$\lambda_{\alpha}^{(\mbox{\scriptsize\boldmath$0$})}\equiv\eta_{\alpha}$ play
a special role. Thinking in the Hamiltonian terms, we stay in the universe
where the wave functions are invariant under the large gauge transformations
[cf. Eq. (2.10)],

$\displaystyle A_{j}\to A_{j}+\frac{2\pi n_{j}}{eL},$
$\displaystyle(s^{f},\psi^{f})\to(s^{f},\psi^{f})e^{2\pi
i\mbox{\scriptsize\boldmath$n$}\cdot\mbox{\scriptsize\boldmath$x$}/L},\quad(t,\xi)\to(t,\xi)e^{-4\pi
i\mbox{\scriptsize\boldmath$n$}\cdot\mbox{\scriptsize\boldmath$x$}/L}\,.$
(6.5)

The zero mode $c_{j}$ of the gauge potential lies in a dual torus,
$c_{j}\in[0,\frac{2\pi}{eL})$, and a characteristic energy of this degree of
freedom is $\sim e^{2}/L$. Most other degrees of freedom have a characteristic
energy $\sim 1/L\gg e^{2}/L$ and are fast, to be integrated out. We will find
then the effective supercharges by averaging the full supercharges over the
ground eigenstate of the fast Hamiltonian:242424In contrast to what we had for
ordinary SQED, we cannot now, even at the leading order, just to cross out the
terms including fast variables in the full Hamiltonian. Well, we can do so for
the terms including higher harmonics of $A_{j}$ and $\lambda_{\alpha}$, but
not for the matter fields.

$\displaystyle\hat{Q}_{\alpha}^{\rm eff}\ =\ \langle\hat{Q}_{\alpha}^{\rm
full}\rangle_{\rm fast\ vacuum},\qquad\hat{\bar{Q}}^{\alpha\,{\rm eff}}\ =\
\langle\hat{\bar{Q}}^{\alpha\,{\rm full}}\rangle_{\rm fast\ vacuum}\,.$ (6.6)

The effective Hamiltonian will then be restored as the anticommutator
$\hat{H}^{\rm eff}=\\{\hat{Q}_{\alpha}^{\rm eff},\hat{\bar{Q}}^{\alpha\,{\rm
eff}}\\}/2$.

However, in contrast to ordinary SQED, the variables $c_{j}$ and their
superpartners $\eta_{\alpha}$ are not the only slow variables that determine
the low-energy dynamics of chiral SQED. There are also the variables
describing the motion along the valley

$\displaystyle\sum_{f=1}^{8}s^{*f}s^{f}-4t^{*}t\ =\ 0\,.$ (6.7)

When the Yukawa coupling $h$ vanishes, the spectrum associated with this
motion is continuous.

If $h$ is nonzero but small, the spectrum is discrete, but the characteristic
excitation energies and level spacings are also small. The valley dynamics is
described in this case by a WZ model with the quartic potential (6.3). By
virial theorem, the characteristic kinetic and potential energies should be of
the same order, which means

$\frac{1}{L^{3}|s|^{2}}\ \sim\ \frac{1}{L^{3}|t|^{2}}\ \sim\
h^{2}L^{3}|s|^{2}|t|^{2}\ \sim\ h^{2}L^{3}|s|^{4}\,,$

giving $|s|^{2}\sim|t|^{2}\sim h^{-2/3}/L^{2}$ and

$\displaystyle E_{\rm char}^{\rm WZ}\ \sim\ \frac{h^{2/3}}{L}\,.$ (6.8)

If $h\ll e^{3}$, this is much less than the characteristic energies in the
sector of the theory associated with the gauge dynamics. We will be interested
in the case where the low-energy dynamics is mainly determined by the gauge
sector. To this end, we must keep $h\gg e^{3}$. Still, as we will see soon,
$h$ cannot be too large. The calculation below is valid in the region

$\displaystyle e^{3}\ \ll\ h\ \ll\ e\,.$ (6.9)

The details of this calculation are spelled out in Ref. [54], but let us give
here some hints explaining how it works. What we should do is to take the fast
Hamiltonian in the leading quadratic approximation, find its spectrum,
treating the slow variables $c_{j},\eta_{\alpha}$ as parameters, and determine
the effective supercharges by calculating the averages of the full
supercharges over the vacuum state of $\hat{H}^{\rm fast}$.

Take one of the multiplets of unit charge, set $L=1$, concentrate on the
values of $c$ close to the origin, neglect the Yukawa terms, and keep, to
begin with, only the zero Fourier modes of the matter fields. In the leading
approximation, the bosonic part of the fast Hamiltonian is then a simple
oscillator,

$\displaystyle\hat{H}^{\rm fast}_{\rm bos}\ =\ -\frac{\partial^{2}}{\partial
s\partial s^{*}}+e^{2}\mbox{\boldmath$c$}^{2}s^{*}\\!s$ (6.10)

[here $s$ is the zero mode of $s(x)$].252525The bosonic fast Hamiltonian
involves also a quartic term $\sim e^{2}(ss^{*})^{2}$, but bearing in mind
(6.11), it is suppressed when we are far enough from the origin,
$e|\mbox{\boldmath$c$}|^{3}\gg 1$. Then

$\displaystyle\langle s^{*}\\!s\rangle_{\rm fast\ vacuum}\ =\
\frac{1}{2e|\mbox{\boldmath$c$}|}\,.$ (6.11)

The part of the full supercharges of the model including only the zero Fourier
modes of one of the matter superfields $S^{f}$ reads

$\displaystyle\hat{Q}_{\alpha}$ $\displaystyle=$
$\displaystyle\left[\hat{P}_{k}(\sigma_{k})_{\alpha}{}^{\gamma}+ies^{*}\\!s\,\delta_{\alpha}^{\gamma}\right]\eta_{\gamma}+\sqrt{2}\left[-i\hat{\bar{\pi}}\delta_{\alpha}^{\gamma}+esc_{k}(\sigma_{k})_{\alpha}{}^{\gamma}\right]\hat{\bar{\psi}}_{\gamma},$
$\displaystyle\hat{\bar{Q}}^{\beta}$ $\displaystyle=$
$\displaystyle\hat{\bar{\eta}}^{\delta}\left[\hat{P}_{k}(\sigma_{k})_{\delta}{}^{\beta}-ies^{*}\\!s\,\delta_{\delta}^{\beta}\right]-\sqrt{2}\,\psi^{\delta}\left[i\hat{\pi}\delta_{\delta}^{\beta}+es^{*}c_{k}(\sigma_{k})_{\delta}{}^{\beta}\right]\,.$
(6.12)

If we add the contribution of higher Fourier harmonics and evaluate the
anticommutator $\\{\hat{Q}_{\alpha},\hat{\bar{Q}}^{\beta}\\}$, the Hamiltonian
thus obtained would match the Lagrangian (2) [a part of it not involving the
mass terms and the fields $t(x)$ and $\xi(x)$].

The second terms in Eqs. (6.1) are fast supercharges describing the dynamics
of the fast variables $s,\psi$. Their anticommutators give the fast
Hamiltonian, so that $\hat{Q}_{\alpha}^{\rm fast}$ and
$\hat{\bar{Q}}^{\beta\,{\rm fast}}$ annihilate its ground state. And the
averages of the first terms (the slow supercharges) over the fast vacuum have
a nontrivial structure. For example, $\hat{Q}_{\alpha}$ includes a term
${ie}s^{*}\\!s\,\eta_{\alpha}$. Its average over the fast vacuum gives the
contribution $\frac{i\eta_{\alpha}}{2|\mbox{\scriptsize\boldmath$c$}|}$ in the
effective supercharge.

And that is not all. The fast Hamiltonian involves also a fermionic part, and
its ground state involves a factor
$w_{\alpha}(\mbox{\boldmath$c$})\psi^{\alpha}$, where

$\displaystyle w_{\alpha}\ =\
\left(\begin{array}[]{c}e^{-i\phi}\cos\frac{\theta}{2}\\\
\sin\frac{\theta}{2}\end{array}\right),$ (6.15)

with $\theta,\phi$ being the polar and azimuthal angles characterizing the
direction of the background field $c$. Then $\langle\hat{P}_{k}\rangle_{\rm
fast\ vacuum}$ does not vanish, giving a “vector potential” ${\cal
A}_{k}(\mbox{\boldmath$c$})$ which should be added to the operator
$\hat{P}_{k}$ in the effective supercharge. We derive

$\displaystyle\hat{Q}_{\alpha}^{\rm eff}\ =\
\left[(\sigma_{k})_{\alpha}{}^{\beta}(\hat{P}_{k}+{\cal
A}_{k})+i\delta_{\alpha}^{\beta}{\cal D}\right]\eta_{\beta},$
$\displaystyle\hat{\bar{Q}}^{\alpha\,{\rm eff}}\ =\
\hat{\bar{\eta}}^{\beta}\left[(\sigma_{k})_{\beta}{}^{\alpha}(\hat{P}_{k}+{\cal
A}_{k})-i\delta_{\beta}^{\alpha}{\cal D}\right],$ (6.16)
$\displaystyle\hat{H}^{\rm eff}\ =\ \frac{1}{2}(\hat{P}_{k}+{\cal
A}_{k})^{2}+\frac{1}{2}{\cal D}^{2}+\mbox{\boldmath$\cal
H$}\,\hat{\bar{\eta}}\mbox{\boldmath$\sigma$}\eta\,,$ (6.17)

where $\hat{P}_{k}=-i\partial/\partial c_{k}$, ${\cal
D}=1/(2|\mbox{\boldmath$c$}|)$ and

$\displaystyle\mbox{\boldmath$\cal H$}\ =\ \nabla\times\mbox{\boldmath$\cal
A$}\ =\ \mbox{\boldmath$\nabla$}{\cal D}\,.$ (6.18)

The vector function ${\cal A}(\mbox{\boldmath$c$})$ depends on field variables
rather than on spatial coordinates and has nothing to do with the vector
potentials $\mbox{\boldmath$A$}(\mbox{\boldmath$x$})$ !

As was also the case for the ordinary SQED [see Eq. (2.23)], the effective
Hamiltonian (6.17) has the benign Wilsonean nature as soon as
$|\mbox{\boldmath$c$}|$ is not too small and

$\displaystyle\kappa\ =\ \frac{1}{e|\mbox{\boldmath$c$}|^{3}}\ \ll\ 1\,.$
(6.19)

But chiral SQED includes also the Yukawa term (6.2), and we want to make sure
that the corresponding corrections to the Hamiltonian $\sim h^{2}(ss^{*})^{2}$
are irrelevant if $\kappa$ is small. We have to compare the characteristic
value of these corrections with the gap in the spectrum of the fast
Hamiltonian, which is of order $e|\mbox{\boldmath$c$}|$. Bearing in mind
(6.11), we derive

$\frac{h^{2}}{e^{2}|\mbox{\boldmath$c$}|^{2}}\ \ll\ e|\mbox{\boldmath$c$}|$

or $h^{2}\kappa\ll e^{2}$. We are on the safe side if

$h\ \ll\ e\,,$

and that is how the upper limit for the allowed values of $h$ in (6.9) was
derived.

Let us now take into account all other Fourier harmonics of all the matter
fields $s^{f}(x),\psi^{f}(x),t(x)$ and $\xi(x)$ and restore the dependence on
$L$. We obtain

$\displaystyle{\cal D}\ =\
4\sum_{\mbox{\scriptsize\boldmath$n$}}\frac{1}{\left|\mbox{\boldmath$c$}-\frac{2\pi\mbox{\scriptsize\boldmath$n$}}{eL}\right|}-\frac{1}{2}\sum_{\mbox{\scriptsize\boldmath$n$}}\frac{1}{\left|\mbox{\boldmath$c$}-\frac{\pi\mbox{\scriptsize\boldmath$n$}}{eL}\right|}$
(6.20)

The Hamiltonian (6.17) with a generic function ${\cal D}(\mbox{\boldmath$c$})$
coincides with the Hamiltonian written in the paper [55]. If ${\cal
D}=-1/(2|\mbox{\boldmath$c$}|)$, the magnetic field

$\displaystyle\mbox{\boldmath$\cal H$}\ =\
\frac{\mbox{\boldmath$c$}}{2|\mbox{\boldmath$c$}|^{3}}$ (6.21)

is the field of a magnetic monopole with the minimal magnetic charge
$g=1/2$.262626Note, however, the presence of the extra scalar potential
$1/(8\mbox{\boldmath$c$}^{2})$, which makes the problem supersymmetric. This
monopole singularity in the configuration space is known in other branches of
physics by the name Pancharatnam-Berry phase [56].

In our case, we are dealing with a cubic lattice of monopoles. In fact, there
are two superimposed cubic lattices: a lattice of monopoles of charge $1/2$
with edge length $\pi/eL$ and the lattice of monopoles of charge $-4$ located
at the nodes of a twice less dense lattice with edge length $2\pi/eL$. The net
magnetic charge density in field space is zero — this is a corollary of the
condition

$\displaystyle\sum_{f}q_{f}^{3}\ =\ 0\,,$ (6.22)

which our chiral model enjoys.

Figure 14: A unit cubic cell of the monopole crystal in field space. The edge
of the cube is $2\pi/(eL)$. The blue blob marks the site with the monopole of
charge $-7/2$. The red blobs mark the sites with the monopoles of charge
$1/2$.

Now let us calculate the Witten index in this system. There are two ways to do
so.

