Non-Abelian Gauged Fractonic Matter Field Theory: New Sigma Models, Superfluids and Vortices

By gauging a higher-moment polynomial global symmetry and a discrete charge conjugation (i.e., particle-hole) symmetry (mutually non-commutative) coupled to matter fields, we derive a new class of higher-rank tensor non-abelian gauge field theory with dynamically gauged matter fields: Non-abelian gauged matters interact with a hybrid class of higher-rank (symmetric or generic non-symmetric) tensor gauge theory and anti-symmetric tensor topological field theory, generalizing [arXiv:1909.13879, 1911.01804]'s theory. We also apply a quantum phase transition similar to that between insulator v.s. superfluid/superconductivity (U(1) symmetry disordered phase described by a topological gauge theory or a disordered Sigma model v.s. U(1) global/gauge symmetry-breaking ordered phase described by a Sigma model with a U(1) target space underlying Goldstone modes): We can regard our tensor gauge theories as disordered phases, and we transient to their new ordered phases by deriving new Sigma models in continuum field theories. While one low energy theory is captured by degrees of freedom of rotor or scalar modes, another side of low energy theory has vortices and superfluids - we explore non-abelian vortices (two types of vortices mutually interacting non-commutatively) beyond an ordinary group structure and their Cauchy-Riemann relation.


