Chern-Simons theory coupled to bifundamental scalars

We study the three-dimensional theory of two Chern-Simons gauge fields coupled to a scalar field in the bifundamental representation of the $SU(N)_k \times SU(M)_{-k}$ gauge group. At small but fixed $M \ll N$, this system approaches the theory of a Chern-Simons field coupled to fundamental matter, conjectured to be dual to a parity-violating version of Vasiliev's higher-spin gauge theory in $AdS_4$. At finite $M/N$ and large 't Hooft coupling this theory (or its SUSY version) is expected to be dual to an Einstein-like gravity. We show at two loops that this theory possesses a line of fixed points at any value of $M/N$. We also prove that turning on a finite but small $M/N$ gaps out the light states that Chern-Simons theory coupled to fundamental matter develops when placed on a torus. We also comment on the higher genus case.

Various aspects of CS theory coupled to fundamental matter have been studied in detail in [30][31][32][33][36][37][38][39][40][41][42][43][44][45][46][47][48]. In Section 2 we study, following [25,31], the non-supersymmetric CS theory coupled to bifundamental scalar matter. We compute the two-loop β-functions of the theory and find two lines of fixed points parametrized by the gauge coupling λ. In Section 3, following [33], we study the CS-bifundamental theories on a spatial torus. Encouragingly, we find that there are no exactly degenerate states in the 't Hooft limit. Instead, we find that the gap is proportional to ξ, when ξ is small, and along the way we develop a straightforward diagrammatic way of arranging the perturbation theory in ξ. Finally, in the concluding section, we point out how these results suggest that bifundamental theories can be used to regulate fundamental matter theories by tuning the ratio ξ of the two ranks in a bifundamental theory. On a spatial torus, by changing ξ one tunes the mass of the gauge field holonomies and, at ξ ∼ 1/N , transitions into a phase that looks like the fundamental theory (i.e. the singlet vector model) on a torus. This indicates that the heretofore mysterious modification of Vasiliev theory that can accommodate for the light states found in [33] should be attainable as a particular limit of the bulk theory studied in [25].
A note on conventions is in order. The two papers on which our analysis rests, [31] and [33], employ different conventions and have slightly different actions. In order to be able to check that our computations correctly reduce to the results of these two papers, we work with different actions in different sections of this paper; in each section we follow the conventions of its guiding reference.

