Most of the terms in (176) obey self-duality because of
$F_{13}^{(0+1(s))}=-F_{24}^{(0+1(s))}$ and
$F_{23}^{(0+1(s))}=F_{14}^{(0+1(s))}$ as is evident from (176), (A.2). The
only terms left to consider are the $12$ and $34$ entries. To study their
self-duality, we write, using the top equation in (A.2)

$\displaystyle F_{12}^{(0+1(s))}-F_{34}^{(0+1(s))}$ $\displaystyle=$
$\displaystyle{\tau^{3}\over 2}\left(-2\pi{L_{3}L_{4}-L_{1}L_{2}\over
V}+\Delta(F_{12}^{s}-F_{34}^{s})\right)$ (179) $\displaystyle=$
$\displaystyle{\tau^{3}\over
2}\left(-2\pi{\Delta\over\sqrt{V}}+2\pi{\Delta\over\sqrt{V}}(1-|F(x,z)|^{2}\right)$
$\displaystyle=$ $\displaystyle-{\tau^{3}\over
2}{2\pi\Delta\over\sqrt{V}}|F(x,z)|^{2}\,.$

This non-selfdual contribution to $F_{mn}^{(0+1(s))}$ is cancelled by the
${\cal{O}}(\Delta)$ contribution of the $W_{n}$ shown in (178).

Displayed in full glory, the Cartan part of the field strength to order
$\Delta$ is given by the sum of (176) and (178):

$\displaystyle F_{mn}^{(0+1)}\big{|}_{Cartan}=$
$\displaystyle\left(\begin{array}[]{cccc}0&-{2\pi\over
L_{1}L_{2}}+\Delta(F_{12}^{s}+{2\pi\over\sqrt{V}}|F(x,z)|^{2})&\Delta
F_{13}^{s}&\Delta F_{14}^{s}\cr{2\pi\over
L_{1}L_{2}}-\Delta(F_{12}^{s}+{2\pi\over\sqrt{V}}|F(x,z)|^{2})&0&\Delta
F_{14}^{s}&-\Delta F_{13}\cr-\Delta F_{13}^{s}&-\Delta
F_{14}^{s}&0&-{2\pi\over L_{3}L_{4}}+\Delta F_{34}^{s}\cr-\Delta
F_{14}^{s}&\Delta F_{13}^{s}&{2\pi\over L_{3}L_{4}}-\Delta
F_{34}^{s}&0\end{array}\right)\leavevmode\nobreak\ .$ (184)

We stress again that $F_{34}^{s}$ is understood to be expressed via
$F_{12}^{s}$ through (A.2), a substitution we have not explicitly done for
lack of space. Further, the above expression is understood to be multiplied by
$\tau^{3}/2$ and the undertermined $F_{mn}^{s}$ are found by solving (161).

The non-Cartan part of the solution, the $W_{n}$ from (A.1) also produce a
non-Cartan field strength. The ${\cal{O}}(\sqrt{\Delta)}$ field strength due
to $W_{n}$ is
$F_{mn}^{(1)}=(\partial_{m}+i\bar{A}_{m}^{3})W_{n}\tau^{+}+(\partial_{m}-i\bar{A}_{m}^{3})W_{n}^{*}\tau^{-}-(m\leftrightarrow
n)$ . It is self-dual by itself and the ${\cal{O}}(\sqrt{\Delta)}$ field
strength of $W_{n}$ is the only non-Cartan contribution to order $\Delta$ (the
next non-Cartan contribution is of order $\Delta^{3/2}$):

$\displaystyle F_{mn}^{(1)}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}0&0&F_{13}^{(1)}&F_{14}^{(1)}\cr
0&0&F_{14}^{(1)}&-F_{13}^{(1)}\cr-F_{13}^{(1)}&-F_{14}^{(1)}&0&0\cr-
F_{14}^{(1)}&F_{13}^{(1)}&0&0\end{array}\right).$ (190)

The nonzero entries of (190) are

$\displaystyle F_{13}^{(1)}$ $\displaystyle=$
$\displaystyle-i\gamma^{*}e^{i\alpha}\sqrt{\omega_{34}\over
2}G(x,z)\tau^{+}+i\gamma e^{-i\alpha}\sqrt{\omega_{34}\over
2}G^{*}(x,z)\tau^{-},$ (191) $\displaystyle F_{14}^{(1)}$ $\displaystyle=$
$\displaystyle\gamma^{*}e^{i\alpha}\sqrt{\omega_{34}\over
2}G(x,z)\tau^{+}+\gamma e^{-i\alpha}\sqrt{\omega_{34}\over
2}G^{*}(x,z)\tau^{-}\leavevmode\nobreak\ ,$

where we shouldn’t forget that $\gamma\sim\sqrt{\Delta}$, as $\gamma={i\over
2}\sqrt{2\pi\Delta\over\sqrt{V}}$, as per (A.1). The function $G(x,z)$ is

$\displaystyle G(x,z)$ $\displaystyle=$
$\displaystyle\sqrt{L_{2}L_{4}}\sum\limits_{n_{1},n_{3}=-\infty}^{\infty}e^{-i2\pi(n_{1}{x_{1}\over
L_{1}}+n_{3}{x_{3}\over L_{3}})}e^{-i{z_{2}\over
L_{2}}(x_{2}-n_{1}L_{2})-i{z_{4}\over L_{4}}(x_{4}-n_{3}L_{4})}$ (192)
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \times\leavevmode\nobreak\
\leavevmode\nobreak\ h_{0}^{12}(x_{2}-(n_{1}-{z_{1}\over
2\pi})L_{2})\;h_{1}^{34}(x_{4}-(n_{3}-{z_{3}\over
2\pi})L_{4})\leavevmode\nobreak\ ,$

where, just like $F(x,z)$, $G$ is dimensionless.

In conclusion, the full field strength tensor of our solution, to order
$\Delta$, is given by the Cartan part (LABEL:Fabelian2) and the non-Cartan
part (190). Owing to the self-duality, the action of the solution is
$S_{0}={4\pi^{2}\over g^{2}}$, as can also be explicitly inferred from the
explicit form of the field strength. These properties of the solutions are
useful in what follows.

###  A.3 The gauge invariants of the fractional instanton background

We now consider the $z_{n}$-dependence of gauge invariants, both local and
nonlocal (i.e. winding Wilson loops) characterizing the solution.

We begin with the $z_{n}$ dependence of local gauge invariants, formed of
traces of powers the field strength tensor. Using the data given above, we now
compute

$\langle
E_{i}^{2}\rangle\equiv\int\limits_{0}^{L_{3}}dx_{3}\int\limits_{0}^{L_{4}}dx_{4}\,{\rm
tr}\,F_{i4}^{2}(x,z),$ (193)

calling it the “$i$-th component of the electric field squared,” averaged over
$x_{3}$ and $x_{4}$.353535The averaging is done solely in order to shorten the
formulae that we display. This was also done in GarciaPerez:2000aiw , when
comparing the analytic solution to the numerical minimization of the action at
given $\Delta$. Omitting overall constants, for $i=1$, we find for the leading
contribution to (193), which is ${\cal{O}}(\Delta)$:

$\displaystyle\langle E_{1}^{2}\rangle$ $\displaystyle\sim$
$\displaystyle\sum\limits_{n_{1},m_{1}}e^{-i2\pi(n_{1}-m_{1}){1\over
L_{1}}({x_{1}}-{z_{2}L_{1}\over 2\pi})}h_{0}^{12}(x_{2}+{z_{1}L_{2}\over
2\pi}-n_{1}L_{2})h_{0}^{12}(x_{2}+{z_{1}L_{2}\over 2\pi}-m_{1}L_{2})$ (194)
$\displaystyle\sim$
$\displaystyle\sum\limits_{n_{1},m_{1}}e^{-i2\pi(n_{1}-m_{1}){1\over
L_{1}}(x_{1}-{z_{2}L_{1}\over 2\pi})}e^{-{\pi\over
L_{1}L_{2}}\left[(x_{2}+{z_{1}L_{2}\over
2\pi}-n_{1}L_{2})^{2}+(x_{2}+{z_{1}L_{2}\over
2\pi}-m_{1}L_{2})^{2}\right]}\,.$

The sum over $m,n$ is rapidly converging and can be seen to produce localized
bumps on ${\mathbb{T}}^{4}$ when plotted, periodic on ${\mathbb{R}}^{2}$ with
period $L_{1},L_{2}$ (usually a limit of $-4\leq m,n\leq 4$ suffices to make
the errors tiny and produces a periodic picture). The point of this discussion
is to illustrate two facts:

  1. 1.

That all local gauge invariants characterizing our fractional instanton depend
on the combinations $z_{2}-{2\pi x_{1}\over L_{1}}$ and $z_{1}+{2\pi
x_{2}\over L_{2}}$ (this also holds for the dependence on $z_{4}-{2\pi
x_{3}\over L_{2}}$ and $z_{3}+{2\pi x_{4}\over L_{4}}$, not shown above).

  2. 2.

That, furthermore, the local gauge invariants have $2\pi$ periodicity in these
variables.

Next, we consider the $z_{n}$ dependence of winding Wilson loops. To order
$\Delta^{0}$, with the abelian background (92), this is a rather
straightforward task. Consider a Wilson loop beginning at some $x$, going
along the $x_{1}$ direction $n_{1}$ times (i.e. from $x$ to
$x+\hat{e}_{1}n_{1}L_{1}$), then $n_{2}$ times in the $x_{2}$ direction,
$n_{3}$ times in the $x_{3}$ direction, and $n_{4}$ times in the $x_{4}$
direction, with the final point being
$x+\sum\limits_{k=1}^{4}\hat{e}_{k}n_{k}L_{k}$. The gauge invariant Wilson
loop along the loop $C_{n_{1},n_{2},n_{3},n_{4}}$, beginning at $x$ and
consisting of the winding straight segments described above, is given by

$\displaystyle W(C_{n_{1},n_{2},n_{3},n_{4}})=\,{\rm
tr}\,\left({\cal{P}}e^{i\int_{C_{n_{1},n_{2},n_{3},n_{4}}}{A}_{k}(x^{\prime})dx^{\prime}_{k}}\;\Omega_{2}^{n_{2}}(x)\;\Omega_{4}^{n_{4}}(x)\right),$
(195)

where the nontrivial transition functions are inserted to ensure gauge
invariance.

The path-ordering can be disregarded for an abelian background ($\bar{A}$ of
(92)) and one can conclude that

$\displaystyle W^{\Delta^{0}}(C_{n_{1},n_{2},n_{3},n_{4}})$ $\displaystyle=$
$\displaystyle 2\cos\left[{1\over 2}\left(n_{1}(z_{1}+{2\pi x_{2}\over
L_{2}})+n_{2}(z_{2}-{2\pi x_{1}\over L_{1}})+n_{3}(z_{3}+{2\pi x_{4}\over
L_{4}})+n_{4}(z_{4}-{2\pi x_{4}\over L_{2}})\right)\right].$

We now observe that the ${\cal{O}}(\Delta^{0})$ Wilson loops
$W^{\Delta^{0}}_{n_{1},n_{2},n_{3},n_{4}}$ are periodic functions of the
variables $\hat{z}_{1}\equiv z_{1}+{2\pi x_{2}\over L_{2}}$,
$\hat{z}_{2}\equiv z_{2}-{2\pi x_{1}\over L_{1}}$, $\hat{z}_{3}\equiv
z_{3}+{2\pi x_{4}\over L_{4}}$, $\hat{z}_{4}\equiv z_{4}-{2\pi x_{3}\over
L_{3}}$. However, in contrast with the local gauge invariants, like the
earlier (194), the periodicity in $\hat{z}_{n}$ is $4\pi$, rather than $2\pi$.
Thus, while local gauge invariants can not distinguish values of $\hat{z}_{n}$
differing by $2\pi$, nonlocal gauge invariant observables distinguish such
values. These values are, therefore, not physically equivalent.

We are led to conclude that the range of values of $\hat{z}_{n}$ which are
distinguished by gauge invariant quantities—and are thus physically
distinct—is given by ${\mathbb{T}}^{4}$. In our description, the
${\mathbb{T}}^{4}$ has “circumference” $4\pi$,
$\hat{z}_{n}\equiv\hat{z}_{n}+4\pi$. This fact is important in the calculation
of the gaugino condensate.

It should be clear that the ${\cal{O}}(\Delta)$ (and higher) contributions to
the Wilson loops (195), which require taking the path ordering into account,
are proportional to the same overall factor as (A.3)—since they come upon
expanding the path-ordered exponential. Thus, eqn. (A.3), is multiplied by
$(1+{\cal{O}}(\Delta))$. The ${\cal{O}}(\Delta)$ terms are $2\pi$ periodic
functions of $\hat{z}_{n}$, even with respect to their simultaneous
reflection, not affecting our conclusion above.

As an explicit simple example, consider the ${\cal{O}}(\Delta)$ contribution
to $W(C_{1,0,0,0})$, obtained by expanding the path-ordered exponential and
using the explicit form of the solution (A.1). Omitting the overall constant,
we obtain363636We indicated that there is an additional contribution of the
Cartan-direction ${\cal{O}}(\Delta)$ component $s_{1}$, which we have not
computed. However, since it is found by solving (161), whose r.h.s. is
(schematically) $\sim F^{*}F$ which has the same properties as the (also
schematically) $|W_{1}|^{2}$ contribution shown in (A.3).

$\displaystyle W(C_{1,0,0,0})\big{|}_{{\cal{O}}(\Delta)-\text{contribution}}$
$\displaystyle\sim$ $\displaystyle\cos{\hat{z}_{1}\over
2}\left(\bigg{|}\sum\limits_{n_{3}}e^{in_{3}\hat{z}_{4}}\;h_{0}^{34}({\hat{z}_{3}L_{4}\over
2\pi}-n_{3}L_{4})\bigg{|}^{2}\right.$ $\displaystyle\left.\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\times\bigg{|}\sum\limits_{n_{1}}e^{-in_{1}\hat{z}_{2}}\;h_{0}^{12}({\hat{z}_{1}L_{2}\over
2\pi}-n_{1}L_{2})\;\int\limits_{0}^{1}dt\;e^{it(\hat{z}_{1}-2\pi
n_{1})}\bigg{|}^{2}+{\text{contribution of }}\;s_{1}\right).$

The expression above illustrates the properties we mentioned earlier: as a
function of $\hat{z}_{n}$, the ${\cal{O}}(\Delta)$ contribution multiplying
the overall $\cos{\hat{z}_{1}\over 2}$ factor is $2\pi$ periodic.

