Chern-Simons-matter dualities with $SO$ and $USp$ gauge groups

In the last few years several dualities were found between the low-energy behaviors of Chern-Simons-matter theories with unitary gauge groups coupled to scalars, and similar theories coupled to fermions. In this paper we generalize those dualities to orthogonal and symplectic gauge groups. In particular, we conjecture dualities between $SO(N)_k$ Chern-Simons theories coupled to $N_f$ real scalars in the fundamental representation, and $SO(k)_{-N+N_f/2}$ coupled to $N_f$ real (Majorana) fermions in the fundamental. For $N_f=0$ these are just level-rank dualities of pure Chern-Simons theories, whose precise form we clarify. They lead us to propose new gapped boundary states of topological insulators and superconductors. For $k=1$ we get an interesting low-energy duality between $N_f$ free Majorana fermions and an $SO(N)_1$ Chern-Simons theory coupled to $N_f$ scalar fields (with $N_f \leq N-2$).


Introduction
Three independent lines of investigation have recently converged on a long list of new dualities, which relate the low-energy (IR) behavior of two different (2 + 1)d field theories.
One source of input came from the condensed matter literature (e.g. [1][2][3][4][5][6]). Another approach was based on the study of Chern-Simons (CS) theories coupled to matter in the fundamental representation with large N and large k with fixed N/k. In some cases two different theories, one of them fermionic and the other bosonic, were argued [7,8,9] to be dual to the same Vasiliev high-spin gravity theory on AdS 4 (see e.g. [10]), and thus dual to each other by a duality exchanging strong and weak coupling. Another approach to finding such dualities with finite N and k is based on starting with a pair of dual N = 2 supersymmetric theories [11][12][13][14][15][16][17][18][19][20][21][22][23] and turning on a relevant operator that breaks supersymmetry. If the flow to the IR is smooth, we should find a non-supersymmetric duality [24,25]. Motivated by this whole body of work, a number of dualities based on unitary gauge groups were conjectured in [26] and elaborated in [27]: for N f ≤ N . All CS theories here and below are viewed as the low-energy limits of the corresponding Yang-Mills-Chern-Simons theories; our conventions and more details on CS couplings and fermion determinants are collected in Appendix A. Here and throughout this paper we take N, k ≥ 0; additional dualities follow from reversing the spacetime orientation in these dualities. The matter fields are in the fundamental representation of the gauge group and it is implicit that the scalars φ are at a |φ| 4 fixed point. The U (L) groups have two levels when L > 1. See [27] for more details. The N = k = N f = 1 versions of these dualities were analyzed and coupled to appropriate background fields in [28,29,30], and their relation to supersymmetric dualities was analyzed in [31,32], thus providing further evidence for their validity.
Our goal in this paper is to extend this line of investigation to orthogonal and symplectic gauge groups. We will conjecture the following IR dualities: 1 Notice that these dualities, as opposed to the ones considered before, involve real scalars and real (Majorana) fermions. The U Sp dualities are conjectured to hold for N f ≤ N . In the 't Hooft limit of large N and k, with fixed N f and N/k, these dualities are supported by the same considerable evidence as their U (N ) counter-parts (since the orthogonal and symplectic theories are just projections of the U (N ) theories at leading order in 1/N ).
As a particularly interesting special case, if we set k = 1 in the SO dualities, we conjecture that the SO(N ) 1 CS theory coupled to N f scalar fields in the vector representation (with any N ≥ N f + 2) flows to N f free Majorana fermions.
In a companion paper [33] we will discuss a number of non-trivial fixed points with enhanced global symmetry. Every one of them has a number of dual descriptions. The full global symmetry appears classically in some descriptions, but it only appears as a quantum enhanced low-energy symmetry in others. For example, taking special cases of (1. The duality between the two theories at the bottom of (1.3) is the simplest example of the U Sp dualities (1.2), thus providing a nontrivial check of them. Also the duality between the two theories at the top of (1.3) appears among our SO dualities for SO(2) = U (1) (1.2), but the SO duality acts on the operators in a different way, that is related to the U (1) duality by a global SU (2) rotation. Thus, the enhanced global symmetry plays a crucial role in the consistency checks of the dualities (1.2).
is normalized such that the Chern-Simons term is k 2·4π Tr AdA + 2 3 A 3 with a trace in the vector representation. As we will discuss below, the SO(N ) k theory with even k is a conventional nonspin topological quantum field theory (it does not require a spin structure), while the theory with odd k is spin.
If we set N f = 0 in the suggested dualities (1.1) and (1.2), we find simple dualities involving topological quantum field theories (TQFTs): Although there exists a large literature about such level-rank dualities, we could not find a precise version of them. In [27] a careful analysis derived the first two lines in (1.4) and clarified that they hold, in general, only when the theories are spin-Chern-Simons theories (see [27] for details). Below we will provide a similar proof of the orthogonal and symplectic dualities in (1.4) and will establish the need for the theories to be spin. 2 The level-rank dualities can be used to show that several TQFTs, although not manifestly so, are time-reversal invariant at the quantum level. We provide a rich set of examples, summarized by the following table: T -invariant theories N property framing anomaly need to add ψ 2 However we will also find that for special values of N, k the dualities (1.4) are valid for conventional TQFTs as well. In particular the first SU/U duality is non-spin for N even and N k = 0 mod 8, the SO duality is non-spin for N, k even and N k = 0 mod 16, and the U Sp duality is non-spin for N k = 0 mod 4.
These theories represent new possible gapped boundary states of topological insulators and topological superconductors.
In Section 2 we analyze in detail the Chern-Simons-matter dualities for U Sp (2N ) gauge groups, and in Section 3 for SO(N ) groups. In Section 4 we comment on the relation to high-spin gravity theories on AdS 4 . Section 5 gives a detailed description of level-rank dualities for orthogonal and symplectic groups, and Section 6 describes their implications for constructing new time-reversal-invariant TQFTs. Appendix A explains our notation and conventions.
After the completion of this work, we received [34] where the duality of free fermions to SO(N ) 1 with scalars is worked out.

