Aspects of 4d supersymmetric dynamics and geometry

As an example of a strong coupling effect one can discuss is the notion of duality. A given CFT $A$ in the UV deformed by a relevant perturbation can flow in the IR to a theory (which can be free, gapped, or interacting CFT) which is equivalent to a theory obtained by staring with another CFT $B$ and some relevant deformation. When an RG flow is involved we will refer to such dualities as IR dualities. In case the deformations preserve conformality, that is they are exactly marginal, we will refer to these effects as conformal dualities. See Figure 1. In the last $25$ years, following in particular Seiberg:1994pq , numerous examples of such dualities have been conjectured using a wide variety of exact supersymmetric techniques. The canonical example is the IR equivalence (in a certain range of parameters) of $SU(N)$ SQCD with $N_{f}$ flavors and $SU(N_{f}-N)$ SQCD with $N_{f}$ flavors, gauge singlet fields and a superpotential. The IR fixed point is fully determined (or labeled) by a choice of a pair, UV CFT and the relevant deformation. The statement of the duality is that this labeling is not unique and moreover might be not unique in an interesting way (e.g. the UV theories having different gauge groups). A canonical example of a conformal duality is the statement that ${\cal N}=4$ SYM with gauge groups $G$ and ${}^{L}G$ (the Langlands dual of $G$) reside at two cusps of the same conformal manifold (see e.g. Kapustin:2006pk for a detailed discussion).

Another interesting strong coupling effect is the emergence of symmetry in the IR. The global symmetry of the IR fixed point can be larger than the one of the UV starting point and the interesting question is whether this can be understood directly in the UV. A simple example is $SU(2)$ with $N_{f}$ flavors and symmetry $SU(2N_{f})$ describing in the IR the $SU(N_{f}-2)$ SQCD with $N_{f}$ flavors with only the $SU(N_{f})^{2}\times U(1)$ symmetry visible in the UV of the latter, see e.g. Leigh:1996ds . Other examples involve emergence of supersymmetry in the flow: e.g. using Intriligator-Seiberg duality Intriligator:1995id ²²2See also Lee:2021crt for recent discussion. the $SU(2)$ ${\cal N}=4$ SYM can be obtained by starting from $SU(2)\times SU(2)$ ${\cal N}=1$ SQCD with the charged matter consisting of three bi-fundamentals.³³3Let us mention in passing here for the experts that this duality has a class ${\cal S}$ Gaiotto:2009hg ; Gaiotto:2009we interpretation. One can take two $T_{N}$ trinions and glue them together with ${\cal N}=1$ vector multiplets; flip the two maximal punctures; close then one of the two punctures completely and the other one to a minimal puncture. This theory has the same geometric data as the one of genus one with a single minimal puncture and ${\cal N}=2$ preserving flux, which is ${\cal N}=4$ SYM with a decoupled hypermultiplet. The $N=2$ case is then the Intriligator-Seiberg duality while for higher $N$ one has novel strongly coupled generalizations of this. In our lectures we will review the notions needed to understand the statements in this footnote.

An interesting question is thus to bundle all of the scattered instances of interesting strong coupling dynamics into some uniform framework which will give some sort of an explanation or an understanding of when and how these phenomena occur. Yet another motivation for such a framework can be found by asking a sort of an inverse question: given a strongly coupled SCFT with given properties to find a weakly couple UV theory which flows to it. The UV theory might exhibit less symmetry than the IR one, and in fact it often has to do so. For example, listing all the 4d ${\cal N}=2$ Lagrangians leading to interacting SCFTs Bhardwaj:2013qia one just does not find candidate descriptions of many of 4d ${\cal N}=2$ SCFTs which can be obtained using a variety of more abstract techniques, e.g. coming from string theory.

Typically one considers dualities between QFTs starting from two theories in the same number of dimensions and ending in the same number of dimensions. However, in recent years, following the seminal work of Gaiotto:2009we , it has been realized that much can be achieved, in particular answering the questions posed above, if one discusses flows across dimensions. Let us first then define the notion of an IR duality across dimensions. We can start from a higher dimensional CFT, in this paper we will start in $\widetilde{D}=6$, and deform the theory by placing it on a compact geometry of dimension $D$, e.g. a Riemann surface with $D=2$. Studying this setup at low energy, i.e. much lower than the scale set by the compact geometry, we will arrive at an effective $\widetilde{D}-D$ dimensional QFT. An interesting question is whether there exists a $\widetilde{D}-D$ dimensional weakly coupled QFT flowing to the same effective theory in the IR. If such a theory exists we will refer to the $\widetilde{D}$ deformed theory and the $\widetilde{D}-D$ one as being IR dual across dimensions. See Figure 2.

