Effective Brane Field Theory with Higher-form Symmetry

We propose an effective field theory for branes with higher-form symmetry as a generalization of ordinary Landau theory, which is an extension of the previous work by Iqbal and McGreevy for one-dimensional objects to an effective theory for $p$-dimensional objects. In the case of a $p$-form symmetry, the fundamental field $\psi[C_p^{}]$ is a functional of $p$-dimensional closed brane $C_p^{}$ embedded in a spacetime. As a natural generalization of ordinary field theory, we call this theory the brane field theory. In order to construct an action that is invariant under higher-form transformation, we generalize the idea of area derivative for one-dimensional objects to higher-dimensional ones. Following this, we discuss various fundamental properties of the brane field based on the higher-form invariant action. It is shown that the classical solution exhibits the area law in the unbroken phase of $\mathrm{U}(1)$ $p$-form symmetry, while it indicates a constant behavior in the broken phase for the large volume limit of $C_p^{}$. In the latter case, the low-energy effective theory is described by the $p$-form Maxwell theory. We also discuss brane-field theories with a discrete higher-form symmetry and show that the low-energy effective theory becomes a BF-type topological field theory, resulting in topological order. Finally, we present a concrete brane-field model that describes a superconductor from the point of view of higher-form symmetry.


Introduction
Symmetry is one of the most important and fundamental concepts in modern physics, and it plays an essential role in classifying phases of vacuum and matter.For instance, various phase transitions can be comprehended through the presence of symmetries and their spontaneous breaking.The Landau theory provides a comprehensive and effective framework for this description [1,2].In the Landau theory and its extensions, an order parameter field ϕ(X), charged under a global symmetry, is introduced, and the theory (i.e., free energy, Hamiltonian, or Lagrangian) is constructed as invariant under the symmetry.Furthermore, it is important to note that the assumption of the conventional Landau theory is that the order parameter ϕ(X) is a local function of the spacetime point.In this sense, the conventional Landau theory describes an effective theory of point-like or zero-dimensional objects, such as particles.
Considering the great success of the Landau theory, a natural question arises: Is it possible to construct effective field theories with extended objects for higher-form symmetries?The purpose of this paper is to explore this possibility, and demonstrate that this framework provides an effective approach to understanding the physics of higher-form symmetry, much as the conventional Landau theory does for 0-form symmetries.The advantage of this approach is that, within the mean-field approximation, it naturally describes the phase transition to topological order by using extended order parameters, which cannot be explained in conventional Landau theory.It should be mentioned that our approach is inspired by Ref. [68], where the effective field theory for 1-form symmetry is introduced as mean string field theory.Generalizing it, we refer to our field theory for p-form symmetry as effective brane field theory.
To construct the brane field theory, we should clarify which type of p-dimensional branes C p should be considered.In this paper, we focus on p-dimensional closed branes C p that extend spatially in a d-dimensional Lorentzian spacetime Σ d 1 .In other words, C p is represented by the spacetime embedding {X µ (ξ)} d−1 µ=0 , where ξ = (ξ 1 , ξ 2 , • • • , ξ p ) denotes the intrinsic coordinates.Then, the brane field ψ[C p ] is no longer just a function of spacetime point, but a functional of {X µ (ξ)} d−1 µ=0 .If we allow any functional forms, there would be little hope that we can obtain controllable brane field theory even at the classical level.Thus, it is natural to impose physically reasonable conditions as in the ordinary quantum field theory: spacetime diffeomorphism invariance and reparametrization invariance.Namely, we assume that ψ[C p ] behaves as a scalar under these transformations.
Not only the concept of the field but also the concept of derivative must be generalized in order to construct a brane-field action invariant under higher-form transformations.In general, the variation of the brane field with respect to a small change of the subspace is described by the functional derivative δψ[C p ]/δX µ (ξ).On a p-dimensional object C p , we can generally consider variations of subspaces of lower dimensions, and the functional derivative contains all such contributions.In this paper, however, we focus on a p-dimensional variation δC p such that the corresponding functional derivative is described by the area derivative, which was originally introduced for one-dimensional objects [69][70][71].We will see that, as the ordinary derivative for p = 0 is given by the one-form ∂ µ ϕ(X)dX µ , the area derivative on the p-dimensional subspace C p is given by a (p + 1)-form functional derivative as shown in Eq. (23).In this sense, the area derivative can be interpreted as one of the natural generalizations of the ordinary derivative ∂ µ ϕ(X).
Following our discussion on the construction of the brane-field theory, we perform a mean-field analysis.First, we show that the classical solution ⟨ψ[C p ]⟩ exhibits the area-law behavior in the unbroken phase of U(1) p-form symmetry, while it is constant in the broken phase in the large volume limit of C p .These behaviors can be naturally interpreted as a generalization of the off-diagonal-long-range order of the two-point correlation function ⟨ϕ † (x)ϕ(y)⟩ for the 0-form symmetries.Second, by considering phase fluctuations of the order parameter, we show that the low-energy effective theory in the broken phase of U(1) p-form symmetry is given by the p-form Maxwell theory, which is a p-form version of the Nambu-Goldstone theorem for 0-form symmetries [10,[72][73][74][75][76].Note that since we are considering theories for extended objects in spacetime, the effective theory can contain many local fluctuations other than the p-form gauge field as a Nambu-Goldstone field.However, these fields typically become massive because they are not protected by the pform symmetry.We will explicitly show this for the spacetime scalar mode as an example.Third, as well as the 0-form case, we can also consider discrete higher-form symmetry and its breaking in the present brane-field theory.By generalizing the discussion of 0-form symmetry in Ref. [77] to the p-form symmetry case, we derive the low-energy effective theory in the broken phase.This theory takes the form of a BF-type topological field theory and exhibits topological order.Finally, we discuss a concrete brane-field model for a superconductor and derive its low-energy effective theory in the superconducting (Higgs) phase.
The organization of this paper is as follows.In Sec. 2, we introduce the p-brane field and the field theory with U(1) p-form symmetry, and discuss several technical aspects, including the generalization of the area derivatives and the construction of the Noether current.In Sec. 3, we focus on the spontaneous breaking of higher-form symmetry.Using the expectation value of the brane field as the order parameter, we discuss the spontaneous breaking of p-form symmetry within the mean-field approximation, the low-energy effective theory, and emergent symmetries in the broken phase.We show that the effective theories for the spontaneous breaking of continuous and discrete higher-form symmetries are the pform Maxwell theory and the BF-type topological field theory, respectively.We also discuss the brane field model for a superconductor and its effective theory.Section 4 is devoted to summary and discussion.The Appendices provide additional details on differential forms, truncated action, and other related calculations.

