A master bosonization duality

HJE master bosonization duality Kristan Jensen 0 1 0 San Francisco , CA 94132 , U.S.A 1 Department of Physics and Astronomy, San Francisco State University We conjecture a new sequence of dualities between Chern-Simons gauge theories simultaneously coupled to fundamental bosons and fermions. These dualities reduce to those proposed by Aharony when the number of bosons or fermions is zero. Our conjecture passes a number of consistency checks. These include the matching of global symmetries and consistency with level/rank duality in massive phases. Duality in Gauge Field Theories; Chern-Simons Theories - A 1 Introduction 2 Mapping out the phase diagram 2.1 Massive phases and critical lines 2.1.1 2.1.2 2.2.1 2.2.2 1, where the subscript indicates the Chern-Simons level. The theories are dual in that the observables of both theories are identical [1{3]. There has been an accumulation of evidence this decade for dualities between nonsupersymmetric Chern-Simons theories coupled to fundamental matter [4{8]. These conjectured dualities may be thought of as taking the level/rank dualities, adding suitable matter content on both sides, and tuning to a conformal eld theory (CFT). The basic sequences of interest in this work were precisely formulated by Aharony and read [7]: Fisher" (WF) scalars, meaning that on the scalar side of the duality one turns on a j j potential and mass term and tunes to criticality. Both dualities require Nf k, although there is a proposal [9] to extend the dualities slightly beyond this \ avor bound." Because the dualities (1.2) and (1.3) relate theories with fundamental fermions to theories with 4 fundamental bosons, they have been dubbed \3d bosonization." These dualities have been the subject of recent attention from a variety of viewpoints. For the special case N = k = Nf = 1, these dualities are related to the surface states of time-reversal invariant topological insulators and the fractional quantum Hall e ect at half lling [10{13], lead to a web of dualities [14{17], and can even be proven on the lattice [18]. They are crucial actors in mapping out the phase diagram of QCD3 as well as some of its cousins. At large N; k, with N=k nite, they are dual to a peculiar theory of gravity known as Vasiliev theory [4{6]. These theories are in fact solvable in this limit, and much is known of their thermal physics and scattering amplitudes [4{6, 19{21]. Away from N = k = Nf = 1, they imply a web of dualities for gauge theories with product gauge groups and (bi)fundamental matter [22], known as quiver gauge theories, and have been embedded intro string theory [23] (see also [24]). For other interesting works see e.g. [25{31]. The dualities (1.2) and (1.3) remain unproven, and in the absence of supersymmetry, it is di cult to envisage a proof. Nevertheless there is signi cant evidence that they are true. The best evidence comes from direct computations at large N; k with N=k nite. The exact global symmetries and their 't Hooft anomalies match [32], as do the quantum numbers of baryon and monopole operators [7]. At large N , these dualities appear to be inherited from a three-dimensional version of Seiberg duality [33, 34], and there is some expectation that the non-supersymmetric bosonization dualities are always the o spring of a parent supersymmetric duality (see e.g. [19, 33{36]). Finally, it is expected that the Chern-Simons-matter theories in (1.2) and (1.3) possess at least one relevant operator, the mass operator for the fundamental fermions or bosons. Deforming by this mass operator triggers a ow to a massive phase, described in the infrared (IR) by a topological eld theory (TFT). The precise low-energy TFT depends on the sign of the mass. One then thinks of the Chern-Simons-matter theories in (1.2) and (1.3) as describing a second order transition separating these two phases. Crucially, the IR TFTs describing the massive phases of one side of the duality match those of the other [7]. For example, deforming SU(N ) k+ N2f theory coupled to Nf fermions by a negative mass leads to a SU(N ) k TFT in the IR, while deforming U(k)N theory coupled to scalars by a positive mass-squared leads to an IR U(k)N TFT. These TFTs are identical by virtue of level/rank duality. A similar computation matches the other phases. In this note we propose a new in nite sequence of bosonization dualities between Chern-Simons-matter theories with both fundamental fermions and bosons.1 It reads 1A related conjecture was made some years ago for Nf = Ns = 1 [33]. ), where the dots indicate a contraction of gauge indices. This last operator is the unsung hero of this note. It is important for the following reason. The theories in (1.4) have a phase in which the gauge group is partially Higgsed, and in this phase this operator gives a mass j j2 to the fermions which are neutral under the unbroken gauge group. To the endpoint of a double trace ow from the free- eld xed point triggered by (j j2)2. The quartic operator picks up a 1=N suppressed anomalous dimension relative to the free- eld xed point, = 3 + O(1=N ), so that it is approximately marginal. In order for the duality to work, we nd that it must be present on both sides of the duality with a coe cient whose sign is that of the Chern-Simons level, i.e. negative for the SU(N ) k+ N2f theory and Ns theory. More precisely, we require these signs in the deep IR of positive for the U(k)N 2 the Higgs phases. Curiously, in supersymmetric Chern-Simons-matter theories with level k and at least N = 2 supersymmetry, the coupling of this operator is xed to be 1=k. As far as we know the sign of the O(1=N ) correction to this dimension has not yet been computed. If it is positive, then this operator is slightly irrelevant yet important in the IR, in which case it is dangerously irrelevant. If the sign is negative so that the operator is relevant, then its coe cient is not a free parameter and must be tuned to realize a CFT. Both possibilities are of interest and warrant future study. In either case it is clear that the sign of this coupling in the