Topological symmetry, spin liquids and CFT duals of Polyakov model with massless fermions
Abstract:
We prove the absence of a mass gap and confinement in the Polyakov model with massless complex fermions in any representation of the gauge group. A topological shift symmetry protects the masslessness of one dual photon. This symmetry emerges in the IR as a consequence of the Callias index theorem and abelian duality. For matter in the fundamental representation, the infrared limits of this class of theories interpolate between weakly and strongly coupled conformal field theory (CFT) depending on the number of flavors, and provide an infinite class of CFTs in dimensions. The long distance physics of the model is same as certain stable spin liquids. Altering the topology of the adjoint Higgs field by turning it into a compact scalar does not change the long distance dynamics in perturbation theory, however, nonperturbative effects lead to a mass gap for the gauge fluctuations. This provides conceptual clarity to many subtle issues about compact QED discussed in the context of quantum magnets, spin liquids and phase fluctuation models in cuprate superconductors. These constructions also provide new insights into zero temperature gauge theory dynamics on and . The confined versus deconfined long distance dynamics is characterized by a discrete versus continuous topological symmetry.
1 Introduction
The Polyakov model, YangMills theory with an adjoint Higgs scalar on , is one of the cornerstones in the study of confinement in gauge theories [1]. Abelian duality is used to show the emergence of a mass gap, and to exhibit linear confinement via the proliferation of the monopoles in the vacuum. Another theory which realizes confinement and a mass gap similarly, i.e, via the proliferation of the flux (or monopoles) is compact lattice QED. These are two different microscopic theories with a different set of symmetries at the cutoff scale. However, at long distances, they are gapped, and they flow to the same theory, constituting a nonperturbative long distance duality.
Although we do not know whether the Polyakov model is relevant in Nature, the lattice QED with fermionic fields appears in two dimensional spin systems, in the spin liquid approach to high superconductivity and in the phase fluctuation model of the cuprate superconductors (See the reviews [2, 3] and [4].) Therefore, the issue of deconfined versus confined long distance characteristic of 2+1 dimensional lattice QED with fermionic matter is experimentally relevant. An important question in this context is the existence (or nonperturbative stability) of the spin liquids, the nonmagnetic Mott insulators with no broken symmetries. In QED, this question translates into whether the strongly coupled fermions and gauge fluctuations remain massless in the long distances, when the nonperturbative effects (consistent with microscopic symmetries) are taken into account. If so, this implies deconfinement and stability. In the literature, a permanent confinement and instability was argued in [5, 6, 7]. Ref. [8, 9] showed that, at least in a large limit where spin symmetry is generalized to , there are some spin liquids which are stable. For small numbers of fermionic flavors, which is experimentally most interesting, this is still an unsettled matter.
In this work, we discuss a variety of related gauge theories, each of which needs to be distinguished very carefully via their microscopic symmetries. For example, consider noncompact continuum QED minimally coupled to flavors of fundamental fermions, and assume one wishes to incorporate the compactness of the gauge field. We show that, common bottomup arguments which claim to account for the compactness of the gauge fields are illdefined, due to nonuniqueness of this procedure. In the continuum, a standard way to obtain compact QED is via the gauge “symmetry breaking” in a parent YangMills adjoint Higgs systems. We show that there are at least two classes of parent theories which differ in the topological structure of their adjoint Higgs field (compact versus noncompact), yet both lead to the desired gauge symmetry breaking and reduce (necessarily) to continuum compact QED. Although indistinguishable in perturbation theory, the nonperturbative behavior of these theories are strikingly opposite: In the theory with noncompact adjoint Higgs scalar (Polyakov model with massless fermions), we demonstrate
(1) 
the existence of a massless photon in the long distances, and the absence of confinement. Of course, the dramatic behavior here is the appearance of a conformal field theory (CFT) in certain cases, to be discussed below. In the theory with compact adjoint Higgs field, the gauge structure reduces at longer distances as (for moderately small number of flavors)
(2) 
The photon gains a mass, and the theory confines. As opposed to the common assertions in the literature, the presence or absence of monopoles has nothing to do with the confining or deconfining behavior of a generic gauge theory. (See the table.1). We introduce a sharp (topological) symmetry characterization to describe the long distance limits (deconfined versus confined, and more delicate refinements) of gauge theories on and small . ^{2}^{2}2 Naively, the (1) seems to be in accord with Ref.[8, 9], and (2) seems to be coinciding with the results of [5, 6, 7]. This is not quite correct. The references [5, 6, 7, 8, 9] study a spin Hamiltonian which, in the flux state, maps into a compact lattice QED with fermions. The global symmetries of this lattice theory is different (although related, see .4.2) from the continuum discussion above. Despite these differences, we will establish precise nonperturbative long distance dualities between spin system and Polyakov model with massless fermions in certain cases.
