (not to be confused with the radius $r$ of
$\mathbb{R}^{3}$) vanishes at $r=0$ so the circle fibration is singular. But
the manifold is nonetheless smooth at $r=0$. To see that, we may look at the
metric close to $r=0$. When $r<<R^{2}$ we may neglect the constant term in $U$
so the metric is

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2r}\left(dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}\right)\right)+2r\left(d\psi+\frac{1}{2}\cos\theta
d\varphi\right)^{2}$

The interpretation of this metric becomes clearer if we define

$\displaystyle r$ $\displaystyle=$ $\displaystyle\frac{\rho^{2}}{2}$

Then the metric becomes

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle
d\rho^{2}+\frac{\rho^{2}}{4}\left(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}\right)+\frac{\rho^{2}}{4}\left(2d\psi+\cos\theta
d\varphi\right)^{2}$ (3.8)

This is now the metric of flat $\mathbb{R}^{4}$. For the details of the
construction of this flat metric we refer to Appendix B. The important point
is that $\psi$ is $2\pi$ periodic.

If we look at the Taub-NUT metric in the other limit when $r>>R^{2}$ we may
approximate $U\approx\frac{1}{R^{2}}$ and the metric is

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{R^{2}}dx^{i}dx^{i}+R^{2}\left(d\psi+\kappa_{i}dx^{i}\right)^{2}$

which describes a cylinder of radius $R$. The Taub-NUT space thus interpolates
between flat $\mathbb{R}^{4}$ at the origin and a cylinder at infinity.

The graviphoton one-form is

$\displaystyle\kappa$ $\displaystyle=$ $\displaystyle\frac{1}{2}\cos\theta
d\varphi$

Its curvature two-form $w=d\kappa$ is

$\displaystyle w$ $\displaystyle=$ $\displaystyle-\frac{1}{2}\sin\theta
d\theta\wedge d\varphi$ (3.9)

that is integrated over $S^{2}$ to

$\displaystyle\oint_{S^{2}}w$ $\displaystyle=$ $\displaystyle-2\pi$

Another way of expressing this curvature two-form is as

$\displaystyle w_{ij}$ $\displaystyle=$
$\displaystyle\epsilon_{ijk}\partial_{k}U$

in flat $\mathbb{R}^{3}$ with metric $\delta_{ij}$ where $\epsilon_{123}=1$
and totally antisymmetric. It is also useful to express this same relation in
a covariant form by using the metric $G_{ij}=U\delta_{ij}$ of the base.
Covariantly we then have

$\displaystyle w_{ij}$ $\displaystyle=$
$\displaystyle\frac{1}{G^{1/6}}{\cal{\varepsilon}}_{ijk}G^{k{{\ell}}}\partial_{{{\ell}}}U$
(3.10)

where we define the covariant tensor
${\cal{\varepsilon}}_{ijk}=\sqrt{G}\epsilon_{ijk}$. We may now confirm the
equivalence of the two expressions (3.9) and (3.10) by choosing polar
coordinates on the base. We then need to study the expression

$\displaystyle w_{\theta\varphi}$ $\displaystyle=$
$\displaystyle\frac{1}{G^{1/6}}{\cal{\varepsilon}}_{\theta\varphi
r}G^{rr}\partial_{r}U$

We start by rewriting everything in terms of $U$ using

$\displaystyle G$ $\displaystyle=$ $\displaystyle U^{3}$ $\displaystyle
G^{rr}$ $\displaystyle=$ $\displaystyle\frac{1}{U}$

Then

$\displaystyle w_{\theta\varphi}$ $\displaystyle=$
$\displaystyle\frac{1}{U\sqrt{U}}{\cal{\varepsilon}}_{\theta\varphi
r}\partial_{r}U$

Next we notice that

$\displaystyle\partial_{r}U$ $\displaystyle=$ $\displaystyle-\frac{1}{2r^{2}}$

We then need to address the quesion of finding an explicit expression for the
antisymmetric tensor component

$\displaystyle{\cal{\varepsilon}}_{\theta\varphi
r}=\sqrt{G}\epsilon_{\theta\varphi r}=U^{3/2}\epsilon_{\theta\varphi r}$

Here $\epsilon_{\theta\varphi r}=r^{2}\sin\theta$ is the determinant of the
Jacobian when we go from Cartesian to Polar coordinates. We now have all
ingredients. Putting them together, we obtain

$\displaystyle w_{\theta\varphi}=-\frac{1}{2}\sin\theta$

which is in perfect agreement with (3.9).

For multi-Taub-NUT we take

$\displaystyle U$ $\displaystyle=$
$\displaystyle\frac{1}{R^{2}}+\frac{1}{2}\sum_{I=1}^{N}\frac{1}{|\vec{x}-\vec{x}_{I}|}$

and the gravi-photon is implicitly defined through

$\displaystyle w_{ij}$ $\displaystyle=$
$\displaystyle{\cal{\varepsilon}}_{ijk}\partial_{k}U$

This is a sum of terms,

$\displaystyle w_{ij}$ $\displaystyle=$ $\displaystyle\sum_{i}w_{ij}^{I}$
$\displaystyle w_{ij}^{I}$ $\displaystyle=$
$\displaystyle{\cal{\varepsilon}}_{ijk}\partial_{k}U^{I}$ $\displaystyle
U^{I}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\frac{1}{|\vec{x}-\vec{x}_{I}|}$

which means that the gravi-photon itself is a sum of terms,

$\displaystyle\kappa_{i}$ $\displaystyle=$
$\displaystyle\sum_{I}\kappa_{i}^{I}$

When $\vec{x}\approx\vec{x}_{I}$ the geometry is locally that of flat
$\mathbb{R}^{4}$ if no other points $\vec{x}_{J}$ coincide with $\vec{x}_{I}$.
If $Q_{I}$ points coincide at $\vec{x}_{I}$, then we have, locally near that
point, the metric

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\frac{Q}{2r}\left(dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}\right)\right)+\frac{2r}{Q}\left(d\psi+\frac{Q}{2}\cos\theta
d\varphi\right)^{2}$

If we put $r=\frac{\rho^{2}}{2}$ we get

$\displaystyle\frac{1}{Q}ds^{2}$ $\displaystyle=$ $\displaystyle
d\rho^{2}+\frac{\rho^{2}}{4}\left(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}\right)+\frac{\rho^{2}}{4Q^{2}}\left(2d\psi+Q\cos\theta
d\varphi\right)^{2}$

If we put $\psi=Q\widetilde{\psi}$ we get

$\displaystyle\frac{1}{Q}ds^{2}$ $\displaystyle=$ $\displaystyle
d\rho^{2}+\frac{\rho^{2}}{4}\left(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}\right)+\frac{\rho^{2}}{4}\left(2d\widetilde{\psi}+\cos\theta
d\varphi\right)^{2}$

where $\widetilde{\psi}\sim\widetilde{\psi}+\frac{2\psi}{Q}$. This is the
metric of the orbifold $\mathbb{C}^{2}/\mathbb{Z}_{Q}$ with the identification
$(z_{1},z_{2})\sim(e^{2\pi i/Q}z_{1},e^{2\pi i/Q}z_{2})$.

We conclude that the multi-Taub-NUT space $TN_{N}$ is everywhere smooth for
$N$ non-coinciding singular points. But when $Q$ singular points coincide we
get an orbifold singularity of the type $\mathbb{C}^{2}/\mathbb{Z}_{Q}$.

Having found these geometries of multi-Taub-NUT, it remains to estanblish the
existence of Killing spinor solutions. To this end, we will simply review an
argument from [5] that shows that such six-manifolds support 8 real
covariantly constant spinors. The existence of a covariantly constant spinor
on $X$ implies that the Ricci tensor must vanish. For hyper-Kahler manifolds
the Ricci tensor vanishes and the generic $SO(4)=SU(2)_{+}\times SU(2)_{-}$
holonomy group is reduced to $SU(2)_{+}$. According to the holonomy principle,
a covariantly constant spinor is a singlet under the holonomy group
$SU(2)_{+}$. Let us represent gamma matrices in the $SO(4)$ tangent space
group of $X$ as

$\displaystyle\gamma^{i}$ $\displaystyle=$
$\displaystyle\sigma^{i}\otimes\sigma^{1}$ $\displaystyle\gamma^{4}$
$\displaystyle=$ $\displaystyle 1\otimes\sigma^{2}$

Then the embedding of the $SU(2)_{\pm}$ generators into $SO(4)$ is done as

$\displaystyle\sigma^{i}P_{\pm}$ $\displaystyle=$
$\displaystyle-\frac{i}{2}\gamma^{i4}\mp\frac{i}{4}{\cal{\varepsilon}}^{ijk}\gamma^{jk}$

where $P_{\pm}=\frac{1}{2}\left(1\otimes 1\pm 1\otimes\sigma^{3}\right)$. From
this, we conclude that spinors that are not rotated by the holonomy group
$SU(2)_{+}$ satisfy $P_{+}\psi=0$ so they are anti-Weyl spinors. The six-
manifold has a tangent space group $SO(1,5)$. A Weyl spinor under this tangent
space group has $4$ complex components and the constraint that it shall be
invariant under $SU(2)_{+}$ imposes another Weyl projection leading to $2$
complex components. For the M5 brane there is in addition an $SO(5)$
R-symmetry group and the spinor has $4$ internal $SO(5)$ R-symmetry spinor
components that leads to in total $2\times 4=8$ complex spinor components, but
there is a Majorana condition that one can impose on a spinor in
$SO(1,5)\times SO(5)\subset SO(1,10)$ resulting in $8$ real components. These
correspond to $8$ real supercharges [5].