Here we thus can also label the IR $\widetilde{D}-D$ dimensional QFT by a pair (CFT, deformation) but this might involve a higher dimensional CFT and a geometric deformation. As we will be mainly interested in 4d physics our starting points will be in 6d (and 5d) where interesting SCFTs are all strongly coupled and do not have a simple field theoretic definition, see e.g. Heckman:2018jxk . As the starting point is strongly coupled, such geometric constructions often lead to 4d theories with properties which are hard or impossible to engineer directly in 4d insisting on all the symmetries being manifest. This in particular led to many such theories being referred to as non-Lagrangian. Across dimensional duals, if existing, thus would provide for a Lagrangian definition of such SCFTs. An example of such a duality is a geometric, class ${\cal S}$ Gaiotto:2009hg ; Gaiotto:2009we , construction as compactification on punctured spheres of a $(2,0)$ 6d SCFT of certain ${\cal N}=2$ Argyres-Douglas SCFTs Argyres:1995jj , and an alternative description of these SCFTs starting with a ${\cal N}=1$ weakly coupled gauge theory in 4d Maruyoshi:2016tqk .

As with in-dimension dualities some of the symmetries of the IR QFT might be explicit in the UV in one description but emergent in the other. Systematically understanding such across dimensional dualities will give us a handle over understanding of emergence of symmetry. In fact the appearance of geometry in the construction gives us a useful knob to start and build a systematic framework to understand emergence of symmetry and duality.

The main idea behind deriving 4d dualities and emergence of symmetry phenomena from 6d stems from the following factorization property of the constructions. One considers compactifying a given 6d SCFT, $T_{6d}$, on surface ${\cal C}$ such that the surface can be written as,

$\displaystyle{\cal C}={\cal C}_{1}\oplus{\cal C}_{2}\oplus\cdots\oplus{\cal C}_{\ell}\,.$

(1)

Here ${\cal C}_{i}$ are punctured surfaces and $\oplus$ is the geometric operation of gluing two surfaces along a puncture. The compactification might be parametrized by additional geometric data, such as flux for a global symmetry supported on the surface. The operation $\oplus$ then associates to the combined surface a sum of these fluxes. We will call a set of surface ${\cal C}_{i}$ complete if any surface ${\cal C}$ can be constructed using these. We then first seek for across dimensional dualities associating a pair of 4d weakly coupled CFTs and deformations $(T_{{\cal C}_{i}},\,\Delta_{i})$ to $(T_{6d},\,{\cal C}_{i})$. Assuming that such a dictionary between a complete set of surfaces ${\cal C}_{i}$ and 4d theories could be found one can find a theory dual across dimensions to $(T_{6d},\,{\cal C})$,

$\displaystyle(T_{\cal C},\Delta)=(T_{{\cal C}_{1}},\,\Delta_{1})\otimes(T_{{\cal C}_{2}},\,\Delta_{2})\otimes\cdots\otimes(T_{{\cal C}_{\ell}},\,\Delta_{\ell})\,.$

(2)

Here $\otimes$ is a field theoretic operation in 4d. This can involve gauging symmetry associated with punctures (in a way we will discuss in detail in the paper), adding fields, and turning on various superpotentials couplings. The precise meaning of $\otimes$ will depend on the 6d SCFT and various other choices. For example, the complete set of ${\cal C}_{i}$ might not be unique; one can have different types of punctures (following from UV dualities exemplified in Figure 1), etc. In the statement (2) we have several RG flows, and thus hidden in it there is an assumption that these types of flows commute. Commutativity of flows is a non-trivial statement (see e.g. Aharony:2013dha and discussion below). However, assuming it and the statement (2) one can arrive at a large web of consistent results which supports the validity of the assumption.

The fact that such a construction exists is highly non-trivial: however as we will discuss in this review in various examples it can be worked out explicitly and by now somewhat systematically. An example of such a construction is the derivation of all compactifications of the $A_{1}$ $(2,0)$ 6d SCFT using tri-fundamentals of $SU(2)$ as across dimensional duals to compactifications on certain three-punctured spheres Gaiotto:2009we . Given such a structure one can then generate various examples of dualities and instances of IR emergence of symmetry systematically. For example, if a given surface can be decomposed in more than one way,

$\displaystyle{\cal C}$

$\displaystyle=$

$\displaystyle{\cal C}^{(1)}_{1}\oplus{\cal C}^{(1)}_{2}\oplus\cdots\oplus{\cal C}^{(1)}_{\ell}={\cal C}^{(2)}_{1}\oplus{\cal C}^{(2)}_{2}\oplus\cdots\oplus{\cal C}^{(2)}_{\ell^{\prime}}\,,$