Dualities between U Sp(2N ) Chern-Simons-matter theories
In Section 5 we will derive and discuss certain dualities of spin-TQFTs that take the form of level-rank dualities: In our conventions U Sp(2N ) = SU (2N ) ∩ Sp(2N, C), and further details are collected in Appendix A. We recall that a spin-TQFT-as opposed to a conventional topological quantum field theory-can only be defined on a manifold with a spin structure, and if multiple spin structures are possible, then the spin-TQFT will depend on the choice. A spin-TQFT always has a transparent line operator of spin 1 2 . In (2.1) the U Sp factors are non-spin, while the SO factors are trivial spin-TQFTs (discussed e.g. in [35]), whose presence is important for the duality to work.
We can then add matter in the fundamental representation, bosonic on one side and fermionic on the other, and conjecture new boson/fermion dualities. This is done in such a way that renormalization group (RG) flows in which all matter becomes massive are consistent with (2.1). We thus propose the following dualities between the low-energy limits of Scalar and Fermionic theories: that makes it into a spin theory and provides a transparent line of spin 1 2 . Theory F is already manifestly spin, however to correctly reproduce its framing anomaly 3 we include SO 4k(N − N f ) 1 . For completeness, let us rewrite the duality including its time-reversed version: where φ i are scalars and ψ i are fermions.
Theory S contains N f complex scalars in the fundamental 2N representation of U Sp(2N ). Since the representation is pseudo-real, we can rewrite them in terms of 4N N f complex scalars subject to the reality condition ϕ ai Ω ab and Ω ab , Ω ij are the corresponding symplectic invariant tensors. This description makes the U Sp(2N f ) flavor symmetry of the theory manifest. There is one quadratic term and two 4 quartic terms one can write that preserve the U Sp(2N f ) flavor symmetry: in terms of the antisymmetric meson matrix M ij = ϕ ai Ω ab ϕ bj they are All these three terms are relevant in a high-energy Yang-Mills-Chern-Simons theory with these matter fields. We turn them on with generic coefficients and make a single finetuning (which we can interpret as the coefficient of the quadratic term). We conjecture that by doing that the long distance theory is at an isolated nontrivial fixed point. It is a generalization of the Wilson-Fisher fixed point, and it has a single U Sp(2N f )-invariant relevant deformation which may be identified with O (2) . Theory S is defined at such a fixed point.
Note that the Z 2 center of the U Sp(2N ) gauge group acts in exactly the same way as the Z 2 center of the U Sp(2N f ) global symmetry group. So local gauge-invariant operators 3 In general, the two sides of the duality have different framing anomalies (see [36]). We fix that by adding a trivial spin-TQFT that has the difference in the anomaly. For our purposes this is the same as a gravitational Chern-Simons term with an appropriate coefficient. 4 For N f = 1 there is only one independent quartic term. Notice that while the spin-TQFTs involved in the level-rank duality (2.1) have a Z 2 one-form global symmetry associated to the center of the group, such a symmetry is broken in the theories with matter because the latter transforms under the center [37]. In fact, the theories in (2.2) do not have any discrete global symmetries.

