QMUL-PH-22-21

Twisted Self-duality

David S<EMAIL_ADDRESS>Tancredi Schettini
<EMAIL_ADDRESS>
Centre for Theoretical Physics, Physical and Chemical Sciences

Queen Mary University of London, 327 Mile End Road, London E1 4NS, UK

Abstract

We examine a generalisation of the usual self-duality equations for Yang-Mills
theory when the colour space admits a non-trivial involution. This involution
allows us to construct a non-trivial twist which may be combined with the
Hodge star to form a twisted self-dual curvature. We will construct a simple
example of twisted self-duality for $su(2)\oplus su(2)$ gauge theory along
with its explicit solutions and then dimensionally reduce from four dimensions
to obtain families of non-trivial non-linear equations in lower dimensions.
This twisted self-duality constraint will be shown to arise in $E_{7}$
exceptional field theory through a Scherk-Schwarz reduction and we will show
how an Eguchi-Hanson gravitational instanton also obeys the twisted self-
duality condition.

##  1 Introduction

The importance of instanton solutions in Yang-Mills theory, described in the
seminal works Belavin et al. (1975); ’t Hooft (1976a); Yang (1977), needs
little introduction. Apart from their central role in understanding the non-
perturbative physics of Yang-Mills theory, the study of self-duality has
provided a fertile ground for a host of mathematical ideas such as
integrabilty and topology, see for instance Atiyah et al. (1978); Ablowitz et
al. (2003). The central mathematical fact is that, in four dimensions with
Euclidean signature, the Hodge Star acting on a two form curvature squares to
one. This allows us to think of the Hodge star as an operator on the two form
curvature with two eigenvalues, $+1$ and $-1$. The self dual sector has the
positive eigenvalue and the anti-self dual sector has the negative eigenvalue.
When we have a similar non-trivial operator acting on the colour space, i.e.
an operator that squares to one, we may then also decompose the colour space
in this way, according to eigenvectors of this involution. Combining the two
operators of Hodge star and the internal involution allows the construction of
a twisted self-duality, a term coined in Cremmer et al. (1998a, b). In fact,
this sort of twisted self-duality has appeared in various formulations of
supergravity and in world sheet duality symmetric formulaitons of the string
theory Tseytlin (1990, 1991); Berman et al. (2008); Berman and Thompson
(2014); Alfonsi and Berman (2021) To be concrete, let us denote with
$\mathcal{F}^{a}$ the curvature two-form that takes values in some Lie algebra
$\bf{g}$. The upper index, $a=1,...,dim(\bf{g})$ is a vector index in a
particular basis for the algebra, $\bf{g}$. Let $\mathcal{J}$ be the operator
that acts on the vector space such that

$\displaystyle(\mathcal{J})^{2}=1\,.$ (1)

Twisted self-duality is then when the Hodge star is combined with
$\mathcal{J}$, so that the curvature obeys the equation:

$\displaystyle\mathcal{F}=*\mathcal{J}\mathcal{F}\,.$ (2)

We will construct an explicit (and probably the simplest) example shortly.
The obvious feature and a necessity for its consistency is that, the combined
operator,

$\displaystyle(*\mathcal{J})^{2}=1\,.$ (3)

This is trivially true in the case described above since both the Hodge star
and $\mathcal{J}$ are involutions. But it does also beg the fascinating
question of just demanding (3) while allowing $*^{2}=-1$ and
$\mathcal{J}^{2}=-1$. In four dimensional Lorenzian space, $*^{2}=-1$, which
rules out real self-dual solutions. Twisted self-dual solutions however are
not ruled out provided one has a twist operator such that
$\mathcal{J}^{2}=-1$. This case is a subject of further investigation, but in
this paper we focus on the Euclidean case described above.
It would be interesting to classify possible operators $\mathcal{J}$ for any
given gauge algebra $\bf{g}$; we will not do so here. Instead, we will try to
be as simple and constructive as possible. First, let us ignore Abelian
groups, where the analysis is simple though certainly not uninteresting.
Indeed the Abelian case which was explored as part of Berman and Rudolph
(2014) was one of the motivations for this paper. The simplest non-abelian
gauge group is $su(2)$, and so a natural way to construct an algebra with an
operator $\mathcal{J}$ is simply to have two copies of $su(2)$ and have
$\mathcal{J}$ map between them. And so we consider the case of
${\bf{g}}=su(2)\oplus su(2)$ and

$\displaystyle\mathcal{J}=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}\,.$ (4)

Of course, one can then use the isomorphism $so(4)=su(2)\oplus su(2)$ and
think of this as an $so(4)$ gauge theory.

In what follows we will first look for solutions of this twisted self-dual
equation. Then we will examine properties of the reduction of this equation to
lower dimensions. This is inspired by the reduction properties of the ordinary
self-dual equations and their relation to integrable systems, as described in
Ablowitz et al. (2003). Following this we will show how the Eguchi-Hanson
instanton originally presented in Eguchi and Hanson (1979a) can be thought of
as a twisted self-dual solution. And finally we will present the derivation of
the twisted self-duality equation as a specific Scherk-Schwarz type reduction
of the equations in $E_{7}$ exceptional field theory (for a review see Berman
and Blair (2020) and references therin). This provided the original
inspiration to study these twisted self-duality equations but we believe they
are interesting enough in their own right that they merit study independent to
the $E_{7}$ exceptional field theory context.

##  2 Instanton Pairs

This section is aimed at presenting the simplest example of twisted self-dual
equation for a non-abelian theory, mentioned above, and constructing natural
solutions by making use of the famous ’t Hooft ansatz. Some exhaustive and
pedagogical reviews on the subject can be found in Vandoren and van
Nieuwenhuizen (2008); Shifman (1994); Actor (1979).

###  2.1 The Standard $SU(2)$ Instanton

The archetype of instanton solution to Yang Mills theory is the t’ Hooft
ansatz constructed more than fifty years ago to solve the $SU(2)$ case in
Euclidean spacetime, which can be found in ’t Hooft (1976b); Jackiw and Rebbi
(1976), for example.
Let us quickly review the Bogomoln’yi bound argument, which neatly shows why
instantons solve the dynamics. Starting from the Yang-Mills action in
Euclidean spacetime, we can complete the square to find a bound for its value:

$\displaystyle S_{Y-M}$ $\displaystyle=\frac{1}{2}\int
d^{4}x\operatorname{Tr}F_{\mu\nu}F^{\mu\nu}$ (5)
$\displaystyle=\frac{1}{4}\int
d^{4}x\operatorname{Tr}\left(F_{\mu\nu}\mp\tilde{F}_{\mu\nu}\right)^{2}\pm
2\operatorname{Tr}F_{\mu\nu}\tilde{F}^{\mu\nu}\geq\frac{1}{2}\int
d^{4}x\operatorname{Tr}F_{\mu\nu}\tilde{F}^{\mu\nu},$ (6)

where
$\tilde{F}_{\mu\nu}=\pm\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}F^{\rho\sigma}$,
with $\epsilon_{\mu\nu\rho\sigma}$ being the Levi-Civita symbol in four
dimensions. The equality holds when $F=\pm\tilde{F}$. Hence, a choice of gauge
field $A_{\mu}$ such that its field strength satisfies
$F_{\mu\nu}=\pm\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}F_{\rho\sigma}$ minimises
the action. If the sign is positive we refer to it as an instanton, while the
minus sign is associated with an anti-instanton. In differential geometry
terms, if we let $*$ be the Hodge star, then the statement above is compactly
written as $F=\pm*F$ and $F$ said to be a (anti) self-dual curvature. The
study of such objects yielded crucial insights on the nature of manifolds and
their classification, see for instance the seminal work of Atiyah et al.
(1978). For the purpose of this article, it is only necessary to introduce the
original ’t Hooft ansatz for the instanton. This provides the connection and
self-dual curvature on the $SU(2)$ bundle over Euclidean four-dimensional
space:

$\displaystyle
A_{\mu}^{a}=2\frac{\sigma_{\mu\nu}^{a}\left(x-x_{0}\right)_{\nu}}{\left(x-x_{0}\right)^{2}+\rho^{2}},$
$\displaystyle
F_{\mu\nu}^{a}=2\partial_{[\mu}A_{\nu]}{}^{a}+\epsilon^{bca}A_{\mu}^{b}A^{c}_{\nu}=-4\frac{\sigma_{\mu\nu}^{a}\rho^{2}}{[\left(x-x_{0}\right)^{2}+\rho^{2}]^{2}},$
(7)

where $\sigma_{\mu\nu}^{a}$ are the self-dual ’t Hooft symbols. They are
defined as:

$\displaystyle\sigma^{a\,\mu\nu}=\begin{cases}\sigma^{a\,44}=0\\\
\sigma^{a\,4\nu}=-\delta^{a\nu}\hskip 11.38092pt\textrm{for}\,\,\,\nu=1,2,3\\\
\sigma^{a\,\mu 4}=\delta^{a\mu}\hskip 11.38092pt\textrm{for}\,\,\,\mu=1,2,3\\\
\sigma^{a\,\mu\nu}=\epsilon^{a\mu\nu}\hskip
11.38092pt\textrm{for}\,\,\,\mu,\nu=1,2,3.\end{cases}$ (8)

The anti self-dual curvature solution is completely analogous, just with
$\sigma_{\mu\nu}^{a}$ replaced by $\bar{\sigma}_{\mu\nu}^{a}$, the anti self-
dual t’ Hooft symbols, which read:

$\displaystyle\bar{\sigma}^{a\,\mu\nu}=\begin{cases}\bar{\sigma}^{a\,44}=0\\\
\bar{\sigma}^{a\,4\nu}=\delta^{a\nu}\hskip
11.38092pt\textrm{for}\,\,\,\nu=1,2,3\\\ \bar{\sigma}^{a\,\mu
4}=-\delta^{a\mu}\hskip 11.38092pt\textrm{for}\,\,\,\mu=1,2,3\\\
\bar{\sigma}^{a\,\mu\nu}=\epsilon^{a\mu\nu}\hskip
11.38092pt\textrm{for}\,\,\,\mu,\nu=1,2,3.\end{cases}$ (9)

###  2.2 Decomposition of the twisted self-dual equation

We will use these tools to build a solution to the the $su(2)\oplus su(2)$
twisted self-dual equation, i.e. equation 2, where $\mathcal{J}$ is given by
4. Explicitly, this reads:

$\displaystyle\left(\begin{array}[]{c}F\\\ \bar{F}\\\
\end{array}\right)=*\begin{pmatrix}0&1\\\
1&0\end{pmatrix}\left(\begin{array}[]{c}F\\\ \bar{F}\\\ \end{array}\right),$
(14)

with $F,\bar{F}$ being $su(2)$ curvature forms associated with $su(2)$
connections $A,\bar{A}$, respectively. We start by using some very basic
techniques and concepts from matrix algebra: eigenvectors and eigenvalues. Let
us think of $F,\bar{F}$ in terms of their colour indices, so that each of them
is a three-vector in colour space. Then, we perform the following
decomposition:

$\displaystyle\left(\begin{array}[]{c}F\\\ \bar{F}\\\
\end{array}\right)=\left(\begin{array}[]{c}F^{1}\\\ \,F^{1}\\\
\end{array}\right)+\left(\begin{array}[]{c}F^{2}\\\ \,-F^{2}\\\
\end{array}\right),$ (21)

where it should be clear that $F^{1},F^{2}$ are also three-vectors. The two
vectors on the RHS are linearly independent. What we are doing is nothing but
a decomposition into eigenvectors of the matrix $\begin{pmatrix}0&1\\\
1&0\end{pmatrix}$.

