Seiberg-like dualities for orthogonal and symplectic 3d $\mathcal{N} = 2$ gauge theories with boundaries

We propose dualities of $\mathcal{N} = (0,2)$ supersymmetric boundary conditions for 3d $\mathcal{N} = 2$ gauge theories with orthogonal and symplectic gauge groups. We show that the boundary 't Hooft anomalies and half-indices perfectly match for each pair of the proposed dual boundary conditions.


Introduction and conclusions
Seiberg duality [1] of 4d N = 1 gauge theories is the duality in the IR which relates an 'electric' SU (N c ) gauge theories with N f flavours of quarks and antiquarks to a 'magnetic' SU (N f − N c ) gauge theories with N f flavours of quarks and antiquarks together with a gauge singlet field coupled through a superpotential (See e.g. [2,3,4,5,6,7,8,9] for various generalisations.). There exists a three-dimensional analogue, aka Seiberg-like duality in 3d N = 2 gauge theories. For the unitary gauge group, the IR duality [10] relates N = 2 gauge theory with gauge group U (N c ), N f fundamental chiral multiplets and N f antifundamental chiral multiplets to N = 2 gauge theory with gauge group U (N f − N c ), N f fundamental chiral multiplets and N f anti-fundamental chiral multiplets as well as additional gauge singlet chiral multiplets and a superpotential. For the symplectic gauge group, the IR duality [10] relates U Sp(2N c ) gauge theory with 2N f fundamental chiral multiplets to U Sp(2N f − 2N c − 2) gauge theory with 2N f fundamental chiral multiplets along with gauge singlets and a superpotential. The dualities are generalised to the theories with a Chern-Simons term [11,12], which can be achieved by adding real masses to flavours, leading to an effective Chern-Simons term at low energy. The 3d Seiberg-like dualities are extended to the SU (N c ) gauge theories [13,14,15,16,17,18,19], the orthogonal gauge theories [20,21,22,23,24,25,26], the G 2 gauge theory [25], the quiver gauge theories [27,28], the gauge theories with arbitrary numbers of fundamental and anti-fundamental matter fields obeying the Z 2 anomaly constraint k + N f −Na 2 ∈ Z [21], with adjoint matter fields [29,30,14,31,16,32,17,19], with other tensor matter fields [33,34,35,36] and with a monopole superpotential [37,38].
In the presence of boundary, the dualities become more elaborate as the bulk fields are subject to certain boundary conditions and they can further couple to additional boundary degrees of freedom. The half-BPS boundary conditions preserving N = (0, 2) supersymmetry in 3d N = 2 gauge theories have been studied in [39,40,41,42,43,44,45,46]. Simple dualities of N = (0, 2) half-BPS boundary conditions of 3d N = 2 gauge theories for Abelian gauge theories were proposed in [40] and various dualities of N = (0, 2) half-BPS boundary conditions for 3d N = 2 gauge theories with unitary gauge groups were proposed in [43]. 3 In this paper, we study N = (0, 2) half-BPS boundary conditions for 3d N = 2 gauge theories with symplectic and orthogonal gauge groups and propose dualities of these boundary conditions. We support our claims by computing boundary 't Hooft anomalies and supersymmetric half-indices which perfectly match for the proposed dual pairs of boundary conditions. For orthogonal gauge groups the global structure of the group is important and has been discussed for 3d Chern-Simons theories in the context of Seiberg-like duality [24] and level-rank duality [51]. We show that with boundary conditions there is a similar set of dualities relating the following groups under Seiberg-like duality: SO ↔ SO, O + ↔ O + , O − ↔ Spin and P in ± ↔ P in ± .
The organisation of this article is as follows. In section 2 we discuss the duality we suggest for N = (0, 2) half-BPS boundary conditions in 3d N = 2 U Sp(2N c ) gauge theories and test it by computing supersymmetric half-indices. In section 3 we propose the dualities of N = (0, 2) half-BPS boundary conditions for 3d N = 2 SO(N c ) gauge theories. We find more extensive identities of half-indices parameterised by two parameters ζ and χ corresponding to the global Z 2 symmetries [24]. In section 4 we discuss the dualities of N = (0, 2) half-BPS boundary conditions for other orthogonal gauge groups. In Appendix A we present the notation and convention of our tools, including boundary anomalies and supersymmetric indices. In Appendix B we show numerical results obtained from Mathematica.

U Sp(2N c ) gauge theories
The quantum dynamics of 3d N = 2 supersymmetric gauge theories with gauge groups G = U Sp(2N c ) has been studied in [52,53,10]. U Sp(2N c ) is the subgroup of SU (2N c ) that keeps an antisymmetric tensor J ab = (I Nc ⊗ iσ 2 ) ab invariant. Since the antisymmetric tensor i 1 ···i 2Nc breaks up into sums of products of the J ab , the U Sp(2N c ) gauge theory has no baryons. Also there is no topological current as the gauge group is simple.
For the symplectic gauge groups, the IR dualities are proposed in [10,53]. We will refer to these dual theories as theory A and theory B, but they are also commonly referred to as the electric and the magnetic theories: • Theory A: U Sp(2N c ) gauge theory with 2N f chiral multiplets Q in the fundamental representation. It contains gauge invariant operators as the meson M = QQ and the monopole operator V .
• Theory B: U Sp(2(N f − N c − 1)) gauge theory with 2N f chiral multiplets q in the fundamental representation, N f (2N f − 1) neutral chiral multiplets M in the rank-2 antisymmetric representation of SU (2N f ) and a chiral multiplet V which has the superpotential W = M qq + VṼ (2.1) whereṼ is the monopole operator.
The charges of the chiral multiplets are given by The quantum numbers crucially depend on the rank of gauge group and the number of flavours.