RG flows
We cannot prove the dualities in (2.2), however we can perform some consistency checks. For instance, we can connect different dual pairs by RG flows triggered by mass deformations. In both Theories S and F, turning on a mass at high energies leads to turning on the unique relevant deformation of the low-energy conformal field theory (CFT). So turning on a bosonic mass-squared m 2 φ in Theory S should have the same effect at low energies as turning on a fermion mass m ψ in Theory F.
In Theory S, if we give a positive mass-squared to one of the complex scalars we simply reduce N f by one unit. However, if we turn on a negative mass-squared, a complex scalar condenses Higgsing the gauge group to U Sp(2N − 2) k , in addition to reducing the number of flavors. In Theory F, when giving mass to one of the complex fermions, the phase of its partition function becomes either e −iπη(A) or 1 in the IR limit, depending on the sign of the mass. In both cases the number of flavors is reduced by one, however in the first case one can use the APS index theorem [38] (see Appendix A) to rewrite the leftover regularization term e −iπη(A) in terms of a shift of the bare gauge and gravitational CS terms. Thus, tuning the mass of the remaining (N f − 1) flavors to zero, the RG flow leads 5 to the following pairs: On the other hand, consider the case N f ≥ N + 1 and turn on generic masses for all the flavors. Theory F flows to a non-trivial topological theory. Let us compare with Theory S. Here generic negative mass-squared for all matter fields Higgses the gauge group completely. The IR theory could be gapped or could have massless Goldstone bosons, but since the gauge group is completely Higgsed, it cannot include a topological sector. Hence, the duality cannot be correct in this case. We conclude that none of the pairs with N f ≥ N + 1 can be dual.

Coupling to background gauge fields
Given that our system has a global U Sp(2N f ) symmetry, we can couple it to background gauge fields for that symmetry. Our goal here is to identify the CS counterterms [39] for these fields that are needed for the duality.
Let us start with the scalar side of the duality. We start with a U Sp(2N ) k CS theory for the dynamical fields and we can also have U Sp(2N f ) k s for some integer k s for the classical fields. If we give masses to all scalars such that the gauge symmetry is not Higgsed, then the low energy theory is purely topological. It is a U Sp(2N ) k ×U Sp(2N f ) k s CS theory, 5 Here we assume that the RG flow still leads to a non-trivial CFT with a single U Sp(2N f − 2)invariant relevant operator.
where the second factor is classical. In the fermionic side of the duality we start with for the dynamical fields and U Sp(2N f ) k f for the classical fields. These mean that the bare CS levels for these two groups are −N + N f and k f + k 2 respectively. Repeating the mass deformation of the bosonic side we find at low energy a topological U Sp(2k) −N as well as a CS counterterm for the classical fields U Sp( . For this to match with the bosonic side we must choose 6 Of course, we have the freedom to add the same CS counterterm on the two sides of the duality. This will add an arbitrary integer to k s and the same integer to k f .
We can repeat the same considerations with an opposite sign for the mass deformations. In the scalar theory, Higgsing occurs and the symmetry group is reduced to where the broken part of the gauge group is identified with the flavor group and this causes the shift of the CS counterterm for the classical fields.
In the fermionic theory we find U Sp(2k As a non-trivial check, equality on the two sides requires the very same relation (2.6).
As we discussed above, the global symmetry that acts faithfully on local operators is U Sp(2N f )/Z 2 and this puts restrictions on the CS counterterms. More precisely, in the bosonic side we would like the bare CS terms to be consistent for U Sp(2N ) k × U Sp(2N f ) k s /Z 2 and the Z 2 quotient is consistent only for In the fermionic side the bare CS terms are U Sp(2k we used (2.6). This is consistent for Fortunately, (2.7) is the same condition as (2.8). In the spirit of 't Hooft anomaly matching, this is a non-trivial consistency check on our duality. The obstruction to the Z 2 quotient is the same in the two sides of the duality.
When the condition (2.7) is not satisfied, we cannot mod out by the Z 2 and fewer backgrounds of the gauge fields are allowed. In those cases it might still be possible to extend the U Sp(2N f ) classical gauge fields to a (3 + 1)d bulk and consistently take the Z 2 quotient there.

Small values of the parameters
It is instructive to look at the dualities (2. 2) for small values of N , k and N f . We scalars confines with a single vacuum: although we have no proof that this is true, it is surely plausible.
As discussed around (1.3), the case N = k = N f = 1 can be derived from the dualities in [26,27], giving us more confidence that the duality is correct.

New fermion/fermion and boson/boson dualities
Combining the dualities for symplectic groups in (2.2) with those for (special) unitary groups in [26,27], we can find new fermion/fermion and boson/boson dualities.
For instance, we can take (2.2) with N = N f = 1 and combine it with the first duality of (5.5) in [27]. This gives us a fermion/fermion duality (To be precise, the theory on the right should include a decoupled SO(2k) 1 factor.) As we discussed, the duality with k = 1 can be derived from [26,27], but the other ones are new.
Note that for all k the theories in (2.9) have a global SU (2) symmetry, which is manifest in the LHS of (2.9). On the RHS of the duality we have a manifest U (1) monopole number symmetry and charge conjugation C, which does not commute with it because it maps the monopole number n to −n. Our duality suggests that this U (1) ⋊ Z C 2 classical symmetry is enhanced in the quantum theory to SU (2). The currents that extend the Abelian symmetry to SU (2) must carry monopole charge. We suggest that they are constructed out of the monopole operator and its conjugate in the U (k) theory. Since these carry charge, each of them should be dressed by a fermion to be gauge-invariant.
Similarly, from (2.2) with k = 1 and [26,27] we can obtain the boson/boson duality Both sides should include SO(0) 1 and be regarded as spin theories. The case N = N f = 1 was already found in [26,27], but the other ones are new. As above, the global symmetry of these theories is U Sp(2N f ), which is manifest in the LHS, thus we conjecture that the manifest U (N f ) and charge conjugation symmetries in the RHS are enhanced to U Sp(2N f ).