And so, we are decomposing the $su(2)\oplus su(2)$ field strength into
components that live in the two eigenspaces defined by the almost product
structure above. Then, we obtain:

$\displaystyle\left(\begin{array}[]{c}F^{1}\\\ \,F^{1}\\\
\end{array}\right)+\left(\begin{array}[]{c}F^{2}\\\ -\,F^{2}\\\
\end{array}\right)=*\begin{pmatrix}0&1\\\
1&0\end{pmatrix}\Bigg{[}\left(\begin{array}[]{c}F^{1}\\\ \,F^{1}\\\
\end{array}\right)+\left(\begin{array}[]{c}F^{2}\\\ -\,F^{2}\\\
\end{array}\right)\Bigg{]}=*\Bigg{[}\left(\begin{array}[]{c}F^{1}\\\
\,F^{1}\\\ \end{array}\right)-\left(\begin{array}[]{c}F^{2}\\\ -\,F^{2}\\\
\end{array}\right)\Bigg{]}.$ (34)

This gives the following two equations (which are in fact six):

$\displaystyle\,F^{1}=*\,F^{1},$ $\displaystyle-\,F^{2}=*\,F^{2}.$ (35)

Hence, we obtained two self-dual equations, which are now untwisted. We know
the solution to those in terms of $F^{1}$ and $F^{2}$: they must be
proportional to the ’t Hooft symbols. Now, we would be tempted to complete the
job by solving for the gauge fields $A^{1}$ and $A^{2}$ that correspond to
$F^{1}$ and $F^{2}$, respectively. Such solutions are to the usual instanton
and anti-instanton. However, the real field strengths in our theory are $F$
and $\bar{F}$, and it is them that need to be written in terms of gauge fields
$A$ and $\bar{A}$. 333Clearly, if we look at the Abelian case, such
consideration is superfluous. However, in the non-Abelian case, the relation
between $A^{1,2}$ and $A,\bar{A}$ is highly non-trivial.

###  2.3 Solutions to the twisted self-dual equation

A very simple solution can be found when we restrict only to one of the two
eigenspaces. Specifically, setting $F^{2}=0$ yields the usual $su(2)$ self-
dual instanton:

$\displaystyle
A_{\mu}^{a}=\bar{A}_{\mu}^{a}=A^{1}_{\mu}{}^{a}=2\frac{\sigma_{\mu\nu}^{a}\left(x-x_{0}\right)_{\nu}}{\left(x-x_{0}\right)^{2}+\rho^{2}},$
$\displaystyle
F_{\mu\nu}^{a}=\bar{F}_{\mu\nu}^{a}=F^{1}_{\mu\nu}{}^{a}=-4\frac{\sigma_{\mu\nu}^{a}\rho^{2}}{[\left(x-x_{0}\right)^{2}+\rho^{2}]^{2}}.$
(36)

This solution has clearly a "instanton number" of 2, since each of the two
$su(2)$ blocks contributes with instanton number 1.
If we restrict to the other eigenspace, things radically change. We need to
solve simultaneously:

$\displaystyle\begin{cases}-F^{2}=*F^{2}\\\ F^{2}=dA-[A,A]\\\
-F^{2}=d\bar{A}-[\bar{A},\bar{A}].\end{cases}$ (37)

The first two equations are satisfied by the anti-instanton solution (i.e.
instanton with instanton number $-1$) and the final equation is for a field
strength with the opposite sign of the anti-instanton. Note that since the
field strength is non-linear in potentials it is not possible to generate the
opposite field strength by scaling the potential by $-1$. We have not be able
to construct solutions for this choice.

Finally, we are left with the most general case of both eigenmodes
contributing to the field strength. If we choose both of them to be
proportional to the same scalar function $f(x^{\mu})$, then we obtain

$\begin{cases}F^{a}_{ij}=\epsilon^{aij}f(x^{\mu})\\\
F^{a}_{i4}=0\end{cases}\begin{cases}\bar{F}^{a}_{ij}=0\\\
\bar{F}^{a}_{i4}=\delta^{a}_{i}f(x^{\mu}).\end{cases}$ (38)

This essentially puts an electric solution in one $su(2)$ and a magnetic
solution in the other $su(2)$. After making some gauge choices, it is easy to
verify that a solution to such system is given by the following non-zero
components of the gauge fields:

$\displaystyle\bar{A}_{4}^{a}=x^{a}f(t),$ $\displaystyle
A^{a}_{i}=\delta^{a}_{i}\sqrt{f(t)},$ (39)

where $f(t)$ is an arbitrary function of time. We will return to these
solutions in section 5, in the context of $E_{7}$ ExFT, where the twisted
self-dual equation will be accompanied by an extra constraint.

##  3 Dimensional Reduction of the twisted self dual equations

In this section, we present some interesting dimensional reductions of the
twisted self-dual equation. We first look at the case where we are dealing
$2n$ copies of Maxwell theory, and find a reduction to the chirality condition
of the duality symmetric sigma model studied in Tseytlin (1991, 1990); Berman
et al. (2008); Kimura et al. (2022). Then, we look at the $su(2)\oplus su(2)$
Yang-Mills theory, providing some generalisations of the work in Ablowitz et
al. (2003).

###  3.1 Dimensional Reduction to the duality symmetric Sigma Model

An $O(d,d)$ symmetric form of the string was introduced in Tseytlin (1991,
1990) where the idea was to double the dimensionality of the target space so
that the $O(d,d)$ T-duality group would act linearly on this doubled space.
Now the target space was doubled the string world sheet theory would need an
additional set of constraints to halve its degrees of freedom to the right
number. This constrained doubled system should then be equivalent to the
ordinary theory in the non-doubled space. This was achieved as follows. The
action of the doubled sigma model with Lorentzian worldsheet reads:

$\displaystyle
S=\frac{1}{4}\int_{\Sigma}\left[\mathcal{H}_{MN}\mathrm{~{}d}\mathbb{X}^{M}\wedge*\mathrm{~{}d}\mathbb{X}^{N}-\Omega_{MN}\mathrm{~{}d}\mathbb{X}^{M}\wedge\mathrm{d}\mathbb{X}^{N}\right]\,,$
(40)

where $\mathbb{X}^{M}$ are from the world sheet perspective 2d scalars where
$(M=1..2d)$. The following twisted self-duality or Chirality relation must
then be imposed:

$\displaystyle\mathrm{d}\mathbb{X}^{M}=\mathcal{J}^{M}{}_{N}*\mathrm{~{}d}\mathbb{X}^{N}.$
(41)

The origin of the Chirality constraint from the perspective of geometric
quantisation is described in Alfonsi and Berman (2021). In what follows, we
will show how the twisted self-dual equation for $2n$ copies of the abelian
Yang-Mills theory can be dimensionally reduced to such condition. Let $F$ be
the collection of $n$ abelian field strengths, and let $\bar{F}$ be an object
of the same kind. Then, the twisted self-dual equation of the form 14 for the
above field strengths living in Euclidean external spacetime reads:

$\displaystyle F_{12}=\bar{F}_{34},\quad F_{13}=\bar{F}_{42},\quad
F_{14}=\bar{F}_{23},$
$\displaystyle\bar{F}_{12}=F_{34},\quad\bar{F}_{13}=F_{42},\quad\bar{F}_{14}=F_{23}.$
(42)

We are suppressing the internal indices for brevity, and it should be
understood that there are $n$ $F$’s and $n$ $\bar{F}$’s, pairwise related as
above.
We start the dimensional reduction by setting $\partial_{3}=\partial_{4}=0$.
Moreover, we also impose $A_{3}=\bar{A}_{3}=A_{4}=\bar{A}_{4}=0$. This
immediately sets $F_{\mu\nu}=0$ (trivially) for $(\mu,\nu)\neq(1,2)$. Hence,
we are left with $F_{12}=0=\bar{F}_{12}$ which needs to be solved for
$A_{1},A_{2},\bar{A}_{1},\bar{A}_{2}$. If we let $\bar{A}_{2}=A_{1}$ and
$\bar{A}_{1}=A_{2}$ we obtain:

$\displaystyle\partial_{1}A_{2}=\partial_{2}A_{1}\,\,\,,\quad\partial_{2}A_{2}=\partial_{1}A_{1}\,.$
(43)

Relabelling $A_{2}=X$ and $A_{1}=\bar{X}$, and reinserting the internal
structure, we get:

$\displaystyle\partial_{1}X^{M}=\partial_{2}\bar{X}^{M}\,\,\,,\quad\partial_{2}X^{M}=\partial_{1}\bar{X}^{M},$
(44)

where $M=1,...,n$. This is nothing but the equation derived in Kimura et al.
(2022) in the Born sigma model context:

$\displaystyle\mathrm{d}\mathbb{X}^{M}=\mathcal{J}^{M}{}_{N}*\mathrm{~{}d}\mathbb{X}^{N}.$
(45)

To unpack it and show the equivalence, we note that we can write the above as:

$\displaystyle\partial_{\alpha}\mathbb{X}^{M}=\mathcal{J}^{M}{}_{N}\varepsilon_{\alpha\beta}\partial^{\beta}\mathbb{X}^{N},$
(46)

where $\varepsilon_{\alpha\beta}$ is the epsilon tensor in the two-dimensional
Lorentzian worldsheet,444One might wonder that, since we are in Lorentzian
space, $*^{2}=-1$, but we are dealing with one-forms in a two-dimensional
manifold, so it does indeed square to one. and we have:

$\displaystyle\mathcal{J}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix},\quad\mathbb{X}^{M}=\begin{pmatrix}X^{\mu}\\\
\bar{X}_{\mu}\end{pmatrix}.$ (47)

.

Before we move to the various non-Abelian models and reductions in the next
section it is worth pondering whether the four dimensional twisted self-
duality might have its origin from a higher dimensional self-dual object. The
natural candidate would be the M-theory five brane. The five-brane has a self-
dual two form whose reduction on a torus leads to a single maxwell field and
as shown in Berman (1997, 1998) different gauge choices in the 6 dimensional
theory lead to two Maxwell fields related by a duality equation. Thinking of
the 4d theory as an $u(1)\oplus u(1)$ gauge theory then the duality constraint
originating from the reduciton of the fivebrane self-duality equation would be
a twisted self-duality equation in 4d. This is all for the Abelian theory,
having a twisted non-Abelian self-duality equation from the fivebrane is tied
to the puzzle of non-Abelian Gerbes.

###  3.2 Dimensional reduction to 2D Integrable Models

The usual (i.e. untwisted) Yang-Mills self-dual equation in Euclidean flat
space is a rich source of integrable equations (see Ward conjecture, Ward
(1985)). We first provide a quick recap on how some of its reductions have
been obtained for the 4d case, following the work of Ablowitz et al. (2003).
We then go on to show that the twisted case encodes all the content of the
untwisted one as a special case, and emphasize how it differs with three
specific reductions.