We first discuss the question of confinement in the Polyakov model with massless fermions, either in real and complex representations. The answer is known for one real representation adjoint Dirac fermion [10]. The fermion number symmetry breaks down spontaneously, and there is a gapless NambuGoldstone boson (the dual photon). The masslessness of dual photon is protected by symmetry breaking order, i.e, Goldstone theorem, and the adjoint fermion acquires a mass. For complex representation fermions, the infrared is more interesting. There are strongly coupled gauge fluctuations and fermions which remain massless in the infrared. The answer entails a different mechanism to keep fermions and a boson massless. It is referred as quantum order (or nonsymmetry breaking order) in condensed matter physics [11, 12]. The appearance of quantum order in the Polyakov model is new. In the first application, the spontaneous breaking of a global symmetry generates and protects a massless boson, in the latter, the unbroken symmetry implies the existence of massless boson and fermions.
The main concept behind the deconfinement in the Polyakov model with massless fermions is a topological symmetry. This symmetry arises in the long distance and protects only one dual photon from acquiring a mass. It relies on the JackiwRebbi zero modes and the index theorem of Callias [13, 14] . Due to the index theorem, a symmetry of the high energy theory transmutes into a shift symmetry for the dual photon. For complex representation fermions, the combination of the topological symmetry and other global symmetries is very powerful, and they severely restrict any perturbative or nonperturbative relevant or marginal operators that may destabilize the masslessness of the strongly interacting photon and fermions. In particular, in theories with fundamental fermions are quantum critical due to the absence of relevant or marginal operators which may destabilize their masslessness. We argue that the strong correlation physics of the fermions and gauge boson at long distance produce a scale invariant, conformal field theory (CFT). In three dimensional nonabelian gauge theories, the earlier examples of infrared strongly coupled CFTs are mostly among extended supersymmetric theories [15, 16]. The nonsupersymmetric gauge theories discussed in this paper provide an infinite class of infrared CFTs which interpolate between weak and strong coupling as the number of flavors is varied, , with a dimensionless coupling constant . The theory turns out to be noncritical, due to the presence of a relevant, nonperturbatively generated flux operator with fermion zero mode insertion.
The existence of the continuous topological shift symmetry is the necessary and sufficient condition to prove that the photon remains massless in the Polyakov model with massless complex fermions.^{3}^{3}3For a real massless Majorana fermion in the adjoint representation, there is no symmetry. Such theories on do indeed confine.[10]. In fact, the fundamental distinction between the theories in (1) and (2) is that, in the latter, the continuous topological shift symmetry for the dual photon is replaced by a discrete one. As opposed to continuous shift symmetry, the discrete shift symmetries cannot prohibit the appearance of a mass term for the scalar. Thus, the photons in the latter case should acquire mass according to symmetry considerations. However, there is the possibility that the monopole fugacity may become irrelevant at large distance in the renormalization group sense. In this case, the long distance theory will exhibit an enhanced topological symmetry relative to the microscopic theory. This implies that the presence of the discrete topological symmetry is necessary, but not sufficient for confining behavior.
