Exotic field theories for (hybrid) fracton phases from imposing constraints in foliated field theory

Fracton phases of matter are gapped phases of matter that, by dint of their sensitivity to UV data, demand non-standard quantum field theories to describe them in the IR. Two such approaches are foliated quantum theory and exotic field theory. In this paper, we explicitly construct a map from one to the other and work out several examples. In particular, we recover the equivalence between the foliated and exotic fractonic BF theories recently demonstrated at the level of operator correspondence. We also demonstrate the equivalence of toric code layers and the anisotropic model with lineons and planons to the foliated BF theory with one and two foliations, respectively. Finally, we derive new exotic field theories that provide simple descriptions of hybrid fracton phases from foliated field theries known to do so. Our results both provide new examples of exotic field theories and pave the way toward their systematic construction from foliated field theories.

A Quantized Periods in the Fractonic Hybrid X-Cube 26 1 Introduction Recent years have seen a sweeping generalization of the concept of global symmetry applied throughout theoretical physics [1][2][3].Approximately in parallel, fracton phases of matter [4,5] emerged as phenomena requiring a framework.Fractons are quasiparticles that cannot move.Depending on the specifics of the model in which they emerge, they might be able to form mobile bound states.These mobile bound states often only have mobility in some directions -hence names such as lineon and planon.Fracton phases of matter are gapped phases of matter that possess these excitations 1 .Like topological phases of matter (see, for example, [7] for a treatment), they have a ground state degeneracy that is robust to local perturbations and reliant on the topology of the manifold on which the system lives (i.e. the presence of non-contractible cycles).Unlike in topological phases of matter, the ground state degeneracy is often subextensive and depends sensitively on the number of edges in the lattice.This, combined with the aforementioned mobility constraints, reflects a peculiar sensitivity of the long distance "universal" physics to short distance data.These features make it clear that fracton phases cannot be straightforwardly described using topological quantum field theory, unlike their comparatively standard counterparts.This raises a natural question: what describes the long distance physics of fractons?Subsystem symmetry [9][10][11] plays a key role in the structure of fracton phases.On the lattice, a subsystem symmetry only acts on parts of the lattice.In the continuum, the defects associated with subsystem symmetries cannot undergo arbitrary (homological) deformations 2 , but must stay pinned to some submanifold of the space on which the system lives.Field theories with subsystem symmetry have been thoroughly examined (c.f.[9][10][11]), inspired by the fact that the effective field theory [8,11] for a prototypical fracton model called the X-Cube [13] possesses subsystem symmetries.This theory is of a BF type, similar to the theories that describe topological phases [7], but the subsystem symmetry is a crucial difference.The restriction on the deformation of symmetry defects is a probe limit of the restricted mobility of the excitations.Moreover, the mixed 't-Hooft anomalies of the subsystem symmetries force the theories to have a non-trivial Hilbert space.After suitable regularization, this accounts for the bizarre ground state degeneracy of fracton phases.These field theories have been dubbed "exotic" field theories by [12].
Another line of work showed that some fracton phases, such as the X-Cube, are only sensible on manifolds admitting a foliation [14].This lead to a low energy description of fracton phases in terms of foliated quantum field theory [15][16][17].The key to foliated quantum field theories is the foliation one form e that is perpendicular to the leaves of the foliation.By coupling fields to e, one can engineer field theories whose defects have similar mobility restrictions.For example, with three foliations, one can have defects that stay on leaves (planons for flat foliations) or defects that must live at the intersection of two leaves (lineons for flat foliations).In [15] the authors demonstrate that the lattice model for the foliated field theory is in the same phase as the lattice model for the exotic field theory for several gapped systems.Moreover, [17] remarks that foliated theories with a flat two form gauge field appear to be related to symmetric, off diagonal tensor gauge theories such as those in the exotic field theories.
Recently, [18] made the correspondence between foliated and exotic theories explicit for some well known models.It did so by specifying which operator in the foliated theory corresponded to which operator in the exotic field theory.This work seeks a constructive approach to the duality outlined therein.Specifically, we show that by integrating out certain fields in the foliated theory, one can address the constraints imposed in such a way that the exotic theory falls out.Our procedure is a continuum version of the map in [15].Before outlining the procedure, let's recapitulate foliated and exotic field theory in some more detail.