### The Usual Self-dual Equation Case

In its most compact form, the Yang-Mills self-dual equation reads:

$\displaystyle F=*F,$ (48)

where $*$ depends only on the choice of signature for a flat metric. For the
Euclidean setting, which is the one that is most often examined, we have that
$*^{2}=1$ and the equations read:

$\displaystyle F_{12}=F_{34},\quad F_{13}=F_{42},\quad F_{14}=F_{23}.$ (49)

Consider the following coordinate change:

$\displaystyle\sigma=\frac{1}{\sqrt{2}}\left(x^{2}+ix^{3}\right),\quad\tau=\frac{1}{\sqrt{2}}\left(x^{1}-ix^{4}\right),$
$\displaystyle\tilde{\sigma}=\frac{1}{\sqrt{2}}\left(x^{2}-ix^{3}\right),\quad\tilde{\tau}=\frac{1}{\sqrt{2}}\left(x^{1}+ix^{4}\right).$
(50)

It induces a field redefinition in both $A$ and $\bar{A}$, according to:

$\displaystyle
A_{4}=\frac{1}{\sqrt{2}}\left(A_{\tau}+A_{\bar{\tau}}\right),\quad
A_{1}=\frac{1}{\sqrt{2}}\left(A_{\sigma}+A_{\bar{\sigma}}\right),$ (51)
$\displaystyle
A_{2}=\frac{i}{\sqrt{2}}\left(A_{\sigma}-A_{\bar{\sigma}}\right),\quad
A_{3}=-\frac{i}{\sqrt{2}}\left(A_{\tau}-A\bar{\tau}\right),$ (52)

which leads to the simpler set of equations:

$\displaystyle F_{\sigma\tau}=0,\quad F_{\bar{\sigma}\bar{\tau}}=0,\quad
F_{\sigma\bar{\sigma}}+F_{\tau\bar{\tau}}=0.$ (53)

This system is the integrability condition for a simple isospectral linear
problem, and its reductions can be obtained by simply choosing a specific
dependence of the fields on the coordinates. For more details, we refer to
Ablowitz et al. (2003).

### The non-linear Schrödinger Equation

With the same choice of coordinates as for the untwisted case, the twisted
self-dual equation become:

$\displaystyle F_{\sigma\tau}+\bar{F}_{\sigma\tau}=0\,\,\,,\hskip
28.45274ptF_{\sigma\bar{\sigma}}+\bar{F}_{\tau\bar{\tau}}=0\,\,\,,\hskip
28.45274ptF_{\sigma\bar{\tau}}-\bar{F}_{\sigma\bar{\tau}}=0\,\,\,,$
$\displaystyle
F_{\bar{\sigma}\bar{\tau}}+\bar{F}_{\bar{\sigma}\bar{\tau}}=0\,\,\,,\hskip
28.45274pt\bar{F}_{\sigma\bar{\sigma}}+F_{\tau\bar{\tau}}=0\,\,\,,\hskip
28.45274ptF_{\bar{\sigma}\tau}-\bar{F}_{\bar{\sigma}\tau}=0\,\,\,.$ (54)

It is easy to see that for $\bar{F}=F$, this reduces to the standard case. We
now proceed to present a specific reduction, which is analogous to the one in
Ablowitz et al. (2003), but for the more general case of the twisted self-dual
equation.

We restrict the dependence of the fields to $x=\sigma+\bar{\sigma}$, $t=\tau$,
and set $A_{\sigma}=0=A_{\bar{\sigma}}$, and we obtain the following six
equations:

$\displaystyle\partial_{x}A_{\tau}+\partial_{x}\bar{A_{\tau}}=0\hskip
75.39963pt\partial_{x}A_{\bar{\sigma}}-\partial_{t}\bar{A}_{\tau}-[\bar{A}_{\tau},\bar{A}_{\bar{\tau}}]=0\hskip
28.45274pt\partial_{x}A_{\bar{\tau}}-\partial_{x}\bar{A}_{\bar{\tau}}=0$
$\displaystyle\partial_{x}A_{\bar{\tau}}-\partial_{t}A_{\bar{\sigma}}-\left[A_{\bar{\sigma}},A_{\bar{\tau}}\right]+\hskip
28.45274pt\partial_{x}\bar{A}_{\bar{\sigma}}-\partial_{t}A_{\tau}-\left[A_{\tau},A_{\bar{\tau}}\right]=0\hskip
28.45274pt\partial_{x}A_{\tau}-\left[A_{\bar{\sigma}},A_{\tau}\right]-$
$\displaystyle\partial_{x}\bar{A}_{\bar{\tau}}-\partial_{t}\bar{A}_{\bar{\sigma}}-\left[\bar{A}_{\bar{\sigma}},\bar{A}_{\bar{\tau}}\right]=0\hskip
170.71652pt\partial_{x}\bar{A}_{\tau}+\left[\bar{A}_{\bar{\sigma}},\bar{A}_{\tau}\right]=0.$
(55)

Now, we set $A_{\tau}=P$ and $\bar{A}_{\tau}=\bar{P}$, with $P,\bar{P}$ being
constant matrices. We also set $A_{\bar{\tau}}=\bar{A}_{\bar{\tau}}=R$,
$A_{\bar{\sigma}}=Q$ and $\bar{A}_{\bar{\sigma}}=\bar{Q}$, so that the above
equations reduce to:

$\displaystyle\partial_{x}R-\partial_{t}Q-[Q,R]+\partial_{x}R-\partial_{t}\bar{Q}-\left[\bar{Q},R\right]=0,$
(56) $\displaystyle\partial_{x}Q-[\bar{P},R]=0,$ (57)
$\displaystyle\partial_{x}\bar{Q}-\left[P,R\right]=0,$ (58)
$\displaystyle\left[P,Q\right]=\left[\bar{P},\bar{Q}\right],$ (59)

since two of them just give $0=0$. P can be diagonalised into the
form:$\left(\begin{array}[]{ll}k&0\\\ 0&-k\end{array}\right)$, while the other
matrices explicitly read:

$\displaystyle\bar{P}=\left(\begin{array}[]{ll}l&s\\\
t&-l\end{array}\right)\,\,,\,\,\,R=\left(\begin{array}[]{ll}a&b\\\
c&-a\end{array}\right)\,\,,\,\,\,Q=\left(\begin{array}[]{ll}p&q\\\
-r&-p\end{array}\right),$ (66)

with $l,s,t$ being constants. It is interesting to note that $P$ and $\bar{P}$
cannot be diagonalised simultaneously, so at this stage one of them needs to
be completely undetermined. Then, equation 58 immediately tells us that
$\bar{Q}$ must be off-diagonal: $\bar{Q}=\left(\begin{array}[]{ll}0&\bar{q}\\\
-\bar{r}&0\end{array}\right)$, for some $\bar{q},\bar{r}$. With this
information, 59 gives us three constraints:

$\displaystyle s\bar{r}=-\bar{q}t\,,\hskip 42.67912ptqk=\bar{q}l\,,\hskip
42.67912ptrk=\bar{r}l\,.$ (67)

If we choose $s=t=0,q=\bar{q},r=\bar{r},k=l$, we have clearly recovered the
untwisted case. Instead, we choose $s=t=0$, but leave $k$ and $l$ unrelated,
with their ratio being the proportionality between $q,r$ and
$\bar{q},\bar{r}$, respectively. This, together with equation 58, gives:

$\displaystyle b=\frac{1}{2l}\partial_{x}q\,,\hskip
42.67912ptc=\frac{1}{2l}\partial_{x}r.$ (68)

Then, the off-diagonal entries of equation 57 are automatically satisfied, and
the diagonal ones yield $\partial_{x}p=0$, which we solve by taking $p=0$.
Whether expected or not, $\bar{P}$ naturally turned out to be of the same form
of $P$, and the same happens here for $Q$, $\bar{Q}$. An interesting choice
would be to force $P$ and $\bar{P}$ to be of a different form, we will return
to this later.
Finally, we are left with 56. Its diagonal part yields:

$\displaystyle a=\frac{1}{4l}(1+\frac{k}{l})(qr).$ (69)

Plugging this into the diagonal part we obtain (assuming $l\neq 0$):

$\displaystyle\partial_{x}^{2}q-l\partial_{t}q+\frac{1}{2}q(1+\frac{k}{l})qr-k\partial_{t}q+\frac{k}{2l}q(1+\frac{k}{l})qr=0$
(70)

from the top-left entry, while

$\displaystyle\partial_{x}^{2}r+l\partial_{t}q+2r\frac{1}{2}r(1+\frac{k}{l})qr+k\partial_{t}r+\frac{k}{2l}r(1+\frac{k}{l})qr=0$
(71)

from the bottom-right one.
By setting $r=q^{*}$, we get:

$\displaystyle\partial_{x}^{2}q-(l+k)\partial_{t}q+\frac{1}{2}q|q|^{2}(1+\frac{k}{l})(1+\frac{k}{l})=0,$
$\displaystyle\partial_{x}^{2}q^{*}+(l+k)\partial_{t}q^{*}+\frac{1}{2}q^{*}|q|^{2}(1+\frac{k}{l})(1+\frac{k}{l})=0,$
(72)

respectively. The choice $k=l=\frac{i}{2}$ recovers the result in Ablowitz et
al. (2003), while any choice of $k,l$ which satisfies $(l+k)^{*}=-(l+k)$ and
$\big{(}(\frac{k+l}{l})^{2}\big{)}^{*}=(\frac{k+l}{l})^{2}$ makes the two
equations degenerate. The latter condition implies that only scalings by a
real constant are allowed on the $|q|^{2}$ term, which can clearly be absorbed
by a redefinition of $q$. Thus, despite the seemingly larger system of
equations, the result of this reduction from the twisted self-dual equation
naturally coincides with the standard (untwisted) self-dual case.

### The Korteweg-de Vries Equation

In the previous subsection, we assumed that $P$ was diagonalisable. When this
is not the case, things radically differ. We will take
$P=\left(\begin{array}[]{ll}0&0\\\ k&0\end{array}\right)$, and
$\bar{P}=\left(\begin{array}[]{ll}0&0\\\ l&0\end{array}\right)$.555Note that
in Ablowitz et al. (2003), $k$ is taken to be $1$. Then, if
$Q=\left(\begin{array}[]{ll}v&z\\\ w&-v\end{array}\right)$ and
$\bar{Q}=\left(\begin{array}[]{ll}v^{\prime}&z^{\prime}\\\
w^{\prime}&-v^{\prime}\end{array}\right)$, equation 59 imposes

$\displaystyle z^{\prime}=\frac{l}{k}z\quad\quad v^{\prime}=\frac{k}{l}v.$
(73)

Taking:

$\displaystyle R=\left(\begin{array}[]{ll}a&b\\\ c&-a\end{array}\right),$ (76)

then equations 57 and 58 tells us that

$\displaystyle
z=z^{\prime}=1\quad\quad\partial_{x}v=-bl\quad\quad\partial_{x}w=2al\quad\quad\partial_{x}v^{\prime}=-bk\quad\quad\partial_{x}w^{\prime}=2ak,$
(77)

which imply also $w^{\prime}=\frac{k}{l}w$. It is now the turn for equation
56, which carries three independent relations. The top-right entry gives:

$\displaystyle 2b_{x}-2vb+2a-2v^{\prime}b+2a=$
$\displaystyle-\frac{2}{l}v_{xx}+\frac{2}{l}vv_{x}+2\frac{w_{x}}{l}+\frac{2k}{l^{2}}vv_{x}=0,$
(78)

from which we infer $w=v_{x}-\frac{1}{2}v^{2}(1+\frac{k}{l})$. Then, the
diagonal part yields:

$\displaystyle 2a_{x}-v_{t}-c+bw-v^{\prime}_{t}-c+bw^{\prime}=$
$\displaystyle\frac{w_{xx}}{l}-v_{t}(1+\frac{k}{l})-\frac{v_{x}}{l}w(1+\frac{k}{l})-2c=0.$
(79)

Using the expression we found above for $w$, we get:

$\displaystyle
c=\frac{1}{2}\Big{[}\frac{v_{xxx}}{l}-\frac{1}{l}(v_{x}^{2}+vv_{xx})(1+\frac{k}{l})-v_{t}(1+\frac{k}{l})-\frac{1}{l}\big{(}v_{x}^{2}-\frac{1}{2}v^{2}v_{x}(1+\frac{k}{l})\big{)}(1+\frac{k}{l})\Big{]}.$
(80)

