# On the removal of the trace mode in lattice ${\cal N}=4$ super Yang-Mills
theory

Simon Catterall<EMAIL_ADDRESS>Department of Physics, Syracuse University,
Syracuse, 13244 NY USA  Joel Giedt<EMAIL_ADDRESS>Department of Physics,
Applied Physics and Astronomy, Rensselaer Polytechnic Institute, 110 8th
Street, Troy NY 12065 USA  Raghav G. Jha<EMAIL_ADDRESS>Department of Physics,
Syracuse University, Syracuse, 13244 NY USA

###### Abstract

Twisted and orbifold formulations of lattice ${\cal N}=4$ super Yang-Mills
theory which possess an exact supersymmetry require a $U(N)=SU(N)\otimes U(1)$
gauge group. In the naive continuum limit, the $U(1)$ modes trivially decouple
and play no role in the theory. However, at non-zero lattice spacing they
couple to the $SU(N)$ modes and can drive instabilities in the lattice theory.
For example, it is well known that the lattice $U(1)$ theory undergoes a phase
transition at strong coupling to a chirally broken phase. An improved action
that suppresses the fluctuations in the $U(1)$ sector was proposed in
Catterall and Schaich (2015). Here, we explore a more aggressive approach to
the problem by adding a term to the action which can entirely suppress the
$U(1)$ mode. The penalty is that the new term breaks the $\mathcal{Q}$-exact
lattice supersymmetry. However, we argue that the term is $1/N^{2}$ suppressed
and the existence of a supersymmetric fixed point in the planar limit ensures
that any SUSY-violating terms induced in the action possess couplings that
also vanish in this limit. We present numerical results on supersymmetric Ward
identities consistent with this conclusion.

##  I Introduction

In recent years a great deal of effort has been devoted to the construction
and numerical studies of lattice formulations of ${\cal N}=4$ super Yang-Mills
theory which retain one exact supersymmetry at non-zero lattice spacing—see
the reviews Giedt (2006); Catterall  _et al._ (2009); Bergner and Catterall
(2016) and references therein. These lattice theories can be derived using
either deconstruction Cohen  _et al._ (2003a, b); Kaplan and Unsal (2005) or
topological field theory methods Catterall (2003, 2005, 2008); Unsal (2006).
In this approach the link fields appearing in the lattice theory take their
values in the algebra of the group, denoted by
$\mathfrak{gl}$($N,\mathbb{C})$.111This restriction is not present for
Sugino’s formulation—see Sugino (2004). Other approaches to studying ${\cal
N}=4$ super Yang-Mills and the AdS/CFT correspondence on a computer include
Honda  _et al._ (2010); Hanada  _et al._ (2009); Anagnostopoulos  _et al._
(2008); Hanada  _et al._ (2011, 2013); Honda  _et al._ (2011); Ishii  _et al._
(2008); Ishiki  _et al._ (2009). This is readily apparent from the (twisted)
scalar supersymmetry (SUSY) transformation

$\displaystyle{\cal Q}{\cal U}_{m}=\psi_{m}$ (I.1)

where $\psi_{m}$ is a twist fermion that transforms as a link variable. Since
it is a fermion, it has an expansion in terms of generators,

$\displaystyle\psi_{m}=\sum_{A=0}^{N^{2}-1}\psi_{m}^{A}t^{A}$ (I.2)

Here, $t^{0}$ is proportional to the unit matrix, and must be included if
(I.1) is to hold, because the link field ${\cal U}_{m}$ on the left-hand side
certainly has an expansion involving the unit matrix, if it is to yield the
usual $a\to 0$ continuum limit

$\displaystyle{\cal U}_{m}(x)=1+a{\cal A}_{m}(x)+\cdots$ (I.3)

(Here, ${\cal A}_{m}(x)$ is a complexification that contains both the gauge
fields and scalars.) On the other hand, SUSY should not convert a group valued
field into a Lie algebra valued field, so in fact ${\cal U}_{m}$ should also
have the expansion

$\displaystyle{\cal U}_{m}=\sum_{A=0}^{N^{2}-1}{\cal U}_{m}^{A}t^{A}$ (I.4)

with the U(1) mode ${\cal U}_{m}^{0}$ fully dynamical. The conclusion of this
argument is that the scalar SUSY ${\cal Q}$ requires the gauge group to be
U(N) and not SU(N), with the bosonic link fields Lie algebra valued.