##  4 Gauge and supersymmetry anomalies

Let us assume that the the circle fiber vanishes on a two-dimensional
submanifold $\Sigma$ with Lorentzian signature. Because the circle fibration
degenerates on $\Sigma$, it is somewhat difficult to analyze what happens
there directly. One way to circumvent this difficulty is by considering a
tubular neighborhood around $\Sigma$. We then consider a five-manifold with a
boundary four-manifold of the form $\Sigma\times S^{2}_{r_{0}}$ where
$S^{2}_{r_{0}}$ is a small sphere that is enclosing $\Sigma$ and then in the
end we shall take the limit $r_{0}\rightarrow 0$. There can be a magnetic flux
through $S^{2}_{r_{0}}$, both for the gauge field and for the gravi-photon
field. Of course one important difference is that for the gravi-photon the
flux is fixed by the geometry, whereas for the gauge field we shall sum over
all possible fluxes. Let us begin by assuming that the gauge group is Abelian.
Then by allowing $r_{0}$ to take any value greater than zero, we find that
these magnetic fluxes are produced by delta functions localized on $\Sigma$.
We have the following modifications of the Bianchi identities,

$\displaystyle\partial_{\mu}F_{\nu\lambda}+\partial_{\lambda}F_{\mu\nu}+\partial_{\nu}F_{\lambda\mu}$
$\displaystyle=$ $\displaystyle 2\pi Q_{F}\delta_{\mu\nu\lambda}$ (4.1)
$\displaystyle\partial_{\mu}w_{\nu\lambda}+\partial_{\lambda}w_{\mu\nu}+\partial_{\nu}w_{\lambda\mu}$
$\displaystyle=$
$\displaystyle-2\pi\sum_{I=I}^{n}Q^{I}\delta^{I}_{\mu\nu\lambda}$ (4.2)

Here $Q^{I}$ denotes the charge at submanifold $\Sigma_{I}$. For the multi-
Taub-NUT, these charges sum up to $N$

$\displaystyle\sum_{I=I}^{n}Q^{I}$ $\displaystyle=$ $\displaystyle N$

where $N=1,2,3,...$ is the integer charactarizing the multi-Taub-NUT. Further,
$\delta^{I}_{\mu\nu\lambda}$ denotes the Poincare dual of $\Sigma_{I}$,
defined as

$\displaystyle\int
d^{5}x\sqrt{-G}{\cal{\varepsilon}}^{\mu\nu\lambda\rho\sigma}\frac{1}{6}\delta^{I}_{\mu\nu\lambda}\frac{1}{2}\omega_{\rho\sigma}$
$\displaystyle=$
$\displaystyle\int_{\Sigma_{I}}d^{2}\sigma\sqrt{-\eta}{\cal{\varepsilon}}^{\alpha\beta}\frac{1}{2}\omega_{\alpha\beta}$
(4.3)

for an arbitrary test-two-form $\omega_{\mu\nu}$. Here $\eta_{\alpha\beta}$
denotes the induced metric on $\Sigma$ that we parametrize by coordinates
$\sigma^{\alpha}$.

Now let us look for a possible six-dimensional origin of the magnetic charge
$Q_{F}$. This would be an integral over a three-cycle,

$\displaystyle Q_{F}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int_{S^{2}_{r_{0}}\times S^{1}}H$

where $S^{1}$ denotes the circle fiber. But since the tubular neighborhood was
inserted by us by hand and can be shrunk to zero size, it does not really
exist in the six-manifold as a genuine cycle. So there is no three-cycle
$S^{2}_{0}\times S^{1}$ in the six-manifold and that means that $Q_{F}=0$.

There are two options for writing the gravi-photon term. Either

$\displaystyle{\cal{L}}_{grav}$ $\displaystyle=$
$\displaystyle\frac{r}{4}{\cal{\varepsilon}}^{\mu\nu\lambda\rho\sigma}\omega(A)_{\mu\nu\lambda}w_{\rho\sigma}$
(4.4)

or

$\displaystyle{\cal{L}}^{\prime}_{grav}$ $\displaystyle=$
$\displaystyle-\frac{r}{8}{\cal{\varepsilon}}^{\mu\nu\lambda\rho\sigma}F_{\mu\nu}F_{\lambda\rho}\kappa_{\sigma}$
(4.5)

The two ways differ by a total derivative,

$\displaystyle{\cal{L}}_{grav}$ $\displaystyle=$
$\displaystyle{\cal{L}}^{\prime}_{grav}+r\nabla_{\rho}\left(\frac{1}{2}{\cal{\varepsilon}}^{\mu\nu\lambda\rho\sigma}\omega(A)_{\mu\nu\lambda}\kappa_{\sigma}\right)$

inconsequential for the equation of motion, but ${\cal{L}}^{\prime}_{grav}$ is
gauge invariant while ${\cal{L}}_{grav}$ is not, so one might prefer to use
${\cal{L}}^{\prime}_{grav}$. But ${\cal{L}}^{\prime}_{grav}$ is not invariant
under a reparametrization of the fiber coordinate. Under a reparametrization
$\psi\rightarrow\psi^{\prime}=\psi+f$ the graviphoton field transforms as
$\kappa_{\mu}\rightarrow\kappa^{\prime}_{\mu}-\partial_{\mu}f$. So this way of
writing the gravi-photon term is not invariant under such a ’geometric’ gauge
transformations.

The gauge field equation of motion is

$\displaystyle\nabla_{\nu}\left(\frac{1}{r}F^{\nu\mu}\right)+\frac{1}{4}{\cal{\varepsilon}}^{\mu\nu\lambda\rho\sigma}F_{\nu\lambda}w_{\rho\sigma}$
$\displaystyle=$ $\displaystyle 0$

This equation of motion holds irrespectively of whether we use the gravi-
photon term (4.4) or (4.5) in the action. If we act by $\nabla_{\mu}$ on the
left-hand side, then we get

$\displaystyle\frac{2}{r}R_{\mu\nu}F^{\mu\nu}+\frac{1}{4}{\cal{\varepsilon}}^{\mu\nu\lambda\rho\sigma}F_{\nu\lambda}\nabla_{\mu}w_{\rho\sigma}$
$\displaystyle=$ $\displaystyle 0$ (4.6)

The first term is zero because the Ricci scalar is symmetric while
$F_{\mu\nu}$ is antisymmetric. But if $Q$ is nonzero, then the second term is
not zero. In this case, the equation of motion (4.6) becomes inconsistent, and
must be modified somehow. Following [13], we make the following ansatz for
such a modified equation of motion,

$\displaystyle\nabla_{\nu}\left(\frac{1}{r}F^{\nu\mu}\right)+\frac{1}{4}{\cal{\varepsilon}}^{\mu\nu\lambda\rho\sigma}F_{\nu\lambda}w_{\rho\sigma}$
$\displaystyle=$ $\displaystyle J^{\mu}$ (4.7)

Now if we act on both sides by $\nabla_{\mu}$ we get

$\displaystyle
2\pi{\cal{\varepsilon}}^{\mu\nu\lambda\rho\sigma}\frac{1}{2}F_{\nu\lambda}\frac{1}{6}\sum_{I=1}^{n}Q^{I}\delta^{I}_{\mu\rho\sigma}$
$\displaystyle=$ $\displaystyle\nabla_{\mu}J^{\mu}$

and if the left-hand side is nonzero, then this shows that 5d SYM can not be
the full story. Something more is needed that can produce $J_{\mu}$.

Let us then use the gravi-photon term (4.4), which is not gauge invariant but
diffeomorphism invariant. If we assume that the gauge group is non-Abelian,
then under a finite gauge transformation

$\displaystyle A^{g}$ $\displaystyle=$ $\displaystyle
g^{-1}A_{\mu}g+ig^{-1}\partial_{\mu}g$ (4.8)

the Chern-Simons three-form transforms as

$\displaystyle\omega(A^{g})_{\mu\nu\lambda}$ $\displaystyle=$
$\displaystyle\omega(A)_{\mu\nu\lambda}+\partial_{\nu}\left(i\partial_{\mu}gg^{-1}A_{\lambda}\right)+\frac{1}{3}g^{-1}\partial_{\mu}gg^{-1}\partial_{\nu}gg^{-1}\partial_{\lambda}g$
(4.9)

The last term gives rise to the following term in the action,

$\displaystyle\frac{1}{16\pi^{2}}\oint_{S^{2}}dx^{i}\wedge
dx^{j}w_{ij}\int_{\mathbb{R}_{+}\times\mathbb{R}^{1,1}}dx^{\mu}\wedge
dx^{\nu}\wedge
dx^{\lambda}\frac{1}{3}{\mbox{tr}}\left(g^{-1}\partial_{\mu}gg^{-1}\partial_{\nu}gg^{-1}\partial_{\lambda}g\right)$

Using that the magnetic charge is $-2\pi Q$ gives

$\displaystyle-\frac{Q}{12\pi}\int_{\mathbb{R}_{+}\times\mathbb{R}^{1,1}}dx^{\mu}\wedge
dx^{\nu}\wedge
dx^{\lambda}{\mbox{tr}}\left(g^{-1}\partial_{\mu}gg^{-1}\partial_{\nu}gg^{-1}\partial_{\lambda}g\right)$

The manifold over which this is to be integrated is
$\mathbb{R}_{+}\times\mathbb{R}^{1,1}$ where $\mathbb{R}_{+}$ is the radial
direction outwards from the $S^{2}$ over which we integrated $w_{ij}$. So this
three-manifold has a boundary $S^{2}$. The metric on this manifold does not
enter since the term is topological. Let us Wick rotate $\mathbb{R}^{1,1}$
into $\mathbb{R}^{2}$ that we subsequently compactify into $S^{2}$. Then the
manifold over which we integrate has turned into $\mathbb{R}^{3}$ where we
have removed a small ball at the center. Expecting nothing particular happens
to the field $g$ at the origin, we can let this ball shrink to zero size and
integrate over the full $\mathbb{R}^{3}$ space. Assuming that $g$ falls off
sufficiently fast at infinity, this amounts to integrating over $S^{3}$ as we
may then identify all points at infinity and make a one-point compactification
of $\mathbb{R}^{3}$ to $S^{3}$ by adding the point at infinity. When
integrating over $S^{3}$ this term is quantized in integer multiples of $2\pi
i$ where the integer is the winding number as we map $S^{3}$ into an $SU(2)$
subgroup of the gauge group by the field $g$.