Finally, equipped with the understanding of the Polyakov models, we turn to the discussion of spin systems. As stated earlier, the spin systems can be mapped into lattice gauge theories in the slave fermion mean field theory. We investigate the relation between the Polyakov model and lattice QED, both with massless fermions, in the long distance limit. These are theories with distinct microscopic symmetries. But, perhaps the most significant distinguishing feature of the lattice QED and continuum Polyakov models is the absence of an analog of the Callias index theorem in lattice QED as shown by Marston [17] , and the analog of a global symmetry in the lattice model. The first is not as severe as it sounds despite the concerns raised in literature [18]. In fact, the latter is the main problem. We will show that, were the global a symmetry of the spin Hamiltonian, the topological symmetry would indeed arise in the infrared despite the absence of an index theorem. If this were the case, we could have carried a precise analogy with the Polyakov model even at small . Unfortunately, only in the sufficiently large limit can we make a reliable statement about the infrared structure of the lattice theory. In particular, we are not able to improve the discussion given in [8, 9]. In the Polyakov model with massless fermions, we are able to sidestep the renormalization group and large analysis of Hermele et.al. [8]. In lattice QED, this analysis seems inevitable. Thus, there is a long distance duality between the spin liquids and Polyakov models with massless fermions in the large limit where both theories flow into the same interacting CFT.
2 Gauge theories in three dimensions
We consider YangMills gauge theory with a noncompact adjoint Higgs scalar on (also known as GeorgiGlashow model) in the presence of massless fermions. The fermions are chosen in complex and real representations such as fundamenal(F) and adjoint(adj). We will label these theories as P(F) and P(adj), respectively. Before discussing them, it is useful to review the basics of the pure Polyakov model [1] and set the notation.
2.1 Polyakov model
The action of gauge theory with an adjoint scalar is
(3) 
is a Lie algebra valued noncompact scalar Higgs field, is the nonabelian field strength, and . The classical potential is chosen such that, at tree level, the theory is in its Higgs regime, . At long distances, only the abelian components are operative. To all orders in perturbation theory, the infrared is a free (noninteracting) Maxwell theory.
The Gaussian fixed point is destabilized due to nonperturbative instanton (monopole) effects. This instability is easiest to see in a dual formulation where the gauge boson is dualized to a scalar, .^{4}^{4}4Our discussion mostly relies on symmetries. Therefore, to lessen the clutter of expressions, we set the dimensionful parameters (e.g. ) to one. These parameters will be restored if necessary. Since an instanton has a finite action, they will proliferate due to entropic effects. This generates nonperturbative effects in the long distance Lagrangian
(4) 
The is a relevant operator which alters the IR physics drastically, and leads to a mass gap .
It is worth nothing that, the dual of the free Maxwell theory, i.e., in the absence of monopoles, described by has a continuous shift symmetry
(5) 
which protects from acquiring mass. The current associated with the shift symmetry is , and its divergence is zero, , reflecting the absence of monopoles and conservation of magnetic flux, hence the name .
In the gauge theory with monopoles, the current is not conserved. Its divergence is where is the monopole charge density. Since the is no longer a symmetry, there is no symmetry reason for the field to remain massless. Indeed, acquires a mass as shown in (4).
: More generally, let the gauge symmetry be broken down to via an adjoint Higgs vacuum expectation value
(6) 
where . There are photons which remain massless to all orders in perturbation theory. Let us dualize them into . Nonperturbatively, there are types of elementary monopoles associated with this pattern, which we label by their magnetic charges where each is an vector with charges under . The antimonopoles carry opposite charges. The monopole operator in a theory without fermions is , and the sum over all elementary monopole effects induce rendering all varieties of photons massive. ^{5}^{5}5We assume, for simplicity, for the elementary monopoles by tuning the potential. This can be relaxed if desired.