Introduction and Overview of Previous Works
Fracton order (see a recent review [1] in condensed matter) concerns new conservation laws imposed on the energetic excitations (such that the particle excitations are known as fractons) of quantum systems which have significant restrictions on their mobility: 1. Excitations cannot move without creating additional excitations (commonly known as fractons), 2. Excitations can only move in certain subdimensional or subsystem directions (for 0-dimensional excitations known as subdimensional particles).
The origins of such constraints are new conservation laws from conserved quantities of higher-moments, including dipole moments [2] (relevant for a vector global symmetry in field theory [3,4]), quadrupole moments, or generalized multipole moment (relevant for the so-called the polynomial global symmetry [4][5][6] or the polynomial shift symmetries [7] in field theory), etc. The composite excitation of each mobilityrestricted excitations are however mobile. The mobility constraint of fracton phases is also related to quantum glassy dynamics [8,9]. Follow the previous work of Ref. [4,6], motivated by the fracton order in condensed matter [1], we continue extending and developing this framework by including the dynamically gauged matter fields in the higher-rank tensor gauge theory in a d + 1 dimensional spacetime (e.g. d + 1d, over a flat spacetime manifold M d+1 , and we should focus on Cartesian coordinates R d+1 ). 1 The important new ingredient in our present work is that the gauge structure can be non-commutative (i.e., the so-called non-abelian), while still coupling to the matter fields -thus a partial goal of our present work is to derive a new non-abelian tensor gauged fracton field theory (that has gauge interactions also coupling with gauged matter fields).
To recall, the field-theoretic models given in Ref. [4,6] offer some unified features that we may summarize via examples that also connect to the literature: 1. An ungauged matter field theory of higher-moment global symmetry without gauge fields (e.g., an ungauged abelian theory with a degree-0 ordinary symmetry that encodes Schrödinger [10] or Klein-Gordon type field theory [11,12] (see Sec. 2.1), with a degree-1 polynomial symmetry pioneered in Pretko's work [2] (see Sec. 2.2) and its general higher-moment degree-(m − 1) polynomial generalization [5,6] (see Sec. 2.3)): For example, there is a field theory captured by the Lagrangian term with a covariant derivate term (P i 1 ,··· ,i m (Φ, · · · , ∂ m Φ)) or P J given in Sec. 2.1 and Sec. 3.1 of Ref. [6]. The schematic path integrals Z are: The dynamical complex scalar fields Φ := Φ(x) = Φ( x, t) and Φ † = Φ † (x) = Φ † ( x, t) ∈ C (1.2) are summed over in a schematic path integral. The Lagrangian term and Z are invariant under the global symmetry transformation.
But a non-abelian version of fractonic theory is not much explored. Recent progress on non-abelian fracton orders from Ref. [18][19][20] are mostly built from lattice models with a discrete gauge group (or a discrete gauge structure in general).
We will take an alternative route to non-abelian fracton via the field theory. A rank-2 non-abelian higherrank tensor gauge theory with a continuous gauge structure is proposed firstly in Ref. [4]. The most general form of rank-m non-abelian higher-rank tensor gauge theory is given by a schematic path integral in Ref. [6]'s Sec. The cocycle ω d+1 ∈ H d+1 ((Z C 2 ) N , R/Z) is a group cohomology data [21] that we can take the continuum topological quantum field theory (TQFT) formulation of discrete gauge theory (see References therein [22][23][24][25] and the overview [4]). The I is an index for specifying the different copies/layers of tensor gauge theories, the cocycle ω d+1 couples different copies/layers of tensor gauge theories together. Thus the cocycle ω d+1 gives rise to the interlayer interaction effects. The index I may be neglected for simplicity below. The real-valued abelian gauge field strength F µ,ν,i 2 ,··· ,i m ∈ R is promoted into a new complexvalued non-abelian gauge field strengthF c µ,ν,i 2 ,··· ,i m ∈ C after gauging a discrete charge conjugation Z C 2 (i.e., particle-hole) symmetry [4,6]: Here are the field contents: • The A ∈ R can be chosen to be a fully-symmetric rank-m real-valued tensor gauge field.
• The B ∈ R is a (d−1)-th Z 2 -cohomology class in terms of Z 2 -discrete gauge theory, or in the continuum formulated as a (d − 1)-form (an anti-symmetric rank-(d − 1) tensor) real-valued gauge field. The B plays the role of a Lagrangian multiplier to set C to be flat.
• The C ∈ R is a Z 2 -cohomology class in terms of Z 2 -discrete gauge theory, or in the continuum formulated as a 1-form (a rank-1 tensor) real-valued gauge field.
3. An abelian gauge theory coupling to gauged matter field : This is pioneered in Pretko's [2] for the rank-2 tensor fields, while we can use the most general form for the rank-m tensor gauge field A = A i 1 ,i 2 ,··· ,i m given in Ref. [6] Sec. 2.1's and Sec. 3.1's schematic path integral: The D A [{Φ}] := R ≡ P − igAQ is defined in Sec. 3.1 of Ref. [6]. Here P and Q are polynomials of Φ and its differential of ∂ Φ for some power of . Here P and Q are uniquely determined by the polynomial Q(x) in the higher-moment global symmetry shown in Ref. [6]. We denote such a polynomial symmetry as U(1) poly following [6], see a review in Sec. 2.3.1 What else topics have not yet been done in the literature but should be formulated? We will focus on these two open issues: 1. A non-abelian gauge theory coupling to gauged matter field : Previous works only did the abelian gauged matter theory, or the non-abelian gauge theory without coupling to (fractonic) matter fields [4,6]. In Sec 2, we provide a systematic framework for non-abelian gauged fractonic matter field theories.
In order to facilitate such a non-abelian gauged matter formulation, we sometimes trade a single complex For the U(1) polynomial symmetry viewpoint, the Φ ∈ C is more natural. The Z C 2 (particle-hole or particle-anti-particle symmetry) transformation acts on Φ → Φ † in the complex U(1) basis, but the Z C 2 acts on the 2-component field naturally as: To introduce the non-abelian gauge coupling to the matter fields, we will need to introduce several new types of gauge derivatives. 2 For example, even for the simplest degree-0 polynomial symmetry with a rank-1 tensor gauge field (1-form gauge field A µ ), we require: The first line D c,Im µ has the C gauge field that only couples to the charged matter Φ Im (the imaginary component) under Z C 2 . The second line D A µ is the gauge covariant derivative of 1-form gauge field A µ after gauging the ordinary 0-form U(1) symmetry (the degree-0 polynomial symmetry). The third line D A,c,Im µ Φ shows the gauge derivative on Φ involving both the U(1) Z C 2 = O(2) gauge fields. The forth and the fith line shows that for the fields charged under Z C 2 (e.g. the rank-m symmetric tensor A ν,i 2 ,··· ,i m → −A ν,i 2 ,··· ,i m is charged under Z C 2 ), then the gauge derivative is D c µ . The g c is a Z C 2 gauge coupling denoted explicitly for the convenience.
We will present explicit examples to gauge both higher-moment and charge conjugation global symmetries including the matter in Sec. 2.