Brane field theory
We explain how to construct field theory for higher-dimensional branes C p .We first introduce the brane field ψ[C p ] by imposing two physically natural conditions: spacetime diffeomorphism invariance and reparametrization invariance.Then, we discuss the relation between the functional derivative and the "area derivative" [69][70][71], which is a natural generalization of the ordinary derivative of a local field ∂ µ ϕ(x).

p-brane field
We discuss how to construct the brane field ψ[C p ].We consider a d-dimensional spacetime manifold Σ d with the metric g µν .We employ the Minkowski metric signature for d-spacetime dimensions as (−, +, +, • • • ).C p is a subspace in Σ d , which can be expressed by an embedding function S p → Σ d , i.e., {X µ (ξ)} d−1 µ=0 , where S p is a p-dimensional space.Therefore, as mentioned in Introduction, ψ[C p ] can be thought of a functional of {X µ (ξ)} d−1 µ=0 .Since we are interested in a brane as a p-dimensional object at a given time of some specific time choice, we will restrict C p to spacelike objects.C p may have a boundary; however, we mainly focus our discussion on the case where C p has no boundary.
In general, ψ[C p ] can take any functionals of {X µ (ξ)} d−1 µ=0 , but we restrict it by imposing the following conditions as in the ordinary field theory: Reparametrization invariance : ψ[C p ] is invariant under the reparametrization on C p : We note that we imposed a scalar condition (1) on the spacetime diffeomorphism to simplify the argument; more generally, it could take a covariant form.Typical examples that satisfy the above conditions are the functionals of various differential forms: where A (a) (X) is a spacetime scalar, and h = det(h ij ) is the determinant of the induced metric, Besides, the index a represents various types of volume integrals.Note that for a given scalar A (a) (X) we can always rewrite the volume integral in Eq. ( 3) by a p-form integration as where Here, η j ) can be explicitly written as where By introducing a p-form, we can also check and where Vol[C p ] is the volume of C p .

Functional derivative and Area derivative
In general, a variation of the brane field ψ[C p ] for an arbitrary change of the manifold δC p = {δX µ (ξ)} d−1 µ=0 is given by the functional derivative as In particular, when δC p has a small support around some point ξ and is given by the boundary of a (p + 1)-dimensional subspace, i.e., ∃ δD p+1 , ∂δD p+1 = δC p (see Fig. 1 as an example), Eq. ( 12) can be also written as where Here, we have used the Stokes theorem in the third line of Eq. ( 14), and introduced the antisymmetrization, where S p is the symmetric group of degree p, and sgn(σ) is the sign of permutations.We call δψ[C p ]/δσ µ 1 •••µ p+1 (ξ) the p-th order area derivative, which is the generalization of the area derivative for one-dimensional objects [68][69][70][71] to higher-dimensional ones.The definition of the are derivative ( 13) is abstract, and we should clarify its relation to the ordinary functional derivative (12).Equation ( 14) can be more explicitly written as which leads to the following expression of δψ[C p ]: This coincides with Eq. ( 12) by the identification Note that for more general variation δC p , the above relation does not necessarily hold, and additional terms can appear on the right-hand side 3 .In particular, the area derivative of the volume integral of a differential p form A (a) p can be calculated in the following way.Under the infinitesimal change C p + δC p , the variation of Eq. ( 5) is where The above equation implies Thus, as long as we consider the brane field whose functional form is given by Eq. ( 3), the area derivative is given by It is also convenient to introduce the (p + 1)-form of the area derivative: Let us see a few important examples below.

