Foliated Quantum Field Theory of Fracton Order

We introduce a new kind of foliated quantum field theory (FQFT) of gapped fracton orders in the continuum. FQFT is defined on a manifold with a layered structure given by one or more foliations, which each decompose spacetime into a stack of layers. FQFT involves a new kind of gauge field, a foliated gauge field, which behaves similar to a collection of independent gauge fields on this stack of layers. Gauge invariant operators (and their analogous particle mobilities) are constrained to the intersection of one or more layers from different foliations. The level coefficients are quantized and exhibit a local duality that spatially transforms the coefficients. This duality occurs because the FQFT is a foliated fracton order. That is, the duality can decouple 2+1D gauge theories from the FQFT through a process we dub exfoliation.

We introduce a new kind of foliated quantum field theory (FQFT) of gapped fracton orders in the continuum. FQFT is defined on a manifold with a layered structure given by one or more foliations, which each decompose spacetime into a stack of layers. FQFT involves a new kind of gauge field, a foliated gauge field, which behaves similar to a collection of independent gauge fields on this stack of layers. Gauge invariant operators (and their analogous particle mobilities) are constrained to the intersection of one or more layers from different foliations. The level coefficients are quantized and exhibit a local duality that spatially transforms the coefficients. This duality occurs because the FQFT is a foliated fracton order. That is, the duality can decouple 2+1D gauge theories from the FQFT through a process we dub exfoliation.
In this work, we focus on gapped 1 type-I [4] fracton models that do not have any gauge-invariant fractal operators [42]. The mobility constraints [43] and other important properties [44,45] of these models have a fundamental dependence on a layering structure of spacetime, known as a foliation structure [46], see Fig. 1. Refs. [47][48][49][50] have shown that these fracton phases can be thought of as a topological quantum field theory (TQFT) that is embedded with stacks of interfaces (also called defects) upon which certain anyons are condensed. These interfaces are the so-called leaves (i.e. layers) of the foliation. Therefore, instead of coupling to a metric g µν , these fracton phase are coupled to one or more foliations. For example, the X-cube model [4] on a simple cubic lattice is coupled to three flat foliations, but more generic foliations are also allowed [46,51]. This is in contrast to TQFT (without interfaces or defects), which does not couple to a metric or foliation.
Previous works have uncovered field theories for the X-cube and other gapped fracton models [50,[52][53][54][55][56][57]. In Ref. [50], the X-cube fracton model was generalized to manifolds with arbitrary curved foliations, but formally quantizing the field theory was left as an open problem. Ref. [53] later showed how to formally treat the X-cube field theory from Ref. [52] as a quantum field theory (QFT) with quantized coefficients.
In this work, we wish to quantize the foliated field theory from Ref. [50]. This task is nontrivial and requires new ideas, such as the introduction of a new kind of foliated gauge field, which behaves like a stack of ordinary gauge fields. We will call a QFT with foliated gauge fields a foliated quantum field theory (FQFT).
We also show that the FQFT is a foliated fracton order [44,46,58,59]. Foliated fracton orders have ground states for which a local unitary transformation can decouple 2D topological orders from the ground state. In the FQFT, this translates to using a local IR duality to decoupled 2+1D gauge theories from the FQFT. This is done by giving a coupling constant a piecewise spatial dependence and showing that this spatial dependence can be manipulated by the duality.
In the following, we begin by reviewing how to mathematically describe a foliation using a 1-form foliation field.
We then introduce the FQFT and discuss its gauge invariant operators, level quantization, and foliated fracton order. Some technical details and extended discussions appear in the Appendix. See Ref. [60] for a recorded talk.