(3)

we will obtain different field theoretic descriptions of $(T_{\cal C},\Delta)$ which should be IR equivalent by construction,

	$\displaystyle(T_{\cal C},\Delta)$	$\displaystyle=$	$\displaystyle(T_{{\cal C}^{(1)}_{1}},\,\Delta^{(1)}_{1})\otimes(T_{{\cal C}^{(1)}_{2}},\,\Delta^{(1)}_{2})\otimes\cdots\otimes(T_{{\cal C}^{(1)}_{\ell}},\,\Delta^{(1)}_{\ell})\,$
		$\displaystyle=$	$\displaystyle(T_{{\cal C}^{(2)}_{1}},\,\Delta^{(2)}_{1})\otimes(T_{{\cal C}^{(2)}_{2}},\,\Delta^{(2)}_{2})\otimes\cdots\otimes(T_{{\cal C}^{(2)}_{\ell^{\prime}}},\,\Delta^{(2)}_{\ell^{\prime}})\,.$

Moreover the geometry ${\cal C}$ might preserve more symmetry than some of the ${\cal C}_{i}$ building blocks. Then the corresponding theories $(T_{{\cal C}_{i}},\,\Delta_{i})$ would have less symmetry than $(T_{{\cal C}},\,\Delta)$. However combining them together using (2) should give in the IR the expected symmetry, if the construction is correct. Building the necessary toolkit to discuss such correspondences and discussing the systematics of these constructions will be the main goal of this review.

We will discuss two explicit instances of collections of across dimension dualities, one starting with $T_{6d}$ which is described by pure $SU(3)$ SYM on its tensor branch and another with $T_{6d}$ being the rank one E-string. These two cases turn out to be rather simple and amenable to a variety of ad hoc techniques, which we will discuss in detail. The systematic treatment of compactifications starting with a large class of more general $T_{6d}$ can be done understanding first compactifications on two punctured spheres. The problem of understanding reductions on such surfaces can be directly related to understanding duality domain walls in 5d Gaiotto:2014ina . We will discuss this procedure using again E-string as an explicit example. Understanding systematically compactifications on surfaces with more punctures can be done studying the interplay between 6d flows and across dimension flows, which we will soon describe.

Of course once we allow for across dimensional dualities we can consider starting and finishing in different combinations of dimensions. There has been a lot of work for example on compactifying 6d theories down to three dimensions and constructing 3d Lagrangians for these, see e.g. Dimofte:2011ju . However, there is another interesting phenomenon that we want to mention here. One can start from two different 6d SCFTs, $T_{6d}^{A}$ and $T_{6d}^{B}$, deform by placing them on different geometries, ${\cal C}^{A}$ and ${\cal C}^{B}$, and consider the resulting four dimensional theories. In certain cases the 4d theories might be IR equivalent and we thus can refer to the pairs $(T_{6d}^{A},{\cal C}^{A})$ and $(T_{6d}^{B},{\cal C}^{B})$ as being 6d dual to each other. By now there are numerous examples of such dualities, though there is no systematic understanding of these. It is likely that to gain such an understanding one would need to exploit various string/M-theoretic constructions. For example, various compactifications of $(2,0)$ theories on spheres with punctures leading to ${\cal N}=2$ theories in 4d turn out to be equivalent to compactifications on tori of different $(1,0)$ theories Ohmori:2015pua ; Ohmori:2015pia ; Baume:2021qho ; compactifications on certain punctured spheres of N5 branes probing $A$-type singularity is 6d dual to compactifications on tori with flux of M5 branes probing $D$-type singularity Kim:2018bpg . We will discuss yet another example of this phenomenon (Rank-one E-string on genus two surface without flux is the same as minimal $SU(3)$ 6d SCFT on a four punctured sphere).