2.1.1 Introducing complex representation fermions
Our goal is to construct the nonperturbative long distance description of Polyakov models with massless fermions. The long distance effective field theory must respect all the (nonanomalous) symmetries of the underlying microscopic theory. In other words, the (perturbative or nonperturbative) operators that can be generated are severely restricted by the microscopic symmetries. Therefore, it is useful to clearly state the symmetries of the microscopic P(F) model. This will also ease the comparison of microscopic and enhanced (emergent) macroscopic global flavor and spacetime symmetries of the theory.
Consider the addition of the massless fermions in the fundamental representation of the gauge group into the Polyakov model. (The generalization to other complex representation fermions is possible.) We interchangeably use the fourcomponent Dirac spinors or two twocomponent Dirac spinors and related to each other via
(7) 
We consider the theories with two component Dirac spinors, or equivalently, four component spinors. The and subscripts are flavor indices. In our conventions, the representations of the two component fermions under the gauge group are where denotes the fundamental representation. These combinations and our subsequent Dirac matrix choices are for later convenience, and will make the Callias index analysis slightly simpler. ^{6}^{6}6In Euclidean space, and should be viewed as independent variables. In particular, they are not related to each other by conjugation The fermions couple to gauge fields and adjoint scalars as
(8) 
where the Euclidean matrices are given by
(9) 
It is also convenient to define
where are the Pauli matrices. The explicit form of the Diraclike operator in this basis is
(10) 
and consequently,
(11) 
In this representation, it is easier to see the global symmetries of the theory. Besides the Euclidean Lorentz symmetry and the discrete charge conjugation, parity and (Euclidean) time reversal symmetries, the theory possesses a discrete
(12) 
and the following global (flavor) symmetries
(13) 
Note that the gauge covariant term possesses a larger global symmetry group. Were the Yukawa’s not present in the theory, the global symmetry would enhance into the . However, the relative sign difference between the covariant derivative and Yukawa couplings prevents this enhancement in the microscopic theory. Since there is no chiral anomaly in dimensions, the symmetry is a true symmetries of the theory. The discrete and symmetries, and continuous flavor symmetry prohibits a fermion mass term. To summarize, the full microscopic symmetry group of the theory is
(14) 
2.1.2 Real representation fermions
We restrict attention to the adjoint representation fermion. Since the adjoint representation is real, the two component (complex) Dirac spinors is appropriate for all circumstances. Thus, . The coupling of fermions to gauge boson and adjoint scalar is
(15) 
The global flavor symmetries of the theory is given by
(16)  
(17) 
Note that, in this case, may be viewed as fermion number symmetry. However, since it does not have the same interpretation in the theories with complex representation fermions, we will not use this nomenclature. Thus, the full symmetry group of the microscopic theory is
(18) 
Remark on QCD: At the classical level, the flavor symmetry group of the Polyakov models with fermions on is the same as the flavor symmetries of the corresponding QCD on or . However, in QCD in four dimensions, the analog of the symmetry that we referred as in (13) and (17) is anomalous. In odd dimensions, there is no chiral anomaly, and the is a true symmetry of the Polyakov model with massless fermions. In four dimensions, due to instanton effects, only a discrete subgroup of survives quantization, where is the number of fermionic zero modes in the background of a four dimensional instanton. The microscopic symmetry will play a major role in the characterization of deconfinement in P() theories.
2.1.3 Perturbative operators and flux operators
In all the P theories, we assume that the theory is always maximally Higgsed, and the long distance is dictated by the maximal abelian subgroup. There are massless bosons whose masslessness is protected to all orders in perturbation theory. Also, there are fermionic zero modes which interact with gauge fluctuations at long distances. Our interest is to determine the stability of such massless fields. There are two categories of operators which may be generated, and alter the long distance physics. These are, following [8],

perturbative (without flux), naturally incorporated in terms of the original variables.

nonperturbative (flux operators), or topological excitations, naturally incorporated in terms of dual photon.