Perturbative fixed points
We consider an SU (N ) k × SU (M ) −k Euclidean CS theory coupled to scalars in the bifundamental representation (N,M ). Without loss of generality we will take M ≤ N . The partition function of our theory is and the action that contains all the marginal terms is (2. 2) The covariant derivative is 3) The bifundamental scalar, φ, is an N × M matrix transforming as φ → U φV † for U ∈ SU (N ), In this section we perform the fixed point analysis of the 't Hooft limit perturbatively in λ = N/k.
We work to two-loop order, closely following the work of [31]. The CS coupling g 2 = 4π/k is quantized and does not run under RG flows; hence we only need to compute the β-functions for g 1 , g 2 , and g 3 . These are found by computing the two-loop amplitude when all external momenta are zero. At tree level, this amplitude is given by, M tree = − g 1 δ i j δ jk δ k l δ lm δ m n δ ni + 11 permutations − − g 2 δ i j δ jk δ k l δ li δ m n δ nm + 17 permutations − − g 3 δ i j δ ji δ k l δ lk δ m n δ nm + 5 permutations . (2.5) Once loop corrections are introduced, the amplitude must be regulated by counterterms δg i (µ) multiplying the above tensor structures. These counterterms depend on the subtraction scale µ and can be used to extract the β-functions β i of all three couplings. We will use dimensional reduction 4 to regulate the UV divergences, and we will work in the minimal subtraction scheme.
Despite the apparent complexity of this computation, Aharony et al. have shown that, remarkably, only eight diagrams need to be computed when dealing with fundamental CS-matter to two-loop order [31]. 5 The same argument goes through for bifundamental CS-matter (Fig. 1). The additional complication in our case is the existence of two gauge fields and two additional multi-trace scalar six-point couplings, so each diagram in the fundamental matter theory can be thought to generate a number of related diagrams in the bifundamental theory, each with the same momentum structure but with different index contractions and multiplicity. These are all straightforward to 4 The CS term cannot be written in general d = 3 dimensions, and so the standard dimensional regularization cannot be applied to CS-matter theories. Instead, one uses the dimensional reduction scheme where the tensor algebra appearing in the Feynman integrals is done in three dimensions and then the resulting scalar integral is analytically continued to general space dimensions. This preserves gauge invariance at least up to two loops. See [40] for details. 5 In an earlier version of this paper, a diagram that is zero for O(N ) was mistakenly concluded to be zero for U (N ), based on the findings in [31]. We thank Guy Gur-Ari and Raghu Mahajan for extensive discussions that have clarified this issue. enumerate and compute. The Feynman rules are given in Appendix A, and applying them gives (2.13) The quantities I Ai label the dimensionally regulated momentum space integrals, which evaluate to Now we can switch to the 't Hooft couplings, 15) and find that the counterterms are given by As a rudimentary check, we notice that taking ξ = 1/N makes the contributions to δλ 1 from diagrams (A2) and (A3) exactly cancel, leaving diagram (A1) as the only process contributing to the β-function for λ 1 ; the same non-trivial cancellation occurs in the fundamental CS-matter model.
Moreover, in this limit, the counterterms are all the same, and are given by This is precisely the leading N behavior found in [31]. This is expected, as each of the three traces in the action (2.2) should collapse to the usual six-point term for fundamental matter, (φ † φ) 3 , and so all three couplings λ i should flow in the same way.
The field strength renormalization can be found by computing two-loop corrections to φ † ii φ jj . These are given by four diagrams on Fig. 2, and these are regulated by the following counterterms Figure 2: A schematic depiction of diagrams contributing to the field strength renormalization to two loops. [40]: The total counterterm needed, expressed in terms of 't Hooft couplings, is This result precisely reduces to the leading N result found in [31] when ξ = 1/N .
The β-functions, β i = µ∂ µ λ i , are now found using standard methods. In dimensional reduction (regularization), the scale independence of the bare six-point couplings requires that This is the renormalization group equation. Differentiating through and setting µ∂ µ λ = −ελ (the CS coupling runs with the scale in d − ε dimensions), we find Knowing that the counterterms are all quadratic or quartic functions of the couplings simplifies these expressions down to Substituting the counterterms (2.16) and (2.22) finally yields In agreement with [31], we find that the β-functions all become suppressed when ξ = 1/N . At finite ξ, we find two lines of fixed points parametrized by λ and given by These two lines of fixed points are shown on Fig. 3.
We have found this two-loop fixed line in the large-N limit, but our analysis holds for any value of the fixed ratio ξ = M/N . It will be interesting to see if this holds to all-loop order, at least in the large-N limit. We can see that as the 't Hooft coupling goes to zero the fixed points approach the trivial value zero, which is consistent with the fact that that the model has no fixed points in the absence of the CS term.
It is easy to see that, at both fixed point lines, the sign of the λ 2 coupling is opposite that of the other two couplings. Both lines could be stable. To answer that question one has to compute the Coleman-Weinberg potential and see if the potential is bounded from below for the specific values of the coupling constants. We will not try to do that in this paper, but an argument may be given which shows that there is no tachyonic mode -at least for small values of the 't Hooft coupling λ.
For example, one could compute the self-energy of the scalar field and look for unphysical poles. In our case the self-energy at least up to two-loop order does not get any contribution from the sixpoint couplings, and so there cannot be a tachyonic pole caused by the negative coupling constants of the six-point interactions.
In addition, the conformal symmetry may be spontaneously broken. For example, the conformal symmetry is spontaneously broken in the vector model with only a φ 6 interaction and no CS term, if the coupling constant λ 6 is greater than 4π 2 . It will be interesting to compute the Coleman-Weinberg potential and study this spontaneous breaking, but we leave that for future research.

No light states on a torus
In this section we change gears and study the low-energy, small-λ limit of the CS-bifundamental theory on the torus in a spacetime with Lorentzian signature. Our goal is to retrieve the spectrum of the theory; our methods closely follow those in [33]. We will see that this spectrum has no state whose energy vanishes in the 't Hooft limit. In principle, one should study the full bifundamental theory with its potential terms. Instead, we will study the toy theory with only the CS terms and the covariant kinetic terms for the scalar field. This captures the essential physics at small 't Hooft coupling and at energy low compared to the inverse size (KK scale) of the spatial torus.

