The Integral Form of D=3 Chern-Simons Theories Probing ${\mathbb C}^n/\Gamma$ Singularities

TL;DR

This paper develops a formalism using integral forms in superspace to analyze D=3 supersymmetric Chern-Simons theories, emphasizing their geometric structure and applications to M2-branes and singularities.

Contribution

It generalizes rheonomic Lagrangians for D=3 matter-coupled gauge theories to arbitrary Kähler manifolds and explores their role in AdS4/CFT3 correspondence involving singularities.

Findings

Generalized rheonomic Lagrangian to arbitrary Kähler manifolds.

Linked geometric data to field content in M2-brane theories.

Outlined scheme for constructing dual Chern-Simons theories for singularities.

Abstract

We consider D=3 supersymmetric Chern-Simons gauge theories both from the point of view of their formal structure and of their applications to the $\mathrm{AdS_4/CFT_3}$ correspondence. From the structural view-point, we use the new formalism of integral forms in superspace that utilizes the rheonomic Lagrangians and the Picture Changing Operators, as an algorithmic tool providing the connection between different approaches to supersymmetric theories. We provide here the generalization to an arbitrary K\"ahler manifold with arbitrary gauge group and arbitrary superpotential of the rheonomic lagrangian of D=3 matter coupled gauge theories constructed years ago. From the point of view of the $\mathrm{AdS_4/CFT_3}$ correspondence and more generally of M2-branes we emphasize the role of the K\"ahler quotient data in determining the field content and the interactions of the Cherns Simons gauge theory when the transverse space to the brane is a non-compact K\"ahler quotient $K_4$ of some flat variety with respect to a suitable group. The crepant resolutions of ${\mathbb C}^n/\Gamma$ singularities fall in this category. In the present paper we anticipate the general scheme how the geometrical data are to be utilized in the construction of the D=3 Chern-Simons Theory supposedly dual to the corresponding M2-brane solution.

Tables

Table 1. Table 1: The homogeneous 7 7 7 -manifolds that admit at least 2 2 2 Killing spinors are all sasakian or tri-sasakian. This is evident from the fibration structure of the 7 7 7 -manifold, which is either a fibration in circles 𝕊 1 superscript 𝕊 1 \mathbb{S}^{1} for the 𝒩 = 2 𝒩 2 \mathcal{N}=2 cases or a fibration in 𝕊 3 superscript 𝕊 3 \mathbb{S}^{3} for the unique 𝒩 = 3 𝒩 3 \mathcal{N}=3 case corresponding to the N 0 , 1 , 0 superscript N 0 1 0 \mathrm{N}^{0,1,0} manifold. Since this latter is also an 𝒩 = 2 𝒩 2 \mathcal{N}=2 manifold, there is in addition the 𝕊 1 superscript 𝕊 1 \mathbb{S}^{1} fibration.

$𝒩$	Name	Coset	$\begin{array}{c}\text{Holon.}\\ 𝔰𝔬\left(8\right)\text{ bundle}\end{array}$	Fibration
8	${𝕊}^7$	$\frac{\mathrm{SO}\left(8\right)}{\mathrm{SO}\left(7\right)}$	1	$\{\begin{array}{c}{𝕊}^7\stackrel{\pi }{⟹}{\mathbb{P}}^3\\ \forall p\in {\mathbb{P}}^3;{\pi }^{-1}\left(p\right)\sim {𝕊}^1\end{array}$
2	$M^{1,1,1}$	$\frac{\mathrm{SU}\left(3\right)\times \mathrm{SU}\left(2\right)\times U\left(1\right)}{\mathrm{SU}\left(2\right)\times U\left(1\right)\times U\left(1\right)}$	$\mathrm{SU}\left(3\right)$	$\{\begin{array}{c}{M}^{1,1,1}\stackrel{\pi }{⟹}{\mathbb{P}}^2\times {\mathbb{P}}^1\\ \forall p\in {\mathbb{P}}^2\times {\mathbb{P}}^1;{\pi }^{-1}\left(p\right)\sim {𝕊}^1\end{array}$
2	$Q^{1,1,1}$	$\frac{\mathrm{SU}\left(2\right)\times \mathrm{SU}\left(2\right)\times \mathrm{SU}\left(2\right)\times U\left(1\right)}{U\left(1\right)\times U\left(1\right)\times U\left(1\right)}$	$\mathrm{SU}\left(3\right)$	$\{\begin{array}{c}{Q}^{1,1,1}\stackrel{\pi }{⟹}{\mathbb{P}}^1\times {\mathbb{P}}^1\times {\mathbb{P}}^1\\ \forall p\in {\mathbb{P}}^1\times {\mathbb{P}}^1\times {\mathbb{P}}^1;{\pi }^{-1}\left(p\right)\sim {𝕊}^1\end{array}$
2	$V^{5,2}$	$\frac{\mathrm{SO}\left(5\right)}{\mathrm{SO}\left(2\right)}$	$\mathrm{SU}\left(3\right)$	$\{\begin{array}{c}{V}^{5,2}\stackrel{\pi }{⟹}{M}_a\sim \text{quadric in }{\mathbb{P}}^4\\ \forall p\in M_a;{\pi }^{-1}\left(p\right)\sim {𝕊}^1\end{array}$
3	$N^{0,1,0}$	$\frac{\mathrm{SU}\left(3\right)\times \mathrm{SU}\left(2\right)}{\mathrm{SU}\left(2\right)\times U\left(1\right)}$	$\mathrm{SU}\left(2\right)$	$\{\begin{array}{c}{N}^{0,1,0}\stackrel{\pi }{⟹}{\mathbb{P}}^2\\ \forall p\in {\mathbb{P}}^2;{\pi }^{-1}\left(p\right)\sim {𝕊}^3\\ \\ N^{0,1,0}\stackrel{\pi }{⟹}\frac{\mathrm{SU}\left(3\right)}{U\left(1\right)\times U\left(1\right)}\\ \forall p\in \frac{\mathrm{SU}\left(3\right)}{U\left(1\right)\times U\left(1\right)};{\pi }^{-1}\left(p\right)\sim {𝕊}^1\end{array}$

Table 2. Table 2: The homogeneous 7 7 7 -manifolds that admit just one Killing spinors are the squashed 7 7 7 -sphere and the infinite family of N p , q , r superscript N p q r \mathrm{N^{p,q,r}} manifolds for p , q , r ≠ 0 , 1 , 0 formulae-sequence 𝑝 𝑞 𝑟 0 1 0 p,q,r\neq 0,1,0 .

$𝒩$	Name	Coset	$\begin{array}{c}\text{Holon.}\\ 𝔰𝔬\left(8\right)\text{ bundle}\end{array}$
1	${𝕊}_{squashed}^7$	$\frac{\mathrm{SO}\left(5\right)\times \mathrm{SO}\left(3\right)}{\mathrm{SO}\left(3\right)\times \mathrm{SO}\left(3\right)}$	$\mathrm{SO}{\left(7\right)}^{+}$
1	$N^{p,q,r}$	$\frac{\mathrm{SU}\left(3\right))\times U(1)}{U\left(1\right)\times U\left(1\right)}$	$\mathrm{SO}{\left(7\right)}^{+}$

Equations

$$\pi :{\mathcal{M}}_7\stackrel{{\mathbb{S}}^1}{⟶}{K}_3$$

$$d{s}_{{\mathcal{M}}_7}^2={\left(d\varphi -\mathcal{A}\right)}^2+g_{i{j}^{\star }}d{z}^i\otimes d{\overline{z}}^{j^{\star }}$$

$$d{s}_{\mathcal{C}\left({\mathcal{M}}_7\right)}^2=d{r}^2+4e^2{r}^{2}d{s}_{{\mathcal{M}}_7}^2$$

$$\pi$$

$$\forall p\in K_3$$

$$V^{I}\,=\,\left\{\begin{array}[]{rcl}V^{0}&=&dr\\ V^{\alpha}&=&e\,r\,\mathcal{B}^{\alpha}\end{array}\right.\quad r\in\mathbb{R}_{+}$$

$$d{V}^I+{\Omega }^{IJ}\wedge V^J=0$$

$${\Omega }^{\alpha \beta }$$

$${\Omega }^{0\beta }$$

$$K_3\stackrel{\pi }{⟵}\mathcal{G}/\mathcal{H}\stackrel{C}{\hookrightarrow }{K}_4\stackrel{A}{\hookrightarrow }{\mathbb{V}}_q$$

$$K_4={\mathbb{C}}^6/{/}_H\mathrm{U}(1)$$

$$\frac{K_3}{\Gamma }\stackrel{\pi }{⟵}\frac{\mathcal{G}/\mathcal{H}}{\Gamma }\stackrel{C}{\hookrightarrow }\frac{K_4}{\Gamma }\stackrel{A}{\hookrightarrow }{\mathbb{V}}_q$$

$$X^{\mathrm{res}}\to \frac{K_4}{\Gamma }$$

$$\pi :\mathbb{W}\to {\mathbb{C}}^n/\Gamma$$

$${\mathcal{O}}_{168}\equiv \frac{{\mathbb{C}}^3}{\mathrm{PSL}(2,{\mathbb{Z}}_7)}$$

$${\mathrm{L}}_{168}\equiv \mathrm{PSL}(2,{\mathbb{Z}}_7)$$

$${\mathcal{M}}_3^{\mathcal{N}}=\frac{\mathrm{ISO}(1,2|\mathcal{N})}{\mathrm{SO}(1,2)}\equiv \frac{Z\left[\mathrm{ISO}(1,2|\mathcal{N})\right]}{\mathrm{SO}(1,2)\times {\mathbb{R}}^{\mathcal{N}(\mathcal{N}-1)/2}}$$

$$\left\{\begin{array}[]{ccl}Q&=&\sqrt{2}q^{-}=(q^{1}-iq^{2})\\ Q^{c}&=&\sqrt{2}iq^{+}=i(q^{1}+iq^{2})\\ Z&=&Z^{12}\end{array}\right.$$

$$\Omega =e^m{P}_m-\frac{1}{2}{\omega }^{mn}{J}_{mn}+\overline{{\psi }^c}Q-\overline{\psi }{Q}^c+\mathfrak{B}Z.$$

$$d\Omega -\Omega \wedge \Omega$$

$$0$$

$$e^m$$

$${\omega }^{mn}$$

$${\psi }_c$$

$$\psi$$

$$\mathfrak{B}$$

$$\begin{array}[]{ccccccc}\gamma^{0}&=&-\,{\rm i}\,\sigma^{2}&;&\eta_{mn}&=&\mbox{diag}\,\left(-,+,+\right)\\ \gamma^{1}&=&\sigma^{3}&;&C\,\gamma^{m}\,C^{-1}&=&-\,\left(\gamma^{m}\right)^{T}\\ \gamma^{2}&=&\sigma^{1}&;&\gamma^{mn}&=&\frac{1}{2}\,\left[\gamma^{m}\,,\,\gamma^{n}\right]\end{array}$$

$$\left\{\begin{array}[]{ccl}D_{m}&=&\partial_{m}\\ D&=&\frac{\partial}{\partial\overline{\theta}}-i\gamma^{m}\theta\partial_{m}\\ D^{c}&=&\frac{\partial}{\partial\overline{\theta}^{c}}-i\gamma^{m}\theta^{c}\partial_{m}\\ \end{array}\right.,$$

$${\psi }_c\equiv C{\overline{\psi }}^T;\overline{\psi }={\psi }^{†}{\gamma }^0;C=\mathrm{i}{\sigma }^2$$

$$\overline{\psi }\wedge {\gamma }_m\psi$$

$$\overline{\psi }\wedge \psi$$

$$\overline{\psi }\wedge {\psi }_c$$

Full text

1. 1 Conceptual and Historical Introduction

2. 1.1 The scope and the goals of the present paper

3. 1.2 The Sasakian structure and the metric cone

4. 1.3 Resurrection of the $AdS_{4}/CFT_{3}$ correspondence and the ABJM setup

5. 1.4 Finite group quotiening

6. 1.5 Crepant resolution of Gorenstein singularities

7. 2 Rheonomic construction of matter coupled $\mathcal{N}=2$ gauge theories in $D=3$