deep IR is not a choice but must be xed by the dynamics, and it is not clear if the dynamics choose the sign we need for the duality to hold. Turning the matter around, if we regard the duality (1.4) as sacrosanct, then we are making an implicit claim about the ow of this operator which would be nice to explicitly check at large N .2 We perform several basic consistency checks on our proposal. The rst is to map out the phase diagram of both sides of (1.4), under the assumption that the fermion and scalar mass operators remain relevant. The theories in (1.4) may be understood as a multi-critical point in which both of these masses are tuned to vanish. We argue that the ensuing two-dimensional phase diagram has ve distinct phases, all visible semiclassically, described by four di erent IR TFTs. These phases are separated by critical lines described by the theories in (1.2) and (1.3), and a critical line described by Nf Ns free fermions and a decoupled TFT. We also discuss the quantum numbers of baryons and monopoles in these theories, nding that, as in the basic bosonization dualities (1.2) and (1.3), the baryons of one side may be consistently matched to monopoles in the other. Finally we deduce the exact global symmetries of both sides of (1.4) and nd that they match. 2We would like to thank O. Aharony and Z. Komargodski for discussions on these points. { 3 { In the title of this note we call the proposal (1.4) a \master" duality. In giving this presumptuous name we have two things in mind. The rst is that this proposal reduces to Aharony's when Ns = 0 or Nf = 0. The second concerns recent works which use the basic dualities as \seed dualities" to generate new ones by gauging global symmetries on both sides of the seed [14{17, 22]. In the context of quiver gauge theories, one of the results of [22] is that one can dualize node-by-node: given a quiver with a SU or U gauge group factor coupled to only bosons or fermions, one can generate a dual quiver by replacing a node and the matter attached to it with its dual according to (1.2) and (1.3). Assuming our conjecture (1.4), one can use the same logic to dualize any node of a quiver with fundamental matter. The remainder of this note is organized as follows. In section 2 we map out the phase diagram of both sides of (1.4), and show that not only can we match the TFTs describing the massive phases, but we can also match the Chern-Simons terms for the global symmetries. We match baryons, monopoles, and global symmetries in section 3. Gauging an appropriate U( 1 ) subgroup of the global symmetry on both sides of our proposal (1.4), we nd that (1.4) also implies a U=U duality which we describe in section 4. In section 5 we comment brie y on an extension of our proposal (1.4) to dualities between Chern-Simons theories with SO and USp gauge groups, and we conclude with some open questions in section 6. Note: while this work was nearing completion F. Benini posted a paper [37] which also conjectures the duality (1.4) as well as extensions for other classical gauge groups. 2 Mapping out the phase diagram We begin with the Lagrangians for the theories on both sides of our proposed duality (1.4). We work in Euclidean signature. The SU(N ) k+ N2f theory is described by a Lagrangian3 LSU = i 4 where a is the SU(N ) gauge eld, Lint describes scalar and fermion interactions, i = 1; : : : ; Nf is a fermion avor index, and m = 1; : : : ; Ns is a scalar avor index. This theory 3In this work we follow [8] and use the convention that the functional determinant of the Dirac operator of a single Dirac fermion coupled to an external gauge eld A and metric g is given by det(D= (A; g)) = jdet(D= (A; g))j exp i (A; g) 2 ; where is the -invariant. On a closed manifold with trivial topology, this phase evaluates to a ChernSimons term with level 12 for A along with a gravitational Chern-Simons term, i (A; g) 2 ! i Z 1 8 AdA + 1916 tr d + the massless theory as U( 1 )k 12 . U(Ns) global symmetry which we impose. To get a handle on the scalar and fermion interactions we consider two limits. The rst is to realize the SU theory as an infrared xed point of a renormalization group ow, starting with a free theory in the UV. For general Nf , Ns, the classically relevant and marginal operators are becomes a \Wilson-Fisher" scalar, and one expects j j4 and j 6j to both be irrelevant with respect to the IR scalings. What of the quartic fermion/scalar operators? Now consider a large N limit. At large N , the fermion and Wilson-Fisher scalar both have dimension 1. The operators j j4, ( )j j2, and j j6 are then \multi-trace" operators whose dimensions by large-N factorization are 4 + O(N 1) and 6 + O(N 1). However it is easy to check that the last quartic operator, HJEP01(28)3 (2.5) (2.7) (2.8) where a0 is the U(k) gauge eld, and again i = 1; : : : ; Nf , m = 1; : : : ; Ns and L0int describes the scalar and fermion interactions. We denote the scalars of the U theory as i and the fermions as m to distinguish them from the bosons and fermions of the SU theory. This theory has a manifest SU(Ns) SU(Nf ) U( 1 )m U( 1 ) global symmetry. The U( 1 )m is a monopole number, while U( 1 ) is carried by the fundamental fermions (with charge +1) and bosons (with charge 1). As in the SU theory, we expect that the operators in L0int are whose coe cients are all tuned to realize a non-trivial IR CFT. As above, the operator O40 = ( m i)( yi m) ; will soon reveal its importance. 