Dualities between SO(N ) Chern-Simons-matter theories
For orthogonal groups there exists a similar level-rank duality of spin-TQFTs (derived in Section 5): As we explain below, the duality can only be true for Moreover, one should remember that Theory S includes the trivial spin-topological sector The matter fields are all in the vector representation. There is considerable evidence for this duality at large N and k; at finite values of N and k we can check its consistency by mass flows, including flows to the level-rank dualities between pure Chern-Simons theories (3.1) described in Section 5.
The two low-energy theories are defined by tuning the masses to zero, and assuming that both sides flow to a fixed point, which has a single relevant operator consistent with the global symmetries of the high-energy theory. For finite values of N , k and N f we do not know when this is true; additional operators could become relevant at low energies, which would prevent the two theories from flowing to the same fixed point, or the fixed point may cease to exist (say because a mass gap develops, or because we spontaneously break some of the flavor symmetries on one side but not the other). The limits on N f above arise because we show that for larger values of N f the duality cannot hold, but we conjecture that it does hold for the values mentioned above.

Flows
We Reversing this logic, if we have a non-trivial duality for some values of (N, k, N f ), we can assume that the duality holds for all ( If for some such value the duality fails but we still have a non-trivial theory with no extra relevant operators, then we expect the duality to fail also for the corresponding higher values (since otherwise we get a contradiction by first flowing to the "higher" IR CFT and then performing the mass flow). and U Sp(2N ) cases.

Global symmetries
The UV Yang-Mills-Chern-Simons theories we start from have an O(N f ) flavor symmetry, as well as two discrete symmetries discussed below. The definition of the IR Theories S and F involves flows from these UV theories that preserve these SO(N f ) and discrete symmetries. The SO(N f ) symmetry allows for a single mass term which needs to be tuned, both on the scalar and on the fermion sides. As in (2.4), for N f > 1 the UV description of Theory S has two possible φ 4 terms. In terms of M ij = φ ai φ aj (i, j are flavor indices and a is a color index) they are O = Tr(M 2 ) . We assume that for generic couplings of these operators they do not lead to any new relevant operator at low energies. 7 For some small values of N and k there are enhanced continuous symmetries, which we will discuss below.
In addition there are two discrete symmetries, that were discussed in detail in [23].
There is a global "charge conjugation" symmetry C, which acts on the matter fields as φ 1i → −φ 1i and all other φ ai are invariant. When C is gauged the gauge group changes from SO(N ) to O(N ). 8 The SO(N ) vector indices may be contracted to form singlets either with δ ab or with ǫ a 1 a 2 ···a N . Operators that involve the latter contraction are odd under C, while all others are even. Since the product of two epsilon symbols may be replaced by a sum of products of δ's, the symmetry is Z 2 .
7 For some values of N , k and N f there are additional fixed points where one or both quartic operators are tuned to zero, and that also have fermionic Gross-Neveu-like duals, but we will not discuss them here. 8 For N even, this symmetry always exchanges the two spinor representations of Spin(N ).
They are in fact complex conjugate representations for N = 2 mod 4, but not for N = 0 mod 4. In particular their classical dimension in the large N limit scales as N 3/2 [42] (and this statement is true also quantum mechanically [43]). Similarly, the lightest baryon operator in the fermionic theory has precisely the same quantum numbers and classical dimension as the lightest monopole operator in the scalar theory.
Above we were not careful about precisely which monopole we choose in the SO (2) theory, and how it transforms under charge conjugation; these issues were discussed in detail in [23], and we review the discussion here. In an SO(2) = U (1) theory, there are monopoles V n that carry n units of the U (1) J magnetic charge (topological charge).
Charge conjugation in this theory takes n to (−n), so we have one lightest monopole subgroup. In order to form a C-odd monopole operator (which was called a monopolebaryon in [23]) we need to take V − and multiply it by a C-odd operator in O(N − 2), namely a product of (N − 2) matter fields contracted with an epsilon symbol (in addition to the extra fields required for canceling the SO(2) charge of the monopole). Repeating the same arguments as above, we find that the lightest monopole-baryon-operator in both theories F and S has a classical dimension scaling as N 3/2 in the large N, k limit with fixed N f , and the operators also lie in identical Lorentz ×SO(N f ) representations in the two theories.
The arguments above strongly suggest that the duality exchanges monopoles and baryons, and takes the monopole-baryon operators to themselves, namely it exchanges the two Z 2 global symmetries C and M. In fact, we can see that this must be the case by performing the mass flow to the pure Chern-Simons theories, and noting (see Section 5) that the level-rank duality in these theories indeed exchanges C with M. So this gives a nice consistency check for the duality. The fact that the classical dimensions on both sides match (at least at large N ) is somewhat surprising, since one would expect their dimension to receive quantum corrections (except for the monopole operator in the fermionic theory which was shown in [43] to receive no quantum corrections to its dimension in the 't Hooft large N limit). This is all very similar to the duality between SU (N ) and U (k) CS-matter theories, which also exchanges baryon number with monopole number [43,26].
By gauging C and/or M we can find related dualities involving O(N ), Spin(N ) and P in ± (N ) gauge theories. These theories can have additional labels, which are the coefficients of terms like w 2 w 1 of the gauge bundle [23]. These can be thought of as CS terms of various discrete gauge fields or as discrete theta parameters analogous to those studied in [44,45,37]. We will not discuss them here.
In the N = 2 supersymmetric version of the Chern-Simons-matter dualities between SO(N ) gauge theories [23], the duality maps C SUSY to itself, while mapping M SUSY to C SUSY M SUSY . Given the fact that one can flow from the supersymmetric theories to the pure Chern-Simons theories, and perhaps also to the non-supersymmetric Chern-Simonsmatter theories along the lines of [24,25], this is confusing. The resolution is that in the

