Dualities between fermionic theories and the Potts model Vladimir Narovlansky Princeton Center for Theoretical Science and Physics Department Princeton University Princeton

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Dualities between fermionic theories and the Potts model

Vladimir Narovlansky

Princeton Center for Theoretical Science and Physics Department, Princeton University, Princeton,

NJ 08544, USA

narovlansky@princeton.edu

Abstract

We show that a large class of fermionic theories are dual to a q→0 limit of the Potts

model in the presence of a magnetic ﬁeld. These can be described using a statistical model

of random forests on a graph, generalizing the (unrooted) random forest description of

the Potts model with only nearest neighbor interactions. We then apply this to ﬁnd a

statistical description of a recently introduced family of OSp(1|2M) invariant ﬁeld theories

that provide a UV completion to sigma models with the same symmetry.

arXiv:2210.01847v1 [hep-th] 4 Oct 2022

Contents

1 Introduction 1

2 Duality between fermionic theories and interacting Potts models 2

3 Theories with OSp(1|2M)symmetry 6

3.1 Fermionic and forest representations . . . . . . . . . . . . . . . . . . . . . . . . 8

3.2 Dual Potts model for the OSp(1|4)theory ..................... 11

4 Forest representations of the OSp(1|4) theory 12

5 Concluding remarks 15

A Continuation of vector models under N→ −N16

1 Introduction

The Potts model [1,2] is an immediate generalization of the Ising model, where at every site

of a lattice there is a degree of freedom that can obtain qpossible states, with q= 2 being

precisely the Ising model. Nearest neighbors interact among themselves according to their state.

For example, whenever they are in the same state, the energy of the system gets a particular

contribution, analogously to the case of spins of particles that point in the same direction.

Given this arguably simplest statistical system one could imagine, it is natural to wonder

how rich its statistical behavior is. That is, it is interesting to know how many diﬀerent phases

can such a simple model capture. In this work, we would like to show that a wide class of

fermionic theories with general interactions are dual to the q→0 limit of the Potts model in

the presence of a magnetic ﬁeld.1This is done in section 2, and the precise relation is given in

Eq. (2.2).

The q→0 limit of the Potts model with only nearest neighbor interactions is known to

be equivalent to a statistical model of unrooted spanning forests on the graph deﬁned by the

lattice on which the Potts degrees of freedom reside. We use a generalization of the spanning

forests theory in order to argue for the duality above. We do this by relating it to fermionic

theories similarly to [3] on the one hand, and to the Potts model generalizing the idea of Fortuin

and Kasteleyn [4,5] on the other hand.

We then apply this equivalence to ﬁnd a statistical description of a class of OSp(1|2M)

invariant theories that were recently introduced [6] in terms of the Potts model as well as span-

ning forests (section 3). In Ref. [6], ﬁeld theories with OSp(1|2M) symmetry were constructed

and were suggested to be the UV completions of sigma models with the same symmetry. The

case M= 1 was studied in [7]. The q→0 limit of the Potts model with nearest neighbor

1We should point out that we use ‘fermions’ to refer to anticommuting degrees of freedom throughout. These

are not necessarily fermions in the usual sense.

interactions, or equivalently the statistical theory of (unrooted) random forests, are known to

correspond to the sigma model with OSp(1|2) symmetry [3,8]. The ﬁeld theory with M= 1,

which is suggested to provide a UV completion of this sigma model, was observed in [7] to have

critical exponents matching to those of spanning forests. Here, we would like to understand

this relation and extend it to general M, with particular emphasis on M= 2. The M= 1 ﬁeld

theory is cubic in the ﬁelds and becomes weakly coupled in six dimensions. This suggests that

the upper critical dimension for unrooted spanning forests is six. In fact, numerical evidence for

this was given in [9] (see also [10]). In a general spanning forest representation, as we describe

below, the roots of the diﬀerent trees can be correlated. We show in section 4how this can

be eliminated, and demonstrate this by analyzing spanning forest descriptions of the M= 2

theory.

An additional motivation to study such vector theories with anticommuting scalars comes

from the de Sitter / Conformal Field Theory (dS/CFT) correspondence. In dS/CFT, gravity

in dS space, having a positive cosmological constant, is related to a non-unitary CFT living

in Euclidean space [11]. This is motivated by the asymptotic symmetries of dS gravity, with

appropriate boundary conditions obtained by analytic continuation of those in AdS [12]. It

was proposed [13] that the dual to the minimal higher spin theory in 4-dimensional de Sitter is

a theory of anticommuting scalars with quartic interactions [14,15], having Sp(N) symmetry.