For example, in the pure Polyakov model, a wouldbe operator of the first category is the relevant ChernSimons term,
(19) 
which would induce a mass term for the photon. However, this operator does not get generated at one loop order (or any order in perturbation theory), because the microscopic theory is parity invariant and the ChernSimons term is parity odd. Thus, this type of instability does not occur.
An operator in the second category is the monopole operator. Indeed, it is allowed by all symmetries and generates the interaction, which, in the deep infrared, is a mass term for the dual photon. This is the type of instability that we will look for in the Polyakov models with massless fermions and some related gauge theories.
We will see that the microscopic symmetries and a topological shift symmetry which arises as a natural consequence of the Callias index theorem very severely restrict the types of operators that can be generated. In some circumstances, the infrared theory is quantum critical, in the sense that there exists no perturbative or nonperturbative operators which may destabilize the masslessness of photons and fermions, and some such theories become conformal field theories.
2.2 Callias index theorem and (continuous) topological symmetry
In the presence of massless (or light) fermions, the monopoles may carry fermionic zero modes attached to them [13]. The number of the fermionic insertions is determined uniquely by the Callias index theorem [14], and matter content of the theory. ^{7}^{7}7 For the relation between the more familiar AtiyahSinger index theorem and Callias index theorem in QCDlike gauge theories on small , see page.37 of [19]. Let denote the index associated with the monopole with charge . The typical form of the monopole operator in the theory with fermions is
(20) 
The number of fermion insertions of each flavor/type, say , in , is determined by the index , by the difference of the dimensions of the zero energy eigenstates:
(21) 
Here, is the Diraclike operator in dimensions in the background of the monopole . In our conventions, the in the monopole operator has only insertions, and an antimonopole operator can only have insertions.
(22) 
This was indeed the reason for the peculiar spinor decomposition (7). For an adjoint fermion, the index is equal to . In the presence of fundamental fermions, the index is where is the monopole that the zero mode is localized into. This is for each flavor of two component Dirac fermion. Since we have even number of fundamental fermions, the number of fermionic zero mode insertion in is always even.
More precisely, for fermions in complex representations, we have two Diraclike operators as seen in (11) and two conjugates,
(23)  
(24)  
(25) 
The total number of fermion zero modes associated with a monopole is .
Symmetry transmutation: The microscopic Polyakov Lagrangian with massless fermions has a global symmetry given in (13) and (17) regardless of whether fermions are in a real or complex representation. Since it is a nonanomalous symmetry, it must be a symmetry of the long distance theory. The transformation,
(26) 
implies . Therefore, the invariance of the monopole operator under (26) necessitates a continuous shift for the dual photons:
(27) 
Since this symmetry originates from the topological index theorem, we will call it a topological shift symmetry, or simply, topological symmetry and refer to it as . Just like the abelian duality transform [1], the topological shift symmetry requires going to sufficiently long distances. In the IR, the symmetry of the original theory intertwines with the shift symmetry for the dual photons (5). This phenomena pervades the physics of all P theories.
More precisely, recall that in the absence of fermions and monopoles, the free Maxwell theory is dual to a free scalar theory with a continuous shift symmetry (5). The presence of monopoles (in the absence of fermions) spoils this symmetry completely. However, in the presence of fermions, the linear combination of the and
(28) 
remains a true symmetry of the theory. ^{8}^{8}8If there was no dual photon field to soakup the phase of the fermionic zero modes, this would indeed imply that must be anomalous, which is incorrect on . Compare this with one flavor QCD on . The instanton vertex also has two fermion insertion and no extra structure to soakup the chiral rotation. Indeed, there is a chiral anomaly on and the is anomalous. Only a subgroup of it is anomalyfree.
