#  Critical Field Theories with OSp$(1|2M)$ Symmetry

###### Abstract

In the paper [L. Fei et al., JHEP 09 (2015) 076] a cubic field theory of a
scalar field $\sigma$ and two anticommuting scalar fields, $\theta$ and
$\bar{\theta}$, was formulated. In $6-\epsilon$ dimensions it has a weakly
coupled fixed point with imaginary cubic couplings where the symmetry is
enhanced to the supergroup OSp$(1|2)$. This theory may be viewed as a “UV
completion” in $2<d<6$ of the non-linear sigma model with hyperbolic target
space H0|2 described by a pair of intrinsic anticommuting coordinates. It also
describes the $q\rightarrow 0$ limit of the critical $q$-state Potts model,
which is equivalent to the statistical mechanics of spanning forests on a
graph. In this letter we generalize these results to a class of OSp$(1|2M)$
symmetric field theories whose upper critical dimensions are
$d_{c}(M)=2\frac{2M+1}{2M-1}$. They contain $2M$ anticommuting scalar fields,
$\theta^{i},\bar{\theta}^{i}$, and one commuting one, with interaction
$g\left(\sigma^{2}+2\theta^{i}\bar{\theta}^{i}\right)^{(2M+1)/2}$. In
$d_{c}(M)-\epsilon$ dimensions, we find a weakly coupled IR fixed point at an
imaginary value of $g$. We propose that these critical theories are the UV
completions of the sigma models with fermionic hyperbolic target spaces H0|2M.
Of particular interest is the quintic field theory with OSp$(1|4)$ symmetry,
whose upper critical dimension is $10/3$. Using this theory, we make a
prediction for the critical behavior of the OSp$(1|4)$ lattice system in three
dimensions.

##  1 Introduction

This letter builds on the paper [1] where the field theory was studied with
the Euclidean action

$S=\int
d^{d}x\bigg{(}\partial_{\mu}\theta\partial^{\mu}\bar{\theta}+\frac{1}{2}\left(\partial_{\mu}\sigma\right)^{2}+g_{1}\sigma\theta\bar{\theta}+\frac{1}{6}g_{2}\sigma^{3}\bigg{)}$
(1.1)

for two anticommuting scalar fields, $\theta$ and $\bar{\theta}$, and one
commuting one, $\sigma$. The global Sp$(2)$ symmetry of this model becomes
enhanced to the supergroup OSp$(1|2)$ because at the IR fixed point in
$6-\epsilon$ dimensions the two coupling constants are imaginary and related
by $g_{2}^{*}=2g_{1}^{*}$. At this weakly coupled fixed point, which is of the
Wilson-Fisher type [2], the interaction becomes proportional to the manifestly
OSp$(1|2)$ invariant form $(\sigma^{2}+2\theta\bar{\theta})^{3/2}$. Using
3-loop calculations, the scaling dimensions of $\sigma$ and $\theta$ in
$6-\epsilon$ dimensions were indeed found to be equal [1]. Since the coupling
is imaginary, the Euclidean path integral associated with (1.1) does not
suffer from the instability encountered for the real $\sigma^{3}$ interaction.
Indeed, the cubic theory of a single scalar field with an imaginary coupling
constant is known [3] to describe the Lee-Yang edge singularity.

Earlier examples of models with OSp$(1|2)$ symmetry are the lattice models and
sigma models describing the spanning forests, or equivalently the
$q\rightarrow 0$ limit of the $q$-state Potts model [4, 5, 6]. In [1] we
showed that the $6-\epsilon$ expansions of the scaling dimensions in our
OSp$(1|2)$ symmetric theory are the same as in the $q\rightarrow 0$ limit of
the $q$-state Potts model [7]. Thus, (1.1) provides an explicit formulation of
the field theory that governs this formal limit and is super-renormalizable in
$d<6$.