Foliated Field Theory
Foliated quantum field theory was introduced in [15] as a continuum description of certain fracton models, called foliated fracton phases.By foliated fracton phase, we mean a fracton phase that can be defined on manifolds that admit a foliation [14].We consider a codimension one foliation of a manifold, that is to say a decomposition along a certain of the manifold into a union of codimension one manifolds called leaves.The foliation is tracked by the foliation one form e = e µ dx µ .The vectors v µ tangent to each leaf of the foliation must satisfy v µ e µ = 0, which is what we mean when we say the foliation one form is perpendicular to the leaves of the foliation.The foliation one form must be supported on the entire manifold, and must satisfy e ∧ de = 0. We will assume that de = 0, though generally de = e ∧ ω, where ω is called the Godbilon-Vey invariant of the foliation.Since we suppose de = 0, we can locally write e = df , for some zero form f .Integrating over e counts the number of leaves in the foliation after one introduces a lattice regularization.Note that the consistency conditions are agnostic to rescaling e i.e. e ∼ γe, for some zero form γ. Thus, we can choose the number of leaves in the foliation, and the IR theory is sensitive to this choice.This sort of UV/IR mixing is ubiquitous in fractonic field theories.We can have multiple foliations, leading to multiple foliation one forms e A .
The foliated field theories we study in this paper are variants of the following BF theory: Let d be the number of spacetime dimensions.Here, a is a one form gauge field (albeit with nonstandard gauge transformations, detailed below), b is a d − 2 form gauge field, A A are 1 form gauge fields, and B A are d − 3 form gauge fields 3 .Inspecting the first term of the above Lagrangian informs us that A A and B A only have components perpendicular to the direction of e, so terms involving foliated gauge fields describe physics on the leaves of a foliation.The third term is (up to the above subtlety in the gauge transformations) the usual BF theory, and the middle term couples the two.While we view A A , B A , a, and b as dynamical fields, we only consider static foliations in this work.The gauge holonomies, which are probe limits of the excitations in the system 4 , are the gauge invariant observables in the theory.They care about the leaves of the foliation, and can naturally live on the leaves or their intersections (considering leaves of different foliations, of course).By our mapping onto an exotic field theory, we will see this explicitly throughout the paper.One can also analyze the gauge holonomies directly in the foliated picture, see [15][16][17][18].To calculate the ground state degeneracy, one can solve Gauss' law as in [18] and counting the number of modes in the solution.Doing so involves counting the number of leaves in the foliation, so we obtain a sub-extensive result.
Let us pause to clarify our notation, especially as compared to the literature.What we denote as A A ∧ e A is denoted A A in [16] and B A in [17].Similarly, what we denote as B A ∧ e A is denoted B A in [16] and A A in [17].In this regard, our notation is most similar to [15,18].Compared to the latter, we will differ by some signs, since we follow the conventions of [15] when writing our actions.The exceptions to this are that we work in Euclidean signature and use {A} to index foliations rather than {k}5 .

Exotic Field Theory
Exotic field theories are field theories that make subsystem symmetries apparent by dint writing fields that transform under discrete subgroups of the rotation group.It is best to illustrate with an example: Here, i, j, k are distinct indices and i and j are symmetric.Moreover, spatial indices label transformation under S 4 , the group orientation preserving rotations of a cube by π/2, rather than transformation under SO(3).Details on the representation theory of S 4 can be found in [11].As discussed in detail below, this theory's gauge invariant operators are gauge holonomies.Thanks to its non-standard gauge transformations, they are constrained to live on lines and planes.Computing the ground state degeneracy amounts the counting the operators, which amounts to counting planes.This diverges, and must be regularized by placing the theory on a cubic lattice.This procedure yields the trademark subextensive ground state degeneracy.Throughout this paper, fields in exotic field theories are written in uppercase letters.This will occassionally result in the need to capitalize fields when mapping from foliated to exotic field theory6 .