Wilson surface
A first important but trivial example is the Wilson surface defined by where A p is a p-form field.As already seen before, the area derivative is given by where F p+1 = dA p is the field strength.

World volume
The next important example is the world volume, This case corresponds to A (a) (X(ξ)) = 1 in Eq. (3).Thus, by using Eq. ( 7), we have

Minimal volume
Another example is the (minimal) volume Vol[M p+1 ] of a (p+1)-dimensional subspace M p+1 enclosed by C p , i.e., ∂M p+1 = C p .See Fig. 2 as an example.In this case, we only regard the boundary subspace C p as a physical variable.Since Vol[M p+1 ] is given by the (p + 1)form integral on M p+1 , this case does not belong to Eq. ( 22), but we can calculate its area derivative as follows.A general variation δC p can be constructed by adding infinitesimal small loop δC Loop p at each point on C p .By representing the bulk of δΓ p as δD p+1 , we have which corresponds to the expression (19).Thus, one can see that E p+1 corresponds to F p+1 in this case, and we have Figure 2: A (p + 1)-dimensional subspace M p+1 with a boundary C p .

Brane field action
Now, we define the brane field action with global U(1) p-form symmetry.At the leading order of the functional-derivative expansion, the action takes the following form, where Vol[C p ] is the world volume (26) and N is a normalization factor determined later.As we mentioned in the previous section, the functional derivative contains various area derivatives of lower degrees in general.In this paper, we focus on the p-th order area derivative and consider the simplified version of Eq. ( 30): where δ(C p ) is defined as Besides, the (path-)integral measure is defined by where T p is the p-brane tension, and DX is the induced measure by the diffeomorphism invariant norm [79], The weight in the (path-)integral measure (33) that we choose is nothing but the p-brane action [80], which suppresses large branes.Besides, when {X µ (ξ)} represents an embedding of a closed subspace C p , its translation {X µ (ξ) + X µ 0 } also represents another closed subspace, which means that there are always zero mode integrations in Eq. (33).Equation ( 31) is a straightforward generalization of the action for mean string-field theory [68] to p-dimensional brane.
The brane action (31) is invariant under the global U(1) p-form transformation, Note that if C p is a boundary, i.e., C p = ∂C p+1 , the contribution of Λ p to the phase vanishes from the Stokes theorem.Even when the topology of space-time is trivial, this transformation is useful as a symmetry, e.g., it leads to Ward-Takahashi identities as in ordinary p = 0 symmetry.We can also promote the global symmetry to the gauge symmetry by introducing a (p + 1)-form gauge field, and replacing the derivative with the covariant derivative, The gauge transformation is given by where we have used Eq.(25).Note that the action (31) is invariant under the spacetime diffeomorphism and the reparametrization on C p by the construction of ψ[C p ].
We should comment on the interactions of the brane field.The potential 31) corresponds to contact interactions in ordinary field theory, but we can also consider more general interactions such as [68] [ which represents the splitting or merging of branes 4 .Such interactions seem to alter the mean-field dynamics of the brane field significantly as the behaviors of phase transition in the ordinary Landau theory change by adding odd potential terms.In this paper, we simply neglect these interactions and focus on the model (31).

