Generalized global symmetries and holography
JJanuary 16, 2018
Generalized global symmetries and holography
Diego M. Hofman ∗ and Nabil Iqbal † Institute for Theoretical Physics, University of Amsterdam,Science Park 904, Postbus 94485, 1090 GL Amsterdam, The Netherlands Centre for Particle Theory, Department of Mathematical Sciences,Durham University, South Road, Durham DH1 3LE, UK
Abstract
We study the holographic duals of four-dimensional field theories with 1-form global symmetries,both discrete and continuous. Such higher-form global symmetries are associated with antisymmet-ric tensor gauge fields in the bulk. Various different realizations are possible: we demonstrate thata Maxwell action for the bulk antisymmetric gauge field results in a non-conformal field theory witha marginally running double-trace coupling. We explore its hydrodynamic behavior at finite tem-perature and make contact with recent symmetry-based formulations of magnetohydrodynamics.We also argue that discrete global symmetries on the boundary are dual to discrete gauge theoriesin the bulk. Such gauge theories have a bulk Chern-Simons description: we clarify the conventional0-form case and work out the 1-form case. Depending on boundary conditions, such discrete sym-metries may be embedded in continuous higher-form symmetries that are spontaneously broken.We study the resulting boundary Goldstone mode, which in the 1-form case may be thought ofas a boundary photon. Our results clarify how the global form of the field theory gauge group isencoded in holography. Finally, we study the interplay of Maxwell and Chern-Simons terms puttogether. We work out the operator content and demonstrate the existence of new backreactedanisotropic scaling solutions that carry higher-form charge. ∗ Electronic address: [email protected] † Electronic address: [email protected] a r X i v : . [ h e p - t h ] J a n ontents I. Introduction
II. Maxwell type Holography
III. Chern-Simons type Holography
IV. Maxwell-Chern-Simons type Holography
V. Conclusion A. Conventions and differential form identities B. MHD diffusion mode at zero magnetic field C. Wilson and t’Hooft lines in U ( N ) gauge theory References . INTRODUCTION
In this paper, we will discuss the manifestation of generalized global symmetries in d -dimensional quantum field theories with holographic duals. In the language of [1], a contin-uous generalized global p -form symmetry is associated with the conservation of an antisym-metric tensor current of rank p + 1. In form language, this conservation law can be writtenwith the help of the Hodge (cid:63) operator as the existence a co-closed p + 1 form Jd (cid:63) J = 0 . (1.1)The existence of such p + 1-forms guarantees the existence of conserved quantities inte-grated over d − p − Q = (cid:90) M d − p − (cid:63)J . (1.2)In this language, 0-form symmetries give rise to standard conserved currents yielding chargesas we integrate them over all of space at fixed time.We can think about these charges is the following way: they count the number of chargedobjects piercing the surface M d − p − . These objects are always p + 1 dimensional. This isconsistent with the fact that, provided a background p + 1-gauge field A for the current J ,charged objects couple to it by introducing a term in the action of the form: δS = iq (cid:90) C p +1 A , (1.3)where C p +1 is the world-volume of the charged objects. For familiar 0-form symmetries theseobjects have a 1-dimensional world-volume and we conclude that particles are the naturalcharged objects of 0-form symmetries. This fact allows for the large number of constructionsof local theories enjoying 0-form symmetries: because particles are the quanta of local fields,there is always a quantum field theory description available for these systems.The situation is manifestly different for higher form symmetries. Consider a 1-form sym-metry. The natural objects charged under the symmetry are strings. An analog descriptionin this case to the quantum field theory would be constituted by a theory where the funda-mental operators live in loop space [2]. A more concrete way of saying this is that operatorsthat create strings on 1 dimensional contour d C are non-local operators Φ[ d C ], where dC isthe boundary of the 2 dimensional string world-volume C . Unfortunately these objects arenotoriously hard to work with and a systematic construction of these theories is not known . See, however, [3] for a recent discussion. N = 4 Super Yang Mills (SYM), which we discuss in section III.Luckily, we know of yet another description of gauge theories: the one given by the holo-graphic principle [4] in terms of a dual gravitational AdS bulk. This description is manifestlygauge invariant and it allows for a more clear interpretation of physical symmetries. Surpris-ingly, to our knowledge, the problem of understanding the precise holographic descriptionof continuous higher form symmetries and their discrete subgroups has not been attackedsystematically. It is the main purpose of this paper to fill this gap in the literature.The recent paper [5] also studies generalized symmetry in holography from the point ofview of magnetohydrodynamic applications.
A. Two concrete examples
In this work, we will mostly consider various realizations of 2-form currents J . While wewill focus mostly on holographic aspects of these theories, it is first useful to discuss somestandard QFT constructions. A thorough discussion can be found in [1]. Formal aspectsrelated to the phase structure and symmetry algebra of the continuous abelian case will bediscussed in a forthcoming publication [6].In this short section we consider two examples.4 . Free electromagnetism Consider first free electromagnetism in four-dimensions, with no matter and a (preciselymarginal) gauge coupling g . This is conventionally written in terms of a 1-form gauge field A e with associated field strength F = dA e . This theory possesses two different 2-formcurrents: J e ≡ g F , J m ≡ g G (1.4)with G = 1 g (cid:63) F . (1.5) J m counts magnetic flux lines; its conservation is equivalent to the Bianchi identity associ-ated with the existence of the potential A e , i.e. the non-existence of dynamical magneticmonopoles. J e counts electric flux lines; its conservation is equivalent to the source-freeMaxwell equation, i.e. the non-existence of dynamical electric charges. Therefore this the-ory has not one but two independent 2-form currents associated to 1-form symmetries. Theobjects that are charged under these symmetries are Wilson lines (electric) and ’t Hooftlines (magnetic).An interesting feature of this theory is now manifest. Note that J e ∼ dA e as a consequenceof the Bianchi identity: the nonlinear realization of the 1-form symmetry associated to the2-form current J e means that it is spontaneously broken, and we may think of the electricphoton A e as being its gapless Goldstone mode [1]. We could equivalently have formulatedthe theory in terms of a magnetic photon A m , in which case we would have concluded that J m ∼ dA m as a consequence of the absence of electric charges; thus in the free photon phaseof Maxwell electrodynamics both generalized global symmetries are spontaneously broken.The presence of a second current and spontaneous symmetry breaking end up beingintimately related. Interestingly a variant of this phenomenon is quite general and it occurs inall dimensions for all generalized global symmetries. Whenever a symmetry is spontaneouslybroken it can be nonlinearly realized as J ∼ dθ for some goldstone θ . But this immediatelyimplies that the current (cid:63) d J is also conserved in this phase. This is just a trivial consequenceof the fact that “monopole” configurations become gapped in the symmetry broken phaseand at low energies they can’t break the emergent conservation of (cid:63) d J . The holographicmanifestation of this point will appear in this paper, and a more complete discussion andthe relation to critical points will be presented in [6].One may now want to introduce sources for the symmetry currents, i.e. to gauge the 1-form global symmetry above with a background 2-form source. As both currents are related A way of making this more manifest corresponds to writing the two currents as self-dual and anti-self-dual (cid:63) , a single background 2-form gauge field b e for the electric current is sufficient, and thesource for the magnetic current is effectively constructed as: b m = 1 g (cid:63) b e (1.6)Now, in a symmetry broken phase the covariant derivative is not linear on the Goldstonefields A e,m . Because of this, the currents must be improved to preserve gauge invarianceas: J e → g ( F − b e ) , J m → g ( G − b m ) . (1.7)The gauge transformations are given for the electric and magnetic symmetries as: e : A e → A e + ϕ e , b e → b e + dϕ e , m : A m → A m + ϕ m , b m → b m + dϕ m . (1.8)It is immediate from the above expressions that electric gauge transformations introducephysical changes of the magnetic gauge potential and vice-versa. This will make the conser-vation equations anomalous as we observe below.We may now write an action for our theory in the presence of background fields in theelectric formulation as S = − g (cid:90) ( F − b e ) ∧ (cid:63) ( F − b e ) (1.9)or equivalently in the magnetic formulation as S = − g (cid:90) ( G − b m ) ∧ (cid:63) ( G − b m ) (1.10)The action above is universal from the low energy point of view: it is the effective actionfor the Goldstones given the symmetries are spontaneously broken. Therefore, regardlessof the UV completion of the theory the Maxwell action is the universal description in thesymmetry broken phase.From this is obvious that the currents (1.7) may be obtained by taking functional deriva-tives with respect to b e and b m appropriately. Notice that independent of the choice offundamental fields ( A e or A m ) the action can be made to depend on either b e or b m , butone cannot write an action that depends on both sources in a way that both symmetries in(1.8) are locally realized on the fundamental fields. components. In this case the currents are independent but shortened by the self-duality constrains. InLorentzian signature this requires complexifying the gauge field. Note this is simply the higher form generalization of the usual form of the superfluid current j = v ( dθ − a )for a spontaneously broken 0-form symmetry, where θ is the Goldstone and a the external source. These two actions are equal using (1.5) and (1.6). However if we perform electric-magnetic duality (1.9)in the usual manner, we obtain (1.10) up to a contact term b m : this is another way to understand thedifference in anomaly structure described below. A e to be the fundamental field weobtain: d (cid:63) g ( F − b e ) = 0 , d (cid:63) g ( G − b m ) = H e , (1.11)with H e = db e . (1.12)In (1.11), the first equation is the equation of motion, while the second equation is just theBianchi identity written in a useful way. One can see immediately that the electric 2-currentremains conserved in this case while the magnetic 2-current conservation is broken by thecurvature of the background gauge field b e . Note also that from the first equation one caninterpret g d (cid:63) b e as (the Hodge dual of) a fixed external electric charge current that actsas a source for the gauge field.Conversely, if one takes the magnetic degrees of freedom to be fundamental, it is theelectric 2-current conservation that is broken: d (cid:63) g ( G − b m ) = 0 , d (cid:63) g ( F − b e ) = − H m , (1.13)with H m = db m . (1.14)In conclusion, in the presence of a source for the magnetic 2-current J m , the electric 2-current J e is no longer quite conserved. As its non-conservation is given by a fixed externalsource (and not by a dynamical operator), this is an anomaly . More precisely, it is a mixedanomaly preventing the simultaneous gauging of the electric and magnetic 2-currents. Notethat this entire structure has an analogue in a massless scalar in two dimensions, which hasconventional momentum and winding 1-currents that have a similar mixed anomaly.