6d theories do not possess interesting supersymmetric relevant or marginal deformations Cordova:2016xhm . However, given a 6d SCFT one can trigger an interesting flow by exploring the moduli space of vacua, i.e. turning on vacuum expectation values for some operators. One then can wonder how flows in six dimensions and across dimensions are interrelated. It turns out that the answer to this question is rather interesting and understanding it gives an explicit tool to derive compactifications on surfaces with more than two punctures. The basic idea is depicted in Figure 5. Let us consider two QFTs in 6d related by a flow triggered by a vev to some operator ${\cal O}_{6d}$, $QFT_{1}$ and $QFT_{2}$. Next we compactify $QFT_{1}$ on some surface ${\cal C}_{A}$ with some value of flux for the 6d global symmetry. Usually one can find the 4d analogue, ${\cal O}_{4d}$, of the operator ${\cal O}_{6d}$, the vev of which connects $QFT_{1}$ and $QFT_{2}$ in 6d. Turning on a vev to ${\cal O}_{4d}$ we flow then to a new QFT in 4d. A natural question is whether there is a surface ${\cal C}_{B}$ (and some value of flux) such that the same 4d QFT can be reached starting with $QFT_{2}$ in 6d. It turns out that the answer is positive but ${\cal C}_{B}$ differs from ${\cal C}_{A}$ by the number of punctures: the difference being determined by the value of flux for symmetries under which ${\cal O}_{6d}$ is charged. Thus understanding say compactifications on two punctured spheres for $QFT_{1}$ we can systematically derive compactifications on spheres with higher number of punctures for $QFT_{2}$. We will discuss in detail this procedure for a sequence of 6d SCFTs called $(D_{N+3},D_{N+3})$ minimal conformal matter theories: $(1,0)$ SCFTs residing on a single M5 brane probing $D_{N+3}$ singularity in M-theory. Models with different values of $N$ are related by RG flows reducing the value of $N$. The simplest model with $N=1$ turns out to be the rank one E-string. In particular this will give us yet another derivation of the three punctured spheres for the E-string theory.

The idea of these lectures is to allow a reader familiar with the basics of supersymmetric dynamics in 4d (e.g. at the level of first chapters of Tachikawa:2018sae ) to familiarize themselves in detail, and in a self contained way, with the techniques and results used to geometrically construct 4d dynamics. We tried to either explicitly review all the needed material, or to cite a paper discussing manifestly and in detail the needed points. Although string/M/F-theoretic considerations are very useful in understanding various aspects of our discussion, we restricted to purely field theoretic exposition for the sake of the lectures being self contained. These lectures are not intended to be an exhaustive review of the subject but rather a pedagogical and self contained exposition of a particular slice of recent developments.

The paper is structured as follows. In section 2 we overview the basic techniques to deal non-perturbatively with ${\cal N}=1$ supersymmetric QFTs in 4d. This involves in particular the review of exact tools to extract information about the IR fixed point in 4d flows. In section 3 we discuss in detail examples of IR dualities, conformal dualities, and IR emergence of symmetry. We discuss several test cases to illustrate the applications of the techniques of section 2. Next in section 4 we analyze in full detail the compactification of 6d minimal $SU(3)$ SCFT to 4d. In particular we will discuss the 6d SCFT itself, its reduction to 5d, and the resulting 4d theories. This is a very tractable case (e.g. the analysis is simplified by the 6d theory not having continuous global (non-R) symmetry). We will illustrate various understandings following from compactifications using this example. In section 5 we discuss in detail compactifications of the rank one E-string. Here we will use non generic but simple techniques to get to the result in a non-cumbersome manner. In section 6 we discuss a systematic approach of studying 5d duality domain walls and commutative diagrams of RG flows. We use these techniques to arrive at the 4d models obtained in compactifications of the E-string theory. Finally in section 7 we discuss some generalities of geometrically engineering 4d SQFTs starting from 6d SCFTs, comment on subjects not covered in the lectures, and make general remarks. Several appendices include pedagogically various topics in 4d/5d/6d supersymmetric physics as well as give more details of some of the computations in the bulk of the paper.

Without simplifying assumptions, such as supersymmetry that we will soon introduce, answering the posed questions is rather hard. Nevertheless, we do have certain tools which we can use quite generally. Not surprisingly they have something to do with symmetry. Let us assume that after deforming $CFT_{UV}$ the global symmetry of the resulting QFT is $\mathscr{G}_{UV}$. Here we will focus on zero-form continuous symmetries, though one can consider generalizations of the discussion to higher form symmetries Gaiotto:2014kfa and higher group structures Benini:2018reh . Typically the deformation breaks explicitly some of the symmetry of $CFT_{UV}$ and some of the symmetry might also be spontaneously broken: lets assume $\mathscr{G}_{UV}$ is the surviving fraction of the symmetry. This symmetry will be preserved during RG flow. Thus, we expect that the symmetry of the theory in the IR should be $\mathscr{G}_{UV}$. There are however two caveats. First, $\mathscr{G}_{UV}$ might not act faithfully in the IR. In particular in an extreme case it might not act at all, for example when the theory develops a gap in the IR. In this case, $\mathscr{G}_{UV}$ will not act manifestly in the IR. Second, the symmetry in the IR can actually be bigger than $\mathscr{G}_{UV}$. Intuitively, some of the interactions/degrees of freedom which break a certain larger symmetry group, of which $\mathscr{G}_{UV}$ is a subgroup, might be washed away/gapped in the IR. In such situations we will say that there is an enhancement (or emergence) of the symmetry in the IR and the symmetry group $\mathscr{G}_{IR}$ might be bigger than $\mathscr{G}_{UV}$. We will denote by $\mathscr{G}_{IR}$ the symmetry group of the IR fixed point if the theory flows to an SCFT, or the symmetry of the weakly coupled gauge theory if it flows to such a theory.