N = (0, 2) half-BPS boundary conditions
We introduce a boundary to the 3d N = 2 theories in such a way as to preserve N = (0, 2) supersymmetry in 2d. For the 3d bulk fields we must impose boundary conditions, and basic N = (0, 2) boundary conditions [40] impose either Neumann or Dirichlet boundary conditions on chiral multiplet (which we denote by N or by D) and either Neumann or Dirichlet boundary conditions on vector multiplet (VM) (which we denote by N or by D) 4 which is compatible with the decomposition of the 3d N = 2 supermultiplets into 2d N = (0, 2) supermultiplets on the boundary. The choice of Neumann or Dirichlet boundary conditions projects out specific N = (0, 2) supermultiplets.
On the 2d boundary we can have anomalies and these give two important constraints. If we have a gauge symmetry, which will be the case for Neumann boundary conditions for bulk vector multiplets, we require gauge anomaly cancellation. On the other hand, for global symmetries, including gauge symmetry broken by Dirichlet boundary conditions for the vector multiplet, we do not require cancellation of the anomalies. Instead, we get constraint on any proposed duality that the 't Hooft anomalies must match, i.e. the anomaly polynomials for the two theories must be equal.
The method to calculate the 2d boundary anomaly polynomial was given in [43]. There are three types of contribution to the anomalies: 3d bulk supermultiplets projected onto the boundary with Neumann of Dirichlet boundary conditions; 2d supermultiplets introduced on the boundary; background Chern-Simons or FI terms. We summarise the general results presented in [43] in appendix A.2. The specific results for Dirichlet boundary conditions. For Neumann boundary conditions the contributions are the same but with opposite sign. The notation s (s) refers to the gauge field strength in theory A (B), x to the field strength for the global SU (2N f ) flavour symmetry, and a and r to the field strengths for the global U (1) a and U (1) R symmetries. We also need 2d boundary matter and the only multiplet required for the examples we consider is a U Sp(2N c ) × U Sp(2(N f − N c − 1)) bifundamental Fermi multiplet, Ψ, which gives anomaly contribution (2.8) Note that this contribution is actually only half of what might be expected for such a Fermi. However, this is precisely the contribution required for anomaly cancellation and we interpret this as due to a reality condition on th Fermi. We will comment again on this when discussing the Fermi contribution to the half-index. For now we consider the 3d dualities with vanishing Chern-Simons level, in which case we have no background Chern-Simons terms.
The boundary conditions we consider are (N , N) in theory A, referring to the choice of Neumann boundary conditions for (VM, Q) together with (D, D, N, D) in theory B for (VM, q, M, V ). With this choice we need to cancel the gauge anomaly in theory A which we can do by including the bifundamental Fermi in theory A. This also leads to anomaly matching with theory B without any further 2d matter. In particular we have and it is easy to see that all dependence on s is cancelled and Taking the opposite choice of boundary conditions for all fields we also get anomaly matching if we add the bifundamental Fermi to theory B instead of theory A. This also cancels the gauge anomaly in theory B.
We therefore have the following proposed dualities

Supersymmetric indices
The supersymmetric "full-index" of 3d N = 2 supersymmetric theories can be defined as a trace over the states on S 2 ×R [55,56,57,58,59]. It can be evaluated as a partition function on S 2 × S 1 from the UV description and the UV formula was obtained in [56] for the theory with canonical conformal dimension and in [57] for the theory with any conformal dimension via the localisation technique. The supersymmetric "half-index" of 3d N = 2 supersymmetric theory T obeying the N = (0, 2) half-BPS boundary condition B can similarly be defined as a trace over the states that correspond to half-BPS local operators on the boundary [39,41,42,43] where F is the Fermion number operator, J is the generator of the Spin(2) ∼ = U (1) J rotational symmetry of the two-dimensional plane where the boundary local operators are supported, R is the R-charge and f is the Cartan generators of the other global symmetry group. The supersymmetric "half-index" of 3d N = 2 supersymmetric theories can be also defined as a trace over the states that corresponds to half-BPS local operators on the boundary [39,41,42,43]. One can compute the half-index as a partition function on HS 2 × S 1 which encodes the N = (0, 2) half-BPS boundary conditions on ∂(HS 2 × S 1 ) = S 1 × S 1 where HS 2 is a hemisphere. The UV formula for the half-index of the Neumann boundary condition for the gauge multiplet was derived in [39,41,42] and the formula for the Dirichlet boundary conditions for the gauge multiplet was proposed in [43]. The half-index can also realise the holomorphic block [60,61], the q-series 3-manifold invariantẐ [62] which has a number of applications to topology and number theory (see e.g. [63,64,65,66,67,68]) and the 4-manifold invariant [69] (see e.g. [70]). .
We summarise the general results here for the full-indices and half-indices, using notation U Sp(2N c ) − [2N f ] A for theory A with gauge group U Sp(2N c ) and 2N f fundamental chiral multiplets, and U Sp(2Ñ c ) − [2N f ] B for the dual theory B.

