The ${\mathcal{N}}_3=3\to {\mathcal{N}}_3=4$ enhancement of super Chern–Simons theories in $D=3$, Calabi HyperKähler metrics and M2-branes on the $\mathcal{C}\left({\mathrm{N}}^{0,1,0}\right)$ conifold

Abstract

Considering matter coupled supersymmetric Chern–Simons theories in three dimensions we extend the Gaiotto–Witten mechanism of supersymmetry enhancement ${\mathcal{N}}_3=3\to {\mathcal{N}}_3=4$ from the case where the hypermultiplets span a flat HyperKähler manifold to that where they live on a curved one. We derive the precise conditions of this enhancement in terms of generalized Gaiotto–Witten identities to be satisfied by the tri-holomorphic moment maps. An infinite class of HyperKähler metrics compatible with the enhancement condition is provided by the Calabi metrics on $T^{\star }{\mathbb{P}}^n$. In this list we find, for $n=2$ the resolution of the metric cone on ${\mathrm{N}}^{\mathrm{0,1,0}}$ which is the unique homogeneous Sasaki–Einstein 7-manifold leading to an ${\mathcal{N}}_4=3$ compactification of M-theory. This leads to challenging perspectives for the discovery of new relations between the enhancement mechanism in $D=3$, the geometry of M2-brane solutions and also for the dual description of super Chern–Simons theories on curved HyperKähler manifolds in terms of gauged fixed supergroup Chern–Simons theories.

Introduction

Matter coupled Chern–Simons gauge theories are of interest both as challenging paradigms in quantum field theory and as theoretical models for the description of certain condensed matter systems.

At the dawn of the new millennium Chern–Simons matter coupled theories rose to prominence in association with the ${\mathrm{AdS}}_4∕{\mathrm{CFT}}_3$ gauge–gravity correspondence. Indeed, after the discovery of the ${\mathrm{AdS}}_5∕{\mathrm{CFT}}_4$ correspondence [20], [21], [22], [31], [36], the program of the ${\mathrm{AdS}}_4∕{\mathrm{CFT}}_3$ was an obvious development where all the results of Kaluza–Klein supergravity, accumulated at the beginning of the eighties could be recycled. In the years 1998–2000 a rush started to complete the derivation of the Kaluza–Klein spectra for all the compactifications of type ${\mathrm{AdS}}_5\times {(\mathrm{G}∕\mathrm{H})}_5$ and ${\mathrm{AdS}}_4\times {(\mathrm{G}∕\mathrm{H})}_7$. The idea was to compare such spectra with the towers of primary conformal fields in the dual gauge theory either in $D=4$, for type IIB D3-branes, or in $D=3$ for M2-branes. The case of the coset ${\mathrm{T}}^{\mathrm{(1,1)}}\equiv \frac{{\mathrm{SU(2)}}_{\mathrm{I}}\times {\mathrm{SU(2)}}_{\mathrm{II}}}{\mathrm{U(1)}}$, where the denominator group is the diagonal of the standard ${\mathrm{U}}_{\mathrm{I,II}\left(1\right)\subset S{U}_{I,II}\left(2\right)}$ was studied and the results were published in the month of May 1999 [12]. The case of the Sasakian homogeneous seven manifolds listed in Table 1 was actively studied and the results were published in [18], [19], [25], [37].

This was one side of the correspondence: that of supergravity. The other side, that of the gauge theory, required the determination of suitable candidates. The first mile-stone in this direction came in 1998 with the paper by Klebanov and Witten [35] where the geometrical description as a Kähler quotient of the metric cone $\mathcal{C}\left({\mathrm{T}}^{\mathrm{(1,1)}}\right)$ on the coset ${\mathrm{T}}^{\mathrm{(1,1)}}$, denominated by them the conifold, was discussed. Indeed, the main point of [35] was the identification of the pivot role of the Kähler quotient in singling out the field content and the interactions of the dual gauge theory on the brane world-volume. In the case of ${\mathrm{T}}^{\mathrm{(1,1)}}$, the metric cone $\mathcal{C}\left({\mathrm{T}}^{\mathrm{(1,1)}}\right)$ can be described as the Kähler quotient of ${ℂ}^2\times {ℂ}^{2\star }$ with respect to a single $\mathrm{U(1)}$. So Klebanov and Witten outlined a pattern that, about one year later and in presence of all the accomplished Kaluza–Klein spectra for the relevant Sasakian manifolds, was generalized to the case ${\mathrm{AdS}}_4∕{\mathrm{CFT}}_3$ in [17].