8. 2.1 The supergeometry of $D=3$, $\mathcal{N}$=2 rigid superspace

9. 2.2 The ingredients

10. 2.2.1 The gauge multiplet

11. 2.2.2 The chiral Wess Zumino multiplets

12. 2.3 The rheonomic lagrangian

13. 3 The space–time Lagrangian of the Maxwell-Chern-Simons theory and some of its applications

14. 3.1 Structure of the scalar potential and of the Lagrangian after the elimination of the auxiliary fields

15. 3.1.1 $\mathcal{N}=2$ Pure Chern Simons Gauge Theory

16. 3.1.2 $\mathcal{N}=2$ Maxwell-Chern-Simons Gauge Theories

17. 3.2 ${\cal N}=3$ Chern Simons gauge theory in three dimensions

18. 3.2.1 The field content and the interactions

19. 3.2.2 The ${\cal N}=3$ gauge theory corresponding to the $N^{0,1,0}$ compactification

20. 4 Integral forms in superspace and three–dimensional Chern–Simons gauge theories

21. 5 Quotient singularities

22. 5.1 The complex Hopf fibration of $\mathbb{S}^{7}$ and quotient singularities $\mathbb{C}^{4}/\Gamma$

23. 5.2 From singular orbifolds to smooth resolved manifolds

24. 6 Conclusions

1 Conceptual and Historical Introduction

The vision of the AdS/CFT correspondence has its starting point in November 1997 with the publication on the ArXive of a paper by Juan Maldacena [1] on the large $N$ limit of gauge theories.

From the viewpoint of the superstring scientific community this was seen as the first explicit example of the long sought duality between gauge theories and superstrings. Yet the scope of this correspondence was destined to be enlarged in many directions and to become, more generically, the gauge/gravity correspondence based on various declinations of the basic idea referred to as the holographic principle. According to this latter, fundamental informations on the quantum behavior of fields leaving on some boundary of a larger space-time can be obtained from the classical gravitational dynamics of fields leaving in the bulk of that space-time. Such wider approach to the AdS/CFT correspondence diminishes the emphasis on strings and brings to higher relevance both supergravity theories and their perturbative and non-perturbative symmetries. In such a framework geometrical issues become the central focus of attention.

It followed immediately, from december 1997 to the late spring of 1998, a series of fundamental papers by Ferrara, Fronsdal, Zaffaroni, Kallosh and Van Proeyen [2],[4],[5],[3], where the algebraic and field theoretical basis of the correspondence was clarified independently from microscopic string considerations.

The AdS/CFT correspondence has a relative simple origin which, however, is extremely rich in ramified and powerful consequences. The key point is the double interpretation of any anti de Sitter group $\mathrm{SO(2,p+1)}$ as the isometry group of the $\mathrm{AdS_{p+2}}$ space or as the conformal group on the $(p+1)$-dimensional boundary $\partial\mathrm{AdS_{p+2}}$. Such a double interpretation is inherited by the supersymmetric extensions of $\mathrm{SO(2,p+1)}$. This is what leads to consider superconformal field theories on the boundary. Two cases are of particular relevance because of concurrent reasons which are peculiar to them: from the algebraic side the essential use of one of the low rank sporadic isomorphisms of orthogonal Lie algebras, from the supergravity side the existence of a spontaneous compactification of the Freund-Rubin type [6]. The two cases are:

A)

The case $p=3$ which leads to $\mathrm{AdS_{5}}$ and to its $4$-dimensional boundary. Here the sporadic isomorphism is $\mathrm{SO(2,4)}\sim\mathrm{SU(2,2)}$ which implies that the list of superconformal algebras is given by the superalgebras $\mathfrak{su}(2,2\mid\mathcal{N})$ for $1\leq\mathcal{N}\leq 4$. On the other hand in Type IIB Supergravity, there is a self-dual five-form field strength. Giving a v.e.v to this latter ($F_{a_{1}a_{2}a_{3}a_{4}a_{5}}\ltimes\epsilon_{a_{1}a_{2}a_{3}a_{4}a_{5}}$), one splits the ambient ten-dimensional space into $5\oplus 5$ where the first $5$ stands for the $\mathrm{AdS_{5}}$ space, while the second $5$ stands for any compact $5$-dimensional Einstein manifold $\mathcal{M}_{5}$. The holonomy of the metric cone on the latter $\mathcal{C}(\mathcal{M}_{5})$ decides the number of supersymmetries and on the $4$-dimensional boundary $\partial\mathrm{AdS_{5}}$ we have a superconformal Yang-Mills gauge theory.

B)

The case $p=2$ which leads to $\mathrm{AdS_{4}}$ and to its $3$-dimensional boundary. Here the sporadic isomorphism is $\mathrm{SO(2,3)}\sim\mathrm{Sp(4,\mathbb{R})}$ which implies that the list of superconformal algebras is given by the superalgebras $\mathrm{Osp}(\mathcal{N}\mid 4)$ for $\mathcal{N}=1,2,3,6,8$. On the other hand in $D=11$ Supergravity, there is a a four-form field strength. Giving a v.e.v to this latter ($F_{a_{1}a_{2}a_{3}a_{4}}\ltimes\epsilon_{a_{1}a_{2}a_{3}a_{4}}$), one splits the ambient ten-dimensional space into $4\oplus 7$ where $4$ stands for the $\mathrm{AdS_{4}}$ space, while $7$ stands for any compact $7$-dimensional Einstein manifold $\mathcal{M}_{7}$. The holonomy of the metric cone on the latter $\mathcal{C}(\mathcal{M}_{7})$ decides the number of supersymmetries and on the $3$-dimensional boundary $\partial\mathrm{AdS_{4}}$ we should have a superconformal gauge theory.

The first case was that mostly explored at the beginning of the AdS/CFT correspondence in 1998 and in successive years. Yet the existence of the second case was immediately evident to anyone who had experience in supergravity and particularly to those who had worked in Kaluza-Klein supergravity in the years 1982-1985. Thus in a series of papers [7],[8],[9],[10],[11],[12],[13], mostly produced by the Torino Group and by the SISSA Group, the $\mathrm{AdS_{4}/CFT_{3}}$ correspondence was proposed and intensively developed in the spring and in the summer of the year 1999. One leading idea, motivating this outburst of activity, was that the entire corpus of results on Kaluza-Klein mass-spectra which had been derived in the years 1982-1986, [6], [14],[15],[16],[18],[19],[20], [21],[23],[24],[25],[26],[27], [28],[29],[30],[31], could now be recycled in the new superconformal interpretation. Actually it was immediately clear that the Kaluza-Klein towers of states, in particularly those corresponding to short representations of the superalgebra $\mathrm{Osp}(\mathcal{N}\mid 4)$, provided an excellent testing ground for the $\mathrm{AdS_{4}/CFT_{3}}$ correspondence. One had to conceive candidate superconformal field theories living on the boundary, that were able to reproduce all the infinite towers of Kaluza Klein multiplets as corresponding towers of composite operators with the same quantum numbers.

In the case the manifold $\mathcal{M}_{7}$ was a coset manifold $\mathcal{G/H}$, an exhaustive list of cases was known since the middle eighties, thanks to the work of Castellani, Romans and Warner [31]. The supersymmetric cases form an even shorter sublist of the main list in [31] and were also classified by the same authors (see table 1 and table 2).

Since it was clear that the theory on the boundary had to be a matter coupled gauge-theory, in three papers [9], [12] and [10], the general form of matter coupled $\mathcal{N}=2,3$ non abelian gauge theories in D=3, with both a canonical kinetic term for the gauge fields and a Chern Simons one, were constructed using auxiliary fields and the rheonomic approach.

In the series of papers [7],[9],[10],[11],[12],[13], it was also conjectured that the gauge theories dual to the supergravity backgrounds of type $\mathrm{AdS_{4}}\times\mathcal{M}_{7}$ have an infrared fixed point where the Yang Mills coupling constant goes to infinity. In this limit the kinetic terms are removed for all the fields in the gauge multiplet. These latter become auxiliary fields and, with the exception of the non abelian gauge one-forms, they can be integrated away leaving, as remnant, a pure Chern Simons gauge theory with a very specific form, that was discussed in the quoted papers.

The question remains how to fill the black box of matter multiplets in the general Chern Simons lagrangian constructed in the way sketched above. We address this issue in the subsection after the next, yet before doing that we clarify the general scope of the present paper.

1.1 The scope and the goals of the present paper

In view of the above considerations, the scope of the present paper is an in depth analysis of matter coupled Maxwell Chern-Simons supersymmetric gauge theories in three space-time dimensions. We aim at a general scheme that encompasses also $\mathcal{N}=3$ and $\mathcal{N}=6$ Chern-Simons theories that are the basis of the ABJM-model.

From the point of view of the contents of the theory we are particularly interested in candidates for the dual CFT.s of M2-brane solutions of D=11 supergravity probing $\mathbb{C}^{n}/\Gamma$ singularities and their resolutions. From the point of view of the constructive principles of supersymmetric field-theories, we are particulary interested in the recently discovered set-up of integral forms in superspace [32, 33, 34, 35] and we plan to explore the properties of the considered class of gauge theories in this respect. We will first provide the appropriate generalization of the rheonomic lagrangian derived in [9] to an arbitrary Kähler manifold with an arbitrary triholomorphic isometry group. Then, according with the general views introduced in [32], we plan to show that the space–time lagrangian and other superfield formulations of the same theory can be obtained by multiplying the rheonomic lagrangian with suitable closed integral forms belonging to the same cohomology class and by integrating on full superspace the result of this wedge multiplication.

The class of considered gauge theories is particularly suited to explore the new view point on superspace since they have a finite set of auxiliary fields and the rheonomic action is an off-shell closed form.

1.2 The Sasakian structure and the metric cone

Coming back to the question how to fill the black box of matter multiplets in the general Chern Simons lagrangian we note that it is in the resolution of this problem that the interplay between the geometry of the compactification manifold $\mathcal{M}_{7}$ and the structure of the $d=3$ superconformal field theory becomes evident.