Full-index
The test of the IR duality for the symplectic gauge group were performed by computing supersymmetric indices in [71,22].
The full-index of theory A takes the form (2.14) where we note that a is the fugacity for the U (1) a symmetry which can be combined with the SU (2N f ) flavour symmetry to form a U (2N f ) flavour symmetry. Here we have set the SU (2N f ) flavour fugacities x α → 1 for simplicity. The full-index of theory B, which is the dual of theory A with gauge group U Sp(2N c ) so has gauge group U Sp(2Ñ c = 2N f − 2N c − 2), is

Half-index
We can construct the half-indices for the 3d N = 2 gauge theories with Dirichlet boundary condition D for the vector multiplet using the expressions in [43]. We briefly review the construction for both Neumann and Dirichlet boundary conditions, and describe some details for symplectic gauge groups in appendix A.3. We find for Dirichlet gauge field boundary conditions in theory A where the effective CS coupling k ef f is determined by the 't Hooft anomaly. As we are not turning on background fluxes, the only term relevant is the gauge field contribution which isÑ c tr(s 2 ) for Dirichlet boundary conditions in theory A. This gives where we note that relative to the unitary cases [43] and the orthogonal cases we discuss later, there is a factor of 2 when mapping the anomaly polynomial terms to the effective Chen-Simons coupling contribution. This factor arises from the conversion of the magnetic charge m ∈ cochar(G) to the electric charge km ∈ weight(G), noting that Cartan-Killing form for G = U Sp(2N c ) differs by a factor of 2 from that for the unitary or orthogonal groups -in particular, the weights of U Sp(2N c ) have length squared 1 2 , not 1. For Neumann gauge field boundary conditions in theory A we have (2.20) In both cases the matter contributions are given by the product of contributions for each chiral multiplet in the theory, depending on the chosen boundary conditions. For the bulk chiral multiplets in theories A and B we find the following contributions to II matter : The bifundamental Fermi would be expected to give contribution but as noted for the anomaly contribution we should impose a reality condition. This will half the number of degrees of freedom so the contribution to the 2d index will be the square root of this quantity. As it is a perfect square this gives a consistent contribution to the index The half-indices for theory B are constructed in exactly the same way but exchanging N c ↔Ñ c except for the matter contributions which are already written specifically for each theory. Note that since the dualities we consider are between theories with opposite boundary conditions for the vector multiplets it is consistent to use fugacities s i for the preserved gauge symmetry in whichever theory has Neumann boundary conditions, and fugacities u i in the other theory with Dirichlet boundary conditions breaking the gauge symmetry to a global symmetry.
We now present some examples which we have checked numerically to find matching indices. For calculational purposes these are all checked with flavour fugacities x α = 1. After presenting these examples we will construct the half-indices with non-zero Chern-Simons level.

2.3
We start with the simplest example with N c = 1, N f = 3, where Theory A is U Sp(2) = SU (2) gauge theory with six fundamental chiral multiplets Q.
The full-index is (2.32) Theory B is U Sp(2) = SU (2) gauge theory with six fundamental chiral multiplets q as well as gauge singlets M and V . The index (2.32) agrees with the index We have checked that the half-index (2.34) agrees with the half-index  The half-index of Dirichlet b.c. (D, D) for Theory A takes the form On the other hand, the half-index of Neumann b.c. (N , N, D, N) plus charged Fermi multiplets for Theory B is Next example is the case with N c = 2, N f = 5, where Theory A has gauge group U Sp(4) and ten fundamental flavours. The full-index of Theory A is (2.38) Unlike the previous example, Theory B has a different gauge group, i.e. U Sp(4) as well as ten fundamental chiral multiplets and neutral chiral multiplets.
We have the full-index The half-index of Dirichlet b.c. (D, D, N, D) for Theory B is We have confirmed that the half-index (2.40) agrees with the half-index

Chern-Simons level k = 0
As shown in [21] we can induce a non-zero Chern-Simon coupling starting from the case of vanishing Chern-Simons level with N F + 2|k| fundamental chirals and giving masses to 2|k| of them. Taking the masses to ±∞ we integrate out these 2|k| chirals and are left with N F flavours and Chern-Simons coupling k = ±|k|. Here the ± signs are the same for the masses and the sign of k. Note that we only require 2|k| ∈ Z so although N F + 2|k| is an even integer (which we can label 2N f as in the case of k = 0), N F is only constrained to be integer. In the dual theory B the corresponding chirals get masses of the opposite sign and consequently theory B gets a Chern-Simons level −k. This produces a duality U Sp(2N c ) k ↔ U Sp(2|k| + N F − 2N c − 2) −k with N F fundamental chirals in both theories. As shown in [43], chiral edge modes are present for bulk chirals with Neumann boundary conditions and positive masses, and with Dirichlet boundary conditions and negative masses. These edge modes introduce additional 2d chiral or Fermi multiplets. On the other hand, with negative masses for Neumann and positive masses for Dirichlet boundary conditions there are no edge modes and the dualities follow through without additional 2d multiplets. We focus on the latter cases for simplicity.
To derive the half-indices with k = 0 we can take limits of the fugacities following the procedure described in [21] for the full indices, and somewhat similar to the process outlined in [43] for unitary half-indices. This will result in the matching of half-indices where we note the the singlet V is also integrated out for k = 0. These half-indices are given by a simple prescription of removing the II V contribution which is present in the half-indices for the k = 0 cases and noting that A derivation follows from the limit taking masses to infinity as explained in [21].
there is no such constraint on the U (2N f ) fugacitiesx α . Now, for the 2|k| values of α corresponding to massive chirals we take the limitx α → 0 for negative mass in theory A andx α → ∞ for positive mass in theory A. For the simple choice of boundary conditions we are considering it is easy to see that this simply removes (set to 1) the contributions to the half-indices from those 2|k| fundamental chirals in theories A and B and at the same time removes the contribution from the corresponding parts of M and the contribution of V . Hence, in the cases we consider, this is a well-defined limit and the matching of half-indices follows for the matching for k = 0, assuming that holds for arbitrary flavour fugacities. We can then write the remaining U (N F ) flavour symmetry as SU (N F ) × U (1) a and replace the remaining N F fugacitiesx α with ax α , now with the constraint on the SU (N F ) fugacities N F α=1 x α = 1. This gives precisely the half-indices following the simple prescription above of simply omitting the contributions from the massive fields. Note that if we took the limits with the 'wrong' boundary conditions, the individual contributions would not have well-defined limits, and indeed we would need to introduce appropriate additional 2d chirals or Fermis to get well-defined matching half-indices.
One additional point to note is that if we calculate the anomaly polynomials for the duals with k = 0 simply by including the CS contribution ±k Tr(s 2 ) for theory A and ∓k Tr(s 2 ) for theory B, we would naively not get matching results. This is because there are background Chern-Simons terms generated by integrating out the massive fields and these contributions only taken into account those associated to the gauge groups. Including these background Chern-Simons levels we do have matching anomaly polynomials and indeed we can see these Chern-Simons levels by simply taking the difference in the anomaly polynomial before and after integrating out the massive fields in each theory. As we have not turned on background fluxes, our halfindices are not sensitive to these other background Chern-Simons levels. In particular we have the following contributions from background Chern-Simons levels, in addition to the ±k Tr(s 2 ) or ∓k Tr(s 2 ) contributions (2.49) In the following, we explicitly show examples of the half-indices with CS level k = 0 for which we have checked the precise agreement. and (2.53)