2. Electromagetism coupled to electrically charged matter
We now turn to a different theory: consider usual Maxwell electrodynamics written interms of an electric gauge potential A , coupled to light electrically charged matter that weschematically represent by φS = − g (cid:90) dA ∧ (cid:63) dA + S [ φ, A ] . (1.15)The action S [ φ, A ] represents the matter action minimally coupled to A in the usual (0-form)gauge invariant manner. 7n this case the A equations of motion are:1 g d (cid:63) F = (cid:63) j [ φ, A ] , dF = 0 , (1.16)with F = dA j [ φ, A ] ≡ δS [ A, φ ] δA . (1.17)The operator j [ φ, A ] denotes the usual electrical 1-current that we gauge in the φ theory tocouple it to A .While J m ∼ (cid:63) dA is still conserved (as there are still no magnetic monopoles), J e isnot, as electric field lines can now end on electric charges. We saw above that spontaneousbreaking of the magnetic current implied conservation of the electric current. Thus we canconclude that the magnetic symmetry is no longer spontaneously broken. Note also that thecoupling to electrically charged matter means that the magnetic presentation of the action(1.10) and realization of the magnetic symmetry (1.8) is no longer simple.Thus to access J m = (cid:63) F we now couple a source minimally in a different way as: S = (cid:90) − (cid:18) g dA ∧ (cid:63) dA + b m ∧ F (cid:19) + S [ φ, A ] . (1.18)As required we recover J m by varying with respect to the source b m . Notice we can integrateby parts the source term to rewrite S = (cid:90) − (cid:18) g dA ∧ (cid:63) dA + A ∧ H m (cid:19) + S [ φ, A ] . (1.19)We see clearly that the field strength for b m is electrically charged under the gauged 0-formsymmetry and introduces a background, modifying the electric current as j [ φ, A ] → j [ φ, A ] = δS [ A, φ ] δA + (cid:63) H m . (1.20)Note that we no longer refer to this as an anomaly as the 1-form electric symmetry wasalready broken explicitly by the φ sector of the theory.Thus, this theory now contains only a single g runs logarithmically. If theIR is weakly coupled and we can ignore electric charges, we will obtain an enhancement ofsymmetry to the electric sector in the infrared, reproducing the above discussion. If the the-ory becomes strongly coupled it could develop a gap (e.g. by developing a superconductingcondensate) in which case the magnetic symmetry is preserved, with all charged excitationsabove the gap. If the theory remains gapless but strongly coupled, we will argue in [6] thatthe electric symmetry is once again emergent at the fixed point. This connection between8he structure of conserved currents and (non)-conformality is borne out in the holographicmodel and it is one of the main points discussed in this work.We thus note that the single characteristic that allows one to identify the set of theoriesthat one might call “U(1) gauge theories coupled to matter in four dimensions” is actually theexistence of a single conserved 2-current representing conserved magnetic flux. No mentionof gauge symmetry is needed in this description. In [7] (see also earlier work in [8, 9]) thehydrodynamic theory of such a system at finite temperature was developed and shown tobe equivalent to a generalized form of relativistic magnetohydrodynamics. See also [10] fora recent discussion of magnetohydrodynamics in the conventional formulation. B. Plan for this paper
Here we outline the contents of the following sections in this paper.In section II we consider a 5 dimensional AdS bulk theory of Maxwell type for a 2-form gauge potential B . We show this theory possesses a single 2-form current dual to B .Furthermore we discuss the identification of sources and responses in this theory. It turnsout that that the source presents a logarithmic ambiguity dual to the renormalization grouprunning of a marginal operator in the dual theory. We also show that physical quantities inthis theory can be defined in terms of a renormalization group invariant scale associated toa Landau pole. This agrees with the comments above: a theory with a single 2-form currentpresents a logarithmic running.Then we consider this theory at finite temperature. We calculate the charge susceptibilityas well as the diffusion constant from quasinormal modes. They are both seen to dependon the Landau pole scale. The resistivity associated to the transport of electric charges isalso computed and found to satisfy an Einstein relation with the above quantities. We lastconsider numerically the emergence of the boundary photon degree of freedom at energiesmuch higher than the temperature.In section III the origin of Chern-Simons theories in the bulk of AdS as a consequenceof symmetry breaking of continuous symmetries down to discrete subgroups is discussed.We first review the situation for usual 0-form symmetries. We make a detailed distinctionbetween spontaneous symmetry breaking and explicit symmetry breaking and explain thatfrom the point of view of the bulk this phenomenon depends only on boundary conditionsand not on the bulk action. We recover the statements made in the previous section:conformal fixed points present in the IR either two spontaneously broken symmetries ortheir disappearance from low energy physics.This discussion is extended to the case of 1-form symmetries and connections with theholographic description of N = 4 Super Yang Mills with U(N) or SU(N) gauge groups are9utlined. The role of discrete symmetries in this case is highlighted.Lastly, in section IV, we combine the elements of previous sections. We discuss theoperator content of the theory. In the case where a relevant operator is present in the theory,in addition to the conserved currents, we obtain new infrared geometries corresponding tofixed points that break Lorentz invariance, enjoy anisotropic scaling and are a generalizationof Lifshitz geometries with a new dynamical scaling exponent ξ . These solutions are ofphysical relevance for N = 4 Super Yang Mills and its N corrections.We end with conclusions where our results our discussed and future directions are sug-gested. Finally we add three appendices with our conventions, a discussion of hydrodynamicresults for diffusive modes using the technology from [7] and a review of the spectrum ofallowed line operators in U ( N ) gauge theory. II. MAXWELL TYPE HOLOGRAPHY
We now turn to holography. Consider an antisymmetric 2-form field B propagatingin a 5d bulk. We would like the bulk action to be invariant under a gauge redundancyparametrized by a 1-form Λ: B → B + d Λ . (2.1)The simplest action one can write for this theory is the Maxwell-type action S [ B ] = 16 γ (cid:90) d x M √− gH MNP H MNP (2.2)where H = dB is the field strength of the 2-form B . In this section we will study the physicsof this system propagating on a fixed asymptotically AdS background.We note that in 5 bulk dimensions this action is the Poincare dual of a conventional1-form gauge field A , related to B via dB ∼ (cid:63) dA . Of course the physics of a conventionalMaxwell gauge field is very well-studied in AdS/CFT. It is well-understood that the gaugefield is dual to a normal one-form current j µ . In the absence of bulk objects that are chargedunder B (or A ) the calculational difference between these two systems results entirely froma difference in boundary conditions at infinity. We will see that this will result in a verydifferent boundary interpretation.We choose coordinates so that r is the holographic direction and the AdS boundary is at r → ∞ . One expects that the boundary value of the B field is related to the field-theorysource as B µν ( r → ∞ ) = b µν (2.3) In this section we use
M, N, P indices for the bulk, µ, ν, ρ for the boundary and i, j, k for the spatialcomponents of the boundary.
10s we will see, this equation is actually ambiguous: logarithmic divergences as we approachthe boundary will require us to interpret it carefully. Nevertheless, by taking functionalderivatives of the on-shell action in the standard manner, we find that the field-theorycurrent is related to the boundary value of the field-strength of H : (cid:104) J µν (cid:105) = − γ lim r →∞ √− gH rµν (2.4)The bulk equation of motion for B is ∂ M (cid:0) √− gH MNP (cid:1) = 0 , (2.5) A. Vacuum correlations and marginal deformations
Consider first a field theory that is Lorentz-invariant but not necessarily conformal, with aconserved antisymmetric current J µν . We begin by studying the theory on flat 4d Euclideanspace ( τ, x i ). Recall that J is a dimension-2 operator. Conservation of the current andanti-symmetry together imply that the vacuum momentum-space correlator must take theform (cid:104) J µν ( k ) J ρσ ( − k ) (cid:105) = (cid:18) − k ( k µ k ρ g νσ − k ν k ρ g µσ − k µ k σ g νρ + k ν k σ g µρ ) + ( g µρ g νσ − g µσ g νρ ) (cid:19) f (cid:18) | k | Λ (cid:19) (2.6)where f is a dimensionless function and Λ is some scale. If we were studying a conformalfield theory, then f would be a constant.With no loss of generality, we may rotate the momentum to point entirely in the τ direction and call it Ω. The only nonzero components of the correlator are now in the tensorchannel, i.e. the full information is captured by (cid:104) J ij (Ω) J ij ( − Ω) (cid:105) = f (cid:18) ΩΛ (cid:19) (2.7)The information of the correlator is captured by the scalar function of momentum f .Let us now turn to holography. We work on pure Euclidean AdS with unit radius: ds = dr r + r (cid:0) dτ + dx i dx i (cid:1) (2.8)We parametrize the bulk field as B ij ( r, τ ) = σ ij β ( r ) e i Ω τ (2.9)with σ ij a constant polarization tensor. (2.5) then becomes simply: ∂ r ( r∂ r β ( r )) − Ω r β ( r ) = 0 (2.10)11et us first study the asymptotic behavior of this equation as r → ∞ . Expanding thesolutions at infinity we find: β ( r → ∞ ) ∼ ˆ b − γ J log r . (2.11)At the moment ˆ b and J are just expansion coefficients, although J is so named becausefrom (2.4) we see that when multiplied by the polarization tensor it is equal to the current J σ ij = J ij . From (2.3) it appears that ˆ b should be interpreted as the source: however wesee that actually its value is ambiguous and runs logarithmically as we take r → ∞ . Notethat J is unambiguous.The running of ˆ b indicates that the physics depends on the value of r at which theboundary condition is applied. Thus the dual theory is not actually conformal. This arisesbecause J is a dimension 2 operator, and the double-trace coupling J in the boundary is marginal but not exactly so: depending on its sign the running is marginally relevant orirrelevant, and it is this logarithmic running that we are seeing here. Precisely the samephenomenon happens for pure Maxwell theory for a 1-form gauge field on AdS (where it isdouble-trace of the normal one-form current j µ that is marginal) and was discussed in detailin [11] (see also [12, 13] for earlier study in the context of a scalar field).We briefly summarize the discussion of [11] here. Consider deforming the CFT by adouble-trace coupling κ J . Via the usual holographic dictionary [12, 14], this modifies therelation between the asymptotic values of the bulk field and the source b to read: β ( r Λ ) − γ Jκ = b (2.12)where the boundary condition is now applied at a particular scale r Λ , and where we havetraded the (ambiguous) expansion coefficient ˆ b for the (well-defined but radially varying)value of the bulk field β ( r Λ ) itself. The boundary condition is thus labeled by two parameters r Λ and κ .However, the boundary condition can equivalently be applied at a different scale r (cid:48) Λ ≡ λr Λ provided we also take the double trace-coupling to transform as1 κ (cid:48) + log λ = 1 κ . (2.13)This is precisely the logarithmic running of a marginal coupling. This means that dimen-sional transmutation should occur, and all observables should depend not on κ and r Λ separately, but rather only on the RG-invariant scale r (cid:63) ≡ r Λ e κ . (2.14)For κ >