2.
A new type of Sigma model : We formulate a new type of Sigma model that can move between the ordered and disordered phases of these higher-rank non-abelian tensor field theories with fully gauged fractonic matter. 3 Similar to the familiar quantum phase transition between insulator v.s. superfluid/superconductivity [27][28][29] (U(1) symmetry disorder described by a topological gauge theory or a disordered Sigma model v.s. U(1) global/gauge symmetry-breaking order described by a Sigma model with a U(1) target space with Goldstone modes), we can regard our tensor gauge theory as a disordered phase, and we drive to its new ordered phase by deriving a new Sigma model in terms of continuum field theory.
Very recently, the superfluid and vortices of an abelian version of pure fractonic theories (without gauge fields) are studied in [30]. Two new ingredients in our work, which are not present in Ref. [30], are the facts that we include the gauge field interactions (thus we include the additional long-range entanglements) and we also include the non-abelian gauge-matter interactions. Suppose we want to construct a field theory that preserves a degree-0 polynomial symmetry with a polynomial Q(x) = Λ 0 . Then a complex scalar field transforms as Φ : while its log transforms as Take the derivative ∂ x i := ∂ i respect to coordinates on both sides, we can eliminate ∂ i Λ 0 thus we get an invariant term: This means ∂ i log Φ is invariant under the global symmetry transformation. We can also define Namely, both the numerator P i (Φ, ∂Φ) and Φ are global-covariant under Φ → e i Q(x) Φ, in order to maintain the ∂ i log Φ to be invariant. Here P i (Φ, ∂Φ) denotes some functional P i that depends on fields Φ or its derivative ∂Φ.
For convenience, we will call such construction a global-covariant 1-derivative to facilitate its further generalization later. We also construct a globally invariant Lagrangian that contains a potential term V (|Φ| 2 ) and a kinetic term 4 In this way, based on the systematic method of Ref. [6], we can re-derive a Lagrangian formulation of Schrödinger equation in 1925 [10] and Klein-Gordon theory in 1926 [11,12] for complex scalar fields.

Gauge-covariant 1-derivative
To gauge a degree-0 polynomial symmetry, we rewrite Q(x) as a local gauge parameter η(x), (2.10) To obtain a gauge invariant term, we can pair the gauge-covariant operator with its complex conjugation so to obtain a gauge invariant Lagrangian (2.14)

Gauge-covariant non-abelian rank-2 field strength
Notice that in the pure matter theory Eq. (2.6) without gauge fields, we already have a degree-0 U(1) global symmetry and a Z C 2 discrete charge conjugation (i.e., particle-hole) symmetry: which makes Eq. (2.6) invariant. It is easy to see that the symmetry group structure is a non-abelian group which acts on the Φ non-commutatively: After we dynamically gauge the U(1) symmetry to obtain Eq. (2.14), we can still keep Z C 2 discrete charge conjugation (i.e., particle-hole) symmetry intact which acts on gauge fields as (2.18) If we fully gauge U(1) Z C 2 = O(2), we get a non-abelian O(2) gauge transformations which acts also on gauge fields non-commutatively: By promoting the global Z C 2 to a local symmetry, we introduce a new 1-form Z C 2 -gauge field C coupling to the 0-form symmetry Z C 2 -charged object A i with a new g c coupling. The Z C 2 local gauge transformation is: Note A i 1 ∈ R is in real-valued, so a generic e i γc(x) complexifies the A i . Thus we restrict gauge transformation to be only Z C 2 -gauged (not U(1) C -gauged) (2.20) so γ c (x) is an integer and A i → ±A i stays in real. In the continuum field theory, the restrict gauge transformation is done by coupling to a level-2 BF theory (the Z C 2 -gauge theory) [4,6]. Thus γ c (x) jumps between even or odd integers in Z, while the Z C 2 -gauge transformation can be suitably formulated on a lattice. We can also directly express the above on a triangulable spacetime manifold or a simplicial complex.
Approach 1: Gauge 0-form Z C 2 -symmetry of 1-form gauge field A Follow Ref. [4], we also define a new covariant derivative with respect to Z C 2 : We obtain a combined O(2) gauge transformation of A j , .
Ref. [4,6] defines a rank-2 non-abelian field strength aŝ The above rank-2 field strength utilizes the viewpoint of gauging 0-form Z C 2 -symmetry of 1-form gauge field A. However, becauseF c i 1 ,i 2 is a O(2) field strength, we can writeF c i 1 ,i 2 in a conventional way as a 2 × 2 matrix like Yang-Mills [31] did. We will puesue this alternative way in the next paragraph.