Small values of N and k
When the theories on both sides are non-Abelian it is difficult to check the dualities.
However, for k = 1, 2, and for N f = N − 1, N , we can (possibly after Higgsing) have Abelian or empty gauge groups, so we can test the dualities in more detail.
So in flows from the UV Yang-Mills-Chern-Simons theory that preserve both SO(N f ) and C we do not need to consider either one of them, and we still require a single fine-tuning at low energies. For N f > 2 there is only one quadratic SO(N f )-invariant operator.

At quartic order there is an SU
When we view the theory as an SO(2) gauge theory, and in particular when we flow to it from higher SO(N ) gauge theories, we only preserve the SO(N f ) symmetry, so the latter operator is also turned on during the flow. We expect this extra operator to be Theory F: N f free Majorana fermions.
As discussed above, the monopole operator of the scalar theory maps to the real fermion.
The lowest case is N = 3 and N f = 1, where we have an SO(3) 1 theory with a single scalar in the vector (adjoint) representation flowing to a free Majorana fermion. Since Theory F in this case has no magnetic symmetry, the duality implies that all baryons and monopole-baryons of Theory S decouple at low energies.

The N = 1 case
In this case, since we should have N f = 1, the scalar theory is just a real Wilson-Fisher scalar. The dual fermionic theory has a real fermion coupled to an SO(k) − 1 2 CS theory. The case k = 1 was already ruled out above. The case k = 2 is related to a U (1) − 1 2 theory, which maps by the dualities of [26,28] to a complex Wilson-Fisher scalar. So, assuming the validity of the U (1) duality, the SO duality cannot hold in this case, and thus also for other cases with k = 2 and N f = N .
For higher values of k and N = N f = 1, the duality may be correct, namely the SO(k) − 1 2 CS theory coupled to a single fermion may flow to the fixed point of a real Wilson-Fisher scalar. Again this implies that all baryonic operators of this theory decouple at low energies.

The k = 2 case
In this case we have a duality between the theory of N f fermions coupled to , and the theory of N f real scalars coupled to SO(N ) 2 . Such a duality implies that the charge conjugation Z 2 symmetry of the scalar theory is enhanced to U (1), and its SO(N f ) flavor symmetry is enhanced to SU (N f ), at low energies.
As discussed above we can rule out the cases with N f = N , so the lowest case is N = 2 and N f = 1. This case is interesting because, as discussed around (1.3), the two dual Abelian theories admit two more non-Abelian descriptions in which the full SU (2) global symmetry of the fixed point is manifest in the UV.
Our duality maps Theory S, a U (1) 2 CS theory coupled to a complex Wilson-Fisher scalar (for N f = 1 there is no difference between the SO(2) and U (1) flows) to Theory F, a U (1) − 3 2 CS theory coupled to a complex fermion. The very same two theories, viewed as U (1) theories, are also mapped to each other by the U (1) duality of [26,28,27]. However, interestingly enough, the operator mappings are not the same in the U (1) duality and the SO(2) duality: the U (1) ↔ U (1) duality preserves the magnetic symmetry and the charge conjugation, while the SO(2) ↔ SO(2) duality exchanges them. Fortunately, this perfectly fits with the enhanced quantum SU (2) global symmetry. In each U (1) CS description there is a manifest U (1) J ⋊ Z C 2 magnetic and charge conjugation symmetry. The U (1) duality trivially maps the two copies of U (1) J ⋊Z C 2 one into the other. The SO(2) duality, instead, maps U (1) J ⋊ Z C 2 in a nontrivial way, which follows from its embedding inside the global SU (2) symmetry: it is an SU (2) rotation.
For k = 2 and N > 2 we obtain more complicated dualities, which we cannot rule out.
The U (1) dualities map the fermionic SO(2) = U (1) theories to SU (N ) 1 theories coupled to N f scalars, so we obtain boson-boson dualities between SU (N ) 1 and SO(N ) 2 theories coupled to N f < N Wilson-Fisher scalars (which are complex and real, respectively).