A continuous shift symmetry can protect a scalar from acquiring a mass. Since there is only one parameter in the transformation (27), only one dual photon is protected by the topological symmetry. At a conceptual level, this shows that one gauge degree of freedom remains massless in the IR of the P() theory regardless of any other detail, so long as the microscopic theory possesses the symmetry. We may call this phase deconfined, since a gauge boson remains infinite ranged. Although this is true, it is a crude characterization. A more refined categorization of the deconfined phases, which can distinguish a free infrared theory (free photon), and a strongly or weakly coupled conformal field theory (CFT) is needed, and will be discussed.
2.3 Revisiting P(adj): Dual scalar as a NambuGoldstone boson
Consider the one flavor P(Adj). (Below is a review and slight refinement of Affleck et.al. [10]). We assume the long distance gauge structure reduces down to . Perturbatively, we have a photon and a neutral fermion, described by
(29) 
a free field theory. Parity forbids relevant perturbative operators such as from being generated [10]. Nonperturbatively, there is only one type of elementary monopole (and its antimonopole.) The index for adjoint fermions. Thus, by (26) and (27), we have
(30) 
There is only one combination of the relevant singlet that one can construct, and which gets induced nonperturbatively:
(31) 
There is also a large class of singlet, but irrelevant multimonopole operators of the form where is some integer. The continuous shift symmetry (30) forbids any kind of potential (such as ), the mass term for the dual photon. This proves that the photon must remain massless nonperturbatively. Affleck et.al. showed that, by expanding the fields around, say, zero, the symmetry is spontaneously broken and the photon is the NambuGoldstone boson. The fermion acquires mass due to breaking. This is the conventional way to have gapless scalars in a gauge field theory. For a fuller discussion, see ref.[10, 20]. For and multiflavor generalizations, see [21].
It is useful to think of the Noether current associated with the symmetry (30) in the flavor theory. It is
(32) 
Recall from .2.1 that the current associated with satisfies where is the magnetic field. Using where is the magnetic charge density, the local current conservation can be reexpressed as
(33) 
which implies the conservation of the current as stated in (28). The final form is the local version of the Callias index theorem, which ties the charge with the charge. Namely, in the presence of adjoint fermions,
(34)  
(35) 
is a conserved charge, where counts the number of the excitations. This means, any perturbative or nonperturbative interaction vertex in the long distance theory preserves . However, the is spontaneously broken by the vacuum, and the photon is a Goldstone boson.
: It is also useful to review the generalization of this theory since it carries important lessons on the interplay of symmetry and dynamics. Due to gauge symmetry breaking down to , there exist photons and massless fermions, the components along the Cartan subalgebra. The infrared Lagrangian in perturbation theory is, therefore,
(36) 
The simplicity of this system relative to the complex representation fermions to be studied in the subsequent section is the electric neutrality of the zero mode fermions. In perturbation theory, there are no relevant or marginal operators which respect the underlying symmetries of the original theory and which may be generated perturbatively. Thus, the Gaussian fixed point is stable to all orders in perturbation theory.
However, there exist a plethora of relevant nonperturbative (flux) operators. The index is for all . The monopole operators are none of which generates a mass term for the dual photons. Notice that each term is manifestly invariant under the topological symmetry (26), (27). In the expansion, at order , there are linearly independent relevant operators, which get generated. Even though there is no fermion zero mode attached to these topological objects, since they are essentially the bound states of a monopole (with charge ) and antimonopole (with charge ), their invariance under the topological symmetry is also manifest. ^{9}^{9}9 A monopole and antimonopole in the presence of massless adjoint fermions interacts logarithmically at large distances in Euclidean , rather than the Coulomb’s law. (Also see [22, 17] for QED, but one needs to be really careful here. See formula (69) and the discussion around it.) The marginally binds a monopole into its antimonopole. The combined state is magnetically neutral, and cannot lead to Debye screening. (A monopoleantimonopole pair is a dipole, and in the long distance, the dipoledipole interaction is , hence the absence of the Debye screening.) In P(adj) with , the presence of the fermion zero modes also leads to bound states of a monopole with charge and antimonopole with charge . The combined topological excitation has a nonzero magnetic charge and at large distances interacts via Coulomb potential, . These excitations are referred to as magnetic bions [21]. The magnetic bions render varieties of photons massive. In QCD(adj) on discussed in Ref.[21], due to an extra elementary monopole, one can form magnetic bions, and the gauge sector is fully gapped. This also has a nice symmetry interpretation. The continuous topological shift symmetry turns into a discrete shift symmetry on small . The discrete shift symmetry cannot prohibit mass term for scalars. Thus, the combined nonperturbative effects up to order is given by
(37) 
This renders varieties of the photons massive leaving the one which is protected by the shift symmetry. As in the case, the breaks down spontaneously and there exist only one Goldstone. The higher order terms in the do not alter this conclusion.