In the continuum limit the entire U(1) sector decouples, and becomes an
uninteresting free theory—all fields are in the adjoint representation and
hence neutral for U(1). However on the lattice this sector is coupled to the
SU(N) part through irrelevant operators, so we cannot completely ignore it. In
fact, it is these irrelevant couplings that can cause various problems. The
first of these was first identified in Catterall  _et al._ (2014) and is
manifested in the appearance of a chirally broken phase for ’t Hooft couplings
$\lambda_{\text{lat}}>1$ (see eqn. II.2 for the definition of the lattice
coupling).

Another way to see that the $U(1)$ mode drives instabilities is to examine the
behavior of the theory under the classical scaling transformation

$\displaystyle{\cal U}_{\mu}$ $\displaystyle\to$ $\displaystyle c{\cal
U}_{\mu}$ (I.5) $\displaystyle{\overline{\cal U}}_{\mu}$ $\displaystyle\to$
$\displaystyle c{\overline{\cal U}}_{\mu}$ $\displaystyle\psi_{\mu}$
$\displaystyle\to$ $\displaystyle c^{\frac{3}{2}}\psi_{\mu}$
$\displaystyle\eta$ $\displaystyle\to$ $\displaystyle c^{\frac{3}{2}}\eta$
$\displaystyle\chi_{\mu\nu}$ $\displaystyle\to$ $\displaystyle
c^{\frac{3}{2}}\chi_{\mu\nu}$

It is trivial to see that the supersymmetric action given in Catterall  _et
al._ (2014) (minus the soft ${\cal Q}$-breaking mass term) is invariant under
this transformation if the Yang-Mills coupling $g^{2}\to c^{4}g^{2}$. This
allows us to write down relations between expectation values of gauge
invariant operators. For example,

$\displaystyle\langle{\rm Tr}\prod_{i=1}^{P}{\cal
U}^{i}\rangle_{g^{2}}=c^{P}\langle{\rm Tr}\prod_{i=1}^{P}{\cal
U}^{i}\rangle_{c^{4}g^{2}}$ (I.6)

in which we have suppressed spacetime coordinates and indices and where ${\cal
U}$ could be replaced by any other appropriately chosen lattice field with a
corresponding change in the multiplicative factor on the RHS. Since the LHS is
independent of $c$ this implies that the expectation value on the RHS must
vary as $c^{-P}$. Note that this rescaling is not allowed if the link
variables ${\cal U}_{\mu}$ are $SL(N,\mathbb{C})$-valued, corresponding to
gauge group SU(N). Thus, it is the U(1) sector that creates this instability.

In Catterall and Schaich (2015) a new supersymmetric term was added to the
lattice action to suppress the $U(1)$ mode fluctuations. This allows for
simulations to be performed out to stronger coupling $\lambda_{\text{lat}}\leq
2$. However, it does not appear sufficient to explore the regime of extreme
strong coupling needed for studies of S-duality Giedt  _et al._ (2016). The
reason for the ineffectiveness of this term at very strong coupling is that it
constrains only the real part of the determinant of the plaquette operator
averaged over all plaquettes associated with a given lattice site.

In this paper, we have attempted to address this problem in a different way by
adding to the lattice action a term which explicitly suppresses the $U(1)$
sector for each link field (we call this the detlink term). We argue that this
term is 1/$N^{2}$ suppressed and hence the exact scalar SUSY ${\cal Q}$ should
be recovered in the large $N$ limit. Furthermore, we show extensive numerical
results that support this conclusion. The existence of this supersymmetry at
large $N$ then guarantees that under renormalization any ${\cal Q}$-breaking
operators that are generated are $1/N^{2}$ suppressed, and the scalar SUSY is
restored without fine tuning as $N\to\infty$. In addition, we show that even
for modest values of $N$ such as $SU(5)$, ${\cal Q}$ invariance is a very good
approximation. Early results for this formulation have appeared in Giedt  _et
al._ (2018).