Outline of the Paper
In this paper, we exploit a simple fact to obtain exotic field theories from foliated field theories -the exotic field theories are foliated theories with the constraints that relate foliated and standard gauge fields imposed.We do so by integrating out the time components of the fields in the middle term of the first Lagrangian above.
The remainder of the paper is structured as follows.We begin with the foliated BF theory in 2+1 dimensions.For 2 foliations, our procedure yields the exotic BF theory in [9] and the correspondence between operators in [18].We then move to the foliated BF theory in 3+1 dimensions.For one foliation, our procedure yields a field theory for a stack of toric codes (2+1 dimensional Z N gauge theories7 ).We have not seen this field theory in the literature before.For two foliations, we find the anisotropic theory with lineons, which has appeared in several places [21][22][23].For three foliations, we obtain the exotic BF theory of the X-Cube from [8,11] and the operator correspondence from [18].We then move to models obtained by coupling the foliated BF theory to additional gauge fields.Our procedure yields novel theories that couple exotic gauge fields to conventional gauge fields and exhibit key physics of hybrid fracton phases introduced in [24], namely that fractonic (reduced mobility) excitations can fuse to mobile excitations and vice-versa.Our work is an important step in constructing a map from foliated to exotic field theories, which could provide a systematic way to uncover further exotic field theories.

Foliated BF Theory in 2+1 Dimensions
In this section, we map the foliated BF theory in 2+1 dimensions to an exotic theory.We obtain the exotic BF theory first described in [9].Moreover, in mapping from the foliated theory to the exotic theory, we identify fields in such a way that we naturally rederive the field dictionary discussed in [18].For the sake of keeping the present work self contained, we discuss the exotic BF theory in detail, describing its gauge invariant operators and deriving and interpreting its ground state degeneracy.Much of this discussion follows [9].We recover the map from fields in the foliated theory to fields in the exotic theory that [18] discusses as a consequence of our procedure.
We consider the foliated BF theory in 2+1 dimensions with two foliations e 1 = dx and e 2 = dy.The Lagrangian is Let us take roll of the fields involved.a is a one form gauge field with gauge redundancy b is a one form gauge field with gauge redundancy A A ∧ dx A is a foliated 1+1 form gauge field with redundancy and B A is a 0 form field with redundancy where m A is an integer valued function of x A alone.The zero form gauge parameters are compact; they are 2π periodic.Intuitively, the first two terms describe decoupled layers of 1+1 dimensional Z N gauge theories, the third term describes 2+1 dimensional Z N gauge theory, and the middle two terms describe the coupling between the two sectors.That BF theory is a Z N gauge theory is made clear in [27,28].One can also gleam this from the quantization of various quantities.Specifically: where C (1) is a closed one-manifold, where C τ is a closed curve around the τ cycle, where S A is a strip whose boundary components are on leaves of the foliation defined by dx A , and (2.9) We now turn to the defects and operators in the theory.The theory has the defect which we interpret as a fracton at (x, y).There is a local operator: and strips: Courtesy of the flux quantization: We now extract the exotic theory from the foliated field theory.Expanding the Lagrangian in components gives The second and fourth collections of terms are interesting -they relate the standard gauge fields to the foliated gauge fields.This is exactly the sort of thing that should be encoded in the exotic field theory.Thus, we integrate out the corresponding Lagrange multipliers A x τ , A y τ , and b τ .Integrating out A A τ simply relates b and B. Integrating out b τ gives the equation Let's define Note that satisfying the above constraint amounts to demanding that the correct behavior for a symmetric tensor gauge field.Upon solving for A i j in terms of A ij , solving for b i in terms of ∂ i B k , relabeling a τ = A τ and defining we obtain which is precisely the BF presentation of the exotic Z N gauge theory in 2+1 dimensions.This theory is discussed in detail in [9,18,30].For the sake of being self-contained, let us examine its main features.The Lagrangian has the gauge redundancy where the gauge parameters are related to those in the foliated theory by One can show the following quantities to be quantized: Let us discuss the global symmetry.There is a defect which we interpret as a fracton located at (x, y).There are two types of global symmetries.The Z N electric symmetry is generated by whereas the Z N magnetic symmetry is generated by As a consequence of the quantized fluxes: Thus, we see that the operators/defects eqs.(2.27) to (2.29) are the exotic counterparts of eqs.(2.10) to (2.12) in the foliated field theory.We remark that these exhaust the symmetries if the periodicity of the torus alligns with the foliations.We only consider such an untwisted torus in this paper.Thorough discussion of twisting the boundary conditions can be found in [29].These symmetries give the ground state degeneracy, which appears in the field theory as the dimension of the Hilbert space.It follows from the canonical commutation relations that8 As detailed in [9], when regularized on an L x × L y square lattice, we obtain the ground state degeneracy Notably, unlike in a topological order, this ground state degeneracy depends on local operators W e .In fact, it is not generally robust, as [9] discusses in detail.This is consistent with no-go theorems for fracton order in 2+1 dimensions [31,32].

