Contents


Aspects of the moduli space of instantons on and its orbifolds


Alessandro Pini111pinialessandro@uniovi.es and Diego Rodriguez-Gomez222d.rodriguez.gomez@uniovi.es



Department of Physics, Universidad de Oviedo, Avda. Calvo Sotelo 18, 33007, Oviedo, Spain


Abstract

We study the moduli space of self-dual instantons on . These are described by an ADHM-like construction which allows to compute the Hilbert series of the moduli space. The latter has been found to be blind to certain compact directions. In this paper we probe these, finding them to correspond to a Grassmanian, upon considering appropriate ungaugings. Moreover, the ADHM-like construction can be embedded into a gauge theory with a known gravity dual. Using this, we realize in (part of) the instanton moduli space providing at the same time further evidence supporting the duality. Moreover, upon orbifolding, we provide the ADHM-like construction of instantons on as well as compute its Hilbert series. As in the unorbifolded case, these turn out to coincide with those for instantons on .


1 Introduction

It is well-known that instantons are very important configurations in gauge theory. For example, the partition function of gauge theories contains contributions from saddle points of all instanton numbers. This can be made fully precise in the case of supersymmetric gauge theories with eight supercharges, when the supersymmetric partition function can be computed exactly thanks to localization (see [1] for a seminal contribution). One can then explicitly see that, in addition to purely perturbative saddle points, the partition function localizes on instantonic configurations, whose contribution one has to sum. On general grounds, such contributions are the one-loop determinants around each instanton saddle point, which can be computed by the so-called Nekrasov instanton partition function. In turn, in the case of pure gauge theories, the latter coincides with the Hilbert series of the instanton moduli space (see e.g. [2, 3]). Therefore, the construction of instanton moduli spaces, as well as the computation of their associated Hilbert series, is of the greatest importance (of course, the reasons alluded before are just a very limited subset of those making of the instanton moduli space a very interesting object).

In the case of instantons on –or its conformal compactification – the problem of constructing instantons of pure gauge theories333We will concentrate on instantons in pure gauge theories with 8 supercharges throughout all the paper. with gauge group was solved long ago by the ADHM construction [4]. Moreover, it turns out that the ADHM construction has a natural embedding into string theory as it arises as the Higgs branch of the Dp-Dp+4 brane system [5, 6, 7, 8]. In this paper we will be interested on the parallel story but for the case of . As opposed to , is a Kähler manifold. This naturally induces a preferred orientation which distinguishes self-dual (SD) from anti-self-dual (ASD) 2-forms. As a result, the construction of gauge connections with ASD and SD curvatures is intrinsically different. In this paper we will concentrate on SD connections on . In the mathematical literature an ADHM-like construction for such gauge bundles has been developed long ago [9, 10, 11, 12, 13]. Very recently, it has been shown that such construction can be embedded into a gauge field theory which, moreover, admits a string/M theory interpretation [14]. Surprisingly, the gauge theories engineering the ADHM construction for instantons on are gauge theories with supersymmetry –that is, 4 supercharges–. Nevertheless, as shown in [15] (see also [16, 17], and [14] for a discussion in the physics context), the Hilbert series and other properties do indeed satisfy properties compatible with the expected hyperKähler condition of the moduli space.

In this paper we study several aspects of these moduli spaces for SD instantons on . As we briefly review, these are relevant in the computation of the partition function of the twisted gauge theory on . The corresponding Hilbert series were computed in [14], where it was shown that they coincide with the Hilbert series of a “parent” instanton on . Nevertheless, being a topologically non-trivial space, it is natural to expect that our instantons are described by extra topological data. Indeed, the dimension of the moduli space seen by the Hilbert series is smaller than the dimension of the actual moduli space. In this paper we explore the “extra directions”, associated to these extra topological data. With hindsight, the Hilbert series misses these directions as they correspond to a non-compact geometry. Indeed, in the case of unitary instantons, the theory describing these directions is a version of the theory in [18] whose moduli space is a (compact) Grassmanian manifold. Upon appropriately ungauging groups we turn them into non-compact by considering the complex cone over the compact base. In this modified scenario the Hilbert series probes the extra directions finding agreement with the expectations.