1. One can use the Cecotti-Girardello method and evaluate the integral (1.7) with the classical counterpart of the Hamiltonian (6.17). Integration over fermion variables reduces to the evaluation of the determinant

$\displaystyle\det\left[\beta(\sigma_{j})_{\alpha}{}^{\beta}{\cal
H}_{j}\right]\ =\ -\beta^{2}\mbox{\boldmath$\cal H$}^{2}\,.$ (6.23)

When we also integrate over momenta, we obtain

$\displaystyle I_{W}\ =\ \lim_{\beta\to
0}\left[-\left(\frac{\beta}{8\pi^{3}}\right)^{1/2}\int
d\mbox{\boldmath$c$}\,(\mbox{\boldmath$\nabla$}{\cal D})^{2}e^{-\beta{\cal
D}^{2}/2}\right],$ (6.24)

where the integral is evaluated over the unit cell of our “crystal” depicted
in Fig. 14. For small $\beta$, the integral is saturated by the regions
adjacent to the singularities of ${\cal D}$ — the sites where the monopoles
sit. The contribution of each such region is equal to $-|g|$, where $g$ is the
charge of the monopole. In our case, the unit cell contains a monopole of
charge $-4+1/2=-7/2$ and seven monopoles of charge $1/2$. The final result is

$\displaystyle I_{W}\ =\ -\frac{7}{2}-\frac{7}{2}\ =\ -7\,.$ (6.25)

This calculation strongly suggests the existence of 7 fermion vacuum states in
this model, but it does not prove it. The matter is that the Cecotti-
Girardello method that reduces the functional integral for the index to an
ordinary one has its applicability limits. First of all, it does not work for
systems with continuous spectrum where the notion of index is ill-defined. In
our case, the motion is finite and the spectrum is discrete, but we a facing
another problem: the Hamiltonian (6.17) has singularities $\sim
1/|\mbox{\boldmath$c$}-\mbox{\boldmath$c$}_{\rm node}|^{2}$ at the nodes of
the lattice. The integral for the index is saturated by the regions of $c$ at
the vicinity of these singularities when
$|\mbox{\boldmath$c$}-\mbox{\boldmath$c$}_{\rm node}|\sim\sqrt{\beta}$. But if
one looks at the original functional integral (1.4), one can observe that the
characteristic values of the higher Fourier harmonics
$\mbox{\boldmath$c$}^{(n)}$ are also of order $\sqrt{\beta}$, and one cannot
be sure that the integral (6.24) not involving higher harmonics gives a
correct value for the index.

Indeed, we will see in the next section that it does not [the value of the
integral (1.7) is not integer] for non-Abelian chiral theories, For SQED it
does, however. We know this because we can confirm the result (6.25) by a
direct analysis of the Schrödinger equation.

2. To do so, we deform the Hamiltonian (6.17) by adding a constant $C$ to $D$. The condition (6.18) is still fulfilled, which assures sypersymmetry. The index does not change under such deformation.

Let $C$ be positive and large, $C\gg 1/(eL)$. The wave functions of the low-
lying states are then localized in the region of small ${\cal
D}^{\prime}={\cal D}+C$, i.e. near the “red” sites with positively charged
monopoles. If $C$ is large, the sites do not “talk” to each other, and we can
pick out one of them, say, the site in the center of the cube with
$\mbox{\boldmath$r$}=\mbox{\boldmath$c$}-(\pi/eL)(1,1,1)=\mbox{\boldmath$0$}$.
Then the problem reduces to the solution of the Schrödinger equation with the
Hamiltonian

$\displaystyle\hat{H}\ =\
\frac{1}{2}(\hat{\mbox{\boldmath$P$}}+\mbox{\boldmath$\cal
A$})^{2}+\frac{1}{2}\left(C-\frac{1}{2r}\right)^{2}+\frac{\mbox{\boldmath$r$}}{2r^{3}}\,\hat{\bar{\eta}}\mbox{\boldmath$\sigma$}\eta\,.$
(6.26)

Zero-energy states are possible only in the $F=1$ sector, where the problem
reduces to the motion of a spin $\frac{1}{2}$ particle in the field of a
monopole and an additional spherically symmetric potential. The angular
variables can then be separated (as is spelled out in Appendix B), and the
lowest harmonic has total angular momentum $j=|g|-1/2=0$. The equation for the
corresponding radial function $\chi(r)$ reads

$\displaystyle-\frac{1}{2r}\frac{\partial^{2}}{\partial
r^{2}}(r\chi)+\frac{1}{2}\left(C^{2}+\frac{C}{r}-\frac{1}{4r^{2}}\right)\chi\
=\ E\chi\,.$ (6.27)

The zero-energy solution of (6.27) is readily found:

$\displaystyle\chi(r)\ \sim\frac{1}{\sqrt{r}}e^{-Cr}\,.$ (6.28)

In a particular gauge, where the “vector potentials” are chosen in the form

$\displaystyle{\cal A}_{1}\ =\
-\frac{c_{2}}{2|\mbox{\boldmath$c$}|(|\mbox{\boldmath$c$}|+c_{3})}\ =\
-\frac{\sin\theta\sin\phi}{2r(1+\cos\theta)},$ $\displaystyle{\cal A}_{2}\ =\
\frac{c_{1}}{2|\mbox{\boldmath$c$}|(|\mbox{\boldmath$c$}|+c_{3})}\ =\
\frac{\sin\theta\cos\phi}{2r(1+\cos\theta)},$ $\displaystyle{\cal A}_{3}\ =\
0\,,$ (6.29)

the full vacuum wave function has the form

$\displaystyle\Psi(r,\theta,\phi;\,\eta_{\alpha})\ =\
\frac{1}{\sqrt{r}}e^{-Cr}\left(\eta_{1}\sqrt{1+\cos\theta}+\eta_{2}\sqrt{1-\cos\theta}\,e^{-i\phi}\right)\,.$
(6.30)

The singularity of this function at $r=0$ is benign — the integral $\int
d\mbox{\boldmath$r$}|\Psi|^{2}$ converges quite well.272727 Note that a
Hamiltonian that contains an attractive singular potential
$V(r)=-\gamma/r^{2}$ with $\gamma>1/8$ exhibits a falling on the center
phenomenon such that the ground state is absent and, which is worse, unitarity
is lost [57]. But in our case $\gamma$ takes the border value $\gamma=1/8$ and
the potential includes also a Coulomb repulsion. Then there is no collapse and
the quantum problem is benign. According to general theorems, $\Psi$ is
annihilated by the action of the supercharges (6.1). A dedicated reader may
check it in explicit calculation.

The unit cell in Fig. 14 contains seven sites with “red” monopoles of charge
$g=1/2$, so that the spectrum contains seven fermionic zero-energy states.

If the constant $C$ is large and negative, the wave function settles on the
blue site with $g=-7/2$. Then the lowest angular harmonic has $j=3$, which
produces a sevenfold angular degeneracy. And the equation for the radial wave
function has only one zero-energy solution, as before.

We calculated the index of the effective Hamiltonian (6.17). This Hamiltonian
has a Wilsonian nature if the Yukawa constant lies in the range (6.9) and if
the conditions

$\displaystyle\kappa_{\mbox{\scriptsize\boldmath$n$}}\ =\
\frac{1}{e|L\mbox{\boldmath$c$}-\pi\mbox{\boldmath$n$}/e|^{3}}\ \ll\ 1\,.$
(6.31)

are all fulfilled so that we are not too close to the corners of the dual
torus where the monopoles sit and where the BP corrections are large. Can we
be sure that our result $I_{W}=-7$ is valid also for the original field theory
(6.1) ?

Well, I can swear that it is, but I would not bet my head on that. The reason
by which I still hesitate to do so is the limited range of the dynamic
variables where the effective Hamiltonian (6.17) is Wilsonean. On the other
hand, the original Witten’s calculation of the index in SYM theories was also
based on the effective Hamiltonian, which was only valid outside the corners
of the dual torus. Hypothetically, these corners could bring about surprises
for SYM theories, and we cannot exclude that they do so for chiral SQED. Of
course, the Hamiltonian (3.18) is much simpler than (6.17) and it has no
singularities. In addition, the result $I_{W}=h^{\vee}$ was independently
confirmed by other methods, as was explained in Sect. 5.

For the chiral SQED, the Hamiltonian is more complicated (the necessity to
introduce a Yukawa coupling being an extra complication), it has singularities
at the corners, and we do not have an independent confirmation of the result
(6.25).

That is where we stand today.

###  6.2 A chiral $SU(3)$ model

The simplest such model includes besides the vector multiplet seven matter
chiral multiplets in the fundamental group representation $\bf 3$ and a
multiplet in the representation $\bar{\bf 6}$. In terms of superfields, the
Lagrangian of the model is

$\displaystyle{\cal L}=\left(\frac{1}{4}{\rm Tr}\int
d^{2}\theta\,\hat{W}^{\alpha}\hat{W}_{\alpha}+{\rm
c.c.}\right)+\frac{1}{4}\int
d^{4}\theta\left(\sum_{f=1}^{7}\overline{S}^{fj}e^{g\hat{V}}S^{f}_{j}+\Phi^{jk}e^{-g\hat{V}}\overline{\Phi}_{jk}\right)$
(6.32)

with $j,k=1,2,3$ and $\Phi^{jk}=\Phi^{kj}$.

This theory is anomaly-free. Indeed, the contribution of the matter fermions
in the non-Abelian anomalous triangle in Fig. 13 is equal in this case to

$\displaystyle\langle A^{a}A^{b}A^{c}\rangle\ \propto\ 7\,{\rm
Tr}\\{t^{(a}t^{b}t^{c)}\\}-{\rm Tr}\\{T^{(a}T^{b}T^{c)}\\}\,,$ (6.33)

where $t^{a}$ and $T^{a}$ are the generators in the triplet and sextet
representations, correspondingly. The latter have the form

$\displaystyle(T^{a})_{ij}^{kl}\ =\
\frac{1}{2}\left[(t^{a})_{i}^{k}\delta_{j}^{l}+(t^{a})_{i}^{l}\delta_{j}^{k}+(t^{a})_{j}^{k}\delta_{i}^{l}+(t^{a})_{j}^{l}\delta_{i}^{k}\right]\,,$
(6.34)

where the factor $1/2$ stems from the requirement that $T^{a}$ satisfy the
same commutation relations, $[T^{a},T^{b}]=if^{abc}T^{c}$, as $t^{a}$.

For any $SU(N)$ group, one can derive

$\displaystyle{\rm Tr}\\{T^{(a}T^{b}T^{c)}\\}\ =\ (N+4)\,{\rm
Tr}\\{t^{(a}t^{b}t^{c)}\\}\,.$ (6.35)

In our case, $N+4=7$ and the net contribution to the anomaly vanishes.

The component Lagrangian of the model includes a quartic scalar potential,
$V(s,\phi)\ =\ \frac{g^{2}}{2}\left(s^{*f}t^{a}s^{f}-\phi
T^{a}\phi^{*}\right)^{2}$, which vanishes on the valley

$\displaystyle s^{*f}t^{a}s^{f}-\phi T^{a}\phi^{*}\ =\ 0\,.$ (6.36)

The motion along this valley brings about a continuous spectrum in the same
way as for chiral SQED. We can also lock this valley by including in the
Lagrangian the Yukawa term,282828When $h=0$, the theory is asymptotically
free: the effective coupling constant $g(\mu)$ runs to zero in the limit
$\mu\to\infty$. But in the theory with a nonzero Yukawa coupling, the latter
runs into a pole in this limit, so that this theory has the same status as QED
and SQED: one can only treat it in the region of energies where both $g(\mu)$
and $h(\mu)$ are small.

$\displaystyle{\cal L}_{\rm Yukawa}^{SU(3)}\ =\
h\int\\!\\!d^{2}\theta\,(S^{f}_{j}S^{f}_{k}\Phi^{jk})\ +\ {\rm c.c.}\,.$
(6.37)

The corresponding scalar potential has the form

$\displaystyle V_{\rm Yukawa}\ \sim\
h^{2}(4|s_{j}^{f}\phi^{jk}|^{2}+|s_{j}^{f}s_{k}^{f}|^{2})\,.$ (6.38)

We repeat what we did for chiral SQED: put the theory in a small spatial box
of length $L$, expand all the fields in the Fourier series, assume the
effective coupling constant $g^{2}(L)$ to be small and assume that the Yukawa
coupling belongs in the range $g^{3}\ll h\ll g$. Under these conditions, the
low-energy spectrum of the theory is determined by the effective Hamiltonian
in the gauge sector — it depends on zero Fourier modes of
$\mbox{\boldmath$A$}^{a}(x)$ belonging to the Cartan subalgebra of $SU(3)$ and
their fermion superpartners, while the Yukawa terms and the whole dynamics
associated with the valley (6.36) can be disregarded. In the following, we
will set $g=L=1$.

The effective Hamiltonian can be found along the same lines as for the Abelian
theory. We have 6 bosonic slow variables that can be presented as

$\displaystyle\hat{\mbox{\boldmath$c$}}\ =\ \frac{1}{2}\,{\rm
diag}(\mbox{\boldmath$a$},\mbox{\boldmath$b$}-\mbox{\boldmath$a$},-\mbox{\boldmath$b$})$
(6.39)

with292929It is the rhombus in Fig. 1. For the reasons to become clear at the
end of this subsection, we do not care now about the Weyl alcove.
$a_{j},b_{j}\in[0,4\pi)$.