I. FOLIATION FIELD
A foliation is a decomposition of a manifold into an infinite number of disjoint lower-dimensional submanifolds called leaves. A common example is to decompose 3+1D spacetime into 3D spacial slices; in this example the codimension-1 leaves can be indexed by the time coordinate. We will describe a codimension-1 foliation using a 1-form foliation field e µ .
The foliation field is analogous to a metric g µν , except e µ describes a foliation geometry instead of a Riemannian geometry. The leaves of the foliation are defined to be the codimension-1 submanifolds that are orthogonal to the foliation field. That is, the tangent vectors v µ of the leaves are in the null space of the foliation field covector: v µ e µ = 0. In order for this definition to work, the foliation field must never be zero [e(x) = 0 ∀x] and it must satisfy the following constraint 2 : More intuition can be obtained by noting that the foliation is invariant under a "gauge transformation" that rescales the foliation field (since this does not affect orthogonality to the leaves): where γ is a scalar function. It is always possible to apply the above transformation such that within an open ball of spacetime, the foliation fields are closed (de = 0) and can be written as the derivative of a scalar function f : e = df . Locally, f can be thought of as a coordinate that indexes the leaves of the foliation, similar to how a time coordinate indexes time slices of spacetime.
In the following, we consider n f ∈ N simultaneous foliations in different directions. We index the different foliations e k by the superscript index k = 1, 2, · · · n f (Fig. 1). Each foliation satisfies Eq. (1) independently: 2 We will use differential form notation throughout this work. In components, Eq. (1) can be written as αβγδ e β ∂γ e δ = 0, where is the Levi-Civita symbol. e k ∧ de k = 0. (We never implicitly sum over repeated foliation k indices; sums over k will always be written explicitly.)

II. FOLIATED QFT
We will study the following foliated QFT (FQFT) 3 : B k and a are 1-form gauge fields. (Note that B k is not a magnetic field in this notation; B k has no dependence on A k .) b is a 2-form gauge field. A k are foliated (1+1)-form gauge fields, which are locally 2-forms that obey the constraint Eq. (4). In Sec. II C, we will show that the physics is equivalent under n k ∼ n k + N and M k , n k , N ∈ Z are quantized level coefficients with m k ≡ n k M k N ∈ Z. (We always assume M k = 0 and N = 0.) n f k=1 sums over the different foliations. Unlike the dynamical gauge fields (A k , B k , a, b), the foliation field e µ is non-dynamical and is not integrated over in the partition function (similar to a static metric g µν ).
If n k = 0, the second term in L describes a 3+1D BF theory (which is a field theory for Z N gauge theory or 3D toric code [63]), while the first term is an FQFT for a stack of infinitesimally-spaced 2+1D BF theories for each foliation, (i.e. a field theory for stacks of Z M k toric codes [64]). When M k = N and n k = 1, the leaves are coupled to the 3+1D BF theory, and the resulting theory describes the ground state Hilbert space 4 of the Z N Xcube model [4,50,52,53] in the limit of infinitesimal lattice spacing. This equivalence can be demonstrated in a number of ways [65] and will be exemplified in Sec. II B. Some intuition from coupled-layer constructions of fracton models applies here as well [66][67][68].

A. Foliated Gauge Field
Foliated Gauge Field -The foliated QFT includes a new kind a gauge field: a foliated (1+1)-form gauge 3 In components, L = field A k for each foliation k. A foliated (1+1)-form gauge field 5 behaves similarly to a stack of independent 1-form gauge fields. This is desirable because when n k = 0, the first term in Eq. (3) should describe a stack of independent 2+1D gauge theories. Locally, a foliated (1+1)-form gauge field A k is a 2-form gauge field that obeys the constraint Eq. (4). Similar to ordinary gauge fields, the exterior derivative dA k is required to be well-defined. Note that this requirement does not put any restriction on the continuity of the foliated gauge field A k between leaves of the foliation. For example if e 1 = dz, then the constraint Eq. (4) implies that A 1 =Ã 1 ∧dz for some 1-formÃ 1 , and A 1 can have arbitrary discontinuities in the z-direction since these discontinuities will not contribute to dA 1 (due to the antisymmetry induced by the wedge product). Furthermore, we allow foliated gauge fields to contain a delta-function onto a leaf. For example if e 1 = dz, then A 1 = x δ(z) dy ∧ dz is allowed. See Appendix B for a more formal definition of foliated gauge fields.
Since the first term in Eq. (3) should describe a stack of 2+1D BF theories for each k with n k = 0, the gauge fields A k and B k should effectively have three components (since the 1-form gauge fields in 2+1D BF theory have three components). Considering again the example e 1 = dz, we indeed see that the constraint Eq. (4) implies that the foliated (1+1)-form has exactly three components: However there is a gauge symmetry B k → α k for an arbitrary foliated (0+1)-form α k , which locally satisfies α k ∧ e k = 0 (i.e. locally α k =α k e k for some scalarα k ); this makes the dz component an unimportant gauge redundancy. Therefore, A k and B k both effectively have 3 components (for each foliation k), as desired.