Besides the $6-\epsilon$ expansion, it is interesting to develop the
$2+\epsilon$ expansion for the critical theory of spanning forests. In [4, 8]
it was argued that it is provided by the OSp$(1|2)$ sigma model [9, 10] with
the action

$S=\frac{1}{2g^{2}}\int
d^{d}x\bigg{(}(\partial_{\mu}\sigma)^{2}+2\partial_{\mu}\theta\partial^{\mu}\bar{\theta}\bigg{)}\
,$ (1.2)

where the constraint $\sigma^{2}+2\theta\bar{\theta}=1$ is imposed. The
constraint has two solutions, $\sigma=\pm(1-\theta\bar{\theta})$, and choosing
one of these solutions breaks the $Z_{2}$ symmetry but preserves the
OSp$(1|2)$; then the global symmetries in the sigma model are the same as in
the cubic theory (1.1). In the recent work [11, 12] the interpretation of the
target space was changed from (half of) the sphere S0|2 to the space H0|2,
which is a fermionic version of the hyperboloid.

Substituting $\sigma=1-\theta\bar{\theta}$ into (1.2), we find the sigma model
with the hyperbolic target space H0|2. It has the following classical action
in terms of the two anticommuting scalar fields:

$S=\frac{1}{g^{2}}\int
d^{d}x\bigg{(}\partial_{\mu}\theta\partial^{\mu}\bar{\theta}-\theta\bar{\theta}\partial_{\mu}\theta\partial^{\mu}\bar{\theta}\bigg{)}\
,$ (1.3)

and it is important to take $g^{2}<0$ so that the model is asymptotically free
in $d=2$ [4, 11, 12]. 111In theories with anticommuting fields, rigorous
approaches to Renormalization Group are sometimes available [13]. Indeed, the
beta function of this theory is the same as for the O$(N)$ non-linear sigma
model continued to $N=-1$, and the theory is asymptotically free for negative
$g^{2}$. Thus, in $d=2+\epsilon$ this theory has a weakly coupled UV fixed
point. The fact that near the upper critical dimension $6$ the model has
another weakly coupled description, involving an extra canonical field
$\sigma$, is analogous to a similar phenomenon in the Gross-Neveu model [14]
near $4$ dimensions [15, 16] (for a more recent discussion, see [17]).

We may interpret the cubic theory (1.1) as a “UV completion” of the H0|2 sigma
model in $2<d<6$: the IR fixed point of the cubic theory presumably describes
the same physics as the UV fixed point of the sigma model. The sense of this
is similar to how the super-renormalizable Gross-Neveu-Yukawa model [15, 16]
provides a UV completion in $2<d<4$ of the non-renormalizable Gross-Neveu
model. Similarly, the $O(N)$ symmetric field theory of $N>2$ scalar fields
with the quartic interaction $\lambda(\phi^{i}\phi^{i})^{2}$ provides a UV
completion in $2<d<4$ of the non-renormalizable $O(N)$ non-linear sigma model.

The OSp$(1|2)$ lattice system [4, 5, 6, 11, 12], which underlies the continuum
descriptions reviewed above, involves introducing on each lattice site $x$ the
vector $u_{x}=(\theta_{x},\bar{\theta}_{x},\sigma_{x})$. The constraint that
it belong to H0|2 implies $\sigma_{x}=1-\theta_{x}\bar{\theta}_{x}$. Then for
each pair of nearest neighbor lattice sites, $x$ and $y$, the factor

$e^{\beta(u_{x}\cdot u_{y}+1)/2}\ ,\qquad u_{x}\cdot
u_{y}=-\sigma_{x}\sigma_{y}-\theta_{x}\bar{\theta}_{y}-\theta_{y}\bar{\theta}_{x}\
,$ (1.4)

is included in the integrand of the partition function. Since all the
integrations are over the Grassmann variables $\theta_{x},\bar{\theta}_{x}$,
they can be performed exactly on a finite lattice. This lattice system has a
particular simplicity and is found to be equivalent to the statistical
mechanics of spanning forests, or alternatively the $q\rightarrow 0$ Potts
model [4, 5, 6]. The Monte Carlo simulations for it were carried out in [18],
indicating the second-order phase transitions in $d=3,4,5$. We will show that
the estimates of critical exponents based on the two-sided Padé
extrapolations, using the $6-\epsilon$ and $2+\epsilon$ expansions, are in
good agreement with the Monte Carlo results.