In paper [11] the authors introduced a systematic bridge between the geometry of $\mathcal{M}_{7}$ and the structure of the boundary gauge theory based on the crucial observation that all the $7$-dimensional cosets with at least two Killing spinors of the $\mathrm{AdS}$-type are sasakian manifolds or tri-sasakian manifolds.

What sasakian means is visually summarized in the following table.

[TABLE]

First of all the $\mathcal{M}_{7}$ manifold must admit an $\mathbb{S}^{1}$-fibration over a complex Kähler three-fold $K_{3}$:

[TABLE]

Calling $z^{i}$ the three complex coordinates of $K_{3}$ and $\phi$ the angle spanning $\mathbb{S}^{1}$, the fibration means that the metric of $\mathcal{M}_{7}$ admits the following representation:

[TABLE]

where the one–form $\mathcal{A}$ is some suitable connection one–form on the $\mathrm{U(1)}$-bundle (1.1).

Secondly the metric cone $\mathcal{C}\left(\mathcal{M}_{7}\right)$ over the manifold $\mathcal{M}_{7}$ defined by the direct product $\mathbb{R}_{+}\otimes\mathcal{M}_{7}$ equipped with the following metric :

[TABLE]

should also be a Ricci-flat complex Kähler $4$-fold. In the above equation $e$ just denotes a constant scale parameter with the dimensions of an inverse length $\left[e\right]=\ell^{-1}$.

Altogether the Ricci flat Kahler manifold $K_{4}$, which plays the role of transverse space to the M2-branes, is a line-bundle over the base manifold $K_{3}$:

[TABLE]

All the manifolds listed in table 1 are sasakian in the sense described above. The $\mathfrak{so}(8)$-holonomy mentioned in this table is the holonomy of the Levi-Civita connection of the metric cone $\mathcal{C}\left(\mathcal{M}_{7}\right)$ which can be easily calculated from that of the $\mathcal{M}_{7}$-manifold relying on the following one-line construction. Define the vielbein of $\mathcal{C}\left(\mathcal{M}_{7}\right)$ in terms of the vielbein of $\mathcal{M}_{7}$ in the following way:

[TABLE]

where $ds^{2}_{\mathcal{M}_{7}}\,=\,\sum_{\alpha=1}^{7}\mathcal{B}^{\alpha}\otimes\mathcal{B}^{\alpha}$. The torsion equation:

[TABLE]

where $\Omega^{IJ}$ is the spin–connection of the metric cone, is solved by:

[TABLE]

having denoted by $\mathcal{B}^{\alpha\beta}$ the spin–connection of $\mathcal{M}_{7}$, namely $d\mathcal{B}^{\alpha}\,+\,\mathcal{B}^{\alpha\beta}\wedge\mathcal{B}^{\beta}\,=\,0$. According to the summary of Kaluza–Klein supergravity presented in [37], $\Omega^{IJ}$ is the $\mathfrak{so}(8)$-connection whose holonomy decides the number of Killing spinor admitted by the $\mathrm{AdS_{4}}\times\mathcal{M}_{7}$ compactification of M-theory. When this holonomy vanishes we have the maximal number of preserved supersymmetries. When it is $\mathrm{SU(3)}\subset\mathrm{SO(8)}$ we have $\mathcal{N}=2$. When it is $\mathrm{SU(2)}\subset\mathrm{SO(8)}$ we might in principle expect $\mathcal{N}=4$, but we actually have only $\mathcal{N}=3$, as firstly remarked by Castellani, Romans and Warner in 1985.

In [11], it was emphasized that the fundamental geometrical clue to the field content of the superconformal gauge theory on the boundary is provided by the construction of the Kähler manifold $K_{4}$ as a holomorphic algebraic variety in some higher dimensional affine or projective space $\mathbb{V}_{q}$, plus a Kähler quotient. The equations identifying the algebraic locus in $\mathbb{V}_{q}$ are related with the superpotential $W$ appearing in the $d=3$ lagrangian, while the Kähler quotient is related with the $D$-terms appearing in the same lagrangian. The coordinates $u,v$ of the space $\mathbb{V}_{q}$ are the scalar fields of the superconformal gauge theory, whose vacua, namely the set of extrema of its scalar potential, should be in one-to-one correspondence with the points of $K_{4}$. Going from one to multiple M2–branes just means that the coordinate $u,v$ of $\mathbb{V}_{q}$ acquire color indices under a proper set of color gauge groups and are turned into matrices. In this way we obtain quivers.

All these conceptual and algorithmic points were enumerated in the set of papers [7],[9],[10], [11],[13], where the cases $Q^{1,1,1}$, $M^{1,1,1}$ and $N^{0,1,0}$ were worked out in detail, finding the algebraic embedding, defining the superpotential and the quiver. Finally the Kaluza–Klein spectrum of supergravity compactified on each of these three spaces was matched with the spectrum of composite conformal operators in the corresponding boundary superconformal theory.

1.3 Resurrection of the $AdS_{4}/CFT_{3}$ correspondence and the ABJM setup

The subject of the $\mathrm{AdS_{3}\times CFT_{3}}$ correspondence was resurrected ten years later in 2007-2009 by the work presented in papers [38],[39],[40] which stirred a great interest in the scientific community and obtained a very large number of citations. We confess that the formalism of three-algebras introduced in [40] is not very clear to us, but we rely on the statement by the authors of [39] that their construction is completely equivalent to the theory presented in [40]. The ABJM-construction of [39] is instead very clear and the attentive reader, making the required changes of notations and names of the objects, can verify that the $\mathcal{N}=3$ lagrangian presented there is just the same as that constructed in papers [7],[9] by letting the Yang-Mills coupling constant go to infinity. What is new and extremely important in the ABJM model is the relative quantization of the Chern Simons levels $k_{1,2}$ of the two gauge groups and its link to a quotiening of the seven sphere by means of a cyclic group $\mathbb{Z}_{k}$. Indeed the theories presented in [39] pertain to the first case in table 1, modified by a finite group quotiening. We just regret that the authors of [39] did not consider it appropriate to quote the papers of ten years before that contain a large part of the ground basis of their results.

For this reason in the first sections of these notes we review the constructions of [7],[9], translated into a more modern notation that refers from the beginning to those geometrical structures, Kähler metrics and moment maps, we will utilize in the sequel. By these means we want to show that the construction method introduced in [39] is just identical to that laid down in 1998-1999 in the several times quoted series of papers; furthermore we want to fix the framework where to discuss and possibly answer a new question which we presently formulate.

1.4 Finite group quotiening

As we emphasized the key guiding item in the construction of the d=3 gauge theory is the $\mathrm{K_{4}}$ manifold and its representation as an algebraic locus in some $\mathbb{V}_{q}$. We can extract the logic which underlies [39], by means of the following arguments. First consider the following projections and embeddings pertaining to the case where $\mathcal{M}_{7}$ is a smooth coset manifold

[TABLE]

In the above formula $\stackrel{{\scriptstyle C}}{{\hookrightarrow}}$ is the embedding map into the metric cone, while $\stackrel{{\scriptstyle A}}{{\hookrightarrow}}$ denotes the algebraic embedding into an affine or projective variety by means of a suitable set of algebraic equations.

For instance in the case of the seven sphere $\mathcal{G/H}\,=\,\mathrm{SO(8)/SO(7)}$, we have $K_{3}\,=\,\mathbb{P}^{3}$ and $K_{4}\,=\,\mathbb{C}^{4}\sim\mathbb{R}^{8}$. Then the algebraic map $\stackrel{{\scriptstyle A}}{{\hookrightarrow}}$ is just the identity map since $\mathbb{V}_{q}\,=\,\mathbb{C}^{4}$.

On the contrary, in the case $N^{0,1,0}$, the base manifold $K_{3}\,=\frac{\mathrm{SU(3)}}{\mathrm{U(1)\times U(1)}}$ is just the $\mathfrak{su}(3)$ flag manifold and $K_{4}$ is obtained as the Kähler quotient of an algebraic locus cut in $\mathbb{V}_{q}\,=\,\mathbb{C}^{6}$ by a quadric equation. In this particular case the entire procedure how to go from $\mathbb{C}^{6}$ to $K_{4}$ can be seen as a HyperKähler quotient with respect to the triholomorphic action of a $\mathrm{U(1)}$ group:

[TABLE]

The quadric constraint is traced back to the vanishing of the holomorphic part of the triholomorphic moment map, while the Kähler quotient encodes the constraint coming from the real part of the same moment map.

Next we consider some finite group $\Gamma\subset\mathcal{G}$ and in eq.(1.8) we replace the homogeneous space $\mathcal{G/H}$ with the orbifold $\frac{\mathcal{G/H}}{\Gamma}$. The finite group quotient extends both to the projection map and to the metric cone enlargment. Thus eq.(1.8) is replaced by:

[TABLE]

Typically the quotient $\frac{K_{4}}{\Gamma}$ is a singular manifold. We need a resolution of the singularities by means of an appropriate resolving map:

[TABLE]

which typically leads to an affine variety $X^{\mathrm{res}}\,\stackrel{{\scriptstyle A}}{{\hookrightarrow}}\,\mathbb{C}^{q}$ embedded by suitable algebraic equations into some $\mathbb{C}^{q}$.

The final outcome is that the coordinates $z^{i}$ of $\mathbb{C}^{q}$ are the matter fields in the $d=3$ conformal field theory, while the embedding equations should determine the superpotential $\mathcal{W}(z)$. The gauging is instead dictated by the final Kähler quotient of the resolved algebraic variety $X^{\mathrm{res}}$ which produces the resolved metric cone $K_{4}^{\mathrm{res}}$.

1.5 Crepant resolution of Gorenstein singularities

It appears from the above discussion that the most fundamental question at stake is a classical problem of algebraic geometry, namely the resolution of singularities, in particular of the quotient singularities. For this there is a well established set of results that were all obtained by the mathematical community at the beginning of 1990.s, under the stimulus of string and supergravity theory.

First of all we fix sum vocabulary.

Definition 1.1

The canonical line bundle $K_{\mathbb{V}}$ over a complex algebraic variety $\mathbb{V}$ of complex dimension $n$ is the bundle of holomorphic $(n,0)$-forms $\Omega^{(n,0)}$ defined over $\mathbb{V}$.

Definition 1.2

An orbifold $\mathbb{V}/\Gamma$ of an algebraic variety modded by the action of a finite group is named Gorenstein if the isotropy subgroup $H_{p}\subset\Gamma$ of every point $p\in\mathbb{V}$ has a trivial action on the canonical bundle $K_{\mathbb{V}}$.

Definition 1.3

A resolution of singularities $\pi\,:\,\,\mathbb{W}\rightarrow\,\mathbb{X}\equiv\mathbb{V}/\Gamma$ is named crepant, if $K_{\mathbb{W}}\,=\,\pi^{\star}K_{\mathbb{X}}$. In particular this implies that the first Chern class of the resolved variety vanishes ($c_{1}\left(T\mathbb{W}\right)\,=\,0$), if it vanishes for the orbifold, namely if $c_{1}\left(T\mathbb{X}\right)\,=\,0$.