##  Appendix B Zero modes of the fractional instanton

###  B.1 Leading-order bosonic zero-modes and measure

Here, we study the leading order bosonic zero modes and construct the
${\cal{O}}(\Delta^{0})$ bosonic zero mode measure. In the following Sections,
we shall argue that the measure remains the same to all orders in $\Delta$.

The bosonic zero modes are related to the dependence of the solution on
$z_{p}$. Their leading-order wave functions are particularly simple

$Z_{n}^{0,(p)}={\partial A_{n}^{cl.}\over\partial z_{p}}={\delta_{np}\over
L_{p}}{\tau_{3}\over 2}+....$ (198)

As usual when performing semiclassical instanton calculations, we add a
background Lorentz gauge fixing term to the bosonic action373737It should be
clear that, to order ${\cal{O}}(\Delta^{0})$, the zero modes (198) already
obey the background gauge condition: the leading-order background is in the
Cartan algebra and thus commutes with $Z_{n}^{0,(p)}$ and, furthermore, the
leading-order zero modes (198) are constant. and expand to second order in
fluctuations $a_{m}$, $A_{m}\equiv A_{m}^{cl}+a_{m}$, $D_{m}\equiv
D_{m}(A^{cl})=\partial_{m}+i[A_{m}^{cl},*]$. We stress that here $A_{m}^{cl}$
is the classical solution on the asymmetric ${\mathbb{T}}^{4}$, taken to the
desired order in $\Delta$. Thus, we obtain the usual action of the bosonic
fluctuations

$\displaystyle S_{b,g.f.}$ $\displaystyle=$
$\displaystyle\int_{T^{4}}\left[{1\over 2g^{2}}\,{\rm
tr}\,F_{mn}F_{mn}+{1\over g^{2}}\,{\rm tr}\,(D_{m}a_{m})^{2}\right]$
$\displaystyle\simeq$ $\displaystyle{4\pi^{2}\over g^{2}}+{1\over
g^{2}}\int_{T^{4}}\,{\rm
tr}\,\left[a_{n}O_{nm}a_{m}\right],\leavevmode\nobreak\
\text{where}\leavevmode\nobreak\ \leavevmode\nobreak\
O_{nm}a_{m}=-D^{2}\delta_{nm}a_{m}-2i[F_{nm},a_{m}].$

We include the rather well-known details that follow in order to motivate the
inner product of modes as well as to follow the factors of 2. Denote by
$Z_{m}^{k}$ the nonzero-eigenvalue eigenfunctions of $O_{nm}$, i.e.
$O_{nm}Z_{m}^{k}=\omega_{k}Z_{n}^{k}$ and expand the nonzero-mode part of the
fluctuation $a_{m}$ (for brevity, using the same letter as the full
fluctuation which includes the zero modes) as
$a_{m}=\sum_{k}\zeta_{k}Z_{m}^{k}$. Then we have that

$S_{b,g.f.}-{4\pi^{2}\over g^{2}}=\sum_{l,k}{\omega_{l}\zeta_{k}\zeta_{l}\over
2}\left({2\over g^{2}}\int_{T^{4}}\,{\rm
tr}\,Z_{m}^{k}Z_{m}^{l}\right)\equiv\sum_{l,k}{\omega_{l}\zeta_{k}\zeta_{l}\over
2}\;U_{kl},$

where the last equation defines the zero-mode norm matrix $U_{kl}$, explicitly
spelled out in (200) below. We diagonalize the matrix $U_{kl}$ with
eigenvalues $u_{k}$ and define the measure of the path integral over the
nonzero modes as $\prod_{k}{d\zeta_{k}\sqrt{u_{k}}\over\sqrt{2\pi}}$, thus
normalizing the path integral to simply produce the product
$\prod_{k}\omega_{k}^{-1/2}$. The upshot is that we defined the inner product
(or moduli space metric)

$U_{kl}={2\over g^{2}}\int_{T^{4}}\,{\rm tr}\,Z_{n}^{k}Z_{n}^{l}$ (200)

which in a diagonal basis is simply $U_{kl}=\delta_{lk}u_{l}$.

We use the same inner product for the zero modes $Z_{n}^{(0),p}$ of eqn.
(198), for which we find, neglecting the ${\cal{O}}(\sqrt{\Delta})$
contributions

$u_{k}^{(0)}={V\over g^{2}L_{k}^{2}}.$ (201)

As before, we expand the gauge field as
$A_{n}=A_{n}^{cl}+\sum_{p=1}^{4}\zeta_{p}^{(0)}Z_{n}^{(0),p}+(\text{nonzero
modes})$, and define the measure over the zero modes as
$\prod_{k=1}^{4}d\zeta_{k}^{(0)}\sqrt{u_{k}^{(0)}\over 2\pi}$, the same as for
the nonzero modes. Taken at face value, this integral is undetermined until we
find the region of integration over $\zeta_{k}^{(0)}$. To change variables
$\zeta_{k}^{(0)}\rightarrow z_{k}$, we note that to leading order in $\Delta$
this is quite straightforward, since the dependence of $A^{cl}(z)$ on $z_{k}$
is linear hence one simply replaces $\zeta_{k}^{(0)}$ by $z_{k}$. These
${\cal{O}}(\Delta^{0})$ considerations allow us to obtain the bosonic zero-
mode measure

$\displaystyle
d\mu_{B}\equiv{d^{4}z\over(\sqrt{2\pi})^{4}}\prod\limits_{l=1}^{4}\sqrt{u_{l}^{(0)}}={V\over
g^{4}}{dz_{1}dz_{2}dz_{3}dz_{4}\over(\sqrt{2\pi})^{4}}\leavevmode\nobreak\ .$
(202)

As described earlier, Section A.3, the $z_{n}$ are integrated over in the
range from $0$ to $4\pi$ modulo an overall reflection.

We end this introductory discussion of bosonic zero modes with two comments:

  1. 1.

Eqn. (202) was obtained by considering only the leading-order solution. Our
next task is to show that the zero modes of the ${\cal{O}}(\sqrt{\Delta})$
(and higher) solution enjoy the same measure. We need to ensure that the
background gauge condition can be satisfied and that the change of variables
from $\zeta_{k}^{(0)}$ to $z_{n}$ results in the same measure.

  2. 2.

An additional question that needs to be discussed is the fact that for
$\Delta=0$, the self-dual abelian solution has extra non-Cartan zero modes, in
addition to the constant modes, as was found long ago vanBaal:1984ar . In
contrast, the asymmetric ${\mathbb{T}}^{4}$ self-dual solution has only the
zero modes discussed above, as we show below.

###  B.2 Fermions and their zero-mode measure

It is well-known (see e.g. Vandoren:2008xg ; Dorey:2002ik ) that there is a
relation between the zero modes of the adjoint Dirac operator and the zero
modes of the bosonic fluctuation operator $O_{mn}$ of eqn. (B.1), to be
exploited later. To this end, as well as because we are interested in the
theory with adjoint fermions, we now consider the adjoint fermions and their
path integral in the fractional instanton background.

The Euclidean space Lagrangian density with the fermions included is

${1\over 2g^{2}}\,{\rm tr}\,F_{mn}F_{mn}+{2\over g^{2}}\,{\rm
tr}\,(\partial_{n}\bar{\lambda}_{\dot{\alpha}}+i[A_{n},\lambda_{\dot{\alpha}}])\bar{\sigma}_{n}^{\dot{\alpha}\alpha}\lambda_{\alpha}),$
(203)

where $\lambda$ and $\bar{\lambda}$ are independent variables.383838All our
notation regarding fermions is as in Dorey:2002ik , save for the fact that
ref. Dorey:2002ik  uses antihermitean gauge fields, necessitating the
replacement $A^{\text{that ref.}}=iA^{\text{this paper}}$. The four-vectors
$\sigma^{n},\bar{\sigma}^{n}$ were already defined in (103). For futher use,
notice, in particular, that
$\sigma_{mn}=(\sigma^{m}\bar{\sigma}^{n}-\sigma^{n}\bar{\sigma}^{m})/4$, and
that these matrices are, explicitly, $\sigma_{12}=\sigma_{34}={i\over
2}\sigma_{3}$, $\sigma_{13}=-\sigma_{24}=-{i\over 2}\sigma_{2}$,
$\sigma_{14}=\sigma_{23}={i\over 2}\sigma_{1}$. As we already noted, we use
$\sigma_{a}$ to denote the usual Pauli matrices, not to be confused with the
components of the four-vector $\sigma^{n}$ of (103). For future use, we note
that it is invariant under the supersymmetry

$\displaystyle\delta A_{n}$ $\displaystyle=$
$\displaystyle\zeta^{\alpha}\;\sigma_{n\;\alpha\dot{\alpha}}\;\bar{\lambda}^{\dot{\alpha}}+\bar{\zeta}_{\dot{\alpha}}\;\bar{\sigma}_{n}^{\dot{\alpha}\alpha}\;\lambda_{\alpha}$
(204) $\displaystyle\delta\lambda_{\alpha}$ $\displaystyle=$
$\displaystyle-\sigma_{mn\;\alpha}^{\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\beta}\;\zeta_{\beta}\;F_{mn}$
$\displaystyle\delta\bar{\lambda}^{\dot{\alpha}}$ $\displaystyle=$
$\displaystyle-\bar{\sigma}^{\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\dot{\alpha}}_{mn\;\;\leavevmode\nobreak\ \leavevmode\nobreak\
\dot{\beta}}\;\bar{\zeta}^{\dot{\beta}}\;F_{mn}$

where the $\sigma$’s are the ones of Dorey:2002ik . As usual for spinors,
$\xi^{1}=\xi_{2},\xi^{2}=-\xi_{1}$ and likewise for dotted.

The procedure for the fermions we shall follow is to again start from the
nonzero modes. We expand the fermions as eigenfunctions of the second order
Hermitean operators

$\displaystyle D\bar{D}$ $\displaystyle=$ $\displaystyle
D^{2}+iF_{mn}\sigma^{mn}\leavevmode\nobreak\ ,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
-(D\bar{D})_{\alpha}^{\leavevmode\nobreak\
\beta}\lambda_{\beta}=\omega^{2}\lambda_{\beta}\leavevmode\nobreak\ $ (205)
$\displaystyle\bar{D}D$ $\displaystyle=$ $\displaystyle
D^{2}+iF_{mn}\bar{\sigma}^{mn}\leavevmode\nobreak\ =D^{2},\leavevmode\nobreak\
\leavevmode\nobreak\
-D^{2}\bar{\lambda}^{\dot{\beta}}=\omega^{2}\bar{\lambda}^{\dot{\beta}}\leavevmode\nobreak\
,$

where, in the second line, we used the self-duality of the background (107).
To discuss the measure, begin by considering the contribution of a single (for
brevity) nonzero eigenvalue $\omega$ to the fermion path integral. Let
$-(D\bar{D})_{\alpha}^{\;\;\beta}\phi_{\beta}^{i}=\omega^{2}\phi_{\alpha}^{i}$,
where $i$ labels the different eigenfunctions, the commuting functions
$\phi_{\alpha}^{i}$, with the same eigenvalue $\omega$ (we note that there are
at least two of them). We expand the nonzero-mode part of the fermion field
(for brevity, denoting it with the same letter $\lambda$, $\bar{\lambda}$)

$\displaystyle\lambda_{\alpha}$ $\displaystyle=$
$\displaystyle\sum_{i}\chi^{i}\;\phi^{i}_{\alpha}$ (206)
$\displaystyle\bar{\lambda}^{\dot{\alpha}}$ $\displaystyle=$
$\displaystyle\sum_{i}\bar{\chi}^{i}\;{1\over\omega}\bar{D}^{\dot{\alpha}\alpha}\phi_{\alpha}^{i},$

where we used the fact that the nonzero eigenfunctions of $D\bar{D}$ and
$\bar{D}D$ are related as shown and we attach the spinor index to the bosonic
solution of the 2nd order equation and not to the Grassmann variable,
$\chi^{i}$ or $\bar{\chi}^{i}$ (the fact that there is more than a single
solution for every $\omega$ is accounted by the index $i$). We also indicate
that the $\lambda$ and $\bar{\lambda}$ expansions have each their separate
Grassmann variables $\chi^{i},\bar{\chi}^{i}$.

Plugging (206) into the fermionic action (203), we obtain after integration of
parts and using the fact that $\phi$ is an eigenvector of $D\bar{D}$:

$\displaystyle S_{F}={2\over g^{2}}\,{\rm
tr}\,(D_{n}\bar{\lambda}_{\dot{\alpha}}\bar{\sigma}_{n}^{\dot{\alpha}\alpha}\lambda_{\alpha})$
$\displaystyle=$ $\displaystyle\sum_{ij}\bar{\chi}^{j}\chi^{i}\omega\;({2\over
g^{2}}\int\,{\rm tr}\,\phi^{i\alpha}\phi^{j}_{\alpha})$ $\displaystyle=$
$\displaystyle\omega\sum_{ij}\bar{\chi}^{j}\chi^{i}\;U_{F}^{ij}\leavevmode\nobreak\
,\leavevmode\nobreak\ $

where the fermion mode inner product matrix is

$\displaystyle U_{F}^{ij}={2\over g^{2}}\int_{T^{4}}\,{\rm
tr}\,(\phi^{i}_{2}\phi^{j}_{1}-\phi^{i}_{1}\phi^{j}_{2}),\leavevmode\nobreak\
U_{F}^{ij}=-U_{F}^{ji},$ (208)

and again we remind ourselves that we are just looking at a single eigenvalue
(one can imagine a sum over them). Then we define the fermion nonzero mode
path integral so that it produces $\omega$ (the minimal number of
eigenfunctions with the same eigenvalue is two, i.e. $i,j=1,2$, with $U_{F}$
generically a $2\times 2$ matrix)

$\int\prod_{i}d\chi^{i}d\bar{\chi}^{i}\;(\det
U_{F})^{-1}e^{-S_{F}}=\omega\leavevmode\nobreak\ .$ (209)

When all nonzero eigenvalues are taken into account, we obtain the square root
of the product over all nonzero eigenvalues of $\bar{D}D$ (or $D\bar{D}$).

The definition of the integrals over the fermion zero modes are done in the
same manner using the same mode normalization matrix, $U_{F}^{ij}$ defined in
(208). We imagine (as we shall argue to be the case in our background) that
only the undotted spinors $\lambda_{\alpha}$ have zero modes, thus we
expand393939Here and below, we use $\phi^{i}_{\alpha}$ to denote the zero-mode
solutions of $D\bar{D}$, obeying
$(D\bar{D})_{\alpha}^{\;\;\beta}\phi_{\beta}^{i}=0$. The reader should forgive
us for using the same letter as in the non-zero mode discussion near (206).