0, this scale is in the UV. To understand this, it is helpful to consider ordinary QEDcoupled to dynamical matter, which is a theory with the same symmetries as the holographic12heory we are studying here and where κ would be identified with e , the electromagneticcoupling. r (cid:63) is then the Landau pole at which the theory breaks down.We now explicitly compute the correlator, which is defined to be the ratio of the sourceand response: f (Ω) ≡ Jb = 1 β ( r Λ ) J − γ κ . (2.15)We now finally need the exact solution to the wave equation (2.10). The solution that isregular as r → β ( r ) = K (cid:18) Ω r (cid:19) . (2.16)Expanding the Bessel function at infinity and performing a short computation we find f (Ω) = 1 γ log (cid:16) Ω¯ r (cid:63) (cid:17) , ¯ r (cid:63) ≡ r (cid:63) e − Γ E , (2.17)where Γ E is the Euler-Mascheroni constant. As claimed, the r Λ and κ dependence hasreassembled into a dependence only on the RG-invariant Landau pole scale ¯ r (cid:63) . Through (2.6)this determines the vacuum correlator; note that the presence of the Landau pole introduceslogarithmic dependence on momenta and spoils conformal invariance, as anticipated. B. Finite temperature
We now consider this system at finite temperature. This is now in the universality classof the hydrodynamic theory studied in [7], except that the background magnetic field is zero;we note that a holographic study of thermodynamics and Kubo formulas with a nonzeromagnetic field was recently performed in [5]. We will study the zero-field system at lowfrequencies and momenta and look for hydrodynamic modes.We consider the system on a general black hole background of the form ds = g tt ( r ) dt + g rr ( r ) dr + g xx ( r ) d(cid:126)x (2.18)We work in Lorentzian signature, so that g tt ( r ) <
0. We assume that the metric has afinite-temperature horizon at r = r h , so that g tt ( r ) ∼ ( r − r h ) and g rr ( r ) ∼ ( r − r h ) − : wealso assume that the metric is asymptotically AdS . The detailed form of the metric willnot be important for our analysis, though for completeness we will sometimes specialize tothe AdS -Schwarzschild metric: ds = r (cid:0) − f ( r ) dt + d(cid:126)x (cid:1) + dr r f f ( r ) = (cid:18) − r h r (cid:19) (2.19)13here the temperature is related to the horizon radius r h by T = r h π . (2.20)We begin by computing the analog of the charge susceptibility, i.e. in other words, we turnon a small constant source b tx and examine the response of J tx , defining the susceptibilityas the ratio of the response to the source. In the absence of any momentum, the equationof motion (2.5) is simply ∂ r (cid:0) √− gH rtx (cid:1) = 0 √− gH rtx = − γ (cid:104) J tx (cid:105) , (2.21)where the last equality follows from (2.4). Working in a gauge B rµ = 0 and imposing theusual horizon boundary condition B tµ ( r h ) = 0, we easily find that B tx ( r ) = − γ (cid:104) J tx (cid:105) (cid:90) rr h dr (cid:48) g rr g tt g xx √− g (2.22)The covariant form of the boundary condition (2.12) is B µν ( r Λ ) − γ J µν κ = b µν (2.23)Using this to relate the field theory source b tx to the value of the bulk field B tx ( r Λ ) we findthat (cid:104) J tx (cid:105) = Ξ b tx Ξ − = γ (cid:18) κ − (cid:90) r Λ r h dr (cid:48) g rr g tt g xx √− g (cid:19) , (2.24)where Ξ is the susceptibility, for which we have now derived an explicit expression in termsof integrals over bulk metric coefficients.Evaluating this on AdS Schwarzschild we findΞ = 1 γ log (cid:0) r (cid:63) πT (cid:1) . (2.25)As claimed, we see that r Λ and κ have reassembled into the RG-invariant scale r (cid:63) .It is instructive to compare this result to the situation in free Maxwell gauge theory withgauge coupling g , in which case we find Ξ free = g . (2.26)This matches nicely with the holographic result: we do not have a precisely marginal pa-rameter g in our computation, but we should instead interpret the logarithm appearing in(2.25) as measuring the running electromagnetic coupling at the scale of interest. Noticethat in the regime of validity of the holographic regime γ (cid:28) J µν . Westudy the correlators at finite frequency ω and spatial momentum k , orienting the spatialmomentum in the z direction. The correlation function can be decomposed into threechannels by their transformation properties under the little group SO (2) of rotations in the xy plane:1. Scalar: (cid:104) J tz J tz (cid:105) . The conservation equation ∂ µ J µν = 0 sets this mode to zero, and wedo not study it any further.2. Vector: (cid:104) J ai J bj (cid:105) where ( a, b ) run over ( t, z ) and ( i, j ) run over ( x, y ). This channel isdetermined by a single scalar function. As we will see, it has a hydrodynamic diffusionmode.3. Tensor: (cid:104) J xy J xy (cid:105) . This channel contains the physics of Debye screening; however ithas no hydrodynamic structure at low frequencies, and thus we will not study it anyfurther in this work.We therefore focus on the vector channel.
1. Hydrodynamics and diffusion
Using the techniques of [15], we can calculate the retarded correlator at small ω, k on anyfinite temperature metric. We define the current J everywhere in the bulk as J µν ( r ) ≡ − γ √− gH rµν ( r ) (2.27)When evaluated at the boundary, this reduces to the field theory current via (2.4). The bulkequations of motion can be conveniently written in terms of J and H as γ ∂ r J zx + iω √− gg tt g zz g xx H tzx = 0 (2.28) − iωJ tx + ikJ zx = 0 (2.29) ∂ r H tzx − γ √− g (cid:0) iωg zz g xx J zx + ikg tt g xx J tx (cid:1) g rr = 0 (2.30)where the first two are dynamical equations of motion and the last is the Bianchi identity.We now evaluate the ratio χ ( r ; ω, k ) ≡ J zx ( r ) − H tzx ( r ) (2.31)as a function of the holographic coordinate r . As explained in [15], this is convenient as ittakes a simple value at the horizon due to infalling boundary conditions. χ ( r h ) = Σ( r h ) Σ( r ) ≡ γ (cid:114) − g − g rr g tt g xx g yy . (2.32)15n the other hand, when evaluated at the boundary it can be related to the field theorycorrelation function. Recall that the retarded correlator can be understood in linear responseas the ratio between the response and the source: (cid:104) J µν ( ω, k ) (cid:105) = − G µν,ρσJJ ( ω, k ) b ρσ ( ω, k ) . (2.33)To understand the precise relationship between G and χ we view (2.23) as an equation onthe 4d boundary and take its 4d exterior derivative. Evaluating the tzx component of theresulting equation, we find H tzx ( r Λ ) − γ κ (cid:0) ikJ tx − iωJ zx (cid:1) = db tzx (2.34)Now using the current conservation equation (2.29) to eliminate J tx and considering a sourcewhere the only component turned on is b zx , we find from (2.33) that G zx,zxJJ ( ω, k ) = − iωχ ( r Λ ; ω, k )1 − γ χ ( r Λ ; ω,k ) κ (cid:0) − k ω (cid:1) iω (2.35)Finally, we now need to evolve χ ( r ) from the horizon at r = r h to the boundary at r Λ .We thus use the bulk equations of motion (2.28) - (2.30) to obtain a flow equation for χ ( r ): ∂ r χ ( r ) = iω (cid:114) g rr − g tt (cid:18) − Σ( r ) + χ Σ( r ) (cid:18) k g xx ω g tt (cid:19)(cid:19) (2.36)In general this non-linear flow equation determines the full frequency dependence of thecorrelator and thus cannot be done analytically. However, it is very simple at low frequen-cies, and allows us to explicitly determine the hydrodynamic behavior in a manner that isindependent of the details of the bulk background.For example, if we assume a frequency and momentum scaling like ω ∼ k , then we canfind the simpler equation ∂ r χχ = − ik ω √− g rr g tt Σ g xx (2.37)which we may now integrate and insert into (2.35) to obtain the following expression for thecorrelator: G zx,zxJJ ( ω, k ) = − iω Σ( r h ) ω + iDk D ≡ Σ( r h ) (cid:20)(cid:90) r Λ r h dr (cid:48) √− g rr g tt g xx Σ + γ κ (cid:21) (2.38)In [7] it was shown that a universal definition of electrical resistivity ρ in a dynamical U (1)gauge theory is given by the Kubo formula: ρ = lim ω → G xz,xz ( ω, k = 0) − iω = Σ( r h ) . (2.39)16ote that the resistivity is given by an expression that depends only on horizon data; thiscan be thought of as a generalization of the usual holographic membrane paradigm [15]to higher-form currents. We see also that there is a hydrodynamic diffusion pole with acalculable diffusion constant D . From the expression for the charge susceptibility (2.24) wesee that the diffusion constant satisfies an Einstein relation ρ = Ξ D . (2.40)This is the diffusion of magnetic flux lines that are extended in the x direction, modulated bya small momentum k in the z direction. The diffusive behavior of magnetic flux in a mediumwith a finite electrical conductivity is of course familiar from elementary electrodynamics:interestingly, here we see it arising in a strongly coupled medium. The existence of thisdiffusion mode follows from the zero-field limit of the hydrodynamic theory developed in [7],as we review in Appendix B.To discuss the numerical values of the transport coefficients, we specialize to the AdS Schwarzschild black hole. We find the resistivity and diffusion constant to be ρ = 1 γ πT D = 1 πT log (cid:16) r (cid:63) πT (cid:17) (2.41)We may compare the resistivity to the (inverse) conductivity of the perturbative QED plasmawith electromagnetic coupling g , computed in [16] to leading order in an expansion in inversepowers of log g to be: σ − = g log g − CT (2.42)where C is a number related to the (electrically) charged particle content. While the grosstemperature dependence is fixed by dimensional analysis, our holographic result for theresistivity does not have any logarithmic dependence on the Landau pole scale: one mayinterpret this as stating that it does not depend on the electromagnetic coupling, and thus ourholographic result for the resistivity (unlike the thermodynamic result (2.25)) is significantlydifferent from the perturbative result. This is a familiar theme in holography as the wellknown result for the shear viscosity of holographic theories makes manifest. Similar to thatcase, the resistivity scales with the number of degrees of freedom (charged under the 1-formsymmetry).