In the case $\mathbb{V}\,=\,\mathbb{C}^{n}$, a resolution of quotient singularity:

[TABLE]

is crepant if the resolved variety $\mathbb{W}$ has vanishing first Chern class, namely if it is a Calabi-Yau $q$–fold.

The Gorenstein condition plus the request that there should be a crepant resolution restricts the possible $\Gamma$.s to be subgroups of $\mathrm{SL(n,\mathbb{C})}$.

Concerning the crepant resolution of Gorenstein singularities $\mathbb{C}^{n}/\Gamma$, what was established in the early 1990s is the following:

For $n=2$ the classification of Gorenstein singularities boils down to the classification of finite Kleinian subgroups $\Gamma\subset\mathrm{SU(2)}$. This latter is just the A-D-E classification and the crepant resolution of singularities is done in one stroke by the Kronheimer construction of ALE-manifolds[41, 42] via an HyperKähler quotient of a flat HyperKähler manifold $\mathbb{H}_{\Gamma}$, whose dimension and structure depends on the group $\Gamma$. 2. 2.

For $n=3$ the classification of finite subgroups of $\mathrm{SL(3,\mathbb{C})}$ was performed at the very beginning of the XX century[43, 44] and it is summarized in [45]. As stressed by Markushevich in [46] in that list there are only two types of groups, either solvable groups or the simple group PSL(2,7) of order 168. For this reason the same Markushevich studied the resolution of the Gorenstein orbifold:

[TABLE]

which corresponds to a unique truely new case. We are going to add several other physical motivations for the study of orbifolds with respect to

[TABLE]

or one of its maximal subgroups. We postpone such discussion to later publications. Here we focus on the general form of $d=3$ Chern Simons gauge theories. 3. 3.

For $n>3$ essentially nothing is known with the exception of those cases that can be reduced to singularities in $n=2,3$.

2 Rheonomic construction of matter coupled $\mathcal{N}=2$ gauge theories in $D=3$

Following the results of [9] and [11] in this section we consider the general form of a matter coupled Maxwell-Chern Simons gauge theory in $D=3$ space-time dimensions. In the quoted references the Kähler manifold spanned by the Wess-Zumino multiplets was considered to be flat and the action of the gauge group on the same was taken to be linear. In the present paper we need to be more general. Hence the formulae of [9] and [11] here are geometrically rewritten in terms of a generic Kähler metric, of Killing vectors and holomorphic moment maps. Supersymmetry is $\mathcal{N}\,=\,2$ in $D=3$ which amounts to the same as $\mathcal{N}=1$ in $D=4$. Just as in [9] and [11] we follow the off-shell approach with auxiliary fields which, in the the final step of the construction, can be eliminated through their own (algebraic) equation of motion leading to the final form of the interactions among physical degrees of freedom.

Furthermore, for all the reasons advocated in the introduction we are interested in the rheonomic construction 444For an overview of the rheonomic approach see the books [49] and [50]. In particular a relatively short modern presentation of the rheonomic principles is presented in chapter 6 of the second volume of [50]. of the theory and in particular in the explicit form of the rheonomic lagrangian which was indeed derived in [9] and [11]. Here that result is generalized to arbitrary Kähler manifolds and to groups with an arbitrary isometric holomorphic action.

We start by fixing our conventions for the geometry of rigid superspace.

2.1 The supergeometry of $D=3$, $\mathcal{N}$=2 rigid superspace

$D=3,~{}\mathcal{N}$–extended superspace is viewed as the following supercoset manifold:

[TABLE]

where $\mathrm{ISO(1,2|\mathcal{N})}$ is the $\mathcal{N}$–extended Poincaré supergroup in three–dimensions. Its superalgebra is the Inonü-Wigner contraction of the superalgebra $\mathrm{Osp(\mathcal{N}|4)}$ spanned by the generators $J_{m}$, $P_{m}$, $q^{i}$. The central extension $Z\left[\mathrm{ISO(1,2|\mathcal{N})}\right]$, which is not contained in the contraction of $\mathrm{Osp(\mathcal{N}|4)}$, is obtained by adjoining to $\mathrm{ISO(1,2|\mathcal{N})}$ the central charges that generate the subalgebra $\mathbb{R}^{\mathcal{N}(\mathcal{N}-1)/2}$. Specializing our analysis to the case $\mathcal{N}\!\!=\!\!2$, we define the new generators:

[TABLE]

The left invariant one–form $\Omega$ on ${\cal M}^{\mathcal{N}}_{3}$ is the following object:

[TABLE]

The superalgebra $\mathrm{ISO(1,2|N)}$ defines all the structure constants apart from those relative to the central charge that are trivially determined. Hence we can write:

[TABLE]

Imposing the Maurer-Cartan equation $d\Omega-\Omega\wedge\Omega=0$ is equivalent to imposing flatness in superspace, i.e. global supersymmetry. So we have

[TABLE]

The above equations are nothing else but the statement that the curvatures of the $\mathcal{N}=2$ super-Poincaré group are zero. The simplest solution for the supervielbein and for the connection satisfying the above structural equations of rigid superspace is the following one:

[TABLE]

where $x^{m}$ are the standard coordinates of flat Minkowsky space and $\theta$ are the anticommuting grassmanian supercoordinates. They form a $D=3$ Dirac spinor corresponding to four independent components. Our conventions for the $D=3$ Dirac matrices are the following ones:

[TABLE]

The superderivatives

[TABLE]

are the vector fields dual to the above one–forms.

Let us observe that by $\psi_{c}$ we denote the conjugate of the spinor $\psi$ according to the following definition in terms of the charge conjugation matrix:

[TABLE]

Relevant Fierz Identities.

Furthermore in the further development of the theory some Fierz identities are particularly useful and relevant:

[TABLE]

2.2 The ingredients

As stated in the introduction we are interested in the general form of an $\mathcal{N}=2,~{}d=3$ super Yang Mills theory coupled to $n$ chiral multiplets arranged into a generic representation $\cal R$ of the gauge group $\cal G$.

In $\mathcal{N}=2,~{}d=3$ supersymmetric theories, two formulations are allowed: the on–shell and the off–shell one. In the on–shell formulation which contains only the physical fields, the supersymmetry transformations rules close the supersymmetry algebra only upon use of the field equations. On the other hand the off–shell formulation contains further auxiliary, non dynamical fields that make it possible for the supersymmetry transformations rules to close the supersymmetry algebra identically. By solving the field equations of the auxiliary fields these latter can be eliminated and the on–shell formulation is thus retrieved. We adopt the off–shell formulation.

2.2.1 The gauge multiplet

The three–dimensional $\mathcal{N}=2$ vector multiplet contains the following Lie-algebra valued fields:

[TABLE]

where ${\cal A}={\cal A}^{\Lambda}t_{\Lambda}$ is the real gauge connection one–form, $\lambda^{\Lambda}$ and $\lambda^{\Lambda}_{c}$ are two complex Dirac spinors (the gauginos), $M^{\Lambda}$ and $D^{\Lambda}$ are real scalars, the latter being the auxiliary field. The capital Greek indices $\Lambda,\Sigma,\dots$ span the adjoint representation of the gauge group $\mathcal{G}$.

The field strength $2$-form is defined below:

[TABLE]

The covariant derivative on any other other field $X$ of the gauge multiplet is defined below:

[TABLE]

From (2.2.1) and (2.15) we obtain the Bianchi identity:

[TABLE]

The vector multiplets contains $4_{B}\oplus 4_{F}$ bosonic and fermionic off–shell degrees of freedom for each generator of the gauge group.

The off-shell rheonomic parametrization of the vector multiplet curvatures, consistent with the Bianchi identities is given below:

[TABLE]

and we also have:

[TABLE]

2.2.2 The chiral Wess Zumino multiplets

The Wess Zumino multiplets have the same structure in $D=3$ as they have in $D=4$.

[TABLE]

The complex scalar fields $z^{i}$ parameterize a Kähler manifold $\mathcal{M}_{K}$ whose geometry is determined by a Kähler potential $\mathcal{K}\left(z,{\overline{z}}\right)$ yielding as usual the metric:

[TABLE]

The continuous isometries (if any) of this metric are generated by holomorphic Killing vectors $k_{\Lambda}^{i}(z)$ according to:

[TABLE]

and the vector multiplets can be used to gauge such symmetries and make them local. One sets:

[TABLE]

If one compares the above equations with the similar ones that appear in the coupling of chiral Wess Zumino multiplets to $\mathcal{N}=1$, $D=4$ supergravity (see for instance [51, 52]), one should notice the absence of the Kähler connection $\mathcal{Q}$ in the covariant derivative of the chiralinos. This is the main structural difference between the case of local and rigid supersymmetry. In the second case, which is that of interest to us in the present paper, there is no $\mathrm{U(1)}$-bundle over the Kähler manifold which is not requested to be Hodge-Kähler, but simply Kähler. Correspondingly the fermions do not transform as sections of the Hodge bundle and there is no Hodge-Kähler connection in their covariant derivative. We have only the gauged version $\widehat{\Gamma}$ of the Levi-Civita holomorphic connection since the chiralinos transform as sections of the tangent bundle $T\mathcal{M}_{K}$.

The off–shell rheonomic parameterizations of the chiral multiplet fields are the following ones:

[TABLE]

and from them we read off the supersymmetry transformation rules.

Additional essential items in the construction of the theory are the moment maps defined by the following equation:

[TABLE]

2.3 The rheonomic lagrangian

Using the rules of the rheonomic approach, the rheonomic lagrangian of a general matter coupled gauge theory in $D=3$ was determined in [9] for the case where the Kähler manifold $\mathcal{M}_{K}$ is flat and the action of the gauge group on the chiral multiplets is linear. The transcription of that result to the general case of an arbitrary $\mathcal{M}_{K}$ is rather straightforward and actually gives rise to more compact and more elegant formulae. We report the result of [9] in its generalized form.

The rheonomic Lagrangian can be organized in the following way:

[TABLE]

Next we display the various addends mentioned in eq.(2.3). We begin with the kinetic part of the gauge lagrangian, named after Maxwell.

[TABLE]

The Chern-Simons part of the vector multiplet rheonomic lagrangian is instead the following one:

[TABLE]

while the Fayet-Iliopoulos addend has the following appearance

[TABLE]

where $\mathfrak{C}_{I}$ denotes a basis of generators of the center $Z\left[\mathbb{G}\right]$ of the gauge Lie algebra $\mathbb{G}$.

The two sectors of the chiral rheonomic lagrangian are displayed next.

[TABLE]

[TABLE]

3 The space–time Lagrangian of the Maxwell-Chern-Simons theory and some of its applications

In the rheonomic approach ([49]), the total three–dimensional $\mathcal{N}\!\!=\!\!2$ rheonomic lagrangian:

[TABLE]

is a closed three–form defined in superspace $d{\cal L}^{N=2}_{rheo}=0$.