The N = 2 case
For N = 2 we have an SO(2) k = U (1) k CS theory coupled to N f charged scalars with Wilson-Fisher couplings; here we can have N f = 1 or N f = 2. The dual for N f = 1 is an 2 theory coupled to a real fermion, and for N f = 2 an SO(k) −1 theory coupled to two real fermions. For k = 1 and k = 2 we have already discussed these theories above.
For k > 2 the dual theory is non-Abelian, and we cannot rule the duality out. Again, it implies that the charge conjugation symmetry of the fermionic theory should be enhanced to U (1) at low energies, and its SO(N f ) flavor symmetry to SU (N f ).
The U (1) duality with N f = 1 maps the same scalar theory to an SU (k) − 1 2 theory coupled to a complex fermion, giving another fermion-fermion duality between SO(k) − 3 2 and SU (k) − 1 2 theories with one real/complex fermion flavor. For N f = 2 the U (1) duality breaks down, but the SO(N ) duality of the previous paragraph may still be valid.

Relation to theories of high-spin gravity
The SO(N ) theories with k = 0 and N f = 1 were the first ones to be suggested to be dual at large N to Vasiliev's high-spin gravity theory on AdS 4 [46,47]; they are dual to the minimal Vasiliev theory, which has only even-spin excitations. There are two versions of this theory, differing by a discrete parameter, that were argued to be dual to theories of N scalars and N fermions, respectively. This was later generalized to U (N ) and SU (N ) theories being dual to non-minimal Vasiliev theories that have excitations of all spins.
There is an obvious generalization of both dualities to higher N f , with the SO(N ) theory containing N f (N f + 1)/2 excitations of even spins, and N f (N f − 1)/2 excitations of odd spins. The Vasiliev theory is only known by its classical equations of motion, so a priori it is not known how to quantize it; the dual field theories with finite N can be viewed as giving a non-perturbative definition of this theory.
For the U (N ) scalar/fermion theories, it was argued that a parameter θ 0 in the Vasiliev theory is related to N/k when the U (N ) Yang-Mills theory is replaced by U (N ) k [8]. This parameter interpolates between the theory dual to parity-invariant scalars, and the one related to parity-invariant fermions, and this led to the conjectured duality between CSscalar and CS-fermion theories. The same parameter exists also in the minimal Vasiliev theories, so it is natural to conjecture that turning it on in the minimal Vasiliev theories corresponds to having SO(N ) k or U Sp(2N ) k CS-matter theories. Again the fact that the Vasiliev theory has two interpretations, as a CS-scalar and as a CS-fermion theory, suggests that at least at large N the SO(N ) and U Sp(2N ) dualities that we discussed above are correct.
At leading order in large N , the orthogonal, symplectic and unitary theories are all the same, consistent with having the same classical equations of motion in the minimal and non-minimal Vasiliev theories (up to having a projection removing half of the fields in the SO and U Sp cases). However, the one-loop corrections should be different. In particular there should be a discrete parameter distinguishing the SO(2N ) and U Sp(2N ) theories, whose effect is to change the sign of all l-loop diagrams with odd values of l. This can be realized by inverting the signs of and of Newton's constant. 9 There should also be discrete parameters on the gravity side distinguishing the different versions of the SO(N ) theories, where one gauges some of the Z 2 discrete symmetries.

Level-rank dualities of 3d TQFTs
Level-rank dualities of 2d chiral algebras can be derived starting from systems of free fermions. For instance, consider a system of N k free real (Majorana) fermions: writing them as ψ aã with a = 1, . . . , N andã = 1, . . . , k one obtains the following conformal embeddings of chiral algebras (see also [50,51]): Here Spin is the standard Kac-Moody chiral algebra, while SO = Spin/Z 2 is the extended chiral algebra [52,53] obtained from Spin by adding a suitable Z 2 generator of spectral flow (see below). The Z 2 quotient in the first line is the extension by the diagonal element.
The centers match on the two sides. A series of equalities of chiral algebras follows: On the right hand sides we have GKO cosets [54]. Moving from two-dimensional chiral algebras to three-dimensional Chern-Simons theories [53], one obtains dualities between the following Lagrangian theories: On the right hand sides, the Lagrangian is the one corresponding to the Lie algebra of the numerator, while the gauge group is the result of the quotient. On the third line, B = Z 2 × Z 2 for k = 0 mod 4 and B = Z 4 for k = 2 mod 4. We stress that these levelrank dualities of 3d TQFTs (or equivalently of 2d chiral algebras) can be rigorously proven.
Similarly, one can start with a system of 4N k 2d real fermions, and writing them as 4N k complex fermions with a symplectic Majorana condition one obtains the conformal embedding This leads to the duality of chiral algebras and in terms of three-dimensional Chern-Simons theories one has the duality between 3d TQFTs.