This application shows that the existence of symmetry provides a characterization for the absence of mass gap in gauge sector and the absence of confinement. The does not imply the absence of monopoles or the irrelevance of monopole operators. And neither the presence of elementary monopoles or magnetically charged bound states of the monopoles implies confinement.
2.4 Complex representation fermions, masslessness and quantum criticality
Let us consider an YangMills noncompact adjoint Higgs system with two component fundamental Dirac fermions on , the P(F) theory. The theory possess the symmetries (14). As always, we assume the gauge structure reduces down to at long distances. The offdiagonal gauge degrees of freedom (bosons) and one component of the fermions in the gauge symmetry doublet, and the scalars acquire masses and decouple from the long distance physics. In perturbation theory, the infrared theory is described by the abelian QED action
(38) 
The action possesses an enhanced (accidental) flavor symmetry group, and a symmetry which is the global part of the gauge symmetry. This enhancement is expected in perturbation theory, because the Higgs scalar acquires mass and disappears from the long distance description. Since the disparity between the gaugekinetic term and Yukawa term in (11) was the source of the lower symmetry, and since there are no Yukawa’s in the long distance limit, there is an enhanced symmetry in perturbation theory.
The nonperturbative effects may in principle be aware of the lower symmetry of the high energy theory, and indeed, they are. Let us first take . As in P(Adj), there is one type of monopole. The index theorem tells us that for each fundamental flavor, the monopole has zero mode. There is one relevant singlet operator which is induced nonperturbatively:
(39) 
The two fermions and the dual photon transform under as
(40) 
The continuous shift symmetry forbids any kind of mass term for the dual photon. In particular, it forbids the operator. Thus, the photon must remain massless nonperturbatively.
In the multiflavor case , the simplest monopole operator has insertion of the fermionic zero modes,
The equality of the number of insertion with the insertion is a consequence of the Callias index theorem and symmetry, i.e, electric charge neutrality. Making the symmetry of the monopole operator manifest gives
(41) 
where are flavor indices. The invariance of the vertex under symmetry necessitates the dual photon to transform as under .
We identified a distinction between the behavior of and theories. In the expansion, the leading nonperturbatively generated flux operator is classically relevant in the case, and irrelevant in the cases. Therefore, the latter class of theories are quantum critical, and will exhibit enhanced symmetry at long distance. For the case, there is one relevant direction and no enhancement of flavor symmetry takes place.
It is again useful to study the Noether currents in the effective long distance theory. Unlike P(Adj), there are two types of conserved currents in the Polyakov model with complex representation fermions. One is associated with symmetry, and the latter is a linear combination of and . These are, in the conventions of .2.1.1,
(42)  
(43)  
(44) 
The conserved charge associated with the current is
(45) 
and the conserved charge associated with is
(46)  
(47) 
Clearly, these symmetries are in accord with the monopole operators and their zero mode structures. In fact, the conservation of the current, is the local reincarnation of the Callias index theorem. We will discuss the infrared limit of these theories after generalizing the basic essentials to gauge theory.
: The difference of long distance physics between and is not special to the P(F) theory. The infrared limit of gauge theory with massless fermion flavors turns out to be rather similar to the flavor theory, as a consequence of the nonperturbative dynamics.