2.1 Massive phases and critical lines At large N the only relevant SU(Nf ) be wrong. The quartic operator O4 has dimension 3 + O(1=N ) at large N and so may be relevant, depending on the sign of the 1=N correction. We also assume that all phases are the ones accessible semiclassically at large jm j; jm2 j when realizing these theories as the { 5 { endpoint of ows starting from a UV free- eld xed point. Finally, for the purposes of a simple presentation, we assume that there are no rst order transitions, with the caveat that there could very well be rst order transitions separating the semiclassical phases we nd at large jm j, jm j2 from the region at small mass. Subject to these assumptions we map out the schematic two-dimensional phase diagram of both theories. Let us begin with the SU theory. We can give the fermions a positive or negative mass, as well as give the scalars a positive or negative mass-squared. Integrating out the massive fermions leads to a one-loop exact shift of the Chern-Simons level. Giving a positive mass-squared simply decouples the scalars, while a negative mass-squared partially Higgses the gauge symmetry. Following previous work, we assume that the interactions prefer to maximally Higgs the gauge theory from SU(N ) down to SU(N Ns), and nd that this is necessary in order for our conjecture to work. In the Higgsed phase, the quartic operators ( compensated for by a suitable shift of the coupling of the fermion mass operator . The second is more interesting. The original N Nf fermions break up into Nf fundamental representations of the unbroken SU(N Ns) gauge symmetry, while the remaining NsNf fermions are gauge-singlets. We refer to these as singlet fermions. Crucially, the operator O4 generates a mass j j2 for the singlet fermions in the Higgs phase. Denote the coe cient multiplying O4 in the Higgs phase as c4. If c4 is positive, then this operator contributes a positive mass to the singlet fermions, while if c4 is negative it contributes a negative mass. In either case, since the fermion mass operator gives a mass to all fermions, it is clear that there are three distinct massive Higgs phases. In one all fermions have a positive mass, in another all fermions have a negative mass, and in the last the singlet fermions have a mass whose sign is opposite those of the remaining charged fermions. We will soon see that our duality requires c4 < 0, so that this new phase exists for m > 0 and that in this phase the singlet fermions have a negative mass. In what follows we must also distinguish between the cases Ns < N and Ns = N . For Ns < N and away from critical lines the Higgsed phase of the SU theory is completely gapped. However for Ns = N the Higgsed phase is gapless with a massless scalar. Correspondingly, in the U theory, for Ns < N both signs of the Dirac mass lead to a non-trivial TFT in the infrared. For Ns = N a negative Dirac mass leaves a U(k)0 or U(k Nf )0 theory (depending on whether or not one is in the Higgsed phase), whose non-abelian part con nes at low energies leaving behind compact electromagnetism in the IR, not a TFT. Ultimately, we will nd that our proposal is still consistent for Ns = N , however only when N . That is we must also have the avor bound (Nf ; Ns) 6= (k; N ) k < N rather than k as advertised in the Introduction. 2.1.1 We begin with the case Ns < N . There are ve distinct massive phases separated by various critical lines, with the critical SU theory living at the original of the phase diagram. { 6 { (a) (b) fermions and Nf scalars. The various phases are described by IR TFTs given in (2.10) and (2.13). and the critical lines are described by the CFTs given in (2.12) and (2.15). However there are only four di erent IR TFTs. See gure 1. The distinct TFTs governing the massive phases are (I) (II) (III) (IV) m m m m Observe that Ns < N is required to have a massive Higgs phase. For k = Nf Phases I and IV are described by SU(N )0 and SU(N Ns)0, which give con ning Yang-Mills theories rather than TFTs. The fourth phase splits into two, one at large positive m where all fermions have a positive mass, and one at intermediate positive m , where the singlet fermions have a negative mass, (IVa) (IVb) m m > 0 ; ms < 0 ; m2 < 0 : SU(N > 0 ; ms > 0 ; m2 < 0 : SU(N Now for the critical lines. There are ve critical lines separating the various phases. Four of the lines are described by SU Chern-Simons theory coupled to either fermions or bosons, and one is the theory of massless singlet fermions. We summarize the theories on the { 7 { critical lines as and IVb only contains the singlet fermions; and the line between phase IVb and I is SU(N ) Yang-Mills coupled to Ns scalars, which we expect to con ne and lead to a gapped theory. By the by, the operator O4 must be present in order that the singlet fermions are gapped on then critical line separating Phases III and IVa. However, this requirement does not constrain the sign of its coupling c4. We do note that if c4 were positive, then the critical line with the massless singlet fermions would run through Phase III rather than Phase IV. Having mapped out the SU phase diagram we move on to consider the U theory. Much of the discussion of the U phase diagram carries over here without modi cation, and so let us simply summarize the salient features. See gure 1. As before, there are ve distinct massive phases described by four di erent IR TFTs, which are given by (I') (II') coe cient of O40, c04, must be positive, so that Phase IV splits into two, with the massive phases are described by If c04 were negative, the critical line with the massless singlet fermions would run through Phase I' rather than Phase IV'. For the special case k = Nf , we see that Phases I' and { 8 { (2.12) (2.13) (2.14) (2.15) IV' are trivial, that the line separating Phases IVa' and IVb' is just given by the massless singlet fermions, and the line between Phase IVb' and I' is trivial. Having assembled all of this information, we may match the phases, lines, and operators of the two theories. Using the basic level/rank duality (1.1) equating we see that the TFTs governing the massive phases of the SU theory (2.10) are precisely those describing the massive phases of the U theory (2.13). Phase I maps to Phase I', and similarly for the others. Comparing the axes on the two phase diagrams in gure 1 then tells us how the mass operators map under the duality, with $ Furthermore, we see that the critical lines in both theories (2.12) and (2.15) match upon using Aharony's dualities (1.2) and (1.3). To match the lines running through Phases IV and IV' we also require level/rank duality to equate the TFTs arising on each line. For the special case k = Nf , the massive phases still match on account of the fact that Phases I, I', IV, and IV' are all trivial. The critical lines also match, upon recalling that the IVb-I and IVb'-I' lines are trivial. We also see that the assumption that our duality holds determines the sign of the quartic couplings c4 and c04. As we mentioned above, if c4 were positive, the \singlet critical line" of the SU theory would run through Phase III, and if c04 were negative, the singlet critical line of the U theory would run through Phase I'. The only consistent possibility is c4 < 0; c04 > 0. An even simpler argument is that Phase IV is the only quadrant of the phase diagram in which both the SU(N ) theory and its U(k) dual are both in a Higgs phase, and so the singlet line must run through it. As we mentioned in the Introduction, we see that the quartic coupling has the same sign as the Chern-Simons level. 