Importantly, we can say something more about the symmetry. If a theory possess a global symmetry with a corresponding conserved current⁴⁴4One can make this discussion more general and abstract by thinking about the conserved charge corresponding to a zero-form symmetry as certain co-dimension one topological operators in the QFT (and higher co-dimension when the symmetry is of a higher form) Gaiotto:2014kfa . $J_{\mu}$, we can turn on background gauge fields $A_{\mu}$, valued in the Lie algebra of some sub-group of the symmetry group, coupled to this conserved current, and compute the effective action $\Gamma[A]$. As the current is conserved the gauge field comes with a gauge symmetry,

$\displaystyle A\to A^{g}\equiv g\cdot A\cdot g^{-1}+(dg)\cdot g^{-1}\,,$

(5)

where $g$ is an element of the symmetry (sub)group. Then we can try to promote $A_{\mu}$ to be dynamical fields. However, there might be an obstruction to doing so, which goes under the name of ‘t Hooft anomaly. The obstruction comes about as the effective action of the theory might or might not be invariant under (5),

$\displaystyle\Gamma\left[A^{g}\right]\overset{?}{=}\Gamma\left[A\right].$

(6)

If the equality does not hold we say that the (sub)group of the symmetry has a ‘t Hooft anomaly. In particular this means that the symmetry cannot be gauged, i.e. the gauge fields $A_{\mu}$ cannot be promoted to dynamical fields. See Bilal:2008qx for a comprehensive introduction to the subject of anomalies and Freed:2014iua for a more recent and general discussion. The important fact about ‘t Hooft anomalies is that they are quantifiable. There are various ways to understand this and let us here mention two of them. First, the anomaly of a continuous symmetry can be captured in $D=2n$ dimensions by an $n+1$ point one loop amplitude involving the conserved currents. In particular, say in $D=4$, this is proportional to,

$\displaystyle a_{G_{1},G_{2},G_{3}}=\mathop{\mathrm{Tr}}\nolimits_{\mathfrak{R}}G_{1}G_{2}G_{3}\,.$

(7)

Here ${\mathfrak{R}}$ is the representation of the chiral fermions of the model (if we have a description of the theory in terms of a Lagrangian) under the $G_{i}$ which are the groups corresponding to the three currents. For example, if $G_{i}$ all correspond to the same $U(1)$ symmetry then the ‘t Hooft anomaly is just given by $\sum_{l=1}^{N}q_{l}^{3}$, where $q_{l}$ are the charges of the $N$ Weyl fermions of the model.

Another way to think about the anomaly is as follows. The failure of the effective action to be invariant cannot take any form and is constrained by Wess-Zumino consistency conditions to be related to what is known as the anomaly polynomial ${\cal I}_{D+2}$ (where $D$ is the number of space-time dimensions which is assumed to be even). ${\cal I}_{D+2}$ is a homogeneous polynomial of degree $D+2$ in characteristic classes built from gauge fields (including gravitational) corresponding to the symmetries of the system. The coefficients of the polynomial encode the ‘t Hooft anomalies of the theory. For example a term of the form,

$\displaystyle n_{123}\,\frac{a_{G_{1},G_{2},G_{3}}}{6}\,\mathop{\mathrm{Tr}}\nolimits\,F_{1}\wedge F_{2}\wedge F_{3}\subset{\cal I}_{6}\,,$

(8)

captures the anomaly $a_{G_{1},G_{2},G_{3}}$ defined in (7). Here $n_{123}$ is a simple numerical factor depending on how many of the three $F_{i}$ correspond to different groups. If all are the same then $n_{123}=1$, if two are the same $n_{123}=3$ and if all are different it is equal to $6$. See Appendix E for more details on the 6d anomaly polynomial.