The space–time lagrangian,

[TABLE]

The first addend is the kinetic part admitting the following explicit form:

[TABLE]

In the above equation $\Pi^{m\mid i}$ and $\overline{\Pi}^{m\mid j^{\star}}$ are zero forms with both a space-time vector index $m$ and a Kähler manifold vector index $i$ (or $j^{\star}$). This denotes that these [math]-forms transform both as sections of the tangent bundle to space-time and as sections of the tangent bundle to the Kähler manifold $T\mathcal{M}_{K}$. From variation in $\delta\,\Pi^{m\mid i}$ and $\delta\,{\overline{\Pi}}^{m\mid i}$ we obtain:

[TABLE]

Similarly for the [math]-forms $\Phi^{m}_{\Lambda}$ which have an index in the adjoint representation of the gauge group $\Lambda$ and a space-time vector index $m$. Their equation of motion is algebraic and identifies them with the space-time derivatives of the corresponding scalar fields $M^{\Lambda}$ belonging to the vector multiplets:

[TABLE]

Finally the [math]-form $F_{pq}$ with two antisymmetric space-time indices has also an algebraic field equation and gets identified with the space-time components of the Yang-Mills curvature:

[TABLE]

The parameter $\mathbf{e}\,=\,\frac{1}{g}$ where $g$ is the Yang-Mills gauge coupling constant, sits in front of all the terms that compose the separately supersymmetric kinetic terms of the vector multiplets. The parameter $\alpha$, instead, sits in front of all the terms that provide the supersymmetrization of the Chern-Simons term. We find these parameters also in the other sectors of the lagrangian. Setting $\mathbf{e}\,=\,0$ we obtain a purely Chern Simons gauge theory, while setting $\alpha\,=\,0$ we suppress the Chern Simons term and all of its supersymmetric partners. In its most general form the lagrangian contains three invariants: the kinetic term associated with $\mathbf{e}$, the Chern-Simons associated with $\alpha$ and the Fayet Iliopoulos term which occurs only in the potential sector.

The next part of the lagrangian is the $2$-fermi part that contains the scalar–field–dependent mass–terms of the fermions:

[TABLE]

In the above equation $\mathbf{\kappa}_{\Lambda\Sigma}$ denotes the Killing metric on the Lie algebra of the gauge group or any other invariant metric if the Lie algebra is not semisimple. Furthermore the holomorphic function $W(z)$ is the so called superpotential which, together with the Kähler metric, determines all self-interactions of the chiral multiplets. Finally both in equation (LABEL:kinosi) and the (3.7) the terms with the parameter $\alpha$ in front are part of the Chern-Simons term supersymmetrization. Together with the residual $\alpha$-terms that we have in the next part of the Lagrangian, namely in the potential part, those above constitute a separate supersymmetry invariant. We can switch on and off the $\alpha$-terms preserving off-shell supersymmetry of the Lagrangian.

The next and last part of the space-time lagrangian is the potential part. It has the following form:

[TABLE]

In the above equation the vectors $\mathfrak{C}_{\Sigma}^{I}$ ($I\,=\,1,\dots,r$) project onto the $r$ independent generators of the center of the gauge Lie algebra $Z(\mathbb{G})$. For each of these generators one can add a separately supersymmetric invariant term, named Fayet Iliopoulos term [53], which is just linear in the corresponding auxiliary fields $D_{I}\,\equiv\,\mathfrak{C}_{\Sigma}^{I}\,D^{\Sigma}$. Namely we have:

[TABLE]

where $\mathfrak{f}_{I}$ are independent constants (the Fayet Iliopoulos constants). Furthermore:

[TABLE]

is the moment map of the gauged holomorphic Killing vectors satisfying the identity:

[TABLE]

3.1 Structure of the scalar potential and of the Lagrangian

after the elimination of the auxiliary fields

Let us first observe that the structure of the theory is substantially different when the standard kinetic term of the gauge fields are included, namely when $\mathbf{e}\neq 0$, and when they are not $\mathbf{e}=0$. Hence we discuss the two cases separately.

3.1.1 $\mathcal{N}=2$ Pure Chern Simons Gauge Theory

When $\mathbf{e}=0$, the lagranian takes the following form:

[TABLE]

where the potential in terms of physical and auxiliary fields is the following one:

[TABLE]

In this case the gauge multiplet does not propagate and it is essentially made of lagrangian multipliers for certain constraints. Indeed the auxiliary fields, the gauginos and the vector multiplet scalars have algebraic field equations so that they can be eliminated through the solutions of such equations of motion. The vector multiplet auxiliary scalars $D^{\Lambda}$ appear only as lagrangian multipliers of the constraint:

[TABLE]

while the variation of the auxiliary fields $\mathcal{H}^{j^{\star}}$ of the Wess Zumino multiplets yields:

[TABLE]

On the other hand, the equation of motion of the field $M^{\Lambda}$ implies:

[TABLE]

which finally resolves all the auxiliary fields in terms of functions of the physical scalars.

Upon use of both constraints (3.14) and (3.15) the scalar potential takes the following positive definite form:

[TABLE]

In a similar way the gauginos can be resolved in terms of the chiralinos:

[TABLE]

In this way if we were able to eliminate also the gauge one form $\mathcal{A}$ the Chern Simons gauge theory would reduce to a theory of Wess-Zumino multiplets with additional interactions. The elimination of $\mathcal{A}$, however, is not possible in the non-abelian case and it is possible in the abelian case only through duality non local transformations. This is the corner where interesting non perturbative dynamics is hidden.

3.1.2 $\mathcal{N}=2$ Maxwell-Chern-Simons Gauge Theories

Of interest are also the mixed Maxwell-Chern-Simons Gauge Theories where both the Maxwell and Chern Simons kinetic terms are included, namely where $\mathbf{e}\neq 0$ and $\alpha\neq 0$. In this case the gauge fields propagate and so do the gauginos and the vector multiplet scalars $M^{\Lambda}$.

At the level of the potential the main difference is the presence of the quadratic term in the vector multiplet auxiliary fields $D^{\Lambda}$. Eliminating these latter through their own field equations and similarly doing with the auxiliary fields of the WZ multiplets we get the following potential for the propagating scalars $M^{\Lambda}$ and $z^{i}$:

[TABLE]

Let us now consider the other terms in the Lagrangian and perform the transition from the first to the second order formalism by eliminating the remaining auxiliary fields. The final form of the second order Lagrangian for the most general Maxwell-Chern-Simons matter coupled gauge-theory is the following one:

[TABLE]

We consider next special instances of theories inside the above described general families.

3.2 ${\cal N}=3$ Chern Simons

gauge theory in three dimensions

In this section we discuss the structure of a three dimensional Chern Simons gauge theory with ${\cal N}=3$ supersymmetry. The starting point is the discussion of a complete $\mathcal{N}=3$ gauge theory in the same dimensions. The ${\cal N}=3$ case is just a particular case in the class of theories described in the previous section since a theory with ${\cal N}=3$ SUSY, must a fortiori be an ${\cal N}=2$ theory. In [9] , the case of ${\cal N}=4$ theories was also considered, within the ${\cal N}=2$ class. These latter are obtained through dimensional reduction of an ${\cal N}_{4}=2$ theory in four–dimensions. Indeed since each $D=4$ Majorana spinor splits, under dimensional reduction on a circle $\mathbb{S}^{1}$, into two $D=3$ Majorana spinors, the number of three–dimensional supercharges is just twice the number of $D=4$ supercharges:

[TABLE]

The ${\cal N}_{3}=3$ case corresponds to an intermediate situation. It is an ${\cal N}_{3}=2$ theory with the field content of an ${\cal N}_{3}=4$ one, but with additional ${\cal N}_{3}=2$ interactions that respect three out of the four supercharges obtained through dimensional reduction. Using an ${\cal N}=2$ superfield formalism and the notion of twisted chiral multiplets it was shown in [54] that for abelian gauge theories these additional ${\cal N}_{3}=3$ interactions are

A Chern Simons term, with coefficient $\alpha$ 2. 2.

A mass-term with coefficient $\mu=\alpha$ for the chiral field $Y^{I}$ in the adjoint of the color gauge group. By this latter we denote the complex field belonging, in four dimensions, to the ${\cal N}_{4}=2$ gauge vector multiplet.

In [12] the authors retrieved for non-abelian gauge theories the same result as that found by the authors of [54] for the abelian theories. In [12] the construction was presented in the component formalism which is better suited to discus the relation between the world–volume gauge theory and the geometry of the transverse cone $\mathcal{C}(\mathcal{M}_{7})$. Let us also remark that the arguments used in[39] are the same which were spelled out ten years earlier in [12]. In this section we summarize in the more general notations based on HyperKähler metric and the triholomorphic moment maps the general form of a non abelian ${\cal N}=3$ Chern Simons gauge theory in three dimensions as it was obtained in [12].

3.2.1 The field content and the interactions

The strategy of [12] was that of writing the ${\cal N}=3$ gauge theory as a special case of an ${\cal N}=2$ theory, whose general form was discussed in previous sections. For this latter the field content is given by:

[TABLE]

and the complete Lagrangian was given in the previous sections. In particular the complete Chern Simons Lagrangian before the elimination of the auxiliary fields was displayed in eq.(3.12).

The Chern Simons ${\cal N}=3$ case is obtained when the following conditions are fulfilled:

• •

The spectrum of chiral multiplets is made of $\mbox{dim}\mathcal{G}\,+\,2n$ complex fields arranged in the following way

[TABLE]

• •

the Kähler potential has the following form:

[TABLE]

where $\widehat{\mathcal{K}}(u,v)$ is the Kähler potential of the Ricci-flat HyperKähler metric of the HyperKähler manifold $\mathrm{HK}_{2n}$. The assumption that $\mathcal{K}(Y,u,v)$ does not depend on $Y^{\Lambda}$ implies that the kinetic term of these scalars vanishes turning them into auxiliary fields that can be integrated away.

• •

The superpotential $W(z)$ has the following form:

[TABLE]

where $\mathcal{P}^{\Sigma}_{+}(u,v)$ denotes the holomorphic part of the triholomorphic moment map induced by the triholomorphic action of the color group on $\mathrm{HK}_{2n}$.

The reason why these two choices make the theory ${\cal N}_{3}=3$ invariant is simple: the first choice corresponds to assuming the field content of an ${\cal N}_{3}=4$ theory which is necessary since ${\cal N}_{3}=3$ and ${\cal N}_{3}=4$ supermultiplets are identical. The second choice takes into account that the metric of the hypermultiplets must be HyperKähler and that the gauge coupling constant was sent to infinity. The third choice introduces an interaction that preserves ${\cal N}_{3}=3$ supersymmetry but breaks (when $\alpha\neq 0$) ${\cal N}_{3}=4$ supersymmetry.