An alternate method to achieve the same result is by truncating the theory
completely to gauge group SU($N$) by having links valued in the group
$SL(N,\mathbb{C})$ rather than algebra $\mathfrak{gl}$($N,\mathbb{C})$.
However, the full truncation (bosonic & fermionic) of the theory from U($N$)
to SU($N$) does not work. A simple way to see this is as follows: Assume a
traceless fermion $\psi_{a}$ which lives on the link in the direction of
$e_{a}$. The gauge invariance acts as

$\psi_{a}(x)\to G(x)\psi_{a}(x)G^{\dagger}(x+e_{a}),$ (I.7)

which yields a $\psi_{a}$ which is not in general traceless. Thus we cannot
eliminate the $U(1)$ mode of the fermion, even under the restriction to
$SU(N)$ gauge group.222This is to be contrasted with Berkowitz  _et al._
(2016); Rinaldi  _et al._ (2018); Berkowitz  _et al._ (2018) where it was
possible to eliminate the $U(1)$ fermion mode. This has the benefit of
improving the condition of the fermion matrix. Note that this is a lattice
effect, since for a site fermion $\eta$, we would have

$\displaystyle\mathop{{\hbox{Tr}\,}}\nolimits
G(x)\eta(x)G^{\dagger}(x)=\eta^{A}(x)\mathop{{\hbox{Tr}\,}}\nolimits
G(x)t^{A}G^{\dagger}(x)=\eta^{A}\mathop{{\hbox{Tr}\,}}\nolimits t^{A}=0$ (I.8)

assuming $\eta(x)$ only involved the generators $t^{A}$ of $SU(N)$, which are
traceless. The distinction between link fermions and site fermions is only
meaningful on the lattice. This same argument does not apply to the link
bosons, since they are valued in the group and the gauge transformation
preserves that feature.

In summary, to maintain lattice gauge invariance, for this hybrid action we
only truncate the bosonic sector down to SU($N$). This construction also
restores $\mathcal{Q}$ supersymmetry in the limit $N\to\infty$. In Table 1, we
show the comparison between these two approaches. This method of maintaining
exact lattice supersymmetry by truncating U(1) sector at large $N$ was
employed in Catterall  _et al._ (2018); Jha  _et al._ (2017) to initiate non-
perturbative checks of gauge/gravity duality at large $N$ in two dimensions.
In this paper, we show detailed numerical results in four dimensions
consistent with the claimed $1/N^{2}$ suppression.

##  II Lattice action

The ${\cal Q}$-exact lattice action takes the form

$\displaystyle S$
$\displaystyle=\frac{N}{4\lambda_{\text{lat}}}\sum_{x}\mbox{Tr}\left[{\cal
Q}\left(\chi_{ab}{\cal D}_{a}^{(+)}{\cal U}_{b}+\eta{\overline{\cal
D}}_{a}^{(-)}{\cal U}_{a}-\frac{1}{2}\eta d\right)\right]+S_{\text{cl}}$
(II.1) $\displaystyle S_{\text{cl}}$
$\displaystyle=-\frac{N}{16\lambda_{\text{lat}}}\sum_{x}\mbox{Tr}\left[\epsilon_{abcde}\chi_{de}(x+e_{a}+e_{b}+e_{c}){\overline{\cal
D}}^{(-)}_{c}\chi_{ab}(x+e_{c})\right],$ (II.2)

where the lattice difference operators take the form of shifted commutators.
For example,

${\cal D}_{a}^{(+)}{\cal U}_{b}(x)={\cal U}_{a}(x){\cal U}_{b}(x+e_{a})-{\cal
U}_{b}(x){\cal U}_{a}(x+e_{b})\equiv{\cal F}_{ab}(x)$ (II.3)

where $e_{a}$ are the principle lattice vectors of the $A_{4}^{*}$ lattice.
The ${\cal Q}$-closed term is still lattice supersymmetric due to the
existence of an exact lattice Bianchi identity,

$\epsilon_{abcde}{\overline{\cal D}}^{(-)}_{c}{\overline{\cal
F}}_{ab}(x+e_{c})=0.$ (II.4)

After we integrate out the auxiliary field $d$, we have

$\displaystyle
S=\frac{N}{4\lambda_{\text{lat}}}\sum_{x}\text{Tr}\left[-{\overline{\cal
F}}_{ab}{\cal F}_{ab}+\frac{1}{2}\left({\overline{\cal D}}_{a}^{(-)}{\cal
U}_{a}\right)^{2}-\chi_{ab}{\cal D}^{(+)}_{[a}\psi_{b]}-\eta{\overline{\cal
D}}^{(-)}_{a}\psi_{a}\right]+S_{\text{cl}}.$ (II.5)

The action also contains a single trace mass term, which helps to lift the
classical flat directions by giving a small mass to the scalar fields:333It
also generates cubic and quartic terms that further stabilize the flat
directions. This mass term has been used for most of our earlier works, and
also appears in Hanada and Kanamori (2011).