Approach 2: Non-abelian O(2) field strength
Earlier from the U(1) symmetry transformation, a single complex component is a more natural view. From the O(2) = SO(2) × Z C 2 view, the 2-component real scalar field (Φ Re ∈ R, Φ Im ∈ R) is natural such that the U U(1) and U Z C 2 symmetry transforms the 2-component field as: Let A gauge field be the generator of U U(1) , and C gauge field be the generator of U Z C 2 . We can write down the non-abelian O(2) gauge field X and field strengthF X with a Lie algebra generator as: 5 Here we use the fact that dC is locally flat.
The Yang-Mills O(2) field strength kinetic term fromF X is proportional to: In comparison, the rank-2 non-Abelian field strengthF c µν defined in Ref. [6] and Eq. (2.23) outputŝ Thus two approaches agree on the Yang-Mills Lagrangian Gauge-covariant non-abelian rank-2 field strength Under the O(2) gauge transformation, wee can explicitly check that the non-abelian rank-2 field strengthF c µν is gauge covariant: where we list down the leading order omitting the potentially higher power of η and γ c terms. Note that: where we need to impose the locally flat condition for the Z C 2 gauge field in the last equality. 6 Thus the gauge covariance is true since we show 5 Readers may wonder whether the O(2) Lie algebra generator needs to be traceless. There are however two facts: (1). It is known that Lie algebra generators of a semi simple Lie algebra must be traceless. A Lie algebra is semisimple if it is a direct sum of simple Lie algebras, i.e., non-abelian Lie algebras g whose only ideals are 0 and g itself. However, a onedimensional Lie algebra (which is necessarily abelian) is by definition not considered a simple Lie algebra, although such an algebra has no nontrivial ideals. Thus, one-dimensional algebras are not allowed as summands in a semisimple Lie algebra. (2). The C is a discrete Z C 2 1-form gauge field so dC is locally flat. Later on we need to impose the condition to show gauge covariance of field strength. (In general we do not have to impose equations of motion to show gauge invariance, although in the case with the Z C 2 -gauge field C, we do require its locally flatness for gauge covariance ofF c i 1 ,i 2 orFX .) Previous work [4,6] does not couple to non-abelian gauge fields to matter field. In this work, we propose a systematic method to generate non-abelian gauged matter theories.

Approach 1: Polynomial invariant method -Covariant derivative on the log as an invariant
For any give complex field N ∈ C, such that N = N Re + iN Im , where the imaginary N Im → −N Im is charged under Z C 2 -symmetry, thus we define a new derivative For example, for the complex scalar field Φ = Φ Re + iΦ Im ∈ C, with the real component Φ Re ∈ R and imaginary component Φ Im ∈ R, . Now based on the same trick in Eq. (2.12), we can absorb the U(1) gauge transformation by introducing the 1-form A gauge field. The A transforms to A under the U(1) part of Eq. (2.22): Here D A ,c,Im µ (iη(x)) = i(D c µ η) because η is not charged under U(1) symmetry but only the U(1) gauge parameter of A itself. The above shows that D A,c,Im is a gauge invariant quantity under U(1) gauge transformation. We can show that it is also gauge invariant quantity under the full O(2) gauge transformation (including Eq. (2.22) and the definition of (±) ∈ {+1, −1} around Eq. (2.22)): Let us focus on the case Φ → e i η(x) Φ c first, we have: Here some of the equalities hold when we focus on the leading order contribution for the gauge trans- We can define a complex conjugation operator of D A,c,Im µ as : this means that the numerators are also gauge-covariant or complex conjugation gauge-covariant. We can construct the gauge invariant quantity via pairing the gauge-covariant term with its complex conjugation, and pairing the complex conjugation gauge-covariant term also with its complex conjugation. So we obtain: 7 Here we are allowed to flip the sign C → −C since C is only a Z2 gauge field. More precisely, C = 2π 2 Z = πZ mod 2π, while C = − C mod 2π. It may also look peculiar that the particle Φ and anti-particle Φ † both couples to the gauge field A with both ±1 couplings. However, we may comfort the readers by reminding the fact that the particle-hole conjugation symmetry Z C 2 is already gauged, thus Z C 2 gauge field couples to both the particle Φ and anti-particle Φ † , via the ΦIm part. Furthermore, the Z C 2 gauge field can flip the sign of their U(1) gauge charge +1 ↔ −1. This seems to suggest that particle and anti-particle may share part of the degree of freedom. This reminds us the famous fact that Majorana fermion has the particle and anti-particle identified as the same, although we should beware that our particle Φ is bosonic instead.
Since all components of A, C, Φ Re , Φ Im ∈ R are reals, we can pair the gauge covariant term with its transpose (the Hodge dual ) to obtain a gauge invariant Lagrangian term (again see footnote 7) Thus we show (again see footnote 7) Similarly (again see footnote 7), Thus we can construct an O(2) gauged matter field theory contains a Lagrangian term Eq. (2.47) and a potential V (|Φ| 2 ) as: (2.53) The theory contains particle Φ and anti-particle Φ † pair together in an intricate way because the particle- ←→ Φ † is also dynamically gauged. If the particle Φ has a gauge charge-1, then the anti-particle Φ † has a gauge charge-(−1) under the U(1) gauge group. (However, see also footnote 7) Follow [4,6], we can consider the N -layers generalization of the theories with (Z C 2 ) N gauged, also by including the O(2)-Yang Mills kinetic term Eq. (2.30) and the level-2 BF theory into the O(2) gauge matter theory Eq. (2.53), we are allowed to introduce the twisted cocycle ω d+1 ∈ H d+1 ((Z C 2 ) N , R/Z) from a group cohomology data [21] to specify the interlayer interactions between N -layers. We can write down a schematic path integral: 2.2 Degree-1 polynomial symmetry to Pretko's field theory and non-abelian generalization