For the second term in (4.9) we applying (4.3) after making an integration by
parts. The final result is that the gravi-photon term transforms as

$\displaystyle S_{grav}(A^{g})$ $\displaystyle=$ $\displaystyle
S_{grav}(A)-\sum_{I=1}^{n}\frac{iQ^{I}}{4\pi}\int_{\Sigma_{I}}dx^{\mu}\wedge
dx^{\nu}{\mbox{tr}}\left(\partial_{\mu}gg^{-1}A_{\nu}\right)-2\pi n_{w}$

where $n_{w}$ is an integer winding number. This integer plays no role in the
quantum theory where one considers the exponentiated action $e^{iS}$ in
Lorentzian signature.

The action is also not supersymmetric. But to study this problem it is
advantagous to first consider a more general setup of a five-manifold with a
generic four-manifold boundary. Then under a supersymmetry variation of the
action, we will pick up the following boundary terms,

$\displaystyle\delta S$ $\displaystyle=$ $\displaystyle\frac{1}{4\pi^{2}}\int
d^{5}x\partial_{\mu}\left(\sqrt{-G}b^{\mu}\right)$ $\displaystyle b^{\mu}$
$\displaystyle=$ $\displaystyle-\frac{1}{r}F^{\mu\nu}\delta
A_{\nu}-\frac{1}{4}{\cal{\varepsilon}}^{\mu\nu\lambda\rho\sigma}A_{\nu}\delta
A_{\lambda}w_{\rho\sigma}$
$\displaystyle-\frac{1}{r}D^{\mu}\phi^{A}\delta\phi^{A}-\frac{i}{2r}\bar{\psi}\Gamma^{\mu}\delta\psi$

Explicitly we get

$\displaystyle b^{\mu}$ $\displaystyle=$
$\displaystyle-\frac{i}{4r}\bar{\psi}\Gamma^{\rho\sigma}\Gamma^{\mu}{\cal{\varepsilon}}\Gamma^{\widehat{\psi}}{\cal{\varepsilon}}F_{\rho\sigma}+\frac{i}{2r}\bar{\psi}\Gamma^{\rho}\Gamma^{\mu}\Gamma^{A}{\cal{\varepsilon}}D_{\rho}\phi^{A}$
$\displaystyle-\frac{1}{4}{\cal{\varepsilon}}^{\mu\nu\lambda\rho\sigma}A_{\nu}i\bar{\cal{\varepsilon}}\Gamma_{\lambda}\Gamma^{\widehat{\psi}}\psi
w_{\rho\sigma}$
$\displaystyle-\frac{i}{2r}\bar{\psi}\Gamma^{\mu}\Gamma^{\rho}\Gamma^{A}{\cal{\varepsilon}}\frac{\partial_{\rho}r}{r}\phi^{A}-\frac{i}{4}\bar{\psi}\Gamma^{\mu}\Gamma^{\rho\sigma}\Gamma^{A}\Gamma^{\widehat{\psi}}{\cal{\varepsilon}}w_{\rho\sigma}\phi^{A}$
$\displaystyle-\frac{1}{4r}\bar{\psi}\Gamma^{\mu}\Gamma^{AB}\Gamma^{\widehat{\psi}}{\cal{\varepsilon}}[\phi^{A},\phi^{B}]$

We would now have liked to proceed along the lines of reference [7] and find
boundary degrees that we add so that the total action becomes supersymmetric
without imposing boundary conditions. This strategy works nicely for low-
dimensional super Yang-Mills. But for 5d SYM this strategy fails. So instead
we will use a different approach. In the end we are not interested in the 4d
boundary theory, but in a 2d submanifold theory. So we want to take the limit
where the radius $r_{0}$ goes to zero. As we are still integrating over
$S^{2}_{r_{0}}$, taking the limit $r_{0}$ to zero, amounts to averaging over
the radial directions. For most terms, such an averaging will produce zero net
result because the fields are not expected to vary very much close to the
singular point. From a 6d viewpoint, this singular point is perfectly regular
and we expect the fields to be smooth close to the singularity. We then expect
that the only terms that will survive the integration over $S^{2}_{r_{0}}$
will be those that arise as magnetic charges when we integrate the two-form
$w_{\mu\nu}$. So we may isolate the terms that involve this two-form and only
consider those terms.

Let us analyze this problem on $\mathbb{R}^{1,1}\times TN$. We use the
relation (A.1) which leads to

$\displaystyle b^{\mu}$ $\displaystyle=$
$\displaystyle\frac{i}{8}\bar{\psi}\Gamma^{A}\Gamma^{\mu}\Gamma^{\rho\sigma}\Gamma^{\widehat{\psi}}{\cal{\varepsilon}}w_{\rho\sigma}\phi^{A}$
$\displaystyle-\frac{1}{4}{\cal{\varepsilon}}^{\mu\nu\lambda\rho\sigma}i\bar{\cal{\varepsilon}}\Gamma_{\lambda}\Gamma^{\widehat{\psi}}\psi
A_{\nu}w_{\rho\sigma}$ $\displaystyle+...$

where we extracted terms proportional to $w_{\mu\nu}$. Now we use the fact
that $w_{\mu\nu}$ has components only in the $\mathbb{R}^{3}$ base of $TN$
where it is a magnetic monopole of strength $-2\pi Q$. The boundary is taken
to be $\mathbb{R}^{1,1}\times S^{2}$ and the normal direction is the radial
direction. Thus we are interested in the radial component

$\displaystyle b^{r}$ $\displaystyle=$
$\displaystyle\frac{i}{4}\bar{\psi}\Gamma^{A}\Gamma^{r\theta\varphi}\Gamma^{\widehat{\psi}}{\cal{\varepsilon}}w_{\theta\varphi}\phi^{A}$
$\displaystyle+\frac{1}{2}{\cal{E}}^{\mu\nu}{\cal{E}}^{r\theta\varphi}i\bar{\psi}\Gamma^{\widehat{\psi}}\Gamma_{\mu}{\cal{\varepsilon}}A_{\nu}w_{\theta\varphi}$
$\displaystyle+...$

Now we need to examine the Weyl projections. On $TN$ we have the Weyl
projection (A.2)

$\displaystyle\Gamma^{123}\Gamma^{\widehat{\psi}}{\cal{\varepsilon}}$
$\displaystyle=$ $\displaystyle{\cal{\varepsilon}}$

By combining that we the 6d Weyl projection

$\displaystyle\Gamma^{0}\Gamma^{123}\Gamma^{4}\Gamma^{\widehat{\psi}}{\cal{\varepsilon}}$
$\displaystyle=$ $\displaystyle-{\cal{\varepsilon}}$

we get

$\displaystyle\Gamma^{04}{\cal{\varepsilon}}$ $\displaystyle=$
$\displaystyle-{\cal{\varepsilon}}$

We define
${\cal{\varepsilon}}^{01234\widehat{\psi}}={\cal{\varepsilon}}^{01234}=1$, and
we put ${\cal{\varepsilon}}^{\mu\nu
ijk}={\cal{\varepsilon}}^{\mu\nu}{\cal{\varepsilon}}^{ijk}$ where we put
${\cal{\varepsilon}}^{04}=-1$ and $\epsilon^{123}=1$. Then we we have

$\displaystyle\epsilon^{\nu\lambda}\Gamma_{\nu}{\cal{\varepsilon}}$
$\displaystyle=$ $\displaystyle\Gamma^{\lambda}{\cal{\varepsilon}}$

Using this relation, we get

$\displaystyle b^{r}$ $\displaystyle=$
$\displaystyle\frac{i}{4}{\cal{E}}^{r\theta\varphi}\bar{\psi}\Gamma^{A}{\cal{\varepsilon}}w_{\theta\varphi}\phi^{A}$
$\displaystyle+\frac{1}{2}{\cal{E}}^{r\theta\varphi}i\bar{\psi}\Gamma^{\widehat{\psi}}\Gamma^{\nu}{\cal{\varepsilon}}A_{\nu}w_{\theta\varphi}$
$\displaystyle+...$

This leads to a variation of the action given by

$\displaystyle\delta S_{5d}$ $\displaystyle=$
$\displaystyle-\sum_{I}\frac{Q_{I}}{8\pi}\int_{\Sigma_{I}}d^{2}xi\bar{\psi}\Gamma^{A}{\cal{\varepsilon}}\phi^{A}$
$\displaystyle-\sum_{I}\frac{Q_{I}}{4\pi}\int_{\Sigma_{I}}d^{2}xi\bar{\psi}\Gamma^{\widehat{\psi}}\Gamma^{\mu}{\cal{\varepsilon}}A_{\mu}$

In a 5d reduced notation

$\displaystyle\delta S_{5d}$ $\displaystyle=$
$\displaystyle\sum_{I}\frac{Q_{I}}{8\pi}\int_{\Sigma_{I}}d^{2}xi\chi\tau^{A}{\cal{E}}\phi^{A}$
(4.11)
$\displaystyle-\sum_{I}\frac{Q_{I}}{4\pi}\int_{\Sigma_{I}}d^{2}x\chi\gamma^{\mu}{\cal{E}}A_{\mu}$

where
${\cal{E}}^{\alpha\dot{\alpha}}={\cal{\varepsilon}}^{\alpha-\dot{\alpha}}$ and
$\chi^{\alpha\dot{\alpha}}=\psi^{\alpha+\dot{\alpha}}$. The 5d conjugate
spinor is defined as
$\bar{\chi}_{\beta\dot{\beta}}=\chi^{\alpha\dot{\alpha}}C_{\alpha\beta}C_{\dot{\alpha}\dot{\beta}}$.
We further reduce to 2d notation by decomposing $\alpha=(u,m)$ and put
$\gamma^{\mu}=(\gamma^{\mu})^{u}{}_{v}\delta^{m}_{n}$. Then
$\chi\gamma^{\mu}{\cal{E}}=\chi_{um\dot{\alpha}}(\gamma^{\mu})^{u}{}_{v}{\cal{E}}^{vm\dot{\alpha}}$.
In the 2d reduced notation, the sum over $m$ is trivial and gives three
identical copies. For more details on our spinor notations, we refer to
Appendix D.