In all cases the (Hyper)-Kähler quotient description of the metric cone on a (tri)-Sasakian manifold is the starting point for the construction of the dual gauge theory on the brane world-volume. The coordinates of the linear space of which we perform the quotient are the (hyper/Wess–Zumino)-multiplets and the color gauge group is accordingly singled out by the quotient. Having singled out the principles for the second side of the correspondence, the explicit construction of the dual gauge theories became possible together with the definition of all the towers of conformal primaries to be compared with the Kaluza–Klein spectra.

The ${\mathcal{N}}_3=4,3$ gauge theories¹ can be identified as special subclasses of ${\mathcal{N}}_3=2,D=3$ gauge theories, whose general form was described in [28], [32] as well as [19] for linear representations, and was generalized to arbitrary Kähler and HyperKähler manifolds in [24], which introduced also a more compact and geometrical notation for the Lagrangian. Utilizing the off-shell formulation of ${\mathcal{N}}_3=2,D=3$ gauge theories of [19], [24], in [7] it was advocated that in the infrared strong coupling limit the gauge coupling constant $g^2$ goes to infinity while the dimensionless Chern–Simons coupling constant $\alpha$ stays finite. In this way all kinetic terms of the fields belonging to the gauge multiplets are suppressed and the latter fields can be integrated out leaving an ${\mathcal{N}}_3=3$ matter coupled Chern–Simons gauge theory whose superpotential has the following very special form:

$$\mathcal{W}=-\frac{1}{8\alpha }{\mathcal{P}}_{\Lambda }^{+}{\mathcal{P}}_{\Sigma }^{+}{\mathfrak{m}}_{\Lambda \Sigma }$$

where ${\mathcal{P}}_{+}^{\Lambda }$ denote the holomorphic part of the moment-maps for the triholomorphic action of the gauge group generators $T^{\Lambda }$ on the HyperKähler manifold ${HK}_{2n}$ spanned by the hypermultiplets. The gauge group is generically denoted $\mathcal{G}$, its Lie algebra is denoted $\mathbb{G}$ and ${\mathfrak{m}}_{\Lambda \Sigma }$ is an invariant non-degenerated quadratic form on $\mathbb{G}$. As we stress later on, ${\mathfrak{m}}_{\Lambda \Sigma }$ is not necessarily the Cartan–Killing form and it is not necessarily positive definite. The full scalar potential for these theories takes the form:

$$V_{scalar}=\frac{1}{6}\left({\partial }_i\mathcal{W}{\partial }_{j^{\star }}\overline{\mathcal{W}}{g}^{i{j}^{\star }}+{\mathbf{m}}^{\Lambda \Sigma }{\mathcal{P}}_{\Lambda }^3{\mathcal{P}}_{\Sigma }^3\right)$$$${\mathbf{m}}^{\Lambda \Sigma }(u,v)\equiv \frac{1}{4{\alpha }^2}{\mathfrak{m}}^{\Lambda \Gamma }{\mathfrak{m}}^{\Sigma \Delta }{k}_{\Gamma }^i{k}_{\Delta }^{j^{\star }}{g}_{i{j}^{\star }}$$

where ${\mathcal{P}}_{\Sigma }^3$ are the real components of the tri-holomorphic moment maps for the action of $\mathcal{G}$ on the HyperKähler manifold ${HK}_{2n}$, while $g_{i{j}^{\star }}$ denotes the components of its HyperKähler metric $\mathbf{g}$ and $k_{\Gamma }^i,k_{\Gamma }^{j^{\star }}$ are the components of the Killing vectors generating $\mathcal{G}$. Indeed the metric $\mathbf{g}$ must admit the gauge group $\mathcal{G}$ as isometry group.

In 2007 Bagger and Lambert presented their version of the ${\mathcal{N}}_3=8$ Chern–Simons theory [4], [5], [42]. Their work allowed us to understand how $\mathcal{N}>3$ enhancements might arise starting from an ${\mathcal{N}}_3=3$ model. Few months after this discovery, all the formulations with $4\le \mathcal{N}\le 8$ were constructed, utilizing the mechanism of gaussian integration of the physical fields of the vector multiplets, originally introduced for the case of the compactification on the tri-Sasakian ${\mathrm{N}}^{\mathrm{0,1,0}}$ manifold in [7]. Supersymmetric Chern–Simons theories were completely classified in the case when the scalar sector parameterizes a flat manifold. The key point was to understand how to specialize the ${\mathcal{N}}_3=3$ theory in order to enhance the R-symmetry.