Conservation law
As with ordinary symmetry, when p-form global symmetry is continuous, we have a current (p + 1)-form J p+1 , which is conserved as The corresponding conserved charge is given as where We can calculate J p+1 in the brane field theory as follows.Instead of the global p-form transformation, we consider an infinitesimal local p-form transformation Then, the variation of the action in the linear order of Λ p should have the form since the action needs to vanish if dΛ p = 0. J p+1 is nothing but the Noether current.From the integrating by part, we obtain If ψ satisfies the equation of motion, the action is stationary, so the divergence of the current vanishes d ⋆ J p+1 = 0. Let us derive the explicit form of J p+1 for the action (31).The variation of the action is calculated as where we have used η ∧ ⋆ω = ω ∧ ⋆η.Comparing Eqs. ( 43) and (45), we obtain This expression shows that the (p + 1)-form current is given by the integral over all the brane configurations.Note that X dependence of the current J p+1 (X) comes from δ(C p ) in Eq. (32).We can also check that Q p generates the p-form transformation (35) in the following way.We formally define the quantum theory of the present brane field by the following path integral: Consider the expectation value of where ". . ." represents other arbitrary operators.We choose Λ p in the field transformation of Eq. ( 42) as for an arbitrary (d − p)-form, f d−p .δ p (D d−p ) can be thought of as a generalization of the delta function.
In this parametrization, the variance of the action ( 43) is while ψ[C p ] transforms as Here, we have defined the linking number as See the left figure in Fig. 3 for an example of the configuration of links, where we take d = 3 and p = 1.We assume other operators in ". . ." do not have support on D d−p , so that ". . ." does not transform under the field transformation with Eq. ( 49).In the path-integral, u / 6 4 p g J R p F X Z g l 8 + / J e W g s v S q a b 4 p T C 6 8 L p Z l 0 q r I 0 T s / p J S b n L Z V o j u a p g u w e 7 d A + H R h n x h f j q / G t 4 2 p k U s 4 Y 3 V v G j 1 t 1 s 7 g H < / l a t e x i t > C d p 1 < l a t e x i t s h a 1 _ b a s e 6 4 = " B x P a 6 / D e c h t u o r R P w m G A q P + 0 R I f l U j o Y L S o p + 3 0 l A i w 6 r O I a B l q B G l q t P r p 9 E 6 6 5 b X t u a T 5 + y K f Y K e l + y O 3 U J R v / H F e V v i W + c 6 u g / O i V b f 0 x X 4 q D 4 B H i F q W 1 s / 6 4 p g J R p F X Z g l 8 + / J e W h s v y y Y y 4 W l 0 m J + f S O d q m G a o T l 6 g c l Z o X V 6 R U W q I H u L L q h D 1 8 Y 7 the field transformation ( 42) is merely a redefinition of the integral variables.Therefore, assuming the path-integral measure is invariant under the field transformation (42), it leads to the identity, We also choose that C p is a p-dimensional subspace on the Cauchy surface at t = t 0 .See the right figure in Fig. 3 for the configuration.In this configuration, the left-hand side in Eq. ( 55) becomes In operator formalism, the ordering of the operator product corresponds to the time ordering, so ; thus, we find that the Noether charge Q p generates the symmetry transformation, Here, I[C d−p−1 , C p ] represents the intersection number between C d−p−1 and C p on the Cauchy surface, which can be obtained by evaluating the right-hand side in Eq. ( 55) with 3 Spontaneous breaking of Higher-form symmetry In this section, we discuss the spontaneous breaking of the higher-form symmetry in the brane field theory.As in the case of 0-form symmetry, gapless modes appear when the continuous p-form symmetry is spontaneously broken [3,75,76,81].For 0-form symmetry, the symmetry breaking is characterized by an order parameter that is the expectation value of a local field ⟨ϕ(x)⟩.This order parameter cannot be directly extended to higher-form symmetries.Alternatively, we can use the off-diagonal long-range order, as the order parameter.Since the two points (x, y) can be written as the boundary of segment M 1 and the distance between x and y can be expressed as the minimal volume |x − y| = Vol(M 1 ), the order parameter can be written in the form: This expression can be naturally extended to accommodate the case of p-form symmetry.
In this context, we can define the order parameter as where ∂M p+1 = C p .We will use Eq. ( 60) as the order parameter of spontaneous breaking of p-form symmetry.In general, the order parameter defined in Eq. ( 60) might vanish in the limit of large M p+1 , depending on C p = ∂M p+1 , i.e., the perimeter law.In such cases, it is necessary to consider a renormalized order operator ⟨Z(C p )ψ[C p ]⟩ with a field-independent functional Z(C p ).If the order operator does not vanish no matter what renormalization is performed, we can say that the symmetry is spontaneously broken.We work within the mean-field approximation.As an ansatz for the solution, we assume that the brane-field configuration ψ[C p ] depends only on the minimum volume with the boundary C p , This corresponds to the truncated treatment in Ref. [68].See also Appendix B for more general truncations.By using Eq. ( 29), the area derivative becomes which leads to Here, we used Eq.(10) to evaluate E p ∧ ⋆E p .Then, the action (31) becomes where is the density of p-brane configurations for a given minimal volume z.The equation of motion for f (z) is By introducing the WKB form f (z) = exp(S(z)), Eq. ( 67) can be also rewritten as As in the usual Landau theory, the vacuum state is determined by the potential V (f 2 ).As a simplest example, let us consider the following potential: In the following, we always assume λ > 0 to guarantee the stability of the system.

Unbroken phase
When m > 0, the minimum of the potential is located at f = 0. Therefore, we can neglect the quartic potential in the equation of motion as Let us find the asymptotic solution for the large volume.For z → ∞, we have g ′ (z)/g(z) ∼ z −1 from dimensional analysis5 and the solution is given by which corresponds to the area law of the brane field where c is a constant that in principle can be determined if we specify the boundary condition for z → 0 (small brane limit).The exponential behavior also justifies neglecting the quartic potential term in the equation of motion.This implies that the order parameter vanishes, indicating the unbroken phase of p-form symmetry.Equation ( 72) should be compared to the correlation function of the ordinary field theory (p = 0): In the p-form case, Eq. ( 72) means within the mean-field approximation.

Broken phase
Let us next consider the case for m < 0. Within the truncated approximation, the equation of motion is given by Eq. ( 67): As in the unbroken case, we focus on the large z behavior.In this case, we can neglect the derivative terms for z → ∞ by dimensional analysis, and the solution is given by 2 is nonvanishing at z → ∞, the p-form symmetry is spontaneously broken.
In ordinary quantum field theory, the non-renormalized order parameter exhibits a perimeter law in the broken phase.However, the order parameter is completely independent of C p in the present model.As already mentioned in Ref. [68], this may be an artifact by having neglected the topology-changing terms such as Eq.(39).It would be interesting to see whether we can actually realize the perimeter law by adding such topology-changing interactions.

