Symplectic gauge group on the Lens Space

We compute the Lens space index for 4d supersymmetric gauge theories involving symplectic gauge groups. This index can distinguish between different gauge groups from a given algebra and it matches across theories related by supersymmetric dualities. We provide explicit calculations for $\mathcal{N}=4$ SYM and for classes of $\mathcal{N}=2$ and $\mathcal{N}=1$ Lagrangian quivers related by S-duality. In these cases the index matches across the S-dual phases, while models in different S-duality orbits have a different Lens index. We provide analogous computations for a 4d $\mathcal{N}=1$ toric quiver gauge theory corresponding to a $\mathbb{Z}_7$ orbifold of $\mathbb{C}^3$. This $SU(n)^7$ gauge theory becomes interesting in the case of $n=2$ because it is conformally dual to other two models, with symplectic and unitary gauge groups, bifundamentals and antisymmetric tensors. We explicitly check this triality at the level of the Lens space index.


Introduction
Higher forms play a crucial role in the modern formulation of symmetries in QFT, because they involve extended object and constrain their charges [1].For example the extended objects charged under a 1-form symmetry are loop operators.The constraints on the charge spectrum of such lines, i.e.Wilson and 't Hooft lines, is reflected in the choice of the gauge group from a given gauge algebra in a gauge theory [2,3].Fixing the global properties of a gauge theory can have important consequences, for example it fixes the periodicity of the theta angle (see for example [4] for the phenomenological implication in the SM).
Another more formal consequence of having different gauge groups for a given gauge algebra is that this difference can be observed on the partition function computed in curved space on a spin manifold [3,5].In general partition functions on curved space are complicated quantities, but the difficulty of such problem is highly simplified for some manifolds in supersymmetric gauge theories, thanks to the help of localization [6].
In general such partition functions can still fail in distinguishing among different global structures because of the presence of further symmetries and dualities relating theories with different gauge groups.This is for example the case of S-duality in N = 4 SYM.In this case many of the choices of the global structure are related to each other by the action of the S-duality group on the spectrum of charges of the line operators.There are nevertheless, depending on the choices of the gauge algebras and the ranks, cases where multiple orbits of the S-duality group are present.The partition functions on the curved space can in principle distinguish such orbits.This picture has been confirmed by explicit calculation from the Lens space index in [5].This index was originally defined in [7] and it corresponds to the superconformal index computed on L(r, 1) × S 1 , where L(r, 1) S 3 /Z r is the three-dimensional Lens space and S 1 is the Euclidean time.
The first explicit calculations of the index on such space have been performed in [5] for the case of N = 4 SU(n).Furthermore Seiberg duality for SO(n) gauge theories with vectors has been analyzed as well.Other calculations of the Lens space index have been performed in [8] for the N = 2 SU (n) quiver corresponding to the non-chiral orbifolds of C 3 (see also [9][10][11] for an analysis for non lagrangian theories).In all the cases the index has been shown to match for models connected by duality while it gave different results for different orbits of the S-duality group.
The 4d Lagrangian SCFTs zoology admits however many other possibile behaviors that have not yet be studied in terms of the Lens space index and that require an investigation.For example models with symplectic gauge groups have not been analyzed so far.This comprises the case of N = 4 where depending on the parity of the gauge rank we have different structure of the S-duality orbits, involving orthogonal groups as well [3].Furthermore symplectic, orthogonal and unitary gauge groups are all involved in the examples of S-duality for N = 2 quivers originally found in [12] that can be studied in terms of the Lens space index.Such S-duality has been recently shown in [13] to persist when breaking N = 2 to N = 1.In this paper we study the Lens space index for these models, showing that they match among the theories in the same S-duality orbit, while they differ for choices of the gauge group in a different orbit.
We conclude our analysis by studying a triality found in [14] that relates three models with either unitary or symplectic gauge groups.It was pointed out in [14] that there are in these cases different choices of the gauge group for each phase and we observe that all these choices give rise to the same Lens space index.On one hand this corroborates the validity of the claim about the triality among these models.On the other hand we explain the absence of multiple orbits by discussing some general expectations from the holographic dual description of one of these three phases in Type IIB string theory.