$\lambda_{\alpha}=\sum_{i}\eta^{i}\phi^{i}_{\alpha}+\text{nonzero modes},$
(210)

where we use $\eta^{i}$ to denote the zero-mode Grassman variable. The fermion
zero-mode measure is then taken to be the “square root” of (209):404040This
definition ensures that, upon perturbing with a zero-mode lifting mass term,
$\delta S_{m}={m\over g^{2}}\,{\rm tr}\,\lambda^{\alpha}\lambda_{\alpha}$, one
obtains $m$ for the zero-mode contribution to the path integral.

$d\mu_{F}=\prod_{i}d\eta^{i}\;(\det U_{F})^{-1/2}\leavevmode\nobreak\
=\prod_{i}d\eta^{i}\;(\text{Pf}U_{F})^{-1}\leavevmode\nobreak\ .$ (211)

####  B.2.1 No zero modes of $D^{2}=D\bar{D}$ on the asymmetric
${\mathbb{T}}^{4}$

The reason to include this Section is that in the $\Delta=0$ abelian self-dual
background, i.e. in the symmetric-${\mathbb{T}}^{4}$ case, the $D^{2}$
$(=\bar{D}D)$ operator has zero modes: $a_{n}=c_{n}\tau^{3}$, with arbitrary
constant $c_{n}$. These zero modes are also zero modes of $O_{mn}$ of (B.1),
owing to the abelian nature of the background. The four Cartan zero modes of
the $\Delta=0$ self-dual solution appear in addition to the two non-Cartan
(complex) zero modes, also obeying $O_{mn}a_{n}=0$ but with $a_{n}$ in the
$\tau^{\pm}$ directions, found in vanBaal:1984ar . Thus the $\Delta=0$ self-
dual abelian background has 8, not 4 bosonic zero modes. While it is expected
that interactions lift half of these zero modes, this has not been shown in
any detail and is a problem for future studies.

In any case, since we find the $\Delta\neq 0$ detuned ${\mathbb{T}}^{4}$ of
greater physical interest, we proceed to show that in our self-dual background
(A.1) the $D^{2}$ operator has no zero modes. Furst, suppose that there exists
an adjoint field $\phi$ (generally complex) obeying $D_{n}D_{n}\phi=0$ in our
background; $\phi$ is, of course, assumed to obey the boundary conditions
appropriate to adjoints, i.e. (89) without the nonhomogeneous term. This
implies that $\int\limits_{{\mathbb{T}}^{4}}\,{\rm
tr}\,\phi^{\dagger}D_{n}D_{n}\phi=0$, or, after integration by parts, noting
that the boundary terms vanish if (89) are obeyed,
$\int\limits_{{\mathbb{T}}^{4}}\,{\rm tr}\,D_{n}\phi^{\dagger}D_{n}\phi=0$.
This is only possible if $D_{n}\phi=0$, or writing explicitly the components
of this equation:

$\displaystyle\partial_{n}\phi^{3}-2iA_{n}^{-}\phi^{+}+2iA_{n}^{+}\phi^{-}$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle(\partial_{n}-iA_{n}^{3})\phi^{-}+iA_{n}^{-}\phi^{3}$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle(\partial_{n}+iA_{n}^{3})\phi^{+}-iA_{n}^{+}\phi^{3}$
$\displaystyle=$ $\displaystyle 0,$ (212)

where $A_{n}^{3,\pm}$ are to be substituted by the ${\cal{O}}(\sqrt{\Delta})$
background fields (A.1). Eqns. (B.2.1) imply that the $D^{2}$ zero mode
$\phi_{0}$ of the abelian background (the $\sim\Delta^{0}$ term with
$A^{\pm}=0$) has components $\phi_{0}^{3}=c,\phi_{0}^{\pm}=0$. One can show
that there are no other zero mode solutions obeying the right boundary
conditions (see vanBaal:1984ar , or perform a SHO analysis of (B.2.1) in the
background (92), similar to the analysis in Section A.1).

We now want to argue that the ${\cal{O}}(\sqrt{\Delta})$ perturbations around
the abelian background of eqn. (A.1) lift the zero eigenvalue of $-D^{2}$. To
this end, we use perturbation theory for the non-negative Hermitean operator
$-D^{2}$ and compute its matrix element in the unperturbed “eigenstate” found
above, $\phi_{0}=c{\tau^{3}\over 2}$. For the shift of the eigenvalue, this
gives $\int_{{\mathbb{T}}^{4}}\,{\rm
tr}\,\phi_{0}^{\dagger}(-D^{2})\phi_{0}={|c|^{2}\over
2}\int_{{\mathbb{T}}^{4}}4A_{n}^{-}A_{n}^{+}$, where only the $n=1,2$
components are nonzero in the order $\sqrt{\Delta}$ solution:
$A_{1}^{+}=W_{1}$ and $A_{2}^{+}=W_{2}=iW_{1}$. The integral is positive
definite, as it is easily seen to be proportional to
$\int_{{\mathbb{T}}^{4}}|F|^{2}$, recall (138), showing that the zero
eigenvalue is lifted in the detuned-${\mathbb{T}}^{4}$ self-dual background.
This is a welcome feature of this background, compared to the one in the
symmetric ${\mathbb{T}}^{4}$ studied in vanBaal:1984ar .

####  B.2.2 The zero modes of $D$ on the asymmetric ${\mathbb{T}}^{4}$ via the
Dirac equation

In this Section, we explicitly study the Dirac equation for the undotted
fermions in the $\sim\sqrt{\Delta}$ background and show that they have two
zero modes. Since, as shown above, $D^{2}=\bar{D}D$ has no zero modes, the
presence of two zero modes of $D\bar{D}$ is guaranteed by the index theorem in
the topological charge $1/2$ background. Thus, we include this discussion only
for completeness. In the later Section, we shall see that these zero modes can
be obtained using supersymmetry.

The Dirac equation for the zero mode is
$\bar{D}^{\dot{\alpha}\alpha}\lambda_{\alpha}=0$ yielding

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\bar{\sigma}^{n\dot{\alpha}\alpha}\left(\partial_{n}\lambda_{\alpha}^{3}-2iW_{n}^{*}\lambda_{\alpha}^{+}+2iW_{n}\lambda_{\alpha}^{-}\right)\,,$
$\displaystyle 0$ $\displaystyle=$
$\displaystyle\bar{\sigma}^{n\dot{\alpha}\alpha}\left(\partial_{n}\lambda_{\alpha}^{-}-i\bar{A}_{n}^{3}\lambda_{\alpha}^{-}+iW_{n}^{*}\lambda_{\alpha}^{3}\right)\,,$
$\displaystyle\ 0$ $\displaystyle=$
$\displaystyle\bar{\sigma}^{n\dot{\alpha}\alpha}\left(\partial_{n}\lambda_{\alpha}^{+}+i\bar{A}_{n}^{3}\lambda_{\alpha}^{+}-iW_{n}\lambda_{\alpha}^{3}\right)\,.$
(213)

In our leading-order background (A.1), these equations give, for the
$\lambda_{\alpha}^{3}$ components:

$\displaystyle(i\partial_{1}+\partial_{2})\lambda_{2}^{3}+(i\partial_{3}-\partial_{4})\lambda_{1}^{3}$
$\displaystyle=$ $\displaystyle
4\gamma^{*}F(x,z)e^{i\alpha}\lambda_{2}^{-}\,,$
$\displaystyle-(i\partial_{1}-\partial_{2})\lambda_{2}^{3}+(i\partial_{3}+\partial_{4})\lambda_{1}^{3}$
$\displaystyle=$ $\displaystyle 4\gamma
F^{*}(x,z)e^{-i\alpha}\lambda_{1}^{+}\leavevmode\nobreak\ .$ (214)

On the other hand, the non-Cartan $\lambda_{\alpha}^{-}$ components satisfy
the equations

$\displaystyle\left[i(\partial_{1}-i{z_{1}\over
L_{1}})+(\partial_{2}-i{z_{2}\over L_{2}})+{2\pi\over
L_{1}L_{2}}x_{2}\right]\lambda_{2}^{-}+\left[i(\partial_{3}-i{z_{3}\over
L_{3}})-(\partial_{4}-i{z_{4}\over L_{4}})+{2\pi\over
L_{3}L_{4}}x_{4}\right]\lambda_{1}^{-}$ $\displaystyle=0\,,$
$\displaystyle\left[i(\partial_{1}-i{z_{1}\over
L_{1}})-(\partial_{2}-i{z_{2}\over L_{2}})+{2\pi\over
L_{1}L_{2}}x_{2}\right]\lambda_{1}^{-}-\left[i(\partial_{3}-i{z_{3}\over
L_{3}})+(\partial_{4}-i{z_{4}\over L_{4}})+{2\pi\over
L_{3}L_{4}}x_{4}\right]\lambda_{2}^{-}$ $\displaystyle=2\gamma
F^{*}(x,z)e^{-i\alpha}\lambda_{1}^{3}\,,$

while the $\lambda_{\alpha}^{+}$ components obey

$\displaystyle\left[-i(\partial_{1}+i{z_{1}\over
L_{1}})-(\partial_{2}+i{z_{2}\over L_{2}})+{2\pi\over
L_{1}L_{2}}x_{2}\right]\lambda_{2}^{+}+\left[-i(\partial_{3}+i{z_{3}\over
L_{3}})+(\partial_{4}+i{z_{4}\over L_{4}})+{2\pi\over
L_{3}L_{4}}x_{4}\right]\lambda_{1}^{+}$
$\displaystyle=2\gamma^{*}F(x,z)e^{i\alpha}\lambda_{2}^{3}\,,$
$\displaystyle\left[i(\partial_{1}+i{z_{1}\over
L_{1}})-(\partial_{2}+i{z_{2}\over L_{2}})-{2\pi\over
L_{1}L_{2}}x_{2}\right]\lambda_{1}^{+}+\left[-i(\partial_{3}+i{z_{3}\over
L_{3}})-(\partial_{4}+i{z_{4}\over L_{4}})+{2\pi\over
L_{3}L_{4}}x_{4}\right]\lambda_{2}^{+}$ $\displaystyle=0\,.$

We now can follow exactly the same steps as in the study of the self-duality
equation for $w_{1}$, see discussion after (109): we introduce $x_{1},x_{3}$
Fourier modes for $\lambda^{\pm}$ and the functions corresponding to (112);
likewise, we can absorb the Wilson lines by a redefinition similar to (118).
Proceeding thus, we can now solve the undotted fermion zero mode equations
(B.2.2,B.2.2, B.2.2) in an expansion in $\sqrt{\Delta}$. Keeping in mind that
$\gamma\sim\sqrt{\Delta}$, it follows that the solution for
$\lambda^{3}_{\alpha}$ is of order $\Delta^{0}$ and is simply given by a two-
component constant Grassmann spinor $\eta_{\alpha}^{3}$. This satisfies the
$\lambda^{3}_{\alpha}$ equation to leading order, since the r.h.s. of (B.2.2)
is of order $(\sqrt{\Delta})^{2}$ as the $\lambda^{\pm}$ solutions are
themselves of order $\sqrt{\Delta}$. Proceeding thus, we find the leading-
order fermion zero-modes:

$\displaystyle\lambda_{\alpha}^{3(0)}$ $\displaystyle=$
$\displaystyle\eta_{\alpha}^{3},$ $\displaystyle\lambda_{1}^{-(0)}$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle\lambda_{2}^{-(0)}$
$\displaystyle=$ $\displaystyle-\eta_{1}^{3}\gamma
e^{-i\alpha}\sqrt{L_{3}L_{4}\over\pi}\sqrt{L_{2}L_{4}}\sum\limits_{n_{1},n_{3}}e^{i2\pi(n_{1}{x_{1}\over
L_{1}}+n_{3}{x_{3}\over L_{3}})}e^{i({z_{2}\over
L_{2}}(x_{2}-n_{1}L_{2})+{z_{4}\over L_{4}}(x_{4}-n_{3}L_{4}))}$
$\displaystyle\qquad\times\qquad h_{0}^{12}(x_{2}-n_{1}+{z_{1}\over
2\pi})h_{1}^{34}(x_{4}-n_{3}+{z_{3}\over 2\pi}),$
$\displaystyle\lambda_{1}^{+(0)}$ $\displaystyle=$
$\displaystyle{\eta_{2}^{3}}\gamma^{*}e^{i\alpha}\sqrt{L_{3}L_{4}\over\pi}\sqrt{L_{2}L_{4}}\sum\limits_{n_{1},n_{3}}e^{-i2\pi(n_{1}{x_{1}\over
L_{1}}+n_{3}{x_{3}\over L_{3}})}e^{-i({z_{2}\over
L_{2}}(x_{2}-n_{1})+{z_{4}\over L_{4}}(x_{4}-n_{3}))}$
$\displaystyle\qquad\times\qquad h_{0}^{12}(x_{2}-n_{1}+{z_{1}\over
2\pi})h_{1}^{34}(x_{4}-n_{3}+{z_{3}\over 2\pi}),$
$\displaystyle\lambda_{2}^{+(0)}$ $\displaystyle=$ $\displaystyle 0.$

That (B.2.2) are solutions to order $\sqrt{\Delta}$ follows by direct
substitution and use of (125). Consider, for example, the
$\lambda_{\alpha}^{-}$ components of the zero modes. That $\lambda_{2}^{-}$
satisfies the top equation in (B.2.2) follows from the fact that it is acted
upon by the lowering operator of the $\omega_{12}$ SHO, while setting
$\lambda_{1}^{-}$ to zero is necessitated by its being acted upon by the
raising operator of the $\omega_{34}$ oscillator as well as by the second
equation in (B.2.2). The lowering operator of the $\omega_{34}$ oscillator
acting on $\lambda_{2}^{-}$ from (B.2.2) can be easily seen to produce the
r.h.s. of the second eqn. in (B.2.2), recalling the definition of $F$ from
(137). One similarly verifies that $\lambda^{+(0)}_{\alpha}$ solve the
(B.2.2).