Then, the bottom-left entry gives the last equation:

$\displaystyle
2c_{x}-w_{t}+2vc-2wa-w^{\prime}_{t}+2v^{\prime}c-2w^{\prime}a=0.$ (81)

Using the expressions found above, we can expand it as:

$\displaystyle
2\Big{[}\frac{v_{xxxx}}{2l}-\frac{v_{x}v_{xx}}{l}(1+\frac{k}{l})-\frac{v_{x}v_{xx}}{2l}(1+\frac{k}{l})-\frac{vv_{xxx}}{2l}(1+\frac{k}{l})-\frac{v_{tx}}{2}(1+\frac{k}{l})-\frac{v_{x}v_{xx}}{l}(1+\frac{k}{l})+$
$\displaystyle+\frac{vv_{x}^{2}}{2l}(1+\frac{k}{l})^{2}+\frac{v^{2}v_{xx}}{4l}(1+\frac{k}{l})^{2}\big{]}-\big{(}v_{xt}-vv_{t}(1+\frac{k}{l})\big{)}(1+\frac{k}{l})+$
$\displaystyle+v\Big{[}\frac{v_{xxx}}{l}-\frac{1}{l}(v_{x}^{2}+vv_{xx})(1+\frac{k}{l})-v_{t}(1+\frac{k}{l})-\frac{1}{l}\big{(}v_{x}^{2}-\frac{1}{2}v^{2}v_{x}(1+\frac{k}{l})\big{)}(1+\frac{k}{l})\Big{]}(1+\frac{k}{l})-$
$\displaystyle-\frac{v_{x}v_{xx}}{l}(1+\frac{k}{l})+\frac{vv_{x}^{2}}{l}(1+\frac{k}{l})^{2}+\frac{v_{xx}v^{2}}{2l}(1+\frac{k}{l})^{2}-\frac{v_{x}v^{3}}{2l}(1+\frac{k}{l})^{3}\,\,\,.$
(82)

Many of the terms cancel, and we are left with:

$\displaystyle\frac{v_{xxxx}}{l}-6\frac{v_{x}v_{xx}}{l}(1+\frac{k}{l})-2v_{tx}(1+\frac{k}{l})=0,$
(83)

which can be rewritten as:

$\displaystyle u_{t}=\frac{u_{xxx}}{2(k+l)}+3\frac{uu_{x}}{l},$ (84)

with $u=-v_{x}$. This is the KdV equation with some constant coefficients,
which for $k=l=1$ trivially reduces to the standard case presented in Ablowitz
et al. (2003). Even for arbitrary $k,l$, there always exist a rescaling of
$x$, $t$ and $u$ which absorbs the constant coefficients. Hence, once again,
the reduction of the twisted self-dual equation presented here leads to the
same integrable model as its untwisted counterpart, up to rescaling.

### The Third Option

In the previous two subsections we studied two dimensional reductions of the
twisted self-dual equation. As we pointed out in both cases, the limit in
which our work reduced to the standard (untwisted) case was evident throughout
our manipulations. Such limit was very straightforward because the choice
$F=\bar{F}$ in 14 immediately removes the twist. And this choice was always
possible in the previous two cases because we took $P$ and $\bar{P}$ to have
the same form (both diagonalisable in the first one, and both lower-diagonal
in the second one). Now we examine the third option: we choose
$P=\left(\begin{array}[]{ll}k&0\\\ 0&-k\end{array}\right)$,
$\bar{P}=\left(\begin{array}[]{ll}0&0\\\ \bar{k}&0\end{array}\right)$.
We start by considering equation 59, with the general ansatz
$Q=\left(\begin{array}[]{ll}p&q\\\ -r&-p\end{array}\right)$ and
$\bar{Q}=\left(\begin{array}[]{ll}\bar{p}&\bar{q}\\\
-\bar{r}&-\bar{p}\end{array}\right)$. This gives

$\displaystyle[P,Q]=\left(\begin{array}[]{ll}0&2qk\\\
2rk&0\end{array}\right)=\left(\begin{array}[]{ll}-\bar{q}\bar{k}&0\\\
2\bar{k}\bar{p}&\bar{q}\bar{k}\end{array}\right)=[\bar{P},\bar{Q}],$ (89)

giving:

$\displaystyle q=0,\quad\quad\bar{q}=0,\quad\quad kr=\bar{k}\bar{p}.$ (90)

Then, taking $R=\left(\begin{array}[]{ll}a&b\\\ c&-a\end{array}\right)$, 57
and 58 read:

$\displaystyle\partial_{x}\left(\begin{array}[]{ll}p&0\\\
-r&-p\end{array}\right)-\left(\begin{array}[]{ll}-b\bar{k}&0\\\
2\bar{k}a&b\bar{k}\end{array}\right)=0,$ (95)
$\displaystyle\partial_{x}\left(\begin{array}[]{ll}\bar{p}&0\\\
-\bar{r}&-\bar{p}\end{array}\right)-\left(\begin{array}[]{ll}0&2bk\\\
-2ck&0\end{array}\right)=0.$ (100)

This gives $b=0\implies p=0$, $\bar{p}=0\implies r=0\implies a=0$ and
$\partial_{x}\bar{r}=2ck$. Hence, so far we achieved the following
simplifications:

$\displaystyle Q=\left(\begin{array}[]{ll}0&0\\\
0&0\end{array}\right),\quad\bar{Q}=\left(\begin{array}[]{ll}0&0\\\
-\bar{r}&0\end{array}\right),\quad R=\left(\begin{array}[]{ll}0&0\\\
-\frac{1}{2k}\partial_{x}\bar{r}&0\end{array}\right).$ (107)

Plugging these in 56 yields:

$\displaystyle\left(\begin{array}[]{ll}0&0\\\
-\frac{1}{k}\partial_{xx}\bar{r}&0\end{array}\right)+\left(\begin{array}[]{ll}0&0\\\
\partial_{t}\bar{r}&0\end{array}\right)=0.$ (112)

Hence, we obtained the diffusion equation in one-dimension (which can always
be viewed as the Schrödinger equation in imaginary time).

##  4 Eguchi-Hanson solutions as twisted self dual solutions

In this section we move to the context of Euclidean gravity, where we found
yet another appearance of the twisted self-dual equation. We present an
example of a well known solution which obeys the twisted self-dual equation:
the Eguchi-Hanson gravitational instanton. It was presented in Eguchi and
Hanson (1979a), and a detailed review can be found in Eguchi and Hanson
(1979b); Eguchi et al. (1980).

###  4.1 Einstein-Cartan Formalism

General Relativity in the Einstein-Cartan formalism shows evident similarities
with Yang-Mills theory. Even more if the underlying spacetime is Euclidean.
The basics are the following.
We define vierbein one-forms as:

$\displaystyle e^{a}=e^{a}{}_{\mu}dx^{\mu}\quad\textrm{s.t.}\quad
g_{\mu\nu}=\delta_{ab}e^{a}{}_{\mu}e^{b}{}_{\nu}\,.$ (113)

Then, the connection one forms are defined by the first Cartan’s structure
equation:

$\displaystyle de^{a}+\omega_{ab}\wedge e^{b}=0,$ (114)

with $\omega_{ab}=-\omega_{ba}$ since they are valued in $so(4)$. Finally, the
curvature reads:

$\displaystyle R_{ab}=d\omega_{ab}+\omega_{ac}\wedge\omega_{cb}.$ (115)

It can be quickly shown that if $w$ is (anti) self-dual in the tangent space,
i.e.

$\displaystyle w_{ab}=\pm\frac{1}{2}\epsilon_{abcd}w_{cd},$ (116)

then so is $R_{ab}$:

$\displaystyle R_{ab}=\pm\frac{1}{2}\epsilon_{abcd}R_{cd}.$ (117)

This equation, together with the Bianchi identity for the curvature (which
holds in virtue of 115), implies that the Einstein equations without matter
are satisfied. The solution that we are about to review is anti self-dual,
i.e. it satisfies the above relation with the minus sign.

###  4.2 The Eguchi and Hanson Solution

To construct the famous instanton solution first derived by Eguchi and Hanson,
it is necessary to define:

$\displaystyle\sigma_{x}=\frac{1}{r^{2}}(xdt-tdx+ydz-zdy)=\frac{1}{2}(\sin\psi
d\theta-\sin\theta\cos\psi d\phi),$
$\displaystyle\sigma_{y}=\frac{1}{r^{2}}(ydt-tdy+zdx-
xdz)=\frac{1}{2}(-\cos\psi d\theta-\sin\theta\sin\psi d\phi),$
$\displaystyle\sigma_{z}=\frac{1}{r}(zdt-tdz+xdy-
ydx)=\frac{1}{2}(d\psi+\cos\theta d\phi).$ (118)

We have that $d\sigma_{x}=2\sigma_{y}\wedge\sigma_{z}$, and cyclic
permutations. Let us take the vierbeins to be

$\displaystyle
e^{a}=\left(f(r)dr,r\sigma_{X},r\sigma_{y},rg(r)\sigma_{z}\right),$ (119)

with

$\displaystyle g(r)=\sqrt{1-(a/r)^{4}}\,\,\,,\quad\quad\quad f(r)=g(r)^{-1}.$
(120)

This gives, according to 114:

$\displaystyle\omega^{1}{}_{0}=\omega^{2}{}_{3}=\left[1-(a/r)^{4}\right]_{x}^{\frac{1}{2}}\sigma_{x}=\left[1-(a/r)^{4}\right]^{\frac{1}{2}}e^{1}/r\,,$
(121)
$\displaystyle\omega^{2}{}_{0}=\omega^{3}{}_{1}=\left[1-(a/r)^{4}\right]^{\frac{1}{2}}\sigma_{y}=\left[1-(a/r)^{4}\right]^{\frac{1}{2}}e^{2}/r\,,$
(122)
$\displaystyle\omega^{3}{}_{0}=\omega^{1}{}_{2}=\left[1+(a/r)^{4}\right]\sigma_{z}=\left[1+(a/r)^{4}\right]e^{3}/\left(r\left[1-(a/r)^{4}\right]^{\frac{1}{2}}\right)\,.$
(123)

The anti self-duality is really encoded in the first equality on each line. As
discussed above, the resulting curvature is also anti self-dual (in the
tangent space). Explicitly, it reads:

$\displaystyle R^{1}{}_{0}=R^{2}{}_{3}=-\frac{2a^{4}}{r^{6}}\left(e^{1}\wedge
e^{0}+e^{2}\wedge e^{3}\right)$ $\displaystyle
R^{2}{}_{0}=R^{3}{}_{1}=-\frac{2a^{4}}{r^{6}}\left(e^{2}\wedge
e^{0}+e^{3}\wedge e^{1}\right)$ $\displaystyle
R^{3}{}_{0}=R^{1}{}_{2}=+\frac{4a^{4}}{r^{6}}\left(e^{3}\wedge
e^{0}+e^{1}\wedge e^{2}\right).$ (124)

Again, the anti-self-duality in the tangent space is encoded in the following
relations:

$\displaystyle R^{1}{}_{0}=R^{2}{}_{3}\,,\hskip
42.67912ptR^{2}{}_{0}=R^{3}{}_{1}\,,\hskip
42.67912ptR^{3}{}_{0}=R^{1}{}_{2}\,.$ (125)