Take a triplet $s_{j}$ and concentrate on its zero Fourier mode. In the
leading approximation, the fast Hamiltonian describing its interaction with
the background gauge field $\hat{\mbox{\boldmath$c$}}$ reads

$\displaystyle H^{\rm fast}_{\rm bos}$ $\displaystyle=$
$\displaystyle-\frac{\partial^{2}}{\partial s^{*}_{j}\partial
s_{j}}+s^{*}(\hat{\mbox{\boldmath$c$}})^{2}s$ (6.40) $\displaystyle=$
$\displaystyle-\frac{\partial^{2}}{\partial s^{*}_{j}\partial
s_{j}}+\frac{1}{4}\left[\mbox{\boldmath$a$}^{2}s_{1}^{*}s_{1}\ +\
(\mbox{\boldmath$a$}-\mbox{\boldmath$b$})^{2}s_{2}^{*}s_{2}\ +\
\mbox{\boldmath$b$}^{2}s_{3}^{*}s_{3}\right]$

The fast ground state wave function (its bosonic part) is

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\Psi(s_{j},s^{*}_{j})=\exp\left\\{-\frac{|\mbox{\boldmath$a$}|}{2}|s_{1}|^{2}\right\\}\exp\left\\{-\frac{|\mbox{\boldmath$a$}-\mbox{\boldmath$b$}|}{2}|s_{2}|^{2}\right\\}\exp\left\\{-\frac{|\mbox{\boldmath$b$}|}{2}|s_{3}|^{2}\right\\}.$
(6.41)

It follows:

$\displaystyle\langle|s_{1}|^{2}\rangle=\frac{1}{|\mbox{\boldmath$a$}|},\qquad\langle|s_{2}|^{2}\rangle=\frac{1}{|\mbox{\boldmath$a$}-\mbox{\boldmath$b$}|},\qquad\langle|s_{3}|^{2}\rangle=\frac{1}{|\mbox{\boldmath$b$}|}\,.$
(6.42)

The corresponding contribution to the slow supercharge [cf. the first terms in
Eq. (6.1)] is

$\displaystyle\hat{Q}_{\alpha}\ =\
\eta_{\gamma}^{a}\left[(\sigma_{k})_{\alpha}{}^{\gamma}\hat{P}^{a}_{k}+i\delta_{\alpha}^{\gamma}s^{*}t^{a}s\right]\,.$
(6.43)

Averaging the last term in (6.43) over the fast vacuum, we extract the
following contribution to the effective supercharge:

$\displaystyle\frac{i\eta^{3}_{\alpha}}{2}[\langle|s_{1}|^{2}\rangle-\langle|s_{2}|^{2}\rangle]+\frac{i\eta^{8}_{\alpha}}{2\sqrt{3}}[\langle|s_{1}|^{2}\rangle+\langle|s_{2}|^{2}\rangle-2\langle|s_{3}|^{2}\rangle]\
=$ (6.44)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\frac{i\eta^{(a)}_{\alpha}}{2}\left[\frac{1}{|\mbox{\boldmath$a$}|}-\frac{1}{|\mbox{\boldmath$a$}-\mbox{\boldmath$b$}|}\right]+\frac{i\eta^{(b)}_{\alpha}}{2}\left[\frac{1}{|\mbox{\boldmath$a$}-\mbox{\boldmath$b$}|}-\frac{1}{|\mbox{\boldmath$b$}|}\right]\equiv
i\left[\eta^{(a)}{\cal D}_{(a)}+\eta^{(b)}{\cal D}_{(b)}\right]\,,$

where303030That means that
$\displaystyle\hat{\eta}_{\alpha}=\eta_{\alpha}^{a}t^{a}\ =\ \frac{1}{2}{\rm
diag}(\eta_{\alpha}^{(a)},\eta_{\alpha}^{(b)}-\eta_{\alpha}^{(a)},-\eta_{\alpha}^{(b)}).$
(6.45)

$\displaystyle\eta^{(a)}_{\alpha}\ =\
\eta^{3}_{\alpha}+\frac{1}{\sqrt{3}}\eta^{8}_{\alpha},\qquad\eta^{(b)}_{\alpha}\
=\ \frac{2}{\sqrt{3}}\eta^{8}_{\alpha}\,.$ (6.46)

Here ${\cal D}_{(a)}$ and ${\cal D}_{(b)}$ are the contributions due to the
zero Fourier mode of the triplet to the induced $D$-terms. If we add the
contributions of the other triplet Fourier modes with the averages
$\langle|s_{1}^{(\mbox{\scriptsize\boldmath$n$})}|^{2}\rangle=1/|\mbox{\boldmath$a$}-4\pi\mbox{\boldmath$n$}|$
etc., and multiply the result by 7 (we have 7 triplets), we derive [cf.
(6.20)]:

$\displaystyle
D^{(a)}=\frac{7}{2}\sum_{\mbox{\scriptsize\boldmath$n$}}\left[\frac{1}{|\mbox{\boldmath$a$}_{2\mbox{\scriptsize\boldmath$n$}}|}-\frac{1}{|(\mbox{\boldmath$a$}-\mbox{\boldmath$b$})_{2\mbox{\scriptsize\boldmath$n$}}|}\right],$
$\displaystyle
D^{(b)}=\frac{7}{2}\sum_{\mbox{\scriptsize\boldmath$n$}}\left[\frac{1}{|(\mbox{\boldmath$a$}-\mbox{\boldmath$b$})_{2\mbox{\scriptsize\boldmath$n$}}|}-\frac{1}{|\mbox{\boldmath$b$}_{2\mbox{\scriptsize\boldmath$n$}}|}\right]\,,$
(6.47)

where
$\mbox{\boldmath$a$}_{\mbox{\scriptsize\boldmath$m$}}=\mbox{\boldmath$a$}-2\pi\mbox{\boldmath$m$},\
\mbox{\boldmath$b$}_{\mbox{\scriptsize\boldmath$m$}}=\mbox{\boldmath$b$}-2\pi\mbox{\boldmath$m$}$.

Consider now the contribution of the antisextet. The part of the fast
Hamiltonian including the zero modes of $\phi_{ij}(\mbox{\boldmath$x$})$ is

$\displaystyle\hat{H}^{\rm fast}\ =\
\hat{\bar{\pi}}^{ij}\hat{\pi}_{ij}+\phi^{ij}\mbox{\boldmath$\cal
X$}^{2}\phi^{*}_{ij},$ (6.48)

where

$\mbox{\boldmath$\cal X$}\ =\
\mbox{\boldmath$A$}^{3}T^{3}+\mbox{\boldmath$A$}^{8}T^{8}\ =\
\mbox{\boldmath$a$}X+(\mbox{\boldmath$b$}-\mbox{\boldmath$a$})Y-\mbox{\boldmath$b$}Z$

with

$\displaystyle X_{ij}^{kl}\ =\
\frac{1}{4}[\delta_{i1}\delta^{k1}\delta_{j}^{l}+\delta_{i1}\delta^{l1}\delta_{j}^{k}+\delta_{j1}\delta^{k1}\delta_{i}^{l}+\delta_{j1}\delta^{l1}\delta_{i}^{k}],$
$\displaystyle Y_{ij}^{kl}\ =\
\frac{1}{4}[\delta_{i2}\delta^{k2}\delta_{j}^{l}+\delta_{i2}\delta^{l2}\delta_{j}^{k}+\delta_{j2}\delta^{k2}\delta_{i}^{l}+\delta_{j2}\delta^{l2}\delta_{i}^{k}],$
$\displaystyle Z_{ij}^{kl}\ =\
\frac{1}{4}[\delta_{i3}\delta^{k3}\delta_{j}^{l}+\delta_{i3}\delta^{l3}\delta_{j}^{k}+\delta_{j3}\delta^{k3}\delta_{i}^{l}+\delta_{j3}\delta^{l3}\delta_{i}^{k}].$
(6.49)

One finds the following scalar averages:

$\displaystyle\langle|\phi^{11}|^{2}\rangle\ =\
\frac{1}{2|\mbox{\boldmath$a$}|},\qquad\langle|\phi^{22}|^{2}\rangle\ =\
\frac{1}{2|\mbox{\boldmath$a$}-\mbox{\boldmath$b$}|},\qquad\langle|\phi^{33}|^{2}\rangle\
=\ \frac{1}{2|\mbox{\boldmath$b$}|},$
$\displaystyle\langle|\phi^{23}|^{2}\rangle\ =\
\frac{1}{|\mbox{\boldmath$a$}|},\qquad\langle|\phi^{12}|^{2}\rangle\ =\
\frac{1}{|\mbox{\boldmath$b$}|},\qquad\langle|\phi^{13}|^{2}\rangle\ =\
\frac{1}{|\mbox{\boldmath$a$}-\mbox{\boldmath$b$}|}.$

The slow supercharge acquires the contribution

$\displaystyle\Delta^{\rm antisextet}\hat{Q}_{\alpha}\ =\
-i{\eta_{\gamma}^{a}}\phi T^{a}\phi^{*}\,.$ (6.51)

Averaging it over the fast vacuum and taking Eq. (6.2) into account, we derive
the contribution

$\displaystyle\frac{i\eta_{\alpha}^{(a)}}{2}\left[\frac{1}{|\mbox{\boldmath$a$}|}-\frac{1}{|\mbox{\boldmath$a$}-\mbox{\boldmath$b$}|}\right]+\frac{i\eta_{\alpha}^{(b)}}{2}\left[\frac{1}{|\mbox{\boldmath$a$}-\mbox{\boldmath$b$}|}-\frac{1}{|\mbox{\boldmath$b$}|}\right]$
(6.52)

to the effective supercharge, which happens to coincide with the contribution
of a triplet. Consider now the contribution of the higher modes
$\phi^{ij}_{(\mbox{\scriptsize\boldmath$n$})}$ in the fast Hamiltonian. An
analysis similar to what we did for the triplets brings about the averages

$\displaystyle\langle|\phi^{11}_{(\mbox{\scriptsize\boldmath$n$})}|^{2}\rangle\
=\
\frac{1}{2|\mbox{\boldmath$a$}_{\mbox{\scriptsize\boldmath$n$}}|},\qquad\langle|\phi^{22}_{(\mbox{\scriptsize\boldmath$n$})}|^{2}\rangle\
=\
\frac{1}{2|(\mbox{\boldmath$a$}-\mbox{\boldmath$b$})_{\mbox{\scriptsize\boldmath$n$}}|},\qquad\langle|\phi^{33}_{(\mbox{\scriptsize\boldmath$n$})}|^{2}\rangle\
=\ \frac{1}{2|\mbox{\boldmath$b$}_{\mbox{\scriptsize\boldmath$n$}}|},$
$\displaystyle\langle|\phi^{23}_{(\mbox{\scriptsize\boldmath$n$})}|^{2}\rangle\
=\
\frac{1}{|\mbox{\boldmath$a$}_{2\mbox{\scriptsize\boldmath$n$}}|},\qquad\langle|\phi^{12}_{(\mbox{\scriptsize\boldmath$n$})}|^{2}\rangle\
=\
\frac{1}{|\mbox{\boldmath$b$}_{2\mbox{\scriptsize\boldmath$n$}}|},\qquad\langle|\phi^{13}_{(\mbox{\scriptsize\boldmath$n$})}|^{2}\rangle\
=\
\frac{1}{|(\mbox{\boldmath$a$}-\mbox{\boldmath$b$})_{2\mbox{\scriptsize\boldmath$n$}}|}.$

Note the different periodicity patterns for the components
$\phi^{11},\phi^{22},\phi^{33}$ vs. the mixed components
$\phi^{12},\phi^{13},\phi^{23}$. Plugging (6.2) into the fast vacuum averages
of the slow supercharge and adding the triplet contribution (6.2), we derive:

$\displaystyle\\!\\!\\!\\!\\!{\cal D}_{(a)}$ $\displaystyle=$
$\displaystyle\frac{9}{2}\sum_{\mbox{\scriptsize\boldmath$n$}}\left[\frac{1}{|\mbox{\boldmath$a$}_{2\mbox{\scriptsize\boldmath$n$}}|}-\frac{1}{|(\mbox{\boldmath$a$}-\mbox{\boldmath$b$})_{2\mbox{\scriptsize\boldmath$n$}}|}\right]\
-\
\frac{1}{2}\sum_{\mbox{\scriptsize\boldmath$n$}}\left[\frac{1}{|\mbox{\boldmath$a$}_{\mbox{\scriptsize\boldmath$n$}}|}-\frac{1}{|(\mbox{\boldmath$a$}-\mbox{\boldmath$b$})_{\mbox{\scriptsize\boldmath$n$}}|}\right],$
$\displaystyle\\!\\!\\!\\!\\!{\cal D}_{(b)}$ $\displaystyle=$
$\displaystyle\frac{9}{2}\sum_{\mbox{\scriptsize\boldmath$n$}}\left[\frac{1}{|(\mbox{\boldmath$a$}-\mbox{\boldmath$b$})_{2\mbox{\scriptsize\boldmath$n$}}|}-\frac{1}{|\mbox{\boldmath$b$}_{2\mbox{\scriptsize\boldmath$n$}}|}\right]\
-\
\frac{1}{2}\sum_{\mbox{\scriptsize\boldmath$n$}}\left[\frac{1}{|(\mbox{\boldmath$a$}-\mbox{\boldmath$b$})_{\mbox{\scriptsize\boldmath$n$}}|}-\frac{1}{|\mbox{\boldmath$b$}_{\mbox{\scriptsize\boldmath$n$}}|}\right].$
(6.54)

The second terms are the contributions of the sextet components
$\phi^{11},\phi^{22},\phi^{33}$. The corresponding lattice is twice as dense
as the lattice for the triplets and for the components
$\phi^{12},\phi^{13},\phi^{23}$. The reason is the same as for chiral SQED.
For a large gauge transformation when, say, $a_{x}$ is shifted by $4\pi$, a
sextet matter fields $\phi^{11},\phi^{22},\phi^{33}$ (but not
$\phi^{12},\phi^{13},\phi^{23}$) have to “make two turns” being multiplied by
$e^{4\pi ix}$ rather than by $e^{2\pi ix}$ to compensate this shift, in the
same way as the fields of double charge did in the Abelian model.