Before we continue, let us write the fermion zero modes (B.2.2) in a more
compact manner, in terms of the function $G(x,z)$ of (192). The fermion zero
modes (B.2.2), that $\omega_{34}=2\pi/(L_{3}L_{4})$, are now written as:

$\displaystyle\lambda_{\alpha}^{(0)}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}\eta_{1}^{3}\cr\eta_{2}^{3}\end{array}\right){\tau^{3}\over
2}+\left(\begin{array}[]{c}\eta_{2}^{3}\cr
0\end{array}\right)\gamma^{*}e^{i\alpha}\sqrt{2\over\omega_{34}}G(x,z)\tau^{+}+\left(\begin{array}[]{c}0\cr-\eta_{1}^{3}\end{array}\right)\gamma
e^{-i\alpha}\sqrt{2\over\omega_{34}}G^{*}(x,z)\tau^{-}$ (224) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}\eta_{1}^{3}\cr\eta_{2}^{3}\end{array}\right){\tau^{3}\over
2}+\left(\begin{array}[]{c}\eta_{2}^{3}\cr
0\end{array}\right)\gamma^{*}e^{i\alpha}{V^{1\over 4}\over\pi^{1\over
2}}G(x,z)\tau^{+}+\left(\begin{array}[]{c}0\cr-\eta_{1}^{3}\end{array}\right)\gamma
e^{-i\alpha}{V^{1\over 4}\over\pi^{1\over 2}}G^{*}(x,z)\tau^{-},$ (231)

with the last equality being true to leading order in $\sqrt{\Delta}$.

####  B.2.3 Zero modes of $D$ and $\bar{D}$ on the symmetric
${\mathbb{T}}^{4}$

For completeness, let us consider the fermionic zero modes of the abelian
self-dual instanton solution of the $\Delta=0$ symmetric ${\mathbb{T}}^{4}$.
For the $\bar{D}\lambda=0$ equation, we can use (B.2.2, B.2.2, B.2.2) with
$\gamma$ set to zero, i.e. without r.h.s. Then, it immediately follows that
the zero modes on the symmetric ${\mathbb{T}}^{4}$ are

$\displaystyle\lambda_{\alpha}^{3(\Delta=0)}$ $\displaystyle=$
$\displaystyle\eta_{\alpha}^{3},$ $\displaystyle\lambda_{1}^{-(\Delta=0)}$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle\lambda_{2}^{-(\Delta=0)}$
$\displaystyle=$
$\displaystyle\eta_{2}^{-}\sqrt{L_{2}L_{4}}\sum\limits_{n_{1},n_{3}}e^{i2\pi(n_{1}{x_{1}\over
L_{1}}+n_{3}{x_{3}\over L_{3}})}e^{i({z_{2}\over
L_{2}}(x_{2}-n_{1}L_{2})+{z_{4}\over L_{4}}(x_{4}-n_{3}L_{4}))}$
$\displaystyle\qquad\times\qquad h_{0}^{12}(x_{2}-n_{1}+{z_{1}\over
2\pi})h_{0}^{34}(x_{4}-n_{3}+{z_{3}\over 2\pi}),$
$\displaystyle\lambda_{1}^{+(\Delta=0)}$ $\displaystyle=$
$\displaystyle\eta_{1}^{+}\sqrt{L_{2}L_{4}}\sum\limits_{n_{1},n_{3}}e^{-i2\pi(n_{1}{x_{1}\over
L_{1}}+n_{3}{x_{3}\over L_{3}})}e^{-i({z_{2}\over
L_{2}}(x_{2}-n_{1})+{z_{4}\over L_{4}}(x_{4}-n_{3}))}$
$\displaystyle\qquad\times\qquad h_{0}^{12}(x_{2}-n_{1}+{z_{1}\over
2\pi})h_{0}^{34}(x_{4}-n_{3}+{z_{3}\over 2\pi}),$
$\displaystyle\lambda_{2}^{+(\Delta=0)}$ $\displaystyle=$ $\displaystyle 0.$

That these are solutions of (B.2.2, B.2.2, B.2.2) with $\gamma=0$ follows from
observing that the annihilation operators of the $\omega_{12}$ and
$\omega_{34}$ SHOs act on $\lambda^{+}_{1}$ and $\lambda^{-}_{2}$ components
only, while the other components are acted upon by creation operators and are
thus set to zero. These four zero modes of $\bar{D}^{\dot{\alpha}\alpha}$ at
$\Delta=0$ combine with the two dotted zero modes, obeying
$D_{\alpha\dot{\alpha}}\bar{\lambda}^{\dot{\alpha}}=0$, which exist due to the
fact that $D^{2}$ has zero modes at $\Delta=0$ (recall Section B.2.1):414141As
in our analysis leading to the bosonic solution of eqn. (136), one can show
that there are no normalizable zero modes of $D$ or $\bar{D}$ except for
(B.2.3,233).

$\displaystyle\bar{\lambda}_{\dot{\alpha}}^{3(\Delta=0)}$ $\displaystyle=$
$\displaystyle\bar{\eta}_{\dot{\alpha}}^{3},$
$\displaystyle\bar{\lambda}_{\dot{\alpha}}^{\pm(\Delta=0)}$ $\displaystyle=$
$\displaystyle 0,$ (233)

The presence of two dotted (233) and four undotted (B.2.3) fermion zero modes
in the self-dual background on the symmetric ${\mathbb{T}}^{4}$ is, of course,
in accord with the index theorem. The presence of extra zero modes is the
fermionic counterpart of the existence of bosonic zero modes of $D^{2}$ on the
symmetric ${\mathbb{T}}^{4}$, as discussed after eqn. (B.2.1) of Section
B.2.1. As mentioned in the main text, the extra zero modes—bosonic and
fermionic—are expected to be lifted once fluctuations around the self dual
background on the symmetric ${\mathbb{T}}^{4}$ and their interactions are
taken into account, but this has not been yet demonstrated.

####  B.2.4 The zero modes of $D$ on the asymmetric ${\mathbb{T}}^{4}$ via
supersymmetry

We now check that the asymmetric ${\mathbb{T}}^{4}$ undotted fermion zero
modes (B.2.2) can be obtained via supersymmetry of the
${\cal{O}}(\sqrt{\Delta})$ bosonic background (92, A.1). Consider the effect
of the SUSY transforms (204) in the gauge field background (92, A.1) with
fermions set to zero, $\bar{\lambda}=\lambda=0$. Since our solution is self-
dual, i.e. obeys $\bar{\sigma}^{mn}F_{mn}=0$, the SUSY transform only produces
$\lambda_{\alpha}$ variations. Computing $\delta\lambda$ we obtain

$\displaystyle\delta\lambda$ $\displaystyle=$
$\displaystyle-(\sigma_{mn}F_{mn}^{(0)}+\sigma_{mn}F_{mn}^{(1)})\zeta+\ldots$
(234)

where $F_{mn}^{(0)}$ is the field strength (97) of the abelian background
(92), and $F_{mn}^{(1)}$ is the order $\sqrt{\Delta}$ contribution from (190,
191). Thus, combining everything, plugging into (234), and recalling Footnote
38, we obtain

$\displaystyle\left(\begin{array}[]{c}\delta\lambda_{1}\cr\delta\lambda_{2}\end{array}\right)$
$\displaystyle=$
$\displaystyle-2iF_{12}^{(0)}\left(\begin{array}[]{c}\zeta_{1}\cr-\zeta_{2}\end{array}\right)+2F_{13}^{(1)}\left(\begin{array}[]{c}\zeta_{2}\cr-\zeta_{1}\end{array}\right)-i2F_{14}^{(1)}\left(\begin{array}[]{c}\zeta_{2}\cr\zeta_{1}\end{array}\right)$
(243) $\displaystyle=$ $\displaystyle
i{4\pi\over\sqrt{V}}\left[\left(\begin{array}[]{c}\zeta_{1}\cr-\zeta_{2}\end{array}\right){\tau^{3}\over
2}+{V^{1\over 4}\over\pi^{1\over
2}}\gamma^{*}e^{i\alpha}G(x,z)\tau^{+}\left(\begin{array}[]{c}-\zeta_{2}\cr
0\end{array}\right)+{V^{1\over 4}\over\pi^{1\over 2}}\gamma
e^{-i\alpha}G^{*}(x,z)\tau^{-}\left(\begin{array}[]{c}0\cr-\zeta_{1}\end{array}\right)\right]$
(250)

On the last line, we again used the small-$\Delta$ relation $\omega_{34}\simeq
2\pi V^{-{1\over 2}}$. Comparing (224) with (LABEL:lambdasusy3), we see that
they are identical provided the supersymmetry parameter in (LABEL:lambdasusy3)
is identified with the Grassmann coefficient in (224) as follows

$\displaystyle\eta_{1}^{3}$ $\displaystyle=$ $\displaystyle 4\pi iV^{-{1\over
2}}\zeta_{1}\leavevmode\nobreak\ ,$ (252) $\displaystyle\eta_{2}^{3}$
$\displaystyle=$ $\displaystyle-4\pi iV^{-{1\over
2}}\zeta_{2}\leavevmode\nobreak\ ,$

showing that all is consistent with SUSY.

###  B.3 The moduli space metric, to any order in $\Delta$

Here we shall study the bosonic and fermionic moduli space to order $\Delta$.
It is well known that for every zero mode solution of the undotted Dirac
equation $\phi_{\alpha}^{(\beta)}$, one can construct two zero mode
eigenvalues of the operator $O_{nm}$ which obey the background gauge condition
$D_{m}(A^{cl})a_{m}=0$.

We begin with our notation for the zero modes (234): we shall denote the
corresponding commuting wave functions by $\phi_{\alpha}^{\beta\;A}$, where
$A=1,2,3$ denotes the $SU(2)$ algebra index. These are the two solutions of
the undotted Dirac equation considered above. Thus, with $(\beta)=1,2$
labeling the two zero modes, we have

$\displaystyle D^{\dot{\alpha}\alpha}\phi_{\alpha}^{(\beta)}$ $\displaystyle=$
$\displaystyle 0\leavevmode\nobreak\ ,$ (253)
$\displaystyle\phi_{\alpha}^{(\beta)}$ $\displaystyle=$
$\displaystyle-(\sigma^{mn})_{\alpha}^{(\beta)}F_{mn}\leavevmode\nobreak\
,\leavevmode\nobreak\ \text{with}\leavevmode\nobreak\
\phi_{\alpha}^{(\beta)}=\sum\limits_{A=1}^{3}\phi_{\alpha}^{\;(\beta)\;A}T^{A},$
(254) $\displaystyle\phi_{\alpha}^{\;(\beta)\;A}$ $\displaystyle=$
$\displaystyle-(\sigma^{mn}F_{mn}^{A})_{\alpha}^{(\beta)}=-2i(\sigma_{3})_{\alpha}^{\;(\beta)}F_{12}^{A}+2i(\sigma_{2})_{\alpha}^{\;(\beta)}F_{13}^{A}-2i(\sigma_{1})_{\alpha}^{\;(\beta)}F_{14}^{A}$
$\displaystyle=$
$\displaystyle-2i\;(\sigma_{a})_{\alpha}^{\;(\beta)}{V}_{a}^{A}\leavevmode\nobreak\
,\text{where ${V}_{a}^{A}$ has
components}\;V_{3}^{A}=F_{12}^{A},V_{2}^{A}=-F_{13}^{A},V_{1}^{A}=F_{14}^{A}.$

In the second equality, we used self-duality of the background and the
explicit form of $\sigma_{mn}$ given in Footnote 38.

####  B.3.1 Fermion zero-mode measure to arbitrary order in $\Delta$

In terms of the zero modes (253) the fermion zero-mode expansion (210),
relabeling the index $i$ (used there to label the different zero modes), $i$
$\rightarrow\beta$, we have

$\displaystyle\lambda_{\alpha}^{A}$ $\displaystyle=$
$\displaystyle\eta^{\beta}\phi_{\alpha}^{(\beta)\;A},\leavevmode\nobreak\
{\text{sum understood over}}\leavevmode\nobreak\ \beta=1,2,$ $\displaystyle
U_{F}^{\beta\gamma}$ $\displaystyle=$ $\displaystyle{1\over
g^{2}}\int_{T^{4}}(\phi^{(\beta)\;A}_{2}\phi^{(\gamma)\;A}_{1}-\phi^{(\beta)\;A}_{1}\phi^{(\gamma)\;A}_{2})\leavevmode\nobreak\
,$ (256)

where again $A=1,2,3$ is the Lie algebra index, summed over the definition of
$U_{F}$. The fermion zero mode norm matrix $U_{F}^{\beta\gamma}$ from (208) is
antisymmetric and we have from (B.3.1) and (253)

$\displaystyle U_{F}^{12}$ $\displaystyle=$ $\displaystyle{1\over
g^{2}}\int_{T^{4}}V_{a}^{A}V_{b}^{A}\;(-4)[(\sigma_{a})_{2}^{\;1}(\sigma_{b})_{1}^{\;2}-(\sigma_{a})_{1}^{\;1}(\tau^{b})_{2}^{\;2}]$
(257) $\displaystyle=$ $\displaystyle{(-4)\over
g^{2}}\int_{T^{4}}V_{a}^{A}V_{b}^{A}\;X^{ab}\,.$

Consider now the matrix
$X^{ab}=(\sigma_{a})_{2}^{\;1}(\sigma_{b})_{1}^{\;2}-(\sigma_{a})_{1}^{\;1}(\sigma_{b})_{2}^{\;2}$
implicitly defined above. We have, from the explicit form of the Pauli
matrices:

$||X^{ab}||=\left(\begin{array}[]{ccc}1&-i&0\cr i&1&0\cr
0&0&1\end{array}\right),$ (258)

thus

$\displaystyle U_{F}^{12}$ $\displaystyle=$ $\displaystyle{-4\over
g^{2}}\int_{T^{4}}V_{a}^{A}V_{a}^{A}=-{4\over
g^{2}}\int_{T^{4}}((F_{12}^{A})^{2}+(F_{14}^{A})^{2}+(F_{13}^{A})^{2})=-{1\over
g^{2}}\int_{T^{4}}(F_{mn}^{A})^{2}$ (259) $\displaystyle=$
$\displaystyle-4\;{4\pi^{2}\over g^{2}}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ (\text{since}\;{1\over
4g^{2}}\int_{T^{4}}(F_{mn}^{A})^{2}={4\pi^{2}\over g^{2}}).$

The Pfaffian of the fermion mode matrix is thus
$\text{Pf}U_{F}=-U_{F}^{12}=4\times{4\pi^{2}\over g^{2}}$ and the fermion
zero-mode measure, defined via the Pfaffian in (211) is

$d\mu_{F}=d\eta^{1}d\eta^{2}\;(\text{Pf}U_{F})^{-1}\leavevmode\nobreak\
=4\times{4\pi^{2}\over g^{2}}d\eta^{1}d\eta^{2}\leavevmode\nobreak\ .$ (260)

To obtain a nonzero result for the fermion zero mode integral, we insert the
gaugino bilinear, $\,{\rm tr}\,\lambda^{\alpha}\lambda_{\alpha}={1\over
2}\eta^{\alpha}\eta^{\beta}\phi^{(\alpha)A}_{\gamma}\phi^{(\beta)A}_{\delta}\epsilon^{\delta\gamma}+$(nonzero
modes), and obtain

$\displaystyle\int d\eta^{1}d\eta^{2}\;({\text{Pf}}U_{F})^{-1}\;\,{\rm
tr}\,\lambda^{\alpha}\lambda_{\alpha}$ $\displaystyle=$
$\displaystyle{g^{2}\over 16\pi^{2}}\int d\eta^{1}d\eta^{2}\;{1\over
2}\;\eta^{\alpha}\eta^{\beta}\phi^{(\alpha)A}_{\gamma}\phi^{(\beta)A}_{\delta}\epsilon^{\delta\gamma}$
(261) $\displaystyle=$ $\displaystyle{g^{2}\over 16\pi^{2}}{1\over
2}\;(\phi^{(2)A}_{\gamma}\phi^{(1)A}_{\delta}\epsilon^{\delta\gamma}-\phi^{(1)A}_{\gamma}\phi^{(2)A}_{\delta}\epsilon^{\delta\gamma})\,,$

where we used (259) and the explicit form of the Pfaffian.