B. Fractons and Gauge Invariant Operators
Fractons and Gauge Invariant Operators -The set of gauge symmetries determines the set of gauge invariant operators. In ordinary topological QFT (e.g. Chern-Simons theory), gauge invariant operators can be smoothly deformed into any shape. However in a foliated QFT, the gauge invariant operators are often constrained to the intersection one or more leaves of different foliations.
Gauge invariant operators can be interpreted as moving topological excitations around in spacetime. Therefore, the rigidity of the gauge invariant operators is analogous to the mobility constraints of the fracton, lineon, and planon particles.
The gauge transformations of the FQFT are where m k ≡ n k M k N . χ k and λ are arbitrary 0-form gauge fields, while µ is an arbitrary 1-form gauge field. ζ k and α k are foliated (0+1)-form gauge fields. Locally, ζ k are 1-form gauge fields that satisfy the constraint ζ k ∧e k = 0, and similar for α k . Now consider the following string operator: where M F 1 is a 1-dimensional manifold described below. Large gauge transformations imply that the charge q is an integer. A nonlocal "equation of motion" (from integrating out b) shows that W = 1 when q is an integer multiple of N ; see Appendix D 1 for details. Therefore W only depends on q modulo N . After a gauge transformation, But locally, ζ k µ =ζ k e k µ for some scalarζ k . Therefore the second term is gauge invariant if for each k with m k = 0, the loop M F 1 is supported on a single leaf of the k th foliation [since then v µ m k ζ k µ ∝ v µ e k µ and v µ e k µ = 0 by the definition of e k above Eq. (1)]. Therefore, if there are n foliations with m k = 0, then the string operator [Eq. (6)] will be bound to the intersection of n leaves.
This constrains the string operator to a codimension-n submanifold (if the foliations are transverse 6 at every point), and the particle transported by this string operator is bound to the same codimension-n submanifold. For example, if there are three spatial 7 foliations (that are transverse at every point) as in Fig. 1, then this string operator can move fractons in time, but it can not not move fractons spatially. An isolated fracton is constrained to the intersection of three leaves, see and M k = N , this fracton is equivalent to the X-cube fracton [4]. Such a foliation (three everywhere-transverse foliations of a 3-manifold) is called a total foliation. It has been proven that all compact orientable 3-manifolds admit a total foliation [69], which implies that all such manifolds admit an FQFT with fractons. Now consider a different string operator: Large gauge transformations imply that the charges shows that M L 1 is supported on the intersection of n leaves, where n is the number of foliations k with nonzero q k = 0. Finally, the B k → B k − n k µ gauge transformation implies that k q k n k = 0. Therefore, the set of allowed charge vectors forms an abelian group G = {q ∈ Z n f | k q k n k = 0}.
A nonlocal "equation of motion" (from integrating out A k ) shows [70] that T = 1 when q k ∈ M k Z (and k q k n k = 0). Thus, the trivial charge vectors form a subgroup N = {q ∈ G | q k ∈ M k Z} G. Since both of these groups (G and N ) are isomorphic to In the Z N X-cube model example with three foliations and n k = 1 and M k = N , the allowed charge vectors are spanned by q (X) k = (0, 1, −1) and q (Y) k = (−1, 0, 1). These particles are bound to a pair of leaves (Fig. 2b) and are therefore restricted to spatially only move along 1D lines (for spatial foliations). These are Z N Xcube lineons. For the standard three flat foliations (e 1 = dx, e 2 = dy, e 3 = dz), q (X) k and q (Y) k can move only in the X and Y directions, respectively; and their sum q 1, 0) can only move in the Z direction. This is analogous to the X-cube model where the composition of an X-axis lineon with a Y-axis lineon results in a Z-axis lineon. Note that even if a charge vector has three nonzero components (q k = 0), it is not a fracton; instead, it is the composition of at most two lineons: q Even for an arbitrary number of foliations with arbitrary coefficients n k , it is always possible to decompose a charge vector q k into lineon and planon charges (which have at most two nonzero elements q k = 0). See Appendix E for a proof. Therefore, the string operator T only describes lineons (or composites of lineons and planons), but never fractons.
Other gauge invariant operators include A k denotes an integral of A k over a 2manifold M P 2 , which can have boundaries but where each boundary must be supported on a single leaf of the foliation k, as in Fig. 2c. T wraps a string excitation around M 2 . In the X-cube model example, T measures the number of fractons. In the X-cube lattice model, this operator would look like a complicated operator that wraps a loop of many lineon excitations around M 2 . 8 In the X-cube example, W moves a pair of X-cube fractons 9 around the top and bottom boundaries of the blue 2manifold M P 2 shown in Fig. 2c. See Appendix C for more general operators and a different approach to understanding the particle mobility constraints.