Nambu-Goldstone modes
What are the low-energy fluctuation modes in the broken phase?As in the case of 0-form symmetry, the phase fields are candidates for low-energy degrees of freedom (d.o.f): Let us see how this d.o.f describes a gapless mode in the effective action.Note that the effective theory has the gauge symmetry because Eq. ( 76) is invariant under this transformation due to the closedness of C p .Here Λ p−1 is (p − 1)-form gauge parameter.For Eq. ( 76) to be invariant, the integral of dΛ p−1 need not vanish, but can be In other words, A p is the U(1) p-form gauge field.Now let us calculate the effective action for A p .By putting Eq. ( 76) into the action (31) and using Eq. ( 25), we have In the following, we consider the flat spacetime g µν = η µν for simplicity.By introducing the Fourier modes, Eq. ( 79) can be written as where There are zero-mode integrations in the above path-integral, and it is convenient to separate them: where X µ NZ (ξ) denotes the nonzero mode.In this expression, it is easy to see that K(k) is proportional to δ (d) (k).Thus, we can choose the normalization N as which leads to which is nothing but the p-form Maxwell theory, and thus, A p is gapless for d > p + 26 .

Other fluctuation modes
What about the other fluctuation modes?In general, they are given by expanding the phase with respect to the derivatives of X µ (ξ): where ϕ(X) is a scalar field and H µν (X) is a symmetric tensor field.These fluctuation modes are typically gapped because they are not protected by spontaneous symmetry breaking.
Here, we actually show that ϕ(X) is gapped as an example.The area derivative is calculated as Then, we have Now, the effective action is written as where The first term in Eq. ( 89) can be expressed as where n µ is the normal vector on C p , and we have used g µν = n µ n ν + h ij e µ i e ν j in the last line.Now let us focus on the flat spacetime g µν = η µν for simplicity.Equation (92) gives the kinetic term where which has to be proportional to η µν as long as the Lorentz symmetry is unbroken. 7As a result, Eq. ( 93) gives the usual kinetic.
For the evaluation of the other terms in Eq. ( 89), note that E p (and correspondingly G µ and M 2 ) does not depend on the spacetime zero mode x µ by definition.Then, the second term becomes the mass term where S p+1 is a closed (p + 1)-dimensional subspace and J p+1 is given in Eq. (46).By substituting Eq. ( 76) into Eq.( 46), we obtain J p+1 = v 2 F p+1 , which can also be derived from the low-energy effective action (85), using the Noether theorem.Consequently, the topological charge (101) becomes The corresponding symmetry operator is U (θ) = e iθQ d−p−2 and the charged object is a (d − p − 2)-dimensional object.For example, when d = 4 and p = 1, the charged object is a worldline of a magnetic particle.Since A p is the U(1) p-form gauge field, it satisfies the Dirac quantization condition, which also leads to

Discrete higher-form symmetry breaking
Up to this point, we have discussed the case with a continuous higher-form symmetry.In general, we can consider a model with a discrete higher-form symmetry and its spontaneous breaking.For example, we can construct a model with Z N p-form symmetry, by adding the following term which explicitly breaks the U(1) p-form symmetry down to Z N , into Eq.( 31).Correspondingly, the VEV is discretized as in a broken phase of Z N p-form symmetry.In this case, the phase degrees of freedom, will no longer be gapless.
The effective theory must be invariant under Z N p-form symmetry corresponding to the shift, M L E / n 5 x e y q e q l 5 z R B r z A 5 b 2 i e F m m J S s j + l l p 0 T with C p Λ p ∈ 2πZ, where n ∈ Z.For example, when p = 0, C p=0 A p=0 = φ(x) is the periodic scalar field, and it has the periodic potential V (φ) = V (φ + 2π/N ).The effective theory must also be invariant under gauge transformation of A p , with C p dΛ p−1 ∈ 2πZ, which is a redundancy in the degrees of freedom of Eq. (107).When the discrete higher-form symmetry is spontaneously broken, it exhibits topological order.The degeneracy of the ground state depends on the topology of the space.We assume that the space manifold has a nontrivial topology such as where C p+1 is a (p + 1)-dimensional subspace with boundaries at infinity C ′ ∞ p and C ∞ p , and D d−p−2 is a (d−p−2)-dimensional subspace.See Fig. 4 as an example.We also assume that C ′ ∞ p and C ∞ p are not contractible, so that the p-form symmetry can act nontrivially.In such a case, there exists a classical static configuration ψ W [C p ] = v exp i C p A W p connecting different ground states.Note that in the case of 0-form symmetry, which exhibits not a topological order but a spontaneous breaking of discrete symmetry, the topological defect connecting the different ground states is nothing but a domain wall.The corresponding topological charge is given by More explicitly, dA W p can be represented as in the thin wall limit, where δ p+1 (D W d−p−1 ) is the Poincaré-dual form defined in Eq. ( 50).Here, D W d−p−1 corresponds to the worldvolume of D d−p−2 .Now, let us study the low-energy effective theory.We generalize the argument of 0-form symmetry discussed in Ref. [77] to the brane field theory.To derive the effective theory, we rewrite Eq. (105) for large v as In the last line, we approximated the cosine by using the Villain formula [82], for large β, and dropped the constant term.Equation ( 112) is gauge invariant under Eq.( 109) accompanied by the shift of n, n → n+N Cp dΛ p−1 /(2π).Similarly, it is invariant under Z N transformation (108).We can replace the integer n in Eq. ( 112) by introducing the flat U(1) gauge field f p , as By performing the same calculations as in Sec.3.3, we have the following effective action of A p : where λN is a coupling constant which includes λ N .See Appendix C for the derivation of the mass term.Here, f p is the U(1) p-form gauge field, and the flatness condition is imposed by the last term by using the Lagrange multiplier B d−p−1 . 8On the other hand, f p can be eliminated using the equation of motion for f p , which leads to For the domain wall configuration (111), the last term becomes which implies that the worldvolume D W d−p−1 couples with the gauge field B d−p−1 .In the low-energy limit, we can neglect higher derivative terms, and we obtain the topological field theory with the action, This effective theory has the following emergent global Z N (d − p − 1)-form symmetry: with respectively.We can also show that V [C d−p−1 ] and W [C p ] correspond to the symmetry operators of the above p and (d − p − 1)-form symmetries.They satisfy where Link[C p , C d−p−1 ] is the linking number defined in Eq. ( 53), and ". . ." denotes other operators that neither link nor intersect C d−p−1 .
As mentioned above, this theory (118) exhibits the grand state degeneracy depending on the topology of the spatial manifold Σ d−1 .Let us look at this in detail using the same argument in Sec.2.4.When Σ d−1 = S p × S d−p−1 , we can choose C p = S p and holds as an operator relation.Since both operators are symmetry operators, we can choose a groundstate |Ω⟩ as an eigenstate of one of the symmetry operators.Here, we take the , where e iθ is the eigevalue.Since W [C p ] is also a symmetry operator, has the same energy as |Ω⟩.But it has a different eigenvalue of Since |Ω⟩ and |Ω ′ ⟩ have different eigenvalues, they are orthogonal, ⟨Ω ′ |Ω⟩ = 0; that is, the ground state is degenerate.