###  4.3 The twisted self-duality

Let us now translate the twisted self-dual equation to the gravitational
context. When dealing with Einstein-Cartan theory, the analogy with Yang-Mills
becomes evident by identifying the curvature two-form $R$ with the field
strength $F$. Hence, the twisted self-dual equation must be a relation for
$R$. The next natural identification is between the internal space of Yang-
Mills and the tangent space of Einstein-Cartan. The "flat" indices $a,b,etc.$
are matrix indices for elements of $so(4)$, hence any pair contains only six
independent combinations. We can therefore replace any pair $ab$ with a single
index $A$ running from $1$ to $6$. With these considerations, 117 can be
written as:

$\displaystyle R^{A}=-J^{A}{}_{B}R^{B},$ (126)

with $J^{A}{}_{B}=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}$. Hence, to recover
the twisted self-dual equation in the context of Einstein-Cartan theory, we
just need to add a Hodge star in the curved space (i.e. acting on the greek
indices):

$\displaystyle R=J*R\iff
R^{A}{}_{\mu\nu}=\frac{g^{1/2}}{2}\epsilon_{\mu\nu\rho\sigma}J^{A}{}_{B}R^{B}{}^{\rho\sigma},$
(127)

where the determinant of the metric $g$ is different from $1$ in this case
(while it was set to $1$ in the Yang-Mills context). Now, the curvature 124
already obeys 126, i.e. anti self-duality in the tangent space (see 125). This
means that the action of $J$ on $R$ just introduces a minus sign. Hence, the
extra requirement from the twisted self-dual equation is anti self-duality,
that is

$\displaystyle
R^{A}{}_{\mu\nu}=-\frac{g^{1/2}}{2}\epsilon_{\mu\nu\rho\sigma}R^{A}{}^{\rho\sigma},$
(128)

for $A=1,2,3$ which correspond to $(a,b)=(1,0),(2,0),(3,0)$, respectively. To
check it, we "flatten" the remaining two greek indices. Omitting the capital
index (which goes along for the ride), we start by contracting the free
indices as:

$\displaystyle
e_{a}{}^{\mu}e_{b}{}^{\nu}R_{\mu\nu}=-e_{a}{}^{\mu}e_{b}{}^{\nu}\frac{g^{1/2}}{2}\epsilon_{\mu\nu\rho\sigma}R^{\rho\sigma},$
(129)

where $e_{a}{}^{\mu}$ is given by
$e_{a}{}^{\mu}e^{a}{}_{\nu}=\delta^{\mu}_{\nu}$. We use such property to bring
the above equation into the form:

$\displaystyle
e_{a}{}^{\mu}e_{b}{}^{\nu}R_{\mu\nu}=-\frac{g^{1/2}}{2}e_{a}{}^{\mu}e_{b}{}^{\nu}e_{c}{}^{\tau}e_{d}{}^{\lambda}\epsilon_{\mu\nu\tau\lambda}\,e^{c}{}_{\rho}e^{d}{}_{\sigma}R^{\rho\sigma},$
(130)

Then, given
$e_{a}{}^{\mu}e_{b}{}^{\nu}e_{c}{}^{\tau}e_{d}{}^{\lambda}\epsilon_{\mu\nu\tau\lambda}=g^{-1/2}$
and $R_{ab}\equiv e_{a}{}^{\mu}e_{b}{}^{\nu}R_{\mu\nu}$, we find:

$\displaystyle R_{ab}=-\frac{1}{2}\epsilon_{abcd}R_{cd}.$ (131)

We use the fact that $R^{A}=R^{A}{}_{ab}e^{a}\wedge e^{b}$ to read off from
124:

$\displaystyle R^{1}\propto\begin{pmatrix}0&-1&0&0\\\ 1&0&0&0\\\ 0&0&0&1\\\
0&0&-1&0\\\ \end{pmatrix},\quad R^{2}\propto\begin{pmatrix}0&0&-1&0\\\
0&0&0&-1\\\ 1&0&0&0\\\ 0&1&0&0\\\ \end{pmatrix},\quad
R^{3}\propto\begin{pmatrix}0&0&0&-1\\\ 0&0&1&0\\\ 0&-1&0&0\\\ 1&0&0&0\\\
\end{pmatrix}.$ (132)

Those are the anti self-dual ’t Hooft symbols (with a sign difference for
$R^{1}$), so they indeed satisfy the anti self-duality relation. Hence, the
Eguchi-Hanson gravitation instanton is twisted self-dual in that it satisfies
equation 127.

###  4.4 Topological Numbers

Now we briefly examine the topological properties of the solution, in the
light of the observations made so far. The bulk contribution to the Euler
character $\chi$ is proportional to (according to Chern’s formula, as
described in Eguchi and Hanson (1979a)):

$\displaystyle\int_{M}\varepsilon_{abcd}R^{a}{}_{b}\wedge
R^{c}{}_{d}=48\pi^{2}.$ (133)

Now, according to the self-duality equation 131, the density
$\varepsilon_{abcd}R^{a}{}_{b}\wedge R^{c}{}_{d}$ is directly related to the
density for the first Pontryagin class $p_{1}\propto R^{a}{}_{b}\wedge
R^{b}{}_{a}$. Specifically, we can use equation for the duality in tnagent
space 131 to quickly compute the integral of the first Pontryagin class from
the Euler class by relating the densities. This then gives:

$\displaystyle p_{1}=-\frac{1}{8\pi^{2}}\int_{M}R^{a}{}_{b}\wedge
R^{b}{}_{a}=-\frac{1}{8\pi^{2}}\cdot 24\pi^{2}=-3.$ (134)

Obviously, this result agrees with the computations described in detail in
Eguchi and Hanson (1979b). For us it is intersting to see how the twisted
self-duality equation in this example ends up relating two topological
indices.

##  5 Twisted self-duality in $E_{7}$ Exceptional Field Theory and Scherk-
Schwarz reductions

The final stage of this journey through the role of the twisted self-dual
equation in physics is its appearance $E_{7}$ exceptional field theory.
We will start by providing an introduction to the theory and outlining the
mechanism of Scherk-Schwarz reduction. This reduction simplifies the dynamics
of the theory, making it simpler to solve. In Berman and Rudolph (2014),
solutions to $E_{7}$ exceptional field theory were found for the case where
the non-abelian features are eliminated via an Abelian reduction. This section
presents a continuation of the above work, since we study the simplest non-
abelian generalisation of such procedure . We will show how the twisted self-
dual equation emerges in the dynamics of the reduced theory, and find a
particular solution.

###  5.1 Introduction to $E_{7}$ Exceptional Field Theory

11-dimensional supergravity is a very special theory, for many reasons. Just
to mention two, 11D sugra is the maximal supergravity and it has the minimum
number of dimensions required to contain the gauge group of the standard model
(see Nahm (1978); Witten (1981)). However, it goes without saying that to
obtain phenomenologically valid models, dimensional reduction must be
performed. It was noticed in Cremmer et al. (1978) that when 11D sugra is
compactified on a n -torus, $E_{n(n)}$ symmetries emerge for $n=6,7,8$. Then,
it is a natural question to ask to what extent these symmetries are present in
the original theory (11D sugra before compactification).
The development of $E_{n}$ Exceptional Field Theory (ExFT) provided an answer
to the above question, showing that the 11D sugra can be reformulated in a way
that makes the $E_{n(n)}$ symmetry manifest prior to any compactification (for
a review see Berman and Blair (2020)).
In this section, we are concerned with the specific case of $E_{7(7)}$ ExFT.
We now briefly summarise the main aspects of such a theory, and refer the
reader to the appendix for more mathematical details and to Hohm and Samtleben
(2014); Hull (2007); Pacheco and Waldram (2008); Berman et al. (2012a) for a
complete introduction.
The spacetime of $E_{7(7)}$ ExFT is $4+56$-dimensional. The first 4
coordinates $x^{\mu}$ parametrise the so-called external spacetime, while the
56 coordinates $Y^{N}$ refer to the internal spacetime. The latter ones sit in
the 56 fundamental representation of $E_{7(7)}$. To recover the usual
spacetime of 11D sugra, we impose the physical section condition. This has the
effect of restricting the dependence of the fields to a subset of the
coordinates: the physical ones. As a final piece of notation, we will use
$\alpha$ for the index corresponding to the adjoint representation of
$E_{7(7)}$, so for instance the generators of the $E_{7(7)}$ algebra are
written as $(t_{\alpha})^{MN}$. Let us now present how this construction is
realised in practice.
The field content of $E_{7(7)}$ ExFT is given by:

$\displaystyle\\{g_{\mu\nu},\mathcal{M}_{MN},\mathcal{A}_{\mu}{}^{M},\mathcal{B}_{\mu\nu\alpha},\mathcal{B}_{\mu\nu
M}\\}.$ (135)

Up to a topological term, the action for this theory takes the form:

$\displaystyle
S=\int\mathrm{d}^{4}x\mathrm{~{}d}^{56}Ye[\hat{R}+\frac{1}{48}g^{\mu\nu}\mathcal{D}_{\mu}\mathcal{M}^{MN}\mathcal{D}_{\nu}\mathcal{M}_{MN}-\frac{1}{8}\mathcal{M}_{MN}\mathcal{F}^{\mu\nu}{}^{M}\mathcal{F}_{\mu\nu}^{N}-V\left(\mathcal{M}_{MN},g_{\mu\nu}\right)],$
(136)

where all the symbols are defined in the appendix. The key observations are
that $\hat{R}$ is built out of $g_{\mu\nu}$ and $\mathcal{F}_{\mu\nu}^{M}$,
while $\mathcal{F}_{\mu\nu}^{M}$ is constructed from $\mathcal{A}_{\mu}^{M}$.
The physical section condition is given by:

$\displaystyle\left(t_{\alpha}\right)^{MN}\partial_{M}\partial_{N}\Phi=0,\quad\left(t_{\alpha}\right)^{MN}\partial_{M}\Phi\partial_{N}\Psi=0,\quad\Omega^{MN}\partial_{M}\Phi\partial_{N}\Psi=0,$
(137)

where $\Phi,\Psi$ are any of the fields above. For the scope of this paper,
the main observations is the following. Once the action has been varied, and
the physical section condition imposed, there still a missing piece: we must
impose the $E_{7}$ twisted self-dual equation

$\displaystyle\mathcal{F}_{\mu\nu}^{M}=\frac{1}{2}e\epsilon_{\mu\nu\rho\sigma}\Omega^{MN}\mathcal{M}_{NK}\mathcal{F}^{\rho\sigma
K}.$ (138)

$\Omega^{MN}$ is the symplectic invariant form, with is invariant under
$E_{7(7)}$, and:

$\displaystyle\mathcal{F}_{\mu\nu}{}^{M}\equiv
F_{\mu\nu}{}^{M}-12\left(t^{\alpha}\right)^{MN}\partial_{N}B_{\mu\nu\alpha}-\frac{1}{2}\Omega^{MK}B_{\mu\nu
K},$ $\displaystyle F_{\mu\nu}^{M}\equiv
2\partial_{[\mu}A_{\nu]}^{M}-\left[A_{\mu},A_{\nu}\right]_{\mathrm{E}}^{M}=2\partial_{[\mu}A_{\nu]}^{M}-2A_{[\mu}{}^{K}\partial_{K}A_{\nu]}^{M}-$
$\displaystyle\frac{1}{2}\left(24\left(t_{\alpha}\right)^{MK}\left(t^{\alpha}\right)_{NL}-\Omega^{MK}\Omega_{NL}\right)A_{[\mu}^{N}\partial_{K}A_{\nu]}^{L}.$
(139)

The reason behind the structure of $\mathcal{F}_{\mu\nu}^{M}$ is outlined in
the appendix.
In recent years, progress has been made in finding solutions for this theory
by eliminating completely the dependence on the external coordinates and
setting the $B$-fields to zero, as described in Berman and Rudolph (2014). It
is easy to see that this immediately reduces $\mathcal{F}_{\mu\nu}^{M}$ to an
Abelian field strength for 56 copies of a Maxwell gauge field
$\mathcal{A}_{\mu}^{M}$. The aim of this section is to present a
generalisation of this result to the case of $\mathcal{F}_{\mu\nu}^{M}$ being
a non-abelian field strength of the usual Yang-Mills type. In order to obtain
such a simplification of the theory, Scherk-Schwarz reduction is employed,
which consists of choosing the ansatz of the schematic form
$\Phi(x,Y)=U(Y)\bar{\Phi}(x)$.666This can be shown to be a consistent
truncation of the theory.
In the search for solutions of $E_{7}$ ExFT, we focus mainly on the self-
duality equation, and view the equations of motion as an extra constraint.