Going back to the off-shell Chern Simons lagrangian given in eq.(3.12) one can perform the elimination of the auxiliary fields that now include $Y^{\Lambda},D^{\Lambda},M^{\Lambda},\mathcal{H}^{i}$ at the bosonic level and the gauginos $\lambda^{\Lambda},\lambda^{\Lambda}_{c},\chi^{\Lambda},\chi^{\Lambda}_{c}$ at the fermionic level (Note that there are two more non propagating gauginos coming from the chiral multiplet in the adjoint representation of the gauge group). We do not enter the details of the integration over the non propagating fermions and we just consider the bosonic lagrangian emerging from the integration over the auxiliary bosonic fields. The first integration to perform is that over the auxiliary field $\mathcal{H}^{\Lambda}$. This is simply the lagrangian multiplier of the constraint:

[TABLE]

Substituting this back into the lagrangian yields a potential with the same structure as that in eq.(3.17) but with a modified superpotential which becomes quadratic in the holomoprhic momentum map:

[TABLE]

3.2.2 The ${\cal N}=3$ gauge theory corresponding to the

$N^{0,1,0}$ compactification

Having clarified the structure of a generic ${\cal N}=3$ gauge theory let us consider, as an illustration, the specific one associated with the $N^{0,1,0}$ seven–manifold following the presentation of [12]. As explained in [11] (see eq.(B.1) of that paper) the manifold $N^{0,1,0}$ is the circle bundle inside $\mathcal{O}(1,1)$ over the flag manifold $\mathbb{F}(1,2;3)$. In other words we have

[TABLE]

where, by definition,

[TABLE]

is the homogeneous space obtained by modding $\mathrm{SU(3)}$ with respect to its maximal torus:

[TABLE]

Furthermore as also explained in [11] (see eq.(B.2)), the base manifold $\mathbb{F}(1,2;3)$ can be algebraically described as the following quadric

[TABLE]

in $\mathbb{P}^{2}\times\mathbb{P}^{2*}$, where $u^{i}$ and $v_{i}$ are the homogeneous coordinates of $\mathbb{P}^{2}$ and $\mathbb{P}^{2*}$, respectively.

Hence a complete description of the metric cone $\mathcal{C}\left(N^{0,1,0}\right)$ can be given by writing the following equations in $\mathbb{C}^{3}\times\mathbb{C}^{3*}$:

[TABLE]

Eq.s (3.33) can be easily interpreted as the statement that the cone ${\cal C}\left(N^{0,1,0}\right)$ is the HyperKähler quotient of a flat three-dimensional quaternionic space with respect to the triholomorphic action of a $\mathrm{U(1)}$ group. Indeed the first two equations in (3.33) can be rewritten as the vanishing of the triholomorphic moment map of a $\mathrm{U(1)}$ group. It suffices to identify:

[TABLE]

Comparing with eq.s (3.28) we see that the cone $\mathcal{C}(N^{0,1,0})$ can be correctly interpreted as the space of classical vacua in an abelian ${\cal N}=3$ gauge theory with $3$ hypermultiplets in the fundamental representation of a flavor group $\mathrm{SU(3)}$. Indeed if the color group is $\mathrm{U(1)}$ there is only one value for the index $\Lambda$. The potential is a positive definite quadratic form in the moment maps with minimum at zero which is attained when the moment map vanishes.

Relying on this geometrical picture of the transverse space to an $M=2$–brane leaving on AdS${}_{4}\times N^{0,1,0}$, in [12] it was conjectured that the ${\cal N}=3$ non–abelian gauge theory whose infrared conformal point is dual to $D=11$ supergravity compactified on AdS${}_{4}\times N^{0,1,0}$ should have the following structure:

[TABLE]

More explicitly and using an ${\cal N}=2$ notation we can say that the field content of the theory proposed in [12] is given by the following chiral fields, that are all written as $N\times N$ matrices:

[TABLE]

and the superpotential before integration on the auxiliary fields $Y$ can be written as follows:

[TABLE]

where $\alpha_{1,2}$ are the Chern Simons coefficients associated with the $\mathrm{SU(N)}_{1,2}$ simple gauge groups, respectively. Setting:

[TABLE]

and integrating out the two fields $Y_{1,2}$ that have received a mass by the Chern Simons mechanism in [12] it was obtained the following effective quartic superpotential:

[TABLE]

The vanishing relations one can derive from the above superpotential are the following ones:

[TABLE]

Consider now the chiral conformal superfields one can write in this theory:

[TABLE]

where the round brackets denote symmetrization on the indices. The above operators have $k$ indices in the fundamental representation of $\mathrm{SU(3)}$ and $k$ indices in the antifundamental one, but they are not yet assigned to the irreducible representation:

[TABLE]

as it is predicted both by general geometric arguments and by the explicit evaluation of the Kaluza Klein spectrum of hypermultiplets [10]. To be irreducible the operators (3.41) have to be traceless. This is what is implied by the vanishing relation (3.40) if we choose the minus sign in eq.(3.38).

In [12] it was noticed that for $N^{0,1,0}$ the form of the superpotential, which is dictated by the Chern-Simons term, is strongly reminiscent of the superpotential considered in [55]. Indeed the CFT theory associated with $N^{0,1,0}$ has many analogies with the simpler cousin $T^{1,1}$ [8]. However it was stressed in [12] that there is also a crucial difference, pertaining to a general phenomenon that was discussed for the case of compactifications on $M^{1,1,1}$ and $Q^{1,1,1}$ in [7] and [11]. The moduli space of vacua of the abelian theory is isomorphic to the cone ${\cal C}\left(N^{0,1,0}\right)$. When the theory is promoted to a non-abelian one, there are naively conformal operators whose existence is in contradiction with geometric expectations and with the KK spectrum, in this case the hypermultiplets that do not satisfy relation (3.42). Differently from what happens for $T^{1,1}$ [55], the superpotential in eq. (3.39) is not sufficient for eliminating these redundant non-abelian operators.

Ten years later in a paper by Gaiotto et al [56], it was advocated that, maintaining the same flavor-group assignments and the same color group, the color representation assignments of the hypermultiplets that lead to the correct dual CFT are slightly different from those shown in eq. (3.36) since in addition to the bi-fundmental representation one needs also the two fundamental ones.

In any case it is appropriate to stress that, on the basis of the general form of the $\mathcal{N}=3$ gauge theory discussed above as a particular case of the general $\mathcal{N}=2$ theory, it was just in [12] that the structure of an $\mathcal{N}=3,D=3$ Chern-Simons gauge theory, corner stone of the famous ABJM model[39], was for the first time derived in the literature. Indeed in [12] it was just conjectured that the gauge coupling constant flows to infinity at the infrared conformal point, so that the effective lagrangian is obtained from the general one by letting $\mathbf{e}\rightarrow 0$. It was in [12] that the conversion of the $Y^{\Lambda}$ field into a lagrangian multiplier was for the first time observed, leading to the generation of an effective superpotential of type (3.39).

In this paper we continue to explore the properties of the general $\mathcal{N}=2$, $D=3$ gauge theory both from the point of view of its formal structure in superspace and as a starting point for the dual gauge theories associated with M–theory probing $\mathbb{C}^{n}/\Gamma$ singularities and their resolutions. The mathematical aspects of $\mathbb{C}^{3}/\Gamma$ resolutions in relation with the construction of $D=3$ Chern Simons gauge theories is the topic of a forthcoming paper by one of us in collaboration with Ugo Bruzzo that is currently in progress [59].

4 Integral forms in superspace and three–dimensional

Chern–Simons gauge theories

In the present section, we reconsider the construction of the action of the Chern Simons gauge theories under investigation by using the method of Integral Forms and of Picture Changing Operators (PCO’s) developed in [32, 33, 34, 35]. For that purpose, we briefly describe the principles of this method and we give some of the relevant results without a complete derivation. The latter will be published elsewhere [60] since the details of those derivations are not important for the scope of the present paper.

The rhenomic action ${\cal L}^{{\cal N}=2}_{rheo}$, decomposed into pieces as in eqs. (2.26–2.30), is our starting point. It is a 3-form on the superspace ${\cal M}^{(3|4)}$ parametrized locally by the variables $(x^{m},\theta,\theta_{c})$ with dimensions $(3|4)$. The superspace is described in sec. 2.1 and the supervielbeins $(e^{m},\psi,\psi_{c})$ form a supervector. The supervielbein, expanded on the anholonomic basis, can be represented by a supermatrix $(3|4)\times(3|4)$. The rigid superspace is flat, but it has torsion.

The geometrical approach to the supersymmetric field theory under consideration is obtained by writing the Lagrangian ${\cal L}^{(3|4)}$ as a $(3|4)$-integral form integrated on the supermanifold ${\cal M}^{(3|4)}$. As explained in [32, 36, 61] the integral form $\omega^{(3|4)}$ carries a form degree and a second quantum number known in the literature as the picture number. The latter denotes how many delta functions of $\psi$ and of $\psi_{c}$ have to be included in order to integrate over the cotangent space.

To form such an integrand we use the Picture Changing Operator ${\mathbb{Y}}^{(0|4)}$. This a closed yet non-exact integral form which is built in terms of differential forms of the supermanifold and in terms of the Dirac delta functions $\delta(\psi)$ and $\delta(\psi_{c})$. In the literature, the integration on supermanifold is discussed and we do not review here, however we would like to point out that being $\psi=d\theta$ and $\psi_{c}=d\theta_{c}$ commuting variables, they need a special prescription for the integration. For that purpose, the integral forms are the central ingredients.

We also need the differential operators ${\boldsymbol{\iota}}_{\psi}$ and ${\boldsymbol{\iota}}_{\psi_{c}}$, they are the contraction operators with respect to the vector fields $D$ and $D_{c}$, dual to the 1-forms $\psi$ and $\psi_{c}$ and they can be viewed as differential operators acting on $\psi$ and $\psi_{c}$. The following general rules are valid

[TABLE]

This means that $\delta(\psi)$ and $\delta(\psi_{c})$ carry no form degree, but they carry picture number (one each delta function), but a derivative of a delta function plays the role of a negative form with positive picture. So a generic form with maximal picture on the supermanifold $(3|4)$ considered in this paper has the generic expression

[TABLE]

where $p=r-l-k$ is the total form degree and $f(x,\theta,\theta_{c})$ is a generic function. The case $p=0$ admits the cases $r=3$ and $l+k=3$, $r=2$ and $l+k=2$, $r=1$ and $l+k=1$ and $r=l=k=0$. A simple example for a $(0|4)$-integral form reads

[TABLE]

It is closed not exact. Furthermore it can also be written in a more covariant way as $\theta^{4}\delta^{4}(\psi)$ by using 4d spinors. Its supersymmetry transformation under both supersymmetries is $d$-exact

[TABLE]

where $\eta^{(-1|4)}$ is a $(-1|4)$-form which can be easily computed, but its explicit form is irrelevant for our purposes. This $(0|4)$-integral form represents the simplest example of a PCO (in the target space, it has been introduced in pure spinor string theory in [62]).

The rheonomic Lagrangian described in (2.26–2.30), ${\cal L}^{{\cal N}=2}_{rheo}$ has the property

[TABLE]

because of the presence of auxiliary fields $D^{\Lambda},\mathcal{H}^{i}$. Then, we can construct the action as follows

[TABLE]

The integrand is a $(3|4)$-integral form, which can be integrated on the supermanifold. It is proportional to the volume form of ${\cal M}^{(3|4)}$. Certainly, we could have looked for something more general, such as ${\cal L}^{(3|4)}$, but in that case we would have lost the contact with the original rheonomic Lagrangian that we want to use. A search for a more general formulation will be presented elsewhere.