Review
Extended operators, such as Wilson and t'Hooft lines, play an important role in the study of Quantum Field Theories.When the spacetime manifold is R 4 the extended operators of the theory do not affect the correlation function of local operators.Nevertheless, two theories that only differ by their extended operators are still distinguished by the correlation functions that involve the extended operators themselves.When the spacetime manifold is non-trivial the extended operators can have a wider impact on the physics.Extended operators can be wrapped on non-trivial cycles of the spacetime providing different backgrounds for the local physics.Furthermore when the theory is compactified to a lower dimension the presence of extended operators can change the spectrum of local operators on the lower-dimensional theory.For example when the spacetime is R 3 × S 1 a Wilson line wrapped around S 1 becomes a local operator in the effective 3-dimensional theory on R 3 .

Line operators in gauge theories
In gauge theories the spectrum of extended operators is closely related to the global structure of the gauge group.The spectrum and correlation functions of local operators only depend on the gauge algebra g associated to the gauge group G while the spectrum of lines depends on the gauge group G and on discrete theta-like parameters.In this paper we will only consider compact gauge groups, therefore we have G = G/H where G is the compact simply connected group with associated Lie algebra g and H ∈ Z( G) is a subgroup of the center of G.The lines can be organized by their electric and magnetic charges (n e , n m ) ∈ Z( G) × Z( G).We always have Wilson lines in every representation of G that belong to the classes (n e , 0) with n e The spectrum of lines is determined by a complete and maximal set of charges (n e , n m ) satisfying (2.1).It turns out that given a choice of gauge group G there still can be different choices for the spectrum of lines.These choices are associated to discrete theta-like parameters that can be introduced in the theory.For example when g = usp(2n) there are three possible choices for the line spectrum, they are depicted in Figure 1.

Lens space index
The Lens space index is a powerful tool for studying the global structure of supersymmetric gauge theories [5,8,11,[15][16][17] .Unlike the supersymmetric index on S 3 × S1 , the Lens space index is sensible to the extended (1 + 1)-dimensional operators of the theory (i.e.Wilson, t'Hooft and dyonic lines) and can be able to distinguish between theories with the same gauge algebra and matter content but with different gauge group.The index is an RG invariant and is expected to match between dual theories as well as being stable under exactly marginal deformations.The Lens space index of a theory can be computed as a supersymmetric partition function on the (Euclidean) spacetime manifold L(r, 1) × S 1 .Here L(r, 1) S 3 /Z r is the three-dimensional Lens space and S 1 is the Euclidean time.The integer r parametrizes different spacetime manifolds, explicitly: where we identify elements related by ∼ r : 3) The fundamental group of the Lens space is: therefore there are two non-contractible 1-cycles: the Euclidean time cycle wrapping S 1 and a 1-cycle γ in L(r, 1) such that γ r is contractible.When a gauge theory is placed on this spacetime one has to sum over all possible gauge bundles.In particular the allowed flat connections for a gauge group G are determined by the holonomies of the gauge field around the non-contractible 1-cycles.We call g and h the holonomies around the cycle γ and the Euclidean time cycle respectively.Two pairs of holonomies (g, h) and (g , h ) are gauge equivalent if g and h can be simultaneously conjugated to g and h by an element of the group G.The sum over flat connection must be performed modulo gauge equivalence.The choice of holonomies is a group homeomorphism G × G → π 1 L(r, 1) × S 1 therefore g and h satisfy the following identities: If G is simply connected eq.