One way of simplifying the solution to the $E_{7}$ ExFT dynamics is to perform
a generalised Scherk-Schwarz reduction Berman et al. (2012b); Aldazabal et al.
(2013). In this section, we will present only the details of such procedure
that are relevant to our discussion.
We take the gauge field $\mathcal{A}_{\mu}{}^{M}$ to be of the form:

$\displaystyle\mathcal{A}_{\mu}{}^{M}(x,Y)=\tilde{U}^{M}{}_{\bar{M}}(Y)\bar{\mathcal{A}}_{\mu}{}^{\bar{M}}(x),$
(140)

where $\tilde{U}^{M}{}_{\bar{M}}(Y)=\rho^{-1}(Y)U^{M}{}_{\bar{M}}(Y)$ will be
referred as "twist matrix"777Note that the word twist here has nothing to do
with the operator $\mathcal{J}$ in the twisted self-dual equation.. As usual,
$Y$ denotes the internal coordinates and $x$ refers to the external ones. Now
we make a couple of key simplifications. Firstly, we assume that the
generalised Lie derivative (given in equation 180) of one vielbein wrt to
another takes the following form:

$\displaystyle\mathbb{L}_{\tilde{U}_{\bar{M}}}\tilde{U}^{N}{}_{\bar{N}}=\tilde{U}^{K}{}_{\bar{M}}\partial_{K}\tilde{U}^{N}{}_{\bar{N}}-\partial_{K}\tilde{U}^{N}{}_{\bar{M}}\tilde{U}^{K}{}_{\bar{N}}+\left(\lambda-\frac{1}{2}\right)\partial_{P}\tilde{U}^{P}{}_{\bar{M}}\tilde{U}^{N}{}_{\bar{N}}-Y^{NM}{}_{KL}\partial_{M}\tilde{U}^{K}{}_{\bar{M}}\tilde{U}^{L}{}_{\bar{N}}$
$\displaystyle\equiv-\tau_{\bar{M}\bar{N}}{}^{\bar{K}}\tilde{U}^{N}{}_{\bar{K}},$
(141)

where $\tau_{\bar{M}\bar{N}}{}^{\bar{K}}$ are constants (yet to be chosen).
Secondly, we set the B-fields in the definition of the field strength (see
194) to zero. With these choices, we can plug this ansatz in 194, which
reduces to 192, and we find:

$\displaystyle\mathcal{F}^{M}_{\mu\nu}=\tilde{U}^{M}{}_{\bar{M}}\big{(}2\partial_{[\mu}\bar{\mathcal{A}}_{\nu]}{}^{\bar{M}}-\mathcal{A}_{[\mu}{}^{\bar{K}}\mathcal{A}_{\nu]}{}^{\bar{P}}\tau_{\bar{P}\bar{K}}{}^{\bar{M}}\big{)}.$
(142)

The full calculation is reported here just for completeness:

$\displaystyle\mathcal{F}_{\mu\nu}^{M}=2\partial_{[\mu}\mathcal{A}_{\nu]}{}^{M}-2\mathcal{A}_{[\mu}{}^{K}\partial_{K}\mathcal{A}_{\nu]}{}^{M}-Y^{MK}{}_{LN}\mathcal{A}_{[\mu}{}^{N}\partial_{K}\mathcal{A}_{\nu]}{}^{L}=$
$\displaystyle
2\tilde{U}^{M}{}_{\bar{M}}\partial_{[\mu}\bar{\mathcal{A}}_{\nu]}{}^{\bar{M}}-2\mathcal{A}_{[\mu}{}^{\bar{K}}\mathcal{A}_{\nu]}{}^{\bar{M}}\tilde{U}^{K}{}_{\bar{K}}\partial_{K}\tilde{U}^{M}{}_{\bar{M}}-Y^{MK}{}_{LN}\mathcal{A}_{[\mu}{}^{\bar{N}}\mathcal{A}_{\nu]}{}^{\bar{L}}\tilde{U}^{N}{}_{\bar{N}}\partial_{K}\tilde{U}^{L}{}_{\bar{L}}=$
$\displaystyle
2\tilde{U}^{M}{}_{\bar{M}}\partial_{[\mu}\bar{\mathcal{A}}_{\nu]}{}^{\bar{M}}-\mathcal{A}_{[\mu}{}^{\bar{K}}\mathcal{A}_{\nu]}{}^{\bar{M}}\big{(}\tilde{U}^{K}{}_{\bar{K}}\partial_{K}\tilde{U}^{M}{}_{\bar{M}}-\tilde{U}^{K}{}_{\bar{M}}\partial_{K}\tilde{U}^{M}{}_{\bar{K}}\big{)}+$
$\displaystyle+\mathcal{A}_{[\mu}{}^{\bar{K}}\mathcal{A}_{\nu]}{}^{\bar{M}}\big{(}-Y^{MK}{}_{LN}\tilde{U}^{N}{}_{\bar{K}}\partial_{K}\tilde{U}^{L}{}_{\bar{M}}\big{)}=$
$\displaystyle
2\tilde{U}^{M}{}_{\bar{M}}\partial_{[\mu}\bar{\mathcal{A}}_{\nu]}{}^{\bar{M}}+\mathcal{A}_{[\mu}{}^{\bar{K}}\mathcal{A}_{\nu]}{}^{\bar{M}}\big{(}\tilde{U}^{K}{}_{\bar{M}}\partial_{K}\tilde{U}^{M}{}_{\bar{K}}-\tilde{U}^{K}{}_{\bar{K}}\partial_{K}\tilde{U}^{M}{}_{\bar{M}}-Y^{MK}{}_{LN}\tilde{U}^{N}{}_{\bar{K}}\partial_{K}\tilde{U}^{L}{}_{\bar{M}}\big{)}=$
$\displaystyle
2\tilde{U}^{M}{}_{\bar{M}}\partial_{[\mu}\bar{A}_{\nu]}{}^{\bar{M}}+\mathcal{A}_{[\mu}{}^{\bar{K}}\mathcal{A}_{\nu]}{}^{\bar{M}}\big{(}-\tau_{\bar{M}\bar{K}}{}^{\bar{P}}\tilde{U}^{M}{}_{\bar{P}}\big{)}=\tilde{U}^{M}{}_{\bar{M}}\big{(}2\partial_{[\mu}\bar{A}_{\nu]}{}^{\bar{M}}-\mathcal{A}_{[\mu}{}^{\bar{K}}A_{\nu]}{}^{\bar{P}}\tau_{\bar{P}\bar{K}}{}^{\bar{M}}\big{)}.$
(143)

Now, defining

$\displaystyle\mathcal{F}_{\mu\nu}^{M}=\tilde{U}^{M}{}_{\bar{M}}\bar{\mathcal{F}}_{\mu\nu}{}^{\bar{M}},$
(144)

we can write such result as:

$\displaystyle\bar{\mathcal{F}}_{\mu\nu}{}^{\bar{M}}=2\partial_{[\mu}\bar{\mathcal{A}}_{\nu]}{}^{\bar{M}}+\tau_{[\bar{N}\bar{K}]}{}^{\bar{M}}\bar{\mathcal{A}}_{\mu}{}^{\bar{N}}\bar{\mathcal{A}}_{\nu}{}^{\bar{K}}+\ldots,$
(145)

where the dots denote terms involving the extra B-fields. Thus, with the
$B-$fields set to zero and the Scherk-Schwarz ansatz, we find that the "flat
field strength", which is a function of the external coordinates only,
recovers the usual Yang-Mills form.
Finally, the Scherk-Schwartz ansatz for the metric reads:

$\displaystyle\mathcal{M}_{MN}=U_{M}{}^{\bar{M}}(y)U_{N}{}^{\bar{N}}(y)\bar{\mathcal{M}}_{\bar{M}\bar{N}}(x).$
(146)

These are all the tools needed to reduce the theory.

###  5.2 Reduction of the $E_{7}$ Twisted Self-dual Equation

We first look at how the above reduction is implemented on the $E_{7}$ twisted
self-dual equation. According to 3, we require the operator
$*\Omega\mathcal{M}$ to square to one. As we anticipated in the introduction,
we restrict ourselves to the case of $*^{2}=1$ and
$\mathcal{J}^{2}=(\Omega\mathcal{M})^{2}=1$.
We present a trivial (i.e. abelian) version of the equation and a non-trivial
(non-abelian) one. The abelian case is obtained by eliminating the dependence
on $Y$, while the non-abelian model that we obtain involves of $su(2)\oplus
su(2)$ non-abelian field strengths.
Also, we only focus on flat metrics. Including curved spacetimes is part of a
future investigation.

We take the x-dependent part of the generalised metric to have the form

$\displaystyle\bar{\mathcal{M}}_{\bar{M}\bar{N}}\left(g_{mn}\right)=g^{1/2}\operatorname{diag}\left[g_{mn},g^{mn,kl},g^{-1}g^{mn},g^{-1}g_{mn,kl}\right],$
(147)

with the same notation as above. Here the internal metric is constrained to be
of Lorentzian signature, i.e. we take
$g_{mn}=\eta_{mn}=\textrm{diag}(-1,1,1,1,1,1,1)$. By considering the possible
values of the indices in $g_{m[n}g_{k]l}$, we find that for the flat case that
we are focusing on, the only non-zero components are of the form:

$\displaystyle\begin{cases}g_{i[i}g_{j]j}=\frac{1}{2}(1-\delta_{ij})\hskip
14.22636pt\textrm{for}\,\,\,i,j=2,...,7\hskip 14.22636pt\textrm{(no sum)}\\\
g_{1[1}g_{j]j}=-\frac{1}{2}\hskip 14.22636pt\textrm{(no sum)}\\\
0\,\,\,\,\,\,\,\,\,\textrm{otherwise}.\end{cases}$ (148)

Hence, we have 21 non-zero components, which we can arrange on the diagonal to
form a 21x21 matrix. Thus, we can write $\bar{\mathcal{M}}_{\bar{M}\bar{N}}$
as

$\displaystyle\left(\begin{array}[]{c | c }\begin{array}[]{c| c c}A&&\\\ \hline\cr&B&\\\ &&\end{array}&\\\ \hline\cr&\begin{array}[]{c| c c}-A&&\\\ \hline\cr&-B&\\\ &&\end{array}\end{array}\right),$ (157)

where $A=\textrm{diag}(1,1,1,1,1,1,-1)$ and
$B=\frac{1}{2}\textrm{diag}(-1,-1,-1,-1,-1,-1,1,...,1)$. With this explicit
expression for $\bar{\mathcal{M}}$, and taking
$\Omega^{MN}=\begin{pmatrix}0&-1\\\ 1&0\end{pmatrix}^{MN}$ After performing
the Scherk-Schwarz reduction, the $E_{7}$ twisted self-dual equation in matrix
notation it reads:

$\displaystyle\underline{\bar{\mathcal{F}}}_{\mu\nu}=\frac{1}{2}e\epsilon_{\mu\nu\rho\sigma}\begin{pmatrix}0&-1\\\ 1&0\end{pmatrix}\left(\begin{array}[]{c | c }\begin{array}[]{c| c c}A&&\\\ \hline\cr&B&\\\ &&\end{array}&\\\ \hline\cr&\begin{array}[]{c| c c}-A&&\\\ \hline\cr&-B&\\\ &&\end{array}\end{array}\right)\underline{\bar{\mathcal{F}}}^{\rho\sigma},$ (166)

Note that now the contraction of the external space (Greek) indices indices
happens via the external Euclidean metric.