$\displaystyle
S_{\text{mass}}=\frac{N}{4\lambda_{\text{lat}}}\mu^{2}\sum_{x,a}\mbox{Tr}\left[\bigg{(}{\cal
U}_{a}^{\dagger}{\cal U}_{a}-\mathbb{I}_{N}\bigg{)}^{2}\right].$ (II.6)

To control the local fluctuations of the $U(1)$ sector we now add a new term
to the action:

$\displaystyle\Delta
S=\frac{N}{4\lambda_{\text{lat}}}\kappa_{\text{link}}\sum_{x,a}|\det{\cal
U}_{a}(x)-1|^{2}$ (II.7)

In the limit $\kappa_{\text{link}}\to\infty$ we can completely remove the
$U(1)$ modes — both gauge and scalar by restricting the links to
$SL(N,\mathbb{C})$. Notice that this term does not break the $SU(N)$
invariance of the action since $\det{\cal U}_{a}(x)$ is invariant under such
transformations. Using a polar decomposition of the link field

$\displaystyle{\cal U}_{a}(x)=(I+h_{a})e^{iB_{a}}$ (II.8)

the determinant can be written for small $h_{a}$ and $B_{a}$ as

$\displaystyle{\rm
det}\left(U_{a}\right)=(1+\frac{1}{\sqrt{N}}h_{a}^{0})e^{i\frac{1}{\sqrt{N}}B_{a}^{0}}$
(II.9)

where the $\frac{1}{\sqrt{N}}$ factor arises from the generators which satisfy
the normalization ${\rm Tr}(T^{a}T^{b})=-\delta^{ab}$ and the superscript
indicates that only the trace mode survives. To quadratic order in the
fluctuations the determinant term becomes

$\displaystyle\Delta
S=\frac{1}{4\lambda_{\text{lat}}}\kappa_{\text{link}}\sum_{x,a}\left(\left(B_{a}^{0}\right)^{2}+\left(h_{a}^{0}\right)^{2}\right)$
(II.10)

The term thus serves to generate masses for the $U(1)$ modes. Additionally,
notice it carries no prefactor of $N$ which then guarantees that it will
generate terms that are $O(1/N^{2})$ suppressed relative to the leading terms
in a perturbative expansion.

This detlink term breaks both the ${\cal Q}$-supersymmetry and the $U(1)$
gauge symmetry. Breaking the $U(1)$ symmetry is likely harmless since the
$U(1)$ sector plays no role in the continuum limit. However breaking the exact
supersymmetry is more problematic since it invalidates the arguments given in
Catterall and Giedt (2014) devoted to the renormalizability of the lattice
theory and specifically the number of counterterms needed to tune to a
supersymmetric continuum limit.

To address this issue, we examine the $N$-dependence of the various terms in
the action. It is clear that the new term being a function of the trace modes
only is suppressed by $1/N^{2}$ as compared to all other terms in the action
which correspond to a sum over all the generators of $U(N)$. If we treat this
term perturbatively, it will yield a subleading contribution to any observable
in the planar limit. Thus, we expect that the exact supersymmetry will be
restored in the large $N$ limit. The presence of an exact supersymmetry at
$N=\infty$ then ensures that any SUSY violating operators appearing at finite
$N$ (and finite $\kappa_{\text{link}}$) are only multiplicatively renormalized
with couplings proportional to positive powers of $1/N^{2}$. In the next
section, we show that these truncated approaches yield stable results for a
range of values of the ’t Hooft coupling $\lambda_{\text{lat}}$ and
measurements of appropriate Ward identities show the expected $1/N^{2}$
behavior.

We perform the numerical simulations with the parallel code presented in
Schaich and DeGrand (2015). Since then, it has been extended to perform
calculations for arbitrary gauge group to access the large $N$ limit and will
be presented in a future publication Schaich  _et al._ (tion). We note that
there is an earlier work that develops a method to have $SU(N)$ gauge group in
supersymmetric lattice gauge theory Kanamori (2012).

##  III Ward identities

We test the restoration of ${\cal Q}$ in the large $N$ limit in two ways.444We
note that while $N=8$ is sufficient for us to see the large $N$ limit in our
four-dimensional lattices, much large $N$ are both necessary and possible in
the case of matrix quantum mechanics Berkowitz  _et al._ (2016); Rinaldi  _et
al._ (2018); Berkowitz  _et al._ (2018). One is via a measurement of the
expectation value of the bosonic action $S_{B}$, which is related to an exact
lattice Ward identity associated with ${\cal Q}$ in the original, unmodified
theory. The results on $8^{4}$ lattice with three different values of
$\lambda_{\text{lat}}=2,3,4$ are shown in Fig. 1, with a normalization such
that $S_{B}=1$ for exact ${\cal Q}$. It can be seen that the restoration is
within 1% in the large $N$ limit, where presumably the small deviation from
$1$ is due to the mass term (II.6) (we take $\mu=0.1$ in our study) and
thermal boundary conditions for the fermions along the temporal direction.