We assume the gauge structure reduces into at long distances. In perturbation theory, the infrared has types of the massless photons, and massless fermions. The other fields acquire masses and decouple from the long distance physics. There are varieties of elementary monopoles. Their Callias indices are given by where without loss of generality, we assumed that the fermion zero mode is localized into the monopole with charge . Thus, the shift symmetry reads
(48)  
(49) 
The symmetries do not forbid the types of monopole operators which do not carry any fermionic zero modes. The first monopole has fermion insertions and is irrelevant for . The list of all the flux operators invariant under the symmetries of the microscopic theory up to order is
(50) 
Hence, out of photons acquire mass due to relevant monopole induced effects. Thus, the P(F) theory undergoes changes in its gauge structure as we consider longer and longer length scales. The first change is perturbative and the latter is nonperturbative as shown in (1). The very long distance theory is quantum critical due to the absence of any relevant or marginal perturbations which may destabilize its masslessness. We will comment on the effects of strong (noncompact) gauge fluctuations in the next section.
Note that regardless of the value of the rank in the original gauge theory, the deep IR of the P(F) theory always reduces to an abelian QED theory with flavors. Below, we discuss the long distance limit of this theory.
2.5 Conformal field theories (CFTs) at long distances
The topological symmetry combined with symmetries such as parity, Lorentz and flavor symmetries forbids any relevant instability that may occur in the infrared limit of our theory. The monopole operators such as , or , where is the dual of the final factor, are forbidden. This means, in the compact continuum QED theory obtained as described above, there are no relevant flux (monopole) operators in the original “electric” theory. Thus, the nonperturbative lagrangian is the same as the perturbative one,
(51) 
where ellipsis stands for irrelevant perturbations consistent with the microscopic symmetries of the underlying theory. This is QED with charged massless fermions, and with an enhanced (accidental) flavor symmetry.
The theory (51) has no dimensionless coupling constant. The expansion parameter is where is some euclidean momentum scale. Thus, perturbative techniques are not useful at low energies. The low energy limit is a strongly correlated system of fermions and gauge fluctuations whose masslessness is protected by . A logical possibility for the infrared theory is a weakly or strongly coupled conformal field theory (CFT) depending on the number of flavors. In order to see this, let us calculate the correction to the photon propagator at one loop order in perturbation theory. Partially integrating out fermions produce the nonanalytic correction to the gauge kinetic term
(52) 
In the large limit, the higher order effects in perturbation theory are suppressed by powers of and the one loop result becomes reliable [23]. The low energy limit is the same as taking to . These changes in the photon propagator can be summarized as
(53) 
Thus, we are left with a theory without any scale in the IR with gauge boson propagator . Using the canonical normalization for the gauge kinetic term, the Lagrangian can be expressed as
(54) 
with a dimensionless expansion parameter . This is a remarkable change in the dynamics.
To appreciate this, let us measure the potential between two external electric charges located at . The Coulomb potential between the two test charges is , in two spatial dimensions, hence marginally confining. The nonperturbative dynamics of the pure Polyakov model alters this potential into a linearly confining one. In the infrared of the theory with massless fundamental fermions, the potential is dictated by conformal behavior. Thus,
(55) 
In some sense, the long distance behavior of the Polyakov model with massless fermions is more drastic than the Polyakov model per se. This example also shows that the presence of a single massless fermion can completely alter the confining property of the gauge theory! However, the main concept here is not really the presence or absence of a fermionic species. Rather, it is the nature (continuous versus discrete) of the topological symmetry, as we will discuss in more detail, especially in connection with QCD* theory.
The microscopic symmetries of the P(F) theory given in (14) enhances and transmutes into
(56) 
in the long distances. In the cases, the relevant respecting operators also individually respects and . The is part of , and is the symmetry associated with conservation of magnetic flux. In the