Being ${\mathbb{Y}}^{(0|4)}$ closed and not-exact, it belongs to the cohomology $H^{(0|4)}$ for which we can choose a representative, i.e. (4.3). Then, choosing a different representative means ${\mathbb{Y}}^{(0|4)}+d\Lambda^{(-1|4)}$ and then we have

[TABLE]

and, since ${\cal L}^{(3|0)}$ is closed, $S$ is invariant. A different representative could have additional properties: for example a supersymmetric invariant PCO (see [34] for an illustration of this case in the context of ${\cal N}=1$ Chern-Simons theories).

To begin with, if we choose the PCO given in (4.3), we get

[TABLE]

which implies that the ${\cal L}^{(3|0)}$ is computed by setting $\theta=\theta_{c}=\psi=\psi_{c}=0$, giving the component action (3.2) and the corresponding equations. So, inserting the easiest PCO, one obtains the pull-back of the action on the sub-manifold ${\cal M}^{(3)}\in{\cal M}^{(3|4)}$ and all the superfields are reduced to their first components coinciding with the physical fields.

Let us now consider a different PCO of the following form:

[TABLE]

(we recall that for commuting spinors we have the following identities $\overline{\psi}\cdot\psi=-\overline{\psi}_{c}\psi_{c}$ and $\overline{\psi}\gamma^{p}\psi=\overline{\psi}_{c}\gamma^{p}\psi_{c}$, those relations are also valid for the contration operators ${\boldsymbol{\iota}}_{\psi}$ and ${\boldsymbol{\iota}}_{\psi_{c}}$, i.e. $\overline{\boldsymbol{\iota}}_{\psi}\cdot{\boldsymbol{\iota}}_{\psi}=\overline{\boldsymbol{\iota}}_{\psi_{c}}\cdot{\boldsymbol{\iota}}_{\psi_{c}}$). To check the closure of the PCO (4.9), we act with the differential $d$ and we have

[TABLE]

If the spinors $\psi$ are free to act on Dirac delta function $\delta^{2}(\psi)\delta^{2}(\psi_{c})$, the $d$ variation vanishes. On the other hand, the contraction operators ${\boldsymbol{\iota}}_{\psi}$ and ${\boldsymbol{\iota}}_{\psi_{c}}$ can act on them and then we have

[TABLE]

where the first term vanishes because of ${\rm Tr}(\gamma^{m})=0$, the second term vanishes because $\epsilon_{mnp}{\rm Tr}(\gamma^{m}\gamma^{p})=0$ and the third term vanishes because of the minus sign between the two pieces. Hence the PCO presented in eq.(4.9 ) is closed and it is not exact. The presence of the explicit $\theta$’s in such a formula implies that it is not manifestly supersymmetric. However, its supersymmetry variation is $d$-exact. This implies that the resulting action will not be manifestly supersymmetric with respect to $\mathcal{N}=2$ supersymmetry, rather only with respect to $\mathcal{N}=1$ supersymmetry.

To show how the superspace action is reproduced, we consider here only the kinetic terms of the matter fields. The other pieces of the action can be derived in the same way and we do not present them here, since they give the usual results in superspace. So, we have (here we display only the terms that give non-trivial contributions)

[TABLE]

In the second line, we have picked up all the terms in the ${\cal L}^{Kalher}_{rheo}$ which are proportional to $\overline{\psi}\wedge\psi\wedge e^{m}$ (there are several terms obtained by expanding all differentials in the action, however, they can be collectively re-expressed in one line as above. In the third line we have used the contraction operators ${\boldsymbol{\iota}}_{\psi}$ and ${\boldsymbol{\iota}}_{\psi_{c}}$ to compute the derivatives with respect to $\psi$ and $\psi_{c}$. Then, we are left with a combination of the fermionic superfields $\chi^{i}$ and $\overline{\chi}^{j^{*}}$ and of the $\theta$’s. We have displaced all differential 1-forms $e^{m}$ and the Dirac delta functions to the end of the integrand. By using a simple Fierz re-arrangment we can re-write the action as in the forth line. There, we also made manifest the Berezin integration over $\theta$ and $\theta_{c}$. In the last line we have used the identity $\overline{\theta}\cdot Dz^{i}=\overline{\theta}\cdot\chi^{i}$ and its conjugate to write the final formula. We have used the symbol $[d^{2}\theta d^{2}\theta_{c}]$ to denote the Berezin integration over $\theta$ and $\theta_{c}$ and the result is the correct $\mathcal{N}=2$ Kähler action for the chiral matter multiplets in $\mathcal{N}=2$ superspace. The rest of the action can be derived analogously.

The present formalism encompasses all possible superspace representations of the action from the component action to the superspace action by changing the PCO in the geometrical action (4.6) whose constant essential ingredient is the rheonomic action constructed according to the principles of rheonomy.

5 Quotient singularities

We come now to the mathematics which is of greatest interest to us, in order to address the physical problem at stake, i.e., the construction of CS theories dual to M2-branes that have the metric cone on orbifolds $\mathbb{S}^{7}/\Gamma$ as transverse space. The first step is to show that such metric cone is just $\mathbb{C}^{4}/\Gamma$. This is a rather simple fact but it is of the utmost relevance since it constitutes the very bridge between the mathematics of quotient singularities, together with their resolutions, and the physics of CS theories. The pivot of this bridge is the complex Hopf fibration of the $7$-sphere.

5.1 The complex Hopf fibration of

$\mathbb{S}^{7}$ and quotient singularities $\mathbb{C}^{4}/\Gamma$

In order to arrive at what is for us most interesting, namely quotient singularities of the type $\mathbb{C}^{4}/\Gamma$ we start from the first of the cases listed in table 1, namely the complex Hopf fibration of the seven sphere:

[TABLE]

We want to establish the following important conclusion. Writing the metric cone over the seven sphere as $\mathbb{C}^{4}$, namely:

[TABLE]

the homogeneous coordinates $Z^{i}$ of $\mathbb{C}\mathbb{P}^{3}$ can be identified with the standard affine coordinates of $\mathbb{C}^{4}$ defined above.

To this purpose we consider the standard definition of the $\mathbb{C}\mathbb{P}^{3}$ manifold as the set of quadruplets $\left\{Z^{1},\dots,Z^{4}\right\}$ modulo an overall complex factor:

[TABLE]

On the other hand we define the $7$-sphere as the locus in $\mathbb{C}^{4}$ cut out by the following constraint:

[TABLE]

Let us define the Kähler metric on the $\mathbb{CP}^{3}$ in terms of the homogeneous coordinates:

[TABLE]

That the above is indeed a metric on $\mathbb{CP}^{3}$ is verified in the following way: if in eq.(5.5) $\mathbf{Z}$ is replaced by $\lambda\mathbf{Z}$ all the factors $\lambda$ and all their differentials cancel identically. If we fix the $\lambda$-gauge by setting $Z_{4}\,=1$ and we rename $Z_{1,2,3}\,=\,Y_{1,2,3}$, then we find that the above metric is identical with the Kähler metric obtained from the Fubini-Study Kähler potential:

[TABLE]

On the other hand if we consider the pull-back of the flat Kähler metric of $\mathbb{C}^{4}$ on the locus (5.3) we obtain the metric of the seven sphere:

[TABLE]

Let us next consider the following 1-form:

[TABLE]

and perform the following two calculations. If we replace $\mathbf{Z}\rightarrow\lambda\mathbf{Z}$, we obtain:

[TABLE]

In particular if $\lambda\,=\,e^{\mathrm{i}\theta}$ we get:

[TABLE]

This shows that $\Omega$ is a $\mathrm{U(1)}$-connection on the principal $\mathrm{U(1)}$-bundle that has $\mathbb{CP}^{3}$ as base manifold and which can be identified with the $7$-sphere. The curvature of this connection is just the Kähler $2$-form on $\mathbb{CP}^{3}$.

On the other hand we have:

[TABLE]

If we restrict the above line element to the locus (5.3) we find:

[TABLE]

In this way we have obtained the desired result: the metric cone over the $7$-sphere is described by the homogeneous coordinates of $\mathbb{CP}^{3}$ interpreted as affine ones on $\mathbb{C}^{4}$:

[TABLE]

Another way of stating the same result is the following one. We can regard $\mathbb{C}^{4}$ as the total space of a line bundle over $\mathbb{CP}^{3}$:

[TABLE]

The form $\Omega$ is a connection on this line-bundle.

The consequence of this discussion is that if we have a finite subgroup $\Gamma\subset\mathrm{SU(4)}$, which obviously is an isometry of $\mathbb{CP}^{3}$ we can consider its action both on $\mathbb{CP}^{3}$ and on the seven sphere so that we have:

[TABLE]

We are therefore interested in describing the theory of M2-branes probing the singularity $\frac{\mathbb{C}^{4}}{\Gamma}$.

5.2 From singular orbifolds to smooth resolved manifolds

The next point which provides an important orientation in addressing mathematical questions comes from physics, in view of the final use of the considered mathematical lore in connection with M2-brane solutions of $D=11$ supergravity and later on in the construction of quantum gauge theories supposedly dual to such M2-solutions of supergravity.

Let us start once again from

[TABLE]

namely from eq. (1.8) that we are rewrite in slightly more general terms. The $\mathrm{AdS_{4}}$ compactification of $D=11$ supergravity is obtained by utilizing as complementary $7$-dimensional space a manifold $\mathcal{M}_{7}$ which occupies the above displayed position in the inclusion–projection diagram (5.16). The metric cone $\mathcal{C}(\mathcal{M}_{7})$ enters the game when, instead of looking at the vacuum:

[TABLE]

we consider the more general M2-brane solutions of D=11 supergravity, where the D=11 metric is of the following form:

[TABLE]

$\eta_{\mu\nu}$ being the constant Lorentz metric of $\mathrm{Mink}_{1,2}$ and:

[TABLE]

being a Ricci-flat metric on an asymptotically locally euclidian $8$-manifold $\mathcal{M}_{8}$. In eq. (5.18) the symbol $H(y)$ denotes a harmonic function over the manifold $\mathcal{M}_{8}$, namely:

[TABLE]

Eq.(5.20) is the only differential constraint required in order to satisfy all the field equations of $D=11$ supergravity in presence of the standard M2-brain ansatz for the $3$-form field:

[TABLE]

In this more general setup the manifold $\mathcal{M}_{8}$ is what substitutes the metric cone $\mathcal{C}(\mathcal{M}_{7})$. To see the connection between the two viewpoints it suffices to introduce the radial coordinate $r(y)$ by means of the position:

[TABLE]

The asymptotic region where $\mathcal{M}_{8}$ is required to be locally euclidian is defined by the condition $r(y)\rightarrow\infty$. In this limit the metric (5.19) should approach the flat euclidian metric of $\mathbb{R}^{8}\simeq\mathbb{C}^{4}$. The opposite limit where $r(y)\rightarrow 0$ defines the near horizon region of the M2-brane solution. In this region the metric (5.18) approaches that of the space (5.17), the manifold $\mathcal{M}_{7}$ being a codimension one submanifold of $\mathcal{M}_{8}$ defined by the limit $r\rightarrow 0$.

To be mathematically more precise let us consider the harmonic function as a map:

[TABLE]

This viewpoint introduces a foliation of $\mathcal{M}_{8}$ into a one-parameter family of $7$-manifolds:

[TABLE]

In order to have the possibility of residual supersymmetries we are interested in cases where the Ricci flat manifold $\mathcal{M}_{8}$ is actually a Ricci-flat Kähler $4$-fold.