2.1.2 Ns = N Now we tackle the case Ns = N . We will be brief and summarize the main features. See gures 2 for the schematic phase diagrams when k < Nf . In the SU theories there are four phases (not all of which are massive). In the Higgs phase we have SU(N ) ! 0, and the only remnant of the scalars is a single compact Goldstone boson. Dualizing it into pure electromagnetism, we write the various phases as (I) (II) (III) m m and Phase III splits into Phases IIIa and IIIb depending on whether the fermions (note that since the gauge group is trivial in the Higgs phase all fermions are \singlets.") get a negative or positive mass. For k = Nf , the theory in Phase I is SU(N )0 which we expect { 9 { (a) (b) coupled to Ns fermions and Nf scalars. The various phases are described by (2.17) and (2.19). and the critical lines are described by the CFTs given in (2.18) and (2.20). The shaded region is gapless. to be massive. The critical lines separating the phases are For k = Nf Phase I and the line separating Phases IIIb and I are trivial. For Nf < k the U theories also have four phases, while for Nf = k they have ve. We have (I') (II') and Phase IV' only exists when Nf = k. For Nf < k, Phase III' splits into Phases IIIa' and IIIb' depending on whether the fermions have a negative or positive mass. For Nf = k, Phase IV' splits into Phases IVa' and IVb' with the same. For Phases III' and, when it exists, IV', we are using the expectation that the non-abelian part of U(k)0 con nes at low energies so that its low energy limit is described by pure electromagnetism. The critical lines when Nf < k are (I'-II') Comparing the phases of the SU theory (2.17) and the phases of the U theory (2.19), we see that the two perfectly match provided that k < Nf . The same is true for the critical lines. However, clearly there is no match when k = Nf , and thus we demand the last of our avor bounds, (Nf ; Ns) 6= (k; N ). In the last subsection we matched the various massive phases and critical lines of the SU and U theories appearing in our proposed duality (1.4). In fact we can perform an even stronger test. We rst couple both sides to slowly varying background gauge elds. (We could also put the theories on a spin manifold with a slowly varying metric, but we do not do so in this work.) The massive phases are then described not only by non-trivial TFTs, but there are also avor Chern-Simons terms whose levels are one-loop exact and which we may match. It is straightforward to deduce the levels for the non-abelian part of the avor symmetry, but as we will see, we must be careful when computing the levels for the abelian part. It is particularly tricky to compute the abelian levels in the SU theories, and we will nd it convenient to represent the SU(N ) theories as U(N ) theories subject to a U( 1 ) quotient. 2.2.1 The U(k) theories We begin with the U(k) theories. The manifest Abelian global symmetry of these theories is U( 1 )m U( 1 )F , where U( 1 )m refers to a monopole number and the U( 1 )F is an ordinary charge under which the bosons carry charge 0 and the fermions charge 1. With this convention, the manifest global symmetry is in fact U(Nf ) elds which couple the manifest symmetry and including carefully chosen Chern-Simons terms for the external elds, the Lagrangian is LU = i Here a0 is a U(k) gauge eld, A0 is a background SU(Nf ) gauge eld, and B0 a background SU(Ns) gauge eld. There are also background abelian gauge elds: the U( 1 )m gauge eld is A~0 1 and the U( 1 )F gauge eld is A~02 . (We can group B0 and A~02 into a U(Ns) gauge (2.21) (2.22) eld if we wish.) Observe that A~01 only appears through a BF coupling to the monopole current ?jm = 21 dtr(a0). The various abelian Chern-Simons levels are subject to quantization conditions, which when violated characterize 't Hooft anomalies for the avor symemtries. For now, we will simply proceed to compute the levels in the various phases without worrying about the precise quantization conditions. Furthermore, we are being a bit sloppy in writing (2.21). For generic parameters we are not allowed to set the various abelian levels to vanish. What we are really doing in this subsection is to compute the jumps in avor Chern-Simons levels from one phase to another, and these jumps of course do not depend on these details. The ve massive phases (2.13) are obtained after turning on fermion and scalar masses. The various Chern-Simons levels receive one-loop shifts after integrating out fermions, as well as additional shifts in the Higgsed phases. Including the avor groups, the massive phases are characterized by (I') : U(k Nf )N where only the rst group is dynamical, and there are 2 2 matrices describing the abelian Chern-Simons levels in each phase, J 0ab 4 A~0adA~0b : JIa'b = N ! (2.23) (2.24) (2.25) (2.26) (2.27) (2.28a) We need to integrate out the dynamical U( 1 ) gauge eld tr(a0) to get these abelian Chern-Simons levels. For example, in Phase I', tr(a0) appears through two terms in the low-energy e ective Lagrangian, Ltr(a0) = i N 4 (k Nf ) tr(a0)dtr(a0) + tr(a0)dA~01 : 1 2 (The rst term is the abelian part of the U(k an equation of motion for tr(a0) which sets Nf ) Chern-Simons term.) There is e ectively so that (2.25) becomes tr(a0) = Ltr(a0) ! N k + Nf