Matching the symmetries
Before going on with the analysis of those dualities, let us fix some notations for orthogonal and symplectic chiral algebras. The center B(G) of the simply-connected group G associated to a Lie algebra g acts on the affine Lie algebra G k as an outer automorphism, and its action is generated by elements σ i of G k via spectral flow. In the corresponding 3d CS theory, B(G) appears as a one-form symmetry [37]  When the generator has half-integer dimension, the chiral algebra can be augmented to a The generator is given by σ = (0, . . . , 0, k) with dimension h(σ) = kN 4 . The action of σ via monodromy is where the congruence class c of a representation [λ] is given by c = N j odd λ j mod 2. In particular the self-parity of σ is (−1) Nk . For N k = 0 mod 4, one can consider the P U Sp(2N ) k ≡ U Sp(2N ) k /Z 2 CS theory; for N k = 2 mod 4, one can consider the P U Sp(2N ) k spin-CS theory. 10 We indicate a highest weight representation by its Dynkin labels (λ 1 , . . . , λ r ) or by its extended Dynkin labels [λ 0 , . . . , λ r ], where r is the rank. In sp(2N ), λ r refers to the long root. In so(2n + 1), λ r refers to the short root, while in so(2n), λ r−1 and λ r refer to the two roots at the "bifurcated tail" of the Dynkin diagram.
In Spin(N ) k with N odd, the Z 2 spectral flow is generated by σ : [λ 0 , λ 1 , λ 2 . . . , λ r ] → [λ 1 , λ 0 , λ 2 , . . . , λ r ] . (5.9) The generator is given by σ = (k, 0, . . . , 0) with dimension h(σ) = k 2 . The action via monodromy is  (5.15) in both N even cases. The one-form symmetry generated by σ can always be gauged to obtain the SO(N ) k CS theory: it is a TQFT for k even, and a spin-TQFT for k odd. Only for N, k both even there is another generator j s that survives the quotient, thus only in this case SO(N ) k has a Z 2 one-form symmetry.
Let us also discuss what type of conventional zero-form symmetries the Chern-Simons theories can have. In Spin(N ) k and SO(N ) k with N even, one defines a "charge conjugation" Z 2 symmetry C that transforms representations as In SO(N ) k gauging this symmetry gives O(N ) k . Counterterms for the classical gauge field of C lead to additional parameters in the O(N ) k theory [23]. In SO(N ) k with k even, we define a "magnetic" Z 2 symmetry M that exchanges the two representations of the extended chiral algebra resulting from a fixed point of the Z 2 spectral flow of Spin(N ) k [52,53]. From the 3d point of view, the magnetic quantum number of a monopole operator is the second Stiefel-Whitney class w 2 of the SO(N ) bundle around its location. This symmetry is gauged when going from SO(N ) k to Spin(N ) k .
Whenever a three-dimensional TQFT has a Z 2 one-form global symmetry generated by σ with spin h(σ) = 1 2 mod 1, we can define a quantum zero-form symmetry K σ acting on the lines in the following way: Having settled the basic definitions, we can analyze the precise mapping of symmetries between the dual theories in (5.3) and (5.6). Consider where the quotient is generated by σ ⊗ σ (which has fixed points). Both sides have a Z 2 one-form global symmetry, and the map of generators is 11 σ ↔ σ ⊗ 1I ∼ 1I ⊗ σ. Both sides have Z 2 zero-form symmetry. On the RHS it is the quantum symmetry K σ . On the LHS it is the magnetic symmetry M Z 2 associated to the fixed points of the Z 2 quotient. The map of generators is K σ ↔ M Z 2 . Consider Both sides have a Z 2 × Z 2 one-form global symmetry for N = 0 mod 4, and Z 4 for N = 2 mod 4. The map of generators is j s ↔ j s ⊗ 1I, σ ↔ σ ⊗ 1I. Both sides have a Z 2 × Z 2 zero-form symmetry, and the map of generators is CK σ ↔ 1I ⊗ M, K σ ↔ C ⊗ 1I. 11 Here and in the following, ∼ means identification by the quotient. Consider where the quotient is generated by j s ⊗ j s (with no fixed points). On both sides the one-form global symmetry is Z 2 and the map of generators is On the RHS, all other generators are projected out by the quotient. The zero-form symmetry is Z 2 × Z 2 , and the map of generators is C ↔ 1I ⊗ M, M ↔ 1I ⊗ C.

Finally, consider
where the quotient is generated by j s ⊗σ (with no fixed points). On both sides the one-form symmetry is Z 2 and the map of generators is The zero-form symmetry is Z 2 for N k = 2 mod 4 and nothing otherwise. The map of generators is K σ ↔ 1I ⊗ K σ .