Brane field model for superconductor
We here discuss a superconducting phase (Higgs phase) in a brane-field model by coupling (p + 1)-form gauge field.We mostly focus on the low-energy degrees of freedom, and leave more detailed studies including the massive degrees of freedom for future investigations.We consider a gauged p-form brane-field model: where g 2 is a gauge coupling whose mass dimension is 2(p + 2) − d and Here, B p+1 is the U(1) (p + 1)-form gauge field.Note that we consider a general charge q ∈ Z compared to Eq. ( 37) 9 .The action is invariant under p-form gauge transformation, where Λ p is p-form normalized as C p+1 dΛ p ∈ 2πZ.In addition to the p-form gauge symmetry, when q > 1, this theory has a global electric Z q (p + 1)-form symmetry: where C p+1 is a (p+1)-dimensional closed subspace.The corresponding symmetry operator and charged objects are 9 One may think that the charge can always be absorbed into the gauge coupling by the field redefinition qB p+1 → B p+1 .However, this is not true since such a redefinition changes the Dirac quantization condition where D d−p−1 is a (d − p − 1)-dimensional subspace with boundary.Using the Stokes theorem and the Maxwell equation, (−1) p d ⋆ H p+2 /g 2 = q ⋆ J p+1 , we obtain Here, q ⋆ J p+1 is the gauge current defined by the variation of B p+1 in the matter part of the action as δS matter = − Σ d δB p+1 ∧ q ⋆ J p+1 .Since the charge is quantized to integer, In addition, the theory has the magnetic U In the following, we consider the Higgs phase, i.e., we assume that there exists a nontrivial minimum ψ[C p ] = v/ √ 2 in the potential V (ψ † ψ).In order to study the low-energy effective theory in a Higgs phase, we focus on the phase modulation in the brane field:10 Then, by repeating the same calculations as before, Eq. (126) becomes where λ is a parameter whose mass dimension is d − 2(p + 1), and F p+1 = dA p .In addition to the original p-form gauge symmetry (128), this effective theory has a (p − 1)-form gauge symmetry given by Equation ( 135) corresponds to the low-energy effective action of the Abelian-Higgs model in the broken phase [7,13].This effective theory has an emergent U(1) (d − p − 2)-form symmetry, whose charge is given by where C p+1 is a (p + 1)-dimensional closed subspace.The charged object is the (d − p − 2)dimensional 't Hooft operator, which is a defect operator formally obtained by excising a codimension (p + 2) dimensional locus from Σ d and imposing a boundary condition on A p around it.Instead, one can express the 't Hooft operator by using a field in the dualized theory.By introducing the dual field of A p as Ãd−p−2 , Eq. ( 135) can be dualized as where Fd−p−1 = d Ãd−p−2 (See Appendix D for the derivation).In the dualized theory, the (d − p − 2)-form symmetry is given by a transformation of Ãd−p−2 as where -dimensional closed subspace.The corresponding charge and charged object for the (d − p − 2)-form symmetry are respectively.Note that there is a correspondence between the dual theory and original theory, ⋆ Fd−p−1 /Λ = dF p+1 .For example, for d = 4 and p = 0, the brane field theory (126) is nothing but the usual Abelian Higgs model, and W [C 1 ] is the Wilson loop while V [C 2 ] is a 2-dimensional surface operator which corresponds to the world surface of the vortex.On the other hand, we can also derive the same effective theory from the brane-field theory with d = 4, p = 1, where the roles of B and Ã are reversed.In this theory, the dual gauge field B1 of the original Abelian-Higgs model appears as a phase d.o.f ψ[C 1 ] ∼ exp i C 1 B1 which corresponds to the 't Hooft operator for the Abelian-Higgs model.More generally, one can see that the gauged (d − p − 3)-form brane field theory gives the same-low-energy effective theory as Eq. ( 138) and that the roles of scalar and gauge fields are exchanged each other.