Another check arises through the supersymmetric Ward identity corresponding to

$\displaystyle\Big{\langle}{\cal Q}\,{\rm Tr}\,\left(\eta{\cal
U}_{a}{\overline{\cal U}}_{a}\right)\Big{\rangle}=0$ (III.1)

This yields

$\displaystyle\Big{\langle}\,{\rm Tr}\,\left(d{\cal U}_{a}{\overline{\cal
U}}_{a}\right)\Big{\rangle}-\Big{\langle}\,{\rm
Tr}\,\eta\psi_{a}{\overline{\cal U}}_{a}\Big{\rangle}=0$ (III.2)

Using the equations of motion to eliminate $d$ we find

$\displaystyle W=\Big{\langle}\,{\rm Tr}\,\left({\overline{\cal D}}_{a}{\cal
U}_{a}{\cal U}_{b}{\overline{\cal
U}}_{b}\right)\Big{\rangle}-\Big{\langle}\,{\rm
Tr}\,\eta\psi_{a}{\overline{\cal U}}_{a}\Big{\rangle}=0$ (III.3)

We further normalize $W$ by the fermion bilinear term appearing on the right
and take the magnitude,

$\displaystyle W=\Bigg{|}\frac{\Big{\langle}\,{\rm Tr}\,\left({\overline{\cal
D}}_{a}{\cal U}_{a}{\cal U}_{b}{\overline{\cal
U}}_{b}\right)\Big{\rangle}-\Big{\langle}\,{\rm
Tr}\,\eta\psi_{a}{\overline{\cal U}}_{a}\Big{\rangle}}{\Big{\langle}\,{\rm
Tr}\,\eta\psi_{a}{\overline{\cal U}}_{a}\Big{\rangle}}\Bigg{|}$ (III.4)

to obtain the quantity shown in Fig. 2. It can be seen that the Ward identity,
which is zero in the limit of exact ${\cal Q}$, is approximately $0.6$% in the
large $N$ limit. Again, we attribute this to the mass term (II.6).

We have also compared these results to the hybrid formulation, where the U(1)
sector is eliminated from the link fields entirely. In Table 1 it can be seen
that the ${\cal Q}$ violation is more for the hybrid than in the detlink
formulation, but with the same 1/$N^{2}$ dependence. The results for the Ward
identity are shown together in Fig. 3. Thus we see that either approach will
restore ${\cal Q}$ in the large $N$ limit, up to the effects of the regulating
mass term.

In Fig. 4 we show the dependence of the Ward identity on the mass term
parameter $\mu$. It can be seen that there is an appreciable decrease in the
Ward identity as $\mu$ is decreased. It is clear that our large $N$
extrapolation will also exhibit the same decrease with $\mu$ and it is
reasonable to assume that the Ward identity will ultimately vanish as
$N\to\infty$ and $\mu\to 0$. For the latter limit it is important that the
spacetime volume is also taken to infinity, since removing $\mu$ at finite $L$
will lead to unstable results as the scalar modes will wander without
restriction.

A final question is the effect of finite volume, given that antiperiodic
boundary conditions are imposed on the fermions. This also violates the ${\cal
Q}$ scalar supersymmetry, so we expect such effects to fall off with the
volume. It can be seen from Table 2 that most of the volume effects are
negligible. Indeed, only at the weakest coupling for the smallest number of
colors is the effect of any significance.