A~01 ; i i.e. an e ective Chern-Simons term for A~01 at level N k+Nf . Accounting for the one-loop shifts to the bare levels (which happen to vanish in this phase), the 2 2 matrix of abelian CS levels is Similar computations in the other phases of the U theory give JIaIb' = JIaIbI' = JIaVba' = JIaVbb' = Nk 0 We continue with the SU(N ) theories. Turning on a background for the manifest U(Nf ) U(Ns) = SU(Nf ) U( 1 )f =ZNf U( 1 )s =ZNs global symmetry, and including Chern-Simons terms for the background elds, the Lagrangian is along with LSU = i i 4 N 4 i a 1 f + A 1 c + where 1 f acts as the identity on avor indices, 1 c as the identity on color indices, and 1 as the identity on all indices. Here we have separated the U( 1 )f U( 1 )s global symmetry into its diagonal part, which couples to A~1, and a scalar part which couples to A~2. Note that diagonal U( 1 ) global symmetry is \baryonic." Our choice of normalization for the conjugate external eld A~1 is such that the baryon charges of gauge-invariant operators are integers. Now consider the ve massive phases (2.10) obtained by turning on the fermion and scalar masses. The avor Chern-Simons levels receive one-loop shifts after integrating out the fermions, and now including the avor groups, the massive phases are described by (I) : SU(N ) k+Nf where again only the rst factor is dynamical and the J 's refer to 2 2 matrices of abelian Chern-Simons levels. Before computing them, observe that the non-abelian Chern-Simons levels in the phases of the SU(N ) theories exactly match the non-abelian levels in the phases of the U(k) theories (2.23), upon identifying the non-abelian avor background of the SU theory with that of the U theory, A = A0 ; B = B0 : We have to work a bit harder to compute the abelian levels. We nd it useful to realize the SU(N ) theories as U(N ) U(1) theories.4 First, begin with U(N ) Simons theory coupled to Nf fermions and Ns scalars. This theory has a manifest SU(Nf ) U( 1 )F global symmetry, where the U( 1 )m is a monopole number. Its 4 i i N 4 4 k + Nf tr ada 2i 3 a 3 + 1 2 Now a~ only appears in the Lagrangian through a BF couplings to tr(a) and so we may integrate it out. This sets the constraint 4Relatedly, it seems that the simplest way to derive the level/rank duality SU(N ) k $ U(k)N is to realize the SU(N ) k theory as a suitable U(N ) U( 1 ) Chern-Simons theory [8]. Here a is a U(N ) gauge eld, the other background elds are as before, and Am is a background eld which couples to monopole number. We have also neglected a matrix of abelian background Chern-Simons terms. Now we gauge the monopole number by promoting Am to a dynamical eld, Am ! a~ ; D D = i(a 1 f + B 1 c)) : and integrating over it in the functional integral. Now there is a new U( 1 ) global symmetry, the monopole number associated with ~a, and we couple said monopole number to a background U( 1 ) eld which we take to be (A~1 + N A~2). The end result is that, with some foresight for the abelian Chern-Simons terms, we rede ne the Lagrangian as along with tr(a) = A~1 + N A~2 ; (2.32) k+ N2f Chernwhich, when inserted back into the Lagrangian, leads to the original SU(N ) theory (2.29). In particular, the covariant derivatives of the matter elds now read D D = = Now let us compute the abelian levels. We illustrate the idea in Phase I. In this phase there is no one-loop shift to the levels from integrating out the fermions, and so plugging the constraint (2.37) into the U( 1 ) U(N ) Chern-Simons term in (2.36) leads to the HJEP01(28)3 combined abelian Chern-Simons terms + k 4 N 2 i k + Nf (A~1 + N A~2)d(A~1 + N A~2) Nf A~1dA~2 + N (k Nf ) A~2dA~2 = i 4 4 N k + Nf A~1dA~1 ; i.e. to a matrix of abelian levels JIab = which happily matches the matrix we obtained in Phase I' of the U(k) theory (2.28a). As for the other phases, a straightforward computation yields JIaIb = JIaIbI = JIaVba = JIaVbb = Nk 0 These matrices precisely match those computed in the corresponding Phases (2.28b) of the U(k) theory, provided that we identify the external U( 1 ) elds as A~1 = A~01 ; A~2 = A~02 : Observe that the baryonic symmetry of the SU(N ) theory, which coupled to A~1, is mapped to the monopole symmetry of the U(k) theory, which coupled to A~0 . Taken together, we 1 see that all avor Chern-Simons terms can be matched across the duality. (2.38) (2.39) (2.40a) (2.40b) (2.41) The point of this section is two-fold. We have already seen that the SU=U duality exchanges the baryon number of the SU theory with the monopole number of the U theory. In the next subsection we see how this works in more detail, matching the quantum numbers of baryons with those of the monopoles. In subsection 3.2 we deduce the faithful subgroup of the manifest SU(Nf ) U(Ns) U( 1 ) global symmetry that acts on both sides of the duality, nding that this faithful global symmetry matches. Our discussion closely imitates that of [7, 32]. In the last section we parameterized the U( 1 )f U( 1 )s global symmetry of the SU(N ) theories with some foresight. We rewrote it in terms of a U( 1 )b U( 1 )F global symmetry, where the rst factor is \baryonic" and the second is an ordinary global symmetry. Under them the fundamental fermions and scalars have charges For simplicity, we take the large N and large k limit with N=k xed. We further take Nf = Ns = 1, although it is straightforward to allow for a more avors. The various gauge-invariant operators of the SU(N ) theory fall into two types. The rst are the mesons Dn ; Dm Dn ; yDn : These operators remain \light" at large N with a dimension that scales as O(N 0). They all have zero baryon charge and only the second type is charged under U( 1 )F . There are also \multi-trace" operators built out of products of the mesons and derivatives. The second class of operators are baryons. The simplest baryons are a product of N fundamental fermions and bosons with the color indices antisymmetrized. Our convention is that they carry charge +1 under U( 1 )b. There must be derivatives acting on the scalars in order to antisymmetrize them, so, schematically the baryons are " : : : |N{zM} | D : : : Dn ; {z M } Besides carrying charge +1 under baryon number, they also carry charge +M under U( 1 )F . There are many such baryons, depending on how we take derivatives. A simple counting exercise at large M with M=N xed [38] reveals that the dimension of the lowest-dimension baryons at large N is approximately given by N 3 M + 23 M 2 . Observe that this is the dimension of a baryon in SU(M ) theory with a fermion plus that of a baryon in SU(N M ) theory with a scalar. There are also multi-trace operators built out of products of the simplest baryons, mesons, and derivatives. (3.1) (3.2) (3.3) Now for the U(k) theories. In these theories there is a U( 1 ) U( 1 ) global symmetry under which the various elds are charged as U( 1 )m U( 1 )F 0 0 0 -1 All fundamental elds are neutral under the monopole number U( 1 )m, and instead the monopole current is given by the U( 1 ) SU(N ) theories there are mesons U(k) gauge eld, jm = 21 " HJEP01(28)3 yDn ; Dm Dn ; Dn ; which remain light at large N with a dimension that scales as O(N 0). All of these operators carry zero monopole number and only the second is charged under U( 1 )F . The second set of operators are monopoles. For a U(k) gauge theory these are characterized by a set of k integers qi (up to permutations by the Weyl group) which give the U( 1 )k U(k) uxes. These integers are called GNO charges, and by our convention the total monopole charge is Pi qi. Monopole operators are not gauge-invariant in U(k)N Chern-Simons theories: in the presence of a monopole with n GNO charges the gauge symmetry is e ectively broken to U( 1 )n U(k n), and due to the Chern-Simons term the monopole carries charge N qi under the ith U( 1 ). These must be canceled by inserting additional elds in the (anti-)fundamental representation of U(k) so as to obtain a gauge-invariant operator. For example, the simplest monopoles carry GNO charges fqig = f1; 0; : : : ; 0g, and these are expected to have the lowest dimension of any monopoles. To make the monopole gauge-invariant we must multiply it by N anti-fundamental elds with the same gauge index. We can make up such an operator out of M scalars and N M fermions, but to do so we must symmetrize the fermions by including appropriate derivatives. So, schematically, these monopoles take the form (GNO ux) y : : : y D : : : Dn : | N{zM } | {z M } Clearly these operators carry monopole number +1 as well as charge +M under U( 1 )F , which coincides with the U( 1 )b U( 1 )F charges of the baryons in (3.3). At large M with M=N xed, the dimension of the lowest-dimension monopoles are given by the same counting argument as for the baryons of the SU(N ) theory, with = N 3 M + 23 M 2 . There are many such operators with various spins. In the monopole background the scalars carry spin 1=2 [39] while the fermions are spin-0, so that the possible spin quantum numbers of the monopoles precisely equals the set of possible spin quantum numbers for the baryons of the SU(N ) theory. In sum, at large N , the quantum numbers of the simplest baryons (3.3) in the SU theory match those of the simplest monopoles (3.6) in the U(k) theory. 3.2 Exact avor symmetries Let us work out the faithfully acting global symmetries that act on the local operators on both sides of our proposed duality (1.4) for generic values of the parameters. (3.4) (3.5) (3.6) The SU(N ) theories have a naive U(Nf ) U(Ns) global symmetry, where the U(Nf ) acts on the Nf fundamental fermions and the U(Ns) on the fundamental scalars. However only a (U(Nf ) U(Ns)) =ZN subgroup of this symmetry acts faithfully on the operator spectrum, where the generator of ZN acts as multiplication by e2 i=N . In physical terms, the gauge-invariant operators charged under the diagonal U( 1 ) symmetry are baryons built C from N fundamental fermions and bosons. There is also a charge conjugation symmetry Z2 which exchanges the fundamental representation with the anti-fundamental representation, so that the total global symmetry is U(Nf ) U(Ns) ZN o Z2C : (3.7) What is the faithful global symmetry that acts on the U(k) theories? Here there is a manifest SU(Nf ) U(Ns) U( 1 )m global symmetry where U( 1 )m is monopole number. As we discussed above, monopoles are characterized by a set of GNO monopole charge is their sum qm = Pi qi. Monopoles carry electric charge by virtue of the bare U(k)N Chern-Simons term, so to render a monopole gauge-invariant it must be dressed with a number of fundamental and anti-fundamental elds. The total number of antifundamental elds minus the number of fundamentals must equal N qm. Given monopoles with U( 1 )F charge M , the monopoles ll out representations of SU(Nf ) SU(Ns) with Nf -ality (N qm M ) mod Nf and Ns-ality M mod Ns. We may then understand U( 1 )m to act as a diagonal U( 1 ), reducing SU(Nf ) U(Ns) U( 1 )m to U(Nf ) U(Ns), subject to an additional ZN quotient. As in the SU(N ) theories there is also a Z2C charge conjugation uxes fqig and the total symmetry, so that the total faithfully acting global symmetry is U(Nf ) U(Ns) ZN o Z2C ; (3.8) which precisely matches (3.7). In this work we do not undertake an analysis of the quantization conditions for avor Chern-Simons terms consistent with the faithfully acting global symmetry (3.7). For typical values of the parameters it will be the case that those avor Chern-Simons terms will necessarily have levels with a fractional part. When it exists this fractional part is an 't Hooft anomaly, and it implies that these theories do not have an intrinsically 3d de nition in a general avor background, and must instead be de ned as living on the boundary of a 4d SPT phase. While we do not deduce the anomalies of the theories in our proposed duality (1.4), we do observe that since the global symmetries match, as do the avor ChernSimons terms in the various phases, then whatever the 't Hooft anomalies are in the SU=U theories, they should match. 