In this way the appropriate rewriting of eq.(1.8-5.16) is as follows:

[TABLE]

The $\mathcal{N}=8$ case with no singularities.

The prototype of the above inclusion–projection diagram is provided by the case of the M2-brane solution with all preserved supersymmetries. In this case we have:

[TABLE]

On the left we just have the projection map of the Hopf fibration of the $7$-sphere. On the right we have the inclusion map of the $7$ sphere in its metric cone $\mathcal{C}(\mathbb{S}^{7})\equiv\mathbb{R}^{8}\sim\mathbb{C}^{4}$. The last algebraic inclusion map is just the identity map, since the algebraic variety $\mathbb{C}^{4}$ is already smooth and flat and needs no extra treatment.

The singular orbifold cases.

The next orbifold cases are those of interest to us in this paper and in paper [59] which will follow. Let $\Gamma\subset\mathrm{SU(4)}$ be a finite discrete subgroup of $\mathrm{SU(4)}$. Then eq.(5.26) is replaced by the following one:

[TABLE]

The consistency of the above quotient is guaranteed by the relation $\mathrm{SU(4)}\subset\mathrm{SO(8)}$. The question marks can be removed only by separating the two cases:

A)

Case: $\Gamma\subset\mathrm{SU(2)}\subset\mathrm{SU(2)_{I}}\otimes\mathrm{SU(2)_{II}}\subset\mathrm{SU(4)}$. Here we obtain:

[TABLE]

and everything is under full control for the Kleinian $\frac{\mathbb{C}^{2}}{\Gamma}$ singularities and their resolution à la Kronheimer in terms of HyperKähler quotients [41, 42],[58].

B)

Case: $\Gamma\subset\mathrm{SU(3)}\subset\mathrm{SU(4)}$. Here we obtain:

[TABLE]

and the study and resolution of the singularity $\frac{\mathbb{C}^{3}}{\Gamma}$ in a physics–friendly way is the main issue in [59] which is currently on preparation. The comparison of case B) with the well known case A) is the main guide in this venture.

Let us begin by erasing the question marks in case A). Here we can write:

[TABLE]

In the last inclusion map on the right, $\mathrm{Id}$ denotes the identity map $\mathbb{C}^{2}\rightarrow\mathbb{C}^{2}$ while $\mathcal{A}_{W}$ denotes the inclusion of the orbifold $\frac{\mathbb{C}^{2}}{\Gamma}$ as a singular variety in $\mathbb{C}^{3}$ cut out by a single polynomial constraint:

[TABLE]

The variables $u,w,z$ are polynomial $\Gamma$-invariant functions of the coordinates $z_{1},z_{2}$ on which $\Gamma$ acts linearly. The unique generator $W_{\Gamma}(u,w,z)$ of the ideal $\mathcal{I}^{W}_{\Gamma}$ which cuts out the singular variety isomorphic to $\frac{\mathbb{C}^{2}}{\Gamma}$ is the unique algebraic relation existing among such invariants. In [58] the relation was discussed between this algebraic equation and the embedding in higher dimensional algebraic varieties associated with the McKay quiver and the HyperKähler quotient.

Let us now consider case B). Up to this level things go in a quite analogous way with respect to case A). Indeed we can write

[TABLE]

In the last inclusion map on the right, $\mathrm{Id}$ denotes the identity map $\mathbb{C}\rightarrow\mathbb{C}$ while $\mathcal{A}_{\mathcal{W}}$ denotes the inclusion of the orbifold $\frac{\mathbb{C}^{3}}{\Gamma}$ as a singular variety in $\mathbb{C}^{4}$ cut out by a single polynomial constraint:

[TABLE]

Indeed as it will be shown in [59] for the cases $\Gamma\,=\,\mathrm{L_{168}}$ discussed by Markushevich [46] and for its subgroups, the variables $u_{1},u_{2},u_{3},u_{4}$ are polynomial $\Gamma$-invariant functions of the coordinates $z_{1},z_{2},z_{3}$ on which $\Gamma$ acts linearly. The unique generator $\mathcal{P}_{\Gamma}(u_{1},u_{2},u_{3},u_{4})$ of the ideal $\mathcal{I}_{\Gamma}$ which cuts out the singular variety isomorphic to $\frac{\mathbb{C}^{3}}{\Gamma}$ is the unique algebraic relation existing among such invariants. As for the relation of this algebraic equation with the embedding in higher dimensional algebraic varieties associated with the McKay quiver, things are now more complicated and a thorough discussion is going to appear in [59].

Finally let us consider the case of smooth resolutions. In case A) the smooth resolution is provided by manifolds $ALE_{\Gamma}$ and we obtain the following diagram:

[TABLE]

In the above equation the map $\stackrel{{\scriptstyle\mathfrak{H}^{-1}}}{{\longleftarrow}}$ denotes the inverse of the harmonic function map on $\mathbb{C}^{2}\times ALE_{\Gamma}$ that we have already discussed. The map $\stackrel{{\scriptstyle\mathrm{Id}\times qK}}{{\longleftarrow}}$ is instead the product of the identity map $\mathrm{Id}\,:\,\mathbb{C}^{2}\rightarrow\mathbb{C}^{2}$ with the Kähler quotient map:

[TABLE]

of an algebraic variety of complex dimension $|\Gamma|+1$ with respect to a suitable Lie group $\mathcal{F}_{|\Gamma|-1}$ of real dimension $|\Gamma|-1$. Finally the map $\stackrel{{\scriptstyle\mathcal{A}_{\mathcal{P}}}}{{\hookrightarrow}}$ denotes the inclusion map of the variety $\mathbb{V}_{|\Gamma|+1}$ in $\mathbb{C}^{2|\Gamma|}$. Let $y_{1},\dots y_{2|\Gamma|}$ be the coordinates of $\mathbb{C}^{2|\Gamma|}$. The variety $\mathbb{V}_{|\Gamma|+1}$ is defined by an ideal generated by $|\Gamma|-1$ quadratic generators:

[TABLE]

Actually the $|\Gamma|-1$ polynomials $\mathcal{P}_{i}(y)$ are the holomorphic part of the triholomorphic moment maps associated with the triholomorphic action of the group $\mathcal{F}_{|\Gamma|-1}$ on $\mathbb{C}^{2|\Gamma|}$ and the entire procedure from $\mathbb{C}^{2|\Gamma|}$ to $ALE_{\Gamma}$ can be seen as the HyperKähler quotient:

[TABLE]

yet we have preferred to split the procedure into two steps in order to compare case A) with case B) where the two steps are necessarily distinct and separated.

Indeed in case B) we can write the following diagram:

[TABLE]

In this case, just as in the previous one, the intermediate step is provided by the Kähler quotient but the map on the extreme write $\stackrel{{\scriptstyle\mathcal{A}_{\mathcal{P}}}}{{\hookrightarrow}}$ denotes the inclusion map of the variety $\mathbb{V}_{|\Gamma|+2}$ in $\mathbb{C}^{3|\Gamma|}$. In this case the definition of the variety $\mathbb{V}_{|\Gamma|+2}$ is a more complicated issue and it will be presented in [59].

6 Conclusions

In the present paper we have considered the general form of D=3 $\mathcal{N}=2$ gauge theories from three point of views:

The structural point of view, meaning with this the application of the various approaches to the construction of supersymmetric field theories and their relation. In this context we have illustrated the use of the integral form formalism and the extraordinary conceptual advances encoded in the notion of the Picture Changing Operators. It appears in general and it was effectively illustrated in the present case that the time-honored rheonomic lagrangian includes in an implicit way all the other formalisms, the component formalism, the various superfield formalisms and so on. One goes from one formalism to other changing representatives of cohomology classes within a new sophisticated setup of cohomological-algebras associated with supermanifolds that was developed in [32, 33, 34, 35]. Establishing algorithmic transitions back and forth from the component-like approach, which is better suited to geometrical visions, to the superfield approach, which is better suited for quantum calculations, is a new added value, possibly quite relevant in relation with the next aspects of the supersymmetric Chern-Simons theories here discussed. 2. 2.

The geometrical point of view meaning with this the upgrading of the rheonomic lagrangian of D=3 matter coupled gauge theories originally constructed in [7]. That lagrangian has now been rewritten in more general terms utilizing an arbitrary Kähler potential $\mathcal{K}(z,{\overline{z}})$ for the Wess-Zumino multiplet, with an arbitrary Lie group of holomorphic isometries and an arbitrary superpotential $W(z)$. This fully general off-shell construction in the rheonomic approach was so far lacking and the present paper fills the gap. In view of what stated in point i) this is particularly important. 3. 3.

The point of view of applications to the $\mathrm{AdS_{4}/CFT_{3}}$ correspondence and the involved algebro–geometric issues. In the context of the general type of theories described above we have reconsidered the issue of the construction of three-dimensional theories dual to $D=11$ supergravity compactified on $\mathrm{AdS_{4}}\times\mathcal{M}_{7}$ or, better said, to $M2$-brane solutions admitting a Kählerian non compact four-fold $K_{4}$ as transverse space to the brane world–volume. Recalling that the particular type of Chern-Simons gauge theories used in the ABJM-like models arises from the gaussian integration of auxiliary fields and of those physical scalars that at the infrared fixed point loose their kinetic terms, according to a mechanism which was discovered much earlier than ABJM in [9, 10, 12] and there fully utilized, we focused on the ample class of cases where $K_{4}$ is obtained as a Kähler or HyperKähler quotient from a larger flat (Hyper)Kähler variety $\mathbb{V}_{q}$. Using as examples the classified 7-dimensional sasakian coset manifolds $\mathrm{G/H}^{sasak}_{7}$, we have stressed that the data for the construction of the searched for world-volume gauge theory are essentially the same as the geometrical data of the Kähler quotient construction of the transverse space.

Apart from the case of metric cones $\mathcal{C}\left(\mathrm{G/H}^{sasak}_{7}\right)$ over sasakian homogeneous spaces another relevant case of Kähler quotients is provided by the crepant resolutions of singular $\mathbb{C}^{n}/\Gamma$ orbifolds where $\Gamma\subset\mathrm{SU(n)}$ is a finite subgroup of the relevant unitary group. In this case a generalization of the Kronheimer construction of ALE manifolds and a generalization of the McKay correspondence play a crucial role in the resolution of the singularity via Kähler quotient and consequently in the construction of the $D=3$ Chern Simons gauge theory. The subtle and exciting mathematical aspects of these construction constitute the target of the forthcoming paper [59]: in the present paper we have anticipated the general scheme how the geometrical data of singularity resolutions are to be utilized within the context of D=3 gauge theory constructions.

Acknowledgements

One of us (P.F.) acknowledges important clarifying discussions with his good friend and coauthor Ugo Bruzzo and with his long time collaborator Aleksander Sorin, particularly at the beginning of the present project. The two of us are grateful to our frequent collaborators and excellent friends Mario Trigiante, Leonardo Castellani, Carlo Maccaferri and Roberto Catenacci for innumerable always precious conversations and discussions.