2. Numerics and an emergent photon
It is not possible to go beyond the hydrodynamic limit analytically. It is however straight-forward to obtain the spectral densities at arbitrary frequencies and momenta numerically.17ere we focus on one particular feature arising from such an investigation and illustrated inFigures 1 - 3.At zero momentum the diffusion pole exhibited above sits at ω diff = 0. Numerically wealso see that at zero momentum there exists a purely overdamped pole . Its precise locationdepends logarithmically on r (cid:63) ; at large r (cid:63) T we find: ω osc πT ≈ − i . (cid:0) r (cid:63) πT (cid:1) , (2.43)where the dependence on r (cid:63) can be extracted analytically from the asymptotic structure of(2.35) at large r Λ , but the prefactor we obtained numerically. This should be considered aheavily damped plasma oscillation; it depends on the model and cannot be obtained fromhydrodynamics. Notice that at very low temperatures compared with the UV scale r (cid:63) ,log (cid:0) r (cid:63) πT (cid:1) becomes large and the pole approaches ω = 0. As we have discussed the effectivegauge coupling is given by Ξ in this theory. At low temperatures it approaches a weaklycoupled regime (although one must remember that γ (cid:28) k . In the remainder of this section wefix r (cid:63) πT = 1000 when quoting all numerical values and plots.As we illustrate in Figure 1, at small k the diffusion pole moves straight down the imag-inary axis following (2.38), and we observe numerically that the overdamped pole movesstraight up the imaginary axis. As the theory is time-reversal invariant, any pole with anonzero value of Re( ω ) must be accompanied by its time-reversal conjugate with − Re( ω ),and thus each isolated pole must remain on the imaginary axis. We focus here on two specific poles near the imaginary axis at k = 0: there also exist other non-hydrodynamic poles that we do not discuss. FIG. 1:
Cartoon illustration of movement of diffusion and plasma oscillation pole in complex fre-quency plane as k is increased from left to right; poles collide on imaginary axis at k (cid:63) πT ≈ .
037 andthen move off of axis symmetrically.
This remains true until the two poles collide at k (cid:63) πT ≈ . k we observe that Im( ω ) remains the same, but that the two poles symmetricallymove off the imaginary axis, developing increasingly larger Re( ω ). Qualitatively similarbehavior involving the merger of poles and subsequent movement off the imaginary axis hasbeen seen in other holographic examples [17–20].As we continue to increase k , eventually the dispersion relation approaches the relativistic ω ∼ k , as seen in Figure 3. In the language of conventional electrodynamics we would callthis high-momentum mode the photon . Indeed, in the QED plasma we expect that atmomenta much larger than the temperature, we expect the screening effects of the plasmato be unimportant, and thus the system should essentially behave as a free photon. Thus alinearly dispersing photon mode must somehow emerge from the hydrodynamic soup.In this holographic model, there is no regime where the system is weakly coupled, butit nevertheless appears that a similar mechanism is at play, resulting in a gapless linearlydispersing mode. In particular, this hydrodynamic to collisionless (i.e. linearly dispersing)crossover is particularly sharp (happening precisely at k (cid:63) ), and the photon mode is actuallycontinuously connected to the hydrodynamic diffusion mode. Note also that the initialpole (2.43) starts out closer to the origin at weaker electromagnetic coupling, and thus thecrossover to the free photon regime happens faster (and the hydrodynamic regime is smaller)in this case.It is an interesting question whether one can precisely interpret this emergent photon asa Goldstone boson of a generalized global symmetry at finite temperature [1, 6].19 .02 0.04 0.06 0.08 0.10 k Π T (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Im (cid:72) Ω (cid:76) Π T FIG. 2:
Movement of imaginary part of diffusion pole (top) and damped plasma oscillation pole(bottom) as a function of momentum k . We have fixed r (cid:63) πT = 1000. Note merger at k (cid:63) πT ≈ . k Π T (cid:45) (cid:45) Re (cid:72) Ω (cid:76) Π T FIG. 3:
Movement of real part of diffusion damped plasma oscillation poles as a function of momen-tum k , with r (cid:63) πT = 1000. For k < k (cid:63) both poles have zero real part; for k > k (cid:63) they symmetricallymove off the imaginary axis, approaching a relativistic linear photon dispersion (dotted line) ω ∼ k at large k . III. CHERN-SIMONS TYPE HOLOGRAPHY
In this section we will study gauge potentials with Chern-Simons couplings in the bulk.The precise Chern-Simons couplings we will study will mix two different gauge potentialstogether and are often called BF theory. It turns out that such Chern-Simons theories inthe bulk are naturally dual to discrete global symmetries on the boundary.20o understand this, we first note that continuous global symmetries in the boundary aredual to continuous gauge symmetries in the bulk. We are not aware of much study of discreteglobal symmetries in the context of holography; however from the reasoning above one mightexpect them to be dual to discrete gauge theories in the bulk. One simple way to understandthis is to artificially construct a discrete global symmetry by breaking a continuous globalsymmetry; the bulk dual of this operation will result in a discrete gauge theory, as we willexplain below.Even away from this example, however, it is generally expected that there can exist noglobal (discrete or otherwise) symmetries in the bulk of a quantum gravity theory [21–23].Thus the constraints arising from a field-theory discrete symmetry can only be encoded in abulk discrete gauge theory. We will see that the machinery of this gauge theory is required toreproduce the boundary physics of a discrete global symmetry. The Chern-Simons theoriesthat we will study are relevant precisely because they provide a continuum description ofsuch discrete gauge theories [21, 24, 25].The essential ideas here are the same both for 0-form and higher form symmetries, so wefirst work out the conventional 0-form case in detail. Here we begin with a system with afamiliar continuous U (1) symmetry and break it down to a Z k ; we find that mixed Chern-Simons terms play an important role, and that the precise realization of the symmetrydepends on boundary conditions. We then move on to the less-familiar case of a 1-formdiscrete symmetry. A. Discrete 0-form global symmetries and holography
Consider a 3d CFT with a continuous U (1) 0-form global symmetry. We will call themicroscopic current for this symmetry j e . We consider also a scalar operator O that hascharge k > φ is the bulk field dual to O , then the current sector of the CFT is representedby the following bulk action: S = (cid:90) M (cid:18) [( d − ikA ) φ ] ∧ (cid:63) (cid:2) ( d + ikA ) φ † (cid:3) − e F ∧ (cid:63) F + · · · (cid:19) , (3.1)where M represents a manifold that is asymptotically AdS . We will now break the U (1)symmetry down to Z k . We will be careful to distinguish two cases:1. Explicit breaking: we do this by adding a term w (cid:82) d x O to the field theory action:in the bulk this corresponds to demanding that φ ( r → ∞ ) ∼ wr ∆ − d (3.2)21here ∆ is the dimension of O , i.e. we turn on the large falloff at infinity and specifyits coefficient. We assume here that O is a relevant operator, ∆ < d , so that we retaincontrol over the UV of the theory. Then in the infrared there should be no remnantof the original continuous U (1) symmetry, and thus this theory should not have aconserved current associated to this charge.2. Spontaneous breaking: here we imagine adding other couplings so that O develops avev without a U (1)-breaking source added. This can be accomplished in a number ofways, e.g. adding a chemical potential as in holographic superfluids [26–28] or moresimply via a symmetry-preserving double-trace coupling O † O as in [29]. In this casewe have φ ( r → ∞ ) ∼ (cid:104)O(cid:105) r − ∆ , i.e the small falloff at infinity.In both cases, however, the bulk field φ ( r ) develops a non-trivial profile. We denote itsmagnitude by ρ ( r ), i.e. φ ( r ) = ρ ( r ) e iθ ( r ) with θ ( r ) a bulk Goldstone mode. The distinctionbetween spontaneous and explicit breaking is contained in the form of ρ ( r ) at large r .The low-energy bulk action then becomes S = (cid:90) M (cid:0) ρ ( dθ − kA ) ∧ (cid:63) ( dθ − kA ) (cid:1) + · · · (3.3)We now dualize the Goldstone θ to a 2-form gauge field B in the usual manner (see e.g.[21, 24]). We find after dualizing that S = (cid:90) M (cid:18) k π A ∧ dB − ρ dB ∧ (cid:63) dB (cid:19) + · · · (3.4)Here B and A have periods (cid:82) M A = 2 π Z , (cid:82) M B = 2 π Z for all closed 1 and 2-cycles M and M . The last term in the action is irrelevant from the point of view of the bulk, andwe will ignore it from now on. Note this implies that the difference between spontaneousand explicit breaking will then be contained in the boundary conditions that we impose onthe other fields at large r .The key physics is in the first term: this Chern-Simons action describes a topologicalfield theory in the bulk, defining a discrete Z k gauge theory. There are no local degrees offreedom, as the equations of motion set both connections to be flat in the bulk: dA = 0 dB = 0 . (3.5) The fact that boundary conditions affect the symmetry algebra of the boundary theory is a well knownfact in the context of the Chern-Simons formulation of
AdS gravity and its higher spin generalizationswhere the Drinfeld-Sokolov reduction is responsible from the reduction of the affine algebra ˆ sl ( N ) downto W N . Z k gauge theory is instead in the braiding of massive excitationsthat are charged under the gauge fields. Here we have unit-charged particles that coupleto the 1-form gauge field as e i (cid:82) C A with C a 1-dimensional worldline: these are excitationscorresponding to quanta of the bulk field dual to a unit-charged operator in the boundary.This operator is uncondensed and thus its quanta must remain massive. On the other hand,recall that the charge- k field φ in the bulk is condensed, and thus its quanta do not coupleto A as massive particles. We also have strings that couple to B as e i (cid:82) W B , with W a 2-dimensional worldsheet: in the original scalar representation of the theory (3.3) these arevortices that carry magnetic flux πk . The non-trivial braiding of particles and strings that iscaptured by the Chern-Simons term in (3.4) is just the Aharonov-Bohm phase of particlesaround flux tubes.We now turn to the holographic interpretation of this bulk theory, which apparentlyshould be dual to a boundary theory with either a global U (1) symmetry spontaneouslybroken to Z k or a theory with only a Z k symmetry, depending on boundary conditions. Wethus study the variation of the action (3.4) in the presence of a boundary.We first study the case that corresponds to the spontaneously broken symmetry. In thiscase we should add boundary terms such that the total action is S tot = k π (cid:90) M B ∧ dA + 12 (cid:18) gk π (cid:19) (cid:90) ∂ M B ∧ (cid:63) B, (3.6)where g is a free parameter that (as we will see) represents non-universal physics, and wherewe have picked its normalization to simplify subsequent equations. On-shell, the variationarises from a boundary term: δS tot = k π (cid:90) ∂ M B ∧ (cid:18) δA + g k π (cid:63) δB (cid:19) (3.7)From this variational principle we conclude that we should take the field theory current j and source a to be j e = k π (cid:63) B (cid:12)(cid:12) ∂ M a = − (cid:20) A + g k π (cid:63) B (cid:21) ∂ M (3.8)Finally, with the benefit of hindsight we define a 2-form j m that is the appropriately nor-malized Hodge dual of j e : j m ≡ g (cid:63) j e (3.9)This structure now captures all of the universal physics of a U (1) symmetry spontaneouslybroken down to Z k : This boundary term is inspired by the 5 dimensional analog of this story discussed in detail in [30]. Wewill discuss their construction in more detail when we move to the case of 1-form global symmetries below.