4 SU=U duality implies a U=U duality The basic sequences of 3d bosonization dualities equate $ $ U(k)N SU(k)N with Nf scalars ; with Nf scalars ; k. There are also time-reversed versions of these conjectures. These dualities are in fact equivalent to each other and yet another duality, U(N ) $ U(k)N;N k with Nf scalars : (4.1) The two subscripts denote independent levels for the non-abelian and Abelian parts of U(N ), U(N )k;k+mN = (SU(N )k U( 1 )kN+mN2 )=ZN : (4.2) HJEP01(28)3 To see that these dualities are equivalent to each other, start with the rst duality, de ning both sides with a suitable and matched choice of background Chern-Simons term for the U( 1 ) global symmetry. Then gauge the U( 1 ) global symmetry on both sides. The same procedure with a slightly di erent choice of background U( 1 ) Chern-Simons level on both sides gives the U=U duality. A completely general choice of U( 1 ) Chern-Simons level leads to yet more dualities between U(N ) theories and U(k) U( 1 ) theories where the U( 1 ) factor is topological [40]. As a side comment, the massive phases of the U=U duality (4.1) match by virtue of a level/rank duality for U gauge groups (see [8]), U(N ) k; k N $ U(k)N;N k ; which also follows from simply setting Nf = 0 in (4.1). Using the same sort of logic, our proposed duality (1.4) implies a U=U duality between U(N ) $ Both sides have a phase diagram that looks identical to those of the SU(N ) and U(k) theories discussed in section 2, and it is easy to see that the massive phases and critical lines all match on account of the level/rank duality (4.3) and U=U bosonization duality (4.1). We also require that the quartic operator ( )( y ) be present on both sides of the duality with the same sign as the non-abelian Chern-Simons level. We start with the Lagrangians for both sides of our proposed SU(N )=U(k) duality, matched so that all avor Chern-Simons levels agree in massive phases. For simplicity, we only turn on a background A~1 for the baryon/monopole U( 1 ) global symmetry. With a useful convention for the bare U( 1 ) Chern-Simons level, we have LSU(N) = LU(k) = i i N 4 4 k + Nf tr ada 3 2i 3 a 3 + 1 2 + 4 N k + Nf A~1dA~1 + iD= tr(a0)dA~1 + jD j2 + iD= D D D i a 1 f + i a 1 f + ia0 1 f ia0 1 f ; : ~ A1 ! a~ ; N Before going on, observe that if we promote A~1 to be a dynamical gauge eld, then in the SU(N ) theory it combines with a into a U(N ) gauge eld a = a + Na~ the SU(N ) theory becomes U(N ) Ns scalars, k+ N2f Chern-Simons theory coupled to Nf scalars and LSU(N) ! LU(N) = i 4 k + Nf tr ada 2i 3 a 3 + iD= + jD j2 + Lint : Meanwhile in the U(k) theory, integrating over a~ enforces tr(a0) = 0 turning it into 2 SU(k)N Ns Chern-Simons theory coupled to Nf scalars and Ns fermions, i N 4 3 LU(k) ! LSU(k) = iD= So we see that the original SU(N )=U(k) duality implies a U(N )=SU(k) duality (4.6) (4.7) (4.8) (4.9) (4.10) U(N ) k+ N2f with Nf ; Ns 2 $ SU(k)N Ns with Ns ; Nf : This is merely the time-reversed version of the original SU(N )=U(k) duality combined with exchanging N $ k, Nf $ Ns. This is yet another consistency check on our proposal (1.4). We can now obtain the U=U dualities (4.4). To the SU(N ) and U(k) Lagrangians in (4.5) we now add a background Chern-Simons term with level 1 for the U( 1 ) baryon/monopole global symmetry, LSU(N) ! LSU(N) LU(k) ! LU(k) i 4 i 4 1 A~1dA~1 ; 1 A~1dA~1 : with a into a U(N ) gauge eld a = a + Na~ the U( 1 ) level by N , giving Now, we promote A~1 to a dynamical gauge eld A~1 ! a~. On the SU(N ) side it combines 1 and the extra U( 1 ) Chern-Simons term shifts k+ N2f with Nf ; Ns ! which is the left side of the U=U duality (4.4). In the U(k) theory, the new eld a~ appears in two terms: It can be integrated out, giving so that La~ = tr(a)da~ a~da~ : 1 2 a~ = tr(a) ; 1 4 La~ ! i 4 1 tr(a)dtr(a) ; 2 2 U=U duality (4.4) as promised. 5 SO and USp dualities Another in nite sequence of level/rank dualities equates [26] which e ectively shifts the U( 1 ) level of the U(k) Chern-Simons term by k. In this way ! ; which is the right side of the U=U duality (4.4). So the SU=U duality (1.4) implies the SO(N ) k USp(2N ) k SO(k)N ; USp(2k)N : One might expect that there are \ avored" versions of these dualities, and indeed there is a natural conjecture for them [26, 27]: SO(N ) k+ N2f with Nf real fermions USp(2N ) k+ N2f with Nf fermions SO(k)N with Nf real scalars ; USp(2k)N with Nf scalars : As before, on the scalar side there is a non-trivial scalar potential tuned to criticality so that the scalars are (real) WF scalars, while the fermions are Majorana. There are also avor bounds Nf for N = 1, Nf k USp for the dualities, while for the SO dualities one requires Nf k 2 k 1 for N = 2, and Nf k for N > 2. Equivalently, the SO dualities simultaneously require Nf The evidence for these dualities is at the same level as for the basic sequence of SU=U bosonization dualities (1.2). To leading order in large N the orthogonal and symplectic theories are just orbifolds of the SU=U theories [26]. The massive phases of both sides of the dualities match, as do the exact global symmetries and 't Hooft anomalies [32]. It is then natural to conjecture a sequence of SO=SO and USp=USp bosonization dualities with both fundamental fermions and bosons. We propose SO(N ) USp(2N ) $ $ 2 2 SO(k)N Ns with Ns ; Nf ; USp(2k)N Ns with Ns ; Nf : (5.3) (5.4) $ $ $ $ (4.12) (4.13) (4.14) (4.15) (5.1) (5.2) For the USp dualities we require Nf N , and for the SO dualities we further require 3 + Ns + Nf k + N . At large N these dualities follow from our original SU=U conjecture (1.4) by suitable orbifold projections. Under the assumption that the only relevant operators in these theories are scalar and fermion mass operators, we may proceed just as in section 2 and map out the phase diagrams of the dual pairs. For general parameters, the phase diagrams look identical to those of the SU=U theories in gure 1, the massive phases match on account of the level/rank dualities (5.1), and the critical lines match on account of (5.2). For the orthogonal sequence, the matching of the critical lines requires the avor bound 3 + Ns + Nf k + N . Let us obtain this last avor bound, starting with the SO(N ) theories. As in our discussion of the SU=U dualities, we require the operator ( i m)( m i) to be present