The gauge theory containing the ADHM construction of unitary instantons admits a large limit where it is dual to an geometry. It is then natural to study the instanton moduli space in the gravity dual. This provides an interesting cross-check of our results as well as more non-trivial evidence of the proposed dualities.

The ADHM construction for instantons on a given space can be used to find the corresponding construction on related spaces obtained by orbifold projections. In this manner, we find the ADHM construction, as well as the Hilbert series, for moduli spaces of instantons on .

The structure of this paper is as follows: in section 2 we briefly review the relevance of SD instantons on in the computation of the partition function for the topologically twisted gauge theory. In particular, we show how SD instantons on arise as the minima of the localization action, as well as (very briefly) review some relevant aspects of the ADHM construction in the mathematical literature. In section 3 we study unitary instantons on , considering in particular the resolution of the extra directions upon ungauging ’s as well as the description of (part of) the instanton moduli space. In section 4 we consider the construction of unitary instantons on the orbifold space. In section 5 we turn to the symplectic case, finding the ADHM construction of their moduli space on . In section 6 we turn to orthogonal instantons, analyzing, very much like in the unitary case, the compact extra directions associated to the non-trivial topology. Moreover, we provide the construction of orthogonal instantons on the orbifolded space. We provide a short summary of the highlights as well as some conclusions in section 7. Finally, we describe some exotic cases as well as compile some figures (relegated to the appendix in order not to clutter the text) in the appendices.

2 Self-dual instanton contributions to supersymmetric gauge theory on

We are interested on pure gauge theories on . Hence our first task would be the construction of the supersymmetric lagrangian for the theory on the curved manifold. To that matter we follow the approach in [19], which amounts to consider the combined system of supergravity plus the gauge theory of interest. Then, a rigid limit freezes the gravitational dynamics so that we are automatically left with the supersymmetric gauge theory on the curved space. Since we are interested on gauge theories, we will use conformal supergravity as in [20].

Recently, the partition function of supersymmetric gauge theories on was considered in [21]. However, in this paper we will be interested on a different version of the gauge theory. Recall that, in order to find the supersymmetric theory, we need to solve the gravitino variation as well as the auxiliary condition in [20]. These provide both the background fields as well as the Killing spinors for the gauge theory on the curved space. A natural solution to these equations is the topological twist [22]. On general grounds, this amounts to redefining the Lorentz group –generically locally – by twisting either with . Nevertheless, as described in e.g. [23], since for Kähler manifolds the holonomy is really , a second version exists whereby one twists the by the Cartan of the (note that in this case one chirality is privileged over the other by the orientation naturally induced by the Kähler form). While in [21] this later choice was considered, in this paper we will focus on the former version of the topological twist, which can be performed both for positive and negative chiralities of the background Killing spinors.

Setting, to begin with, all supergravity fields other than metric and gauge field to zero, the equations defining the supersymmetric backgrounds are defined by the conformal Killing spinor equation [20] (we refer to this reference for details)

(1)

where the covariant derivative acting on the background Killing spinors is

(2)

while is the gauge field and is the covariant derivative acting on spinors including the spin connection. Moreover the metric of the is

(3)

With hindsight, in this paper we will be interested on keeping the positive chirality spinors. Choosing then

(4)

where is the ’t Hooft symbol and the are the Pauli matrices, we have that the spin connection part in the covariant derivative is cancelled, so that the Killing spinors are simply444We choose a chiral representation for the Dirac algebra, so that .

(4)

Furthermore, one can check that the remaining supergravity equation is solved upon appropriately tuning the supergravity scalar [22].