As in the Abelian model, the effective supercharges acquire besides the
induced ${\cal D}$ terms the “vector potential” contributions coming from the
averages $\langle\partial/\partial\mbox{\boldmath$a$}\rangle$ and
$\langle\partial/\partial\mbox{\boldmath$b$}\rangle$ over the fermion factor
in the fast ground state wave function. As a result, we derive the following
expressions for the effective supercharges:

$\displaystyle\hat{Q}_{\alpha}^{\rm eff}\ $ $\displaystyle=$ $\displaystyle\
{\eta^{(a)}_{\beta}}\left[(\sigma_{k})_{\alpha}{}^{\beta}(\hat{P}_{k}^{(a)}+{\cal
A}^{(a)}_{k})+i\delta_{\alpha}^{\beta}{\cal D}_{(a)}\right]$ $\displaystyle+$
$\displaystyle\
{\eta^{(b)}_{\beta}}\left[(\sigma_{k})_{\alpha}{}^{\beta}(\hat{P}_{k}^{(b)}+{\cal
A}^{(b)}_{k})+i\delta_{\alpha}^{\beta}{\cal D}_{(b)}\right],$
$\displaystyle\hat{\bar{Q}}^{\alpha\,{\rm eff}}\ $ $\displaystyle=$
$\displaystyle\
{\hat{\bar{\eta}}^{(a)\beta}}\left[(\sigma_{k})_{\beta}{}^{\alpha}(\hat{P}_{k}^{(a)}+{\cal
A}^{(a)}_{k})+i\delta_{\beta}^{\alpha}{\cal D}_{(a)}\right]$ (6.55)
$\displaystyle+$ $\displaystyle\
{\hat{\bar{\eta}}^{(b)\beta}}\left[(\sigma_{k})_{\beta}{}^{\alpha}(\hat{P}_{k}^{(b)}+{\cal
A}^{(b)}_{k})+i\delta_{\beta}^{\alpha}{\cal D}_{(b)}\right],$

where $\hat{P}_{k}^{(a)}=-i\partial/\partial a^{k}$ and
$\hat{P}_{k}^{(b)}=-i\partial/\partial b^{k}$ and $\mbox{\boldmath$\cal
A$}^{(a,b)}$ and ${\cal D}_{(a,b)}$ are related so that

$\displaystyle{\cal H}^{(a)}_{j}$ $\displaystyle\equiv$
$\displaystyle\varepsilon_{jkl}\,\partial_{k}^{a}{\cal A}^{(a)}_{l}\ =\
\partial_{j}^{a}{\cal D}_{(a)}\,,$ $\displaystyle{\cal H}^{(b)}_{j}$
$\displaystyle\equiv$ $\displaystyle\varepsilon_{jkl}\,\partial_{k}^{b}{\cal
A}^{(b)}_{l}\ =\ \partial_{j}^{b}{\cal D}_{(b)}\,,$ $\displaystyle{\cal
H}^{(ab)}_{j}$ $\displaystyle\equiv$
$\displaystyle\varepsilon_{jkl}\,\partial_{k}^{a}{\cal A}^{(b)}_{l}\ =\
\varepsilon_{jkl}\,\partial_{k}^{b}{\cal A}^{(a)}_{l}\ =\
\partial_{j}^{a}{\cal D}_{(b)}\ =\ \partial_{j}^{b}{\cal D}_{(a)}\,.$ (6.56)

Bearing in mind that

$\displaystyle\\{\hat{\bar{\eta}}^{(a)\beta},\eta^{(a)}_{\alpha}\\}=\\{\hat{\bar{\eta}}^{(b)\beta},\eta^{(b)}_{\alpha}\\}=\frac{4}{3}\delta_{\alpha}^{\beta},\quad\\{\hat{\bar{\eta}}^{(a)\beta},\eta^{(b)}_{\alpha}\\}=\\{\hat{\bar{\eta}}^{(b)\beta},\eta^{(a)}_{\alpha}\\}=\frac{2}{3}\delta_{\alpha}^{\beta}\,,$

we derive the effective Hamiltonian:313131The effective supercharges (6.2) and
the Hamiltonian (6.2) were derived in Ref. [58]. Unfortunately, the
expressions for the induced ${\cal D}$ terms quoted in that paper involved a
mistake. As a result, the value of the phase space integral for the Witten
index was also not correct. But we will confirm in the calculations below the
main conclusion of that work: the phase space integral (1.7) evaluated for the
Hamiltonian (6.2) has a fractional value, so that the CG method to evaluate
the index does not work in this case.

$\displaystyle\hat{H}^{\rm eff}\ =\ \frac{2}{3}\left[(\hat{P}_{k}^{(a)}+{\cal
A}^{(a)}_{k})^{2}+(\hat{P}_{k}^{(b)}+{\cal
A}^{(b)}_{k})^{2}+(\hat{P}_{k}^{(a)}+{\cal
A}^{(a)}_{k})(\hat{P}_{k}^{(b)}+{\cal A}^{(b)}_{k})\right.$
$\displaystyle+\left.{\cal D}_{(a)}^{2}+{\cal D}_{(b)}^{2}+{\cal D}_{(a)}{\cal
D}_{(b)}\right]$ $\displaystyle+\mbox{\boldmath$\cal
H$}^{(a)}\,\hat{\bar{\eta}}^{(a)}\mbox{\boldmath$\sigma$}\eta^{(a)}+\mbox{\boldmath$\cal
H$}^{(b)}\,\hat{\bar{\eta}}^{(b)}\mbox{\boldmath$\sigma$}\eta^{(b)}+\mbox{\boldmath$\cal
H$}^{(ab)}(\hat{\bar{\eta}}^{(a)}\mbox{\boldmath$\sigma$}\eta^{(b)}+\hat{\bar{\eta}}^{b}\mbox{\boldmath$\sigma$}\eta^{(a)})\,.$
(6.58)

####  6.2.1 Phase space integral

The Hamiltonian (6.2) is essentially more complicated than (6.17), and it is
difficult in this case to find the vacuum states explicitly, as we did for
SQED. So we use the Cecotti-Girardello method and evaluate the integral (1.7)
for the classical version of the Hamiltonian (6.2). This calculation is more
complicated than in the Abelian case, but is still feasible.

Note first of all that the integral (1.7) was written in the assumption that
the bosonic and fermion variables form a canonical basis with the ordinary
Poisson brackets

$\displaystyle\\{p_{j},q_{k}\\}_{\rm P.B.}\ =\
\delta_{jk},\qquad\\{\psi_{a},\bar{\psi}_{b}\\}_{\rm P.B.}\ =\
i\delta_{ab}\,.$ (6.59)

In our case, it is so for bosonic variables, but not so for the fermion
variables whose Poisson brackets are nontrivial, as follows from (6.2). The
Jacobian of the transformation from the canonical variables
$\eta_{\alpha}^{3},\eta_{\alpha}^{8},\bar{\eta}^{\beta 3},\bar{\eta}^{\beta
8}$ to
$\eta_{\alpha}^{(a)},\eta_{\alpha}^{(b)},\bar{\eta}^{\alpha(a)},\bar{\eta}^{\alpha(b)}$
gives the factor $(2/\sqrt{3})^{4}=16/9$. Integrating over the fermions, we
derive

$\displaystyle I_{W}=\ \lim_{\beta\to
0}\frac{16}{9}\beta^{4}\int\frac{d\mbox{\boldmath$a$}\,d\mbox{\boldmath$P^{(a)}$}d\mbox{\boldmath$b$}\,d\mbox{\boldmath$P^{(b)}$}}{(2\pi)^{6}}\left|\begin{array}[]{cc}\mbox{\boldmath$\cal
H$}^{(a)}\mbox{\boldmath$\sigma$},&\mbox{\boldmath$\cal
H$}^{(ab)}\mbox{\boldmath$\sigma$}\\\ \mbox{\boldmath$\cal
H$}^{(ab)}\mbox{\boldmath$\sigma$},&\mbox{\boldmath$\cal
H$}^{(b)}\mbox{\boldmath$\sigma$}\end{array}\right|$ (6.62)
$\displaystyle\cdot\exp\left\\{-\frac{2}{3}\beta\left[(\mbox{\boldmath$P$}^{(a)}+\mbox{\boldmath$\cal
A$}^{(a)}_{k})^{2}+(\mbox{\boldmath$P$}^{(b)}+\mbox{\boldmath$\cal
A$}^{(b)}_{k})^{2}+(\mbox{\boldmath$P$}^{(a)}+\mbox{\boldmath$\cal
A$}^{(a)}_{k})(\mbox{\boldmath$P$}^{(b)}+\mbox{\boldmath$\cal
A$}^{(b)}_{k})\right.\right.$ $\displaystyle\left.\left.+{\cal
D}_{(a)}^{2}+{\cal D}_{(b)}^{2}+{\cal D}_{(a)}{\cal D}_{(b)}\right]\right\\}.$
(6.63)

Calculating the determinant and integrating over momenta, we obtain

$\displaystyle I_{W}=\lim_{\beta\to 0}\frac{\beta}{4\pi^{3}\sqrt{3}}\int
d\mbox{\boldmath$a$}d\mbox{\boldmath$b$}\left[\|\mbox{\boldmath$H$}^{(ab)}\|^{4}+2\|\mbox{\boldmath$H$}^{(ab)}\|^{2}(\mbox{\boldmath$H$}^{(a)}\cdot\mbox{\boldmath$H$}^{(b)})\right.$
$\displaystyle\left.+\
\|\mbox{\boldmath$H$}^{(a)}\|^{2}\|\mbox{\boldmath$H$}^{(b)}\|^{2}-4(\mbox{\boldmath$H$}^{(a)}\cdot\mbox{\boldmath$H$}^{(ab)})(\mbox{\boldmath$H$}^{(b)}\cdot\mbox{\boldmath$H$}^{(ab)})\right]$
$\displaystyle\times\exp\left\\{-\frac{2}{3}\beta\left[D_{(a)}^{2}+D_{(b)}^{2}+D_{(a)}D_{(b)}\right]\right\\}.$
(6.64)

The integral runs over the fundamental cell $a_{j},b_{j}\in[0,4\pi)$ of our
“crystal” (recall that the gauge invariant slow wave functions are periodic in
the space of weights with the period $4\pi$ for each component of $a$ and
$b$). If $\beta$ is small, the integral is saturated by the regions near the
singularities of ${\cal D}_{(a)}$ and ${\cal D}_{(b)}$. There are 64 such
regions in the fundamental cell:

(i) its corner $\mbox{\boldmath$a$}\sim\mbox{\boldmath$b$}\sim 0$;

(ii) 42 symmetric regions of the type
$\mbox{\boldmath$a$}\sim\mbox{\boldmath$a$}_{0}=2\pi(1,0,0),\mbox{\boldmath$b$}\sim\mbox{\boldmath$b$}_{0}=2\pi(0,1,0)$
(so that all three vectors
$\mbox{\boldmath$a$}_{0},\mbox{\boldmath$b$}_{0},\mbox{\boldmath$a$}_{0}-\mbox{\boldmath$b$}_{0}$
are nonzero); and

(iii) 21 asymmetric regions of the type $\mbox{\boldmath$a$}\sim
2\pi(1,0,0),\mbox{\boldmath$b$}\sim 0$ where one of these vectors is equal to
zero.

Consider first the region around the corner. The angular integration gives

$\displaystyle d\mbox{\boldmath$a$}d\mbox{\boldmath$b$}\ =\
8\pi^{2}\rho_{a}d\rho_{a}\rho_{b}d\rho_{b}\rho_{ab}d\rho_{ab}$ (6.65)

with
$\rho_{a}=|\mbox{\boldmath$a$}|,\,\rho_{b}=|\mbox{\boldmath$b$}|,\rho_{ab}=|\mbox{\boldmath$a$}-\mbox{\boldmath$b$}|$.
The induced ${\cal D}$ terms are

$\displaystyle{\cal D}_{(a)}\ =\
4\left(\frac{1}{\rho_{a}}-\frac{1}{\rho_{ab}}\right),\qquad{\cal D}_{(b)}\ =\
4\left(\frac{1}{\rho_{ab}}-\frac{1}{\rho_{b}}\right).$ (6.66)

Substituting this into (6.2.1) and bearing in mind (6.2), we obtain the
following expression for the corner contribution:323232The integral does not
depend on $\beta$ and we have set $\beta=1/16$.

$\displaystyle I_{W}({\rm corner})=\
\frac{32}{\pi\sqrt{3}}\iiint_{0}^{\infty}\frac{d\rho_{a}d\rho_{b}d\rho_{ab}}{\rho_{a}^{3}\rho_{b}^{3}\rho_{ab}^{3}}\left[\rho_{a}^{4}+\rho_{b}^{4}+\rho_{ab}^{4}+\right.$
$\displaystyle\left.\rho_{a}\rho_{b}(\rho_{ab}^{2}-\rho_{a}^{2}-\rho_{b}^{2})+\rho_{a}\rho_{ab}(\rho_{b}^{2}-\rho_{a}^{2}-\rho_{ab}^{2})+\rho_{b}\rho_{ab}(\rho_{a}^{2}-\rho_{b}^{2}-\rho_{ab}^{2})\right]$
$\displaystyle\cdot\exp\left[-\frac{2}{3}\left(\frac{1}{\rho_{a}^{2}}+\frac{1}{\rho_{b}^{2}}+\frac{1}{\rho_{ab}^{2}}-\frac{1}{\rho_{a}\rho_{b}}-\frac{1}{\rho_{a}\rho_{ab}}-\frac{1}{\rho_{b}\rho_{ab}}\right)\right]$
(6.67)

with the constraint $|\rho_{a}-\rho_{b}|\leq\rho_{ab}\leq\rho_{a}+\rho_{b}$.