SO(N c ) gauge theories
The quantum dynamics of 3d N = 2 supersymmetric gauge theories with orthogonal gauge groups which is more subtle than with the unitary or symplectic gauge groups has been studied in [23,24]. The IR dualities associated with the Lie algebra g = so(N c ) of gauge symmetry are derived in [24] by taking an appropriate limit of the 4d N = 1 gauge theories on a circle. There are three distinct 4d gauge theories for g = so(N c ) with gauge group 2 , a Z 2 global symmetry that changes the sign of the non-trivial line operator [72].
The 4d duality between SO(N c ) + gauge theories leads to the 3d IR duality [24]: in the vector representation, N f (N f + 1)/2 neutral chiral multiplets M in the rank-2 symmetric representation of SU (N f ) and a chiral multiplet V which has the superpotential The charges of the chiral multiplets are Here Z C 2 (resp. ZC 2 ) is the charge conjugation symmetry in theory A (resp. theory B). Z M 2 (resp. ZM 2 ) is the magnetic symmetry in theory A (resp. theory B) [51].

N = (0, 2) half-BPS boundary conditions
The requirement of gauge anomaly cancellation and 't Hooft anomaly matching are the same as for symplectic gauge theories. Again, using the general results presented in [43] and summarised in appendix A.2 we find the following specific results for the for Dirichlet boundary conditions. For Neumann boundary conditions the contributions are the same but with opposite sign. As for the symplectic case, the notation s (s) refers to the gauge field strength in theory A (B), x to the field strength for the global SU (N f ) flavour symmetry, and a and r to the field strengths for the global U (1) a and U (1) R symmetries. We also need 2d boundary matter and the only multiplet required for the examples we consider is an SO(N c )×SO(N f −N c +2) bifundamental Fermi which gives anomaly contribution (3.8) Note that as for the symplectic case this contribution is actually only half of what might be expected for such a Fermi. However, this is precisely the contribution required for anomaly cancellation and we interpret this as due to a reality condition on the Fermi. We will comment again on this when discussing the Fermi contribution to the halfindex. For now we consider the 3d dualities with vanishing Chern-Simons level, in which case we have no background Chern-Simons terms.
The boundary conditions we consider are (N , N) in theory A, referring to the choice of Neumann boundary conditions for (VM, Q) together with (D, D, N, D) in theory B for (VM, q, M, V ). With this choice we need to cancel the gauge anomaly in theory A which we can do by including the bifundamental Fermi in theory A. This also leads to anomaly matching with theory B without any further 2d matter. In particular we and it is easy to see that all dependence on s is cancelled and Taking the opposite choice of boundary conditions for all fields we also get anomaly matching if we add the bifundamental Fermi to theory B instead of theory A. This also cancels the gauge anomaly in theory B.
We therefore have the following proposed dualities  and in all cases for theory B we haveζ = ζ andχ = ζχ.