Foliated BF Theory in 3+1 Dimensions
In this section, we obtain exotic field theories from the foliated BF theory in 3+1 dimensions with one, two, and three foliations.For one foliation, we find that the theory is equivalent to the toric code layers.Our procedure yields a presentation of this theory that we have not seen in literature, and we analyze it in detail.For two foliations, the theory is equivalent to the anisotropic theory with lineons and planons.We find the presentation of the theory studied in [21,22,30] and analyze it in detail.For three foliations, the theory is equivalent to the exotic presentation of the X-Cube model studied in [8,11].We discuss this theory in detail.It is worth noting that we recover the map from fields in the foliated theory to fields in the exotic theory that [18] discusses as a consequence of our procedure.These equivalences between foliated and exotic theories were shown on the lattice in [15].Since we will examine multiple choices of foliation(s), we begin by examining the foliated BF theory in 3+1 dimensions generally.The Lagrangian is As before, we take note of the fields present.a is a one form gauge field with the gauge redundancy b is a two form gauge field with the gauge redundancy A A ∧ e A are foliated 1+1 form gauge fields with redundancy and B A are foliated 1 form gauge fields with redundancy As before, the zero form gauge parameters λ 0 and λ A are compact by dint of being 2π periodic.β A is a zero form gauge parameter compactified by the identification β A ∼ β A + 2π.µ 1 is a one form gauge parameter.It has its own gauge redundancy, since we can redefine it by an exact one form dµ 0 .We compactify µ 0 by forcing it be 2π periodic.
Intuitively, the first two terms describe decoupled layerings of 2+1 dimensional Z N gauge theories, the third term describes 3+1 dimensional Z N gauge theory, and the middle two terms describe the coupling between the two sectors.One can show the following quantities to be quantized: where C (2) is a closed two-manifold, where C L is a one-cycle on the intersection of leaves of different foliations, where S A is a strip whose boundary components are on different leaves of the foliation defined by e A , and where C I is a one-cycle in the intersection of leaves in each foliation for which q A = 0. Let us now discuss some of the operators/defects in the theory.The first is For one foliation in ,e = dz, C L is in the τ − x − y space.For two foliations :e 1 = dx and e 2 = dy, C L is in the τ − z plane.For three foliations: e 1 = dx, e 2 = dy, and e 3 = dz, C L is a in τ i.e. we have a fracton.There are electric holonomies: For one foliation, this holonomy is the identity.For two and three foliations, we obtain lineons.There are also magnetic holonomies (written below for flat foliations e A = dx A ): where S A is a strip whose boundary is the disjoint union of two leaves of the A th foliation.Thus, this is a planon.The quantization conditions imply: We remark that lattice models corresponding to this foliated field theory for the foliation choices we discuss below were constructed in [15].Those authors then obtained more familiar lattice theories by acting with a finite depth local unitary circuit (we note that this keeps the Hamiltonian in the same phase [20]) and treating local stabilizers as constraints.The constraints we enforce by integrating out Lagrange multipliers are continuum versions of those constraints.We also remark that this is the first section in which novel exotic theories are obtained, since [17,18] only discuss the relationship between foliated and exotic field theories for 3 foliations in 3+1 dimensions.