It is convenient to make a variable change

$\displaystyle\rho_{a}=\mu\rho,\quad\rho_{b}=\nu\rho,\quad\rho_{ab}=(1-\mu-\nu)\rho\,,$
(6.68)

after which the integral over $\rho$ can be easily done, and we derive

$\displaystyle I_{W}({\rm corner})\ =\
\frac{8\sqrt{3}}{\pi}\int_{0}^{1/2}d\mu\int_{1/2-\mu}^{1/2}d\beta\,\frac{A(\mu,\nu)}{\mu\nu\kappa
B(\mu,\nu)}\,,$ (6.69)

where $\kappa=1-\mu-\nu$ and

$\displaystyle\\!\\!\\!\\!\\!\\!\\!A(\mu,\nu)$ $\displaystyle=$
$\displaystyle\mu^{4}+\nu^{4}+\kappa^{4}+\mu\nu(\kappa^{2}-\mu^{2}-\nu^{2})+\mu\kappa(\nu^{2}-\mu^{2}-\kappa^{2})+\nu\kappa(\mu^{2}-\nu^{2}-\kappa^{2})\,,$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!B(\mu,\nu)$ $\displaystyle=$
$\displaystyle\mu^{2}\nu^{2}+\mu^{2}\kappa^{2}+\nu^{2}\kappa^{2}-\mu\nu\kappa\,.$
(6.70)

Mathematica calculated for us the integral (6.2.1) numerically, and this
calculation produced an integer answer:

$\displaystyle I_{W}({\rm corner})\ =\ 16\,.$ (6.71)

So far so good. Consider now the contributions of 42 symmetric regions. The
corresponding ${\cal D}$ terms have the same form as in Eq. (6.73) with
$\rho_{a}=|\mbox{\boldmath$a$}-\mbox{\boldmath$a$}_{0}|$ etc., but with the
factor $1/2$ rather than 4. Also the “magnetic fields” $\mbox{\boldmath$\cal
H$}^{(a)},\mbox{\boldmath$\cal H$}^{(b)}$ and $\mbox{\boldmath$\cal
H$}^{(ab)}$ entering the preexponential in (6.2.1) are 8 times smaller than in
the corner. As a result, the contribution of a symmetric region in the index
is equal to $16/64=1/4$. And the contribution of all such regions is

$\displaystyle I_{W}^{\rm sym}\ =\ \frac{42}{64}\times 16\ =\
10\frac{1}{2}\,.$ (6.72)

It is fractional.

Consider now the 21 asymmetric regions. For example, a region around the
singularity at
$\mbox{\boldmath$a$}_{0}=2\pi(1,0,0),\mbox{\boldmath$b$}_{0}=0$. The $\cal D$
terms are

$\displaystyle{\cal D}_{(a)}\ =\
\frac{1}{2}\left(\frac{1}{\rho_{ab}}-\frac{1}{\rho_{a}}\right),\qquad{\cal
D}_{(b)}\ =\ -\frac{1}{2\rho_{ab}}-\frac{4}{\rho_{b}}$ (6.73)

with
$\rho_{a}=|\mbox{\boldmath$a$}-\mbox{\boldmath$a$}_{0}|,\rho_{b}=|\mbox{\boldmath$b$}|$,
and
$\rho_{ab}=|\mbox{\boldmath$a$}-\mbox{\boldmath$b$}-\mbox{\boldmath$a$}_{0}|$.
The contribution of this region to the index is333333We have now set
$\beta=4$.

$\displaystyle I_{W}^{\rm asym}=\
\frac{1}{2\pi\sqrt{3}}\iiint_{0}^{\infty}\frac{d\rho_{a}d\rho_{b}d\rho_{ab}}{\rho_{a}^{3}\rho_{b}^{3}\rho_{ab}^{3}}\left[64(\rho_{a}^{4}+\rho_{ab}^{4})+\rho_{b}^{4}\right.$
$\displaystyle\left.8\rho_{a}\rho_{b}(\rho_{a}^{2}+\rho_{b}^{2}-\rho_{ab}^{2})+64\rho_{a}\rho_{ab}(\rho_{b}^{2}-\rho_{a}^{2}-\rho_{ab}^{2})+8\rho_{b}\rho_{ab}(\rho_{b}^{2}+\rho_{ab}^{2}-\rho_{a}^{2})\right]$
$\displaystyle\cdot\exp\left[-\frac{2}{3}\left(\frac{1}{\rho_{a}^{2}}+\frac{1}{\rho_{ab}^{2}}+\frac{64}{\rho_{b}^{2}}-\frac{1}{\rho_{a}\rho_{ab}}+\frac{8}{\rho_{a}\rho_{b}}+\frac{8}{\rho_{b}\rho_{ab}}\right)\right].$
(6.74)

A numerical calculation gives the value $3$ for this integral.343434The fact
that a numerical calculation of these complicated integrals gives nicely
looking simple fractions or even integers looks as a small mathematical
miracle, of which I have no explanation. Moreover, consider a set of integrals
$\displaystyle I_{q}=\
\frac{1}{2\pi\sqrt{3}}\iiint_{0}^{\infty}\frac{d\rho_{a}d\rho_{b}d\rho_{ab}}{\rho_{a}^{3}\rho_{b}^{3}\rho_{ab}^{3}}\left[q^{2}(\rho_{a}^{4}+\rho_{ab}^{4})+\rho_{b}^{4}\right.$
$\displaystyle\left.q\rho_{a}\rho_{b}(\rho_{ab}^{2}-\rho_{a}^{2}-\rho_{b}^{2})+q^{2}\rho_{a}\rho_{ab}(\rho_{b}^{2}-\rho_{a}^{2}-\rho_{ab}^{2})+q\rho_{b}\rho_{ab}(\rho_{a}^{2}-\rho_{b}^{2}-\rho_{ab}^{2})\right]$
$\displaystyle\cdot\exp\left[-\frac{2}{3}\left(\frac{1}{\rho_{a}^{2}}+\frac{1}{\rho_{ab}^{2}}+\frac{q^{2}}{\rho_{b}^{2}}-\frac{1}{\rho_{a}\rho_{ab}}-\frac{q}{\rho_{a}\rho_{b}}-\frac{q}{\rho_{b}\rho_{ab}}\right)\right]$
(6.75) with integer $q$. The “experimental” values of these integrals are
$I_{q\leq 0}=\frac{1-q}{3},\qquad I(1)=\frac{1}{4},\qquad
I_{q>1}=\frac{2(q-1)}{3}\,.$ I wonder if one can derive these amusing results
analytically. The contribution of all 21 asymmetric regions is

$\displaystyle I_{W}^{\rm asym}\ =\ 63\,.$ (6.76)

Adding all the contributions, we obtain

$\displaystyle I_{W}\ =\ 89\frac{1}{2}\,.$ (6.77)

Actually, the notation $I_{W}$ is not correct here, because the Witten index
cannot be fractional in a system with discrete spectrum (the range of bosonic
dynamic variables $a$ and $b$ in the Hamiltonian (6.2) is finite, and the
spectrum should be discrete). We have not calculated the index, but just
quoted the value of the integral (6.62). The fractional value of the integral
is not a paradox by the reasons clarified on p. 6.1: the effective Hamiltonian
(6.2) has singularities so that the Cecotti-Girardello method by which the
index was evaluated above may not work. Unfortunately, we cannot say why it
works for chiral SQED, but does not work for chiral SQCD.353535Were the result
of the calculation of the phase space integral integer, we would have to
recall that not all the eigenfunctions of the Hamiltonian (6.2) contribute in
the index of the original theory, but only the functions invariant under Weyl
permutations:
$\mbox{\boldmath$a$}\leftrightarrow\mbox{\boldmath$b$}-\mbox{\boldmath$a$}$,
etc. But the fractional result makes the discussion of this subtle question
pointless.

###  6.3 Minimal $SU(5)$ model

The simplest chiral anomaly-free $SU(5)$ supersymmetric model includes a
quintet $S_{j}$ and an antidecuplet $\Phi^{[jk]}$. The generators in the
antisymmetric representation of an $SU(N)$ group read

$\displaystyle(T^{a})_{ij}^{kl}\ =\
\frac{1}{2}\left[(t^{a})_{i}^{k}\delta_{j}^{l}-(t^{a})_{i}^{l}\delta_{j}^{k}-(t^{a})_{j}^{k}\delta_{i}^{l}+(t^{a})_{j}^{l}\delta_{i}^{k}\right]\,.$
(6.78)

One can derive that

$\displaystyle{\rm Tr}\\{T^{(a}T^{b}T^{c)}\\}\ =\ (N-4)\,{\rm
Tr}\\{t^{(a}t^{b}t^{c)}\\}\,,$ (6.79)

and, for $N=5$, the contributions of the quintet and antidecuplet in the
anomalous triangle cancel, indeed.

The Lagrangian reads

$\displaystyle{\cal L}=\left(\frac{1}{4}{\rm Tr}\int
d^{2}\theta\,\hat{W}^{\alpha}\hat{W}_{\alpha}+{\rm
c.c.}\right)+\frac{1}{4}\int d^{4}\theta\left(\overline{S}e^{g\hat{V}}S+\Phi
e^{-g\hat{V}}\overline{\Phi}\right).$ (6.80)

An important distinction of this model compared to the models considered in
the previous sections is the absence of the scalar valley. In other words, the
equation system

$\displaystyle\phi^{ij}(T^{a})_{ij}{}^{kl}\phi^{*}_{kl}-s^{*j}(t^{a})_{j}{}^{k}s_{k}\
=\ 0$ (6.81)

with $T^{a}$ written in Eq. (6.78) has no solutions.

One can be convinced in that quite explicitly.363636See Appendix A of the
review [37]. Introduce the matrices $A_{k}{}^{j}=s_{k}s^{*j}$ and
$B_{k}{}^{j}=\phi^{*}_{kl}\phi^{lj}$. Then (6.81) boils down to
Tr$\\{(A+2B)t^{a}\\}=0$, i.e. $A+2B=C\cdot\mathbb{1}$. By a gauge rotation,
$\phi^{ij}$ can be brought to the following canonical form:

$\displaystyle\phi\ =\ \left(\begin{array}[]{ccccc}0&x&0&0&0\\\ -x&0&0&0&0\\\
0&0&0&y&0\\\ 0&0&-y&0&0\\\ 0&0&0&0&0\end{array}\right).$ (6.87)

This gives

$B=\phi^{*}\phi\ =\ -{\rm diag}\,(|x|^{2},|x|^{2},|y|^{2},|y|^{2},0)\,.$

Hence

$s_{j}s^{*k}\ =\ {\rm
diag}\,(C+2|x|^{2},C+2|x|^{2},C+2|y|^{2},C+2|y|^{2},C)\,,$

and this equation has no solutions.

Thus, we do not need here to introduce a Yukawa coupling to lock a nonexistent
scalar valley (besides, we are not able to do so: in this case, one cannot
write a trilinear gauge-invariant term in the superfield Lagrangian). The
theory has only the gauge coupling and is asymptotically free. To study its
vacuum structure, it suffices to study the gauge branch of classical vacua.

Take a constant Abelian background

$\displaystyle\hat{\mbox{\boldmath$c$}}\ =\ \frac{1}{2}\,{\rm
diag}(\mbox{\boldmath$a$},\mbox{\boldmath$b$}-\mbox{\boldmath$a$},\mbox{\boldmath$c$}-\mbox{\boldmath$b$},\mbox{\boldmath$d$}-\mbox{\boldmath$c$},-\mbox{\boldmath$d$})\,.$
(6.88)

The fast Hamiltonian describing its interaction with the zero Fourier modes of
the scalar quintet components is

$\displaystyle H^{\rm fast}_{\rm quint}\ =-\frac{\partial^{2}}{\partial
s^{*j}\partial
s_{j}}+\frac{1}{4}\left[\mbox{\boldmath$a$}^{2}|s_{1}|^{2}\right.$
$\displaystyle\left.+(\mbox{\boldmath$b$}-\mbox{\boldmath$a$})^{2}|s_{2}|^{2}+(\mbox{\boldmath$c$}-\mbox{\boldmath$b$})^{2}|s_{3}|^{2}+(\mbox{\boldmath$d$}-\mbox{\boldmath$c$})^{2}|s_{4}|^{2}+\mbox{\boldmath$d$}^{2}|s_{5}|^{2}\right].$
(6.89)

The scalar averages are

$\displaystyle\langle|s_{1}|^{2}\rangle=\frac{1}{|\mbox{\boldmath$a$}|},\qquad\langle|s_{2}|^{2}\rangle=\frac{1}{|\mbox{\boldmath$b$}-\mbox{\boldmath$a$}|},\qquad\langle|s_{3}|^{2}\rangle=\frac{1}{|\mbox{\boldmath$c$}-\mbox{\boldmath$b$}|},$
$\displaystyle\quad\langle|s_{4}|^{2}\rangle=\frac{1}{|\mbox{\boldmath$d$}-\mbox{\boldmath$c$}|},\qquad\langle|s_{5}|^{2}\rangle=\frac{1}{|\mbox{\boldmath$d$}|}\,.$
(6.90)

The fast Hamiltonian describing the interaction of the gauge background with
the zero Fourier modes of the scalar decuplet components is