Following [20], negative chirality spinors could be included choosing a Killing vector of as upon turning on . Let us stick however to the topological case. Then, since the theory is invariant under the supersymmetry generated by the above , we could add to the action the -invariant term , being . The standard argument suggest then that the action is -invariant. A straightforward calculation gives (we set )

(5)

where we have imposed the reality condition [21]. Since eq.(5) is strictly positive, in the classical limit the theory localizes on configurations such that the scalar in the vector multiplet is constant and lies along the Cartan of the gauge group while . Note that, had we chosen to keep negative chirality spinors, we would have obtained . Being more explicit, the condition is, in the conventions of [20], equivalent to555Here .

(6)

that is, must be self-dual (SD). Since, for the standard orientation of the , the Kähler form is also self-dual, we have that the relevant gauge configurations in this case are instantons of the same duality type of the Kähler form. This is precisely the type of instantons described in [14] using the King and Bryan-Sanders constructions in [12, 13] elaborating on [9, 10, 11].

2.1 The construction of self-dual instantons on

While we are interested on constructing self-dual instantons on , it is however more convenient to regard them, upon orientation reversal of the base manifold, as anti-self dual (ASD) instantons on (the opposite-oriented ). Then, we can directly borrow the construction of their moduli spaces from King [12] and Bryan-Sanders [13]. Let us give a lightning overview of the relevant ingredients of the construction and defer to [9, 10, 11, 12, 13] for the detailed account (see also [14] for more references).

On very general grounds, there is a correspondence between the moduli space of instantons on projective algebraic surfaces and the moduli space of (stable) holomorphic bundles which goes under the name of Hitchin-Kobayashi correspondence. In this context, the ADHM construction can be regarded as a device to construct holomorphic bundles over the appropriate manifold.

An alternative version of the Hitchin-Kobayashi correspondence, more useful for our purposes, was proven by Donaldson by using the so-called Ward correspondence, which associates an anti-self-dual (ASD) connection –that is, a connection whose curvature is ASD– on a (not complex) manifold to a holomorphic bundle on an related manifold . Roughly speaking, one regards as a conformal compactification of some underlying complex manifold . Since both the Yang-Mills equations and the self-duality constraints are conformally invariant, solutions with definite duality properties (say ASD) on can be naturally extended into solutions on . Note that, in doing this, the behaviour of the gauge field at the added point must be specified, that is, a framing must be chosen. In particular, we choose a trivial framing, where the gauge transformations become the identity at infinity.

On the other hand, it is well-known that connections with an ASD curvature on a complex manifold are in one-to-one correspondence with holomorphic bundles on .666Roughly speaking, this is due to the fact that the ASD condition on a connection is equivalent to the integrability condition of , hence defining a holomorphic bundle on through the Newlander-Nirenberg theorem. See [9, 10, 11, 12, 13] and [14] for more references. Since the moduli space of the latter is a rather sick notion, being a non-compact space, we can considering a holomorphic compactification of into whereby we add the complex line at infinity and demand the holomorphic bundle to be trivial over there. Hence, all in all, the problem of constructing trivially framed ASD connections on is mapped to the construction of holomorphic bundles over trivial over . The ADHM construction is precisely the device constructing such bundles.

In the case at hand we consider , the blow-up of at a point defined as

(7)

Then, on one hand we can find a conformal compactification of into –the opposite-oriented – as follows

(8)

Note that is not really a complex manifold, as the orientation does not follow from the Kähler form.

On the other hand, we can find a holomorphic compactification by adding which compactifies into blown up at a point, that is, Hirzebruch’s first surface . Hence we have that framed ASD connections over are in one-to-one correspondence with holomorphic bundles over which are trivial over . Since upon orientation reversal, ASD connections on become SD connections on , it follows that the desired moduli spaces are in correspondence with holomorphic bundles over . Then, the ADHM construction is precisely the device to construct such bundles.

While here we will not dive into more details, an instrumental notion in arriving to the actual ADHM construction, from this point of view, is the associated twistor space, which takes into account the sphere bundle of compatible complex structures over . Instead of delving into more intricacies, here we will describe the ADHM-like description of instantons for unitary, orthogonal and symplectic gauge groups embedded in a gauge theory as in [14], and refere to [9, 10, 11, 12, 13] for the details of their construction along the lines outlined here.