We stress that the fermion zero mode measure (211) as well as (261) hold to
arbitrary order in $\Delta$. Furthermore, the measure and the result (261) are
independent on $\Delta$.

####  B.3.2 Bosonic zero-modes and moduli-space metric to any order in
$\Delta$

Now to the wave functions of the bosonic zero modes obtained from the
fermionic modes and automatically obeying the gauge condition. For every
fermionic zero mode $\phi_{\alpha}^{\;(\beta)}$, $\beta=1,2$, there are two
bosonic zero modes. Thus, in total there are four independent bosonic zero
modes. The advantage of the discussion that follows is that the bosonic zero
modes thus obtained automatically obey the gauge condition and, furthermore,
that their construction holds to arbitrary orders in $\Delta$.

The four-vector expressions for the bosonic zero modes thus obtained are
denoted by $Z_{n}^{(\beta)\;A}$ and $Z_{n}^{(\beta\;^{\prime})\;A}$, where
$\beta={1,2}$, $A=1,2,3$. These modes are determined as follows (see e.g.
Vandoren:2008xg ; Dorey:2002ik ). First one forms the quaternions made out of
the zero-mode solutions of the undotted Dirac equation, explicitly:

$\displaystyle\sigma^{n}Z_{n}^{(\beta)\;A}=\left(\begin{array}[]{cc}Z_{4}^{(\beta)}+iZ_{3}^{(\beta)}&Z_{2}^{(\beta)}+iZ_{1}^{(\beta)}\cr-
Z_{2}^{(\beta)}+iZ_{1}^{(\beta)}&Z_{4}^{(\beta)}-iZ_{3}^{(\beta)}\end{array}\right)^{A}$
$\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\phi_{1}^{(\beta)}&-\phi_{2}^{(\beta)\;*}\cr\phi_{2}^{(\beta)}&\phi_{1}^{(\beta)\;*}\end{array}\right)^{A},$
(266)
$\displaystyle\sigma^{n}Z_{n}^{(\beta\;^{\prime})\;A}=\left(\begin{array}[]{cc}Z_{4}^{(\beta\;^{\prime})}+iZ_{3}^{(\beta\;^{\prime})}&Z_{2}^{(\beta\;^{\prime})}+iZ_{1}^{(\beta\;^{\prime})}\cr-
Z_{2}^{(\beta\;^{\prime})}+iZ_{1}^{(\beta\;^{\prime})}&Z_{4}^{(\beta\;^{\prime})}-iZ_{3}^{(\beta\;^{\prime})}\end{array}\right)^{A}$
$\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}i\phi_{1}^{(\beta)}&i\phi_{2}^{(\beta)\;*}\cr
i\phi_{2}^{(\beta)}&-i\phi_{1}^{(\beta)\;*}\end{array}\right)^{A}\leavevmode\nobreak\
.$ (271)

Thus, each zero mode $\phi_{\alpha}^{(\beta)}$ of the undotted Dirac equation
can be used to build two four-vector bosonic zero modes, denoted by
$Z_{n}^{(\beta)}$ and $Z_{n}^{(\beta\;^{\prime})}$. Their four-vector
components are then inferred from (266):

$\displaystyle Z_{n}^{(\beta)\;A}$ $\displaystyle=$
$\displaystyle\left\\{\Im\phi_{2}^{(\beta)\;A},-\Re\phi_{2}^{(\beta)\;A},\Im\phi_{1}^{(\beta)\;A},\Re\phi_{1}^{(\beta)\;A}\right\\},$
$\displaystyle Z_{n}^{(\gamma\;^{\prime})\;A}$ $\displaystyle=$
$\displaystyle\left\\{\Re\phi_{2}^{(\gamma)\;A},\Im\phi_{2}^{(\gamma)\;A},\Re\phi_{1}^{(\gamma)\;A},-\Im\phi_{1}^{(\gamma)\;A}\right\\},$
(272)

Now, knowing the four-vector components of the zero modes, we can use (266)
and (253) to find $\det U_{kl}$ and argue for its $\Delta$-independence. We
need to compute the $4\times 4$ matrix of different overlaps (200)

$\displaystyle U^{\beta,\gamma}$ $\displaystyle=$ $\displaystyle{2\over
g^{2}}\int_{T^{4}}\,{\rm tr}\,Z_{n}^{(\beta)}Z_{n}^{(\gamma)}={1\over
g^{2}}\int_{T^{4}}Z_{n}^{(\beta)\;A}Z_{n}^{(\gamma)\;A}$ $\displaystyle
U^{\beta,\gamma+2}=U^{\gamma+2,\beta}$ $\displaystyle=$ $\displaystyle{2\over
g^{2}}\int_{T^{4}}\,{\rm
tr}\,Z_{n}^{(\beta)}Z_{n}^{(\gamma\;^{\prime})}={1\over
g^{2}}\int_{T^{4}}Z_{n}^{(\beta)\;A}Z_{n}^{(\gamma\;^{\prime})\;A},$
$\displaystyle U^{\beta+2,\gamma+2}$ $\displaystyle=$ $\displaystyle{2\over
g^{2}}\int_{T^{4}}\,{\rm
tr}\,Z_{n}^{(\beta\;^{\prime})}Z_{n}^{(\gamma\;^{\prime})}={1\over
g^{2}}\int_{T^{4}}Z_{n}^{(\beta\;^{\prime})\;A}Z_{n}^{(\gamma\;^{\prime})\;A},$
(273)

where the trace is now in the Lie-algebra generator space. The four-vector
inner products are

$\displaystyle
Z_{n}^{(\beta)}Z_{n}^{(\gamma)}=Z_{n}^{(\beta\;^{\prime})}Z_{n}^{(\gamma\;^{\prime})}$
$\displaystyle=$
$\displaystyle\Im\phi_{2}^{(\beta)}\Im\phi_{2}^{(\gamma)}+\Re\phi_{2}^{(\beta)}\Re\phi_{2}^{(\gamma)}+\Im\phi_{1}^{(\beta)}\Im\phi_{1}^{(\gamma)}+\Re\phi_{1}^{(\beta)}\Re\phi_{1}^{(\gamma)}$
$\displaystyle=$ $\displaystyle\sum\limits_{\alpha=1}^{2}{1\over
2}(\phi_{\alpha}^{(\beta)}\phi_{\alpha}^{(\gamma)\;*}+\phi_{\alpha}^{(\beta)\;*}\phi_{\alpha}^{(\gamma)})\,,$
$\displaystyle Z_{n}^{(\beta)}Z_{n}^{(\gamma\;^{\prime})}$ $\displaystyle=$
$\displaystyle\Im\phi_{2}^{(\beta)}\Re\phi_{2}^{(\gamma)}-\Re\phi_{2}^{(\beta)}\Im\phi_{2}^{(\gamma)}+\Im\phi_{1}^{(\beta)}\Re\phi_{1}^{(\gamma)}-\Re\phi_{1}^{(\beta)}\Im\phi_{1}^{(\gamma)}$
(274) $\displaystyle=$ $\displaystyle\sum\limits_{\alpha=1}^{2}{i\over
2}(\phi_{\alpha}^{(\beta)}\phi_{\alpha}^{(\gamma)\;*}-\phi_{\alpha}^{(\beta)\;*}\phi_{\alpha}^{(\gamma)})\leavevmode\nobreak\
,$

where momentarily we omitted the group index $A$ (to be restored below). Thus,
using (253), with the shorthand $\sigma.F^{A}\equiv\sigma^{mn}F_{mn}^{A}$, we
find

$\displaystyle U^{\beta,\gamma}=U^{\beta+2,\gamma+2}$ $\displaystyle=$
$\displaystyle{1\over
2g^{2}}\int_{T^{4}}\sum\limits_{\alpha=1}^{2}(\phi_{\alpha}^{(\beta)\;A}\phi_{\alpha}^{(\gamma)\;*\;A}+\phi_{\alpha}^{(\beta)\;*\;A}\phi_{\alpha}^{(\gamma)\;A})$
$\displaystyle=$ $\displaystyle{1\over
2g^{2}}\int_{T^{4}}\sum\limits_{\alpha=1}^{2}((\sigma.F^{A})_{\alpha}^{\;(\beta)}(\sigma.F^{A})_{\alpha}^{\;(\gamma)\;*}+(\sigma.F^{A})_{\alpha}^{\;(\beta)\;*}(\sigma.F^{A})_{\alpha}^{\;(\gamma)})\,,$
$\displaystyle U^{\beta,\gamma+2}=U^{\gamma+2,\beta}$ $\displaystyle=$
$\displaystyle{i\over
2g^{2}}\int_{T^{4}}\sum\limits_{\alpha=1}^{2}(\phi_{\alpha}^{(\beta)\;A}\phi_{\alpha}^{(\gamma)\;*\;A}-\phi_{\alpha}^{(\beta)\;*\;A}\phi_{\alpha}^{(\gamma)\;A})$
$\displaystyle=$ $\displaystyle{i\over
2g^{2}}\int_{T^{4}}\sum\limits_{\alpha=1}^{2}((\sigma.F^{A})_{\alpha}^{\;(\beta)}(\sigma.F^{A})_{\alpha}^{\;(\gamma)\;*}-(\sigma.F^{A})_{\alpha}^{\;(\beta)\;*}(\sigma.F^{A})_{\alpha}^{\;(\gamma)}).$

Let us now study the matrix

$A_{\beta\gamma}={1\over
2g^{2}}\int_{T^{4}}\sum\limits_{\alpha=1}^{2}(\sigma.F^{A})^{\;(\beta)}_{\alpha}(\sigma.F^{A})^{\;(\gamma)\;*}_{\alpha}.$
(276)

Using the notation of eqn. (253), we rewrite it as

$\displaystyle A_{\beta\gamma}$ $\displaystyle=$ $\displaystyle{1\over
2g^{2}}\int_{T^{4}}V^{A}_{a}V^{A}_{b}\;4\;(\sigma_{a})_{\alpha}^{\leavevmode\nobreak\
\leavevmode\nobreak\ (\beta)}((\sigma_{b})^{*})_{\alpha}^{\leavevmode\nobreak\
\leavevmode\nobreak\ (\gamma})={2\over
g^{2}}\int_{T^{4}}V^{A}_{a}V^{A}_{b}\;(\sigma_{2}\sigma_{a}\sigma_{b}\sigma_{2})_{(\beta)}^{\leavevmode\nobreak\
\leavevmode\nobreak\ (\gamma)}$ (277) $\displaystyle=$
$\displaystyle\delta_{\beta}^{\gamma}\;{2\over
g^{2}}\int_{T^{4}}V^{A}_{a}V^{A}_{a}={1\over
2g^{2}}\int_{T^{4}}(F_{mn}^{A})^{2}\,.$

Thus, the $4\times 4$ matrix $U$ with matrix elements (B.3.2), expressed
through $A$ of (276) is of the form

$\displaystyle U$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}A_{11}+A_{11}^{*}&A_{12}+A_{12}^{*}&i(A_{11}-A_{11}^{*})&i(A_{12}-A_{12}^{*})\cr
A_{12}+A_{12}^{*}&A_{22}+A_{22}^{*}&i(A_{12}-A_{12}^{*})&i(A_{22}-A_{22}^{*})\cr
i(A_{11}-A_{11}^{*})&i(A_{12}-A_{12}^{*})&A_{11}+A_{11}^{*}&A_{12}+A_{12}^{*}\cr
i(A_{12}-A_{12}^{*})&i(A_{22}-A_{22}^{*})&A_{12}+A_{12}^{*}&A_{22}+A_{22}^{*}\end{array}\right)$
(282) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}2A_{11}&0&0&0\cr 0&2A_{22}&0&0\cr
0&0&2A_{11}&0\cr 0&0&0&2A_{22}\end{array}\right)$ $\displaystyle=$
$\displaystyle{\text{diag}}(1,1,1,1)\;{1\over
g^{2}}\int_{T^{4}}(F_{mn}^{A})^{2}$ $\displaystyle=$
$\displaystyle{\text{diag}}(1,1,1,1)\;{4}\;{4\pi^{2}\over
g^{2}}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
(\text{since}\;{1\over 4g^{2}}\int_{T^{4}}(F_{mn}^{A})^{2}={4\pi^{2}\over
g^{2}})\leavevmode\nobreak\ .$ (288)

where we used the diagonal form of $A$ form (277).