$\displaystyle H^{\rm fast}_{\rm dec}\
=-2\frac{\partial^{2}}{\partial\phi^{[jk]}\partial\phi^{*}_{[jk]}}+\frac{1}{2}\left[\mbox{\boldmath$b$}^{2}|\phi^{12}|^{2}+(\mbox{\boldmath$a$}+\mbox{\boldmath$c$}-\mbox{\boldmath$b$})^{2}|\phi^{13}|^{2}\right.$
$\displaystyle+(\mbox{\boldmath$a$}+\mbox{\boldmath$d$}-\mbox{\boldmath$c$})^{2}|\phi^{14}|^{2}+(\mbox{\boldmath$a$}-\mbox{\boldmath$d$})^{2}|\phi^{15}|^{2}+(\mbox{\boldmath$a$}-\mbox{\boldmath$c$})^{2}|\phi^{23}|^{2}$
$\displaystyle+(\mbox{\boldmath$b$}-\mbox{\boldmath$a$}+\mbox{\boldmath$d$}-\mbox{\boldmath$c$})^{2}|\phi^{24}|^{2}+(\mbox{\boldmath$b$}-\mbox{\boldmath$a$}-\mbox{\boldmath$d$})^{2}|\phi^{25}|^{2}+(\mbox{\boldmath$c$}-\mbox{\boldmath$b$}+\mbox{\boldmath$d$}-\mbox{\boldmath$c$})^{2}|\phi^{34}|^{2}$
$\displaystyle\left.+(\mbox{\boldmath$c$}-\mbox{\boldmath$b$}-\mbox{\boldmath$d$})^{2}|\phi^{35}|^{2}+\mbox{\boldmath$c$}^{2}|\phi^{45}|^{2}\right].$
(6.91)

The scalar averages are

$\displaystyle\langle|\phi^{12}|^{2}\rangle=\frac{1}{|\mbox{\boldmath$b$}|},\quad\langle|\phi^{23}|^{2}\rangle=\frac{1}{|\mbox{\boldmath$a$}-\mbox{\boldmath$c$}|},\quad\langle|\phi^{15}|^{2}\rangle=\frac{1}{|\mbox{\boldmath$a$}-\mbox{\boldmath$d$}|},\quad\langle|\phi^{45}|^{2}\rangle=\frac{1}{|\mbox{\boldmath$c$}|},$
$\displaystyle\langle|\phi^{13}|^{2}\rangle=\frac{1}{|\mbox{\boldmath$a$}+\mbox{\boldmath$c$}-\mbox{\boldmath$b$}|},\
\langle|\phi^{14}|^{2}\rangle=\frac{1}{|\mbox{\boldmath$a$}+\mbox{\boldmath$d$}-\mbox{\boldmath$c$}|},\
\langle|\phi^{24}|^{2}\rangle=\frac{1}{|\mbox{\boldmath$b$}-\mbox{\boldmath$a$}+\mbox{\boldmath$d$}-\mbox{\boldmath$c$}|},$
$\displaystyle\langle|\phi^{25}|^{2}\rangle=\frac{1}{|\mbox{\boldmath$b$}-\mbox{\boldmath$a$}-\mbox{\boldmath$d$}|},\
\langle|\phi^{34}|^{2}\rangle=\frac{1}{|\mbox{\boldmath$b$}-\mbox{\boldmath$d$}|},\
\langle|\phi^{35}|^{2}\rangle=\frac{1}{|\mbox{\boldmath$c$}-\mbox{\boldmath$b$}-\mbox{\boldmath$d$}|}.$

Taking into account also higher Fourier harmonics of the matter fields and
proceeding in the same way as we did for the $SU(3)$ model, we can find the
effective supercharges and Hamiltonian [58].

$\displaystyle\hat{Q}_{\alpha}^{\rm eff}\ $ $\displaystyle=$ $\displaystyle\
{\eta^{(a)}_{\beta}}\left[(\sigma_{k})_{\alpha}{}^{\beta}(\hat{P}_{k}^{(a)}+{\cal
A}^{(a)}_{k})+i\delta_{\alpha}^{\beta}{\cal D}_{(a)}\right]$ (6.93)
$\displaystyle+$ $\displaystyle\
{\eta^{(b)}_{\beta}}\left[(\sigma_{k})_{\alpha}{}^{\beta}(\hat{P}_{k}^{(b)}+{\cal
A}^{(b)}_{k})+i\delta_{\alpha}^{\beta}{\cal D}_{(b)}\right],$ $\displaystyle+$
$\displaystyle{\eta^{(c)}_{\beta}}\left[(\sigma_{k})_{\alpha}{}^{\beta}(\hat{P}_{k}^{(c)}+{\cal
A}^{(c)}_{k})+i\delta_{\alpha}^{\beta}{\cal D}_{(c)}\right]$ $\displaystyle+$
$\displaystyle\
{\eta^{(d)}_{\beta}}\left[(\sigma_{k})_{\alpha}{}^{\beta}(\hat{P}_{k}^{(d)}+{\cal
A}^{(d)}_{k})+i\delta_{\alpha}^{\beta}{\cal D}_{(d)}\right]$

and similarly for $\hat{\bar{Q}}^{\alpha\,{\rm eff}}$ expressed via
$\bar{\eta}^{(a)\alpha}$ etc. Here

$\displaystyle\eta_{\alpha}^{(a)}\ $ $\displaystyle=$ $\displaystyle\
\eta_{\alpha}^{3}+\frac{\eta_{\alpha}^{8}}{\sqrt{3}}+\frac{\eta_{\alpha}^{15}}{\sqrt{6}}+\frac{\eta_{\alpha}^{24}}{\sqrt{10}}\,,$
$\displaystyle\eta_{\alpha}^{(b)}\ $ $\displaystyle=$ $\displaystyle\
2\left(\frac{\eta_{\alpha}^{8}}{\sqrt{3}}+\frac{\eta_{\alpha}^{15}}{\sqrt{6}}+\frac{\eta_{\alpha}^{24}}{\sqrt{10}}\right),$
$\displaystyle\eta_{\alpha}^{(c)}\ $ $\displaystyle=$ $\displaystyle\
3\left(\frac{\eta_{\alpha}^{15}}{\sqrt{6}}+\frac{\eta_{\alpha}^{24}}{\sqrt{10}}\right),$
$\displaystyle\eta_{\alpha}^{(d)}\ $ $\displaystyle=$ $\displaystyle\
\frac{4\eta_{\alpha}^{24}}{\sqrt{10}}\,.$ (6.94)

The fermion variables have the following anticommutators:

$\displaystyle\\{\hat{\bar{\eta}}^{(a)\beta},\eta^{(a)}_{\alpha}\\}=\\{\hat{\bar{\eta}}^{(d)\beta},\eta^{(d)}_{\alpha}\\}=\frac{8}{5}\delta_{\alpha}^{\beta},\quad\\{\hat{\bar{\eta}}^{(b)\beta},\eta^{(b)}_{\alpha}\\}=\\{\hat{\bar{\eta}}^{(c)\beta},\eta^{(c)}_{\alpha}\\}=\frac{12}{5}\delta_{\alpha}^{\beta}\,,$
$\displaystyle\\{\hat{\bar{\eta}}^{(a)\beta},\eta^{(b)}_{\alpha}\\}=\\{\hat{\bar{\eta}}^{(b)\beta},\eta^{(a)}_{\alpha}\\}=\\{\hat{\bar{\eta}}^{(c)\beta},\eta^{(d)}_{\alpha}\\}=\\{\hat{\bar{\eta}}^{(d)\beta},\eta^{(c)}_{\alpha}\\}=\frac{6}{5}\delta_{\alpha}^{\beta},$
$\displaystyle\\{\hat{\bar{\eta}}^{(a)\beta},\eta^{(c)}_{\alpha}\\}=\\{\hat{\bar{\eta}}^{(c)\beta},\eta^{(a)}_{\alpha}\\}=\\{\hat{\bar{\eta}}^{(b)\beta},\eta^{(d)}_{\alpha}\\}=\\{\hat{\bar{\eta}}^{(d)\beta},\eta^{(b)}_{\alpha}\\}=\frac{4}{5}\delta_{\alpha}^{\beta},$
$\displaystyle\\{\hat{\bar{\eta}}^{(a)\beta},\eta^{(d)}_{\alpha}\\}=\\{\hat{\bar{\eta}}^{(d)\beta},\eta^{(a)}_{\alpha}\\}=\frac{2}{5}\delta_{\alpha}^{\beta}\,\quad\\{\hat{\bar{\eta}}^{(b)\beta},\eta^{(c)}_{\alpha}\\}=\\{\hat{\bar{\eta}}^{(c)\beta},\eta^{(b)}_{\alpha}\\}=\frac{8}{5}\delta_{\alpha}^{\beta}\,.$

The supercharges $\hat{Q}_{\alpha}^{\rm eff}$ and $\hat{\bar{Q}}^{\alpha\,{\rm
eff}}$ look similar to (6.2), but they now include extra dynamic variables
($\mbox{\boldmath$c$},\mbox{\boldmath$d$}$ and their superpartners) and
represent sums of four rather than two terms. The induced ${\cal D}$ terms are

$\displaystyle{\cal
D}_{(a)}=\frac{1}{2}\left(\langle|s_{1}|^{2}\rangle-\langle|s_{2}|^{2}\rangle-\langle|\phi_{13}|^{2}\rangle-\langle|\phi_{14}|^{2}\rangle-\langle|\phi_{15}|^{2}\rangle+\langle|\phi_{23}|^{2}\rangle+\langle|\phi_{24}|^{2}+\langle|\phi_{25}|^{2}\rangle\right),$
$\displaystyle{\cal
D}_{(b)}=\frac{1}{2}\left(\langle|s_{2}|^{2}\rangle-\langle|s_{3}|^{2}\rangle-\langle|\phi_{21}|^{2}\rangle-\langle|\phi_{24}|^{2}\rangle-\langle|\phi_{25}|^{2}\rangle+\langle|\phi_{31}|^{2}\rangle+\langle|\phi_{34}|^{2}+\langle|\phi_{35}|^{2}\rangle\right),$
$\displaystyle{\cal
D}_{(c)}=\frac{1}{2}\left(\langle|s_{3}|^{2}\rangle-\langle|s_{4}|^{2}\rangle-\langle|\phi_{31}|^{2}\rangle-\langle|\phi_{32}|^{2}\rangle-\langle|\phi_{35}|^{2}\rangle+\langle|\phi_{41}|^{2}\rangle+\langle|\phi_{42}|^{2}+\langle|\phi_{45}|^{2}\rangle\right),$
$\displaystyle{\cal
D}_{(d)}=\frac{1}{2}\left(\langle|s_{4}|^{2}\rangle-\langle|s_{5}|^{2}\rangle-\langle|\phi_{41}|^{2}\rangle-\langle|\phi_{42}|^{2}\rangle-\langle|\phi_{43}|^{2}\rangle+\langle|\phi_{51}|^{2}\rangle+\langle|\phi_{52}|^{2}+\langle|\phi_{53}|^{2}\rangle\right)\\!,$
(6.96)

which gives (when we take into account the contributions of all the Fourier
harmonics)