##  5 WZW theories on $\Sigma_{I}$

A supersymmetric WZW theory has been constructed in [14] using a superfield
formulation. Here we will use a component formulation instead. Let us start by
analyzing the following supersymmetric WZW Lagrangian (where
${\cal{\varepsilon}}^{01}=1$),

$\displaystyle{\cal{L}}_{WZW}(g,A_{\mu})$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi}{\mbox{tr}}\left(g^{-1}\left(\partial_{\mu}-iA_{\mu}\right)g\right)^{2}-\frac{i}{4\pi}{\cal{\varepsilon}}^{\mu\nu}{\mbox{tr}}\partial_{\mu}gg^{-1}A_{\nu}$
$\displaystyle-\frac{i}{8\pi}{\mbox{tr}}\left(\bar{\lambda}\gamma^{\mu}\partial_{\mu}\lambda\right)$
$\displaystyle+\frac{1}{12\pi}{\cal{\varepsilon}}^{\mu\nu\lambda}{\mbox{tr}}\left(g^{-1}\partial_{\mu}gg^{-1}\partial_{\nu}gg^{-1}\partial_{\lambda}g\right)$

To get the action, we should integrated this over $\Sigma_{I}$, except for the
last term that should be integrated over some three-manifold whose boundary is
$\Sigma_{I}$. This Lagrangian is invariant under the supersymmetry variations

$\displaystyle g^{-1}\delta g$ $\displaystyle=$
$\displaystyle-\bar{\cal{\varepsilon}}\lambda$ $\displaystyle\delta\lambda$
$\displaystyle=$
$\displaystyle-i\gamma^{\mu}{\cal{\varepsilon}}g^{-1}\partial_{\mu}g$

provided we impose the following chiral projection on the supersymmetry
parameter,

$\displaystyle\gamma^{01}{\cal{\varepsilon}}$ $\displaystyle=$
$\displaystyle{\cal{\varepsilon}}$

Under a variation of $A_{\mu}$, keeping $g$ and $\lambda$ fixed, we have

$\displaystyle\delta{\cal{L}}_{WZW}(g,A_{\mu})$ $\displaystyle=$
$\displaystyle-\frac{i}{4\pi}\partial_{\mu}gg^{-1}\left(\delta
A^{\mu}+{\cal{\varepsilon}}^{\mu\nu}\delta
A_{\nu}\right)-\frac{1}{4\pi}A_{\mu}\delta A^{\mu}$

If we make the specific variation

$\displaystyle\delta A_{\mu}$ $\displaystyle=$
$\displaystyle-\chi\gamma^{\mu}{\cal{E}}$

as induced from 5d, then we get

$\displaystyle\delta{\cal{L}}_{WZW}(g,A_{\mu})$ $\displaystyle=$
$\displaystyle\frac{i}{4\pi}\partial_{\mu}gg^{-1}\chi\left(\gamma^{\mu}+{\cal{\varepsilon}}^{\mu\nu}\gamma_{\nu}\right){\cal{E}}+\frac{1}{4\pi}A_{\mu}\chi\gamma^{\mu}{\cal{E}}$

The first term vanishes by the projection $\gamma^{01}{\cal{E}}={\cal{E}}$. So
if we add the following 2d action,

$\displaystyle S_{WZW}$ $\displaystyle=$
$\displaystyle\sum_{I}Q_{I}\int_{\Sigma_{I}}d^{2}x{\cal{L}}_{WZW}(g_{I},A_{\mu})$
(5.1)

then we cancel the second term in (4.11).

The resulting Lagrangian is also not fully gauge invariant, where the gauge
variation acts on the fields as

$\displaystyle A_{\mu}$ $\displaystyle\rightarrow$ $\displaystyle
hA_{\mu}h^{-1}-i\partial_{\mu}hh^{-1}$ (5.2) $\displaystyle g$
$\displaystyle\rightarrow$ $\displaystyle hg$ (5.3)

for a gauge parameter $h$. One obviously gauge noninvariant term is
${\cal{L}}_{non}=-\frac{i}{4\pi}{\cal{\varepsilon}}^{\mu\nu}{\mbox{tr}}\partial_{\mu}gg^{-1}A_{\nu}$,
which transforms into

$\displaystyle{\cal{L}}_{non}$ $\displaystyle\rightarrow$
$\displaystyle{\cal{L}}_{non}-\frac{i}{4\pi}{\cal{\varepsilon}}^{\mu\nu}{\mbox{tr}}\left(h^{-1}\partial_{\mu}hA_{\nu}\right)$
$\displaystyle-\frac{1}{4\pi}{\cal{\varepsilon}}^{\mu\nu}\left(\partial_{\mu}hh^{-1}\partial_{\nu}hh^{-1}+\partial_{\mu}gg^{-1}h^{-1}\partial_{\nu}h\right)$

The other gauge noninvariant term is the WZ term
${\cal{L}}_{WZ}=\frac{1}{12\pi}{\cal{\varepsilon}}^{\mu\nu\lambda}{\mbox{tr}}\left(g^{-1}\partial_{\mu}gg^{-1}\partial_{\nu}gg^{-1}\partial_{\lambda}g\right)$
that transforms into

$\displaystyle{\cal{L}}_{WZ}$ $\displaystyle\rightarrow$
$\displaystyle{\cal{L}}_{WZ}+\frac{1}{12\pi}h^{-1}\partial_{\mu}hh^{-1}\partial_{\nu}hh^{-1}\partial_{\lambda}h$
$\displaystyle+\frac{1}{4\pi}{\cal{\varepsilon}}^{\lambda\mu\nu}\partial_{\lambda}\left(\partial_{\mu}hh^{-1}\partial_{\nu}hh^{-1}+\partial_{\mu}gg^{-1}h^{-1}\partial_{\nu}h\right)$

We now see that many terms cancel between $\delta{\cal{L}}_{non}$ and
$\delta_{WZ}$ and we are left with the gauge variation

$\displaystyle\delta S_{2d}$ $\displaystyle=$
$\displaystyle-\sum_{I}\frac{iQ_{I}}{4\pi}\int dx^{\mu}\wedge
dx^{\nu}{\mbox{tr}}\left(h^{-1}\partial_{\mu}hA_{\nu}\right)$

In order to match the gauge variation (5.3) with the gauge transformation
(4.8) we shall substitute $h$ here by $g^{-1}$, in which case we get

$\displaystyle\delta S_{2d}$ $\displaystyle=$
$\displaystyle\sum_{I}\frac{Q_{I}}{4\pi}\int dx^{\mu}\wedge
dx^{\nu}{\mbox{tr}}\left(i\partial_{\mu}gg^{-1}A_{\nu}\right)$

that is cancelling the gauge variation of $S_{5d}$.

##  6 The WZW current

If we would vary the gauge potential in the WZW theory, then we would get the
equation of motion

$\displaystyle\partial_{\mu}gg^{-1}-{\cal{\varepsilon}}_{\mu\nu}\partial^{\nu}gg^{-1}-iA_{\mu}$
$\displaystyle=$ $\displaystyle 0$ (6.1)

This equation is not gauge invariant. But that is not so surprising. The gauge
potential in the WZW theory is a background field and we are not supposed vary
a background field. However, it is a dynamical field in the 5d SYM theory and
so if we vary the gauge potential in the combined system of 5d SYM plus the
WZW theory, then we should recover a gauge invariant equation of motion. The
gauge invariant completion of the left-hand side in (6.1) is

$\displaystyle D_{\mu}gg^{-1}-{\cal{\varepsilon}}_{\mu\nu}D^{\nu}gg^{-1}$
(6.2)

Thus we are looking for a missing term $i{\cal{\varepsilon}}_{\mu\nu}A^{\nu}$
that should come from the 5d SYM upon variation of the gauge potential, so
that (6.1) is completed into (6.2). Let us examine the gravi-photon term, and
the following term

$\displaystyle\frac{1}{16\pi^{2}}A_{\mu}\partial_{\nu}\delta
A_{\lambda}w_{\rho\sigma}$

when we vary the gauge potential. To derive the Euler-Lagrange equation of
motion, we would make an integration by parts. But here, when we make an
integration by parts we must be careful because of (4.2). It is not too hard
to see that this will exactly produce our missing 2d term that will complete
(6.1) into (6.2).