The idea is that higher spin theory in dS is obtained by the one in AdS by continuing the

cosmological constant Λ to −Λ. Using the relation N∼1

GNΛof the boundary central charge

to Newton’s constant GN, this means taking N→ −N. The transformation N→ −Nis

implemented by turning the commuting variables into anticommuting. This is seen using a

Hubbard–Stratonovich ﬁeld since each anticommuting loop gives a minus sign and comes with

a factor of N. Therefore, going from bosons where each loop is assigned a value of Nto fermions

implements taking N→ −N. We give an alternative non-perturbative argument for this claim

in appendix A.

A UV completion of the quartic vector theory in dimension beyond four dimensions is

the theory with M= 1 potential that we discuss in section 3, with an arbitrary number of

anticommuting ﬁelds. This was suggested in [7] to provide a higher spin theory in dS in higher

dimension.

2 Duality between fermionic theories and interacting Potts

models

Consider a many-body system (or a regularized ﬁeld theory) that we describe using an undi-

rected graph Gwith vertices Vand edges E. Let the graph be edge-weighted so that to every

edge ewe assign a weight we. We can assume that every two vertices are connected by at most

one edge.2We can also denote by wij =wji the weight of the edge connecting vertices iand

jin case they are connected by an edge, and zero otherwise. The Laplacian matrix of Gis a

symmetric |V|×|V|matrix with entries Lij =−wij for i6=jand Lii =Pk6=iwik, so that each

row and column sum to zero. For example, the graph Gcan be a bounded cubic lattice with

nearest neighbors connected by edges. For unit weights, the Laplacian matrix is simply the

discretized Laplacian.

A rather general fermionic theory is deﬁned by providing a lattice Gand placing Grassmann

variables θiand ¯

θion each vertex i∈V. The interactions we consider are speciﬁed by subgraphs

of Glabeled by Γ with vertex sets VΓ, so that the partition function is given by

ZDθD¯

θ e¯

θLθ+PΓtΓQi∈VΓ(¯

θiθi)(2.1)

where tΓare the coupling constants. As explained below, without loss of generality, we will

take the Γ’s to have no edges for the purpose of this formula.

We would like to show that this theory is equivalent to the Potts model with interactions

beyond nearest neighbors. The Potts model can be thought of simply as a generalization of the

Ising model, where instead of two states, the degrees of freedom have qstates. In terms of the

graph description above, we place variables σion every vertex i, taking values σi∈ {1,··· , q}.

Similarly to the Ising model, we can also introduce a ﬁxed external magnetic ﬁeld with direction

h∈ {1,··· , q}. Nearest neighbor interactions are given simply by Kronecker delta symbols

δσi,σj, where we consider iand jto be nearest neighbors if they are connected by an edge in the

graph. We will introduce higher interactions by coupling not only nearest neighbor pairs, but

also next-to-nearest neighbor terms and beyond (if necessary) with Kronecker delta symbols.3

We will ﬁnd it useful to couple them to the external magnetic ﬁeld.

Speciﬁcally, the claim is that the fermionic theory (2.1) is equivalent to a q→0 limit of a

Potts model, such that

ZDθD¯

θ e¯

θLθ+PΓtΓQi∈VΓ(¯

θiθi)= lim

q→0q−|V|/2X

σi∈{1,··· ,q}

e−H.(2.2)

The Hamiltonian of the theory is

H=−X

e=hiji

log(1 + √qwe)δσi,σj−X

log 1 + tΓq|VΓ|/2Y

i∈VΓ

δσi,h (2.3)

where in the ﬁrst sum we go over nearest neighbors.4

In the remainder of this section, we prove (2.2). There are two steps: ﬁrst, we express the

2This is so, because only the sum of weights of edges connecting two vertices enters in the Laplacian matrix

below. Note also that there are no edges connecting a vertex to itself.

3This can be thought of as placing the Potts model on a hypergraph rather than a graph.

4Note that we can replace each of the three explicit occurrences of √qin the formula by qαfor any 0 < α < 1

and the formula will still be valid.

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DualitiesbetweenfermionictheoriesandthePottsmodelVladimirNarovlanskyPrincetonCenterforTheoreticalScienceandPhysicsDepartment,PrincetonUniversity,Princeton,NJ08544,USAnarovlansky@princeton.eduAbstractWeshowthatalargeclassoffermionictheoriesaredualtoaq!0limitofthePottsmodelinthepresenceofamagneticeld...

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Dualities between fermionic theories and the Potts model Vladimir Narovlansky Princeton Center for Theoretical Science and Physics Department Princeton University Princeton

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