Figure 1: The bosonic action, normalized such that it should be equal to $1$ if the ${\cal Q}$ symmetry is fully restored (exact). It can be seen that the $N$ dependence falls of as $1/N^{2}$, as expected. The difference from $1$ in the large $N$ limit is anticipated from the presence of the small mass term (II.6) with $\mu=0.1$. Fits to $A+B/N^{2}$ are also shown in the plot. For these runs we take $\kappa_{\text{link}}=5,5,10$ for the three values of $\lambda_{\text{lat}}$ respectively.  Figure 2: The Ward identity (III.4) for the $8^{4}$ lattice with detlink action, $\lambda_{\text{lat}}=2,3,4$, $\mu=0.1$ and $\kappa_{\text{link}}=5,5,10$ respectively. Fits to $A+B/N^{2}$ are also shown.  Figure 3: The comparison between the Ward identity results for the hybrid and detlink cases on $8^{4}$ lattice for $\lambda_{\text{lat}}=2$. In the large $N$ limit, the difference is negligible. Figure 4: Ward identity dependence on the mass parameter $\mu$.  $\lambda_{\text{lat}}$ | $N$ |  $\Delta S_{B}$ (detlink) | W (detlink) |  $\Delta S_{B}$ (hybrid) | W (hybrid)
---|---|---|---|---|---
2.0 | 3 | 0.0606(1) | 0.0373(8) | 0.1238(4) | 0.0684(1)
| 4 | 0.0426(2) | 0.0273(7) | 0.0753(2) | 0.0491(0)
| 5 | 0.0311(1) | 0.0204(4) | 0.0505(1) | 0.0328(0)
| 6 | 0.0239(1) | 0.0159(4) | 0.0362(1) | 0.0233(0)
| 7 | 0.0192(1) | 0.0131(4) | 0.0276(1) | 0.0184(0)
| 8 | 0.0159(1) | 0.0110(3) | 0.0218(1) | 0.0141(0)
Table 1:  The comparison between the supersymmetry breaking observables using the detlink and the hybrid formulations on $8^{4}$ lattice for $\lambda_{\text{lat}}=2$. $\Delta S_{B}$ denotes the deviation from the supersymmetric value. See Fig. 3 for details. $\lambda_{\text{lat}}$ | $N$ |  $8^{4}$ |  |  $16^{4}$ |
---|---|---|---|---|---
|  | $\Delta S_{B}$ | W | $\Delta S_{B}$ | W
2.0 | 3 | 0.0606(1) | 0.0373(8) | 0.0407(18) | 0.207(19)
| 4 | 0.0426(2) | 0.0273(8) | 0.0425(0) | 0.0281(2)
| 5 | 0.0311(1) | 0.0204(8) | 0.0310(1) | 0.0202(3)
3.0 | 4 | 0.0413(2) | 0.0216(7) | 0.0420(2) | 0.0218(3)
| 5 | 0.0309(1) | 0.0166(4) | 0.0336(1) | 0.0174(4)
4.0 | 3 | 0.0781(3) | 0.0336(14) | 0.0788(1) | 0.0357(3)
| 4 | 0.0528(2) | 0.0254(11) | 0.0521(1) | 0.0230(4)
Table 2:  The comparison between supersymmetry breaking observables using the
detlink code on $8^{4}$ and $16^{4}$ lattices. The volume effects are small in
comparison at fixed $N$.

##  IV Conclusions

We have shown that simulations of lattice ${\cal N}=4$ SYM targeting the
$SU(N)$ rather than the $U(N)$ theory are possible at moderately strong
coupling $\lambda_{\text{lat}}\leq 4$. This is a stronger coupling than has
been achieved with the improved action described in Catterall and Schaich
(2015), where only $\lambda_{\text{lat}}\leq 3$ was possible. In the case of
gauge group $SU(2)$ simulations have even been performed at
$\lambda_{\text{lat}}=6$. However, unfortunately so far, we have not been able
to extend this to even stronger couplings. Instead we observe the system
appears to undergo a crossover or phase transition to a regime in which the
fermion operator develops very many small eigenvalues. We attribute this to
the presence of residual supersymmetry breaking associated with the
determinant term. Work is underway to develop a supersymmetric link based
determinant term which may allow us to bypass these problems and access yet
stronger couplings. The improvement that we do see is reflective of control
over the instabilities associated with the flat direction exhibited in the
scaling (I). The corresponding U(1) fluctuations are much more dangerous than
the SU(N) related flat directions because they allow the theory to wander into
regimes associated with coarser lattice spacings, where confinement is a
generic feature. In the future, we will present results where further
improvements can be obtained by preserving ${\cal Q}$ exactly while still
controlling this U(1) sector in a rather aggressive way.

## Acknowledgements

SC and RGJ were supported by the US Department of Energy (DOE), Office of
Science, Office of High Energy Physics, under Award Number DE-SC0009998. JG
was supported in part by the Department of Energy, Office of Science, Office
of High Energy Physics, Grant No. DE-SC0013496. Numerical calculations were
carried out on the Center for Computational Innovations at Rensselaer and DOE-
funded USQCD facilities at Fermilab. We thank David Schaich for comments and
discussions.

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