On word of caution is in order. Even though in the following we will loosely refere to instantons on , the previous description of the precise construction should be borne in mind –that is, we are describing SD instantons on or equivalently ASD instantons on –.

3 instantons on

As described in [14], the King construction [12] for unitary instantons on can be embedded into a quiver gauge theory. The theory in question is a gauge theory whose quiver is in the left panel of fig.1, supplemented with the superpotential

(9)
Figure 1: Quiver diagram for instantons on (on the left) and for instantons on (on the right).

Note that the chiral nature of the theory demands, because of the parity anomaly, the gauge nodes to have a non-vanishing Chern-Simons level and respectively, where are integers including zero. In the following we will concentrate on the case .

As a gauge theory, it has been argued [24, 25] that the theory flows to an IR fixed point, where the charges of the fields are listed in table 1.

Fields
1/2
1/2
1/4
1-1/4
1/4
term 1
Table 1: Transformations of the fields for the quiver gauge theory. Here is an unknown real parameter whose value, nevertheless, does not affect subsequent results.

For the paticular case , as argued in [25], the mesonic moduli space (excluding “Higgs-like” directions where fundamental fields take a VEV) of the theory is the direct product of a conifold times the complex line. In general, as is increased, this geometric branch of the moduli space becomes an increasingly more involved toric manifold (see [25]).

The instanton moduli space of interest is that of instantons on , denoted as . It arises as a Higgs-like branch of the full moduli space of the gauge theory dubbed as instanton branch where fundamental fields take a VEV. Note that the instanton gauge group appears as the flavor symmetry of the ADHM construction. Note as well that, in order to specify the instanton, in general a set of numbers including the instanton number is required. We will come back to this issue below.

More precisely, as described in [14], the instanton branch of the moduli space arises when we set (as well as all monopole operators, typically denoted by ) to zero. It is important to note that the truncation is consistent with the quantum constraint on the moduli space introduced in [25]. Then, the only relevant F-term arises from the superpotential and reads

(10)

Together with the field content and gauge groups of the gauge theory, this constraint precisely realizes the King construction. Note that, even though the flavor symmetry is , the part is really gauged. Hence we can think of our instantons as instantons of (even though, as we will review below, we should really think of ).

In the following we will be interested on the Hilbert series of the instanton moduli space. The ADHM construction just introduced (and the corresponding orthogonal and symplectic versions in addition to their orbifoldings to be described below) allows to compute it using by now standard methods as in e.g. [14, 26, 27, 28] (see also [29] for the study of instantons on ). Let us pause to make a point on notation. Through all the paper we will denote the Hilbert series of the instantons moduli space as , being the integers characterizing instanton, which appears as the date of gauge group of the ADHM construction; those characterizing the instanton gauge group appearing as flavour group in the ADHM construction and the ambient manifold of the instanton.

As anticipated, in order to specify a particular instanton on a set of quantum numbers is required. It is clear that one such integer is the instanton number. However, since is a topologically non-trivial manifold, it is natural to expect that instantons on might carry extra quantum numbers. Indeed, as reviewed in [14] following [15], we can characterize the instanton by its first Chern number and its instanton number . Using the correspondence between ASD connections on and holomorphic bundles on , these can be written as

(11)

being the class inside –recall that in this case and –. These, in turn, are related to the quiver data as follows

(12)

As an algebraic variety, can be mapped into the moduli space of a related instanton on –described by the Higgs branch of the theory on the right panel of fig.1– in the following way

(13)

being the fields of the quiver diagram for theory. Indeed, if we multiply the F-term relation (10) by and we apply the map (13) we recover the F-term for instantons on

(14)

In turn, the inverse map can also be defined as

(15)

Let us momentarily consider the case where , which corresponds to and . From the construction in eq.(13), it is clear that the integer in the quiver on the right panel of fig.1 is identified with . Thus, we have that as an algebraic variety, the moduli space of instantons on is identified with the moduli space of instantons on . Consistently, the Hilbert series of these instantons coincide, from which it follows that .