The OSp$(1|2M)$ lattice systems may be constructed analogously: in this case
there are $M$ pairs of Grassmann variables on each lattice site,
$\theta_{x}^{i},\bar{\theta}_{x}^{i},i=1,\ldots M$, and

$\displaystyle\sigma_{x}=\left(1-2\sum_{i=1}^{M}\theta_{x}^{i}\bar{\theta}_{x}^{i}\right)^{1/2}\
,\qquad u_{x}\cdot
u_{y}=-\sigma_{x}\sigma_{y}-\sum_{i=1}^{M}(\theta_{x}^{i}\bar{\theta}_{y}^{i}+\theta_{y}^{i}\bar{\theta}_{x}^{i})\
.$ (1.5)

It is of obvious interest to study possible critical behavior in such lattice
systems for $M>1$. In this paper, we propose an extention of the field
theoretic approach (1.1) to these systems. We are led to consider theories
with interactions of order $2M+1$, i.e. $\sigma^{2M+1}$ plus terms involving
the anticommuting fields.222Field theories with interactions involving all
even powers of the field were studied long ago starting with [19]. Such
theories have the upper critical dimensions

$d_{c}(M)=2\frac{2M+1}{2M-1}\ .$ (1.6)

We use the results [20] for the O$(N)$-invariant field theories with
interactions of order $2M+1$, which are renormalizable at the upper critical
dimensions $\frac{10}{3}$, $\frac{14}{5}$, etc., and substitute $N=-2M$ to
account for the anticommuting nature of the $N$ scalar fields. Then we find
OSp$(1|2M)$ invariant IR fixed points where the interaction term is
proportional to
$\left(\sigma^{2}+2\theta^{i}\bar{\theta}^{i}\right)^{(2M+1)/2}$ with an
imaginary coefficient. These critical theories appear to be non-perturbatively
well-defined, and it would be very interesting to compare the continuum
results with those in the OSp$(1|2M)$ lattice systems.

##  2 Scaling dimensions for the OSp$(1|2)$ model

The one-loop beta functions and anomalous dimensions for the theory (3.7) are
[1]

$\displaystyle\beta_{1}$
$\displaystyle=-\frac{g_{1}\epsilon}{2}-\frac{1}{12(4\pi)^{3}}g_{1}(10g_{1}^{2}+12g_{1}g_{2}-g_{2}^{2})\
,$ $\displaystyle\beta_{2}$
$\displaystyle=-\frac{g_{2}\epsilon}{2}+\frac{1}{4(4\pi)^{3}}(8g_{1}^{3}-2g_{1}^{2}g_{2}-3g_{2}^{3})\
,$ $\displaystyle\gamma_{\theta}$
$\displaystyle=\frac{g_{1}^{2}}{6(4\pi)^{3}}\
,\qquad\gamma_{\sigma}=\frac{g_{2}^{2}-2g_{1}^{2}}{12(4\pi)^{3}}\ .$ (2.1)

There is an OSp$(1|2)$ invariant IR fixed point where 333There is another
fixed point where the couplings have the opposite sign. The two are physically
equivalent because they are related by the transformation
$\sigma\rightarrow-\sigma$ in the path integral.