with a coe cient with the same sign as the Chern-Simons level. For general parameters, there are then ve distinct phases and four di erent TFTs, given by (5.5) (5.6) (5.7) (5.8) (5.9) (I) (II) (III) (IV) m m m m > 0 ; m2 > 0 : SO(N ) k+Nf ; < 0 ; m2 > 0 : SO(N ) k ; < 0 ; m2 < 0 : SO(N > 0 ; m2 < 0 : SO(N Phase IV splits into two, The critical lines are described by (IVa) (IVb) m m > 0 ; ms < 0 ; m2 < 0 : SO(N > 0 ; ms > 0 ; m2 < 0 : SO(N (II-III) SO(N ) k with Ns ; (III-IVa) SO(N Ns) (IVb-I) SO(N ) k+Nf with Ns : The corresponding phase diagram is identical to that on the left side of gure 1. Meanwhile, there are generally ve phases of the SO(k) theories described by four di erent TFTs, We require the coe cient of the ( m so that Phase IV' splits into two, i)( i m) operator to be nonzero and positive, (I') (II') < 0 ; ms < 0 ; m2 < 0 : SO(k < 0 ; ms > 0 ; m2 < 0 : SO(k The critical lines are (I'-II') SO(k)N with Nf ; (II'-III') SO(k)N Ns with Ns ; 2 (III'-IVa') SO(k)N Ns with Nf ; NsNf singlet 's + SO(k and the phase diagram coincides with the right side of gure 1. The massive phases of the SO(N ) theory (5.5) clearly match those of the SO(k) theory (5.8) upon using the SO level/rank duality (5.1). Similarly the theories arising on the critical lines match by virtue of the SO bosonization duality (5.2), and for the case of the IVa-IVb and IVa'-IVb' lines one must also use the SO level/rank duality. However, the avor bounds on the SO dualities (5.2) Nf k + N only hold for lines III-IVa and III'-IVa' if 3 + Ns + Nf k + N ; (5.10) (5.11) is satis ed, which originates that bound. anomalies for future work. 6 We leave a careful computation of the exact global symmetries and their 't Hooft In this work we have conjectured new in nite sequences of dualities between nonsupersymmetric Chern-Simons-matter theories with fundamental bosons and fermions. The three inequivalent dualities are the \basic" SU=U sequence (1.4), a real SO=SO sequence (5.3), and a USp=USp sequence (5.4). We performed some basic consistency checks on our proposal, including the matching of phase diagrams and, for the SU=U dualities, the matching of global symmetries. We conclude with a short list of open questions. 1. Komargodski and Seiberg have recently suggested [9] that the basic bosonization duality (1.2) and its cousins may be extended beyond the avor bound Nf k. Their proposal is that new \quantum" phases open up in the range k < Nf < N (N; k) where N (N; k) is some presently unknown function of N and k. Their logic has also been useful in a proposal to map out the phase diagram of Chern-Simons theory with a single adjoint fermion [29]. It would be interesting to understand to what extent our conjecture (1.4) can also be extended beyond the avor bounds Nf k and Ns N . 2. There are a host of supersymmetric dualities between 3d eld theories with at least N = 2 supersymmetry (SUSY), including Seiberg duality (sometimes called Giveon/Kutasov duality [41] in three dimensions) and mirror symmetry [42]. At large N there is signi cant evidence that the basic 3d bosonization dualities are inherited from a Seiberg duality between certain N = 2 supersymmetric Chern-Simons theories with unitary gauge group coupled to chiral multiplets [33, 34] (although it seems [43] that the particular ow studied in [33] does not work as advertised). After turning on a deformation which completely breaks SUSY, the low-energy theory on both sides of the duality ows to a product of bosonic and fermionic Chern-Simons-matter theories. The underlying Seiberg duality then exchanges the bosonic half of the electric theory with the fermionic part of the magnetic one, and the fermionic half of the electric theory with the bosonic part of the magnetic one. However there are more general Giveon/Kutasov dualities with a single gauge group as studied in [44]. These equate a N = 2 SUSY-Chern-Simons theory with Nf chiral and Nf0 anti-chiral multiplets (meaning matter in both the fundamental and antifundamental representations) and a SU(Nf ) SU(Nf0 ) global symmetry with another N = 2 theory with chiral and anti-chiral multiplets and, reminiscent of 4d Seiberg duality, gauge-neutral mesons which are bifundamental under the avor symmetry. It would be nice to understand if our proposed duality (1.4) is inherited from these more general dualities along the lines of [33, 34], and, if so, to understand what happens to the mesons. (Indeed one of the original motivations behind the present work was to identify a non-SUSY bosonization duality with mesons, although, as we found, a duality with fermions and bosons does not seem to be allow for such gauge-neutral elds.) 3. Chern-Simons theory with fundamental matter is analytically tractable in the large N and k limit with N=k xed, and indeed, the best evidence for the non-supersymmetric bosonization dualities comes from explicit computations of correlation functions, the thermal partition function, and scattering amplitudes in that limit. It would be nice to extend those computations to allow for multiple bosons and fermions. In particular, one ought to be able to address the perplexing questions related to the to the quartic operator ( )( y ) which played an important role in our proposed duality. In section 2 we saw that if this operator was tuned away, or if its coe cient had the wrong sign, then the duality (1.4) was inconsistent. But this sign (or vanishing) is likely determined by the underlying dynamics, which are yet unsolved. Relatedly, it is not yet known if this operator is irrelevant (in which case it is in fact dangerously irrelevant) or relevant at large but nite N . We intend to return to these questions soon. 4. Finally, recall the conjectured duality between the Chern-Simons theories with one boson or one fermion at large N; k with N=k nite. A natural question is then if there is a Vasiliev-like theory dual to Chern-Simons theory with both Nf fermions and Ns scalars, and if so, if our proposed duality is consistent with it. Acknowledgments We are especially grateful to A. 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