(2.5) implies that g and h can be simultaneously conjugated to a maximal torus of G.The same is not true if G is not simply connected.The problem of commuting pairs 1 has been studied for example in [15,18,19], the solution for the groups associated to the algebra su(n) and for Spin(n), SO(n) ± together with the constraint coming from (2.6) has been used in [5] to compute the Lens space index of theories with the corresponding gauge group.The commuting pairs for symplectic groups has been studied e.g. in [18], in [20] they have been used to compute the elliptic genus of two-dimensional theories with symplectic gauge group.In Appendix A we summarize these results and show how to compute the Lens space index for U Sp(2n) gauge theories.
A supersymmetric theory can be placed on the Lens space L(r, 1) × S 1 while preserving N = 1 supersymmetry.The corresponding partition function localizes on the flat connections of the gauge group and reduces to an effective matrix model.The infinite dimensional path integral reduces to a sum/integral over the holonomies g and h.Each multiplet contributes to the integrand in the following way.Suppose that on the background of a specific choice of holonomies a field with R-charge2 R acquire a phase e 2πim r when is rotated around the cycle γ and a phase u when is rotated around the euclidean time cycle.Then the contribution from a chiral multiplet is: where p and q are fugacities associated to the spacetime symmetries SU (2) L ×SU (2) R and is the contribution of the multiplet to the Casimir energy.The contribution of a vector multiplet is: where: . (2.10) The elliptic Gamma function Γ(z; p, q) is defined by the infinite series: where the series converges and by analytic continuation elsewhere.In this paper we will take p = q = x in order to simplify the computation of the indices.Each of the indices presented in this paper can be further refined by considering different fugacities.The check of a specific duality through the Lens space index consist in an identity between two such sum/integrals.We lack the tools to prove these identities analytically; what we do instead is to expand the indices in a Taylor series for small spacetime fugacities (small x) and match the indices order by order in this expansion.
The holonomies can be organized by uplifting them to the universal cover group G.The uplifted holonomies g and h satisfy: where µ and ν range over the possible uplifts of 1 ∈ G.For g = so(2n + 1) or usp(2n) this means that there are four holonomy sectors Z µ,ν labelled by (µ, ν) = (±1, ±1).
The Lens space index of the gauge theories with g = so(2n + 1) is [5]: (2.14) Similarly for g = usp(2n): (2.15) 3 N = 4 S-duality with U Sp(2n) gauge group S-duality maps N = 4 SYM with gauge algebra g and gauge coupling τ to SYM with gauge algebra g and gauge coupling τ .The dual algebra g can be either g itself or the GNO dual algebra g ∨ .In this paper we consider the S-duality orbits for N = 4 SYM with gauge algebras B n = so(2n + 1) and C n = usp(2n) 3 .The full duality group is the subgroup of SL(2, R) generated by: which act on the gauge coupling τ by modular linear transformation: The S-duality orbits of N = 4 SYM with gauge algebra usp(4k) and so(4k + 1), reproduced from [3,21].Additionally, the S element of the duality group maps a theory with g = so(2n + 1) to a theory with g = g ∨ = usp(2n) while the T element of the duality group leaves the gauge algebra invariant.
The S-duality group forms different orbits depending on whether n is even (Figure 2) or odd (Figure 3).In this section we perform a precision test of the S-duality orbits by computing the Lens space index of these theories for low values of n and small fugacities.We will see that the index is the same between theories that lie in the same orbit and is different between theories in different orbits.