Global-covariant 2-derivative
Now we construct a field theory that preserves a degree-1 polynomial symmetry with a polynomial Q(x) = (Λ k x k + Λ 0 ). A degree-1 polynomial symmetry transforms Φ and log Φ as (2.56) Take ∂ x i ∂ x j := ∂ i ∂ j on both sides, we construct a globally invariant term, We also define as a global-covariant 2-derivative term, in order to maintain the ∂ i ∂ j log Φ to be invariant. The gaugeinvariant Lagrangian contains In this way, based on the systematic method of Ref. [6], we can re-derive a Lagrangian formulation of Pretko in 2018 [2], which are recently revisited in [3,5] and [4] from other field theory perspectives.

Gauge-covariant non-abelian
Follow Ref. [4,6], we define a non-abelian rank-3 field strengtĥ Under the gauge transformation we can again showF c µνξ is gauge-covariant: The (µ ↔ ν) are the term exchanging µ and ν respect to the previous term. The non-abelian field strength has firstly appeared in Ref. [4,6]. The gauge-invariant non-abelian gauge field kinetic Lagrangian term corresponds to: with a covariant factor of power 2 as e i 2η(x) , we may call this as "2-covariant" for the convenience.

Non-abelian
In Sec. 2.2.2, we had learned that for the abelian gauge sector, we require to introduce a symmetric tensor gauge field A µ,ν in order to cancel the gauge transformation ∂ i ∂ j η between Eq. (2.63) and Eq. (2.64). Thus, we also symmetrize the above equation 9 in order to naturally couple to a symmetric tensor gauge field later:  Similarly, we have: (2.80) (2.81) (2.83) Follow [4,6], we can consider the N -layers generalization of the theories with (Z C 2 ) N gauged, also by including the [U(1) x (d) Z C 2 ]-gauge kinetic term Eq. (2.71) and the level-2 BF theory into the gauge matter theory, again we are allowed to introduce the twisted cocycle ω d+1 ∈ H d+1 ((Z C 2 ) N , R/Z) from a group cohomology data [21] to specify the interlayer interactions between N -layers. We can write down a schematic path integral (see also footnote 7): The new ingredient in our present work beyond the previous Ref. [4,6] is that now the matter fields directly interact with non-abelian gauge fields.

Degree-(m-1) polynomial symmetry to non-abelian higher-rank tensor gauged matter theory
In this subsection, we outline a generalization of previous Sec. 2.1 and Sec. 2.2 to a general degree-(m-1) polynomial symmetry and by gauging it and the particle-hole Z C 2 symmetry to obtain a non-abelian higher-rank tensor gauged matter field theory.