Conserved currents and Goldstone mode : the bulk equations of motion (3.5) implythe following equations for j e and j m : d (cid:63) j e = 0 d (cid:63) j m = da (3.10)We have a locally conserved 1-form current j e , arising from the original spontaneouslybroken symmetry. We also have a 2-form current j m that is conserved up to a localfunction of the applied source: though this language is not usually used for a superfluid,this should be thought of as a mixed anomaly as in (1.11) or (1.13). Thus we haveboth a U (1) 0-form and a 1-form symmetry: the 0-form symmetry is the originalmicroscopic U (1), but the 1-form symmetry is emergent in the infrared and measuresvorticity. Here we will mostly focus on the realization of the 0-form symmetry.The second equation implies that j e can locally be written as j e = 1 g ( dψ − a ) (3.11)with ψ a 0-form. Note that g − plays the role of the superfluid stiffness. The conser-vation equation for j e then implies that d (cid:63) ( dψ − a ) = 0 , (3.12)which is precisely the equation of motion for a U (1) Goldstone mode ψ in the presenceof an external source a .2. Ward identities in the presence of charged operators : we now consider adding chargedobjects in the bulk. We study a minimally electrically charged particle moving along abulk curve C that intersects the boundary at two points x and x : this is holograph-ically dual to an insertion of the unit-charged operator Ψ and its conjugate at x and x . We thus add a Wilson line term (cid:82) C A to the action to find that the equation ofmotion for B and thus the conservation equation for j is modified to read k π dB = − δ ( C ) d (cid:63) j e = δ ( x ) − δ ( x ) (3.13)where δ ( C ) is a delta function along the curve C . The resulting non-conservationof j is precisely the Ward identity for the current in the presence of the unit-chargedoperator Ψ( x ) at the points x and x .The other possible source is a 2-dimensional worldsheet coupling to B in the bulk as (cid:82) W B , intersecting the boundary along a 1-dimensional curve ∂W . In this case it is Note that if a bulk differential form vanishes, its projection down to the boundary also vanishes. A and d (cid:63) j m that are modified: k π dA = − δ ( W ) d (cid:63) j m = da − πk δ ( ∂W ) (3.14)In the field theory, the fact that j m has a source is usually interpreted as a unit-vorticity vortex along the curve ∂W . To understand this, first consider setting theexternal source a to 0 and integrating j e over the boundary of an arbitrary large disc D that includes ∂W : (cid:82) ∂D j e = (cid:82) D dj e = − g (cid:82) D d (cid:63) j m = πg k . We see that there isa net long-distance circulation of the microscopic current around ∂W , as we expectfor a superfluid vortex. On the other hand, if we now turn on a , then this circulationcan be stopped provided the net flux in a is (cid:82) ∂W a = πk , which is the expected fluxquantization for a Z k vortex. This is the Ward identity for the 1-form symmetry. † ( x ) ( x ) FIG. 4:
Computing two-point function of Ψ holographically: as the bulk charged worldline cannotbreak, the answer always depends strongly on separation between endpoints. Z k order parameters : finally, we look for order parameters for the broken symmetry.As the unit-charged operator Ψ transforms under the unbroken Z k , it does not developa vev and thus the two-point function (cid:104) Ψ † ( x )Ψ( x ) (cid:105) should vanish at large separation,lim | x − x |→∞ (cid:104) Ψ † ( x )Ψ( x ) (cid:105) = 0 (3.15)As described above, this computation of this two-point function requires us to add aWilson line (cid:82) C A to the action. By assumption the quanta of Ψ are massive in the bulkand such a Wilson line will also come with a term m (cid:82) C ds that measures the length L along the bulk worldline. A single Wilson line cannot break, and thus the geometriclength grows with distance and will always suppress the correlator as exp( − mL ) atlarge spacelike separation, as in Figure 4.Consider now the operator Ψ k . This is invariant under the unbroken Z k and so In the following discussion the properties of O should be completely analogous to those of Ψ k . We choose | x − x |→∞ (cid:104) (Ψ k ( x )) † Ψ k ( x ) (cid:105) = |(cid:104) Ψ k (cid:105)| (3.16) ( k ( x )) † k ( x ) FIG. 5:
Once k Wilson lines can end on a bulk monopole, the 2-point function of widely separatedinsertions of Ψ k ( x ) will be dominated by configurations like this, where the answer is independent ofseparation. In the bulk we now have k Wilson lines. The key fact here is now that k Wilson linesactually can end in the bulk, provided that they end on a monopole event in B , i.e at a bulkpoint X where d B ( X ) (cid:54) = 0, or more properly where (cid:90) S ( X ) dB = 2 π, (3.17)where the integral is taken over a small S surrounding X . To understand this, let usconsider the bulk action with k Wilson lines ending on a point X which we excise from themanifold: S = k (cid:90) C A + k π (cid:90) M A ∧ dB (3.18)Now we perform a gauge transformation A → A + d Λ that vanishes at the boundary.The variation of the action receives, then, only contributions from the region around themonopole as δ Λ S = k Λ( X ) − k π (cid:90) S ( X ) Λ dB (3.19)where the last term is the boundary term from the gauge variation of the Chern-Simonsterm. Thus we see that the termination of k worldlines is consistent with gauge invarianceprovided that they end on a monopole in B . Once k worldlines can break, the correlator will to discuss Ψ k as it makes it manifest that it represents a source for k Wilson lines associated to Ψ. D C k ( x ) k ( x ) C FIG. 6:
When k bulk Wilson lines can end on a monopole event in B in the bulk, a Dirac string C emerges from the monopole: we will distinguish the case where the string stays in the bulk (left) andthe case where the string hits the boundary at x D (right) eventually saturate at a value independent of separation as we expect for a broken symmetry,as we see in Figure 5.We see that a very important role is played by the monopole events. In this case themonopole event is just an insertion of the original UV-complete field φ (as one can verify bytracing back through the duality and noting that the field e iθ carries the correct monopolecharge), but in general in quantum gravity we expect that such objects will always existsuch that the charge lattice is filled (see e.g. [21–23]).We now turn to the case of the explicitly broken symmetry. Our discussion mathematicallyparallels that of [31], though our interpretation is slightly different, as we focus on therole played by the conserved currents. To understand this, we first note that we havegiven physical importance to the 2-form gauge field B : from (3.8), it defines the field-theory current. We have also explained that we should allow monopole events in B . Thiscombination may seem somewhat dangerous, as in the presence of a monopole, the field B is not well-defined. Said differently, from the location of the monopole X emerges a “Diracstring” (as shown in Figure 6) which in this case is a 1-dimensional worldline C around whichwe have (cid:82) S ( C ) B = 2 π . In general Dirac strings are thought to be completely unobservable,as they have trivial braiding with any charged excitations and can be moved around by bulkgauge transformations.In the presence of a boundary, however, this is not true. Indeed, the distinction betweenthe spontaneous and explicitly broken symmetry depends on whether or not the Dirac stringis allowed to intersect the boundary.If the Dirac string is not allowed to intersect the boundary, then the discussion of the pre-27ious few paragraphs applies: B evaluated at ∂ M remains well-defined as it never intersectsthe Dirac string. There is no subtlety in the definition of j (3.8) and thus the conserved j e implies that we have a continuous symmetry.If the Dirac string is allowed to intersect the boundary, then the the charge defined asan integral on a 2-manifold M Q = (cid:90) M (cid:63)j e ≡ k π (cid:90) M B (3.20)jumps discontinuously by k as M is dragged across the Dirac string. Thus the intersectionof the Dirac string with the boundary corresponds to the boundary insertion of a k -chargedoperator. In the presence of such insertions we no longer have a continuously conservedcurrent, but the Z k valued object Q = exp (cid:18) πik Q (cid:19) (3.21)still defines a conserved charge, in that its value does not change as M is moved throughthe (end of the) Dirac string, and the preserved symmetry is Z k .Let us now examine whether the boundary conditions discussed above actually allow theDirac string to intersect the boundary. Our discussion here will be mostly heuristic.The boundary term in (3.6) associates an action cost to the existence of a Dirac string.Indeed, given the quantization condition (cid:82) S B = 2 π Z for a sphere surrounding the end ofthe string on the boundary we know the boundary term in the action (3.6) scales as: S Dirac ∼ g Λ (3.22)where we have only kept track of the g -dependence; Λ is a UV cutoff, and the answer is UVdivergent due to the divergence of the Goldstone mode near the core of the charge. TheUV divergence indicates the configuration is not normalizable and, therefore, not allowedwithout the inclusion of further boundary terms that would cancel it. These terms would bedirectly responsible for the disappearance of the continuous symmetry. They are howevernot available to us in the effective low energy description (3.6). In the absence of these termswe conclude that for any finite g , UV divergences prohibit these Dirac strings, and we finda continuous U (1) symmetry spontaneously broken down to Z k .On the other hand, as g →
0, Dirac strings are energetically allowed. Each intersectionof the Dirac string with the boundary corresponds to the insertion of a charged field theoryoperator; thus these boundary conditions correspond in the field theory to having a non-trivial charged source turned on as in (3.2). From the point of view of (3.6) this is somethingof a singular point, as the boundary conditions degenerate to A = 0 at the boundary. Theonly information that remains in this theory is associated with topological objects such as28he charge operator (3.21) and Wilson lines (3.13). Such topological objects are the onlyuniversal information that we expect from the holographic representation of a Z k discretesymmetry. In particular, the vortex configurations (3.14) violate the boundary conditionson A and are no longer allowed.We summarize:1. Boundary conditions (3.6) with finite g correspond to the case of a continuous U (1)symmetry spontaneously broken down to Z k with g − corresponding to the superfluidstiffness for the associated gapless mode.2. Boundary conditions (3.6) with g = 0 correspond to a field theory with only a Z k symmetry, dual to a completely topological theory encoding the algebra of Z k charges. B. Discrete 1-form global symmetries and holography