$g_{1}=i\sqrt{(4\pi)^{3}\epsilon\over 5}\ ,\qquad g_{2}=2g_{1}\
,\qquad\gamma_{\theta}=\gamma_{\sigma}=-\frac{\epsilon}{30}\ .$ (2.2)

These results can be extended to the 4-loop order using the formulae from [21]
for the $O(N)$ invariant cubic model [22, 23], and then setting $N=-2$. 444The
5-loop renormalization of cubic theories in $6-\epsilon$ dimensions was
carried out recently [24, 25], but we will not include 5-loop corrections
here. Then we find

$\Delta_{\theta}=2-\frac{8}{15}\epsilon-\frac{7}{450}\epsilon^{2}-\frac{269-702\zeta(3)}{33750}\epsilon^{3}-\frac{207313-4212\pi^{4}+936\zeta(3)+907200\zeta(5)}{24300000}\epsilon^{4}+\mathcal{O}(\epsilon^{5})\
.$ (2.3)

This expansion coincides with the corresponding one in the formal
$q\rightarrow 0$ limit of $q$-state Potts model.

The $2+\epsilon$ expansion for the $O(N)$ sigma model is [26]

$\Delta_{\theta}=\frac{1}{2}\epsilon+\frac{1}{2(N-2)}\epsilon-\frac{N-1}{2(N-2)^{2}}\epsilon^{2}+\mathcal{O}(\epsilon^{3})\
,$ (2.4)

which for $N=-1$ becomes

$\Delta_{\theta}=\frac{1}{3}\epsilon+\frac{1}{9}\epsilon^{2}+\mathcal{O}(\epsilon^{3})\
.$ (2.5)

Using the “two-sided” $(4,3)$ and $(6,1)$ Padé approximations, which utilize
both the $6-\epsilon$ and $2+\epsilon$ expansions, we obtain the estimates
$\Delta_{\theta}\approx 1.46$ in $d=5$; $\Delta_{\theta}\approx 0.92$ in $d=4$
and $\Delta_{\theta}\approx 0.415$ in $d=3$. They are in very good agreement
with the Monte Carlo simulations of the $q=0$ Potts model. Using the values
for $\gamma/\nu$ given in Table I of [18], we find that

$\Delta_{\theta}=\frac{d}{2}-\frac{\gamma}{2\nu}$ (2.6)

are very close to our Padé estimates in $d=3,4,5$.

The next important operator in the sigma model has dimension [26]

$\Delta_{+}=d-\nu^{-1}=2-\frac{1}{N-2}\epsilon^{2}+\mathcal{O}(\epsilon^{3})\
.$ (2.7)

For $N=-1$ this becomes

$\Delta_{+}=2+\frac{1}{3}\epsilon^{2}+\mathcal{O}(\epsilon^{3})$ (2.8)

The expansion of $\Delta_{+}$ in $d=6-\epsilon$ may be found using the theory
(1.1), where it corresponds to the OSp$(1|2)$ invariant primary operator
$\sigma^{2}+2\theta\bar{\theta}$. Using the 4-loop results [21], we find

$\Delta_{+}=4-\frac{2}{3}\epsilon+\frac{1}{30}\epsilon^{2}+\frac{173-864\zeta(3)}{27000}\epsilon^{3}+\frac{51683-1296\pi^{4}+140400\zeta(3)+272160\zeta(5)}{4860000}\epsilon^{4}+\mathcal{O}(\epsilon^{5})\
.$ (2.9)

Performing the two-sided Padé approximations, we find $\Delta_{+}\approx 3.36$
in $d=5$; $\Delta_{+}\approx 2.8$ in $d=4$ and $\Delta_{+}\approx 2.2$ in
$d=3$. They are in good agreement with the Monte Carlo results for $\nu$ given
in Table I of [18] for $q=0$.