Z C 2 charge-conjugation (particle-hole) symmetry
In addition to the polynomial symmetry in Sec. 2.3.1, as noticed in [4,6], we have a Z C 2 charge-conjugation (particle-hole) symmetry. It acts on the complex scalar Φ switching from a particle to an anti-particle. The Z C 2 symmetry persists even after we gauge the abelian polynomial-symmetry, which also acts on the rank-m abelian symmetric tensor A i 1 ,··· ,i m and the gauge parameter η v (x) for Φ → e i ηv Φ: The U(1) degree (m-1)-polynomial symmetry does not commute with Z C 2 symmetry. Abbreviate Eq. (2.86)'s as U(1) poly symmetry So we have indeed a non-abelian/non-commutative global symmetry structure: (2.89) 2.3.3 Non-Abelian/non-commutative gauge structure: Even after we gauge the U(1) polynomial symmetry, we can still observe the gauge transformation of [U U(1) poly ] does not commute with the Z C 2 global symmetry transformation, which both can act on the Φ and the rank-m symmetric tensor A respectively: By gauging the degree-(m-1) polynomial symmetry and keep only the rank-m symmetric tensor A i 1 ,··· ,i m , we are left with a non-abelian/non-commutative gauge structure We propose a polynomial invariant method to generalize the procedure of Sec. 2.2.4 from a degree-1 polynomial to a generic degree (here a degree-(m-1) polynomial). First, we determine, For example, the degree-1 case is obtained in Eq. (2.72). The degree-2 case is obtained in Ref. [6]. 11 The functional P c should be a generalization of the result obtained in Ref. [6]. The derivative (D c,Im ) involves the coupling to a 1-form C gauge field. Moreover, when we turn off the C gauge field, we reduce D c,Im i 1 D c,Im i 2 . . . D c,Im i m log Φ to a previous formula obtained in Ref. [6]: 11 For a degree-2 polynomial symmetry: we construct a covariant 3-derivative (triplederivative) below. First, log Φ → log Φ + i Q(x) = log Φ + i (Λi,jxixj + Λixi + Λ0). we take ∂x i ∂x j ∂x k := ∂i∂j∂ k on both sides → ∂i∂j∂ k log Φ, which is globally invariant under the degree-2 polynomial symmetry: We use the symmetrized tensor notation: T (i 1 i 2 ···i k ) = 1 k! σ∈S k Ti σ1 i σ2 ···i σk , with parentheses (ijk) around the indices being symmetrized. The S k is the symmetric group of k elements. Since the denominator Φ 3 → e i 3Q(x) Φ 3 , so does the numerator P i,j,k (Φ, · · · , ∂ 3 Φ) → e i 3Q(x) P i,j,k (Φ, · · · , ∂ 3 Φ), which we call the denominator and numerator are 3-covariant. Lagrangian thus contains |P i,j,k | 2 := P i,j,k (Φ)P i,j,k (Φ † ) [6].
. . ) yields a symmetrization over the subindices under the lower bracket (i 1 , · · · , i m ), the permutation (m!)-terms. The P c i 1 ,··· ,i m (Φ, · · · , (D c,Im ) m Φ) is not gauge covariant under the generic gauge transformation. But we can append the A gauge field to make it gauge covariant: where we implicitly sum over all possible indices as {i 1 ,··· ,i m} over both the left and right hand sides. The special case when m = 1 is given in Eq. (3.14) and m = 2 is given in Eq. (2.78). The non-abelian gauge covariant rank-(m+1) field strength is already obtained and defined in Ref. [6]:

New Sigma Models in a Family and Two Types of Vortices
Section 2 proposes a family of non-abelian gauged matter field theories. In this section, we study the "dualized" theory -instead of using the matter field degrees of freedom, we try to incorporate the vortex degrees of freedom into the field theory.
To start with, there are at least two types of vortex degrees of freedom that we can identify.
1. The complex scalar matter field can be written as: 2. The anti-symmetric tensor TQFT sector (the level-2 BF theory as a Z 2 gauge theory twisted by Dijkgraaf-Witten group cohomology topological terms) can also be regarded as the disordered phase of a sigma model given by another scalar field θ I . The derivations of sigma models governing the ordered-disorder phases relevant for twisted Dijkgraaf-Witten type TQFTs 13 had been studied in [32,33,35], here we will implement the procedure done in [32]. We write θ I = θ s,I + θ v,I , (3.10) 13 To recall the approach of Ref. [32,33] and their generalization: • The ordered phase of sigma models describes the weak fluctuations around the symmetry-breaking phases.
• The disorder phases of sigma models describes the strong fluctuations around the symmetry-restored phases as continuum formulations of TQFTs, SETs or SPTs of Dijkgraaf-Witten type.
The U(1) spontaneously symmetry breaking phase has a superfluid ground state, which is an ordered phase respect to φ with an order parameter exp( i θ) = 0.
It is well-known that if we disorder the U(1) spontaneously symmetry breaking (superfluid) state, we can obtain an disordered phase known as a gapped insulator [27][28][29]. Our approach is basically along this logical thinking, except that we generalize the approach by: • Disordering the ordered phase (U(1) symmetry breaking superfluid) to a disordered phase of gapped topological order (e.g. the ZN-gauge theory, where the Z1-gauge theory means a trivial gapped insulator, and the Z2-gauge theory means the low energy theory of deconfined Z2-toric code, Z2-spin liquid or Z2-superconductor). (Beware that Ref. [32,33] only consider the case of a superfluid-insulator transition for N = 1, here we consider a superfluid-topological-order transition for a generic N.
• Follow [32,33], introducing additional topological multi-kink Berry phase specified by the cocycle of cohomology group to the superfluid.
To comprehend our formalism, here we overview this approach using field theory [32]. We start from the superfluid state in a d-spacetime dimension described by a bosonic U(1) quantum phase θ kinetic term and a superfluid compressibility coefficient χ, the partition function Z is: The θ = θs + θv with a smooth piece θs and a singular vortex piece θv for the bosonic phase θ. We emphasize that the θv is essential to capture the vortex core, see Sec. 4. We introduce an auxiliary field j µ and apply the Hubbard-Stratonovich technique [34], By integrating out the smooth part [Dθs], we obtain a constraint δ(∂µj µ ) into the path integral measure. Naively, in the anti-symmetric tensor differential form notation, the constraint seems in disguise and the solution in disguise is j = 1 2π ( dB). (Here we choose a normalization convention.) However, we imagine the procedure is the N-fold vortex of superfluid becomes a trivial object (instead of a 1-fold vortex of superfluid ) that can be created or annihilated for free from the ZN-gauge theory vacuum. Instead we may impose a revised constraint 2π d( j) = NZ. (3.6) Note that the 2πN on the right hand side means that N-fold of 2π vortices become to be identified as a trivial zero vortex (none vortex). The solution is, with the Hodge star, where θ s,I describes the smooth (s) part while the θ v,I describes the singular vortex (v) part. See the footnote 13 and [32], the exterior derivative of the vortex field should be identified as the 1-form C gauge field as: dθ v = C. (3.11) In this way, the TQFT sector of the theory (as a disordered phase of some sigma model) can be re-written as a sigma model with the vortex field θ v degree of freedom: where ω d+1 ({dθ v,I }) is mapped to a multi-kink Berry phase topological term exp(i [32,33].
We replace and redefine a new derivative on the right hand side by substituting

13)
We can define a generic form j µ = N 2π(d−2)! µµ 2 ...µ d ∂µ 2 Bµ 3 ...µ d , with an anti-symmetric tensor B with a total spacetime dimension d (most conveniently, we may consider 2d space or 2+1d spacetime in order to implement a winding number in Sec. 4), to satisfy this constraint. To disorder the superfluid, we have to make the θv-angle strongly fluctuates -namely we should take the χ → ∞ limit to achieve large |δθv| 2 1, the disordered limit of superfluid. Plug in Eq. (3.7), the partition function becomes: Hereafter we may compensate the dropped ±-sign by a field-redefinition. Although naively d 2 = 0, due to the singularity core of θv, the ( d 2 θv) can be nonzero, see Sec. 4, which implies that (at least for the 2-dimensional space mapping to a deformed S 1 -circle as a target space): 1 2π d 2 θv = n mod N, thus n ∈ ZN. (3.8) Thus, ( d 2 θv) describes the vortex core density and the vortex current, which we denote In addition, Noether theorem leads to the conservation of the vortex current: the continuity equation for some 1-form gauge field C. We can thus define the singular part of bosonic phase dθv = C as a 1-form gauge field, to describe the vortex core. The partition function in the disordered state away from the superfluid, now becomes that of an gapped insulator (for N = 1) or topologically ordered state with a topological level-N BF action as a ZN-gauge theory:  14 We also have the gauge transformation descended from 1-form C gauge field, ∂ν θv,I → ∂ν θv,I + 1 gc ∂ν γc,I (x), We can either include or omit the I index for these operators.
By combining two kinds of vortex degrees of freedom from the vortex 1 of φ and the vortex 2 of θ, thus we can rewrite Eq. (2.96) into a sigma model-like expression for a non-abelian gauged fractonic matter theory: Here B is only a Lagrange multiplier. Here we also have not yet replaced the symmetric tensor gauge field A to any kinds of vortex degrees of freedom in Eq. (3.17). As some of the readers may wonder, and it is tempting to ask this question: whether the gauge field A can be "dualized" into some new vortex degrees of freedom. However, we will not attempt to attack this issue and leave this as an open question for future work.