In the previous section, we extensively discussed a mixed Chern-Simons term (3.4) ina four-dimensional bulk and showed that it represents the physics of a discrete Z k Z k was actually embedded into a spontaneously broken U (1) symmetry.Now we turn to the higher-form analog of this story: in other words, we introduce two2-forms B and C and study the mixed Chern-Simons action S CS [ B, C ] = k π (cid:90) M C ∧ dB (3.23)where B and C are invariant under separate 1-form gauge invariances: B → B + d Λ C → C + d Γ (3.24)where Λ and Γ are 1-forms. Invariance of the (exponential of the) action under large gaugetransformations requires k to be integer. The theory described by (3.23) defines a Z k gaugetheory in the bulk, describing the braiding statistics of string worldsheets that couple to B and C . From the arguments in the previous section, we expect that this theory should bedual to a discrete Z k symmetry that may (depending on boundary conditions) be embeddedinside a spontaneously broken higher-form U (1). A (mostly psychological) difference fromthe previous section is that it is not simple to fully UV complete this theory in the bulk:the analog of (3.1) is not straightforward. This is because the fundamental objects chargedunder the action of 1-form symmetries are not particles but extended objects. Thus ourdiscussion will be purely in the Chern-Simons formulation.29e begin by noting that this theory has been extensively studied for a variety of purposes[30, 32–34]. A careful quantum treatment has been performed in [35]. Here we will essentiallyre-cast existing results in the language used in the previous section. We will discuss theresults purely in terms of objects appearing in the low-energy action: however this preciseterm appears in Type IIB on AdS × S , and we will discuss the connection to N = 4 SYMat the end.We begin by studying the theory on a manifold with a four-dimensional boundary. Westudy the action with the following boundary terms [30, 36]: S tot [ B, C ] = S CS [ B, C ] + 12 (cid:18) gk π (cid:19) (cid:90) ∂M C ∧ (cid:63) C (3.25)Here g is a free parameter whose normalization we have picked so that it turns out to be theboundary Maxwell coupling. The bulk term in the variation is proportional to the equationsof motion, which simply require that B and C be flat: dB = 0 dC = 0 . (3.26)The boundary variation is δS tot [ B, C ] = k π (cid:90) ∂M C ∧ (cid:18) δB + g k π (cid:63) δC (cid:19) (3.27)We can now identity the field theory current J e and 2-form source b e : J e ( x ) = (cid:63) k π C (cid:12)(cid:12)(cid:12)(cid:12) ∂M b e ( x ) = − (cid:18) B + g k π (cid:63) C (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) ∂M . (3.28) J e is the microscopic U (1) 2-form current; following (1.5), we also define a magnetic currentas J m ≡ g (cid:63) J e (3.29)We may now study all of the same considerations as in the previous section:1. Conserved currents and Goldstone photon:
We begin with a discussion of the localconservation equations. From the bulk flatness conditions (3.26) and the definition ofthe source (3.28) we find d (cid:63) J e = 0 d (cid:63) J m = db e (3.30)These conservation laws are the higher-form analog of (3.10), describing two conserved2-form currents J e and J m . From the point of view taken here, J m is an emergentsymmetry. These are also precisely equivalent to the conservation laws derived fromthe action for a free U (1) Maxwell gauge field (1.11). In (3.10) we showed that the30urrent could be written in terms of the action of a free Goldstone: here we can followprecisely the same logic to conclude that J e = 1 g ( dA e − b e ) d (cid:63) ( dA e − b e ) = 0 (3.31)with A e an arbitrary 1-form that is the higher-form Goldstone mode. It is also theMaxwell photon. Thus the Maxwell photon can be thought of as the Goldstone modeof a spontaneously broken 1-form symmetry [1].2. Ward identities in the presence of charged operators:
The two possible charged oper-ators are string worldsheets W that couple to B and C respectively. For strings thatcouple to B we add a term (cid:82) W B to the action to find k π dC = − δ ( W ) d (cid:63) J e = δ ( ∂W ) (3.32)where ∂W is the one-dimensional intersection of the string worldsheet with the bound-ary: we see that in the field theory this boundary represents a line-like operator L B ( ∂W ) that is charged under the electric symmetry J e .We now add a string that couples to C : we find k π dB = − δ ( W ) d (cid:63) J m = (cid:18) db e − πk δ ( ∂W ) (cid:19) (3.33)Thus the line-like operator L C ( ∂W ) living on ∂W is charged under J m . From thepoint of view of the boundary photon, ∂W is the worldline of a magnetic monopole.Note that it appears to have fractional charge; we will return to this point later.3. Z k order parameters: We first recall what it means for a line-like operator to be“condensed”: as described in [1], we say that a line-like operator is condensed if itobeys a perimeter law, as this is the higher-form generalization of the factorization oflocal operators (3.16). Any dependence on the geometric data characterizing the loopthat is stronger than this (e.g. an area law) is the analog of the uncondensed result(3.15).Now consider a single bulk string coupled to B , dual to the insertion of the lineoperator L B ( ∂W ). As the string cannot end in the bulk, its bulk tension will resultin an expectation value (cid:104) L B ( ∂W ) (cid:105) that depends on its radius more strongly than aperimeter law (though, depending on the IR geometry, perhaps less strongly than anarea law). Thus we say that L B ( ∂W ) is uncondensed.We now turn to L B ( ∂W ) k , dual to k bulk strings. Following arguments preciselyanalogous to those around (3.18), k bulk strings can end on a monopole in C , i.e. on31 one-dimensional worldline P around which we can integrate dC : (cid:90) S ( P ) dC = 2 π (3.34)This monopole will generally have mass, but the tension in the strings will genericallypull this object towards the boundary, effectively localizing the worldsheet of the B -string near the boundary and resulting in a true perimeter law for (cid:104) L B ( ∂W ) k (cid:105) . Thus L B ( ∂W ) k is condensed and we see the U (1) symmetry generated by J is spontaneouslybroken to Z k .Precisely the same arguments apply for k copies of the object charged under C , where k C -strings are allowed to end on a monopole in B . Thus if all monopole events areallowed, the symmetry generated by (cid:63) J is also spontaneously broken down to Z k .We now turn to the issue of spontaneous versus explicit breaking of the U (1) (cid:48) s : again, ourdiscussion parallels that around the lower-form case. The monopole in C has a Dirac string(which is a 2-dimensional object around which (cid:82) S C = 2 π ): if this C -type Dirac string isallowed to intersect the boundary then the current J e ceases to be well-defined and we canonly consider the exponential of the integrated charge Q e ≡ exp (cid:18) πik (cid:90) M (cid:63) J e (cid:19) (3.35)which is only defined modulo k , breaking the symmetry down to Z k .Similarly, the monopole in B can have a Dirac string: here the situation is slightlydifferent, as we see from (3.28) that actually the definition of the source b itself has become-ill defined. If we assume that the source is trivial, then we can conclude that the gauge-invariant charge is again the Z k -valued object Q m ≡ exp (cid:18) ik (cid:90) M (cid:63) J m (cid:19) (3.36)As discussed around (3.22), the energetics of the intersection of the Dirac strings withthe boundary depends on the value of g . A computation paralleling (3.22) shows that theboundary action of a C -type Dirac string (i.e. a boundary charge for J e ) scales as g Λ L andthat of a B -type Dirac string (i.e. a boundary charge for J m ) scales as g − Λ L , with Λ theUV cutoff and L the boundary length of the intersection. Thus we conclude that1. For finite g , neither type of Dirac string is permitted: the boundary symmetry is U (1) e × U (1) m spontaneously broken down to Z k × Z k , and g is the gauge coupling ofthe boundary photon. 32. For g = 0, C -type Dirac strings are permitted. This breaks U (1) e explicitly down to Z k : note that the boundary conditions become simply B = 0, which prohibits themagnetic charges (3.33) and thus we have only the single electric Z k .3. For g = ∞ , B -type Dirac strings are permitted, breaking U (1) m explicitly down to Z k . The boundary conditions are now C = 0, prohibiting the electric charges (3.32)and leaving only the single magnetic Z k .Up till now, our discussion has been purely in terms of the objects appearing in thelow-energy action. We now discuss the connection with SU ( N ) N = 4 Super-Yang-Mills.In particular, since the early days of AdS/CFT, it has been known that the usual action ofType IIB string theory compactified on AdS × S has precisely such a Chern-Simons term,with k = N , B being the NS-NS 2-form, and C the R-R 2-form [32, 33, 37]. Thus our resultsmay immediately be taken over. The objects coupling to B are fundamental strings, andthose coupling to C are D1 branes; they are dual respectively to Wilson and t’Hooft lineson the boundary. Monopoles in B are D S (i.e. Witten’s baryonvertex [38]), and monopoles in C are N S S .Note now that the global form of the gauge group determines the spectrum of allowedline operators [39–41] and thus is relevant for the structure of generalized symmetries [1, 42].The three cases above seem to realize U ( N ), SU ( N ), and SU ( N ) / Z N , as we now discuss .Case 1 corresponds to U ( N ) gauge theory. The full generalized symmetry group of U ( N )gauge theory is U (1) e × U (1) m . As we briefly review in Appendix C, in U ( N ) gauge theoryone is allowed both Wilson and t’Hooft lines. From the point of view of the continuous U (1)generalized symmetry currents, minimally charged t’Hooft lines appear to have magneticcharge that is 1 /N -th the minimum U (1) Dirac quantum; this is precisely what we see in(3.33). The “singleton” boundary photon identified above can be thought of as the U (1)factor of the U ( N ) gauge group; as expected, it lives on the boundary and does not interactwith the bulk except through charged objects.Case 2 corresponds to SU ( N ) gauge theory, where we have Z N Wilson lines (i.e. funda-mental strings coupled to B ) but t’Hooft lines are not allowed.Case 3 corresponds to SU ( N ) / Z N gauge theory, where we have Z N t’Hooft lines (i.e. D1strings coupled to C ) but Wilson lines are not allowed.As far as we understand the precise classification above is novel but is broadly consistentwith the existing literature on this subject. It would be instructive to subject this pictureto more detailed tests. We are very grateful to D. Tong for instructive discussions about the contents of this section. V. MAXWELL-CHERN-SIMONS TYPE HOLOGRAPHY