‘t Hooft anomalies are useful for us since they don’t change during the RG-flow and thus are the same in the UV and the IR: this fact goes under the name of ‘t Hooft anomaly matching condition. There are various ways to see this. First, we can add to the theory a number of Weyl fermions (called spectators), such that the combined anomaly of the theory of interest and the additional fermions is zero. Then we can weakly gauge the symmetry. Assuming we can always stay in a regime where the gauge coupling is small, we can flow to the IR where we decouple the spectators. As there is no obstruction to the gauging in the IR, and we decouple the same fermions, the anomaly should not change in the process. Yet another argument for the non-renormalization is through what is called anomaly inflow. One way to render the symmetry non-anomalous is to realize the theory on the boundary of a $D+1$ dimensional space with a background field $A$ living in $D+1$ dimensions. Then, we can add a Chern-Simons term in the bulk that cancels the anomaly of the boundary theory. This will again remove the obstruction from gauging the symmetry which should hold in the UV as well as in the IR.

Thus, we learn that the anomaly polynomial ${\cal I}_{6}$ computed in the UV should be the same as the one computed in the IR, no matter what is the IR behavior of our deformation. This is true for the symmetries we can identify in the UV. At the IR fixed point new symmetries can emerge and as we cannot identify them in the UV their anomalies do not have to be zero. One way to understand this is that we can move out of the IR fixed point by irrelevant deformations which explicitly break the emergent symmetries and thus invalidate the ‘t Hooft anomaly matching argument. On the other hand, if a sub-group of the symmetry group does not act faithfully in the IR its anomaly has to be zero also in the UV due to the ‘t Hooft anomaly matching argument⁵⁵5Note that this is not true for the case of spontaneous symmetry breaking. A symmetry with a non-trivial ‘t Hooft anomaly in the UV can be spontaneously broken in the IR..

Until now our discussion did not involve supersymmetry at all. Indeed, without supersymmetry we do not have at the moment useful robust tools beyond matching symmetries and anomalies to understand the physics in the IR. However with supersymmetry there are more things that we can say. Let us start the discussion of supersymmetric theories in $D=4$ with what the interplay between the supersymmetry and ‘t Hooft anomalies can give us.

We will assume throughout these lectures that we are dealing with ${\cal N}=1$ supersymmetric theories in 4d . Moreover we will assume that these theories possess an R-symmetry. A $U(1)$ R-symmetry is a necessary ingredient of the ${\cal N}=1$ superconformal group, as for example it appears on the right-hand side of anti-commutation relations between supercharges $Q_{\alpha}$ and their superconformal counterparts $S_{\alpha}$. See Appendix A for a summary of the ${\cal N}=1$ superconformal group in 4d. The SCFT in the UV thus has an R-symmetry which is part of the superconformal group. We will only discuss deformations which preserve ${\cal N}=1$ supersymmetry. We will also assume that some combination of this R-symmetry and an abelian subgroup of the global symmetry group of $CFT_{UV}$ is not broken by the deformation, though of course as we introduce a scale the conformal symmetry is broken. In the IR, if we arrive to a conformal fixed point, we again acquire the superconformal R-symmetry. However, the superconformal symmetry in the UV and in the IR might not be the same symmetry. Nevertheless, under certain assumptions that we will detail, the fact that the R-symmetry is intimately related to the superconformal group allows us to determine it in the IR.

Any conformal theory, supersymmetric or not, in four dimensions has two important numbers associated to it: these are referred to as the $a$ and the $c$ conformal anomalies. The conformal anomalies measure, among other things, the failure of the expectation value of the trace of the stress-energy tensor to vanish when the theory is placed on a curved background with metric $g_{\mu\nu}$,

$\displaystyle\langle{T^{\mu}}_{\mu}\rangle=\frac{c}{16\pi^{2}}W^{2}-\frac{a}{16\pi^{2}}E_{4}\,,$

(9)

where $W$ is the Weyl tensor and $E_{4}$ is the Euler density, both built from certain combinations of the metric $g_{\mu\nu}$ and its derivatives Deser:1993yx .⁶⁶6In any conformal theory the $a$ conformal anomaly is smaller at the IR fixed point relative to the UV fixed point Cardy:1988cwa ; Komargodski:2011vj . In superconformal theories, as the stress energy tensor and the R-symmetry are part of the same symmetry algebra the various anomalies are interrelated. It is possible for example to utilize these relations to arrive at the following extremely useful statements Anselmi:1997am ,

$\displaystyle a=\frac{9}{32}\mathop{\mathrm{Tr}}\nolimits R^{3}-\frac{3}{32}\mathop{\mathrm{Tr}}\nolimits R\,,\qquad c=\frac{9}{32}\mathop{\mathrm{Tr}}\nolimits R^{3}-\frac{5}{32}\mathop{\mathrm{Tr}}\nolimits R\,.$

(10)

Here it is important that $R$ is the R-symmetry in the superconformal group. $\mathop{\mathrm{Tr}}\nolimits R^{3}$ is the ‘t Hooft anomaly corresponding to $\mathop{\mathrm{Tr}}\nolimits F^{3}$ term in the anomaly polynomial we have discussed above, with $F$ being the field strength for the R-symmetry background gauge field. Whereas $\mathop{\mathrm{Tr}}\nolimits\,R$ is the mixed R-symmetry gravity anomaly, which in the anomaly polynomial appears as a coefficient of a term involving $\mathop{\mathrm{Tr}}\nolimits F$ and a certain Pontryagin class computed using the background metric.