Cauchy-Riemann Relation, Winding Number and Topological Degree Theory
Here we derive a relation used in the previous section, relating the vortex degrees of freedom to a winding number, via the Cauchy-Riemann relation and topological degree theory, at least for the 2-dimensional space mapping to a deformed S 1 -circle as a target space. On the complex plane z := x + iy = r exp(iϕ), (4.1) we define the Hodge star operator (for the differential form, this is the Hodge dual) on the 1-form as (f dx + g dy) = (−g dx + f dy), (4.2) for some generic functions f and g.
Then we have dz = −i dz with dz = dx + i dy. We can compute that d df = ∆f dx ∧ dy, (4.3) with the Laplacian operator ∆.
If f = u + iv is holomorphic dependent on z independent ofz (namely the Cauchy-Riemann equation), df = f dz and also i df = f dz with f = df dz , so we see that dv = du.
Take f = log z, then u = log r, v = ϕ, where z = re i ϕ is the polar coordinate, then we have ddv = d du = ∆ log r dx ∧ dy = 2πδ 0 (4.4) where δ 0 is the delta function at the origin 0 of the polar coordinate. Hence we derive that For a general S 1 valued function φ defined outside a singular point p, we may assume p = 0 at the origin (without loss of generality).
Applications to two types of vortices: 1. For the fractonic phase field φ: We can always write the phase of the fractonic matter field in Eq. (3.1) as where n is the winding number. In terms of the degree theory, we only focus on Σ 2 → S 1 , specifically here we consider Σ 2 = R 2 − {0}, as a punctured 2-plane mapping to a circle S 1 (or the U(1) target space). We can possibly generalize this result to other target spaces. The first term φ s extends smoothly over 0 is known as the smooth fluctuation. Then using the previous result we see that 1 2π ddφ = 1 2π dd(φ s + φ v ) = 1 2π dd(φ v ) = n 1 2π dd(ϕ) = n ∈ Z. (4.7) Thus importantly, we can identify the solution of vortex core equation 1 2π d 2 φ v = n ∈ Z as the winding number.

2π
B I d(dθ v,I ) = B I · n θv to a term associated to the winding number n θv .

Conclusions
We have proposed a systematic framework to obtain a family of non-abelian gauged fractonic matter field theories in Sec. 2. We have derived a new family of Sigma models with two types of vortices in Sec. 3 that can interplay and transient between the disordered phase (with higher-rank tensor non-abelian gauge theories coupled to fractonic matter) and ordered phase (with superfluids and vortex excitations) of Sigma models. We formulate two types of vortices, one is associated to the fractonic matter fields (d(dφ)), the other is associated to the 1-form C gauge field (d(dθ v )). These two types of vortices mutually interact non-commutatively when they communicate via the higher-rank tensor gauge field A as a propagator. We apply the Cauchy-Riemann relation and topological degree theory to capture the winding number for the two types of vortices in Sec. 4.
Here we make some extended comments and also list down some open questions.
5. In the present literature, there are three different routes to obtain non-abelian fracton orders: (1) Gauge the charge conjugation (i.e., particle-hole) Z C 2 -symmetry and U(1) poly polynomial symmetry [4,6], (2) Gauge the permutation symmetry of N -layer systems [19,20], (3) Couple to non-abelian TQFT/topological order [18,42,43] and [4,6]. Given a N -layer systems, there is a larger non-abelian group structure that we can explore. In the previous work [4,6] and our present work, we focus on the finite abelian group (Z C 2 ) N by gauging N -layer of particlehole symmetries, and consider a non-abelian gauge structure: U(1) poly (Z C 2 ) N . In fact, a natural larger group is including also the S N permutation symmetry group of N -layers [19,20]. The (Z C 2 ) N and S N form a short exact sequence via a group extension: The G nAb is related to the hyperoctahedral group in mathematics. An overall larger non-abelian group structure including both G nAb and U(1) poly , that mutually are non-commutative, possibly can be studied also via field theories or lattice models in the future.
6. Ref. [6] points out possible proper tools for studying these field theories include algebraic variety and affine-geometry/manifold. One motivation is to explore these theories and more general models on general affine manifolds beyond the Euclidean spacetime. This is left for future work.
7. Quantization and quantum path integral: The most challenging question may be that to explore the full quantum nature of the path integral we proposed, or study the quantization of these field theories.