One Foliation: Toric Code Layers
Let e = dz.In components, the Lagrangian is As usual, we integrate out fields that relate foliated and standard gauge fields, that is we integrate out A z τ , b τ x , and b τ y .Since nothing is treated symmetrically, we simply solve for the usual fields in terms of the foliated fields and plug in, giving for i ∈ {x, y}, and We can now rewrite the Lagrangian in the following suggestive form Note that, upon dimensional reduction in z, we obtain the usual BF theory in 2+1 dimensions, as expected.One can show the following periods to be quantized: Let us examine the global symmetries of the theory.It has a defect and a magnetic symmetry, generated by the operators It also has an electric symmetry, generated by We note that the above holonomies are in fact gauge invariant.µ z has the gauge symmetry µ z ∼ µ z + ∂ z β, where β is a zero form.Thus, the argument of the exponent reduces to 2π times the difference between winding numbers at different values of z, which give a trivial holonomy.There is an electric defect that can be defined as expected: It follows from the quantized periods that This is consistent with the results from the foliated side of the duality.9 We use these symmetries to determine the ground state degeneracy of the system on T 3 , represented in the field theory by the dimension of the Hilbert space.It follows from the canonical commutation relations that the symmetry operators satisfy: and similarly for x ↔ y.This implies if we discretize the z axis.This is the result we would expect from dimensional reduction.We note that fracton orders based on layered toric code appear in [33][34][35], which analyze them in terms of infinite component K-matrix Chern-Simons theory.

Two Foliations: Anisotropic Theory with Lineons and Planons
Let e 1 = dx and e 2 = dy.In component form, the Lagrangian is We integrate out b τ x , b τ y , A x τ , and A y τ .Let's examine the consequences.Integrating out whereas integrating out A y τ imposes We do not do any more with these constraints for the time being.Integrating out b τ z gives We note this is solved by where the gauge parameters in the exotic field theory are related to those in the foliated theory as we obtain Let us analyze this theory.The following periods are quantized: The theory has the defects: It has a magnetic symmetry, generated by which is a lineon in the z direction and providing planons in the y-z and x-z planes, respectively 10 .It also has an electric symmetry, generated by giving a lineon in the z direction and providing planons in the y-z and x-z planes, respectively 11 .The quantized periods imply: consistent with results on the foliated side of the duality.
One can obtain the ground state degeneracy on T 3 from the symmetries of the theory.The canonical commutation relations imply: When accounting for constraints (detailed in [30]) and regularizing the x-y plane on on an L x × L y square lattice gives L x + L y − 1 operators.Note that we could have done this calculation exchanging e and m, resulting in another L x + L y − 1 operators and giving a ground state degeneracy of GSD = N 2Lx+2Ly−2 . (3.59)

Three Foliations: The X-Cube Model
Let e 1 = dx, e 2 = dy, and e 3 = dz.In components, the Lagrangian is As before, we use this to solve for b and plug directly into the Lagrangian.Integrating out b τ a gives (∂ We introduce the symmetric gauge field as before, defining and demanding that A ij = A ji .We further define These fields have the gauge redundancy where the gauge parameters in the exotic field theory can be written in terms of those in the foliated theory as Writing a τ = A τ , solving for b ij and A A i as indicated above, and plugging into the remaining Lagrangian gives Note that i and j are symmetric and all values of the indices must be distinct.The following quantities are quantized: Let us discuss this theory in some detail.It has a magnetic defect: which cannot be deformed into spatial directions.We interpret it as a fracton.The theory has a magnetic dipole symmetry in the yz plane generated by where C yz is a 1-cycle in the yz plane.These operators can be deformed in a gauge invariant way in the yz plane, so we identify them with planons.The theory also has electric defects: . Finally, the theory has an electric tensor symmetry in the x direction generated by which gives a lineon in the x direction, since we cannot deform it into other directions in a gauge invariant way.We can construct similar operators out of Âxy and Âzx , yielding lineons in the y and z directions, respectively.It follows from the quantized periods that where C ij i is a 1-cycle in the ij plane that winds once in i.When noting a constraint in each sector (discussed in [11]), this gives L x + L y − 1 N dimensional spaces from the operators in the x-y plane.Working similarly in the other planes gives when the theory is regularized on an L x × L y × L z cubic lattice with periodic boundary conditions in all three directions.