An interesting construction is that presented by Gaiotto and Witten in [27], which was later further developed also in [29]. Their starting point is an ${\mathcal{N}}_3=1$ theory with the field content of an ${\mathcal{N}}_3=3$ one. Adding a suitable superpotential the theory becomes ${\mathcal{N}}_3=3$ supersymmetric. By means of a restriction imposed on the superpotential one obtains an ${\mathcal{N}}_3=4$ supersymmetric theory. Further restrictions lead to higher $\mathcal{N}$-extended supersymmetric theories, see e.g. [30]. An important feature is that these restrictions are equivalent to suitable choices of the gauge group and of the matter representation.

The setup of [7] shows that for general groups and general couplings the Chern–Simons interactions break R-symmetry from $SO\left(4\right)$ to $SO\left(3\right)$ and consequently also supersymmetry from ${\mathcal{N}}_3=4$ to ${\mathcal{N}}_3=3$, as we already explained.

Yet one can try to specialize the theory in order to recover $SO\left(4\right)$ R-symmetry and this is the main issue of the present paper.

Another important discovery was made by Gaiotto and Witten, in particular dealing with the case when the hypermultiplets span a flat target manifold. They found that the enhancement to $\mathcal{N}\ge 4$ supersymmetry implies also the existence of a Lie super-algebra $\mathfrak{G}$ whose bosonic part is the Lie algebra $\mathbb{G}$ of the gauge group $\mathcal{G}$. This issue was thoroughly investigated in [14]. The authors of this paper worked directly with the formulation of the super Chern–Simons matter coupled theories obtained after the elimination of the non-dynamical fields and with the final superpotential written in terms of dynamical fields. They showed that the crucial issues for the supersymmetry enhancement are the following:

• suitable choices of the gauge group $\mathcal{G}$ with its related Lie algebra $\mathbb{G}$ which is not necessarily semisimple, rather it typically also involves abelian $\mathfrak{u}\left(1\right)$ factors,

• suitable choices of complex or symplectic linear representations $\mathcal{D}\left(\mathbb{G}\right)$ to which the scalar multiplets are assigned,

• a suitable choice of a non-degenerate, yet not positive definite $\mathcal{G}$-invariant metric ${\mathfrak{m}}_{\Lambda \Sigma }$ on the Lie algebra $\mathbb{G}$.

In all instances classified in [14], the above enumerated choices correspond to the embedding $\mathbb{G}\hookrightarrow \mathfrak{G}$ of the bosonic Lie algebra into a super-Lie algebra, the representations of the scalar multiplets being the same of the fermionic generators of $\mathfrak{G}$. In certain cases the metric $\mathfrak{m}$ is the restriction to the bosonic generators of the super Cartan–Killing metric of $\mathfrak{G}$.

This provides a challenging Occam’s razor in the classification of supersymmetric Chern–Simons theories. Indeed, this brings us to another interesting feature discovered by Kapustin and Saulina [33]. These authors showed that the same Lie super-algebra $\mathfrak{G}$ can be used to construct a Chern–Simons supergauge theory, namely a pure Chern–Simons theory whose gauge group is the supergroup $\mathfrak{G}$. Quantizing such a topological theory à la BRST and introducing the ghosts for the fermionic part of the supergauge symmetry, after topological twist, these latter can be identified with the matter multiplets of the standard supersymmetric Chern–Simons theory of the bosonic subalgebra $\mathbb{G}\subset \mathfrak{G}$.² This relation between the ${\mathcal{N}}_3=4$ supersymmetric Chern–Simons theory and the supergroup Chern–Simons one, described by Kapustin and Saulina, is somehow reminiscent of the relation between the Neveu–Schwarz and the Green–Schwarz formulations of superstrings, where one trades world volume supersymmetry for supersymmetry in the target space. Kapustin and Saulina advocated that the supergroup formulation is helpful to build supersymmetric Wilson-loops [15]. Similar topics were also investigated later in [38] and [39].

The supergroup Chern–Simons formulation is well established in the flat scalar manifold case. Instead, what might be the relevant supergroup $\mathfrak{G}$ and what might be its role in ${\mathcal{N}}_3=4$ enhanced super Chern–Simons theories with curved HyperKähler target spaces is not clear yet. This issue will be addressed in future publications [1].