Let us follow [3] and put

$\displaystyle J_{\mu}$ $\displaystyle=$ $\displaystyle
D_{\mu}gg^{-1}-{\cal{\varepsilon}}_{\mu\nu}D^{\nu}gg^{-1}$

Then the 5d SYM equation of motion is modified to

$\displaystyle
D^{\nu}\left(\frac{1}{r}F_{\nu\mu}\right)+\frac{1}{4}{\cal{\varepsilon}}^{\mu\nu\lambda\rho\sigma}F_{\nu\lambda}w_{\rho\sigma}$
$\displaystyle=$ $\displaystyle\sum_{I}i\pi Q^{I}\delta^{I}_{123}J_{\mu}$
(6.3)

Let us now assume the gauge group is Abelian for simplicity. Nothing essential
changes in the argument we will make below when the gauge group is non-
Abelian. When the gauge group is Abelian, we put

$\displaystyle g$ $\displaystyle=$ $\displaystyle e^{i\phi}$

and then $\phi$ will be a $2\pi$ periodic scalar field, and the WZW action is
given by

$\displaystyle S_{WZW}$ $\displaystyle=$ $\displaystyle-\frac{Q}{8\pi}\int
d^{2}x\left((\partial_{\mu}\phi-A_{\mu})^{2}+{\cal{\varepsilon}}^{\mu\nu}\phi
F_{\mu\nu}+i\bar{\lambda}\gamma^{\mu}\partial_{\mu}\lambda\right)$

that is invariant under the supersymmetry variations

$\displaystyle\delta\phi$ $\displaystyle=$ $\displaystyle
i\bar{\cal{\varepsilon}}\lambda$ $\displaystyle\delta\lambda$ $\displaystyle=$
$\displaystyle\gamma^{\mu}{\cal{\varepsilon}}(\partial_{\mu}\phi-A_{\mu})$

for $\gamma{\cal{\varepsilon}}={\cal{\varepsilon}}$. Under the gauge variation

$\displaystyle\delta\phi$ $\displaystyle=$ $\displaystyle\Lambda$
$\displaystyle\delta A_{\mu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\Lambda$

we have the variation

$\displaystyle\delta S_{WZW}$ $\displaystyle=$
$\displaystyle\frac{Q}{8\pi}\int
d^{2}x\Lambda{\cal{\varepsilon}}^{\mu\nu}F_{\mu\nu}$

that is canceling against the gauge variation of the 5d SYM action,

$\displaystyle\delta S_{5d}$ $\displaystyle=$
$\displaystyle-\frac{Q}{8\pi}\int
d^{2}x\Lambda{\cal{\varepsilon}}^{\mu\nu}F_{\mu\nu}$

The equation of motion for $\phi$ is given by

$\displaystyle\nabla^{2}\phi-\nabla^{\mu}A_{\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}{\cal{\varepsilon}}^{\mu\nu}F_{\mu\nu}$ (6.4)

For the gauge field, we should consider the combined system of 5d SYM coupled
to 2d WZW, and then we find the equation of motion

$\displaystyle\nabla_{\nu}\left(\frac{1}{r}F^{\nu\mu}\right)+\frac{1}{4}{\cal{\varepsilon}}^{\mu\nu\lambda\rho\sigma}F_{\nu\lambda}w_{\rho\sigma}$
$\displaystyle=$ $\displaystyle-\pi
Q\delta_{123}\left(\nabla^{\mu}\phi-{\cal{\varepsilon}}^{\mu\nu}\nabla_{\nu}\phi-A^{\mu}+{\cal{\varepsilon}}^{\mu\nu}A_{\nu}\right)$

Acting on both sides by $\nabla_{\mu}$ we get

$\displaystyle\frac{1}{4}{\cal{\varepsilon}}^{\mu\nu\lambda\rho\sigma}F_{\nu\lambda}\partial_{\mu}w_{\rho\sigma}$
$\displaystyle=$ $\displaystyle-\pi
Q\delta_{123}\left(\nabla^{2}\phi-\nabla_{\mu}A^{\mu}+\frac{1}{2}{\cal{\varepsilon}}^{\mu\nu}F_{\mu\nu}\right)$

Consistency with (6.4) implies that

$\displaystyle\partial_{i}w_{jk}+\partial_{k}w_{ij}+\partial_{j}w_{ki}$
$\displaystyle=$ $\displaystyle-2\pi Q\delta_{123}$

which is a consistency check that everything fits together nicely.

In addition to this WZW theory, if we also add a mass term for the five scalar
fields that is localized to $\Sigma_{I}$,

$\displaystyle S_{mass}$ $\displaystyle=$
$\displaystyle-\frac{Q_{I}}{16\pi}\int d^{2}x\phi^{A}\phi^{A}$

then that has the supersymmetry variation

$\displaystyle\delta S_{mass}$ $\displaystyle=$
$\displaystyle-\frac{Q_{I}}{8\pi}\int d^{2}xi\chi\tau^{A}{\cal{E}}\phi^{A}$

that cancels the first term in (4.11).

##  7 Discussion

Our proposal is that we need to add additional degrees of freedom on
$\Sigma_{I}$ that are not present in 5d SYM. A different viewpoint appears to
have been taken in [13] and [12] where it appears to have been suggested that
we may obtain the WZW degrees directly from the degrees of 5d SYM. This latter
viewpoint would then be similar with the idea that the KK modes are not
supposed to be added but are already present in 5d SYM as solitonic solutions
(instanton particles).

An on-shell solution for the gauge field was constructed in [13] by making use
of the harmonic two-form on Taub-NUT. The strategy was to first solving the 6d
equations of motion

$\displaystyle dH$ $\displaystyle=$ $\displaystyle 0$ $\displaystyle H$
$\displaystyle=$ $\displaystyle*H$

for the selfdual tensor field of the Abelian M5 brane on
$\mathbb{R}^{1,1}\times TN$. The solution is given by

$\displaystyle H$ $\displaystyle=$ $\displaystyle h_{\mu}dx^{\mu}\wedge\Omega$

Here $\Omega$ is the unique antiselfdual harmonic two-form on $TN$,

$\displaystyle\Omega$ $\displaystyle=$
$\displaystyle\frac{\partial_{k}U}{U^{2}}\left(-e^{k}\wedge
e^{4}+\frac{1}{2}{\cal{\varepsilon}}^{ijk}e^{i}\wedge e^{j}\right)$

where

$\displaystyle e^{i}$ $\displaystyle=$ $\displaystyle\sqrt{U}dx^{i}$
$\displaystyle e^{4}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{U}}\left(d\psi+\kappa^{i}dx^{i}\right)$

This solution was subsequently generalized to multi-Taub-NUT and shown to be
supersymmetric in [12].

Let us here return to the Taub-NUT space and let us take the gauge group to be
Abelian, and let us make the following ansatz for an on-shell solution

$\displaystyle F_{i\mu}$ $\displaystyle=$
$\displaystyle\frac{\partial_{i}U}{U^{2}}h_{\mu}$

with all other components vanishing. To see whether this satisfies the
equation of motion, we shall start by computing

$\displaystyle\nabla^{i}\left(\frac{1}{r}F_{i\mu}\right)+\frac{1}{2}{\cal{\varepsilon}}_{\mu}{}^{i\nu
jk}F_{i\nu}w_{jk}$ $\displaystyle=$
$\displaystyle-\frac{1}{U^{3/2}}\frac{1}{U^{2}}\partial_{i}U\partial_{i}U\left(h_{\mu}+{\cal{\varepsilon}}_{\mu\nu}h^{\nu}\right)+\frac{1}{U}\frac{1}{U^{3/2}}\partial_{i}\partial_{i}Uh_{\mu}$

The first term vanishes if we demand that

$\displaystyle h_{\mu}+{\cal{\varepsilon}}_{\mu\nu}h^{\nu}$ $\displaystyle=$
$\displaystyle 0$

The second term is identically zero. First we note that
$\frac{1}{U^{3/2}}\partial_{i}\partial_{i}U$ is a delta function with respect
to the integartion measure $d^{3}xU^{3/2}$. Second, $1/U$ evaluated at $|x|=0$
is zero, which is killing the whole thing. Thus we find that this solves the
equation of motion (6.3) with $J_{\mu}=0$. From the equation of motion
$\partial^{\mu}F_{\mu i}=0$ we find $\partial^{\mu}h_{\mu}=0$. By combining
this with $h_{\mu}=-{\cal{\varepsilon}}_{\mu\nu}h^{\nu}$ we get
$\partial_{\mu}h_{\nu}-\partial_{\nu}h_{\mu}=0$ that is locally solved by
$h_{\mu}=\partial_{\mu}\varphi$ for some scalar field $\varphi$. Our
computation shows a different result from [13] where it was found that
$J_{\mu}\sim h_{\mu}$, from which one would conclude that the scalar field
$\phi$ in the WZW theory would be the same as the component $\varphi$ that
sits in the super Yang-Mills field $F_{\mu i}$ rather than a new degree of
freedom. We notice that there is no contradiction having $J_{\mu}=0$ here
since by acting on the equation of motion by $\partial^{\mu}$ we get zero on
the left-hand side because
$\partial_{\mu}w_{ij}+\partial_{i}w_{j\mu}+\partial_{j}w_{\mu i}=0$ so we not
generate a delta function from the left-hand side by acting by
$\partial_{\mu}$ there. On the right-hand side we get of course zero too, if
$J_{\mu}=0$.