Low-energy effective Hamiltonian
As usual, we canonically quantize in the gauge A 0 = B 0 = 0, where 0 denotes the time component.
We are interested in low-energy degrees of freedom only, and so we disregard the spatially varying (non-zero momentum) modes of all of the fields, as these necessarily have a gap set by the size of the torus. 6 This is just dimensional reduction onto a single spatial point; naturally, the dimensionally reduced theory is just a form of quantum mechanics. Its Lagrangian can be written as Note that the minus signs in the φ 2 A 2 terms come from our choice of the metric convention (mostly minus). From here on, we treat all variables (A i , B i , and φ, with i = 1, 2 and 2 ) as matrices of c-or q-numbers, and we drop the indices on the traces. Moreover, to compactify notation, we write the sum of traces as a trace of a sum, even when the matrices in this sum are not all of the same dimension.
In [33], a simpler version of this model -the one arising from the theory of fundamental scalars coupled to one gauge field -was studied by canonical quantization, treating the φ † A 2 i φ term perturbatively in 1/N . This method readily gives the spectrum of the theory, and we will follow the same approach here. We will also develop a formal justification for using perturbation theory.
Letting g 2 ≡ 4π/k, we choose the canonical variables defined by We use a, b, etc. to denote generators of SU (N ), and α, β, etc. to denote generators of SU (M ).

The Hamiltonian is
It is useful to switch to variables that will give the ladder operators upon quantization, and so we let The Hamiltonian becomes Notice that the choice of a negative level for the SU (M ) CS action translates into the last term of the Hamiltonian above. This term allows for simultaneous creation of SU (N ) and SU (M ) holonomies.
In other words, the conserved quantum number of "particles" created by the c a 's and d α 's will be the difference (rather than the sum) of numbers of "particles" of each species.
To quantize the holonomy degrees of freedom, we impose Using T a T a = C 2 (N ) for SU (N ) (or for any simple group) and likewise for SU (M ), the normalordered Hamiltonian becomes and we see that the scalar fields will acquire a mass due to the vacuum energy of the holonomies.
This mass is set by The mass of the holonomy excitations will be set by the vacuum energy of the scalar excitations, and to find it we must quantize the scalars as well. We let where a and b are, respectively, N × M and M × N q-number matrices. The conjugation operation † acts on such a matrix by transposing it and taking a Hermitian conjugate of each q-number element.
We impose cannonical commutation relations on each of the N M elements of a or b, and so instead of the standard cyclicity of the trace we have Similarly, for an M × M matrix T , we find Tr aT a † = Tr T a † a + N Tr T (3.15) and, for an N × N matrix S, These identities allow us to find the normal-ordered form of the Hamiltonian, which is where, using Tr

The unperturbed spectrum
The spectrum is now easily found using perturbation theory. We shift the ground state energy to zero and choose the unperturbed Hamiltonian to be This Hamiltonian can be exactly diagonalized in the 't Hooft limit. The Hilbert space is spanned by the standard SHO eigenstates of the quadratic part of H 0 , and we will later show that the quartic term merely provides a correction to some of the eigenenergies. Hence, we will from now on adopt the usual language of creation/annihilation operators. The term that we have to treat We approach the regime of CS coupled to fundamental matter by taking ξ → 0, and in this limit the holonomy states become light, as found in [33]. On the other hand, the states of the other holonomy have bare mass g 2 N C(M )/2m, and in the 't Hooft limit this is These states remain massive as we approach the fundamental representation by letting ξ become The physical eigenstates of the unperturbed Hamiltonian must be invariant under the remaining SU (N ) × SU (M ) gauge transformations. Thus, if |Ω is the vacuum of the theory, c † a |Ω is not a physical state, but Tr c † 2 is, and so is any state created by a gauge-invariant combination of creation operators. 8 The operators we consider transform as The vectors k and have n components each, and it is understood that permuting the entries of either one does not generate a physically new state. Furthermore, due to the relations between traces of matrix powers, not all single-trace operators are independent. For instance, out of the holonomy degrees of freedom, only C 2 , . . . , C N and D 2 , . . . , D M are independent; the others can all be written as multi-trace combinations of these operators. The interdependence of these operators is a finite-N or high-energy effect, and it will not figure in our analysis of the low-energy spectrum.
The states created by these operators must all have unit norm. In the 't Hooft limit, one can 7 As already hinted, this mass will be corrected by some of the quartic terms, and hence we refer to the coefficients of the quadratic terms in H0 as bare masses. 8 The state Tr c † |Ω is not physical because the generators of SU (N ) are traceless.
check that the correct normalization is