The Z N 2 Magnetic Model
The theories we have examined so far all relate the two form b to the foliated one form B k .For this reason, they are called magnetic models.One can extend Z N by other groups and couple to a G/Z N gauge field to produce more general magnetic models [17].We label these theories by the group G.One example is the G = Z N 2 magnetic model,which we examine in this section.We apply the same procedure as before -integrate out the Lagrange multipliers that impose constraints that relate foliated gauge fields to standard gauge fields -to the theory with one, two, and three flat foliations.For one foliation, we find an exotic field theory for the hybrid toric tode layers.For two foliations, we find an exotic field theory for a hybrid version of the anisotropic theory with lineons and planons.
For three foliations, we find an exotic version of the fractonic hybrid X-Cube model.All of the systems were introduced on the lattice in [24].Their characteristic behavior is that they contain both mobile and fractonic excitations, and these two types of excitations fuse into each other.The relationship between the foliated theory and these systems is documented in [17].We have not seen the exotic field theories we uncover in the literature before.Aside from being an exhibition of our map from foliated to exotic field theories, they are interesting because they provide a simple and generalizable way to capture hybrid fracton physics in exotic field theory.We detail this, in addition to other features of these theories, on a case by case basis.Since we will make various choices of foliation(s) throughout this section, give general details about the theory here.Its Lagrangian is Let's note the gauge fields present.a is a one form with gauge field with the redundancy b is a two form gauge field with the redundancy a ′ is a one form gauge field with the gauge redundancy b ′ is a two form gauge field with the redundancy A A ∧ e A are foliated 1+1 form gauge fields with the redundancy and B A are foliated gauge fields with the redundancy As usual, the zero form gauge parameters λ 0 , λ ′ 0 , λ A ,β A , and χ A are compactified by making them 2π periodic.The one form parameters µ 1 , µ ′ 1 , and λ ′ 1 can all be shifted by exact one forms dµ 0 , dµ ′ 0 , and dΛ 0 , respectively.All of the zero forms so defined are similarly compactified.One can show the following periods are quantized: C and All of the manifolds over which these are integrated are as defined above.The quantities eqs.( are still gauge invariant.However, the above quantization conditions imply that: Moreover, we also have the line a ′ ] (4.15) and the surface Courtesy of the quantization conditions, these satisfy

One Foliation: Hybrid Toric Code Layers
Let e = dz.In components, the Lagrangian is As in section 3.1, we integrate out A z τ ,b τ x , and b τ y and define for i ∈ {x, y}, and so that these fields have the redundancy in eqs.(3.18) and (3.19).Solving for A z k and b xy using the constraints and plugging the results into the leftover Lagrangian yields The following periods are quantized: C This theory has more or less the same operators discussed in section 3.1, with the qualifier stemming from the fact that the quantized periods imply that Moreover, it has a two form symmetry, under which the charged operators are B ′ ] ( where C (2) is a two cycle.It also has a one form symmetry, under which the charged operators are A ′ ], ( where C (1) is a one cycle.Courtesy of the quantization conditions, these obey just as in the foliated field theory.Moreover, the canonical commutation relations imply 1) , C (2) )]W (2)  m (C (2) )W (1)  e (C (1) ) (4.30) where I(C (1) , C (2) ) is the intersection number between the two arguments.This contributes to the ground state degeneracy on T 3 .Noting that C (1) can be generated by a cycle winding x, a cycle winding y, and a cycle winding z.Moreover, C (2) can be generated by a cycle wrapping x-y, a cycle wrapping y-z, and a cycle wrapping x-z.This provides three more N dimensional spaces, so that we have The terms with prefactor − i 2π do not contribute to the ground state degeneracy.What they do contribute are the fusion rules that map N electric planons to a mobile particle and N loops to a magnetic planon.As an example, we demonstrate the former.The Âzτ equation of motion is Now, imagine fusing N of the line operators: dA ′ ] = exp[i Here, d is the exterior derivative in the spatial directions, C xy is a closed curve in the xy plane, D xy is a region diffeomorphic to a disk in the xy plane such that ∂D xy = C xy , and D (2) is a region diffeomorphic to a disk such that ∂D (2) = C (1) .The second equality and fourth equalities are from Stokes' theorem and the third equality follows from the equation of motion.The upshot is that fusing N electric planons gives a mobile particle 12 .This field theory, which we have not seen in this form in the literature, encapsulates the essential physics of the hybrid toric code layers introduced in [24].