Exploiting the interplay between supersymmetry and the R-symmetry in a conformal theory one can also arrive at the conclusion that the mixed ‘t Hooft anomalies between any global $U(1)$ symmetry and the superconformal R-symmetry are also related to the mixed gravitational-$U(1)$ anomaly Intriligator:2003mi ,

$\displaystyle\mathop{\mathrm{Tr}}\nolimits\,U(1)=9\mathop{\mathrm{Tr}}\nolimits\,U(1)\,R^{2}\,.$

(11)

Moreover, the positivity of the two point function of the currents of the global $U(1)$ symmetry can be related to the negativity of yet another ‘t Hooft anomaly,

$\displaystyle\mathop{\mathrm{Tr}}\nolimits U(1)^{2}\,R<0\,.$

(12)

Combining all these observations Intriligator and Wecht arrived at a very simple procedure to determine the R-symmetry of the IR fixed point by knowing all the abelian symmetries and the R-symmetry preserved along the RG-flow Intriligator:2003mi . One defines the following trial $a$ anomaly,

$\displaystyle a(\{\alpha\})=\frac{9}{32}\mathop{\mathrm{Tr}}\nolimits(R+\alpha_{i}U(1)^{(i)})^{3}-\frac{3}{32}\mathop{\mathrm{Tr}}\nolimits(R+\alpha_{i}U(1)^{(i)})\,.$

(13)

Here $\alpha_{i}$ are arbitrary real numbers associated to the $i$-th $U(1)$ symmetry: the summation over $i$ is implied in the equation. Now we notice that,

$\displaystyle\frac{32}{3}\,\frac{\partial}{\partial\alpha_{j}}a(\{\alpha\})=9\mathop{\mathrm{Tr}}\nolimits(R+\alpha_{i}U(1)^{(i)})^{2}U(1)^{(j)}-\mathop{\mathrm{Tr}}\nolimits U(1)^{(j)}=0\,.$

(14)

The last equality holds if $(R+\alpha_{i}U(1)^{(i)})$ is the superconformal R-symmetry following (11). On the other hand,

$\displaystyle\frac{16}{27}\,\beta_{j}\frac{\partial}{\partial\alpha_{j}}\beta_{k}\frac{\partial}{\partial\alpha_{k}}a(\{\alpha\})=\mathop{\mathrm{Tr}}\nolimits(R+\alpha_{i}U(1)^{(i)})\beta_{j}U(1)^{(j)}\beta_{k}U(1)^{(k)}<0\,,$

(15)

where $\beta_{i}$ are arbitrary real numbers and the last inequality follows from (12) assuming again $R+\alpha_{i}U(1)^{(i)})$ is the superconformal R-symmetry. This thus implies that:

The superconformal R-symmetry maximizes the trial $a(\{\alpha\})$.

The procedure of obtaining the superconformal R-symmetry discussed here is called $a$-maximization. This statement however has an important caveat. We assume that we have identified all the $U(1)$ symmetries that can possibly mix with the R-symmetry to produce the superconformal one. However, as we already discussed some of these symmetries can only emerge in the IR. This possibility should always be kept in mind, as it is usually hard to rule it out. In some cases, certain indications that this has to be the case can be derived. For example, using certain unitarity bounds, superconformal representation theory implies that the R-charge of any chiral operator in the theory has to be bigger or equal to $\frac{2}{3}$. The R-charge saturates this bound only if the operator is a free chiral field. If using $a$-maximization leads to certain chiral operators violating these bounds, then some of the assumptions going into the computation have to be wrong. A natural way for this to happen is if some of the abelian symmetries in the IR have been missed Kutasov:2003iy .