27)
Above we use s to denote the "symmetry factor," with s = n if i = /n and k i = k/n for all i, and with s = 1 otherwise. These normalizations can be derived using planar diagram techniques that we will introduce below, and then we will also explain why one may think of each gauge boson as Similarly, |E 1,0 1 is an eigenstate of Tr c † cb † b with the same eigenvalue, and so In general, it is true that The 1/N corrections come from the multi-trace operators generated by the action of the quartics on some of the n > 1 states. For instance, take n = 2 and = (1, 1); we find The upshot of these calculations is that the unperturbed Hamiltonian at large N and M has the same eigenstates as the quadratic, SHO Hamiltonian, with unperturbed energies

Perturbative corrections
The remaining quartic couplings, assembled in (3.22), can be treated perturbatively. We wish to find corrections E n to energies of the non-degenerate eigenstates of H 0 . Calculating E n 's at n > 2 can in principle be done by the usual methods of quantum-mechanical perturbation theory. However, studying all possible contractions between traces quickly becomes a very involved task.
We will now show that, fortunately, perturbative calculations drastically simplify by using a diagrammatic technique that automatically keeps track of index contractions in expressions like s 1 |V |s 2 s 2 |V |s 3 s 3 |V |s 1 . We do not rederive quantum mechanical perturbation theory; we merely note that known expressions for perturbative corrections can be efficiently encoded using diagrams. The "Feynman rules" for computing energy corrections are as follows: The scalar propagator has no corresponding prefactors, so we can just contract scalars with each other without further worries. We must only contract particles of one type with other These rules allow for a quick estimate of the size of the effect of each order in perturbation theory. To do so, notice that each energy is of order m ∼ √ λ, while each vertex contributes a factor of order √ λ/N . At order n, the product of these factors gives √ λ/N n . On the other hand, a connected diagram in double-line notation can only have up to n loops in total. 9 Any diagrams with fewer than n loops will be suppressed in the 't Hooft limit, so they may be discarded. If a diagram has exactly n loops, some of these loops will be M -loops and some will be N -loops. In the case of interest to us (i.e. when computing corrections to E(C 2 ), which we will do in the next section in order to find the gap in the theory), studying the allowed vertices shows that a diagram of order 2n will have n loops of each type, while a diagram of order 2n + 1 will have n + 1 Mloops and n N -loops. Thus, perturbation theory gives an expansion in powers of ξ ≡ M/N , with perturbations of orders 2n − 1 and 2n both being suppressed by ξ n . This shows that perturbation theory is well-defined at low energies (where the 1/N corrections cannot be compensated for by pure combinatorics, i.e. where n N ). In particular, in the case of fundamental matter coupled to CS theory, ξ becomes of order 1/N , and we recover the perturbation in powers of 1/N found in [33].

State normalizations and vacuum amplitudes
The rules above are a bit formal and perhaps unintuitive. As already mentioned, the double-line formalism can be used to simply derive the normalizations of physical states, as given in eqs. that are planar or that can be made planar by cyclic permutations contribute, and there are exactly s of these. In short, we conclude that the vacuum amplitude is