Two Foliations: Hybrid Anisotropic Model with Lineons and Planons
Let e 1 = dx and e 2 = dy.The Lagrangian is As in section 3.2, we integrate out b τ x , b τ y , b τ z , A x τ , and A y τ .The constraints from integrating out A x τ and A y τ are addressed as before.The novelty is the constraint from integrating out b τ z , which is We incorporate this by defining and demanding that it be symmetric in its indices.We treat the constraints from integrating out b τ x and b τ y as in section 3. Written in terms of these fields, the Lagrangian is This is not quite the desired result, as Ãxy does not have the gauge redundancy of a symmetric hollow gauge field.To rectify this, define in terms of which the Lagrangian is The following periods are quantized: This theory has the same operators discussed in the section 3.2 are more or less present here, with the qualifier stemming from the fact that the quantized periods now imply: Thus, these operators and defects are the exotic counterparts of those in the foliated field theory.Moreover, it has the one and two form symmetries discussed in eqs.(4.27), (4.28) and (4.30).Just as before, this provides three more N dimensional spaces to the calculation of the ground state degeneracy on T 3 , so that we have The second set of terms do not contribute to the ground state degeneracy.What they do contribute are the fusion rules that map N magnetic lineons to a mobile electric particle and N loops to an electric lineon.As an example, we demonstrate the former.The Âz equation of motion is Now, consider fusing N lineons: A ′ ] = W (1)  e , (4.54) so fusing N lineons gives a mobile particle.Here, C τ z is a 1-cycle in τ z plane, D τ z is a surface diffeomorphic to a disk in the τ z plane such that ∂D τ z = C τ z , and d ′ is the exterior derivative in the τ z plane.C (1) and D (2) are as defined below eqs.(4.28) and (4.33).
The second and fourth equalities are from Stokes' theorem and the third equality is from equation (4.54).The upshot is that fusing N magnetic lineons gives a mobile electric particle.

Three Foliations: Fractonic Hybrid X-Cube Model
Let e 1 = dx = dx 1 , e 2 = dy = dx 2 , and e 3 = dz = dx 3 .The Lagrangian is As in section 3.3, we integrate out b τ i and A i τ for all i.The constraints from integrating out b τ i are treated as in section 3.3, whereas we treat the constraints from integrating out A i τ as in section 4.2.That is, we define Moreover, as in section 3.3, we define: These gauge fields have the gauge redundancy in eqs.(3.66) to (3.68) with the exotic gauge parameters related to the foliated gauge parameters as in equation (3.69).In terms of these, the Lagrangian is Of course, just as in section 4.2, we are note quite done.To rewrite the theory in terms of the desired gauge fields, we need to define so that the Lagrangian takes the form One can show the following periods to be quantized: The defects and operators discussed in the section 3.3 are still present here.However, since the periods are quantized differently, they fuse differently.In particular, we have: Thus, these operators/defects are the exotic counterparts to the holonomies in the foliated field theory.Moreover, it has the one and two form symmetries discussed in eqs.(4.27), (4.28) and (4.30).Just as before, this provides three more N dimensional spaces to the calculation of the ground state degeneracy on T 3 , so that we have The second set of terms do not contribute to the ground state degeneracy.What they do contribute are fusion rules mapping N fractons to a mobile particle and N loops to a lineon.
As an example, we demonstrate the former.Since planons are fracton dipoles, we work with those, expecting a dipole of mobile particles.The Âk(ij) τ equation of motion is Now, imagine fusing N planons (in the x-y plane for definiteness): (dxA ′ x + dyA ′ y )], (4.72) where C xy , D xy , D (2) , and C (1) are defined below equations eqs.(4.28) and (4.33).The second and fourth equalities follow from Stokes' theorem and the third equality follows from the above equation of motion.We see that fusing N planons results in a dipole of mobile particles.Therefore, this simple field theory, which we have not seen in this form in the literature, encapsulates the key physics of the fractonic hybrid X-Cube introduced in [24].