In the reference [12] it was argued that we should use the gauge invariant
gravi-photon term (4.5) rather than the Chern-Simons-like gravi-photon term
(4.4) that we have used in this paper. But using (4.5) has the problem that it
is not reparametrization invariant. If the 5d SYM would have no gauge anomaly,
then there would be no need to add a gauged WZW theory and one could think
this could then be the end of the story. However, the problem with
supersymmetry would remain exactly the same as we have presented here. This is
true regadless of what form of the gravi-photon term we choose as our
preference. This makes us believe that it would be rather difficult to come up
with some alternative construction that is based on (4.5). It should probably
be possible, but one would then need to deal with a reparametrization anomaly
and find some way to cancel this anomaly. The reference [12] also makes a
couple of interesting observations. First, the solution found in [13] was
shown to be supersymmetric. It preserves all supersymmetries. There are no
fermionic zero modes and no broken supersymmetries. The intersection brane was
shown to carry an electric charge and it has a tension, both of which are
expressed in terms of the function $h_{\mu}$ (denoted $\nu_{+}$ in [12]). This
has a non-Abelian generaltzaion and also a generalization to multi-Taub-NUT
[12]. It would be interesting to show that the mass saturates some BPS bound
determined by some central charge. Presumably that central charge would be
proportional to the electric charge.

In [12] it was objected that the WZW for multi-Taub-NUT could not be the usual
WZW with one three-manifold with $N$ different boundaries whenever $Q>1$. But
there is no need for the three-manifold to connect all the $N$ different
intersection branes. Some three-manifold could extend from one intersection
brane out to infinity. Then if $Q$ intersection branes would coincide, then we
just take the WZW level to be equal to $Q$ and extend the three-manifold to
infinity, and it would not affect the WZW theories on the other intersection
branes if we extend the three-manifold to infinity in such a way that it does
not cross some of the other submanifolds. So by allowing the three-manifold to
extend to infinity rather than joining different intersection branes, we seem
to be able to avoid the problem that was raised in [12]. We notice that there
is plenty of room to draw three-manifolds that extend to infinity. These
three-manifolds are lines in $\mathbb{R}^{3}$ and extended along
$\mathbb{R}^{1,1}$. Singular points where the circle fiber vanishes in multi-
Taub-NUT corresponds to points in $\mathbb{R}^{3}$. So we can always find a
line from any singular point in $\mathbb{R}^{3}$ that extends to infinity.
There is no need to connect the different singular points with lines. These
lines may instead extend to infinity and then we have genuinely different WZW
theories for each singular point. But there appears to be some ambiguity in
how we may choose to draw these lines though. We expect that different choices
will lead to equivalent physical descriptions.

## Acknowledgements

This work was supported in part by NRF Grant 2020R1A2B5B01001473 and NRF Grant
2020R1I1A1A01052462.

##  Appendix A Differential geometry

###  A.1 A formula for the spin connection

Given a vielbein $e^{a}{}_{\mu}$, the spin connection
$(\omega_{\mu})^{a}{}_{b}$ is implicitly defined by two equations. It is
covariantly constant, and the torsion is vanishing,

$\displaystyle\partial_{\mu}e^{a}{}_{\nu}+(\omega_{\mu})^{a}{}_{b}e^{b}{}_{\nu}-e^{a}{}_{\rho}\Gamma^{\rho}_{\mu\nu}$
$\displaystyle=$ $\displaystyle 0$ $\displaystyle\Gamma^{\rho}_{[\mu\nu]}$
$\displaystyle=$ $\displaystyle 0$

Then we get

$\displaystyle(de^{a})_{\mu\nu}+(\omega_{\mu})^{a}{}_{b}e^{b}{}_{\nu}-(\omega_{\nu})^{a}{}_{b}e^{b}{}_{\mu}$
$\displaystyle=$ $\displaystyle 0$

We contract this equation by $e^{c\nu}$,

$\displaystyle(de^{[d})_{\mu\nu}e^{c]\nu}+(\omega_{\mu})^{[dc]}+(\omega_{\nu})^{a[d}e^{c]\nu}e^{a}{}_{\mu}$
$\displaystyle=$ $\displaystyle 0$
$\displaystyle\frac{1}{2}(de^{a})_{\mu\nu}e^{c\mu}e^{d\nu}e^{a}{}_{\rho}+(\omega_{\mu})^{a[d}e^{c]\mu}e^{a}{}_{\rho}$
$\displaystyle=$ $\displaystyle 0$

Subtracting these equations leaves us with

$\displaystyle(\omega_{\mu})^{dc}$ $\displaystyle=$ $\displaystyle
e^{\nu[d}(\partial_{\mu}e^{c]}_{\nu}-\partial_{\nu}e^{c]}_{\mu})-\frac{1}{2}e^{d\kappa}e^{c\tau}(\partial_{\kappa}e^{a}_{\tau}-\partial_{\tau}e^{a}_{\kappa})e^{a}_{\mu}$

This is a formula to compute the spin connection directly from the vielbein.

###  A.2 Covariant derivatives on a circle bundle

We denote 6d objects with hats, 5d base manifold objects without the hat. So
the 6d metric is

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\widehat{G}_{MN}dx^{M}dx^{N}$

We assume this metric has the circle-bundle form

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle
G_{\mu\nu}dx^{\mu}dx^{\nu}+r^{2}\left(d\theta+\kappa_{\mu}dx^{\mu}\right)^{2}$

The manifest Killing vector is $\widehat{v}=\partial_{\theta}$. For its lower
components we have $\widehat{v}_{\theta}=r^{2}$ and
$\widehat{v}_{\mu}=r^{2}\kappa_{\mu}$. The vielbein is

$\displaystyle\widehat{e}^{\widehat{\theta}}$ $\displaystyle=$ $\displaystyle
r\left(d\theta+\kappa_{\mu}dx^{\mu}\right)$
$\displaystyle\widehat{e}^{\widehat{\mu}}$ $\displaystyle=$ $\displaystyle
e^{\widehat{\mu}}{}_{\nu}dx^{\nu}$

The inverse vielbein is

$\displaystyle\widehat{e}_{\widehat{\theta}}$ $\displaystyle=$
$\displaystyle\frac{1}{r}\partial_{\theta}$
$\displaystyle\widehat{e}_{\widehat{\mu}}$ $\displaystyle=$ $\displaystyle
e^{\nu}{}_{\widehat{\mu}}\left(\partial_{\nu}-\kappa_{\nu}\partial_{\theta}\right)$

The inverse metric is

$\displaystyle\widehat{G}^{MN}\partial_{M}\phi\partial_{N}\phi$
$\displaystyle=$
$\displaystyle\frac{1}{r^{2}}\partial_{\theta}\phi\partial_{\theta}\phi+G^{\mu\nu}(\partial_{\mu}\phi-\kappa_{\mu}\partial_{\theta}\phi)(\partial_{\nu}\phi-\kappa_{\nu}\partial_{\theta}\phi)$

If we perform dimensional reduction along the fiber, then we put
$\partial_{\theta}=0$ and we get

$\displaystyle\widehat{G}^{MN}\partial_{M}\phi\partial_{N}\phi$
$\displaystyle=$ $\displaystyle
G^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi$

This corresponds to the metric

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle
G_{\mu\nu}dx^{\mu}dx^{\nu}$

on the base-manifold. The simple form of this dimensionally reduced metric is
why this form of the fiber-bundle metric is a preferred choice when we perform
dimensional reduction.

The covariant derivatives of a vector field
$\widehat{\nabla}_{M}\widehat{v}_{N}=\partial_{M}\widehat{v}_{N}-\widehat{\Gamma}_{MN}^{P}\widehat{v}_{P}$
are

$\displaystyle\widehat{\nabla}_{\theta}v_{\theta}$ $\displaystyle=$
$\displaystyle\partial_{\theta}v_{\theta}-r\nabla^{\rho}rv_{\rho}-r\kappa_{\rho}\nabla^{\rho}rv_{\theta}$
$\displaystyle\widehat{\nabla}_{\theta}v_{\mu}$ $\displaystyle=$
$\displaystyle\partial_{\theta}v_{\mu}-\frac{1}{2}r^{2}w_{\mu}{}^{\rho}v_{\rho}+r\kappa_{\mu}\nabla^{\rho}rv_{\rho}$
$\displaystyle+\frac{1}{2}r^{2}w_{\mu}{}^{\rho}\kappa_{\rho}v_{\theta}-r\kappa_{\mu}\kappa_{\rho}\nabla^{\rho}rv_{\theta}-\frac{1}{r}\nabla_{\mu}rv_{\theta}$
$\displaystyle\widehat{\nabla}_{\mu}v_{\nu}$ $\displaystyle=$
$\displaystyle\nabla_{\mu}v_{\nu}+r^{2}w^{\rho}{}_{(\mu}\kappa_{\nu)}v_{\rho}+r\nabla^{\rho}r\kappa_{\mu}\kappa_{\nu}v_{\rho}$
$\displaystyle-\nabla_{(\mu}\kappa_{\nu)}v_{\theta}-r^{2}\kappa_{\rho}w^{\rho}{}_{(\mu}\kappa_{\nu)}v_{\theta}-r\kappa_{\rho}\nabla^{\rho}r\kappa_{\mu}\kappa_{\nu}v_{\theta}-\frac{2}{r}\nabla_{(\mu}r\kappa_{\nu)}v_{\theta}$

The covariant derivatives of a spinor field
$\widehat{\nabla}_{M}\psi=\partial_{M}\psi+\frac{1}{4}\omega_{M}^{\widehat{M}\widehat{N}}\Gamma_{\widehat{M}\widehat{N}}\psi$
are

$\displaystyle\widehat{\nabla}_{\theta}\psi$ $\displaystyle=$
$\displaystyle\partial_{\theta}\psi-\frac{r^{2}}{8}\Gamma^{\mu\nu}\psi
w_{\mu\nu}-\frac{1}{2}\Gamma^{\mu}\Gamma^{\widehat{\theta}}\psi\partial_{\mu}r$
$\displaystyle\widehat{\nabla}_{\mu}\psi$ $\displaystyle=$
$\displaystyle\nabla_{\mu}\psi+\kappa_{\mu}\left(\widehat{\nabla}_{\theta}\psi-\partial_{\theta}\psi\right)+\frac{r}{4}\Gamma^{\nu}\Gamma^{\widehat{\theta}}\psi
w_{\mu\nu}$

Here we define $\Gamma_{\mu}=\Gamma_{\widehat{\nu}}e^{\widehat{\nu}}{}_{\mu}$
and
$\widehat{\Gamma}_{M}=\Gamma_{\widehat{N}}\widehat{e}^{\widehat{N}}{}_{M}$.