Supersymmetric indices 3.2.1 Full-index
For the orthogonal gauge groups the Seiberg-like dualities have been tested from the evaluation of supersymmetric indices in [22,73,24]. The index depends on the global structure of the gauge group. As shown in [24] all the indices can be constructed from the SO(N c ) indices provided we include discrete fugacities ζ and χ for the Z M 2 and Z C 2 groups. For χ = +1 or N c being odd the full-index of theory A with gauge group G = SO(N c = 2N + ) reads For χ = −1 and N c even the full-index of theory A is In theory B there are additional gauge singlets M and V . As shown in (3.2), the chiral superfield V is odd under Z M 2 so that the index of V depends on the fugacity ζ. For χ = +1 or N c being odd the full-index of theory B is For χ = −1 and N c even the full-index of theory B is 6 The dualities lead to the identities of full-indices (3.13)-(3.16) [24]: Using the expressions in [43] for general gauge group and choosing group SO(N c ) (see appendix A.3 for further details) where N c = 2N + for ∈ {0, 1} since we often have to distinguish between N c even or odd, we have for Dirichlet gauge field boundary conditions in the cases with N c being even with χ = +1 or N c being odd with χ = ±1

Half-index
where the effective CS coupling k ef f is determined by the 't Hooft anomaly and the indices i and j take values from 1 to N . The relevant anomaly contribution isÑ c Tr(s 2 ) giving In the case χ = −1 with N c even we have instead where now i and j take values from 1 to N − 1.
For Neumann gauge field boundary conditions we have, excluding the case of N c even with χ = −1 Note that the symmetry factor is 2 N −1+ N !, the order of the Weyl group for SO(N c ), except for the case of N c even and χ = −1. In the latter case it is 2 N −1 (N − 1)! which is twice the order of the Weyl group of SO(N c − 2). This result can be found in [74], although note that they refer to the 'Weyl group' of O(N c ) which have an order twice that of the Weyl groups for SO(N c ). The half-indices for theory B are given by the same expressions after replacing N c = 2N + withÑ c = 2Ñ +˜ .
For the 3d fields in theory A with gauge group SO(N c = 2N + ) and theory B with gauge group SO(Ñ c = 2Ñ +˜ = N f − N c + 2) with Neumann or Dirichlet boundary conditions we have the following contributions to the half-indices where for χ = −1 and N c even we send s N → 1 and s −1 N → −1 in the contributions from Q, while forχ = −1 andÑ c even we send sÑ → 1 and s −1 N → −1 in the contributions from q.
The bifundamental Fermi, again with a reality condition as in the symplectic case, gives contribution to the 2d index where F 10 is included only if N c is odd, F 01 is included only ifÑ c is odd and F 11 is included only if both N c andÑ c are odd. In the case of N c even and χ = −1 we replace z N → 1 and z −1 We now list some explicit examples using notation SO(N c ) − [N f ] A ζχ to refer to theory A with gauge group SO(N c ) with N f fundamental chirals and discrete fugacities ζ and χ. This is dual to theory B with gauge group SO(Ñ c ) with N f fundamental chirals and discrete fugacitiesζ = ζ andχ = ζχ which we label ζχ . We first consider some examples with gauge group SO(1) in theory A. These are free theories with N f chirals but they are dual to interacting SO(N f + 1) − [N f ] theories. The indices and half-indices for N f free chirals are very simple so this leads to an interesting set of q-series identities (assuming the duality holds). In some examples the (half-)index of the dual theory also takes a simple form and the matching of (half-)indices can easily be checked analytically. Following these SO(1) examples we consider some cases with gauge group SO(2), equivalent to U (1) with N f chirals of charge +1 and N f of charge −1, or gauge group SO(3). The latter differs from SU (2) examples in that the matter is in the triplet representation corresponding to adjoint, not fundamental, of SU (2). As there are significant differences between SO(N c ) with N c odd or even, our examples cover all combinations with (N c ,Ñ c ) both even, both odd or one even and one odd.
For theory A we have the full-indices (3.38) (3.39) These indices agree with the indices of theory B Note that in these examples, since the half-index of theory A is not sensitive to the value of ζ we have This is a particular example of the more general case We will see the cases of N f = 2, 3 below but not specifically comment again on this point.
We now present the half-indices with Neumann boundary conditions in theory A (3.51) These half-indices agree with the theory B half-indices Note that,unlike for the full-indices, these theory A half-indices are sensitive to the value of ζ due to the bifundamental Fermi multiplet.
Instead, taking Dirichlet boundary conditions in theory A we have These agree with the theory B half-indices Note that, as for the full-indices, the half-indices of theory A are not sensitive to the value of ζ so we have This is a particular example of the more general case (3.65) We will see the cases for N f = 2, 3 below.
In theory A we have the full-indices which agree with the indices of theory B (3.75) These agrees with the theory B half-indices The theory A Dirichlet half-indices are For theory A the full-indices agree with the full-indices in theory B given by (N, N, N

) + Fermis
The half-indices with Neumann boundary conditions in theory A are and the matching theory B half-indices are  With Dirichlet conditions the theory A half-indices are The matching theory B half-indices are We now consider an example with SO(2) gauge group in theory A. This is equivalent to U (1) with each fundamental chiral of SO(2) corresponding to two chirals of U (1) with charges ±1. Note that the cases with χ = −1 in theory A are not sensitive to the value of ζ so we have Here we present only the cases with N f = 2 which gives gauge group SO(2) also in theory B.
The theory A full-indices are 108) The matching theory B indices are In the indices (3.110) and (3.112) the contribution from the gauge singlet V is cancelled by that from the meson M . Accordingly, the index (3.110) can be alternatively interpreted as the index of the mirror theory [75,76,57] for the SQED 2 , i.e. the U (1) gauge theory with two pairs of chiral multiplets of charge ±1 as well as gauge singlets. Note that such a cancellation does not occur and the indices of theory B and that of the mirror theory are distinguished when one turns on the fugacities for the flavour and topological symmetries.

SO(2) − [2] with N , (N, N) + Fermis
The theory A half-indices are They agree with theory B indices  As for the full-indices, for χ = −1 the theory A Dirichlet index is not sensitive to the value of ζ so we have identities (3.122) The matching theory B half-indices are We now consider examples with non-Abelian gauge groups. In the first case with SO(3) − [3] A the dual theory B is still Abelian with gauge group SO(2). In the following section we will look at the case of N f = 4 where theory B also has gauge group SO(3).