##  3 Field theory for $M>1$

The 2-d sigma model with target space H0|2M may be defined by picking one of
the two solutions of the constraint

$\sigma^{2}+2\sum_{i=1}^{M}\theta^{i}\bar{\theta}^{i}=1\ .$ (3.1)

The sigma model classical action is

$\displaystyle S=\frac{1}{2g^{2}}\int
d^{2}x\bigg{(}(\partial_{\mu}\sigma)^{2}+2\sum_{i=1}^{M}\partial_{\mu}\theta^{i}\partial^{\mu}\bar{\theta}^{i}\bigg{)}\
,$ (3.2)

with the substitution of

$\sigma=\left(1-2\sum_{i=1}^{M}\theta^{i}\bar{\theta}^{i}\right)^{1/2}\ .$
(3.3)

For example, for $M=2$

$\sigma=1-\theta^{1}\bar{\theta}^{1}-\theta^{2}\bar{\theta}^{2}-\theta^{1}\bar{\theta}^{1}\theta^{2}\bar{\theta}^{2}\
.$ (3.4)

In general, the expansion of the square root in (3.3) terminates with the term
of order $2M$ proportional to $\prod_{i=1}^{M}\theta^{i}\bar{\theta}^{i}$.
Related to this fact, we will propose a critical field theory with
interactions of order $2M+1$.

The H0|2M sigma model may be thought of as the O$(N)$ sigma model with
$N=1-2M$. For $g^{2}<0$ it is asymptotically free in $d=2$ and therefore has a
UV fixed point in $d=2+\epsilon$. An interesting question is how to continue
this theory to the dimension slightly below the upper critical one. For $M=1$
its upper critical dimension is $6$, and in $d<6$ we can view it as Euclidean
field theory (1.1) with interaction
$\left(\sigma^{2}+2\theta^{i}\bar{\theta}^{i}\right)^{3/2}$. However, such a
description cannot be applicable to $M>1$. Indeed, already for $M=2$ the
potential would contain the term
$\sim\theta^{1}\bar{\theta}^{1}\theta^{2}\bar{\theta}^{2}/\sigma$ which is not
admissible in renormalizable field theory. We propose that the proper
generalization of our $M=1$ construction to higher $M$ involves higher powers
in the OSp$(1|2M)$ invariant potential, so that its expansion in the
anticommuting variables does not contain any terms with negative powers of
$\sigma$; namely,

$\left(\sigma^{2}+2\theta^{i}\bar{\theta}^{i}\right)^{(2M+1)/2}=\sigma^{2M+1}+(2M+1)\sigma^{2M-1}\sum_{i=1}^{M}\theta^{i}\bar{\theta}^{i}+\ldots+(2M+1)!!\
\sigma\prod_{i=1}^{M}\theta^{i}\bar{\theta}^{i}\ .$ (3.5)

In $d_{c}(M)-\epsilon$ there is a weakly coupled IR fixed point of the
interacting field theory

$S=\int
d^{d}x\bigg{(}\partial_{\mu}\theta^{i}\partial^{\mu}\bar{\theta}^{i}+\frac{1}{2}\left(\partial_{\mu}\sigma\right)^{2}+g\left(\sigma^{2}+2\theta^{i}\bar{\theta}^{i}\right)^{(2M+1)/2}\bigg{)}\
,$ (3.6)

where at the fixed point $g$ is imaginary and $\sim\sqrt{\epsilon}$.

The $M=2$ model is particularly interesting, since its critical dimension
$\frac{10}{3}$ is above $3$. In the $\frac{10}{3}-\epsilon$ expansion, we
expect good results since $\epsilon=1/3$ is small. This was indeed the case
for the model which is described by the $i\sigma^{5}$ theory. The
$\frac{10}{3}-\epsilon$ expansion for this model was obtained in [27], where
it was called the Blume-Capel or the tricritical Lee-Yang universality class.
555Quintic field theories with $S_{q}$ symmetry were similarly studied in
[28].