Summary and discussion
We have proposed an effective brane field theory with higher-form symmetry by generalizing the previous work for a mean string field theory [68].As a generalization of the ordinary field ϕ(x) for p = 0, the fundamental field ψ[C p ] that is charged under the p-form transformation is defined as a functional of the p-dimensional brane C p .We constructed an action that is invariant under the higher-form transformation using the area derivative acting on higher-dimensional objects.Furthermore, we have discussed the spontaneous breaking of both U(1) and discrete higher-form symmetries and studied their low-energy effective theories, which are p-form Maxwell and the BF-type topological field theories, respectively.
There are several issues to be addressed.First, while we have focused on closed subspaces in this paper, we can generalize to branes with boundaries.In this case, the area derivatives need to be treated carefully since we have contributions from both the bulk and the boundary.Compared to the closed-manifold case, one of the crucial differences is that low-energy effective theory typically contains other higher-form fields originating from the boundary d.o.f as well as the bulk ones.Such an effective theory might have emergent gauge symmetry as well as emergent higher-form global symmetry.
Second, we have considered an effective theory for a single type of extended object, but it would be interesting to consider a theory in which objects of different dimensions interact.Additionally, a theory that includes objects constrained on an extended object or on the intersection of extended objects can also be considered.Symmetries of such a theory could be described by higher groups or, more generally, non-invertible symmetries.It is possible that a theory exhibits anomalies where symmetries are broken by quantum corrections.It would be interesting to consider whether an anomaly specific to brane field theory could exist.
Finally, it is interesting to study a brane field theory without Lorentz invariance.In the case of 0-form symmetry without Lorentz invariance, there exist two types of Nambu-Goldstone modes, and unlike in Lorentz-invariant systems, there is no one-to-one correspondence between the generators of the broken symmetry and the Nambu-Goldstone modes [83][84][85][86][87].The complete relation can be understood by considering the expectation value of the commutation relation of broken generators [88][89][90][91].This concept has been extended to the case with p-form symmetry without Lorentz invariance using a low-energy effective theory [81].It is interesting to study how the low-energy effective theory is derived from the perspective of the brane field theory.
We would like to investigate these problems in our future work.

A Differential forms
We summarize the basics of differential forms.We consider a d-dimensional spacetime Σ d .The totally anti-symmetric tensor is represented by ϵ µ 1 •••µ d .In particular, we have We also define On a p-dimensional subspace C p , we have Let be a general p-form.Then, the Hodge dual is defined by For a Lorentzian spacetime Σ d , we have As usual, we can construct the integral over Σ d by However, what we want is an integration over C p .To construct it, we define which leads to

B Truncated action
When the brane field ψ[C p ] is given by a functional as Eq. ( 3), the action in Eq. ( 31) becomes The truncated action in Eq. ( 153) can be interpreted as a field theory on a curved manifold, whose background metric is determined by the brane configurations in Eqs. ( 154) and (155).

C Calculation of mass term
The effective action for broken Z q p-form symmetry discussed in Sec.(3.6) contains where δA p = N A p − f p .Following the same procedures as Sec.3.3, this can be estimated as where Assuming the spacetime symmetry is not broken, we have where c 0 (k 2 ) and c 1 (k 2 ) are functions of k 2 in general.In the low-energy limit, however, we can neglect the k dependence, and the first term gives which corresponds to the mass term in Eq. (114).
Inserting Eq. (164) into Eq.( 163), the action reduces to which coincides with Eq. (135) except the kinetic term of B p .One can see that 2πλ := e 2 corresponds to the gauge coupling and reproduces the same normalization as in Ref. [7] for d = 4.