135)
with the matching theory B half-indices given by

SO(3) − [3] with D, (D, D, D) For Dirichlet conditions we have theory A half-indices
(3.149) These match the theory B half-indices In the set of examples we have non-Abelian gauge group in both theories. The theory A full-indices are

) + Fermis
The theory A half-indices with Neumann boundary conditions are We find that these half-indices agree with those for theory B (3.173) We find agreement with the theory B half-indices As shown in [21] we can induce a non-zero Chern-Simon coupling starting from the case of vanishing Chern-Simons level with N f + |k| fundamental chirals and giving masses to |k| of them. The process is almost identical to that for symplectic gauge groups described in section 2.5. Taking the masses to ±∞ we integrate out these |k| chirals and are left with N f flavours and Chern-Simons coupling k = ±|k|. Here the ± signs are the same for the masses and the sign of k. Note that for orthogonal gauge groups k ∈ Z. In the dual theory B the corresponding chirals get masses of the opposite sign and consequently theory B gets a Chern-Simons level −k. This produces a duality SO(N c ) k ↔ SO(|k| + N f − N c + 2) −k with N f fundamental chirals in both theories. The matching with Z 2 fugacities ζ and χ follows in the same way as for k = 0. Again for simplicity to avoid chiral edge modes [43] we focus on the cases with negative masses for Neumann and positive masses for Dirichlet boundary conditions which don't require additional 2d multiplets other than those already present for k = 0. This results in the matching of half-indices for • (N , N) b.c. in theory A with gauge group SO(N c ) k ↔ (D, D, N) b.c. in theory B with gauge group SO(Ñ c ) −k with k < 0.
where we note that again the singlet V is also integrated out for k = 0.
These half-indices are given by a simple prescription of removing the II V contribution which is present in the half-indices for the k = 0 cases and noting that A derivation again follows from the limit taking masses to infinity as explained in section 2.5 following the procedure for the full index [21].
Again as we have not turned on background fluxes, our half-indices are not sensitive to other background Chern-Simons levels, but we note that these again arise from the process of integrating out the chirals. In particular, in addition to the ±k Tr(s 2 ) or ∓k Tr(s 2 ) terms, we have the following contributions from background Chern-Simons r)))ar Below we have k ∈ {1, 2}. For theory A we have (3.182) (3.184) (3.186) (3.188) and while for theory B we have

Checks on matching of indices
We have performed checks on the claimed matching half-indices using Mathematica to expand the q-series to at least order q 5 after first defining a fugacity y = q r−1/2 to replace all terms with dependence on the parameter r in the R-charge assignment. Details of the expansion to this order are listed in appendix B.2. Some of the identities can also be checked analytically. The cases where the indices do not include any integration or summation over monopole charges are particularly straightforward. This will occur in full-indices and half-indices with gauge group SO(1) as well as in the case of gauge group SO(2) with χ = −1. The full-indices and half-indices in both theories will satisfy this property for precisely the following cases which we have presented and the cases When checking the cases with SO(2) and χ = −1 it is useful to note that setting x = −q in the first of these identities and then using the second with x = q gives We give just one explicit example of SO(2) − [2] +− including the flavour fugacities.
where for the theory B half-index we used x 1 x 2 = 1 for the SU (2) flavour symmetry to see that the contribution for V cancels part of the contribution from M .