The quintic model with four anticommuting scalars and $Sp(4)$ symmetry has the
Euclidean action

$S=\int
d^{d}x\bigg{(}\partial_{\mu}\theta^{i}\partial^{\mu}\bar{\theta}^{i}+\frac{1}{2}\left(\partial_{\mu}\sigma\right)^{2}+\frac{g_{1}}{6}\sigma\left(\theta^{i}\bar{\theta}^{i}\right)^{2}+\frac{g_{2}}{6}\sigma^{3}\theta^{i}\bar{\theta}^{i}+\frac{g_{3}}{120}\sigma^{5}\bigg{)}\
.$ (3.7)

Since the one-loop $\beta$ and $\gamma$ functions were calculated in [20] for
the quintic model with $N$ additional scalar fields and O$(N)$ symmetry, we
may set $N=-4$ to obtain the corresponding results for the theory (3.7). After
performing a suitable multiplicative redefinition of the couplings,
$g_{j}\rightarrow Ag_{j}$, which simplifies the formulae, we find the one-loop
beta functions for the theory (3.7):

$\displaystyle\beta_{1}=-\frac{3g_{1}\epsilon}{2}-\frac{1}{80}$
$\displaystyle(1864g_{1}^{3}+10080g_{1}^{2}g_{2}+8664g_{1}g_{2}^{2}+10800g_{2}^{3}-480g_{1}g_{2}g_{3}+1620g_{2}^{2}g_{3}-3g_{1}g_{3}^{2})\
,$ $\displaystyle\beta_{2}=-\frac{3g_{2}\epsilon}{2}+\frac{1}{80}$
$\displaystyle(1120g_{1}^{3}+2888g_{1}^{2}g_{2}+10800g_{1}g_{2}^{2}-15192g_{2}^{3}$
$\displaystyle-$ $\displaystyle
80g_{1}^{2}g_{3}+1080g_{1}g_{2}g_{3}-8640g_{2}^{2}g_{3}-1251g_{2}g_{3}^{2})\
,$ $\displaystyle\beta_{3}=-\frac{3g_{3}\epsilon}{2}+\frac{1}{16}$
$\displaystyle(640g_{1}^{2}g_{2}-4320g_{1}g_{2}^{2}+23040g_{2}^{3}+8g_{1}^{2}g_{3}+10008g_{2}^{2}g_{3}-1377g_{3}^{3})\
.$ (3.8)

The anomalous dimensions are

$\displaystyle\gamma_{\theta}=\frac{1}{20}(-2g_{1}^{2}+3g_{2}^{2})\
,\qquad\gamma_{\sigma}=\frac{1}{80}(8g_{1}^{2}-72g_{2}^{2}+3g_{3}^{2})\ .$
(3.9)

We find an IR stable fixed points where

$(g_{1},g_{2},g_{3})=\frac{i}{3}\sqrt{\epsilon\over 139}(3,2,8)\ .$ (3.10)

The two anomalous dimensions are equal at this fixed point,
$\gamma_{\theta}=\gamma_{\sigma}=\frac{1}{4170}\epsilon$, suggesting that the
Sp$(4)$ symmetry is enhanced to OSp$(1|4)$. This is indeed the case, since the
interaction in (3.7) combines at the fixed point into $\sim
i\left(\sigma^{2}+2\theta^{1}\bar{\theta}^{1}+2\theta^{2}\bar{\theta}^{2}\right)^{5/2}$.

The IR scaling dimension is

$\Delta_{\theta}=\Delta_{\sigma}=\frac{d-2}{2}+\gamma_{\theta}=\frac{2}{3}-\frac{1042}{2085}\epsilon+O(\epsilon^{2})\
.$ (3.11)

Substituting $\epsilon=1/3$ gives the answer
$\Delta_{1-loop}=\frac{3128}{6255}\approx 0.50008$. This is extremely close to
the free dimension $1/2$ due to the smallness of the coefficient of $\epsilon$
in the anomalous dimension. We can try to improve on this estimate using the
two-sided Padé including the $2+\epsilon$ expansion for the $N=-3$ sigma
model:

$\Delta(2+\epsilon)=\frac{2}{5}\epsilon+\frac{2}{25}\epsilon^{2}+\mathcal{O}(\epsilon^{3})\
.$ (3.12)