r=2
The contributions from the Z µ,ν sectors for the theories with gauge algebras so (7) and usp(6) up to O(x 4 ) are: The Lens space indices for the six possible theories are: (3.20) We see that the Lens space index of theories that lie in the same orbit matches up to this order while the indices of theories that lie in different orbits are different.

r=4
Up to O(x 2 ) the indices are: Up to O(x 2 ) the indices are: (3.27) The duality orbits are given in Figure 2 with k = 2.

r=2
Up to O(x 2 ) the indices are: (3.31) Up to O(x 2 ) the indices are: 1 + 3x  The duality orbits are given in Figure 3 with k = 2.

r=2
Up to O(x) the indices are:

N = 2 S-duality of elliptic models with orientifolds
In this section we consider N = 2 quiver theories that contain orthogonal or symplectic gauge groups.This theories were studied in [12] where they arose as low energy gauge theories of elliptic models with orientifolds in Type IIA string theory.The brane construction involves D4 branes wrapped on a compact dimension as well as N S5 branes and O6 ± orientifold planes.The configuration preserves d = 4 N = 2 supersymmetry and the low energy theory on the stack of D4 branes is a quiver gauge theory.Due to the presence of the orientifold planes the quiver can contain real (orthogonal and/or symplectic) gauge groups as well as two-index tensorial representations of unitary gauge groups.These theories have a Z 2 subgroup of the center that is not broken by the matter content, therefore we can consider different global structures.In particular there are three choices of global structures, analogously to the case of N = 4 with gauge algebras so(2n+1) and usp(2n) studied in the previous section.
In this paper we consider the case of quivers with two gauge nodes of the families ii) and iii) of [12].The quivers in N = 1 notation are shown in (4.1).The fields have the following representations: X vec There are three choices of global form for each of these theories, namely: and: In principle these theories can be engineered in Type IIB by probing the nonchiral C 3 /Z 3 orbifold with 2n D3 branes and introducing an O3 plane on top of the stack of D3 branes.The number of D3 branes plus their orbifold images is 6n, therefore we expect that the S-duality orbits will be the same as the ones of N = 4 SYM with gauge algebras usp(6n) and so(6n + 1) shown in Figure 5.
The indices of the two theories are given by: In this section we compute the Lens space index for these theories with n = 2 and r = 2, 4 and with n = 3 and r = 2.We find that the indices match for theories that lie in the same orbit and is different between theories that lie in different orbits.The smallest value of n for which all the groups have positive ranks is n = 2. Then the gauge algebra for the two theories are so(5) × su(3) and usp(2) × su(4).We notice that for this value of n we could regard usp(2) as su(2) and all the gauge groups would be either orthogonal or special unitary.The index could be computed without the technology developed in this paper for symplectic gauge groups.This is only true for n = 2, while for higher n the technology for computing the index in the presence of symplectic gauge groups is needed.

r=2
The contributions to the indices up to O(x 2 ) are: The indices are: The contributions to the indices up to O(x 2 ) are: The indices are:

−
(2) = 8 + 13x 2/3 + 68x 4/3 + 81x 2 (4.13) 4.2 SO(7) × SU (5) and U Sp(4) × SU (6) In this section we consider the case of n = 3.Then the gauge algebras are so(7)×su (5) and usp(4) × su(6). r=2 The contributions to the indices up to O(x 2 ) are: The indices are: 5 N = 1 inherited S-duality of elliptic models with orientifolds In this section we consider N = 1 models that can be obtained from the models studied in Section 4 by adding a mass deformation for the adjoints.The mass deformation breaks supersymmetry to N = 1.When the massive adjoints are integrated out we obtain the gauge theories described by the quivers in (5.1).
The superpotentials are, respectively: The fields have the following representations: A large family of theories that includes these models where first studied in [22], arising from orbifold projection of toric theories.The authors also presented the duality webs associated to those theories.These dualities where later understood as inherited (in the sense of [23,23]) from the N = 2 models of [12] in [13].In this paper we focus on a specific duality among the ones presented in [13,22], namely the duality that relates the two quiver theories above.Similarly to their N = 2 counterparts, these theories have a Z 2 subgroup of the center that is not broken by the matter content and therefore can have different global structures.With an abuse of notation we use the same names for the N = 1 theories that we used in the N = 2 case, namely (4.3) and (4.4).The duality orbits are shown in Figure 5.
In this section we provide a check of these duality orbits by computing the Lens space index for small n and small fugacities.In particular we perform the check with n = 2 and r = 2, 3 and with n = 3 and r = 2.We find that the indices of theories in the same orbit are the same and the indices of theories in different orbits are different.The contributions to the indices up to O(x 2 ) are: (5.4) (5.5) The indices are:

A conformal triality
In this section we consider the conformal triality introduced in [14].We compute the Lens space index for the three gauge theories involved.Each theory has a Z 2 subgroup of the center that is not broken by the matter content and has three possible choices of global structure.We find that at small fugacities all nine theories have the same index, this suggests that they are all dual to each other.We argue that this is true by exploiting the fact that one of the frames describes the low energy theory of D3 branes on the tip of an orbifold C 3 /Z 7 .The holographic picture [21] then suggests that the three choices of global structure for this frame are dual to each other.

Frame A
The first frame is a N = 1 gauge theory with gauge algebra usp(6) described by the quiver in (6.1).4) SU( 2) USp( 6) (6.1)where 14 is the totally antisymmetric representation of usp (6).The theory is conformal, all the matter fields have R-charge assignment 2  3 .With this assignment the one-loop beta function vanishes.The center of the gauge algebra Z 2 is not broken by the matter content and the theory has three possible global structures: The holonomy sectors can be organized by (µ, ν) = (±1, ±1) similarly to the cases studied in the previous sections.

r=2
The contributions from the Z µ,ν sectors up to O(x 2 ) are: 3) The indices are: The contributions from the Z µ,ν sectors up to O(x 2 ) are: The indices are:

Frame B
The second frame is a SU (2) 7 gauge theory described by the quiver in (6.7).