Other orthogonal gauge theories
For the Lie algebra so(N c ), there exist other possibilities of the gauge groups, O(N c ) ± , Spin(N c ) and P in(N c ) ± , which lead to distinct gauge theories [24,51]. Another possibility is SO(2N )/Z 2 but we do not consider this case. While the SO(N c ) gauge theory has the monopole operators carrying weights as well as roots of the dual magnetic group, the Spin(N c ) gauge theory only contains the monopole operators carrying roots. In the Spin(N c ) gauge theory the minimal monopole operator which turns on one unit of magnetic flux and parametrises the Coulomb branch looks semi-classically V Spin ≈ exp( 2σ 1 g 2 3 + 2iγ 1 ) where σ i and γ i are the adjoint scalar fields in the vector multiplet and the dual photons. It can be also described as in terms of the semi-classical Coulomb branch coordinates whereĝ 2 3 = g 2 3 /4π and g 3 is the 3d gauge coupling constant. In other words, it is the square of the minimal monopole operator V ≈ exp( σ 1 (4.4) One can also consider the gauge theories with disconnected orthogonal gauge groups O(N c ) and P in(N c ) which include the reflections along the line bundle or determinant bundle. While the baryon B = Q Nc is an independent operator that is charged under Z C 2 for SO(N c ) and Spin(N c ), it is not for O(N c ) and P in(N c ) since Z C 2 is gauged. We can construct O(N c ) + theories from SO(N c ) theories by gauging the charge conjugation symmetry Z C 2 of the SO(N c ). Alternatively, gauging Z MC 2 , the diagonal [24,51]. At the level of the Lagrangian description, they are distinguished by a discrete theta angle which is proportional to w 1 ∧ w 2 where w i ∈ H i (X, Z), i = 1, 2 are Z 2 -valued Stiefel-Whitney characteristic classes of O(N c ) bundle on a 3-manifold X. Both w 1 and w 2 , i.e. theta angle are non-zero only for O(N c ) but not for SO(N c ), Spin(N c ) or P in(N c ) ± [77,78].
The O(N c ) + gauge theories and their dualities were discussed in [20,21,22,23,73]. In particular, the matching of the indices for the O(N c ) + gauge theories was tested in [22,73]. Similarly to the SO(N c ) gauge theory, the O(N c ) + gauge theory has the minimal monopole operator V = exp( σ 1 g 2 3 + iγ 1 ) which is charge-conjugation-even and gauge invariant.
On the other hand, in the O(N c ) − gauge theory the monopole operator V is chargeconjugation-odd and not gauge invariant. Instead, the product of V with the baryon B as well as the monopole operator V Spin and the baryon-monopole operator β = are the SO(2) monopoles which are even and odd under Z C 2 . Furthermore, one can obtain the P in(N c ) gauge theories by gauging the global symmetry Z M 2 × Z C 2 , which can also be viewed as gauging the Z C 2 of Spin(N c ) or the [24,51]. More precisely this gives P in(N c ) + but a modification of the gauging process produces P in(N c ) − [51].
The 4d duality between Spin(N c ) and O(Ñ c ) − gauge theories gives rise to the 3d IR dualities [24]: • Theory A: Spin(N c ) gauge theory with N f chiral multiplets Q in the vector representation.
• Theory B: O(Ñ c = N f − N c + 2) − gauge theory with N f chiral multiplets q in the vector representation, N f (N f + 1)/2 neutral chiral multiplets M in the rank-2 symmetric representation of SU (N f ) and a chiral multiplet V which has a superpotential. and similarly with O(N c ) − for theory A and Spin(Ñ c ) for theory B.
For the P in(N c ) + gauge theories, the duality can be derived by gauging Z M 2 of the O(N c ) + duality [20,21,22,23,73]. It gives rise to the duality between the P in(N c ) + gauge theory and P in(Ñ c = N f − N c + 2) + gauge theory.
The identities (3.17) imply the identities of the O(N c ) ± , Spin(N c ) and P in(N c ) + full-indices [24] as well as P in(N c ) − full-indices: where 7 the indices can be constructed from the SO(N c ) ζχ indices by summing over ζ to gauge Z M 2 , χ to gauge Z C 2 , or χ together with a change in sign of ζ to gauge Z MC 2 . The result is where χ = ±1 corresponds to a projection onto even or odd states under, Z C 2 for O(N c ) + and P in(N c ) + or Z MC 2 for O(N c ) − , and ζ = ±1 determines a projection onto even or odd states under Z M 2 for Spin(N c ) and P in(N c ) + . Such an interpretation of ζ and χ is less straightforward for P in(N c ) − . These parameters can be viewed as discrete theta angles.
Similarly to the equalities (4.5)-(4.7) of the full-indices, we find the half-index version of the identities: (4.17) The equalities (4.11)-(4.14) indicate the dualities of the basic N = (0, 2) half-BPS boundary conditions for 3d N = 2 theories with orthogonal gauge groups; O(N c ) ± , Spin(N c ) and P in(N c ) ± . These can easily be extended to include Chern-Simons levels for the gauge groups by integrating out some of the fundamental chirals as explained for the SO(N c ) half-indices in section 3.9.
The conventions are chosen so that a left-handed 2d complex fermion with charge c under a U (1) symmetry with field strength f contributes +(cf ) 2 to the anomaly polynomial. This is therefore the contribution from Fermi multiplet charged under the U (1). A right-handed fermion, and hence a chiral multiplet, contribute with the opposite sign, i.e. −(cf ) 2 .
A key result is that 3d chiral multiplets with Dirichlet (Neumann) boundary conditions will project out a the right-handed (left-handed) fermion, leaving a boundary left-handed (right-handed) fermion, but that the anomaly contribution will be half that of a 2d fermion. Therefore a 3d chiral multiplet with charge c under the U (1) symmetry will contribute 1 2 (cf ) 2 for Dirichlet and − 1 2 (cf ) 2 for Neumann boundary conditions. One slight subtlety is that for the U (1) R-symmetry the fermions have a shifted R-charge so if a chiral multiplet has R-charge ρ there will be anomaly contributions ± 1 2 ((ρ − 1)r) 2 . If we have several U (1) factors we add up the contributions of the form cf before squaring.
These results generalise to irreducible representations R of non-Abelian simple compact groups G by setting c = 1 and replacing f 2 with Tr R (f 2 ). The trace in representation R is given by More precisely the results for SO(N ) apply to N ≥ 4 but we choose the same normalisation for SO(2) and SO (3).
In addition, a 3d Chern-Simons coupling at level k gives a contribution kf 2 .