In the first part of this paper we considered the Maxwell term alone for a 2-form gaugefield, and in the second we considered a mixed Chern-Simons term for two 2-form gaugefields. We now consider combining these ingredients by studying both together, i.e. westudy the action S = (cid:90) d x √− g (cid:18) − γ √ gH MNP H MNP − γ (cid:48) G MNP G MNP + k π (cid:15) MNP QR B MN G P QR (cid:19) (4.1)where H = dB and G = dC .Note that this is the most general quadratic action for the fields ( B, C ). However, from thepoint of view of the bulk, the Maxwell terms are irrelevant perturbations to the long-distancephysics described by the Chern-Simons term. In addition to the physics of flat connectionsdescribed in the previous section, we will now have an extra topologically massive mode forthe gauge fields [43]. Similar topologically massive bulk gauge fields have been studied inthe context of AdS /CFT in [44, 45].Note that we may apparently remove a parameter from the problem by rescaling C toobtain the same normalization for the two bulk Maxwell terms S = 1 γ (cid:90) d x √− g (cid:18) − H MNP H MNP − G MNP G MNP + λ(cid:15) MNP QR B MN G P QR (cid:19) (4.2)where λ ≡ kγ (cid:48) γ π · . The quantum physics still depends on γ (cid:48) as the rescaling modifies thequantization conditions on the periods of C [44]. However in this section our considerationswill be purely classical in the bulk, and we can express all of our results in terms of λ and γ . A. Operator content
We begin by describing the operator content of the dual theory. Consider first setting thecoefficient of the Chern-Simons mixing term λ to 0. This results in two copies of the theorystudied in Section II, which will have two decoupled boundary currents, each of dimension2. If we now turn on the Chern-Simons coupling, the IR structure of the bulk theory isstrongly modified: as described in detail in Section III B, the flat part of both ( B, C ) isnow dual to a single tensor operator J . As the number of degrees of freedom should remainthe same in the presence of the mixing term , the non-flat part of ( B, C ) must contribute An interesting subtlety is that in the theory with no mixing term, we have two separately conservedcurrents. However in the theory with the mixing term, we have a single conserved current J that obeys ∇ M G MNP − λ(cid:15) ABCNP H ABC = 0 , (4.3) ∇ M H MNP + λ(cid:15) ABCNP G ABC = 0 . (4.4)It is often convenient to assemble these into a single complex 2-form Z and its field strength W : Z ≡ B + iC W ≡ dZ = H + iG (4.5)in which case the two equations of motion can be combined into one, which we write in formnotation as d (cid:63) W = − λiW . (4.6)Following a treatment of a lower-dimensional problem in [45], we would now like to separate Z into a flat part Z and a non-flat part ζ : Z = Z + ζ dZ = 0 (4.7)where Z is flat and presumably ζ contains the massive mode that we are interested in. Ofcourse this split is ambiguous, as we can always transfer more flat parts of the connectioninto ζ . To fix this ambiguity, we first note that ζ satisfies the equation d (cid:63) dζ = − λidζ . (4.8)We may now choose ζ such that it satisfies the following equation: ζ = i λ (cid:63) dζ (4.9)(4.9) implies that (4.8) is satisfied: it is however not the most general solution to (4.8), andthe choice of this particular ζ amounts to a particular division of the connection into flatand non-flat pieces.The physics stored in the flat part Z was described in the previous section: we wouldnow like to study the physics in ζ . To that end, we study linearized perturbations aroundLorentzian AdS , written as ds = dr r + r (cid:0) − dt + dx + dy + dz (cid:1) . (4.10) two separate conservation equations (for J and (cid:63) J ), and another higher-dimension operator that obeys noconservation law at all: thus the number of independent components is preserved though the constraintsare redistributed.
35e would first like to determine the conformal dimensions; thus we study solutions to (4.9)that are independent of the field theory directions. Notice that this implies ζ rµ = 0 as oursolution satisfies dζ | boundary = 0. ζ is, therefore, a 2-form with components only in the fieldtheory directions. The first order equation (4.9) becomes: r∂ r ζ = 6 iλ (cid:63) ζ (4.11)where (cid:63) is the 4d Hodge star with respect to the flat Lorentzian metric ds = − dt + dx i dx i .It is now useful to introduce the projectors onto self-dual and anti-self-dual 2-forms in 4d: P ± ≡
12 (1 ± i(cid:63) ) i (cid:63) P ± = ± P ± P ± = P ± P + P − = 0 (4.12)Defining a basis of definite chirality boundary 2-forms using ζ ± = P ± ζ , we see that (4.11)becomes r∂ r ζ ± = ± λζ ± (4.13)and thus the general solution takes the form ζ ( r ) = r λ ζ + + r − λ ζ − (4.14)Thus we see the expected two falloffs at infinity, where the corresponding polarization tensorsobey a certain projection condition. This is the usual structure at infinity for a first-orderdynamical system in AdS/CFT (see e.g. the well-studied case of fermions [46, 47]).Via the usual rules we expect that if λ > ζ + is the source and ζ − is the response.Note that as λ →
0, the two solutions coincide and we obtain the logarithm seen in (2.11).To find the dimension ∆ of the dual operator, we note that regardless of the spin of theoperator, the difference between the two exponents is always equal to the difference between∆ and 4 − ∆, which means that ∆ = 2 + 6 | λ | (4.15)The dimension is always given by the expression above, though the choice of which of thetwo falloffs is normalizable depends on the sign of λ so that we remain above the unitaritybound for a conserved current.The existence of this operator is somewhat interesting: as it arises from the quadraticpart of the bulk action, it is a generic feature of any holographic theory. We also note thatin the dual to a large N gauge theory, λ ∼ N − [33] and thus the dimension is very closeto that of a conserved current. It would be interesting to understand if this operator has aclean interpretation in the dual theory.Finally, we note that our treatment is incomplete: technically speaking, a careful identi-fication of sources and vevs requires that we holographically renormalize the theory definedby (4.2). We leave such an analysis for future study.36 . Backreacted scaling solutions In this section we couple the above system to gravity and demonstrate the existence ofnew anisotropic scaling solutions. We study stationary points of the following action: S = (cid:90) d x √− g (cid:20) κ ( R + 12) + 1 γ (cid:18) − H MNP H MNP − G MNP G MNP + λ(cid:15) MNP QR B MN G P QR (cid:19)(cid:21) , (4.16)where we are working in units where this system admits an AdS vacuum with unit AdSradius. The full equations of motion are those for the gauge fields (4.3) and (4.4), togetherwith those arising from varying the metric:12 κ (cid:18) R AB − g AB R − g AB (cid:19) + 16 γ (cid:18) g AB ( H + G )2 − H ANP H NPB − G ANP G NPB (cid:19) = 0(4.17)Note that the Chern-Simons term does not directly contribute to the gravitational equationsof motion as it is topological: however, as it affects the dynamical equations for the gaugefields it dramatically changes the character of the allowed gravitational solutions.We first briefly discuss known solutions when λ = 0. In this case we have two decoupled2-form gauge fields coupled to gravity. In the 5d bulk these 2-forms can be dualized to 1-formvector fields, and we are thus simply discussing solutions to the very well-studied Einstein-Maxwell theory in AdS in a different bulk duality frame. If these solutions carry electriccharge, then we have the well-known AdS-Reissner-Nordstrom black branes [48], which haveAdS IR asymptotics at zero temperature (see e.g. [49, 50] for reviews). On the other hand,if they have a nonzero magnetic field along (say) the x direction, then an asymptoticallyAdS solution is not analytically known, but there exists an exact IR scaling solution that isAdS × R , where the AdS is made out of ( t, r, x ) [51, 52]. Returning to the duality frameused in this paper, such solutions correspond to having a nonzero boundary J tx and havebeen studied from the point of view of generalized symmetries and magnetohydrodynamicsin [5].We now return to finite λ . Somewhat surprisingly, we can still find exact scaling solutionsto the backreacted system, though we have not been able to analytically construct a fullbulk RG flow to AdS in the UV. We expect that such RG flows could be found numerically.The IR solution is a product of AdS and a shrinking R . It is similar to a Lifshitzgeometry [53], in that the dimensions ( t, x ) scale at a different rate from ( y, z ). From thisperspective they represent the emergence of a (deformed) CF T in the IR living on theworldsheet of magnetic flux tubes in the boundary CF T . The solution is: B = b u dt ∧ dx C = c u ξ dy ∧ dz ds = L du − dt + dx u + 1 u ξ (cid:0) dy + dz (cid:1) , (4.18)37hich is invariant under the scaling isometry u → σu t → σt x → σx y → σ ξ y z → σ ξ z (4.19)and thus ξ plays a role analogous to the Lifshitz dynamical exponent z . As usual for scalinggeometries, the solution is unique in that the parameters appearing in the solution arecompletely fixed in terms of bulk coupling constants. A solution of the equations of motionis found provided that L = (cid:114) (cid:115) − λ + √ − λ − λ (4.20) b = L κ √ − L + 81 L λ (4.21) c = b λL (4.22) ξ = (3 Lλ ) − (4.23)If we imagine taking λ →
0, then we see that L → √ and ξ → ∞ : the C field becomes puregauge and decouples, and the ( y, z ) directions cease to shrink. The bulk geometry becomesthe magnetic brane AdS × R of [51].For nonzero λ this is a novel solution. We note also that at λ = these solutions becomeonce again AdS backgrounds: ξ → L →
1. The gauge fields turn offand we recover the purely gravitational solution. Beyond this point solutions cease to exist,as they can no longer be supported by fluxes. While it is hard to interpret this fact withoutknowing the exact interpolating solutions from
AdS to the IR, it is interesting to noticethat the solutions exactly disappear when the massive mode found in (4.14) becomes dualto marginal boundary operators (4.15). One might expect that once that value is crossedno deformation caused by such operator can affect the IR, so new scaling solutions wouldnot be available at λ > / solution, and the resulting spacetime is dual to a particular stateof N = 4 SYM, presumably corresponding to color flux tubes oriented in the x direction. Itwould be very interesting to understand the physics described here from the field-theoreticalpoint of view. V. CONCLUSION