$\displaystyle\\!\\!\\!\\!{\cal
D}_{(a)}=\frac{1}{2}\sum_{\mbox{\scriptsize\boldmath$n$}}\left[\frac{1}{|\mbox{\boldmath$a$}_{2\mbox{\scriptsize\boldmath$n$}}|}+\frac{1}{|(\mbox{\boldmath$a$}-\mbox{\boldmath$c$})_{2\mbox{\scriptsize\boldmath$n$}|}}+\frac{1}{|(\mbox{\boldmath$b$}-\mbox{\boldmath$a$}+\mbox{\boldmath$d$}-\mbox{\boldmath$c$})_{2\mbox{\scriptsize\boldmath$n$}}|}+\frac{1}{|(\mbox{\boldmath$b$}-\mbox{\boldmath$a$}-\mbox{\boldmath$d$})_{2\mbox{\scriptsize\boldmath$n$}}|}\right.$
$\displaystyle\left.-\frac{1}{|(\mbox{\boldmath$a$}-\mbox{\boldmath$b$})_{2\mbox{\scriptsize\boldmath$n$}}|}-\frac{1}{|(\mbox{\boldmath$a$}+\mbox{\boldmath$c$}-\mbox{\boldmath$b$})_{2\mbox{\scriptsize\boldmath$n$}}|}-\frac{1}{|(\mbox{\boldmath$a$}+\mbox{\boldmath$d$}-\mbox{\boldmath$c$})_{2\mbox{\scriptsize\boldmath$n$}}|}-\frac{1}{|(\mbox{\boldmath$a$}-\mbox{\boldmath$d$})_{2\mbox{\scriptsize\boldmath$n$}}|}\right],$
$\displaystyle\\!\\!\\!\\!{\cal
D}_{(b)}=\frac{1}{2}\sum_{2\mbox{\scriptsize\boldmath$n$}}\left[\frac{1}{|(\mbox{\boldmath$a$}-\mbox{\boldmath$b$})_{2\mbox{\scriptsize\boldmath$n$}}|}+\frac{1}{|(\mbox{\boldmath$b$}-\mbox{\boldmath$d$})_{2\mbox{\scriptsize\boldmath$n$}|}}+\frac{1}{|(\mbox{\boldmath$a$}+\mbox{\boldmath$c$}-\mbox{\boldmath$b$})_{2\mbox{\scriptsize\boldmath$n$}}|}+\frac{1}{|(\mbox{\boldmath$b$}-\mbox{\boldmath$c$}+\mbox{\boldmath$d$})_{2\mbox{\scriptsize\boldmath$n$}}|}\right.$
$\displaystyle\left.-\frac{1}{|\mbox{\boldmath$b$}_{2\mbox{\scriptsize\boldmath$n$}}|}-\frac{1}{|(\mbox{\boldmath$b$}-\mbox{\boldmath$c$})_{2\mbox{\scriptsize\boldmath$n$}}|}-\frac{1}{|(\mbox{\boldmath$b$}-\mbox{\boldmath$a$}-\mbox{\boldmath$d$})_{2\mbox{\scriptsize\boldmath$n$}}|}-\frac{1}{|(\mbox{\boldmath$b$}-\mbox{\boldmath$a$}+\mbox{\boldmath$d$}-\mbox{\boldmath$c$})_{2\mbox{\scriptsize\boldmath$n$}}|}\right],$
$\displaystyle\\!\\!\\!\\!{\cal
D}_{(c)}=\frac{1}{2}\sum_{\mbox{\scriptsize\boldmath$n$}}\left[\frac{1}{|\mbox{\boldmath$c$}_{2\mbox{\scriptsize\boldmath$n$}}|}+\frac{1}{|(\mbox{\boldmath$b$}-\mbox{\boldmath$c$})_{2\mbox{\scriptsize\boldmath$n$}|}}+\frac{1}{|(\mbox{\boldmath$a$}-\mbox{\boldmath$c$}+\mbox{\boldmath$d$})_{2\mbox{\scriptsize\boldmath$n$}}|}+\frac{1}{|(\mbox{\boldmath$b$}-\mbox{\boldmath$a$}+\mbox{\boldmath$d$}-\mbox{\boldmath$c$})_{2\mbox{\scriptsize\boldmath$n$}}|}\right.$
$\displaystyle\left.-\frac{1}{|(\mbox{\boldmath$d$}-\mbox{\boldmath$c$})_{2\mbox{\scriptsize\boldmath$n$}}|}-\frac{1}{|(\mbox{\boldmath$a$}-\mbox{\boldmath$c$})_{2\mbox{\scriptsize\boldmath$n$}}|}-\frac{1}{|(\mbox{\boldmath$a$}+\mbox{\boldmath$c$}-\mbox{\boldmath$b$})_{2\mbox{\scriptsize\boldmath$n$}}|}-\frac{1}{|(\mbox{\boldmath$c$}-\mbox{\boldmath$b$}-\mbox{\boldmath$d$})_{2\mbox{\scriptsize\boldmath$n$}}|}\right],$
$\displaystyle\\!\\!\\!\\!{\cal
D}_{(d)}=\frac{1}{2}\sum_{\mbox{\scriptsize\boldmath$n$}}\left[\frac{1}{|(\mbox{\boldmath$d$}-\mbox{\boldmath$c$})_{2\mbox{\scriptsize\boldmath$n$}}|}+\frac{1}{|(\mbox{\boldmath$d$}-\mbox{\boldmath$a$})_{2\mbox{\scriptsize\boldmath$n$}|}}+\frac{1}{|(\mbox{\boldmath$c$}-\mbox{\boldmath$b$}-\mbox{\boldmath$d$})_{2\mbox{\scriptsize\boldmath$n$}}|}+\frac{1}{|(\mbox{\boldmath$b$}-\mbox{\boldmath$a$}-\mbox{\boldmath$d$})_{2\mbox{\scriptsize\boldmath$n$}}|}\right.$
$\displaystyle\left.-\frac{1}{|\mbox{\boldmath$d$}_{\mbox{\scriptsize\boldmath$2n$}}|}-\frac{1}{|(\mbox{\boldmath$b$}-\mbox{\boldmath$d$})_{2\mbox{\scriptsize\boldmath$n$}}|}-\frac{1}{|(\mbox{\boldmath$a$}+\mbox{\boldmath$d$}-\mbox{\boldmath$c$})_{2\mbox{\scriptsize\boldmath$n$}}|}-\frac{1}{|(\mbox{\boldmath$b$}-\mbox{\boldmath$a$}+\mbox{\boldmath$d$}-\mbox{\boldmath$c$})_{2\mbox{\scriptsize\boldmath$n$}}|}\right].$

In this case, the elementary cell of our crystal has only one node. The
induced vector potentials are related to ${\cal D}_{(a,b,c,d)}$ by the
relations (6.2) to which one should add similar relations involving ${\cal
D}_{(c,d)}$ and $\mbox{\boldmath$\cal A$}^{(c,d)}$:

$\displaystyle{\cal H}^{(c)}_{j}$ $\displaystyle\equiv$
$\displaystyle\varepsilon_{jkl}\,\partial_{k}^{c}{\cal A}^{(c)}_{l}\ =\
\partial_{j}^{c}{\cal D}_{(c)}\,,$ $\displaystyle{\cal H}^{(ad)}_{j}$
$\displaystyle\equiv$ $\displaystyle\varepsilon_{jkl}\,\partial_{k}^{a}{\cal
A}^{(d)}_{l}\ =\ \varepsilon_{jkl}\,\partial_{k}^{d}{\cal A}^{(a)}_{l}\ =\
\partial_{j}^{a}{\cal D}_{(d)}\ =\ \partial_{j}^{d}{\cal D}_{(a)},$

etc. Calculating the anticommutator $\\{\hat{\bar{Q}}^{\alpha\,{\rm
eff}},\hat{Q}_{\alpha}^{\rm eff}\\}$, one can derive an expression for the
effective Hamiltonian. It represents a rather obvious generalization of (6.2).
For example, bearing in mind (6.3), one can write a contribution

$\displaystyle\hat{H}^{eff}_{\cal D}\ =\ \frac{1}{5}\left[4\left(({\cal
D}_{(a)}^{2}+{\cal D}_{(d)}^{2}\right)+6\left({\cal D}_{(b)}^{2}+{\cal
D}_{(c)}^{2}\right)+6\left({\cal D}_{(a)}{\cal D}_{(b)}+{\cal D}_{(c)}{\cal
D}_{(d)}\right)\right.$ $\displaystyle\left.+\ 4\left({\cal D}_{(a)}{\cal
D}_{(c)}+{\cal D}_{(b)}{\cal D}_{(d)}\right)+2{\cal D}_{(a)}{\cal
D}_{(d)}+8{\cal D}_{(b)}{\cal D}_{(c)}\right]\,.$ (6.99)

Another contribution represents an analogous quadratic form of
$\hat{P}_{k}^{(a,b,c,d)}+{\cal A}^{(a,b,c,d)}_{k}$. And there are 10 bifermion
terms including $\mbox{\boldmath$\cal H$}^{(a)},\mbox{\boldmath$\cal
H$}^{(bc)}$, etc.

One can also write and probably, applying certain efforts and using a clever
numerical method,373737The problem boils down to calculating a rather
complicated 10-dimensional integral, which is a nontrivial task even for
Mathematica. calculate numerically the phase space integral (1.7) for this
model in an attempt to evaluate the index of the effective Hamiltonian.
Bearing in mind the absence of the scalar vacuum valleys, it should coincide
with the index of the original field theory (6.80). It cannot be excluded,
however, that this complicated calculation would give a meaningless fractional
result, as a similar calculation did for $SU(3)$…

###  6.4 Spontaneous supersymmetry breaking

The main physical question to which the Witten index analysis helps to answer
is whether supersymmetry is broken spontaneously in a given theory or not. In
the gauge theories discussed so far, supersymmetry was not broken. But there
are also models involving chiral matter multiplets where supersymmetry is
broken. We are sure of that because the breaking occurs in the weak coupling
regime where everything is under control!

The simplest such model (the so-called 3-2 model) is based on the gauge group
$SU(3)\times SU(2)$ [47] (see also the reviews [37, 38, 59]). It includes the
following chiral matter multiplets: a multiplet $Q_{j=1,2,3;f=1,2}$ in the
representation (3, 2), two multiplets $U^{j}$ and $D^{j}$ in the
representation ($\bar{\bf 3}$, 1), and a multiplet $L_{f}$ in the
representation (1, 2).

This matter content is similar to the fermion matter content of one generation
of the Standard Model. The fields $Q$ (their fermion components) may be
associated with left quarks, the fields $U$ and $D$ with right antiquarks
$\bar{u}_{R}$ and $\bar{d}_{R}$, and the fields $L$ with left electron and
neutrino. There is no $U(1)$ group and no “right electron”. The model is
anomaly-free in the same way as the Standard model is383838Of course, we do
not mean that this model has some phenomenological significance. Probably not.
But for a physicist familiar with SM, this remark may have a mnemonic value.

The component Lagrangian of the model includes the quartic scalar interaction
term [cf. Eqs. (2), (5.2)]:

$\displaystyle V\ \sim\ ({\cal D}^{a=1,\ldots,8})^{2}+(\tilde{\cal
D}^{A=1,2,3})^{2},$ (6.100)

where

$\displaystyle{\cal D}^{a}\ $ $\displaystyle=$ $\displaystyle\
(q_{jf})^{*}(t^{a})_{j}{}^{k}q_{kf}-u^{j}(t^{a})_{j}{}^{k}u^{*}_{k}-d^{j}(t^{a})_{j}{}^{k}d^{*}_{k},$
$\displaystyle\tilde{\cal D}^{A}\ $ $\displaystyle=$ $\displaystyle\
(q_{jf})^{*}(t^{A})_{f}{}^{g}q_{jg}+(l_{f})^{*}(t^{A})_{f}{}^{g}l_{g}\,.$
(6.101)

Here $t^{a}$ are the generators of $SU(3)$ and $t^{A}$ are the generators of
$SU(2)$.

The important observation is that this model admits a vacuum valley — there
are classical field configurations for which the ${\cal D}$ terms (6.4) and
the potential (6.100) vanish. Up to gauge rotations, a general solution to the
equations ${\cal D}^{a}=\tilde{\cal D}^{A}=0$ is

$\displaystyle q_{jf}\ =\ \left(\begin{array}[]{cc}\tau_{1}&0\\\ 0&\tau_{2}\\\
0&0\end{array}\right),\qquad l_{f}\ =\
\left(0,\sqrt{|\tau_{1}|^{2}-|\tau_{2}|^{2}}\right),$ (6.105) $\displaystyle
u^{j}\ =\ (\tau_{1},0,0),\qquad\qquad d^{j}\ =\ (0,\tau_{2},0),$ (6.106)

where $\tau_{1,2}$ are complex parameters ($|\tau_{1}|\geq|\tau_{2}|$).

We have already dealt with the vacuum valleys. For the simplest nonchiral
$SU(2)$ model, the valley equations were written in Eq. (5.14). But the
solution to these equations,

$\displaystyle s_{j}\ =\ \left(\begin{array}[]{c}\tau\\\
0\end{array}\right),\qquad\qquad t^{j}\ =\ (\tau,0)\,$ (6.109)

was not stable in the massless case due to the presence of the nonperturbative
instanton-generated superpotential (5.10).

The same happens in the 3-2 model. The instantons generate the following
superpotential:

$\displaystyle{\cal W}^{\rm inst}_{3-2}\ =\
\frac{\Lambda^{7}}{Q_{jf}Q_{kg}U^{j}D^{k}\varepsilon^{fg}}.$ (6.110)

As was the case in the nonchiral SQCD, the valley solution (6.105) is not
stable under the scale transformations $\tau\to\lambda\tau$, and we are facing
the phenomenon of run-away vacuum. In nonchiral SQCD, the vacuum was
stabilized by giving a mass to the matter fields. In the chiral 3-2 model, we
cannot do so, but we can instead include a Yukawa term

$\displaystyle{\cal L}_{\rm Yukawa}\ =\
h\,\varepsilon^{fg}Q_{jf}U^{j}L_{g}\,,$ (6.111)

which plays the same role — it locks the valley and fixes the values of
$\tau_{1,2}$ at

$\displaystyle\tau_{1,2}\ =\ c_{1,2}\frac{\Lambda}{h^{1/7}},$ (6.112)

where $c_{1,2}$ are some particular irrelevant for us numerical constants.

And here comes an essential difference between the chiral and nonchiral
models. In the nonchiral massive SQCD, supersymmetry is not broken and the
energy of the vacuum states is zero. But in the 3-2 model, the vacuum energy
density ${\cal E}$ is not zero. To estimate it, it suffices to substitute the
characteristic value (6.112) of the scalar fields in the quartic WZ potential:

$\displaystyle{\cal E}\sim\left(\frac{\partial{\cal W}_{\rm
Yukawa}}{\partial\phi}\right)^{2}\sim h^{2}\phi_{\rm
char}^{4}=c_{3}h^{10/7}\Lambda^{4}\,.$ (6.113)

As the breaking occurs in the weak coupling regime, the numerical factor
$c_{3}$ can also be evaluated [47].

The fact that supersymmetry is broken in this case can be explained as
follows. Both in nonchiral SQCD and in the 3-2 model, instanton effects bring
about a nonzero gluino condensate $\langle\lambda\lambda\rangle$. In nonchiral
SQCD, the gluino condensate was related to the scalar vev,
$\langle\lambda\lambda\rangle\propto m\tau^{2}$, in such a way that the vacuum
average of the RHS of the Konishi identity (5.21) vanished. This implied that
the average of the anticommutator in the LHS of (5.21) vanished too, which
meant that supersymmetry was not broken.

But there is no mass in the 3-2 model, and nonvanishing
$\langle\lambda\lambda\rangle$ means that the anticommutator on the left-hand
side of the corresponding Konishi identity (which is valid for all gauge
supersymmetric theories including matter multiplets) has a nonzero vev. Hence
$\hat{Q}_{\alpha}|{\rm vac}\rangle\neq 0$ and supersymmetry is spontaneously
broken. The coupling constants in this theory are small and everything
(including the vacuum energy) can be perturbatively calculated.