with
∂D d−p = Σ d−p−1 , and an infinitesimal parameter ϵ.Here, δ p (D d−p ) is the Poincarédual form of D d−p such that s 4 u n b o i l g y d p U z n v T 1 D w w O P c 0 P F 0 Z G x 8 Y n n q 3 G Q T N y e N 0 J 3 C B q 2 F b M X e H z u h T S 5 Y 0 w 4 p Z n u 3 z N 3 l t W 7 2 s t H s U i 8 N / J d s g 3 P G v H F 9 v C s S S g i t w c L 7 I S 0 2 u q 1 z B T o 0 j p K g c T u Q K 9 p y 0 K y K E m e c T J J w n b J Y t i 7 H U y i V E I b I M S Y B E s o d 8 5 7 V M e 3 C a 8 O D w s o H s 4 d / C 3 n q 7 / L O P 6 D y 4 e t T I = < / l a t e x i t > t < l a t e x i t s h a 1 _ b a s e 6 4 = " j h 1 L R + H x 8 k T r b 9 H Z M W r i P S M S n 5 I = " > A A A D N n i c h V I 9 S x x R F L 0 7 i R 8 x x o / Y B G y W L E p A W O 6 K J s F K T J N S 1 + y 6 o L L M j E 9 9 O F / M v N 2 w D v 4 B S 5 s U p k k g R f A P 2 N v 4 B y y 2 S R e C 2 A Q M p E m R 8 9 6 O B h U 3 7 z H z 7 p y 5 5 7 x 7 D 9 e J P J k o 5 n b O e v C w p 7 e v / 9 H A 4 8 E n Q 8 g c W I p P k v a I 8 G w G 0 g S y D D B r q D 9 x a + V j M 0 w L f W T A z b x S 0 e n h j M P E 3 w G X / l S z 7 l I z 7 n P / d q p U Z D 1 9 L C 6 X S 4 I q o P 7 z 9 b / v 1 f l o 9 T 0 f Y / V h e G g + x u P U k o 2 e h A 9 7 + B 9 y a 27 i u C f g s M j S d d F N L r a h R c 0 F 5 0 y 9 Y e K t z w 2 t w h 4 W V k E O 2 q 2 + m n u f v h c n m u P J F O 8 m e + g J + f u M 0 n c D R o / n K / L I n y o V E P w H l v 3 P d N B Q G q T 4 H H U N 0 z 0 V V d M a L U o K g Ls 1 S 6 P T l 3 g + p 0 s f S y O L s 0 U 5 h f y K a q n 8 b p O b 3 A 5 L y i e X p L i 1 Q x E 3 N A h / T R O r a + W d + t H 5 1 U K 5 d x x u j G s n 7 + B b i i t 8 g = < / l a t e x i t > t 0 + ⌘ < l a t e x i t s h a 1 _ b a s e 6 4 = " F 2 v k r 3 e 1 I O i j B S d I g v b B a r 2 i R P M = " > A A A D N n i c h V I 9 T x t B E B 1 f w k c I 4 S M 0 k W i s W K A 0 W G M E S U S F S J M S T G w s A b L u j g V W 3 J f u 1 o 7 M i T 9 A S Z O C N I m U I u I P 0 N P w B y j c p I s i R B O J S G l S 5 O 3 6 I A K E s 6 u 7 n X s 3 7 + 3 M 0 z i R J x P F 3 M 5 Z D x 7 2 9 P b 1 P x p 4 1 0 z g 9 p x e Y n F c 0 T 2 9 p k S p m Y g 7 o k D 5 a x 9 Y 3 6 7 v 1 o 5 N q 5 T L O G N 1 Y 1 s + / v k C 3 y g = = < / l a t e x i t > t 0 ⌘ < l a t e x i t s h a 1 _ b a s e 6 4 = " X L u b T A J

C d p 1 Figure 3 :
Figure 3: (Left) Configuration of the link between C p and Σ d−p−1 for d = 3 and p = 1.(Right) Configuration of C p and C d−p−1 to obtain the commutation relation in operator formalism.The arrows indicate the orientation of the subspace.
This is the relation in the path integral.To derive the relation in the operator formalism, consider the Cauchy surface labeled by time t.Let C d−p−1 (t 0 ) be (d − p − 1)-dimensional subspace on the Cauchy surface at t = t 0 .We choose that Σ d−p−1 = C d−p−1 (t 0 + η) ∪ C d−p−1 (t 0 −η) with an infinitesimal parameter η.Here, C d−p−1 is the (d−p−1)-dimensional subspace with the opposite orientation of C d−p−1 .
in addition to the original Z N p-form symmetry (108), where C d−p−1 is a (d − p − 1)dimensional closed subspace.The charged objects for p-and (d − p − 1)-form symmetries are the Wilson surfaces: in the operator formalism at time t.The ordering of the operator product corresponds to the time ordering.That is, the pair of symmetry operators V [C d−p−1 ] and V −1 [C d−p−1 ], corresponds to the operator on C d−p−1 (t + η) ∪ C d−p−1 (t − η) in the path integral formalism.Here η is an infinitesimal parameter and C d−p−1 is the subspace with the opposite orientation of C d−p−1 (t).In this case, C p (t) and C d−p−1 (t + η) ∪ C d−p−1 (t − η) can be linked in space-time.This means that

C p+2 H
p+2 ∈ 2πZ.In other words, for a given quantization condition, the charge is determined up to Z. respectively, where C d−p−2 is a (d − p − 2)-dimensional closed subspace.The discrete symmetry means that U [C d−p−2 ] is a topological operator, which can be checked as follows.By deforming C d−p−2 by C d−p−2 + ∂D d−p−1 , we obtain