C. Level Quantization
Level Quantization -Now we study the quantization of the level coefficients M k , n k , and N . First note that m k ≡ n k M K N and n k appear as coefficients in the gauge transformations [Eq. (5)] of compact gauge fields (a and B k ); this implies that m k , n k ∈ Z.
Consider how the Lagrangian transforms under the gauge transformations [Eq. (5)]: Locally, the new terms are total derivatives. But since these are derivatives of gauge fields, their integral over a closed manifold can be nonzero. However the integral is quantized such that the change in the action is an integer multiple of 2πM k plus an integer multiple of 2πN . Therefore, the partition function Z = e i L is gauge The equations of motion that result from integrating out a and B k imply [70] that locally db = dA k = 0 and globally the operators in Eq. (8) are quantized: Together, these local and global equations of motion show that the b ∧ A k term in the FQFT action [Eq. (3)] is quantized as follows: This implies that the action is invariant under the following identification n k ∼ n k + N .

III. EXFOLIATION
Ref. [46] showed that a finite-depth local unitary transformation can map between the ground states of (1) an X-cube model of lattice length L 0 in one direction, and (2) an X-cube model of lattice length L 0 − 1 in the same direction along with a decoupled layer of toric code (and some trivial decoupled qubits). We will refer to this process as exfoliation. In high-energy terminology, exfoliation corresponds to a local IR duality that decouples 2+1D gauge theories from a 3D FQFT. A fracton order that admits exfoliation is said to be a foliated fracton order [46,58,59]. The X-cube model is a foliated fracton order that is foliated by toric code layers [46,71].
We now show that the FQFT is a foliated fracton order by exfoliating 2+1D BF theories. For simplicity, consider a flat foliation e 1 = dz (which may coexist with other foliations e k ). We want to demonstrate a duality from an FQFT with constant n 1 ∈ Z to an FQFT with a spatiallydependentñ 1 (z) that is zero within z 1 < z < z 2 : On the right-hand-side of the duality, the A 1 and B 1 fields within z 1 < z < z 2 are decoupled from the rest of the fields. The equations of motion for A 1 and B 1 are dA 1 = dB 1 ∧ e 1 = 0 within z 1 < z < z 2 . These equations of motion do not contain z-derivatives ∂ z [recall A 1 ∧ e 1 = 0 from Eq. (4)], which shows that A 1 and B 1 at different z are completely decoupled. These decoupled fields constitute an exfoliated stack of infinitesimallyspaced 2+1D BF theories. The duality results from the following transformation: We are using a notation where the integrals above are defined as 10 In order for this definition to make sense, we have implicitly chosen a flat connection to parallel transport the gauge fields. B 1 (z 2 ) is shorthand for B 1 (t, x, y, z 2 ), just as B 1 is shorthand 10 In Eq. (13), we use the convention that integrals do not pick up delta functions on their end points; e.g. 1 0 δ(x)dx = 0.
for B 1 (t, x, y, z). In Appendix G, we show that the above transformation transforms the equations of motion according to Eq. (12), which demonstrates the exfoliation duality. See Fig. 3 for an example.

IV. CONCLUSION
We have introduced a generic foliated QFT (FQFT) that is capable of describing a large class of foliated gapped fracton models on foliated manifolds. We also demonstrated a novel duality that spatially transforms the level coefficients, which shows that the FQFT is a foliated fracton order [46,58,59].
We thank Shu-Heng Shao, Nathan Seiberg, Ho Tat Lam, Pranay Gorantla, Po-Shen Hsin, Anton Kapustin, Xie Chen, and Wilbur Shirley for helpful discussion. K.S. is supported by the Walter Burke Institute for Theoretical Physics at Caltech. Here, we review the definition of a 1-form gauge field. Mathematically, a 1-form gauge field with gauge group G = U (1) is a connection on M × G (or more generally a G-bundle E → M), where M is the spacetime manifold. [92,93].
This can be made more explicit by considering a good open cover of the spacetime manifold M; i.e. consider a collection of sets U i ⊂ M that cover M (i.e. ∪ i U i = M) such that finite intersections U i1 ∩ U i2 ∩ · · · ∩ U in are diffeomorphic to an open ball. A 1-form gauge field can then be specified by the following data and constraints [94]: (1) The gauge field is locally defined on each U i by a 1-form A (i) . 11 (2) On nonempty overlaps U i ∩ U j (depicted in yellow below), the two locally defined fields A (i) and A (j) must be equal up to a gauge transformation: where g (ij) are called transition functions. (3) On nonempty triple-overlaps U i ∩ U j ∩ U k (depicted in yellow below), the transition functions must satisfy the cocycle condition up to an integer multiple of 2π: A 0-form gauge field θ can be similarly defined by a 0form θ (i) : U i → R on each U i where θ (i) − θ (j) ∈ 2πZ on overlaps U i ∩ U j . Thus, θ could alternatively be defined as a U (1)-valued functionθ : M → U (1).
See also the beginning of Ref. [94] for another a review of q-form gauge fields and Section 2.1 of Ref. [95] for an explicit example of how to define integrals of gauge fields.