Perturbative calculation of the gap
As an application of interest for studying the low-energy behavior of the system, we now show how the diagrammatic rules above can be used to calculate the first correction to the gap of the system.
The ground state only receives corrections from disconnected (bubble) diagrams. As usual, these do not affect the gap calculation; they merely renormalize the ground state energy which we may always shift to zero. (This is why we only need to compute connected diagrams, as per rule 2.) Thus, we take E(Ω) = E 0 (Ω) = 0. The only corrections to the gap come from the corrections to E(C 2 ), the energy of the first excited state.
The state |C 2 has energy E 0 (C 2 ) = 2m SU (N ) , and so the unperturbed gap is The first order corrections is trivially  Figure 6: The three diagrams that contribute to E(C 2 ) up to second order in perturbation theory, shown in single-line notation (above) and double-line notation (below). The first diagram is the zero-vertex (unperturbed) energy, and the other two are the only two-vertex diagrams that contribute in the 't Hooft limit.
The second order correction is non-trivial, and hence we resort to the Feynman rules developed in the previous section.
We have already argued that E 2 (C 2 ) scales like ξ, and is hence of the same order as the un- , with s = 1 and E 0 (E 2,0 1 ) − E 0 (C 2 ) = 2(m + m SU (N ) ), while the in-and out-states have s = 2. Hence, the two-vertex contribution, or the second order term in perturbation theory, is .
We are working up to first order in ξ, and in this regime the second-order-corrected gap is Higher order corrections can be found in a similarly straightforward way. Note that this answer reduces to the answer for CS-fundamental on a torus when ξ = 1/N .
In this paper we have studied the Chern-Simons theory coupled to bifundamental scalar fields.
At two-loop order, in the 't Hooft limit, the theory has two lines of fixed points parametrized by the 't Hooft coupling. These lines exist for all values of the fixed ratio M/N . When this ratio is zero the theory goes over to the CS theory coupled to the fundamental matter which has a dual description in terms of a parity-violating version of the Vasiliev higher-spin gauge theory in AdS 4 .
When the ratio is small the dual gravitational theory should be some deformation of the Vasiliev theory, as has been conjectured in [25]. When this ratio approaches unity, the field theory is some kind of a non-supersymmetric version of the ABJM theory. The dual gravitational theory should be an Einstein gravity theory in the large 't Hooft coupling limit. It will be fascinating to better understand this bosonic theory in the bulk.
We have also studied the low-lying spectrum of this theory placed on a torus. CS theory coupled to fundamental matter has a set of low lying states whose energy goes like √ λ/N and so in the strict classical limit, N = ∞, they are exactly degenerate zero energy states. Our analysis shows that the bifundamental matter has no such states on the torus. The energy in this case goes like √ λ M/N for small value of M/N , and this gap stays nonzero even when N = ∞. This is encouraging and leads to the picture where one can think of the bifundamental theory as a regulator of the fundamental theory which regulates the singular low-energy states of the fundamental theory on a torus whose bulk dual is still mysterious.
Let us now place the theory on a genus g ≥ 2 Riemann surface, Σ g . The number of states in the Hilbert space of pure CS theory on Σ g with gauge group U (N ) and level k goes like k (g−1)N 2 for large k [34]. So, for the bifundamental theory, this will go like k (g−1)(N 2 +M 2 ) . These states have exactly zero energy in the pure CS theory. It will be very interesting to know the fate of these large number of exactly degenerate states once we add matter to it. If we add fundamental matter, then the degeneracy is not lifted at least in the very weakly coupled regime. However, for the bifundamental matter things could be different. We saw in the case of torus that adding bifundamental matter lifts the degeneracy of pure CS states even in the classical limit in the bulk. The same thing could happen for higher genus surfaces and if this is the case then it will be fascinating. We leave this question for future study.
We have left untouched many important things in this paper. For example, we have not provided any argument for the all-loop existence of the fixed line. One can also compute the anomalous dimensions of the operators, and it is known that the currents acquire non-zero anomalous dimensions in the bifundamental theory. It would be good to have an expression for that as a function of the 't Hooft coupling and the fixed ratio M/N . Another important thing is to compute the free energy of this theory. It will give us a wealth of information about the bulk gravity theory, in particular about black holes in the bulk. In passing we would like to mention that the bifundamental fermions coupled to CS gauge theory may be simpler in this respect, because by standard arguments the fermionic theory is a conformal field theory for all values of the 't Hooft coupling and the ratio M/N .
We have not touched upon the issue of duality [36][37][38] in these bifundamental theories. For that one has to study the scalar and fermion theories in much more detail. We leave that for future research.

Acknowledgement
We would like to thank Shiraz Minwalla for suggesting this problem and Eric Perlmutter for comments on the previous version of the paper. We would also like to thank Guy Gur-Ari, Raghu

A Feynman rules for CS-bifundamental theories
The action of the CS-bifundamental theory is given by eq. (2.2). The Feynman rules are easily obtained if the notation is decompactified and all indices are explicitly written. This gives S = S A + S B + S matter , with (A.1) where the latter equality for C 2 (N ) ≡ T a T a comes as a special case of the choice Note that the gluon propagators must be assigned a direction. Taking into account the overall minus sign from the Boltzmann factor e −S , the vertices are found to be: φ † ii (p 1 )φ jj (p 2 )φ † kk (p 3 )φ ll (p 4 )φ † mm (p 5 )φ nn (p 6 ) 0 = −g 1 δ i j δ jk δ k l δ lm δ m n δ ni + 11 other permutations (2π) 3 δ 3 p i − − g 2 δ i j δ jk δ k l δ li δ m n δ nm + 17 other permutations (2π) 3 δ 3 p i − − g 3 δ i j δ ji δ k l δ lk δ m n δ nm + 5 other permutations (2π) 3 δ 3 p i , (A.12) These can be depicted using a variant of the usual double-line notation with two types of lines. We introduce these diagrams in Section 3, Fig. 5. This notation has the usual advantage of allowing one to quickly estimate which diagrams dominate in the 't Hooft limit. Note that we do not include any Feynman rules for ghosts because these do not appear in any of the diagrams relevant for our purposes.