SU(2)
SU( 2) SU( 2) All the bifundamental matter fields have R-charge 2  3 .There is a diagonal Z 2 subgroup of the center that is not broken by the matter content.There are three possible .Each node in the quiver diagram corresponds to a SU (n) gauge group, where n is the number of D3 branes that prove the singularity.choices for the global structure: The conformal manifold has complex dimension 21 [14].On a specific point of the conformal manifold the theory describes the low energy theory of two D3 branes in Type IIB probing the tip of the orbifold C 3 /Z (1,2,4) 7 .
In general the theory of n D3 branes that probe this singularity is a SU (n) 7  toric theory described by the dimer in Figure 6.For general n the quiver is oriented while for n = 2 the bifundamental representations are pseudoreal and the theory is described by the unoriented quiver above.In the holographic description the choice of global structure is encoded in the boundary conditions for the Type IIB two-form fields B 2 and C 2 [21,24,25].These are constrained by the five-dimensional CS action: where X 5 is the Sasaki-Einstein manifold whose real cone is . We notice that this is the same CS action as the one for N = 4 SU (n) SYM, therefore the duality orbits of the SU (2) 7 theory will be mapped to the ones of N = 4 SU (n) SYM.In the n = 2 case that we are interested in the three choices of global structure for the N = 4 theory are all dual, therefore we expect that T (B) , T  The contributions to the Z µ,ν sectors of the Lens space index up to O(x 2 ) are: The indices are: The contributions to the Z µ,ν sectors of the Lens space index up to O(x 2 ) are: The indices are: + 63x 4/3 + 70x 2 (6.13)

Frame C
The second frame is a SU (2) 2 ×SU (4) gauge theory described by the quiver in (6.14).4) SU( 2) SU( 2) USp( 6) (6.14)where 6 is the two-index antisymmetric representation of su(4).All the matter fields hare R-charge 2  3 .The theory has three possible global structures: where Z 2 is the diagonal subgroup of the center that is not broken by the matter content. r=2 The contributions to the Z µ,ν sectors of the Lens space index up to O(x 2 ) are: The Lens space indices for the nine theories in the three frames match al low fugacities.This is a nontrivial check of the conformal triality conjectured in [14].Furthermore we propose that once the global properties of the three frames are taken into account all the nine theories that can be built lie on the same orbit.We argued that this can be understood from an holographic point of view in Frame B. It would be interesting to investigate this phenomenon in the other frames, A and C, we leave this discussion to future work.

Conclusions
In this paper we computed the Lens space for a series of models involving U Sp(2n) gauge theories and various degrees of supersymmetry.We found the expected matching among models related by supersymmetric dualities and we also showed that the index does not coincide among different S-duality orbits.In the analysis of the various models involving U Sp(2n) gauge groups we encountered dual phases with orthogonal gauge groups with odd rank.While for the odd case the situation is quite under control in the even case the center is in general of order four and more care is needed in the calculation of the almost commuting holonomies.This problem does not emerge in the models studied in [5] because the matter content breaks the center to an Z 2 subgroup that exchanges the two spinorial representations.The cases where the whole center or a different subgroup is preserved by the matter content has not be studied yet and it requires a separate analysis.The simplest examples corresponds to SO(2n) N = 4 SYM.There are also interesting examples with lower supersymmetry, like the N = 1 S-dual models proposed in [26].Useful hits in this direction can be found in [27].Other N = 4 models that deserve an analysis are the one with exceptional gauge group and without a trivial center, i.e. the one with E 7 and E 8 algebra.A last comment is related to the SU (n) 7 orbifold that gives origin the triality for n = 2.This model for generic n corresponds to a toric quiver gauge theory originating from a stack of D3 branes probing a Calabi-Yau toric threefold.The lattice of charges for the line operators for models of this type (i.e. for toric quiver gauge theories) corresponds to the one obtained for SU (n) N = 4 SYM.This can by shown by explicit analysis on the charge spectrum, as discussed in [28].It should be interesting to explain this behavior from a purely type IIB perspective, along the lines of [21].

Figure 1 :
Figure 1: The three possible choices of line charges for gauge theories with gauge algebra usp(2n).The orange dots represent the charges of the line operators that are included in the corresponding theory.