Level-rank dualities of spin-TQFTs
So far we have discussed level-rank dualities between TQFTs. We can obtain simpler dualities if we consider spin-TQFTs. As we explain below, we obtain the following: We recall that SO(N ) 1 is a trivial spin-TQFT with two transparent lines of spins 0, 1 2 and with framing anomaly c = N 2 (see e.g. [35]). Before deriving (5.25) and (5.26), let us discuss the symmetries and their map, starting with the orthogonal case (5.25). As we discussed after (5.15), SO(N ) k has a Z 2 one-form global symmetry for N, k both even, and not otherwise. Thus the one-form symmetries match. Moreover, SO(N ) k has a charge conjugation Z 2 zero-form symmetry C for N even, and a magnetic Z 2 symmetry M for k even. 12 Those two symmetries are exchanged in the duality (5.25), as it also follows from the derivation of the duality that we give below.
In the symplectic case (5.26), on both sides there is a Z 2 one-form global symmetry generated by σ, and a quantum zero-form symmetry K σ for N k = 2 mod 4.
Note that in the CS theories with matter in the fundamental representation discussed in the main text, both for gauge group SO and U Sp, the possible one-form global symmetry is broken by the presence of matter [37].
Next, we derive the dualities (5.25) and (5.26). The simplest case is to start with Spin(N ) k ↔ Spin(N k) 1 × SO(k) −N with N even, k odd. We can gauge the Z 2 one-form symmetry generated by σ ↔ σ ⊗ 1I to directly obtain (5.25). Since k is odd, SO(N ) k is a spin theory and hence adding SO(0) 1 does not change it. Another simple case is to start and it preserves the Z 2 generator σ ↔ σ ⊗ 1I ∼ 1I ⊗ σ. If we gauge the latter one-form symmetry as well, we obtain which is precisely (5.25).
To get the other cases, we make use of the following identity of spin-TQFTs: 29) discussed in [35]. Starting from U Sp(2N ) k ↔ Spin(4N k) 1 ×U Sp(2k) −N /Z 2 , we multiply both sides by SO(0) 1 making them into spin theories, then apply (5.29) and obtain The theory (Z 2 ) −2Nk is a TQFT with Z 2 × Z 2 one-form symmetry. The Z 2 quotient is generated by pairing σ in U Sp(2k) −N , whose spin is h = − Nk 4 mod 1, with a generator in (Z 2 ) −2Nk that has opposite spin. In the quotient SO(4N k) 1 remains as a spectator. The product of U Sp(2k) −N by (Z 2 ) −2Nk gives four times as many fields, however the freelyacting quotient by Z 2 reduces to the original ones (one can check that the surviving states have the same dimensions as the original ones). Thus one has a simple duality of TQFTs: This leads to the duality in (5.26). Exactly the same reasoning can be applied to the case SO(N ) k ↔ Spin(N k) 1 ×SO(k) −N /Z 2 with N, k even. We multiply both sides by SO(0) 1 , then we apply (5.29), and finally observe that the quotient can be "unfolded" by the simple duality: (Z 2 ) −2nm ×SO(2m) −2n /Z 2 ↔ SO(2m) −2n . Here we set N = 2n and k = 2m. This leads to (5.25).
The last case is SO(N ) k ↔ Spin(N k) 1 × Spin(k) −N /B, with N odd, k even. After multiplication by SO(0) 1 and application of (5.29), we obtain This case is a little bit more intricate. For N k = 0 mod 4, 2 . The first factor is obtained by pairing j s in Spin(k) −N , whose spin is h(j s ) = − Nk 16 mod 1, with a generator in (Z 2 ) − N k 2 that has the opposite spin. Therefore the first quotient is nonspin. The second factor is generated by W 1,0 ⊗ σ (in the notation of [35]) and it is a spin quotient. We can first use (Z 2 ) − N k 2 to unfold the quotient by Z  In the next section we will make use of the non-spin level-rank dualities (5.33) and (5.34) to find new T -invariant TQFTs.

T -invariant TQFTs from level-rank duality
It is of considerable interest to find topological field theories that are time-reversal invariant (T -invariant) up to an anomaly, because they can lead to gapped boundary states of topological insulators or topological superconductors. Some known examples are based on Chern-Simons theories with various product groups and possibly appropriate quotients [57][58][59][60][61][62][63]35]. It turns out that level-rank duality is a powerful tool to find new examples. 13 Specifically, a level-rank duality that exchanges N ↔ k, when applied to a theory with N = k, shows that such a theory is T -invariant quantum mechanically. 13 We thank E. Witten for a useful discussion about this point.
In some cases the theory involved is already a spin theory. In some other cases the theory is not spin but the level-rank duality, and therefore T -invariance, only holds after we tensor with a trivial spin theory, which we denote by ψ. Second, from the dualities of non-spin TQFTs (4.18) in [27] and ( The special case SO(3) 3 appears in the literature as SU (2) 6 /Z 2 [57]. The special case U (1) 2 is known as the "fermion/semion theory" [57] and was discussed recently in [35,28]. where δk = x r for complex or pseudo-real representations, and δk = 1 2 x r for real representations.