A Padé approximant for the scaling dimension as a function of $d$,

$\Delta_{P}(d)={16(d-2)(8360-2091d)\over 476500+d(-192316+18813d)},$ (3.13)

is consistent with the expansions near $10/3$ and $2$ dimensions. It gives
$\Delta_{P}(3)\approx 0.485$ which is somewhat lower than $\Delta_{1-loop}$.
It would be interesting to calculate the $O(\epsilon^{2})$ correction to
$\Delta$ using the 2-loop diagrams in the quintic theory (3.7). We expect this
to reduce the uncertainty in estimating the scaling dimension in $d=3$. It
would be also useful to study the critical exponents using the Functional
Renormalization Group approach [29] to theory (3.7).

Let us also discuss the theory with Sp$(6)$ symmetry and seventh-order
interactions:

$S=\int
d^{d}x\bigg{(}\sum_{i=1}^{2}\partial_{\mu}\theta^{i}\partial^{\mu}\bar{\theta}^{i}+\frac{1}{2}\left(\partial_{\mu}\sigma\right)^{2}+\frac{g_{1}}{90}\sigma\left(\theta^{i}\bar{\theta}^{i}\right)^{3}+\frac{g_{2}}{36}\sigma^{3}\left(\theta^{i}\bar{\theta}^{i}\right)^{2}+\frac{g_{3}}{120}\sigma^{5}\theta^{i}\bar{\theta}^{i}+\frac{g_{4}}{5040}\sigma^{7}\bigg{)}\
.$ (3.14)

Using Gracey’s results [20] for the $O(N)$ symmetric theory with seventh-order
interactions, and continuing them to $N=-6$, we find an IR fixed point in
$\frac{14}{5}-\epsilon$ dimensions. In the normalization of couplings such
that

$\displaystyle\gamma_{\theta}$
$\displaystyle=\frac{1}{3024}\left(24g_{1}^{2}-200g_{2}^{2}+45g_{3}^{2}\right)\
,$ (3.15) $\displaystyle\gamma_{\sigma}$
$\displaystyle=\frac{1}{6048}\left(-48g_{1}^{2}+1800g_{2}^{2}-1350g_{3}^{2}+15g_{4}^{2}\right)\
,$ (3.16)

it is located at

$(g_{1},g_{2},g_{3},g_{4})=i\cdot 0.00392514\sqrt{\epsilon}(15,6,8,48)\ .$
(3.17)

At this fixed point $\gamma_{\theta}=\gamma_{\sigma}\approx-5.50241\cdot
10^{-6}\epsilon$, and we observe the enhancement of the Sp$(6)$ symmetry to
OSp$(1|6)$.

Our calculations provide evidence for the consistency of our proposal for
$M>1$, but a lot remains to be done. In particular, it would be interesting to
formulate a Monte Carlo approach to the OSp$(1|4)$ lattice model, where all
the integrations in the partition function are over the Grassmann variables
$\theta_{x}^{1},\bar{\theta}_{x}^{1},\theta_{x}^{2},\bar{\theta}_{x}^{2}$. The
results could be then compared with our prediction that the scaling exponents
have exactly mean field values in four dimensions, but exhibit small
deviations from them in three dimensions. Also, perhaps a conformal bootstrap
approach to the critical exponents can be attempted along the lines of the
method [30], which is applicable to non-unitary theories.

## Acknowledgments

I am very grateful to Roland Bauerschmidt and Tom Spencer for the important
recent discussions and explanations that have rekindled my interest in field
theories with OSp$(1|2M)$ symmetry, and for their comments on a draft of this
paper. I am also grateful to Simone Giombi and Grisha Tarnopolsky for very
useful discussions, and to them and Lin Fei for an earlier collaboration. I
thank John Gracey for useful correspondence. Finally, I would like to thank
Sergio Caracciolo and Giorgio Parisi for sharing their wisdom about the
statistical mechanics of spanning forests. This work was supported in part by
the US NSF under Grant No. PHY-1914860.

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