Spontaneous supersymmetry breaking may occur also in other models if

  * •

The model admits a vacuum valley.

  * •

Instanton-generated superpotential pushes the system along the valley.

  * •

The model is chiral and the mass term cannot be written. To lock the valley,
one has to introduce the Yukawa terms.

In Sects. 6.1, 6.2, 6.3, we considered three different chiral models, but
neither of them satisfied all the conditions above.

  * •

The chiral SQED involves a valley, the equation (6.7) has a plenty of
solutions, but there are no instantons and no instanton-generated
superpotential.

  * •

The minimal chiral $SU(3)$ model with 7 triplets and an antisextet has a
valley, has instantons, but the latter fail to generate the required
superpotential [47].

  * •

The minimal chiral $SU(5)$ model has no valleys.

A next in complexity model, where supersymmetry is spontaneously broken in the
weak coupling regime, is the $SU(5)$ chiral model with two quintets
$S^{f=1,2}$ and two antidecuplets $\Phi^{f}$. In this case, the expression for
${\cal D}^{a}$ includes twice as much variables as in the minimal $SU(5)$
model and solutions to the equations ${\cal D}^{a}=0$ can be found though
their explicit form is not simple [60]. The instanton superpotential is
generated in this valley. It reads

$\displaystyle{\cal W}^{\rm inst}_{SU(5)}\ =\
\frac{\Lambda^{11}}{B^{fg}B_{fg}},$ (6.114)

where
$B_{fg}=\varepsilon_{ff^{\prime}}\varepsilon_{gg^{\prime}}B^{f^{\prime}g^{\prime}}$
and
$B^{fg}=\varepsilon_{ijklp}(\Phi^{f})^{ij}(\Phi^{h})^{kl}(\Phi_{h})^{pq}S^{g}_{q}$.
To lock the valley, one introduces the Yukawa term
$\propto(\Phi^{1})^{ij}S^{f}_{i}S^{g}_{j}\varepsilon_{fg}$.

Many more models with spontaneous SUSY breaking have been studied in the
literature. We address the reader to the original paper [47], to the reviews
[37, 38] and to many papers written afterwards for further details.

## Appendix A. Nuts and bolts of group theory.

We quote here some very basic facts and definitions of the theory of simple
Lie groups and Lie algebras.

For a Lie algebra of rank $r$, we can introduce the Cartan-Weyl basis, which
includes $r$ elements of the Cartan subalgebra $\mathfrak{C}$ and a set of
root vectors $E_{\alpha_{j}}$. Then for any $h\in\mathfrak{C}$,

$\displaystyle[h,E_{\alpha_{j}}]\ =\ \alpha_{j}(h)E_{\alpha_{j}}\,.$ (A.1)

The functions $\alpha_{j}(h)$ are linear forms on the Cartan subalgebra, which
are called the roots. If we choose in $\mathfrak{C}$ an orthonormal basis the
roots are represented as $r$-dimensional vectors.

The root system has the following properties:

$\bullet$ For any root $\alpha$, $-\alpha$ is also a root. The corresponding
root vectors are Hermitially conjugated to each other:
$E_{-\alpha}=E^{\dagger}_{\alpha}$.

$\bullet$ Let $\alpha,\beta$ be two arbitrary roots. Then, if $\alpha+\beta$
is also a root,

$\displaystyle[E_{\alpha},E_{\beta}]\ \propto\ E_{\alpha+\beta}\,.$ (A.2)

If $\alpha+\beta$ is not a root, the commutator vanishes.

$\bullet$ In the set of all roots, one can distinguish a subset of $r$ simple
roots. All other roots represent linear combinations of simple roots with
integer coefficients, which either are all positive (and in this case, we are
dealing with the positive roots) or all negative for negative roots.

$\bullet$ The notation $\alpha$ is convenient to reserve for positive roots
and denote the negative roots by $-\alpha$. For the positive roots, the sum of
the integer coefficients of their expansion is called the level of the root. A
root with the maximal level is called the highest root. The coefficients of
the expansion of the highest root are called Dynkin labels of the algebra.

$\bullet$ The commutator $[E_{\alpha},E_{-\alpha}]$ belongs to $\mathfrak{C}$.
A coroot corresponding to the root $\alpha$ and denoted as $\alpha^{\vee}$ is
proportional to this commutator, $\alpha^{\vee}=c[E_{\alpha},E_{-\alpha}]$,
with the coefficient $c$ determined from the condition

$\displaystyle e^{2\pi i\alpha^{\vee}}\ =\ \mathbb{1},\qquad{\rm but}\qquad
e^{i\phi\alpha^{\vee}}\ \neq\ \mathbb{1}\quad{\rm if}\ \phi<2\pi\,.$ (A.3)

In the convenient Chevalley normalization of the root vectors, $c=1$.

$\bullet$ Let $E_{\alpha}$ be a simple root vector. A fundamental coweight
$\omega_{\alpha}$ is an element of the Cartan subalgebra that commutes with
all simple root vectors besides $E_{\pm\alpha}$, while
$[\omega_{\alpha},E_{\alpha}]=E_{\alpha}$.

$\bullet$ For a generic Lie algebra, the roots (the $r$-dimensional vectors
that represent them) may have different length. If the length of the longer
roots is normalized to 1, the lengths of the shorter roots may be $1/\sqrt{2}$
or $1/\sqrt{3}$ (for $g_{2}$). Two different long simple roots may be
orthogonal or form the angle $2\pi/3$ between them. The same concerns two
short simple roots. The angle between a long simple root and a short simple
root of length $1/\sqrt{2}$ may be $\pi/2$ or $3\pi/4$. The algebra $g_{2}$
involves a long simple root and a short simple root of length $1/\sqrt{3}$.
They form the angle $5\pi/6$.

$\bullet$ A convenient way to represent a generic simple algebra
$\mathfrak{g}$ [or a generic simple group ${\cal G}=\exp(\mathfrak{g})$] is to
draw a Dynkin diagram. Many such diagrams were drawn in Sect. 4. The large
circles there are the long simple roots and smaller circles represent short
simple roots. If the roots are orthogonal, they are not related by a line. A
single line means the angle $2\pi/3$, a double line the angle $3\pi/4$ and a
triple line the angle $5\pi/6$. The dashed circles stand for $-\theta$, where
$\theta$ is the highest root (that is why the diagrams drawn in Fig. 3 etc.
are extended; in the ordinary Dynkin diagrams, the highest root is not shown).
The numbers in the circles are the Dynkin labels.

To illustrate this, consider the algebra $spin(7)$. It has rank 3 and 3 simple
roots: $\alpha,\beta$ and $\gamma$, two long and one short. Its Dynkin diagram
was drawn in Fig. 3. There are 6 other positive roots: four long roots
$\alpha+\beta,\beta+2\gamma,\alpha+\beta+2\gamma$ and the highest root
$\theta=\alpha+2\beta+2\gamma$, and two short roots $\beta+\gamma$ and
$\alpha+\beta+\gamma$. The simple coroots were written in Eq. (4). The basis
of $spin(7)$ includes 9 positive and 9 negative root vectors and 3 independent
elements of $\mathfrak{C}$. The dimension of $spin(7)$ is $9\cdot 2+3=21$.

We give here three more definitions.

(i) The maximal torus is an Abelian subgroup of $\cal G$ generated by its
Cartan subalgebra.

(ii) The centralizer of an element $\Omega\in{\cal G}$ or of a set of elements
$\\{\Omega_{i}\in{\cal G}\\}$ is the subgroup of $\cal G$ that commutes with
$\Omega$ (resp. with all the elements $\Omega_{i}$ in the set).

(iii) We define the Dynkin index $I(R)$ of a representation $R$ in group $\cal
G$ as393939Sometimes, it is defined with an extra factor 2 [61].

$\displaystyle{\rm Tr}\\{T^{a}T^{b}\\}\ =\ I(R)\delta^{ab}\,,$ (A.4)

where $T^{a}$ are the generators in the representation $R$ normalized in a
standard way so that $[T^{a},T^{b}]=if^{abc}T^{c}$, with $f_{abc}$ being the
structure constants of the group.

In any simple group, the Dynkin index of the adjoint representation coincides
with the adjoint Casimir eigenvalue. Another name for the latter important
invariant is dual Coxeter number. $I=1/2$ for the fundamental representation
in $SU(N)$, and $I=2$ for the vector representation in $SO(N)$.

### Octonions

Figure 15: Fano plane

In Sect. 5.3, we were discussing a sypersymmetric gauge theory based on the
group $G_{2}$. This group can be defined as the group of automorphisms of
octonion algebra. An octonion is an object $a_{0}+a_{\alpha}e_{\alpha}$, where
$a_{0}$ and $a_{1,\ldots,7}$ are real numbers and $e_{\alpha}$ satisfy the
algebra

$\displaystyle e_{\alpha}e_{\beta}\ =\
-\delta_{\alpha\beta}+f_{\alpha\beta\gamma}e_{\gamma}\,.$ (A.5)

One of the possible choices for the antisymmetric tensor
$f_{\alpha\beta\gamma}$ is

$\displaystyle f_{165}=f_{341}=f_{523}=f_{271}=f_{673}=f_{475}=f_{246}=1$
(A.6)

and all other nonzero components are restored by antisymmetry. The values
(A.6) can be mnemonized by looking at the Fano plane depicted in Fig. 15.
$f_{\alpha\beta\gamma}$ is nonzero only for the indices lying on the same
line, with the arrows indicating the order of indices when
$f_{\alpha\beta\gamma}$ is positive.

## Appendix B. The derivation of Eq. (6.27)

The Schrödinger equation for a scalar charged particle in a monopole field was
solved by Igor Tamm404040He was visiting Cambridge at that time and learned
about Dirac’s work directly from Dirac. back in 1931 [62]. Let the electric
charge of the particle be $e=1$. The angular momentum operator that commutes
with the Hamiltonian

$\displaystyle\hat{H}_{0}\ =\
\frac{1}{2}\left(\hat{\mbox{\boldmath$P$}}+\mbox{\boldmath$\cal
A$}\right)^{2}\,,$ (B.1)

where $\cal A$ is the vector potential of a monopole414141Dirac and Tamm used
an expression for $\cal A$ that involved a singularity on the Dirac string,
but later it was realized that the projection $\mbox{\boldmath$\cal
A$}\cdot\mbox{\boldmath$n$}$ represents a fiber bundle on $S^{2}$, which can
be described by nonsingular expressions on two overlapping charts [63]. with
the minimal magnetic charge $g=1/2$, has the form

$\displaystyle\hat{\mbox{\boldmath$L$}_{0}}\ =\
\mbox{\boldmath$r$}\times\left(\hat{\mbox{\boldmath$P$}}+\mbox{\boldmath$\cal
A$}\right)+\frac{\mbox{\boldmath$r$}}{2r}\,.$ (B.2)

The second term is the angular momentum of electromagnetic field. The
eigenvalues of $\hat{\mbox{\boldmath$L$}_{0}}^{2}$ are $l(l+1)$ with integer
or half-integer $l$. The eigenvalues of $\hat{L}_{3}^{(0)}$ are
$m=-l,-l+1,\ldots,l$. The corresponding eigenfunction are the so-called
monopole harmonics [63].

We have

$\displaystyle\hat{H}_{0}\ =\ -\frac{1}{2r^{2}}\frac{\partial}{\partial
r}\left(r^{2}\frac{\partial}{\partial
r}\right)+\frac{1}{2r^{2}}\left[\mbox{\boldmath$r$}\times(\hat{\mbox{\boldmath$P$}}+\mbox{\boldmath$\cal
A$})\right]^{2}=-\frac{1}{2r^{2}}\frac{\partial}{\partial
r}\left(r^{2}\frac{\partial}{\partial
r}\right)+\frac{\hat{\mbox{\boldmath$L$}_{0}}^{2}-\frac{1}{4}}{2r^{2}}\,.$
(B.3)

For the states with definite $l$, this gives

$\hat{H}_{0}\ =\ -\frac{1}{2r^{2}}\frac{\partial}{\partial
r}\left(r^{2}\frac{\partial}{\partial r}\right)\ +\
\frac{l(l+1)-\frac{1}{4}}{2r^{2}}\,.$

The full Hamiltonian (6.17) commutes with

$\displaystyle\hat{\mbox{\boldmath$L$}}\ =\
\hat{\mbox{\boldmath$L$}}_{0}+\frac{1}{2}\hat{\bar{\eta}}\mbox{\boldmath$\sigma$}\eta.$
(B.4)

In fact, (B.4) may be interpreted as a Hamiltonian describing the interaction
of an electron or other spin $1/2$ particle with a monopole, 424242One can
also consider the interaction of a $W$ boson, which has spin 1 [64]. A bound
state of a monopole and a $W$ boson is a dyon [65]. and the second term in
(B.4) is the spin part of the angular momentum. The matrix
$\mbox{\boldmath$\sigma$}\cdot\mbox{\boldmath$n$}$ has the eigenvalues
$\lambda=\pm 1$. Choosing $\lambda=-1$ (this corresponds to the total angular
momentum $j=0$), we are led to the radial Schrödinger equation

$\displaystyle-\frac{1}{2r}\frac{\partial^{2}}{\partial
r^{2}}(r\psi)+\frac{l(l+1)-\frac{5}{4}}{2r^{2}}\psi\ =\ E\psi\,.$ (B.5)

Taking $l=1/2$ and adding the term $D^{2}/2$ in the potential, we reproduce
(6.27).

I am indebted to Ken Konishi and Misha Vysotsky for valuable comments.

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