Conclusion and Discussion
In this paper, we introduced a simple recipe for extracting exotic field theories from foliated field theories -integrate out the Lagrange multipliers that relate foliated gauge fields to the standard gauge fields.We applied this recipe to a variety of foliated field theories, uniting a mixture of old and new results.We began with the 2 + 1 dimensional foliated BF theory and map it to the 2 + 1 dimensional exotic BF theory, rediscovering the map between gauge fields in [18].We then moved to the 3 + 1 dimensional foliated BF theory, which we analyzed for 1, 2, and 3 flat foliations.For one foliation we obtain an exotic field theory for the toric code layers that we have not encountered previously in the literature.For two foliations, we obtain the exotic theory for the anisotropic theory with lineons and planons.For three foliations, we obtain the exotic BF theory for the X-Cube model and recreate the map between operators in [18].In all three cases, our map is the natural continuum version of the work done on the lattice in [15].Our next targets were foliated field theories shown in [17] to contain hybrid fracton phenomenology.Our procedure supplies new exotic field theories for the hybrid phases discussed in [24] that give a transparent way to account for hybrid fracton behavior in exotic field theory that begs to be generalized by future work.
Our results lead to a plethora of directions for further work: • Throughout this paper, we examined a number of foliations less than or equal to the number of spatial dimensions.It would be interesting to apply the construction herein to theories with more foliations than spatial directions.In particular, Chamon's model [36] is a four foliated fracton phase [37] that is related to the four foliated X-Cube.Examining four-foliated theories and comparing the result to the field theories discussed in [38,39] would be fruitful.
• The foliations discussed in this paper are all along the x,y, or z direction.It would be interesting to discuss foliations with curved leaves, or foliations with nontrivial Godbillon-Vey invariant.
• In addition to the magnetic models with a closed two form b, [17] discusses electric models with a closed one form a. Obtaining exotic field theories for electric models would expand the scope of foliated field theories from which wee know how to obtain an exotic field theory.[17] uncovers a duality between G electric models and G/Z N magnetic models.It would be fruitful to see how this duality appears in exotic field theory.
• In [32], the authors capture the mobility constraints of the X-Cube by embedding a network of condensation defects in the three dimensional toric code.It might be illustrative to connect that viewpoint to quantum field theory, particularly since defect networks can also encapsulate the mobility constraints of type II fractons.
• The map from foliated to exotic field theory presented in the paper is only true if integrating by parts simply transports partial derivatives and gives minus signs.Generally, this is not true if the manifold on which the field theory lives is not closed.Thus, one should ask which boundary conditions are suitable for the duality in the paper.This could be particularly subtle in light of the fact that BF theories are not gauge invariant on manifolds with a boundary [7,23,38].
• We do not discuss gapless theories at all in this paper, despite them appearing in both foliated [17] and exotic [9,10] settings.Understanding the relationship between gapless foliated and gapless exotic theories is an out standing problem.
• Our recipe for hybrid fracton phenomenology is straightforward and generalizablesimply couple the exotic theory and the BF theory to a theory that doesn't contribute to the ground state degeneracy and enforces the appropriate fusion rules as equations of motion.It would be intriguing to see what hybrid phases one can write down this way that do not involve the X-Cube.Also, [17] contains more complicated magnetic models whose exotic field theory remains to be uncovered.Some of these correspond to non-Abelian hybrid fracton phases, introduced in [25].Moreover, coupling exotic field theories to more familiar field theories seems to be a frontier ripe for exploration.

. 34 )
provided we demand A xy = A yx .This field has the same gauge redundancy as in(2.20).Integrating out b τ x gives ∂ y a z − ∂ z a y − A y z = 0 (3.35) and integrating out b τ y gives ∂ z a x − ∂ x a z + A x z = 0. (3.36) Upon solving for A x y , A y x , A x z , A y z , b zx , and b yz using the constraints, rewriting a z = A z and defining the following: Âxy τ = B x τ − B y τ (3.37) Â = ∂ x B x y − ∂ y B y x − b xy ( , y) N = 1.(3.80) Thus, we see that eqs.(3.76) to (3.79) are the exotic counterparts to eqs.(3.10), (3.11) and (3.13) in the foliated field theory.Let us now discuss the calculation of ground state degeneracy on T 3 from global symmetries.The canonical commutation relations imply fields have the gauge redundancy in eqs.(3.40), (3.41) and (3.68) as before, where the exotic gauge parameters are related to the foliated gauge parameters by (3.43).