In the general case , one finds that the above construction still holds upon setting . Consistently, as described in [14], the Hilbert series corresponding to the instanton branch of the quiver on the left panel in fig.1 coincides with the Hilbert series of the Higgs branch of the quiver on the right panel of fig.1, that is

(16)

where is the fugacity of the R-charge, the fugacity associated with the global symmetry and y are the fugacities associated with the global symmetry. Note that the fugacity associated to -charge is re-scaled from in the case into in the case.

Naively, eq.(16) suggests that the dimension of the moduli space of unitary instantons on is

(17)

Note that, even though the quiver is specified by three integers , eq.(17) is only sensitive to two of them. However, it is possible to consider an extended notion of the moduli space where the extra directions associated to all the three quantum numbers specifying the instanton are taken into account. This is the so-called resolved (as the extra directions are discerned) moduli space, denoted as , whose dimension is [15, 16, 17]

(18)

Note that for the dimension of is equal to the dimension of . This suggests that is really a modulo quantity corresponding to an instanton gauge group which is really . We warn the reader that, while in the following we will not clutter notation by supresing the , the global properties of the gauge group must be kept on mind.

3.1 The resolved moduli space and the Grassmanian

In order to explore the resolved moduli space it is instructive to first consider the simplest case where . The theory simplifies into a one-noded quiver flavored only with fundamental fields (and not antifundamentals) shown in fig.2. Recall that the CS level is adjusted so as to cancel the parity anomaly, and, furthermore, there is no superpotential.

Figure 2: Quiver diagrams for Grassmanian (we show the dual pair –see text–).

The leftover theory in this particular case corresponds to a version of the theory considered in [18]. Then, as argued in that reference, the moduli space is a complex Grassmanian (compact) manifold, consistently with the expectations in [15, 16, 17].

We can now understand why is insensitive to these extra directions, as, forming a compact Grassmanian manifold, the Hilbert series is blind to them. Indeed, since in the theory in fig.2 the gauge group is , the Higgs-like moduli space is empty, as no gauge-invariant can be constructed out of fundamental fields. Consistently, formula (17) gives a zero-dimensional moduli space. However, as in [30], we can consider a version of the theory where only the non-abelian part of is gauged, while the is kept as a global baryonic symmetry (alternatively, we could think of this as the master space [31] of the theory). In this case we can form baryon-like gauge-invariant operators, thus finding a non-empty moduli space which in fact is a complex cone over the Grassmanian. It is straightforward to compute the Hilbert series and read-off the dimension of the moduli space from the pole at , finding

(19)

Recalling that the is due to the over which we are not integrating over –resulting in moduli space which is a complex cone over the Grassmanian–, we find a result in accordance with eq.(18).

Eq.(19) is invariant under the exchange . Indeed, one can explicitly check that the Hilbert series of the theories with gauge group and are identical, thus suggesting a duality among these theories. Such duality is also suggested by the brane construction in [18]. In that reference, in a IIA system consisting on an brane and a - D4 branes intersection, D2 branes are stretched along direction between the and the -D4 intersection. Then, the D4’s can be broken on the and, say, the lower part of them can be sent to infinity. As argued in [18], the gauge theory on the D2’s is precisely the version of the gauge theory in the first panel in fig.2. Upon T-duality along , this system engineers the actual gauge theory of interest, namely that in the first panel of fig.2. Explicitly, the system contains

  • An brane along .

  • A braneweb with an brane along meeting D5 branes along and emanating a fivebrane.

  • D3 branes along , starting at the braneweb junction and ending on the .