###  A.3 Reducing the conformal Killing spinor equation

Let us now analyze the conformal Killing spinor equations

$\displaystyle\widehat{\nabla}_{\mu}{\cal{\varepsilon}}$ $\displaystyle=$
$\displaystyle\widehat{\Gamma}_{\mu}\eta$
$\displaystyle\widehat{\nabla}_{\theta}{\cal{\varepsilon}}$ $\displaystyle=$
$\displaystyle\widehat{\Gamma}_{\theta}\eta$

We expand the covariant derivatives and the gamma matrices in 5d quantities,

$\displaystyle\nabla_{\mu}{\cal{\varepsilon}}+\kappa_{\mu}\left(\widehat{\nabla}_{\theta}{\cal{\varepsilon}}-\partial_{\theta}{\cal{\varepsilon}}\right)+\frac{r}{4}\Gamma^{\nu}\Gamma^{\widehat{\theta}}{\cal{\varepsilon}}w_{\mu\nu}$
$\displaystyle=$
$\displaystyle\Gamma_{\mu}\eta+r\kappa_{\mu}\Gamma_{\widehat{\theta}}\eta$
$\displaystyle\partial_{\theta}{\cal{\varepsilon}}-\frac{r^{2}}{8}\Gamma^{\mu\nu}{\cal{\varepsilon}}w_{\mu\nu}-\frac{1}{2}\Gamma^{\mu}\Gamma^{\widehat{\theta}}{\cal{\varepsilon}}\partial_{\mu}r$
$\displaystyle=$ $\displaystyle r\Gamma_{\widehat{\theta}}\eta$

We put $\partial_{\theta}{\cal{\varepsilon}}=0$ and get

$\displaystyle\nabla_{\mu}{\cal{\varepsilon}}+\frac{r}{4}\Gamma^{\nu}\Gamma^{\widehat{\theta}}{\cal{\varepsilon}}w_{\mu\nu}$
$\displaystyle=$ $\displaystyle\Gamma_{\mu}\eta$
$\displaystyle-\frac{r^{2}}{8}\Gamma^{\mu\nu}{\cal{\varepsilon}}w_{\mu\nu}-\frac{1}{2}\Gamma^{\mu}\Gamma^{\widehat{\theta}}{\cal{\varepsilon}}\partial_{\mu}r$
$\displaystyle=$ $\displaystyle r\Gamma_{\widehat{\theta}}\eta$

We get

$\displaystyle\nabla_{\mu}{\cal{\varepsilon}}+\frac{r}{4}\Gamma^{\nu}\Gamma^{\widehat{\theta}}{\cal{\varepsilon}}w_{\mu\nu}$
$\displaystyle=$
$\displaystyle-\frac{r}{8}\Gamma_{\mu}\Gamma^{\rho\sigma}\Gamma^{\widehat{\theta}}{\cal{\varepsilon}}w_{\rho\sigma}+\frac{1}{2r}\Gamma_{\mu}\Gamma^{\rho}{\cal{\varepsilon}}\partial_{\rho}r$

that we may also write as

$\displaystyle\nabla_{\mu}{\cal{\varepsilon}}$ $\displaystyle=$ $\displaystyle
M_{\mu}{\cal{\varepsilon}}$

where we define

$\displaystyle M_{\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{2r}\Gamma_{\mu}\Gamma^{\rho}\partial_{\rho}r-\frac{r}{8}\Gamma_{\mu}\Gamma^{\rho\sigma}\Gamma^{\widehat{\theta}}w_{\rho\sigma}-\frac{r}{4}\Gamma^{\nu}\Gamma^{\widehat{\theta}}w_{\mu\nu}$

###  A.4 Covariantly constant spinors on Taub-NUT

The conditions that both the derivatives
$\widehat{\nabla}_{\theta}{\cal{\varepsilon}}=0$ and
$\partial_{\theta}{\cal{\varepsilon}}=0$ are vanishing imply that

$\displaystyle\frac{r^{2}}{4}\Gamma^{\mu\nu}{\cal{\varepsilon}}w_{\mu\nu}+\Gamma^{\mu}\Gamma^{\widehat{\theta}}{\cal{\varepsilon}}\partial_{\mu}r$
$\displaystyle=$ $\displaystyle 0$ (A.1)

Let us study this condition in the context of Taub-NUT. There we have

$\displaystyle\frac{1}{2}\Gamma^{ij}{\cal{\varepsilon}}w_{ij}=\frac{1}{U^{1/2}}\Gamma^{i}\Gamma^{\widehat{\theta}}{\cal{\varepsilon}}\partial_{i}U$

We may express $w_{ij}=\epsilon_{ijk}\partial_{k}U$ in a covariant way as

$\displaystyle w_{ij}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{U}}G^{k{{\ell}}}{\cal{\varepsilon}}_{ijk}\partial_{{{\ell}}}U$

where ${\cal{\varepsilon}}_{ijk}=\sqrt{G}\epsilon_{ijk}$ is the covariant form
of the antisymmetric tensor where ${\cal{\varepsilon}}_{123}=1$ and totally
antisymmetric. Then

$\displaystyle\frac{1}{2}\Gamma^{ij}{\cal{\varepsilon}}G^{k{{\ell}}}{\cal{\varepsilon}}_{ijk}\partial_{{{\ell}}}U$
$\displaystyle=$
$\displaystyle\Gamma^{{{\ell}}}\Gamma^{\widehat{\theta}}{\cal{\varepsilon}}\partial_{{{\ell}}}U$

Let us cancel out $\partial_{{{\ell}}}U$ on both sides and use
$\Gamma_{k}\Gamma^{k}=3$ to get

$\displaystyle\frac{1}{6}\Gamma^{ijk}{\cal{\varepsilon}}{\cal{\varepsilon}}_{ijk}$
$\displaystyle=$ $\displaystyle\Gamma^{\widehat{\theta}}{\cal{\varepsilon}}$
(A.2)

This is a Weyl projection condition that reduces the amount of supersymmetry
by half and assures that ${\cal{\varepsilon}}$ is invariant under the
$SU(2)_{+}$ holonomy group.

Let us notice that for

$\displaystyle U$ $\displaystyle=$
$\displaystyle\frac{1}{R^{2}}+\frac{1}{2|\vec{x}|}$

we have

$\displaystyle w=-\frac{x_{k}}{4|\vec{x}|^{3}}\epsilon_{ijk}dx^{i}\wedge
dx^{j}$

which is a monopole with charge $\oint_{S^{2}}w=-2\pi$.

##  Appendix B Flat metric on $\mathbb{R}^{4}$

The flat metric on $\mathbb{R}^{4}=\mathbb{C}^{2}$ is
$ds^{2}=|dz_{1}|^{2}+|dz_{2}|^{2}$. We may parametrize the space by Euler
anges as

$\displaystyle z_{1}$ $\displaystyle=$
$\displaystyle\rho\cos\frac{\theta}{2}e^{-\frac{i}{2}(2\psi+\varphi)}$
$\displaystyle z_{2}$ $\displaystyle=$
$\displaystyle\rho\sin\frac{\theta}{2}e^{-\frac{i}{2}(2\psi-\varphi)}$

where $\psi\sim\psi+2\pi$. These coordinates can be obtained by acting on the
spin-up state with the Euler rotation

$\displaystyle g$ $\displaystyle=$ $\displaystyle
e^{-\frac{i}{2}\phi\sigma_{3}}e^{-\frac{i}{2}\theta\sigma_{2}}e^{-\frac{i}{2}2\psi\sigma_{3}}$
(B.1)

In this form it is clear that $g^{-1}=g^{{\dagger}}$ so the rotation operator
is unitary. The Maurer-Cartan forms are

$\displaystyle g^{-1}dg$ $\displaystyle=$
$\displaystyle-\frac{i}{2}\sigma_{a}\omega^{a}$

where

$\displaystyle\omega^{1}$ $\displaystyle=$ $\displaystyle\sin 2\psi
d\theta-\sin\theta\cos 2\psi d\phi$ $\displaystyle\omega^{2}$ $\displaystyle=$
$\displaystyle\cos 2\psi d\theta+\sin\theta\sin 2\psi d\phi$
$\displaystyle\omega^{3}$ $\displaystyle=$ $\displaystyle 2d\psi+\cos\theta
d\phi$

##  Appendix C Gauge group normalization

We assume the gauge group Lie algebra is

$\displaystyle[T_{a},T_{b}]$ $\displaystyle=$ $\displaystyle
if_{ab}{}^{c}T_{c}$

with the metric

$\displaystyle{\mbox{tr}}(T_{a}T_{b})=h_{ab}=\frac{1}{2}\delta_{ab}$

We write the gauge potential as $A_{\mu}=A^{a}_{\mu}T_{a}$. The field strength
is defined as

$\displaystyle F_{\mu\nu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}-i[A_{\mu},A_{\nu}]$

or in component form

$\displaystyle F^{a}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}A^{a}_{\nu}-\partial_{\mu}A^{a}_{\nu}+f_{bc}{}^{a}A^{b}_{\mu}A^{c}_{\nu}$

We fix the normalization by taking $SU(2)$ gauge group with group element $g$
as in (B.1) parametrized by the Euler angles. Then we get

$\displaystyle{\mbox{tr}}(g^{-1}dg)$ $\displaystyle=$
$\displaystyle\frac{3}{2}\omega^{1}\omega^{2}\omega^{3}$ $\displaystyle=$
$\displaystyle-3\sin\theta d\theta d\varphi d\psi$

which gives

$\displaystyle\int{\mbox{tr}}\left(g^{-1}dg\right)^{3}$ $\displaystyle=$
$\displaystyle-24\pi^{2}$

and consequently we shall normalize the Wess-Zumino term as

$\displaystyle\frac{1}{12\pi}\int{\mbox{tr}}\left(g^{-1}dg\right)^{3}$
$\displaystyle=$ $\displaystyle\frac{1}{12\pi}\int
d^{3}x\sqrt{-G}{\cal{\varepsilon}}^{\mu\nu\lambda}{\mbox{tr}}\left(g^{-1}\partial_{\mu}gg^{-1}\partial_{\nu}g^{-1}\partial_{\lambda}g\right)$

and it will be quantized in units of $2\pi$ where the integer measures the
winding number as we map $S^{3}$ into $SU(2)$.