Thus, we have for the norm matrix (200) of the bosonic zero modes (to any
arbitrary order in $\Delta$ to which the solution has been find):

$U^{kl}=\delta^{kl}\;4\;{4\pi^{2}\over g^{2}}\leavevmode\nobreak\
,\leavevmode\nobreak\ k,l=1,2,3,4.$ (289)

The point, so far, is to argue that the inner-product of the bosonic zero
modes obeying the gauge condition—or the moduli space metric (289)—are
$\Delta$-independent.424242The reader may notice that while we use the same
letter, the moduli space metric $U^{kl}$ of (289) does not equal the one
constructed earlier from the leading-order derivatives of the classical
solution w.r.t. $z_{n}$, the metric $U^{kl}$ of eqns. (200, 201). In fact, we
have $g^{8}\det U^{kl}_{eq.(\ref{ugeneral})}=(4\pi)^{8}$ while $g^{8}\det
U^{kl}_{eq.(\ref{zeromodenorm})}=V^{2}$. This is accounted for by the
difference in normalization of the respective zero modes; see Section B.3.3
for explicit expressions. This difference contributes an extra factor of
$({4\pi\over\sqrt{V}})^{8}\prod_{k}(L_{k}^{2})={(4\pi)^{8}\over V^{2}}$ to the
determinant of eq. (201).

The careful reader may remark that the fact that the bosonic zero mode metric
is proportional to the classical action, as in (289), is well-known, hence the
$\Delta$-independence follows from the fact that the action of the self-dual
solution is $\Delta$-independent. However, we presented the steps outlined
above, especially the explicit relation between the undotted Dirac equation
zero modes and the bosonic zero modes of eqn. (B.3.2), in order to use them to
verify that the derivatives of the ${\cal{O}}(\sqrt{\Delta})$ classical
solution we found with respect to $z_{n}$ are equal to the modes (B.3.2)
obeying the background gauge condition, after an appropriate gauge
transformation. This somewhat more involved procedure, compared to the similar
task for the BPST instanton, is what we discuss next.

####  B.3.3 The derivatives of the ${\cal{O}}(\sqrt{\Delta})$ solution and the
gauge condition

To this end, let us compute the zero modes (B.3.2) in the $\Delta$ expansion
and compare to the derivatives of the classical solution, $\partial
A_{n}^{cl}/\partial{z_{k}}$. The zero modes (B.3.2) obeying the gauge
condition434343As these can be constructed from $F_{mn}^{cl.}$ to any order in
$\Delta$, we shall call these “exact” zero modes. are determined by the
matrices (253)

$\displaystyle||\Re\phi_{\alpha}^{(\beta)\;A}||$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}0&2F_{13}^{A}\cr-2F_{13}^{A}&0\end{array}\right)\,,$
(292) $\displaystyle||\Im\phi_{\alpha}^{(\beta)\;A}||$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}-2F_{12}^{A}&-2F_{14}^{A}\cr-2F_{14}^{A}&2F_{12}^{A}\end{array}\right)\,.$
(295)

This yields for $Z_{n}^{(\beta)},Z_{n}^{(\beta)^{\prime}}$ of (B.3.2),

$\displaystyle Y_{n}^{(3)\;A}\equiv Z_{n}^{(1)\;A}$ $\displaystyle=$
$\displaystyle\left\\{\Im\phi_{2}^{(1)\;A},-\Re\phi_{2}^{(1)\;A},\Im\phi_{1}^{(1)\;A},\Re\phi_{1}^{(1)\;A}\right\\}=\left\\{-2F_{14}^{A},2F_{13}^{A},-2F_{12}^{A},0\right\\},$
$\displaystyle Y_{n}^{(1)\;A}\equiv-Z_{n}^{(2)\;A}$ $\displaystyle=$
$\displaystyle\left\\{-\Im\phi_{2}^{(2)\;A},\Re\phi_{2}^{(2)\;A},-\Im\phi_{1}^{(2)\;A},-\Re\phi_{1}^{(2)\;A}\right\\}=\left\\{-2F_{12}^{A},0,2F_{14}^{A},-2F_{13}^{A}\right\\},$
$\displaystyle Y_{n}^{(4)\;A}\equiv-Z_{n}^{(1\;^{\prime})\;A}$
$\displaystyle=$
$\displaystyle\left\\{-\Re\phi_{2}^{(1)\;A},-\Im\phi_{2}^{(1)\;A},-\Re\phi_{1}^{(1)\;A},\Im\phi_{1}^{(1)\;A}\right\\}=\left\\{2F_{13}^{A},2F_{14}^{A},0,-2F_{12}^{A}\right\\},$
$\displaystyle Y_{n}^{(2)\;A}\equiv-Z_{n}^{(2\;^{\prime})\;A}$
$\displaystyle=$
$\displaystyle\left\\{-\Re\phi_{2}^{(2)\;A},-\Im\phi_{2}^{(2)\;A},-\Re\phi_{1}^{(2)\;A},\Im\phi_{1}^{(2)\;A}\right\\}=\left\\{0,-2F_{12}^{A},-2F_{13}^{A},-2F_{14}^{A}\right\\}.$

We relabelled the four zero modes $Z_{n}^{(1)}$, $Z_{n}^{(2)}$,
$Z_{n}^{(1)^{\prime}}$, $Z_{n}^{(2)^{\prime}}$ by $Y_{n}^{(k)}$, where $k$
indicates that in our $\Delta$-expanded solution these correspond to
derivatives of the classical background with respect to $z_{k}$ (as we shall
see shortly). The functions $Y_{n}^{(k)}$ calculated in the leading
$\sim\sqrt{\Delta}$ solution are explicitly presented below.

Let us now also write our $\sim\sqrt{\Delta}$ self dual classical solution
(A.1):

$\displaystyle A_{n}^{cl}(x,z)$ $\displaystyle=$ $\displaystyle({2\pi
n_{nm}x^{m}\over L_{n}L_{m}}+{z_{n}\over L_{n}}){\tau^{3}\over
2}+{\tau^{+}}(\delta_{n1}+i\delta_{n2})W+{\tau^{-}}(\delta_{n1}-i\delta_{n2})W^{*},$
$\displaystyle W$ $\displaystyle=$
$\displaystyle-{i\sqrt{\pi\Delta}\over\sqrt{2V^{1/2}}}e^{i\alpha}F(x,z),$
(297)

where for brevity we introduced the (not anti-symmetric!) tensor $n_{nm}$
where only $n_{12}=n_{34}=1$ are nonzero. Consider now the derivative of the
classical solution w.r.t. $z_{p}/L_{p}$

$\displaystyle L_{p}{\partial A_{n}^{cl}\over\partial
z_{p}}=\delta_{np}{\tau^{3}\over
2}+{\tau^{+}}(\delta_{n1}+i\delta_{n2}){L_{p}\partial W\over\partial
z_{p}}+{\tau^{-}}(\delta_{n1}-i\delta_{n2}){L_{p}\partial W^{*}\over\partial
z_{p}}).$ (298)

The Cartan term is the one we considered before, see Section B. To find the
zero mode wave functions, an explicit computation using the SHO $h_{0}$ and
$h_{1}$ properties from (125) shows that

$\displaystyle L_{1}{\partial F\over\partial z_{1}}$ $\displaystyle=$
$\displaystyle-\sqrt{L_{1}L_{2}\over 4\pi}\tilde{G}\,,$ $\displaystyle
L_{2}{\partial F\over\partial z_{2}}$ $\displaystyle=$ $\displaystyle
i{L_{2}z_{1}\over 2\pi}F-i\sqrt{L_{1}L_{2}\over 4\pi}\tilde{G}\,,$
$\displaystyle L_{3}{\partial F\over\partial z_{3}}$ $\displaystyle=$
$\displaystyle-\sqrt{L_{3}L_{4}\over 4\pi}{G}\,,$ $\displaystyle
L_{4}{\partial F\over\partial z_{4}}$ $\displaystyle=$ $\displaystyle
i{L_{4}z_{3}\over 2\pi}F-i\sqrt{L_{3}L_{4}\over 4\pi}{G}\,.$ (299)

Here, we defined, in addition to $F(x,z)$ (137) and $G(x,z)$ (192), a new
function $\tilde{G}(x,z)$, similar to $G(x,z)$ but where the 1st-excited state
is in the $\omega_{12}$ SHO instead:

$\displaystyle\tilde{G}(x,z)$ $\displaystyle=$
$\displaystyle\sqrt{L_{2}L_{4}}\sum\limits_{n^{1},n^{3}=-\infty}^{\infty}e^{-i2\pi(n_{1}{x_{1}\over
L_{1}}+n_{3}{x_{3}\over L_{3}})}e^{-i{z_{2}\over
L_{2}}(x_{2}-n_{1}L_{2})-i{z_{4}\over L_{4}}(x_{4}-n_{3}L_{4})}$ (300)
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \times\leavevmode\nobreak\
\leavevmode\nobreak\ h_{1}^{12}(x_{2}-(n_{1}-{z_{1}\over
2\pi})L_{2})\;h_{0}^{34}(x_{4}-(n_{3}-{z_{3}\over
2\pi})L_{4})\leavevmode\nobreak\ .$

Using these expressions, we find the derivatives of the classical solution
w.r.t. $z_{p}$:

$\displaystyle L_{1}{\partial A_{n}^{cl}\over\partial z_{1}}$ $\displaystyle=$
$\displaystyle\delta_{n1}{\tau^{3}\over
2}+\left[{\tau^{+}}(\delta_{n1}+i\delta_{n2}){i\sqrt{\Delta}\over
2\sqrt{2}}e^{i\alpha}\tilde{G}+h.c.\right]\,,$ $\displaystyle L_{2}{\partial
A_{n}^{cl}\over\partial z_{2}}$ $\displaystyle=$
$\displaystyle\delta_{n2}{\tau^{3}\over
2}+\left[{\tau^{+}}(\delta_{n1}+i\delta_{n2})(-{i\sqrt{\pi\Delta}\over\sqrt{2V^{1/2}}}e^{i\alpha})(i{L_{2}z_{1}\over
2\pi}F-i\sqrt{L_{1}L_{2}\over 4\pi}\tilde{G})+h.c.\right]\,,$ $\displaystyle
L_{3}{\partial A_{n}^{cl}\over\partial z_{3}}$ $\displaystyle=$
$\displaystyle\delta_{n3}{\tau^{3}\over
2}+\left[{\tau^{+}}(\delta_{n1}+i\delta_{n2}){i\sqrt{\Delta}\over
2\sqrt{2}}e^{i\alpha}{G}+h.c.\right]\,,$ $\displaystyle L_{4}{\partial
A_{n}^{cl}\over\partial z_{4}}$ $\displaystyle=$
$\displaystyle\delta_{n4}{\tau^{3}\over
2}+\left[{\tau^{+}}(\delta_{n1}+i\delta_{n2})(-{i\sqrt{\pi\Delta}\over\sqrt{2V^{1/2}}}e^{i\alpha})(i{L_{4}z_{3}\over
2\pi}F-i\sqrt{L_{3}L_{4}\over 4\pi}{G})+h.c.\right]\,.$

The expressions for the exact zero modes $Y_{n}^{(k)}$ from (B.3.3) can also
be computed using our knowledge of the field strength to order
$\sqrt{\Delta}$, $F_{mn}^{(0)},F_{mn}^{(1)}$ of eqns. (97) and (191). Next, we
use this information and (B.3.3) to determine the gauge transformations needed
to bring the $L_{k}{\partial A_{n}^{cl}\over\partial z_{k}}$ zero modes into
the background-Lorentz gauge. We shall see that on the ${\mathbb{T}}^{4}$ with
twists, this is slightly different from the usual BPST instanton. To find
these gauge transformations, we now consider the derivatives of the classical
solution w.r.t. each $z_{k}$ in turn.

Zero mode $\mathbf{\partial\over\partial z_{1}}$ vs $Y^{(1)}$

: Here, we compare $Y_{n}^{(1)}$ to the derivative w.r.t. $z_{1}$. From the
above we find that the four-vectors of the exact zero mode and the derivative
of the classical solution are

$\displaystyle Y_{n}^{(1)}$ $\displaystyle=$
$\displaystyle\left\\{{4\pi\over\sqrt{V}}{\tau^{3}\over
2},0,\tau^{+}(-i{\sqrt{2\Delta}\pi\over\sqrt{V}}e^{i\alpha}G(x,z))+h.c.,\tau^{+}{\sqrt{2\Delta}\pi\over\sqrt{V}}e^{i\alpha}G(x,z)+h.c.\right\\}\,,$
$\displaystyle L_{1}{\partial A_{n}^{cl}\over\partial z_{1}}$ $\displaystyle=$
$\displaystyle\left\\{{\tau^{3}\over 2}+(\tau^{+}{i\sqrt{\Delta}\over
2\sqrt{2}}e^{i\alpha}\tilde{G}+h.c.),-(\tau^{+}{\sqrt{\Delta}\over
2\sqrt{2}}e^{i\alpha}\tilde{G}+h.c.),0,0\right\\}.$ (302)

Their difference is

$\displaystyle Y_{n}^{(1)}-{4\pi\over\sqrt{V}}L_{1}{\partial
A_{n}^{cl}\over\partial z_{1}}$ $\displaystyle=$
$\displaystyle\tau^{+}{\pi\sqrt{2\Delta}e^{i\alpha}\over
V^{1/2}}\left(\begin{array}[]{c}{-i}\tilde{G}(x,z)\cr\tilde{G}(x,z)\cr-
iG(x,z)\cr G(x,z)\end{array}\right)+h.c.$ (307) $\displaystyle=$
$\displaystyle
D_{n}(A^{cl.})\Lambda^{(1)}=\partial_{n}\Lambda^{(1)}+i[A_{n}^{cl.}\big{|}_{\Delta=0},\Lambda^{(1)}].$
(308)

After some algebra, we find that the gauge transformation making the
derivative of the classical solution obey the gauge condition is
${\cal{O}}(\sqrt{\Delta})$:

$\displaystyle\Lambda^{(1)}(x,z)=-{\sqrt{2\Delta\pi}\over V^{1\over
4}}e^{i\alpha}F(x,z)\tau^{+}+h.c.\,.$ (309)

Zero mode $\mathbf{\partial\over\partial z_{2}}$ vs $Y^{(2)}$

: Here we have the four-vectors

$\displaystyle Y_{n}^{(2)}$ $\displaystyle=$
$\displaystyle\left\\{0,{4\pi\over\sqrt{V}}{\tau^{3}\over
2},\tau^{+}{\sqrt{2\Delta}\pi\over\sqrt{V}}e^{i\alpha}G(x,z)+h.c.,\tau^{+}{i\sqrt{2\Delta}\pi\over\sqrt{V}}e^{i\alpha}G(x,z)+h.c.\right\\}\,,$
$\displaystyle L_{2}{\partial A_{n}^{cl}\over\partial z_{2}}$ $\displaystyle=$
$\displaystyle\left\\{\tau^{+}{\sqrt{\Delta\pi}e^{i\alpha}\over\sqrt{2}V^{1\over
4}}\left[{L_{2}z_{1}\over 2\pi}F(x,z)-\sqrt{L_{1}L_{2}\over
4\pi}\tilde{G}(x,z)\right]+h.c.,\right.$ (310)
$\displaystyle\left.{\tau^{3}\over
2}+\tau^{+}i{\sqrt{\Delta\pi}e^{i\alpha}\over\sqrt{2}V^{1\over
4}}\left[{L_{2}z_{1}\over 2\pi}F(x,z)-\sqrt{L_{1}L_{2}\over
4\pi}\tilde{G}(x,z)\right]+h.c.,0,0\right\\}\,.$