Note that the -D4 intersection in the IIA system becomes a braneweb in the IIB system, as D5-branes meeting an give rise to a fivebrane. In fact, it is precisely this bending what gives the expected CS level in the gauge theory [32, 33]. In this it is important to recall that the D3’s meet the fivebranes right at the junction, as this is what makes the theory to contain only fundamental (and not antifundamental) matter [18] which in turn generates the CS level.

We can now imagine crossing the to the other side. Then, due to the Hanany-Witten effect, the final configuration contains D3 branes but is otherwise identical, consistently with our finding that the two theories in fig.2 yield to the same Hilbert series (for a more detailed account of the duality in the case we refere to [18]).

Coming back to the general discussion, in view of the case it is natural to guess that ungauging the abelian part of the largest gauge symmetry will allow us to resolve the extra directions in . To that matter, let us now consider the case . Writing the remaining gauge group as , we can compute the Hilbert series upon integration only over the non-abelian part. Reading the dimension of the moduli space from the order of the pole at , from explicit computations for and and , we find

(20)

which is precisely the expected result (18). Unfortunately, explicitly checking higher rank cases is technically challenging. Nevertheless, it would be very interesting to perform further checks for higher ranks.

3.2 Rank one and AdS/CFT

In the particular case of , upon setting and for , the theory engineering the moduli space of unitary instantons on becomes exactly that found in [25] to describe M2 branes probing , the direct product of a conifold times the complex line. The metric of the cone can be written as

(21)
(22)

Then, on general grounds, the near-brane geometry for a stack of M2 branes probing this cone is , which, in global coordinates, can be written as

(23)

being the radius of the space. Besides, there is a 6-form flux whose field strength integrates to on . Hence, in the large limit, the gauge theory is holographically dual to with units of flux through . It is thus natural to wonder whether, at least partially, the moduli space of unitary instantons on can be geometrically realized in this context.

As discussed in [25], the gauge theory contains a mesonic branch of the moduli space which realizes the dual geometry. In general, it is natural to expect that the holographic dual captures gauge theory operators made out of bifundamental fields, while those corresponding to fundamental matter would require extra multiplets on top of the to account for the “flavor brane open string” degrees of freedom. Hence, it is natural to expect that the subbranch of the instanton branch involving just fields is visible in the geometry. This is indeed analogous to the cases discussed in [28, 34], where only the “closed string fields” in the quiver are captured by the gravity dual.

More explicitly, following [28, 34], it is natural to expect that this subbranch of the instanton branch is captured by dual giant graviton branes moving in the appropriate subspace corresponding to the instanton branch. To that matter, we consider a probe M2 brane wrapping , where is the sphere inside the . Moreover we assume that and , while

(24)

The action for such probe brane is

(25)

which becomes

It is easy to convince oneself that the equations of motion fix (for simplicity, from now on we set ). Then, Legendre transforming to the Hamiltonian we obtain

The minimum energy configurations are

(26)

for which

(27)

Both configurations are degenerated in energy, one corresponding to pointlike gravitons and the other to true dual giant gravitons. The energy is

(28)

Coming back to the solution in eq.(26), we can parametrize the phase space of the spinning M2 as a dynamical system by the coordinates and the conjugated momenta . Moreover the conjugated momenta must obey the following constraints

As usual, the matrix encodes the symplectic form associated to the phase space of our dynamical system as ( stands for Dirac brackets). Deleting the row and column corresponding to the trivial coordinate, we find

Therefore the symplectic structure reads

Integrating we obtain

(29)

Hence, upon introducing , we just recover the data of . Following [28, 34], we can do symplectic quantization of this dynamical system. On general grounds, that amounts to identify the holomorphic functions on the phase space –in this case – with the allowed wavefunctions. These can easily be counted, simply obtaining the Hilbert series for .

Let us now turn to the gauge theory. As discussed, we expect our probe branes to be dual to operators on the instanton branch not containing fundamental fields. These are of the schematic form

(30)

Note that the F-terms imply that the indices are completely symmetrized; that is, the operators are in a spin representation of the global symmetry rotating the ’s. Hence, for a fixed charge , the number of operators is , and the corresponding generating function is just , which is precisely the Hilbert series –here is a generic fugacity–.