Indeed, as already noticed, supersymmetric Chern–Simons theories were mostly constructed assuming that the scalar sector parameterizes a flat Kähler manifold which, in the $\mathcal{N}\ge 3$ has to be HyperKähler. More general cases with curved HyperKähler manifolds were only sketched. In the formulation of [24] the scalar fields parameterize a generic Kähler or HyperKähler manifold and the gauge group is the isometry group of such a manifold. In addition, one has suitable superpotential functions.

The goal of the present paper is to show that, within the more general setup of [24], where the hypermultiplets span generic HyperKähler manifolds $H{K}_{2n}$, Chern–Simons ${\mathcal{N}}_3=3$ gauge theories are enhanced to ${\mathcal{N}}_3=4$ supersymmetry, if and only if the tri-holomorphic moment-maps ${\mathcal{P}}_{\Lambda }^{\pm ,3}$ of the $H{K}_{2n}$ isometry group $\mathcal{G}$ (the gauge group),³ satisfy the following differential–algebraic constraints:

$${\partial }_i\left({\mathcal{P}}^{+}\cdot {\mathcal{P}}^{+}\right)={\partial }_{\overline{\ell }}\left({\mathcal{P}}^{-}\cdot {\mathcal{P}}^{-}\right)=0$$$${\partial }_i\left({\mathcal{P}}^{+}\cdot {\mathcal{P}}^3\right)={\partial }_{\overline{\ell }}\left({\mathcal{P}}^{-}\cdot {\mathcal{P}}^3\right)=0$$$${\partial }_{\overline{\ell }}\left({\mathcal{P}}^{+}\cdot {\mathcal{P}}^3\right)={\partial }_i\left({\mathcal{P}}^{-}\cdot {\mathcal{P}}^3\right)=0$$

together with

$${\partial }_i{\partial }_{\overline{\ell }}\left(2{\mathcal{P}}^3\cdot {\mathcal{P}}^3-{\mathcal{P}}^{+}\cdot {\mathcal{P}}^{-}\right)=0.$$

In the above formulae the scalar product is taken with respect to the previously mentioned non-degenerate invariant metric ${\mathfrak{m}}_{\Lambda \Sigma }$, whose signature is not necessarily positive (or negative) definite.

These constraints are our main result and they are supposed to generalize Section 3.2.2 of the Gaiotto–Witten paper [27], which analyzes a Chern–Simons coupling to a ${\mathcal{N}}_3=4$ non-linear sigma model (NLSM) with a generic (i.e. curved) HyperKähler target space. Such a coupling generically breaks supersymmetry to ${\mathcal{N}}_3=3$, however if the above constraints are satisfied, ${\mathcal{N}}_3=4$ supersymmetry remains unbroken. These constraints are a weaker formulation of the conditions introduced by Gaiotto and Witten, since they are their derivatives. For flat target spaces (of the matter system coupled to Chern–Simons), which is the main focus of [27], the stronger Gaiotto–Witten constraints are valid. However, when the HyperKähler target space of the NLSM is curved, the above weaker formulation of the constraints is sufficient to guarantee $\mathrm{SO}\left(4\right)$ R-symmetry of the Lagrangian and hence ${\mathcal{N}}_3=4$ supersymmetry. Of course, this does not mean that there are no curved target spaces that would satisfy the stronger version of the constraints. In fact, examples of such target spaces were provided in Section 3.2.3 of [27]. However, we will show shortly that there are also examples of target spaces that do not satisfy the stronger Gaiotto–Witten constraints, but they do satisfy our weaker formulation (even if the violation of the strong constraints is mild in all cases known to us). Nevertheless, such models still have ${\mathcal{N}}_3=4$ supersymmetry.