We again consider the difference

$\displaystyle Y_{n}^{(2)}-{4\pi\over\sqrt{V}}L_{2}{\partial
A_{n}^{cl}\over\partial z_{2}}$ $\displaystyle=$
$\displaystyle\tau^{+}{\pi\sqrt{2\Delta}e^{i\alpha}\over\sqrt{V}}\left(\begin{array}[]{c}-{2\sqrt{\pi}\over
V^{1\over 4}}\left[{L_{2}z_{1}\over 2\pi}F(x,z)-\sqrt{L_{1}L_{2}\over
4\pi}\tilde{G}(x,z)\right]\cr-i{2\sqrt{\pi}\over V^{1\over
4}}\left[{L_{2}z_{1}\over 2\pi}F(x,z)-\sqrt{L_{1}L_{2}\over
4\pi}\tilde{G}(x,z)\right]\cr G(x,z)\cr iG(x,z)\end{array}\right)+h.c.$ (315)
$\displaystyle=$ $\displaystyle
D_{n}(A^{cl.})\Lambda^{(2)}=\partial_{n}\Lambda^{(2)}+i[A_{n}^{cl.}\big{|}_{\Delta=0}+A_{n}^{cl.}\big{|}_{{\cal{O}}(\sqrt{\Delta})},\Lambda^{(2)}].$
(316)

Here, in contrast with (309), we find that the gauge transformation making the
zero mode obey the background condition also has a Cartan-subalgebra piece
$\sim\Delta^{0}$, in addition to an ${\cal{O}}(\sqrt{\Delta})$ piece which is
proportional to (309):

$\Lambda^{(2)}(x,z)={2L_{2}z_{1}\over\sqrt{V}}{\tau^{3}\over
2}+(i\Lambda^{(1)+}(x,z)\tau^{+}+h.c.)\leavevmode\nobreak\ .$ (317)

Naturally, only terms of order $\sqrt{\Delta}$ are to be kept on the r.h.s. of
(315).

Zero mode $\mathbf{\partial\over\partial z_{3}}$ vs $Y^{(3)}$

: Remarkably, here we find

$Y_{n}^{(3)}-{4\pi\over\sqrt{V}}L_{3}{\partial A_{n}^{cl}\over\partial
z_{3}}=0\implies\Lambda^{(3)}=0,$ (318)

hence no compensating gauge transform is needed. This follows from the four-
vector expressions for the zero modes

$Y_{n}^{(3)}=\left\\{i{\sqrt{2\Delta}\pi\over\sqrt{V}}e^{i\alpha}G\tau^{+}+h.c.,-{\sqrt{2\Delta}\pi\over\sqrt{V}}e^{i\alpha}G\tau^{+}+h.c.,{4\pi\over\sqrt{V}}{\tau^{3}\over
2},0\right\\}\,,$ (319)

and

$\displaystyle L_{3}{\partial A_{n}^{cl}\over\partial
z_{3}}=\left\\{i{\sqrt{\Delta}\over
2\sqrt{2}}e^{i\alpha}G\tau^{+}+h.c.,-{\sqrt{\Delta}\over
2\sqrt{2}}e^{i\alpha}G\tau^{+}+h.c.,{\tau^{3}\over
2},0\right\\}\leavevmode\nobreak\ .$ (320)

Notice that $x_{3}$ is the direction where $A^{cl}$ depends on
$x_{4}+z_{3}L_{4}/(2\pi)$ and where the classical $\sqrt{\Delta}$ solution
vanishes.

Zero mode $\mathbf{\partial\over\partial z_{4}}$ vs $Y^{(4)}$

: Here, the four-vectors of the exact zero mode and the derivative of the
classical solution are

$\displaystyle
Y_{n}^{(4)}=\left\\{-{\sqrt{2\Delta}\pi\over\sqrt{V}}e^{i\alpha}G\tau^{+}+h.c.,-i{\sqrt{2\Delta}\pi\over\sqrt{V}}e^{i\alpha}G\tau^{+}+h.c.,0,{4\pi\over\sqrt{V}}{\tau^{3}\over
2}\right\\}\,,$ (321)

and

$\displaystyle L_{4}{\partial A_{n}^{cl}\over\partial z_{4}}$ $\displaystyle=$
$\displaystyle\left\\{{\sqrt{\pi\Delta}\over\sqrt{2V^{1/2}}}e^{i\alpha}({L_{4}z_{3}\over
2\pi}F-\sqrt{L_{3}L_{4}\over 4\pi}{G})\tau^{+}+h.c.,\right.$ (322)
$\displaystyle\left.i{\sqrt{\pi\Delta}\over\sqrt{2V^{1/2}}}e^{i\alpha}({L_{4}z_{3}\over
2\pi}F-\sqrt{L_{3}L_{4}\over 4\pi}{G})\tau^{+}+h.c.,0,{\tau^{3}\over
2}\right\\}\leavevmode\nobreak\ .$

Their difference is

$\displaystyle Y_{n}^{(4)}-{4\pi\over\sqrt{V}}L_{4}{\partial
A_{n}^{cl}\over\partial z_{4}}$ $\displaystyle=$
$\displaystyle-\tau^{+}{\sqrt{2\pi\Delta}\over
V^{1/2}}e^{i\alpha}{L_{4}z_{3}\over V^{1\over
4}}\left(\begin{array}[]{c}F(x,z)\cr iF(x,z)\cr 0\cr 0\end{array}\right)+h.c.$
(327) $\displaystyle=$ $\displaystyle
D_{n}(A^{cl.})\Lambda^{(4)}=\partial_{n}\Lambda^{(4)}+i[A_{n}^{cl.}\big{|}_{\Delta=0}+A_{n}^{cl.}\big{|}_{{\cal{O}}(\sqrt{\Delta})},\Lambda^{(4)}].$
(328)

Here, we have that

$\Lambda^{(4)}(x,z)={2L_{4}z_{3}\over\sqrt{V}}{\tau^{3}\over 2}$ (329)

only has a $\Delta=0$ part. The claim that $\Lambda^{(4)}$ obeys (327) is
straightforwardly verified

Summary:

in each case we have verified that to order $\sqrt{\Delta}$,

${4\pi\over\sqrt{V}}L_{k}{\partial A_{n}^{cl}\over\partial
z_{k}}+D_{n}(A^{cl})\Lambda^{(k)}=Y_{n}^{(k)}$ (330)

obeys the background gauge condition, where $\Lambda^{(k)}$, for $k=1,2,3,4$,
are given in (309, 317, 318, 329), respectively. As follows from these
explicit expressions, the gauge transformation $\Lambda^{(k)}$ which makes the
derivative of the classical solution obey the background-gauge condition is
$\Omega$-periodic (recall the definition after (90)), i.e. obeys the same
periodicity conditions (89) as the classical solution. In fact, the
$x$-dependent part of $\Lambda^{(k)}$ is expressed through the
${\cal{O}}(\sqrt{\Delta})$ component of the classical solution $A_{k}^{cl}$,
similar to the BPST case. The $\Omega$-periodicity of $\Lambda^{(k)}$ is
important in what follows.

We shall next argue that the measure in terms of the $z_{n}$ variables remains
the one we found earlier by studying the leading-order zero modes, eqn. (202).
For use below, we also rewrite the zero modes (330) as

${\partial A_{n}^{cl}\over\partial
z_{k}}+D_{n}(A^{cl})\tilde{\Lambda}^{(k)}={\sqrt{V}\over 4\pi
L_{k}}Y_{n}^{(k)}\leavevmode\nobreak\ .$ (331)

The motivation for the rescaling evident in (331) is that the zero modes are
now directly proportional to the derivatives of $A_{n}^{cl}$ with respect to
$z_{k}$. The compensating gauge transformation
$\tilde{\Lambda}^{(k)}={\sqrt{V}\over 4\pi L_{k}}\Lambda^{(k)}$ is trivially
related to $\Lambda^{(k)}$ appearing in (330) and appearing in (309, 317, 318,
329).

####  B.3.4 The Jacobian and the all-order bosonic measure

We begin by shifting the bosonic field $A_{n}(x)$, obeying (89), to be
integrated over in the path integral by the classical solution
$A_{n}^{cl.}(x,z)$ of (B.3.3). We can choose to expand the fluctuation in
terms of a complete set of eigenfunctions of the hermitean operator
$O_{mn}(x,z)$—the zero modes (331) and the nonzero modes $Z_{n}^{q}$:444444Of
eigenvalues $\omega_{q}$, using the same notation as after eqn. (B.1):
$O_{mn}Z_{n}^{(q)}=\omega_{q}Z_{n}^{(q)}$.

$\displaystyle A_{n}(x)-A_{n}^{cl.}(x,z)$ $\displaystyle=$
$\displaystyle\sum\limits_{k=1}^{4}\zeta_{k}^{(0)}{\sqrt{V}\over 4\pi
L_{k}}\leavevmode\nobreak\
Y_{n}^{(k)}(x,z)+\sum_{q}\zeta_{q}\;Z_{n}^{(q)}(x,z).$ (332)

The coefficients $\zeta_{k}^{(0)}$ and $\zeta_{q}$ are the projections of the
gauge field fluctuation $A_{n}(x)-A_{n}^{cl.}(x,z)$ onto the $Y^{(k)}(x,z)$
and $Z_{q}(x,z)$ directions in field space, thus they have implicit
$z_{k}$-dependence. In what follows, we trade the integration over
$\zeta_{k}^{(0)}$ for integration over the $z_{k}$. In our subsequent
discussion we shall not explicitly write the contribution of the nonzero modes
to the measure of the path integral, due to their cancellation with the
contribution of the nonzero modes of the fermions and ghosts.

With the expansion (332), the measure of the bosonic zero modes now takes the
form, using (289)454545Recalling that (289) calculated $U_{kl}={2\over
g^{2}}\int_{{\mathbb{T}}^{4}}\,{\rm
tr}\,Y_{n}^{(k)}Y_{n}^{(l)}={16\pi^{2}\over g^{2}}\delta_{kl}$. and taking
into account the normalization of the zero modes

$\displaystyle d\mu_{B}$ $\displaystyle=$ $\displaystyle\left(\det{V\over
16\pi^{2}L_{k}L_{l}}\;U_{kl}\big{|}_{eq.(\ref{ugeneral})}\right)^{1\over
2}\;\prod\limits_{k=1}^{4}\left[{d\zeta_{k}^{(0)}\over\sqrt{2\pi}}\right]={V\over
g^{4}}\;\prod\limits_{k=1}^{4}\;{d\zeta_{k}^{(0)}\over\sqrt{2\pi}}\leavevmode\nobreak\
.$ (333)

In order to integrate over fields orthogonal to the $Y_{n}^{(k)}(x,z)$ zero
modes, we consider their inner product with (332)

$\displaystyle f_{k}$ $\displaystyle=$ $\displaystyle{2\over g^{2}}\,{\rm
tr}\,\int_{{\mathbb{T}}^{4}}(A_{n}(x)-A_{n}^{cl.}(x,z))\;Y_{n}^{(k)}(x,z).$
(334)

We then insert unity in the path integral, in the form

$\displaystyle 1=\prod_{k}dz_{k}\prod_{p}\delta(f_{p})\;|\det{\partial
f_{k}\over\partial z_{l}}|\leavevmode\nobreak\ ,$ (335)

where $\partial f_{k}\over\partial z_{l}$ is evaluated at the value of $z_{n}$
making delta function vanish.464646 Evaluating
$\int_{{\mathbb{T}}^{4}}A^{cl.}_{n}Y^{(k)}_{n}$ using the explicit expressions
of Section B.3.3 and plugging into (334), we find that to leading-order in
$\Delta$ the zero of $f_{k}$ occurs for $z_{k}=z_{k}^{*}$, where
$z_{k}^{*}=-\pi(\delta_{k3}+\delta_{k4})+L_{k}\tilde{A}_{k}^{3}+{\cal{O}}(\sqrt{\Delta})$.
Here, $\tilde{A}_{k}^{3}$ is the constant ${\mathbb{T}}^{4}$-mode of the
Cartan component of $A_{n}$.  Then, using (332), the orthogonality of the zero
and nonzero modes, we have that (289), that

$\displaystyle f_{p}={\sqrt{V}\over 4\pi L_{p}}{16\pi^{2}\over
g^{2}}\zeta_{p}^{(0)}\leavevmode\nobreak\ ,$ (336)

so that the $\delta$-function from (335) sets $\zeta_{k}^{(0)}=0$ after
integrating over $\zeta_{k}^{(0)}$ with the measure (333) (to avoid confusion,
recall that $\zeta_{k}^{(0)}$ has implicit $z$-dependence).