We can explicitly compare the gauge theory operators with our probe brane configurations in the gravity side. To that matter, let us first note that exactly the same configuration in the gravity side would have been obtained fixing and having our brane orbiting respectively. Hence, in all our formulas we can trade for . In particular, eq.(28) becomes .

In order to compare our probe branes with the gauge theory operators we need to identify charges. It is reasonable to guess that the momentum along is proportional to the R-symmetry. Hence let us identify , being (not to be confused with the arbitrary integer in table 1) proportional to the charge under the in way to which we will shortly come back. Moreover, in order to understand the momenta, it is instructive to consider momentarily removing the quarks from the gauge theory. It then exhibits an global symmetry, rotating respectively the and fields. Then, the quark multiplets break the down to a , while the rotating the ’s remains as a global symmetry. We identify the charge, denoted as , with as . With no loss of generality, let’s assume , which corresponds to the choice . Then translates into . Analogously, we identify with the Cartan of the denoted .

Note that eq.(26) translates into , and therefore . Let us compare this with the gauge theory operators (30). Using table 1, the charges of the operators in the expression (30) are and . As expected, being chiral operators, they satisfy the usual relation . Moreover, it is clear that , so that . Comparing the ranges for in gravity and field theory we find the identification

(31)

Turning now to the energy for our branes, we find , which, upon using eq.(31), becomes , precisely as expected for chiral operators.

Moreover, we can explicitly fix the value of . To that matter, let us turn to the field theory operators and consider the highest weight state, which corresponds to . For this one , while . In turn, from the gravity side, the brane with highest is . Since this must correspond to , we find . Hence, this implies .

We can offer an alternative test of our identifications. To that matter, let us consider metric fluctuations polarized along the internal manifold. On general grounds, these fluctuations correspond to operators of the schematic form , being the stress-energy tensor of the theory. Note that, for the particular case when the inserted operator is one of those in eq.(30), we expect that the dimension is . In turn, these fluctuations satisfy the Klein-Gordon equation in . For a of the form this problem was considered in [35], where it was shown that the dimension of the dual operators can be written in terms of the eigenvalues of the scalar laplacian on . In turn, borrowing the results from [36], the eigenvalues of the scalar laplacian on the conifold are

(32)

where are respectively, the total spin and the charge along the direction. For the operators in eq.(30) we have that . In turn, the charge must satisfy . Focusing on the highest weight state, we would require , which is nothing but as seen before. Then, using [35],

(33)

This precisely coincides with our expectations upon identifying . This can be written as , which becomes upon using the identification advocated above.

So far we have considered the case . It is natural to expect that can be accommodated into the gravity dual by adding non-vanishing flat over a 2-cycle in the internal manifold [37]. Nevertheless, such modification of the background would not change our computation. Hence we would find the same result even for the case , in agreement with the field theory result where the Hilbert series only depends on .

4 instantons on

A natural generalization of the ADHM construction of instantons on is to consider orbifolding the ambient manifold upon quotienting by a subgroup of its symmetries. In particular, since is invariant under a action corresponding to the coordinates in eq.(3), it is natural to consider quotienting such symmetry by some discrete subgroup of it. Note that the spinors in eq.(4) are constant and morever annihilated by ( are the Lorentz generators in tangent space indices ). Therefore we can consider a orbifold of the direction whereby we restrict . In the rest of the paper we will be interested on the ADHM construction of instantons on these orbifolded spaces. To that matter, we will take as starting point the ADHM construction in the unorbifolded case, on which we will implement the orbifold by standard methods [8].

Let us consider the case of unitary instantons presented above. In order to find the orbifolded theory we first need to identify the transformation properties of the fields. These read:

  • The fields (with ) in the bifundamental representation.

    (34)
  • The fields and in the bifundamental representation.

    (35)
  • The fields and .

    (36)

where the matrices , and are given by