In this work we have studied various aspects of generalized global symmetries in quantumfield theories with holographic duals, focusing on 1-form symmetries in four-dimensional38uantum field theories. We briefly summarize the main points of our analysis below.We began with a study of a single continuous conserved 2-form current J , dual to anantisymmetric tensor field with a Maxwell action in a five-dimensional bulk. We showed thatthis field theory is not conformal: instead the double-trace coupling J runs logarithmicallyand the theory has a Landau pole in the UV. We further studied this theory at finitetemperature, computing transport coefficients and showing the existence of a diffusion modethat is compatible with the hydrodynamic analysis of [7].We then turned to a study of discrete symmetries. We began with the case of a 0-formdiscrete symmetry: this is just a conventional discrete symmetry in field theory (i.e. wherethe charge is defined on a codimension-1-manifold, or a “time-slice”). We argued that ingeneral, a field-theoretical discrete symmetry is holographically dual to a discrete gaugetheory in the bulk. It is well-known that such gauge theories have a low-energy descriptionin terms of a mixed Chern-Simons theory, and we explained in detail how to understand theuniversal physics of the discrete symmetry from the bulk topological theory. We also showedthat this discrete symmetry may be embedded inside a continuous symmetry which can bespontaneously broken; in the Chern-Simons description, the associated Goldstone bosoncan be thought to live on the boundary. The distinction between explicit and spontaneousbreaking arises from different boundary conditions on the Chern-Simons gauge fields.Next, we studied a 1-form discrete symmetry, which has a similar Chern-Simons descrip-tion in terms of 2-form antisymmetric tensor gauge fields. This case is relevant for the studyof N = 4 super-Yang-Mills theory, which is expected to realize (at least) a discrete higherform symmetry. The precise symmetry structure and spectrum of charged line operators de-pends on the precise presentation of the gauge group: in particular, we clarify the distinctionbetween the holographic duals of the U ( N ), SU ( N ), and SU ( N ) / Z N gauge theories and ex-plain the holographic boundary conditions that realize the generalized symmetry structureexpected for the three different cases. In the U ( N ) case there is a continuous Abelian global1-form symmetry that is spontaneously broken down to a discrete subgroup: we identify theboundary photon (i.e. the “U(1)”) as the Goldstone mode of the symmetry breaking.Finally, we studied the bulk theory with both the Maxwell and Chern-Simons terms forthe 2-form gauge fields. Here the higher-derivative Maxwell terms result in new massivemodes in the bulk which are dual to higher-dimension tensor operators in the boundary. Weperform a preliminary analysis of this theory, computing the dimension of the new operator.We also study gravitationally backreacted solutions to this theory, finding an exact IR scalingsolution that appears to be dual to color flux tubes extended in one of the spatial directions.There are many directions for future research. We expect that the detailed understand-ing of the implementation of discrete symmetries (both conventional and higher-form) inAdS/CFT will have holographic applications. In particular, it would be interesting to un-39erstand if the tools of holography can be helpful in recent efforts to understand topologicalphases of matter (see e.g. [54]) and the phase structure of non-Abelian gauge theories fromthe point of view of generalized symmetry. More concretely, the existence of the higher-dimension tensor operator alluded to above is somewhat mysterious from the field-theoreticalpoint of view. As it arises from the most general possible quadratic action in the bulk, weexpect it to have an interpretation in the field theory. It would also be interesting to connectthe scaling solution found above to an asymptotically AdS solution and interpret it fromthe point of view of color flux tubes in gauge theory.Finally, the analyses (both holographic and otherwise) performed here indicate an inter-esting structure involving the interplay between conformality, spontaneously broken gener-alized p -form symmetry, and emergent d − p − Acknowledgements
We thank N. Bobev, B. Craps, S. Cremonesi, X. Dong, S. Hartnoll, E. Katz, P. Koroteev,A. Potter, N. Poovuttikul, S. Ross, E. Shaghoulian and D. Tong for illuminating discussions,and S. Grozdanov for collaboration on related issues and for sharing [5] prior to publication.NI would like to thank Delta ITP at the University of Amsterdam and the Aspen Centerfor Physics (which is supported by National Science Foundation grant PHY-1607611) forhospitality during the completion of this work. NI is supported in part by the STFC underconsolidated grant ST/L000407/1. This work is part of the Delta ITP consortium, a programof the NWO that is funded by the Dutch Ministry of Education, Culture and Science (OCW).
Appendix A: Conventions and differential form identities
In this work we normally use
M, N to refer to 5d bulk indices, µ, ν to refer to 4d fieldtheory bulk indices, and i, j to refer to 3d spatial indices. Section III A involves a 4d bulkand a 3d boundary; however we write that section entirely using index-free differential forms.Our our conventions for differential forms are those of [55], and we record some usefulidentities below: d ( ω p ∧ η q ) = dω p ∧ η q + ( − p ω p ∧ dη q (A1) ω p ∧ η q = ( − pq η q ∧ ω p (A2) ω p ∧ (cid:63)η p = η p ∧ (cid:63)ω p (A3)40he square of the Hodge star acting on a p form in n dimensions on a metric with s minussigns in its eigenvalues is (cid:63) = ( − s + p ( n − p ) . (A4)In particular, in Lorentzian signature in 4d acting on a 2-form, we have (cid:63) = − π . We pick conventions where electric andmagnetic charges satisfying Dirac quantization satisfy Q e ≡ (cid:82) (cid:63) J e = Z but Q m ≡ (cid:82) (cid:63) J m =2 π Z . Appendix B: MHD diffusion mode at zero magnetic field
A theory of magnetohydrodynamics from the point of view of generalized symmetries wasdeveloped in [7]. Here we specialize that theory to the case with zero background magneticfield, ending with the derivation of the diffusion mode obtained holographically in (2.38). Ifthe background field is zero, then the fluctuations of the 2-form current J µν and the stresstensor decouple, and we thus consider only J µν .In ideal hydrodynamics we have J µν (0) = 2 ρu [ µ h ν ] h = 1 u = − u µ the fluid velocity and h µ the direction of the background field, where ρ is its mag-nitude. To take the zero-field limit smoothly, it is convenient to define the un-normalizedvector B µ ≡ ρh µ and work to first order in B µ . Note that in this limit the symmetry of thebackground is enhanced from SO (2) to SO (3), as the special direction picked out by h µ islost. In particular, note that the transverse SO (2) invariant projector used in [7]∆ µν ≡ g µν + u µ u ν − h µ h ν = g µν + u µ u ν − B µ B ν B (B2)is not analytic in B ; thus we expect that it actually cannot explicitly appear in the zero-fieldlimit. This enforces some restrictions on the form of the hydro theory. E.g. from [7] we havethe following form for the first-order dissipative correction to J µν : J µν (1) = − r ⊥ h [ ν ∆ µ ] β h ρ ∇ [ β (cid:18) h ρ ] µT (cid:19) T − r (cid:107) ∆ µρ ∆ νσ ∇ [ ρ (cid:18) µh σ ] T (cid:19) T (B3)where we have set the background sources to zero and rewritten the last term slightly forlater convenience. r ⊥ , (cid:107) are resistivities that are parallel and perpendicular to the backgroundfield; however as the background field is taken to zero, the enhanced symmetry means thatthese two should coincide, i.e. r ⊥ = r (cid:107) ≡ r . We then find J µν (1) = − rσ ρ [ µ σ ν ] β T ∇ ρ (cid:18) µh β T (cid:19) (B4)41here σ µν is the SO (3) invariant projector: σ µν ≡ g µν − u µ u ν (B5)and ∆ µν as defined in (B2) no longer makes an appearance. Now in the small ρ limit wemay rewrite ρ = Ξ µ (B6)where Ξ is the susceptibility ∂ρ∂µ , and assembling together (B1) and (B4) the current takesthe form J µν = 2 u [ µ B ν ] − rσ ρ [ µ σ ν ] β T ∇ ρ (cid:18) B β Ξ T (cid:19) , (B7)which is manifestly smooth in B µ .We now consider a linear perturbation around a fluid at rest (i.e. u µ = δ µt ). We workin Fourier space, and give the perturbation spacetime dependence e − iωt + ikz . We consider amagnetic field perturbation where only B x (cid:54) = 0 and where the temperature is held fixed. Wefind J tx = B x J zx = ikr Ξ B x (B8)Current conservation ∂ µ J µν = 0 immediately gives us the dispersion relation ω = − iDk D ≡ r Ξ (B9)which is precisely the mode found holographically in (2.38), modulo the fact that in thissection we refer to the resistivity as r (to avoid confusion with the magnetic field density ρ ) whereas in the main text we refer to the resistivity as ρ . Note that dispersion relationcannot be found from taking a direct zero-field limit of the dispersion relations presentedin [7], as the hydrodynamic limit taken in that work assumes that the background field isnonzero. Appendix C: Wilson and t’Hooft lines in U ( N ) gauge theory For completeness, here we review the spectrum of Wilson and t’Hooft lines in U ( N ) gaugetheory. This question is well-studied; recent works include [39, 40]. We found [41] (whichstudied a similar problem in the context of the Standard Model) particularly helpful andour discussion will follow the approach taken there. Recall first: U ( N ) = U (1) × SU ( N ) Z N (C1)Wilson lines are labeled by ( q, z e ), where q is their electric charge under the U (1) and z e = 0 , , · · · N − ∈ Z N is their center-valued non-Abelian electric charge. The Z N quotient42n the definition of U ( N ) tells us that allowed Wilson lines have q = z e + N k , with k ∈ Z .t’Hooft lines are labeled by ( g, z m ), where g is their magnetic charge under the U (1) and z m ∈ Z N .Mutual locality requires that the Dirac quantization condition between ( q, z e ) and ( g (cid:48) , z (cid:48) m )be satisfied: qg (cid:48) − πN z e z (cid:48) m = 2 π Z (C2)If we consider z e = q = 1, we find that g (cid:48) = 2 πN z (cid:48) m + 2 πp with p ∈ Z (C3)Now we can consider the more general case and check that there are no further restrictions: qg (cid:48) + 2 πN z e z (cid:48) m = 2 π ( pz e + z (cid:48) m k + pN k ) = 2 π Z (C4)In other words, from the point of view of the U (1) factor alone, minimally quantizedt’Hooft lines appear as magnetic-monopoles that carry 1 /N -th the charge of the Diracmonopole. 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