Now, when the constraints (1.3)–(1.4) have been established the obvious question is which examples do we know of non-trivial HyperKähler manifolds endowed with continuous isometries whose moment maps satisfy these constraints? The first example was noted by Kapustin and Saulina in [33] and it is provided by the Eguchi–Hanson space (in the same paper a generalization is proposed, where the target space takes the form $T^{\ast }(G∕T)$ with $T$ the maximal torus of $G$). This HyperKähler manifold is $T^{\star }{\mathbb{P}}^1$, namely the total space of the cotangent bundle to the one-dimensional complex projective space: ${\mathbb{P}}^1\sim {\mathbb{S}}^2$. The isometry group acting tri-holomorphically on the corresponding Ricci flat HyperKähler metric is $\mathrm{SU(2)}$ and in Appendix A we review the appropriate calculation of its moment maps, showing that they satisfy the necessary constraints for enhancement. To be precise, the HyperKähler quotient construction (at level zero) yields the singular space ${ℂ}^2∕{\mathbb{Z}}_2$. This space satisfies the strong version of the Gaiotto–Witten constraints. As a next step, ${ℂ}^2∕{\mathbb{Z}}_2$ can be blown-up to the Eguchi–Hanson space $T^{\ast }{\mathbb{P}}^1$ by choosing a non-zero level $\kappa$. Then the constraints (1.3) still remain valid in the strong version (i.e. without derivatives) but the strong version of the constraint (1.4) no longer vanishes, it is rather proportional to $\kappa$. However, it clearly vanishes after applying the derivatives in (1.4) and hence the NLSM with target space $T^{\ast }{\mathbb{P}}^1$ coupled to a Chern–Simons term still has ${\mathcal{N}}_3=4$ supersymmetry.

Actually the Eguchi–Hanson manifold is the first in an infinite series of HyperKähler manifolds, i.e. the $T^{\star }{\mathbb{P}}^n$ manifolds, endowed with the Calabi HyperKähler metrics that were explicitly constructed in [13], using a Maurer–Cartan differential form approach. Such a construction is reviewed and applied to the case of interest to us in Section 4. Indeed we make the conjecture that the enhancement constraints (1.3)–(1.4) hold true for the $\mathrm{SU(n\; +\; 1)}$ isometry of the Calabi HyperKähler metric on $T^{\star }{\mathbb{P}}^n$ for all values of $n\in \mathbb{N}$ and in Appendix C.1 we explicitly prove our conjecture for the case $n=2$.

In contrast to [27], we do not have an interpretation of the constraints (1.3)–(1.4) in terms of Lie superalgebras at the moment. Note, that for our class of examples $T^{\ast }{\mathbb{P}}^n$ (and in particular the two special cases – the Eguchi–Hanson space and $T^{\ast }{\mathbb{P}}^2$ – for which we perform computations in detail) we need to gauge a simple group, namely $\mathrm{SU}(n+1)$, in order to satisfy the constraints (1.3)–(1.4) and thus achieve supersymmetry enhancement. This simple group is not the bosonic subgroup of any non-trivial Lie supergroup. This is unlike the scenario in the work of Gaiotto–Witten, where the gauge group is the bosonic subgroup of the supergroup series $\mathrm{SU}\left(n\right|m)$ or $\mathrm{OSp}\left(n\right|m)$, i.e. it is formed by two simple factors (up to a possible $\mathrm{U}\left(1\right)$).

The Calabi metric on $T^{\star }{\mathbb{P}}^2$ is not a randomly chosen case rather it has a profound physical relevance. Indeed it corresponds to the resolution of the conic singularity at the tip of the metric cone $\mathcal{C}\left({\mathrm{N}}^{\mathrm{0,1,0}}\right)$. As displayed in Table 1, the coset manifold ${\mathrm{N}}^{\mathrm{0,1,0}}$ is the unique tri-holomorphic, homogeneous Sasaki–Einstein manifold that exists in $7$-dimensions. Somehow, as we already remarked above, ${\mathrm{N}}^{\mathrm{0,1,0}}$ is the $7$-dimensional analogue of the Sasaki–Einstein homogeneous manifold ${\mathrm{T}}^{\mathrm{1,1}}$ in $5$-dimensions. It leads to a compactification of M-theory on

$${ℳ}_{11}={\mathrm{AdS}}_4\times {\mathrm{N}}^{\mathrm{0,1,0}}$$

which preserves ${\mathcal{N}}_4=3$ supersymmetry and whose Kaluza–Klein spectrum was explicitly calculated and organized into $\mathrm{OSp(3|4)}\times \mathrm{SU(3)}$ supermultiplets in [7], [25]. The metric cone $\mathcal{C}\left({\mathrm{N}}^{\mathrm{0,1,0}}\right)$ has a description as a HyperKähler quotient of ${ℂ}^3\times {ℂ}^{3^{\star }}$ with respect to the tri-holomorphic action of a $\mathrm{U(1)}$ group. The correct dual ${\mathcal{N}}_3=3$ gauge theory on a single M2-brane (or $N$ M2-branes if $\mathrm{U}\left(1\right)\to \mathrm{U}\left(N\right)$) probing the transverse space $\mathcal{C}\left({\mathrm{N}}^{\mathrm{0,1,0}}\right)$ was proposed in [26]. All this is just synoptic with the Klebanov–Witten construction of the conformal gauge theory dual to the ${\mathrm{AdS}}_5\times {\mathrm{T}}^{1,1}$ compactification of type IIB supergravity [35]. There the metric cone $\mathcal{C}\left({\mathrm{T}}^{\mathrm{1,1}}\right)$ is described as the Kähler quotient of ${ℂ}^2\times {ℂ}^{2^{\star }}$ with respect to the holomorphic action of a $\mathrm{U(1)}$ group. In this synopsis the smooth Calabi HyperKähler metric on $T^{\star }{\mathbb{P}}^2$ is the analogue of the Ricci flat Kähler metric on the conifold resolution constructed and discussed in [6], [40], [41].