On the other hand, evaluating the derivative of (334) , we find

$\displaystyle{\partial f_{k}\over\partial z_{l}}$ $\displaystyle=$
$\displaystyle{2\over g^{2}}\int_{{\mathbb{T}}^{4}}\,{\rm
tr}\,\left({\partial(A_{n}-A_{n}^{cl})\over\partial
z_{l}}\;Y_{n}^{(k)}+(A_{n}-A_{n}^{cl}){\partial Y_{n}^{(k)}\over\partial
z_{l}}\right)$ $\displaystyle=$ $\displaystyle{2\over
g^{2}}\int_{{\mathbb{T}}^{4}}\,{\rm tr}\,\left(-{\partial
A_{n}^{cl.}\over\partial z_{l}}\;Y_{n}^{(k)}+(A_{n}-A_{n}^{cl}){\partial
Y_{n}^{(k)}\over\partial z_{l}}\right).$

Next, we use (331) to replace ${\partial A_{n}^{cl.}\over\partial z_{l}}$ by
$Y_{n}^{(k)}$. Using the fact that $\tilde{\Lambda}^{(k)}$ is
$\Omega$-periodic allows us to integrate by parts on ${\mathbb{T}}^{4}$
without a boundary term, i.e. set $\int_{{\mathbb{T}}^{4}}\,{\rm
tr}\,D_{n}\tilde{\Lambda}^{(l)}\;Y_{n}^{(k)}=0$, since $Y_{n}^{(k)}$ obeys the
background gauge condition. We thus find

$\displaystyle{\partial f_{k}\over\partial z_{l}}$ $\displaystyle=$
$\displaystyle{2\over g^{2}}\int_{{\mathbb{T}}^{4}}\,{\rm
tr}\,\left(-{\sqrt{V}\over 4\pi
L_{l}}Y_{n}^{(l)}\;Y_{n}^{(k)}+(A_{n}-A_{n}^{cl}){\partial
Y_{n}^{(k)}\over\partial z_{l}}\right)$ (337) $\displaystyle=$
$\displaystyle-{\sqrt{V}\over 4\pi L_{l}}{16\pi^{2}\over
g^{2}}\delta_{kl}+{2\over g^{2}}\int_{{\mathbb{T}}^{4}}\,{\rm
tr}\,(A_{n}-A_{n}^{cl}){\partial Y_{n}^{(k)}\over\partial
z_{l}}\leavevmode\nobreak\ .$

Thus, the unity insertion (335) becomes

$\displaystyle 1=\prod_{k}dz_{k}\;\delta(\zeta_{k}^{(0)}{\sqrt{V}\over 4\pi
L_{k}}{16\pi^{2}\over g^{2}})\;\bigg{|}\det\left({\sqrt{V}\over 4\pi
L_{l}}{16\pi^{2}\over g^{2}}\delta_{kl}-{2\over
g^{2}}\int_{{\mathbb{T}}^{4}}\,{\rm tr}\,\zeta_{q}Z_{n}^{(q)}{\partial
Y_{n}^{(k)}\over\partial z_{l}}\right)\bigg{|},$ (338)

where we used (332) and took the liberty to set $\zeta_{k}^{(0)}=0$ in the
determinant, due to integrating the delta function with the measure (333). The
nonzero mode part of the fluctuations can be ignored to leading order. Thus,
collecting everything, we find that the Jacobian factors from the delta
function and the determinant in (338) cancel out and the bosonic measure (333)
becomes, after inserting (338) and integrating over $\zeta_{k}^{(0)}$,

$\displaystyle d\mu_{B}$ $\displaystyle=$ $\displaystyle{V\over
g^{4}}\prod\limits_{k=1}^{4}{dz_{k}\over\sqrt{2\pi}}\leavevmode\nobreak\ .$
(339)

The bosonic zero mode measure is thus equal to the leading-order measure
constructed earlier in (202).

## References

  * (1) M. Garcia Perez, A. Gonzalez-Arroyo, and C. Pena, Perturbative construction of selfdual configurations on the torus, JHEP 09 (2000) 033, [hep-th/0007113].
  * (2) D. Gaiotto, A. Kapustin, N. Seiberg, and B. Willett, Generalized Global Symmetries, JHEP 02 (2015) 172, [arXiv:1412.5148].
  * (3) D. Gaiotto, A. Kapustin, Z. Komargodski, and N. Seiberg, Theta, Time Reversal, and Temperature, JHEP 05 (2017) 091, [arXiv:1703.00501].
  * (4) Y. Tanizaki, Anomaly constraint on massless QCD and the role of Skyrmions in chiral symmetry breaking, JHEP 08 (2018) 171, [arXiv:1807.07666].
  * (5) M. M. Anber and E. Poppitz, On the baryon-color-flavor (BCF) anomaly in vector-like theories, JHEP 11 (2019) 063, [arXiv:1909.09027].
  * (6) S. Bolognesi, K. Konishi, and A. Luzio, Gauging 1-form center symmetries in simple $SU(N)$ gauge theories, JHEP 01 (2020) 048, [arXiv:1909.06598].
  * (7) M. M. Anber and E. Poppitz, Generalized ’t Hooft anomalies on non-spin manifolds, JHEP 04 (2020) 097, [arXiv:2002.02037].
  * (8) T. Sulejmanpasic, Y. Tanizaki, and M. Ünsal, Universality between vector-like and chiral quiver gauge theories: Anomalies and domain walls, JHEP 06 (2020) 173, [arXiv:2004.10328].
  * (9) M. M. Anber, Self-conjugate QCD, JHEP 10 (2019) 042, [arXiv:1906.10315].
  * (10) N. Lohitsiri and T. Sulejmanpasic, Comments on QCD3 and anomalies with fundamental and adjoint matter, JHEP 10 (2022) 081, [arXiv:2205.07825].
  * (11) S. Bolognesi, K. Konishi, and A. Luzio, Dynamics from symmetries in chiral $SU(N)$ gauge theories, JHEP 09 (2020) 001, [arXiv:2004.06639].
  * (12) M. M. Anber, S. Hong, and M. Son, New anomalies, TQFTs, and confinement in bosonic chiral gauge theories, JHEP 02 (2022) 062, [arXiv:2109.03245].
  * (13) A. Kapustin and N. Seiberg, Coupling a QFT to a TQFT and Duality, JHEP 04 (2014) 001, [arXiv:1401.0740].
  * (14) G. ’t Hooft, A Property of Electric and Magnetic Flux in Nonabelian Gauge Theories, Nucl. Phys. B 153 (1979) 141–160.
  * (15) G. ’t Hooft, Some Twisted Selfdual Solutions for the Yang-Mills Equations on a Hypertorus, Commun. Math. Phys. 81 (1981) 267–275.
  * (16) E. Witten, Constraints on Supersymmetry Breaking, Nucl. Phys. B 202 (1982) 253.
  * (17) V. A. Novikov, M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, Instanton Effects in Supersymmetric Theories, Nucl. Phys. B 229 (1983) 407.
  * (18) G. C. Rossi and G. Veneziano, Nonperturbative Breakdown of the Nonrenormalization Theorem in Supersymmetric QCD, Phys. Lett. B 138 (1984) 195.
  * (19) D. Amati, G. C. Rossi, and G. Veneziano, Instanton Effects in Supersymmetric Gauge Theories, Nucl. Phys. B 249 (1985) 1–41.
  * (20) I. Affleck, M. Dine, and N. Seiberg, Supersymmetry Breaking by Instantons, Phys. Rev. Lett. 51 (1983) 1026.
  * (21) V. A. Novikov, M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, Supersymmetric Instanton Calculus (Gauge Theories with Matter), Nucl. Phys. B 260 (1985) 157–181.
  * (22) D. Amati, K. Konishi, Y. Meurice, G. C. Rossi, and G. Veneziano, Nonperturbative Aspects in Supersymmetric Gauge Theories, Phys. Rept. 162 (1988) 169–248.
  * (23) M. A. Shifman and A. I. Vainshtein, Instantons versus supersymmetry: Fifteen years later, pp. 485–647, 2, 1999.  hep-th/9902018.
  * (24) M. A. Shifman, Exact results in gauge theories: Putting supersymmetry to work. The 1999 Sakurai Prize Lecture, Int. J. Mod. Phys. A 14 (1999) 5017–5048, [hep-th/9906049].
  * (25) N. Dorey, T. J. Hollowood, V. V. Khoze, and M. P. Mattis, The Calculus of many instantons, Phys. Rept. 371 (2002) 231–459, [hep-th/0206063].
  * (26) S. Vandoren and P. van Nieuwenhuizen, Lectures on instantons, arXiv:0802.1862.
  * (27) J. Terning, Modern supersymmetry: Dynamics and duality.  2006\.
  * (28) A. A. Belavin, A. M. Polyakov, A. S. Schwartz, and Y. S. Tyupkin, Pseudoparticle Solutions of the Yang-Mills Equations, Phys. Lett. B 59 (1975) 85–87.
  * (29) S. F. Cordes, The Instanton Induced Superpotential in Supersymmetric QCD, Nucl. Phys. B 273 (1986) 629–648.
  * (30) D. Finnell and P. Pouliot, Instanton calculations versus exact results in four-dimensional SUSY gauge theories, Nucl. Phys. B 453 (1995) 225–239, [hep-th/9503115].
  * (31) N. M. Davies, T. J. Hollowood, and V. V. Khoze, Monopoles, affine algebras and the gluino condensate, J. Math. Phys. 44 (2003) 3640–3656, [hep-th/0006011].
  * (32) A. Kovner and M. A. Shifman, Chirally symmetric phase of supersymmetric gluodynamics, Phys. Rev. D 56 (1997) 2396–2402, [hep-th/9702174].
  * (33) A. Ritz and A. I. Vainshtein, Instantons at strong coupling, averaging over vacua, and the gluino condensate, Nucl. Phys. B 566 (2000) 311–328, [hep-th/9909073].
  * (34) F. Cachazo, M. R. Douglas, N. Seiberg, and E. Witten, Chiral rings and anomalies in supersymmetric gauge theory, JHEP 12 (2002) 071, [hep-th/0211170].
  * (35) C. Csaki and H. Murayama, Discrete anomaly matching, Nucl. Phys. B 515 (1998) 114–162, [hep-th/9710105].
  * (36) T. J. Hollowood, V. V. Khoze, W.-J. Lee, and M. P. Mattis, Breakdown of cluster decomposition in instanton calculations of the gluino condensate, Nucl. Phys. B 570 (2000) 241–266, [hep-th/9904116].
  * (37) N. M. Davies, T. J. Hollowood, V. V. Khoze, and M. P. Mattis, Gluino condensate and magnetic monopoles in supersymmetric gluodynamics, Nucl. Phys. B 559 (1999) 123–142, [hep-th/9905015].
  * (38) B. J. Harrington and H. K. Shepard, Periodic Euclidean Solutions and the Finite Temperature Yang-Mills Gas, Phys. Rev. D 17 (1978) 2122\.
  * (39) T. C. Kraan and P. van Baal, Periodic instantons with nontrivial holonomy, Nucl. Phys. B 533 (1998) 627–659, [hep-th/9805168].
  * (40) M. F. Atiyah, N. J. Hitchin, V. G. Drinfeld, and Y. I. Manin, Construction of Instantons, Phys. Lett. A 65 (1978) 185–187.
  * (41) K.-M. Lee and P. Yi, Monopoles and instantons on partially compactified D-branes, Phys. Rev. D 56 (1997) 3711–3717, [hep-th/9702107].
  * (42) K.-M. Lee and C.-h. Lu, SU(2) calorons and magnetic monopoles, Phys. Rev. D 58 (1998) 025011, [hep-th/9802108].
  * (43) G. V. Dunne and M. Ünsal, New Nonperturbative Methods in Quantum Field Theory: From Large-N Orbifold Equivalence to Bions and Resurgence, Ann. Rev. Nucl. Part. Sci. 66 (2016) 245–272, [arXiv:1601.03414].
  * (44) E. Poppitz, Notes on Confinement on $R^{3}\times S1$: From Yang-Mills, Super-Yang Mills, and QCD (adj) to QCD(F), Symmetry 14 (2022), no. 1 180, [arXiv:2111.10423].
  * (45) P. van Baal, Some Results for SU(N) Gauge Fields on the Hypertorus, Commun. Math. Phys. 85 (1982) 529.
  * (46) P. van Baal, SU($N$) Yang-Mills Solutions With Constant Field Strength on $T^{4}$, Commun. Math. Phys. 94 (1984) 397.
  * (47) A. Gonzalez-Arroyo and P. Martinez, Investigating Yang-Mills theory and confinement as a function of the spatial volume, Nucl. Phys. B 459 (1996) 337–354, [hep-lat/9507001].
  * (48) A. Gonzalez-Arroyo, P. Martinez, and A. Montero, Gauge invariant structures and confinement, Phys. Lett. B 359 (1995) 159–165, [hep-lat/9507006].
  * (49) A. Gonzalez-Arroyo and A. Montero, Do classical configurations produce confinement?, Phys. Lett. B 387 (1996) 823–828, [hep-th/9604017].
  * (50) M. Garcia Perez, A. Gonzalez-Arroyo, and A. Sastre, From confinement to adjoint zero-modes, eCONF C0906083 (2009) 06, [arXiv:1003.5022].
  * (51) M. Luscher, Some Analytic Results Concerning the Mass Spectrum of Yang-Mills Gauge Theories on a Torus, Nucl. Phys. B 219 (1983) 233–261.
  * (52) P. van Baal, QCD in a finite volume, hep-ph/0008206.
  * (53) M. Garcia Perez, A. Gonzalez-Arroyo, A. Montero, and P. van Baal, Calorons on the lattice: A New perspective, JHEP 06 (1999) 001, [hep-lat/9903022].
  * (54) P. J. Braam and P. van Baal, Nahm’s Transformation for Instantons, Commun. Math. Phys. 122 (1989) 267.
  * (55) M. Ünsal, Strongly coupled QFT dynamics via TQFT coupling, JHEP 11 (2021) 134, [arXiv:2007.03880].
  * (56) E. Cohen and C. Gomez, Chiral Symmetry Breaking in Supersymmetric Yang-Mills, Phys. Rev. Lett. 52 (1984) 237.
  * (57) A. A. Cox, E. Poppitz, and F. D. Wandler, The mixed 0-form/1-form anomaly in Hilbert space: pouring the new wine into old bottles, JHEP 10 (2021) 069, [arXiv:2106.11442].
  * (58) A. González-Arroyo, Constructing SU(N) fractional instantons, JHEP 02 (2020) 137, [arXiv:1910.12565].
  * (59) A. Gonzalez-Arroyo and A. Ramos, Expansion for the solutions of the Bogomolny equations on the torus, JHEP 07 (2004) 008, [hep-th/0404022].
  * (60) Y. Tanizaki and M. Ünsal, Center vortex and confinement in Yang–Mills theory and QCD with anomaly-preserving compactifications, PTEP 2022 (2022), no. 4 04A108, [arXiv:2201.06166].
  * (61) A. Gonzalez Arroyo and C. P. Korthals Altes, The Spectrum of Yang-Mills Theory in a Small Twisted Box, Nucl. Phys. B 311 (1988) 433–449.
  * (62) G. ’t Hooft, Computation of the Quantum Effects Due to a Four-Dimensional Pseudoparticle, Phys. Rev. D 14 (1976) 3432–3450. [Erratum: Phys.Rev.D 18, 2199 (1978)].
  * (63) M. M. Anber and E. Poppitz, Anomaly matching, (axial) Schwinger models, and high-T super Yang-Mills domain walls, JHEP 09 (2018) 076, [arXiv:1807.00093].
  * (64) G. ’t Hooft, Aspects of Quark Confinement, Phys. Scripta 24 (1981) 841–846.