A.3 General expressions for half-indices
Here we summarise the general results presented in [39,41,42,43] (also see [79,80,81] for 2d indices) to calculate the half-indices. For a 2d chiral multiplet with R-charge ρ and other U (1) charges giving fugacity x the contribution to the half-index is while for a Fermi multiplet we have A 3d chiral with these charges will give contributions for Neumann or Dirichlet boundary conditions. If we have a representation R of a non-Abelian simple group G we take the product of similar expressions over the weights of R. So for a 3d chiral with Neumann boundary conditions we have For example, the s α could be labelled s 1 , s 2 , . . . , s N with the constraint s 1 s 2 · · · s N = 1 for the fundamental representation of SU (N ) -for U (N ) we simply drop the constraint s 1 s 2 · · · s N = 1. These fugacities s i correspond to the fundamental weights µ i , with the constraint i µ i = 0 for SU (N ). One point to note is that the weights of the adjoint representation include zero weights and the vector multiplet fermions have opposite chirality to those from chiral multiplets for the same boundary condition. So, for U (N ), the contribution of the vector multiplet with Neumann boundary conditions is given by the gauginos (with R-charge 1) in the surviving 2d vector multiplet Instead, for Dirichlet boundary conditions the the contribution is from the surviving 2d chiral multiplet (with R-charge 0) giving contribution (A. 17) and now the gauge symmetry is broken so G is a global symmetry, and there is no projection onto gauge invariant states. However, as explained in [43] the complete non-perturbative half-index with Dirichlet boundary conditions for the vector multiplet is given after including a sum over monopole charges and including factors for the effective Chern-Simons levels. Specifically, there is a sum over all magnetic charges m ∈ cochar(G). Each term in this sum is given by the same half-index but with the fugacities s i shifted by a factor q m i reflecting the spin induced by magnetic charge m i and a factor for the monopole contribution. This monopole factor is [43] ( where k ef f is a bilinear form defined by the anomaly polynomial. The interpretation is that for U (1) at Chern-Simons level k we would have contribution (−q 1/2 ) km 2 s km reflecting the induced electric charge km of the monopole due to the Chern-Simons level and consequently the induced spin km 2 /2. Note that our convention is that the indices involve tracing (−1) F , whereas in [43] the convention was (−1) R . These conventions are simply related by q 1/2 ↔ (−q) 1/2 and can lead to some sign differences when comparing explicit expressions for indices and half-indices. In particular, for the Dirichlet half-indices in our convention we may get sign factors depending on the magnetic charges in the sum, of the form (−1) km 2 as objects with spin km 2 /2 are fermions for odd km 2 . Such terms indeed appear for the orthogonal half-indices, but not for the symplectic case where in all cases (−1) k ef f [m,m] = 1.
The results above in terms of rank(G) and product over roots hold also for SU (N ), U Sp(2N ) and SO(N c ).
The details for the symplectic case are very similar to the unitary case. The fundamental weights for U Sp(2N ) are ±µ 1 , ±µ 2 , . . . , ±µ N giving fugacities s ±1 i , while the roots are µ i − µ j ∀i = j and ±(µ i + µ j ) ∀i ≤ j.
For SO(N c ) we need to consider separately the cases of N c even or odd. For the even case we write N c = 2N and we have fundamental weights ±µ 1 , ±µ 2 , . . . , ±µ N and the roots are ±µ i ± µ j ∀i < j where all four sign combinations are taken.
For the case of odd N c we write N c = 2N + 1 and we have one additional fundamental weight µ 0 = 0 giving additional roots ±µ i + µ 0 = ±µ i ∀i.
The above results, along with details of the monopole sums and effective Chern-Simons levels define the half-indices, although we found that the effective Chern-Simons levels need a different normalisation factor for symplectic gauge groups. However, there is a further complication for the SO(N c ) indices when the discrete fugacity χ for the Z C 2 global symmetries is included. For the case of odd N c if we have χ = +1 we can consider that we have a gauge fugacity s 0 = 1 arising from the fundamental weight µ 0 = 0. If instead we have χ = −1 we set s 0 = −1. So, in general we have s 0 = χ meaning that the fugacities associated to the roots ±µ i + µ 0 are χs ±1 i . Note that these four different sectors for SO(N c ), labelled by ζ and χ arise in 3d and so are inherited on the boundary since we have bulk gauge fields with boundary conditions. In 2d theories on T 2 there are additional sectors as there is a different classification of flat connections on T 2 as clearly explained in [74]. The point to note here is that this means the half-index for orthogonal groups is quite different from the 2d elliptic genus.
For even N c = 2N if we have χ = −1 we must replace the gauge fugacities corresponding to fundamental weights µ N and −µ N with s N → 1 and s −1 N → −1. In cases of ambiguity, it is necessary to check whether the factor of s N originated from fundamental weight µ N or −µ N . One point to note in this case is that in the derivation of the vector multiplet contribution to the index one of the factors of (q) ∞ (for Neumann or Dirichlet boundary conditions) arose from the adjoint weight µ N + (−µ N ) = 0 giving factor (qs N s −1 N ; q) ∞ = (q) ∞ . In the case of χ = −1 this is replaced, since s N s −1 N → 1 × (−1) = −1 by the factor (−q; q) ∞ . I.e. in the half-index we replace (q) Finally, our notation for fugacities is to label gauge fugacities s i as above for halfindices with Neumann boundary conditions for the vector multiplet and instead u i if we have Dirichlet boundary conditions. This applies to both theories A and B in the proposed dualities and is unambiguous as all these dualities involve Neumann for one theory and Dirichlet for the other. Of course, gauge invariance is imposed by integrating over the s i so the results will depend on the u i only.
Our notation for other fugacities is a for U (1) a and x α for global SU (N F ) flavour symmetries. In numerical checks we often set x α = 1 for simplicity. Note that α x α = 1 but we could also use fugacitiesx α which don't obey such a constraint for the global U (N f ) flavour symmetry by combining SU (N F ) and U (1) a . We comment on this when discussing dualities for non-zero Chern-Simons levels.

B Series expansions of indices
We show several terms in the expansions of indices obtained by using Mathematica.
We show the expansions of full-indices and the half-indices in powers of q by defining a fugacity y := q r− 1 2 . We find the matching at least up to q 5 .