In the HyperKähler quotient the level $\kappa$ of the moment map plays the role of resolution parameter. For $\kappa =0$ we have the singular metric cone, while for $\kappa \ne 0$ we obtain the Calabi metric on the smooth manifold $T^{\star }{\mathbb{P}}^2$. From the M2-brane gauge theory viewpoint the $\mathrm{U(1)}$ gauge group (which becomes $\mathrm{U(N)}$ for $N$ M2-branes) is the color group, while the $\mathrm{SU(3)}$ part of the isometry group of $T^{\ast }{\mathbb{P}}^2$ is the global flavor group that emerges at the IR fixed point (the $\mathrm{SO}\left(3\right)$ part of the isometry is the R-symmetry of the dual gauge theory).

The ${\mathcal{N}}_3=3$ SCFT (i.e. with $\mathrm{OSp}\left(3\right|4)$ space–time symmetry) on M2-branes probing the metric cone $\mathcal{C}\left({\mathrm{N}}^{\mathrm{0,1,0}}\right)$ (or its resolution $T^{\ast }{\mathbb{P}}^2$) flows on the Coulomb branch (after taking quantum corrections into account) to a ${\mathcal{N}}_3=4$ NLSM with target space the metric cone $\mathcal{C}\left({\mathrm{N}}^{\mathrm{0,1,0}}\right)$ (or $T^{\ast }{\mathbb{P}}^2$). In this paper we show that when we take this resulting ${\mathcal{N}}_3=4$ NLSM and couple it to a Chern–Simons term (by gauging the $\mathrm{SU}\left(3\right)$ part of the isometry group of its target space), the final outcome will be a gauged NLSM with ${\mathcal{N}}_3=4$ supersymmetry (unlike the original SCFT on M2-branes that has ${\mathcal{N}}_3=3$ supersymmetry).

Our paper is organized as follows: Section 2 summarizes the general structure of ${\mathcal{N}}_3=3$ Chern–Simons gauge theory on curved scalar manifolds as geometrically formulated in [24]. Section 3 is the main core of the present article. Utilizing the appropriate quaternionic vielbein formalism for HyperKähler and Quaternionic Kähler manifolds introduced in [2] and systematically reviewed in [23] we show that we can rewrite the ${\mathcal{N}}_3=3$ Chern–Simons theory in a manifestly ${\mathcal{N}}_3=4$ form à la Gaiotto–Witten whenever the weak constraints (1.3)–(1.4) are satisfied. Section 4 deals with the case of the HyperKähler Calabi metric on $T^{\star }{\mathbb{P}}^2$ and its relation with the $N^{0,1,0}$-compactification of M-theory. In Section 4.1 we recall the HyperKähler quotient construction of the metric cone $\mathcal{C}\left({\mathrm{N}}^{\mathrm{0,1,0}}\right)$. In Section 4.2 we discuss the resolution of the conifold singularity which we do in two different but equivalent ways: in Section 4.3 we resolve the singularity uplifting the moment map to a non vanishing level $\kappa$ in the HyperKähler quotient, while in Section 4.4, following the approach of [13], we perform the direct construction of the Calabi HyperKähler metric utilizing the Maurer–Cartan forms of $\mathrm{SU(3)}$ on the coset ${\mathrm{N}}^{\mathrm{0,1,0}}$. In particular in Eqs. (4.38), (4.41) we present the intrinsic components of the Riemann tensor and of the $\mathrm{USp(4)}$ curvature 2-form ${\mathbb{R}}^{\alpha \beta }$ that, up to our knowledge, were not yet explicitly available in the literature. Finally Section 5 contains our conclusions.

The several appendices contain the details of lengthy calculations, in particular those of the moment maps on curved and flat spaces.