Example
Here, we review a simple example of a field configuration for a trivial 2π flux. Consider BF theory on a 2+1D torus: L = N 2π B ∧ dA where A and B are 1-form gauge fields. Decompose the 3-torus as M t × M x × M y with lengths l t , l x , and l y , where M x is a circle with coordinate x ∈ [0, l x ), and similar for M t and M y . Then a 2π flux that is evenly spread throughout space will have dA = 2π l x l y dx ∧ dy (A3) 11 The parenthesis in A (i) are used to emphasize that i is an index for a spacetime patch U i ; i is not a coordinate index µ = 0, 1, 2, 3. To formally specify the gauge field A, first choose an open cover 12 given by (Fig. 4): . Now the gauge field can be defined by with transition functions g (12) = g (14) = g (32) = 0 0 < x < lx 2 2π ly y lx 2 < x < l x and 0 ≤ y < ly 2 2π ly (y − l y ) lx 2 < x < l x and ly 2 < y ≤ l y g (13) = g (24) = 0 Note that A (i) and g (ij) are continuous and satisfy Eq. (A1) and (A2). Also note that if we rescale A (i) and g (ij) by some constant α ∈ R, then Eq. (A2) will only be satisfied if α ∈ Z. Therefore, the total flux dA must be an integer multiple of 2π, which is physically trivial.

Appendix B: Foliated Gauge Fields
Here, we provide a more formal definition of foliated gauge fields. See Appendix A for a review of ordinary gauge fields.
We will provide two definitions, which we believe are equivalent. The first definition is that a foliated gauge field A is given by an ordinary gauge field A on each leaf of a foliation. Then the integral of a foliated (q + 1)form gauge field A over a foliated (q + 1)-dimensional manifold M is given by the infinite sum of integrals over each leaf ⊂ M of the foliation: M A ≡ A . We now provide a second definition, which avoids the infinite summation over leaves. This definition is also simpler locally (as it reduces to just a constrained 2-form gauge field). We use this second definition throughout the rest of this text. Consider a good open cover of sets U i ⊂ M [as defined above Eq. (A1)] that cover the spacetime manifold M, which is foliated using a foliation field e, as defined in Sec. I. A foliated (1+1)-form gauge field is defined by the following data and constraints: (1) The foliated gauge field is locally defined on each U i by a 2-form field A (i) that obeys the constraint A (i) ∧ e = 0 [as in Eq. (4)]. (2) On nonempty overlaps U i ∩ U j , the two locally defined fields A (i) and A (j) must be equal up to a gauge transformation: where g (ij) is a foliated (0+1)-form transition function that obeys g (ij) ∧ e = 0. (3) On nonempty triple-overlaps U i ∩ U j ∩ U k , these transition functions must satisfy a foliated cocycle condition: where s is any 1D manifold (possibly with boundaries) transverse 13 to the foliation. An example is depicted in the graphic, with s drawn as a blue line.

Example
Here, we demonstrate a foliated analog of the example in Appendix A 1. That is, we wish to describe a field configuration for a trivial 2π flux on a single leaf of a foliation. We will consider the following FQFT on a 3+1D torus: where A is a foliated (1+1)-form gauge field and B is a 1-form gauge field. This FQFT describes a foliation of 2+1D BF theories.
For the first definition, a 2π flux that is evenly spread throughout a leaf 0 of the foliation will have: where indexes the different leaves of the foliation. A 0 can then be defined as in Appendix A 1. Now consider the second foliated gauge field definition. For simplicity, consider a flat foliation with e = dz. Decompose the 4-torus as M t × M x × M y × M z with lengths l t , l x , l y , and l z , where M x is a circle with coordinate x ∈ [0, l x ), and similar for M t , M y , and M z . A 2π flux that is evenly spread throughout a leaf (at z = z 0 ) of the foliation will have: To formally specify the foliated gauge field A, first choose an open cover given by (Fig. 4): Now the foliated gauge field can be defined by 13 A 1-dimensional manifold is transverse to a foliation if it is never tangent to a leaf.