#### Vanilla case: No Dependence on Y

The choice of the twist matrices determines the form of $\bar{\mathcal{F}}$
according to 141 and 145, so that choosing them to be constant yields the
usual abelian field strength. Then, 166 gives the twisted self dual
formulation of Maxwell theory, presented in Bunster and Henneaux (2011). Thus,
the above equation describes 28 copies of Maxwell theory.

#### Non-trivial case: $su(2)\oplus su(2)$

To produce a twisted self-dual equation for $su(2)\oplus su(2)$ field
strength, we need to define twist matrices that give some non-zero components
of $\tau_{[\bar{N}\bar{K}]}{}^{\bar{M}}$ (see 145). We need to do this in
accordance to 141. The choice we make is:

$\displaystyle\tau_{\bar{N}\bar{K}}{}^{\bar{M}}=\,\epsilon_{\bar{N}\bar{K}\bar{M}}\hskip
14.22636pt\textrm{for }\hskip 14.22636pt\bar{N},\bar{K},\bar{M},=1,2,3\,,$
(167)

and

$\displaystyle\tau_{\bar{N}\bar{K}}{}^{\bar{M}}=\epsilon_{\bar{N}\bar{K}\bar{M}}\hskip
14.22636pt\textrm{for }\hskip 14.22636pt\bar{N},\bar{K},\bar{M},=29,30,31.$
(168)

This allows us to turn off all the gauge field and field strength components
but the ones labelled by the internal index with value $1,2,3,29,30,31$. In
other words, we are left with two standard $su(2)$ field strengths, which we
will denote by $F$ and $\bar{F}$, respectively. Note that we have used bars to
denote quantities depending only on the external coordinates in the Scherk-
Schwarz reduction, but this has nothing to do with our choice here.
After having performed Scherk-Schwarz reduction of the twisted-self dual
equation, we end up with:

$\displaystyle\left(\begin{array}[]{c}F\\\ \bar{F}\\\
\end{array}\right)=*\begin{pmatrix}0&1\\\
1&0\end{pmatrix}\left(\begin{array}[]{c}F\\\ \bar{F}\\\ \end{array}\right),$
(173)

with

$\displaystyle F=dA-[A,A],\hskip
14.22636pt\bar{F}=d\bar{A}-[\bar{A},\bar{A}],\hskip 14.22636ptA,\bar{A}\in
su(2).$ (174)

It is understood that the Hodge star involves the Euclidean four-dimensional
metric, so that this is nothing but 14.

###  5.3 Reduction of the Kinetic Term for $\mathcal{F}$

It is evident from 136 that to solve the dynamics for $\mathcal{A}_{\mu}^{M}$,
we need to consider the equations of motion coming from the "kinetic" term
$-\frac{1}{8}\mathcal{M}_{MN}\mathcal{F}^{\mu\nu}{}^{M}\mathcal{F}_{\mu\nu}^{N}$.888The
field $\mathcal{A}_{\mu}^{M}$ is also involved in the first term, but the
equations of motion descending from it are trivially satisfied if we take the
external metric to be flat. The beauty of the Scherk-Schwarz reduction is
that, as for the $E_{7}$ twisted self-dual equation, we can simply put bars on
everything. Then, it is the specific choice of twist matrices and
corresponding "structure constants" (141) which determines the exact form of
the equations of motion.

#### Vanilla case: No Dependence on Y

For this case, the equations of motion are the same that we get for n copies
of Maxwell theory. The generalised (internal) metric is what gives the trace
structure to the kinetic term, so we might be worried that factors of $\pm 1$
and $\frac{1}{2}$ could make a difference. But, since the theory is abelian
and $\bar{\mathcal{M}}$ is diagonal, gauge fields with different internal
indices are completely uncoupled. Thus, an independent scaling of each term in
the sum
$-\frac{1}{8}\bar{\mathcal{M}}_{\bar{M}\bar{N}}\bar{\mathcal{F}}^{\mu\nu}{}^{\bar{M}}\bar{\mathcal{F}}_{\mu\nu}^{\bar{N}}$
is irrelevant. The equation of motion will still be the usual

$\displaystyle\partial_{\mu}\bar{\mathcal{F}}^{\mu\nu}=0.$ (175)

#### Non-trivial case: $su(2)\oplus su(2)$

In this case, we (consciously) decided to work with only 6 space-time gauge
fields out of $\bar{\mathcal{A}}^{\bar{M}}_{\mu}$. Specifically, we chose
$\bar{M}=1,2,3,29,30,31$. The blocks of $\bar{\mathcal{M}}_{\bar{M}\bar{N}}$
corresponding to $\bar{M},\bar{N}=1,2,3$ and $\bar{M},\bar{N}=29,30,31$ are
two identity matrices, which means that we have a genuine trace structure for
the kinetic term in this case. Moreover, the barred and unbarred field
strengths (we decompose $\mathcal{F}$ into $F,\bar{F}$, as in the previous
section) are decoupled, leading to two sets of standard equations of motion
for $SU(2)$ Yang-Mills:

$\displaystyle\partial_{\mu}F^{\mu\nu}-[A_{\mu},F^{\mu\nu}]=0\,,\hskip
28.45274pt\partial_{\mu}\bar{F}^{\mu\nu}-[\bar{A}_{\mu},\bar{F}^{\mu\nu}]=0\,.$
(176)

###  5.4 The Full Dynamics

Solutions to $E_{7}$ Exceptional Field Theory must satisfy both the $E_{7}$
twisted self-dual equation and the standard equations of motion descending
from the kinetic term. Both of these have been Scherk-Schwartz reduced in the
previous subsections, so that we are now in the position of proposing
solutions to the (reduced) theory. Section 2 was dedicated to studying the
relation 173, so the natural course of action is to check if the solutions
developed in 2.3 also satisfy the equations of motion from the kinetic sector.
It is evident that the first solution (i.e. two identical $su(2)$ instantons)
does satisfy the equations of motion, since they decouple for each $su(2)$
block and each instanton individually solves the $su(2)$ Yang-Mills theory.
However, a quick computation shows that, even though it has zero stress-energy
tensor, the second solution does not satisfy the equations of motion. Hence,
we have that the first solution solves $E_{7}$ exceptional field theory under
the Scherk-Schwarz reduction considered, while the second one does not.

##  6 Discussion

There are a clear set remaining questions to be answered that are posed by
this paper. First, as discussed at the outset, what are the possible
Lorentzian equations and their solutions? This is particularly interesting
since it should allow twisted self-duality where ordinary self-duality is
forbidden. Is there a more interesting group that admits an appropriate
involution? In terms of the exceptional field theory, can we find the full
solutions once we couple to gravity including both the internal generalised
metric and the external metric. Is there are interpretation in E7 ExFT of the
Eguchi Hanson solution? It is tempting to see it as a gravitational instanton
in the “off-diagonal" space. Can the dimensional reductions described in the
paper be useful for constructing solutions in lower dimensional ExFT’s. Beside
this set of questions inspired by ExFT it is interesting to consider other
theories where twisted self-duality plays a key role. Various constructions
come to mind for topological theories or generalised BF type of theories.
There is much in this direction left to explore.

##  7 Appendix: $E_{7}$ ExFT

We now present the highlights in the construction of $E_{7}$ ExFT. We first
sketch the structure of the "generalised" symmetry of the theory and then
gauge it in order to obtain the covariant theory. Finally, we present the
building blocks of the corresponding action. For properties of $E_{7}$, the
reader is referred to appendix B of Cremmer and Julia (1979) and appendix B of
Pacheco and Waldram (2008).
This part is mainly taken from Hohm and Samtleben (2014).
We define the projector onto the adjoint rep of $E_{7}$ as:

$\displaystyle\mathbb{P}^{K}{}_{M}{}^{L}{}_{N}\equiv\left(t_{\alpha}\right)_{M}^{K}\left(t^{\alpha}\right)_{N}^{L}=\frac{1}{24}\delta_{M}^{K}\delta_{N}^{L}+\frac{1}{12}\delta_{M}^{L}\delta_{N}^{K}+\left(t_{\alpha}\right)_{MN}\left(t^{\alpha}\right)^{KL}-\frac{1}{24}\Omega_{MN}\Omega^{KL}.$
(177)

We have that $\mathbb{P}^{P}{}_{K}{}^{K}{}_{P}=133$. With this, we proceed to
define the (exceptional) Lie derivative wrt the (generalised vector) $V^{M}$
of weight $\lambda$:

$\displaystyle\delta
V^{M}=\mathbb{L}_{\Lambda}V^{M}\equiv\Lambda^{K}\partial_{K}V^{M}-12\mathbb{P}^{M}{}_{N}{}^{K}{}_{L}\partial_{K}\Lambda^{L}V^{N}+\lambda\partial_{P}\Lambda^{P}V^{M}.$
(178)

This can be generalised straighforwardly to an arbitrary (generalised) tensor.
It follows that $\Omega^{MN}$ is invariant:

$\displaystyle\mathbb{L}_{\Lambda}\Omega^{MN}=0.$ (179)

Explicitly, the exceptional Lie derivative reads:

$\displaystyle\delta_{\Lambda}V^{M}=\mathbb{L}_{\Lambda}V^{M}=$
$\displaystyle\Lambda^{K}\partial_{K}V^{M}-\partial_{K}\Lambda^{M}V^{K}+\left(\lambda-\frac{1}{2}\right)\partial_{P}\Lambda^{P}V^{M}-12\left(t_{\alpha}\right)^{MN}\left(t^{\alpha}\right)_{KL}\partial_{N}\Lambda^{K}V^{L}-\frac{1}{2}\Omega^{MN}\Omega_{KL}\partial_{N}\Lambda^{K}V^{L}.$
(180)

We have implicitly used the fact that $\Omega_{KL}=-\Omega_{LK}$ and
$(t_{\alpha})_{KL}=(t_{\alpha})_{LK}$. We can make this expression, and the
ones that are about to follow, less cumbersome by defining the "Y-tensor":

$\displaystyle
Y^{MN}{}_{KL}=12\left(t_{\alpha}\right)^{MN}\left(t^{\alpha}\right)_{KL}+\frac{1}{2}\Omega^{MN}\Omega_{KL}.$
(181)

Then,

$\displaystyle\mathbb{L}_{\Lambda}V^{M}=\Lambda^{K}\partial_{K}V^{M}-\partial_{K}\Lambda^{M}V^{K}+\left(\lambda-\frac{1}{2}\right)\partial_{P}\Lambda^{P}V^{M}-Y^{MN}{}_{KL}\partial_{N}\Lambda^{K}V^{L}.$
(182)

There are some gauge parameters that give trivial transformations. They are of
the form:

$\displaystyle\Lambda^{M}\equiv\left(t^{\alpha}\right)^{MN}\partial_{N}\chi_{\alpha},\quad\Lambda^{M}=\Omega^{MN}\chi_{N},$
(183)

where $\chi_{N}$ satisfies the

$\displaystyle\left(t_{\alpha}\right)^{MN}\chi_{M}\partial_{N}=\left(t_{\alpha}\right)^{MN}\chi_{M}\chi_{N}=0,\quad\Omega^{MN}\chi_{M}\partial_{N}=0.$
(184)

We can directly verify that

$\displaystyle[\delta_{\Lambda_{1}},\delta_{\Lambda_{2}}]=\delta_{[\delta_{\Lambda_{2}},\delta_{\Lambda_{1}}]_{\mathrm{E}}},$
(185)

where $[\cdot,\cdot]_{\mathrm{E}}$ is the E-bracket, which reads:

$\displaystyle\left[\Lambda_{2},\Lambda_{1}\right]_{\mathrm{E}}^{M}=2\Lambda_{[2}^{K}\partial_{K}\Lambda_{1]}^{M}+12\left(t_{\alpha}\right)^{MN}\left(t^{\alpha}\right)_{KL}\Lambda_{[2}^{K}\partial_{N}\Lambda_{1]}^{L}-\frac{1}{4}\Omega^{MK}\Omega_{NL}\partial_{K}\left(\Lambda_{2}^{N}\Lambda_{1}^{L}\right).$
(186)

The Jacobiator reads (we skip many steps presented in Hohm and Samtleben
(2014))

$\displaystyle J^{M}\left(V_{1},V_{2},V_{3}\right)=$
$\displaystyle-\frac{1}{2}\left(t_{\alpha}\right)^{MK}\partial_{K}\left(\left(t^{\alpha}\right)_{PL}\left(V_{1}^{P}\left[V_{2},V_{3}\right]_{\mathrm{E}}^{L}+\text{
cycl. }\right)\right)$ (187)
$\displaystyle+\frac{1}{12}\Omega^{MK}\Omega_{NL}\left(V_{1}^{N}\partial_{K}\left[V_{2},V_{3}\right]_{\mathrm{E}}^{L}+\left[V_{1},V_{2}\right]_{\mathrm{E}}^{N}\partial_{K}V_{3}^{L}+\mathrm{cycl}.\right).$
(188)

So, it is trivial in the sense that it does not generate any transformation.
We can now construct a theory that is manifestly invariant under the
generalised diffeomorphisms defined above. To do that, we introduce gauge
fields $\mathcal{A}_{\mu}^{M}$ and define the associated covariant derivative
as:

$\displaystyle\mathcal{D}_{\mu}\equiv\partial_{\mu}-\mathbb{L}_{\mathcal{A}_{\mu}}.$
(189)

Then, it follows that:

$\displaystyle\mathcal{D}_{\mu}V^{M}\equiv
D_{\mu}V^{M}-\lambda\partial_{K}\mathcal{A}_{\mu}{}^{K}V^{M}\equiv\partial_{\mu}V^{M}-\mathcal{A}_{\mu}{}^{K}\partial_{K}V^{M}+V^{K}\partial_{K}\mathcal{A}_{\mu}{}^{M}+\frac{1-2\lambda}{2}\partial_{K}\mathcal{A}_{\mu}{}^{K}$
$\displaystyle V^{M}$
$\displaystyle+12\left(t_{\alpha}\right)^{MN}\left(t^{\alpha}\right)_{KL}\partial_{N}\mathcal{A}_{\mu}{}^{K}V^{L}+\frac{1}{2}\Omega^{MN}\Omega_{KL}\partial_{N}\mathcal{A}_{\mu}{}^{K}$
$\displaystyle V^{L},$ (190)

where we implicitly defined $D_{\mu}V^{M}$. If we require the correct
transformation for the covariant derivative, we find:

$\displaystyle\delta\mathcal{A}_{\mu}^{M}=\partial_{\mu}\Lambda^{M}-\mathcal{A}_{\mu}^{K}\partial_{K}\Lambda^{M}+\Lambda^{K}\partial_{K}\mathcal{A}_{\mu}^{M}+12\left(t_{\alpha}\right)^{MN}\left(t^{\alpha}\right)_{KL}\Lambda^{L}\partial_{N}\mathcal{A}_{\mu}{}^{K}+\frac{1}{2}\Omega^{MN}\Omega_{KL}\Lambda^{L}\partial_{N}\mathcal{A}_{\mu}^{K}$
$\displaystyle=D_{\mu}\Lambda^{M}-\frac{1}{2}\left(\partial_{K}A_{\mu}^{K}\right)\Lambda^{M}\equiv\mathcal{D}_{\mu}\Lambda^{M}.$
(191)

Hence, we find that $\Lambda^{M}$ has weight $\lambda=\frac{1}{2}$.
It is now the turn for the field strength:

$\displaystyle F_{\mu\nu}^{M}\equiv
2\partial_{[\mu}\mathcal{A}_{\nu]}^{M}-\left[\mathcal{A}_{\mu},\mathcal{A}_{\nu}\right]_{\mathrm{E}}^{M}=2\partial_{[\mu}\mathcal{A}_{\nu]}^{M}-2\mathcal{A}_{[\mu}{}^{K}\partial_{K}\mathcal{A}_{\nu]}^{M}-\overbrace{\frac{1}{2}\left(24\left(t_{\alpha}\right)^{MK}\left(t^{\alpha}\right)_{NL}-\Omega^{MK}\Omega_{NL}\right)}^{Y^{MK}{}_{LN}}\mathcal{A}_{[\mu}^{N}\partial_{K}\mathcal{A}_{\nu]}^{L}.$
(192)

This is not covariant wrt generalised diffeomorphisms (and this is a
consequence of the non-vanishing Jacobiator, see Hohm and Samtleben (2014)).
The variation is given by:

$\displaystyle\delta
F_{\mu\nu}{}^{M}=2D_{[\mu}\delta\mathcal{A}_{\nu]}^{M}-\partial_{K}\mathcal{A}_{[\mu}{}^{K}\delta\mathcal{A}_{\nu]}^{M}-12\left(t_{\alpha}\right)^{MK}\left(t^{\alpha}\right)_{NL}\partial_{K}\left(A_{[\mu}^{N}\delta\mathcal{A}_{\nu]}^{L}\right)-$
$\displaystyle\frac{1}{2}\Omega^{MK}\Omega_{LN}\left(\mathcal{A}_{[\mu}^{N}\partial_{K}\delta\mathcal{A}_{\nu]}^{L}-\partial_{K}\mathcal{A}_{[\mu}^{N}\delta\mathcal{A}_{\nu]}^{L}\right).$
(193)

We recover covariance by including two additional fields, $B_{\mu\nu\alpha}$
and $B_{\mu\nu K}$, in the definition of the field strength:

$\displaystyle\mathcal{F}_{\mu\nu}{}^{M}\equiv
F_{\mu\nu}{}^{M}-12\left(t^{\alpha}\right)^{MN}\partial_{N}B_{\mu\nu\alpha}-\frac{1}{2}\Omega^{MK}B_{\mu\nu
K}.$ (194)

This works as follows. The variation of the extended field strength reads:

$\displaystyle\delta\mathcal{F}_{\mu\nu}^{M}=2\mathcal{D}_{[\mu}\delta
A_{\nu]}^{M}-12\left(t^{\alpha}\right)^{MN}\partial_{N}\Delta
B_{\mu\nu\alpha}-\frac{1}{2}\Omega^{MK}\Delta B_{\mu\nu K},$ (195)

with

$\displaystyle\Delta B_{\mu\nu\alpha}$ $\displaystyle\equiv\delta
B_{\mu\nu\alpha}+\left(t_{\alpha}\right)_{KL}\mathcal{A}_{[\mu}^{K}\delta\mathcal{A}_{\nu]}^{L}$
(196) $\displaystyle\Delta B_{\mu\nu K}$ $\displaystyle\equiv\delta B_{\mu\nu
K}+\Omega_{LN}\left(A_{[\mu}^{N}\partial_{K}\delta\mathcal{A}_{\nu]}^{L}-\partial_{K}\mathcal{A}_{[\mu}^{N}\delta\mathcal{A}_{\nu}]^{L}\right).$
(197)

Then, with the choices

$\displaystyle\delta_{\Lambda}\mathcal{A}_{\mu}^{M}$
$\displaystyle=\mathcal{D}_{\mu}\Lambda^{M}$ (198)
$\displaystyle\Delta_{\Lambda}B_{\mu\nu\alpha}$
$\displaystyle=\left(t_{\alpha}\right)_{KL}\Lambda^{K}\mathcal{F}_{\mu\nu}^{L}$
(199) $\displaystyle\Delta_{\Lambda}B_{\mu\nu M}$
$\displaystyle=-\Omega_{KL}\left(\mathcal{F}_{\mu\nu}{}^{K}\partial_{M}\Lambda^{L}-\Lambda^{L}\partial_{M}\mathcal{F}_{\mu\nu}{}^{K}\right)$
(200)

we obtain the covariant transformation of the field strength:

$\displaystyle\delta_{\Lambda}\mathcal{F}_{\mu\nu}^{M}=\Lambda^{K}\partial_{K}\mathcal{F}_{\mu\nu}^{M}-12\mathbb{P}^{M}{}_{N}{}^{K}{}_{L}\partial_{K}\Lambda^{L}\mathcal{F}_{\mu\nu}{}^{N}+\frac{1}{2}\partial_{K}\Lambda^{K}\mathcal{F}_{\mu\nu}{}^{M}.$
(201)

Finally, we now give explicit expressions for the terms that appear in the
action 136. We have:

$\displaystyle\widehat{R}_{\mu\nu}^{ab}$ $\displaystyle\equiv
R_{\mu\nu}^{ab}[\omega]+\mathcal{F}_{\mu\nu}^{M}e^{a\rho}\partial_{M}e_{\rho}^{b},$
$\displaystyle V=$
$\displaystyle-\frac{1}{48}\mathcal{M}^{MN}\partial_{M}\mathcal{M}^{KL}\partial_{N}\mathcal{M}_{KL}+\frac{1}{2}\mathcal{M}^{MN}\partial_{M}\mathcal{M}^{KL}\partial_{L}\mathcal{M}_{NK},$
$\displaystyle-\frac{1}{2}g^{-1}\partial_{M}g\partial_{N}\mathcal{M}^{MN}-\frac{1}{4}\mathcal{M}^{MN}g^{-1}\partial_{M}gg^{-1}\partial_{N}g-\frac{1}{4}\mathcal{M}^{MN}\partial_{M}g^{\mu\nu}\partial_{N}g_{\mu\nu}.$
(202)

where $R_{\mu\nu}^{ab}$ is the spin connection for the vierbein $e^{a\mu}$ and
$g=e^{2}=\textrm{det}(e_{a}{}^{\mu})^{2}=\textrm{det}(g_{\mu\nu})$. We do not
want to dwell too much on details that are not relevant for our discussion, so
we refer the reader to Hohm and Samtleben (2014) for more details on the
topological term and for why the action takes this specific form.

## Acknowledgements

The authors would like to thank Costis Papageorgakis, Malcom Perry and Chris
White for discussions and comments on the manuscript and in particular Rod
Halburd for early discussions on self-duality and dimensional reduction. TSG
would like to thank Emma Albertini, Pietro Capuozzo and Emanuele Panella for
their invaluable suggestions. TSG was supported by an STFC studentship. DSB
thanks Pierre Andurand for his generous donation supporting this research.

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