##  Appendix D Gamma matrices

###  D.1 Two dimensions

In 2d we have the Majorana representation

$\displaystyle\gamma^{0}$ $\displaystyle=$ $\displaystyle i\sigma^{2}$
$\displaystyle\gamma^{1}$ $\displaystyle=$ $\displaystyle\sigma^{1}$

The chirality matrix is then $\gamma=\gamma^{01}=\sigma^{3}$. The charge
conjugation matrix is $C=\gamma^{0}=i\sigma^{2}={\cal{\varepsilon}}$. The
Dirac conjugate spinor is $\bar{\psi}=\psi^{{\dagger}}\gamma^{0}$. Writing out
the spinor components, we have the spinor $\psi^{u}$ acted on by the gamma
matrices $(\gamma^{\alpha})^{u}{}_{v}$, however, the charge conjugatation
matrix is ${\cal{\varepsilon}}_{uv}$ with component
${\cal{\varepsilon}}_{+-}=1$. The Dirac conjugate is
$\bar{\psi}_{v}=(\psi^{u})^{*}i(\sigma^{2})^{u}{}_{v}$. We note that
$(\psi^{u})^{*}$ transforms under Lorentz rotations like
$\psi_{u}=\psi^{v}{\cal{\varepsilon}}_{vu}$. For the components, we have
$\bar{\psi}_{+}=-(\psi^{-})^{*}$ and $\bar{\psi}_{-}=(\psi^{+})^{*}$. We may
impose the Majorana condition
$\bar{\psi}_{u}=\psi^{v}{\cal{\varepsilon}}_{vu}$. In the Majorana
representation it amounts to a spinor with real components,
$(\psi^{+})^{*}=\psi^{+}$ and $(\psi^{-})^{*}=\psi^{-}$. We note that
$C\gamma^{\mu}$ are symmetric.

###  D.2 Five dimensions

We define 5d gamma matrices as

$\displaystyle\gamma^{0}$ $\displaystyle=$
$\displaystyle(\gamma^{0})^{u}{}_{v}\delta^{m}_{n}$ $\displaystyle\gamma^{4}$
$\displaystyle=$ $\displaystyle(\gamma^{1})^{u}{}_{v}\delta^{m}_{n}$
$\displaystyle\gamma^{i}$ $\displaystyle=$
$\displaystyle\gamma^{u}{}_{v}(\sigma^{i})^{m}{}_{n}$

The charge conjugation matrix is

$\displaystyle C$ $\displaystyle=$
$\displaystyle(\gamma^{1})^{u}{}_{v}{\cal{\varepsilon}}_{mn}$

We have

$\displaystyle\gamma^{04}$ $\displaystyle=$
$\displaystyle\gamma^{u}{}_{v}\delta^{m}_{n}$ $\displaystyle\gamma^{01234}$
$\displaystyle=$ $\displaystyle-i\delta^{u}_{v}\delta^{m}_{n}$

We note that both $C$ and $C\gamma^{\mu}$ are antisymmetric, consistent with
the decomposition of a product of two spinors into a scalar, a vector and an
antisymmetric tensor,

$\displaystyle 4\otimes 4$ $\displaystyle=$ $\displaystyle 1_{anti}\oplus
5_{anti}\oplus 10_{symm}$

###  D.3 Eleven dimensions

We define the gamma matrices as

$\displaystyle\Gamma^{\mu}$ $\displaystyle=$
$\displaystyle\gamma^{\mu}\otimes\sigma^{1}\otimes 1$
$\displaystyle\Gamma^{\psi}$ $\displaystyle=$ $\displaystyle
1\otimes\sigma^{2}\otimes 1$ $\displaystyle\Gamma^{A}$ $\displaystyle=$
$\displaystyle 1\otimes\sigma^{3}\otimes\tau^{A}$

where $A=6,7,8,9,10$ and $\tau^{12345}=1$. The 6d chirality matrix is

$\displaystyle\Gamma=\Gamma^{0}\Gamma^{123}\Gamma^{4}\Gamma^{\psi}=1\otimes\sigma^{3}\otimes
1$

The charge conjugatation matrix is

$\displaystyle C$ $\displaystyle=$ $\displaystyle
C_{5}\otimes{\cal{\varepsilon}}\otimes C^{\prime}_{5}$

where ${\cal{\varepsilon}}=i\sigma^{2}$. Thus $C$ is antisymmetric while
$C\Gamma^{M}$ for $M=(\mu,\psi)$ and $C\Gamma^{A}$ are symmetric. The Majorana
condition is

$\displaystyle\bar{\psi}$ $\displaystyle=$ $\displaystyle\psi^{T}C$

where $\bar{\psi}=\psi^{{\dagger}}\Gamma^{0}$.

##  Appendix E 5d SYM in reduced notation

Expressed in terms of 5d gamma matrices and 5d spinors, we have

$\displaystyle\delta\phi^{A}$ $\displaystyle=$
$\displaystyle-i{\cal{E}}\tau^{A}\chi$ $\displaystyle=$ $\displaystyle
i\chi\tau^{A}{\cal{E}}$ $\displaystyle\delta A_{\mu}$ $\displaystyle=$
$\displaystyle{\cal{E}}\gamma_{\mu}\chi$ $\displaystyle=$
$\displaystyle-\chi\gamma_{\mu}{\cal{E}}$ $\displaystyle\delta\chi$
$\displaystyle=$
$\displaystyle-\frac{i}{2}\gamma^{\mu\nu}{\cal{E}}F_{\mu\nu}-\gamma^{\mu}\tau^{A}{\cal{E}}{\cal{D}}_{\mu}\phi^{A}$
$\displaystyle-\frac{ir}{2}\gamma^{\mu\nu}\tau^{A}{\cal{E}}w_{\mu\nu}\phi^{A}-\frac{1}{2}\tau^{AB}{\cal{E}}[\phi^{A},\phi^{B}]$

## References

  * [1] A. Sen, “Dynamics of multiple Kaluza-Klein monopoles in M and string theory,” Adv. Theor. Math. Phys. 1, 115-126 (1998) [arXiv:hep-th/9707042 [hep-th]].
  * [2] H. Baum, F. Leitner, “The twistor equation in Lorentzian spin geometry,” Math. Z. 247 (2004) 795.
  * [3] H. Linander and F. Ohlsson, “(2,0) theory on circle fibrations,” JHEP 01 (2012), 159 [arXiv:1111.6045 [hep-th]].
  * [4] F. Ohlsson, “(2,0) theory on Taub-NUT: A note on WZW models on singular fibrations,” [arXiv:1205.0694 [hep-th]].
  * [5] E. Witten, “Geometric Langlands From Six Dimensions,” [arXiv:0905.2720 [hep-th]].
  * [6] dos Santos, Wytler Cordeiro. (2020). “Notes on the Weyl tensor, decomposition of Riemann tensor, Ruse-Lanczos identity and duality of the curvature tensor,” Zenodo. https://doi.org/10.5281/zenodo.3814369
  * [7] D. V. Belyaev and P. van Nieuwenhuizen, “Rigid supersymmetry with boundaries,” JHEP 04 (2008), 008 [arXiv:0801.2377 [hep-th]].
  * [8] E. Witten, “On Holomorphic factorization of WZW and coset models,” Commun. Math. Phys. 144 (1992), 189-212
  * [9] E. Witten, “Nonabelian Bosonization in Two-Dimensions,” Commun. Math. Phys. 92, 455-472 (1984)
  * [10] R. Dijkgraaf, L. Hollands, P. Sulkowski and C. Vafa, “Supersymmetric gauge theories, intersecting branes and free fermions,” JHEP 02, 106 (2008) [arXiv:0709.4446 [hep-th]].
  * [11] E. Witten, “Five-brane effective action in M theory,” J. Geom. Phys. 22, 103-133 (1997) [arXiv:hep-th/9610234 [hep-th]].
  * [12] N. Lambert and M. Owen, “Charged Chiral Fermions from M5-Branes,” JHEP 04 (2018), 051 [arXiv:1802.07766 [hep-th]].
  * [13] F. Ohlsson, “(2,0) theory on Taub-NUT: A note on WZW models on singular fibrations,” [arXiv:1205.0694 [hep-th]].
  * [14] E. Abdalla and M. C. B. Abdalla, “Supersymmetric Extension of the Chiral Model and Wess-Zumino Term in Two-dimensions,” Phys. Lett. B 152, 59-62 (1985)
  * [15] C. S. Chu and D. J. Smith, “Multiple Self-Dual Strings on M5-Branes,” JHEP 01, 001 (2010) [arXiv:0909.2333 [hep-th]].