Appendix C: Mobility Constraints and Currents
In Sec. II B, we studied the rigidity of the gauge invariant operators. This rigidity is analogous to the particle mobility constrains characteristic of fracton models. Consider a more general operator of the form e i L where: J k and i are 2-forms; I k is a (2+1)-form (i.e. I k ∧e k = 0); and j is a 3-form. J k , I k , j, and i can be thought of as current sources that parameterize the generic operator e i L . e i L is only gauge invariant if the following mobility constraints are satisfied: These five constraints result from imposing gauge invariance under the ζ k , χ k , α k , λ, and µ transformations in Eq. (5), respectively. The local foliation field constraint A k ∧ e k = 0 [Eq. (4)] also results in the following redundancy: J k → J k + φ k ∧ e k , where φ k is an arbitrary 1-form. This gives J k the same number of degrees of freedom as a (2+1)-form. [96] When the FQFT describes Z N X-cube (i.e. when n k = 1 and M k = N with three foliations): j is the fracton current, J is a fracton dipole current, linear combinations of I currents result in lineons, and i is a current for string excitations which do not appear in the X-cube model 14 .
Eq. (C2) tells us that any j current that passes through a leaf of a foliation must be compensated by 14 However, it is possible to map the i current to a string of many lineons using the mappings in Sections 3.3.2 and 3.3.3 of Ref. [50].
the divergence of J current. This is analogous to the Xcube model where moving a fracton (j current) requires creating fracton dipoles (J divergence). Eq. (C5) implies that the i current describes string excitations. If there are no string excitations (i.e. if di = 0), then Eq. (C5) implies that k I k = 0. This implies that I current must come in pairs. I k ∧ e k = 0 [Eq. (C3)] implies that the I current can only move along a leaf of the k th foliation. But since I current must come in pairs for different foliations k, a particle of I current must be bound to two leaves for two different foliations, which implies that I describes currents of lineons.

Appendix D: Equations of Motion
The equations of motion for the FQFT Lagrangian coupled to source currents, L + L [Eqs. (3) and (C1)], are given below:

Quantized Integrals
Since the gauge fields are compact, there are also nonlocal "equations of motion" that result in quantized integrals.
For example, let us derive the following quantized period from the main text [Eq. (10)]: where M 2 is a closed 2-manifold. If M 2 is contractible, then M2 b = 0 by the local equation of motion db = 0 (D3). Consider a simple example of non-contractible M 2 on a spacetime manifold that is an l t × l x × l y × l z 4-torus. Let M 2 be a tz-plane. Now consider summing over field configurations with flux for all Q ∈ Z (similar to the example in Appendix A 1). Summing over this subset of field configurations shows that the partition function is zero unless the following integral is quantized: where the last equality follows because the integral of b over any tz-plane will be equal due to the equation of motion db = 0 (D3). This demonstrates the quantization (D5).
We now derive the other quantized period in Eq. (10): Consider the simple but nontrivial example where M P 2 is a tz-plane of a spacetime 4-torus, and suppose that the first foliation field is e 1 = dz. Then similar to Eq. (D6), we can sum over fluxes dB 1 = Q 2π lxly dx ∧ dy for all Q ∈ Z and apply dA 1 = 0 [Eq. (D2)] to derive Eq. (D7) with k = 1.
We now derive the following quantized period 15 : where M F 1 is supported on a single leaf for each foliation with n k = 0 [as in Eq. (6)]. Consider the simple but nontrivial example where M F 1 is a loop around a periodic time direction and centered at the origin of the spatial manifold R 3 . Then Eq. (D8) will result from summing over fluxes db = Q 2π δ 3 (x) dx ∧ dy ∧ dz for all Q ∈ Z and choosing B k such that dB k + n k b = 0. These fluxes can be realized by b = −Q 1 2 d(cos θ) ∧ dφ and B k = n k Q 1 2 (cos θ − 1) dφ in spherical coordinates. Summing over this subset of field configurations shows that the partition function is zero unless L ∈ 2πZ, where This demonstrates the quantization Eq. (D8).
Finally, we derive the following quantized period 15 : 15 We will only demonstrate quantization of Eq. (D8) and (D9) when M F 1 and M L 1 are removed from the respective spacetimes. This means that the gauge fields will not have to be well-defined or continuous on M F 1 or M L 1 . This is sufficient for demonstrating the operator quantization in Sec. II B of the main text.
where k q k n k = 0 and M L 1 is supported on a single leaf for each foliation with q k = 0 [as in Eq. (7)]. Consider the simple but nontrivial example where M L 1 is a loop around a periodic time direction and centered at the origin of the spatial manifold R 3 . Then Eq. (D9) will result from summing over fluxes dA k = q k M k F Q where F Q = Q 2π δ 3 (x) dx ∧ dy ∧ dz for each Q ∈ Z with a µ chosen such that da + k m k A k = 0. To realize the flux dA k , consider the example foliation e 1 = dz; N F Q k q k n k = 0; therefore, it is possible to choose a µ such that da + k m k A k = 0. Summing over this subset of field configurations shows that the partition function is zero unless the following integral is quantized:

Appendix E: Lineon Operator
Consider the string operator T in Eq. (7) with a charge vector q k such that k q k n k = 0. Below, we prove that this charge vector can always be decomposed into a sum of lineon and planon charge vectors (which have at most two nonzero elements).
To prove this, first extract all planon charges 16 q (k ) k ≡ q k δ k,k from the charge vector q k : k ∈KP sums over all foliations k such that q k = 0 and n k = 0. We are left with a new charge vector q k , such that q k = 0 for all k with n k = 0. Next, we show that q k can be decomposed into lineons.
If q k has at most two nonzero components, then q k is a lineon and the proof is complete. Otherwise, without loss of generality (by reordering the foliations k), assume that q 1 = 0. Next, we show that q k can be decomposed into q k and lineon charge vectors q (1,k ) k : k ∈KL only sums over foliations k = 2, 3, · · · , n f such that q k = 0. q (1,k ) k = 0 only for k = 1 and k = k . Also note that all charge vectors in this proof (q k , q k , q k , q (k ) k , and q (1,k ) k ) are valid charge vectors (i.e. k q k n k = 0, and similar for the other charge vectors). We want to choose the q (1,k ) k such that q 1 = 0 and q k = 0 for 16 δ k,k = 1 if k = k else δ k,k = 0. each k with q k = 0. Then q k will have at least one more zero element than q k . Thus, we can complete the proof by repeatedly reapplying the logic of this paragraph (with q k → q k ) until q k is a lineon with two nonzero components.
We now just need to show that the decomposition in Eq. (E2) is possible. Without loss of generality, assume n k = 0 and q k = 0 for all foliations k (by just ignoring foliations k for which this is not true). Let r k,k ≡ n k gcd(n k , n k ) (E4) for some integers Q (k ) . gcd denotes the greatest common divisor.
Appendix F: Entanglement RG Entanglement RG [97] studies coarse graining by using a local unitary transformation to decouple degrees of freedom from a ground state.
For example, a local unitary can be used to coarsegrain the ground state (GS) of toric code on a periodic 2L × 2L lattice to the toric code GS on a periodic L × L lattice along with decoupled qubits [98]: In a 3D foliated fracton order, the local unitary decouples 2D topological orders in addition to decoupled qubits. For example, slightly coarse-graining the X-cube model in one direction exfoliates a decoupled layer of toric code [46,99]: U |L x × L y × L z X-cube GS = (F2) |L x × L y × (L z − 1) X-cube GS ⊗ |L x × L y toric code GS ⊗ | ↑ · · · ↑ ×LxLy Entanglement RG is convenient for exactly solvable lattice models since the RG can often be done exactly using a simple formalism. Entanglement RG is also useful because it only discards degrees of freedom after they have been explicitly decoupled. This is in contrast to Wilsonian RG, where one could (in principal) accidentally integrate out important degrees of freedom. For z < z 1 or z > z 2 , the equations of motion do not transforms since the duality transformation is trivial in this region of spacetime. Therefore, we have shown that the duality transformation (13) transforms the equations of motion (D1)-(D4) by n 1 ↔ñ 1 (z) [Eq. (12)].