Using the relation (10) we can compute the conformal anomalies of free fields, which will be useful for us in what follows. A free chiral superfield $Q$ has an R-charge of $\frac{2}{3}$. This is the R-charge of the scalar component of the superfield. The R-charge of the Weyl fermion is shifted⁷⁷7In superspace notations this is due to the fact that the superspace coordinates have a unit charge under the R-symmetry. by $-1$ and thus the conformal anomalies of the free chiral field are,

$\displaystyle a=\frac{9}{32}(\frac{2}{3}-1)^{3}-\frac{3}{32}(\frac{2}{3}-1)=\frac{1}{48}\,,\qquad c=\frac{9}{32}(\frac{2}{3}-1)^{3}-\frac{5}{32}(\frac{2}{3}-1)=\frac{1}{24}\,.$

(16)

For the free vector superfield the R-symmetry is zero, and thus the R-symmetry of the gaugino is $+1$. This implies the following anomalies,

$\displaystyle a=\frac{9}{32}-\frac{3}{32}=\frac{3}{16}\equiv a_{v}\,,\qquad c=\frac{9}{32}-\frac{5}{32}=\frac{1}{8}\equiv c_{v}\,.$

(17)

For future convenience we also define the contributions to the conformal anomalies of chiral fields of general R-charge $r$ to be,

$\displaystyle a_{R}(r)=\frac{9}{32}(r-1)^{3}-\frac{3}{32}(r-1)\,,\qquad c_{R}(r)=\frac{9}{32}(r-1)^{3}-\frac{5}{32}(r-1)\,.$

(18)

We assume that a gauge group ${\cal G}$ is given, the number of chiral fields is $\text{dim}\,{\mathfrak{R}}$, and that the assignment of R-charge $\frac{2}{3}$ to all chiral superfields is consistent with superpotentials and anomalies. We also assume that the theory has $L$ $U(1)_{\ell}$ symmetries under which the $i$-th chiral superfield has charges $Q^{\ell}_{i}$. Then we define a trial superconformal R-symmetry to be,

$\displaystyle R^{tr}_{i}(\{\alpha\})=\frac{2}{3}+Q^{\ell}_{i}\alpha_{\ell}\,,$

(19)

with $\alpha_{\ell}$ arbitrary real parameters. Computing the trial $a$-anomaly,

	$\displaystyle\widetilde{a}(\{\alpha\})$	$\displaystyle\equiv$	$\displaystyle\frac{32}{3}a(\{\alpha\})-a_{v}\,\text{dim}\,{\cal G}=\sum_{i=1}^{\text{dim}\,{\mathfrak{R}}}\left(3(\frac{2}{3}+\alpha_{\ell}Q^{\ell}_{i}-1)^{3}-(\frac{2}{3}+\alpha_{\ell}Q^{\ell}_{i}-1)\right)$
		$\displaystyle=$	$\displaystyle\frac{2}{9}\,\text{dim}\,{\mathfrak{R}}+3\sum_{i=1}^{\text{dim}\,{\mathfrak{R}}}\left(\alpha_{\ell}\,\alpha_{\ell^{\prime}}\,\alpha_{\ell^{\prime\prime}}Q_{i}^{\ell}Q_{i}^{\ell^{\prime}}Q_{i}^{\ell^{\prime\prime}}-\alpha_{\ell}\,\alpha_{\ell^{\prime}}Q_{i}^{\ell}Q_{i}^{\ell^{\prime}}\right)\,,$

we immediately see that the term linear in $\alpha_{\ell}$ cancels out. This implies that taking $\alpha_{\ell}=0$ is a stationary point and it is then trivial to show that it is a local maximum. One way to see this is to take some arbitrary direction in $\{\alpha\}$ space parametrized as $\alpha_{\ell}=n_{\ell}\,t$ and see how $\widetilde{a}(\{\alpha\})$ changes with $t$,

$\displaystyle\widetilde{a}(\{\alpha\})=\frac{2}{9}\,\text{dim}\,{\mathfrak{R}}+3\left(\sum_{i=1}^{\text{dim}\,{\mathfrak{R}}}\Delta R_{i}^{3}\right)\,t^{3}-3\left(\sum_{i=1}^{\text{dim}\,{\mathfrak{R}}}\Delta R_{i}^{2}\right)t^{2}\,,$

(21)

where we defined $\Delta R_{i}=Q^{\ell}_{i}n_{\ell}$. Obviously then $t=0$ is a maximum for any choice of direction $n_{\ell}$ unless $\forall\,i$ $\Delta R_{i}=0$. However the latter case implies that $R^{tr}_{i}(\{n\,t\})=\frac{2}{3}$, and thus the R-symmetry is still the free one.

$\square$

An important question regarding CFTs is what is the collection of relevant and exactly marginal deformations of a model leading to an inequivalent fixed point. In particular for SCFTs we will be interested in such deformations which preserves ${\cal N}=1$ supersymmetry.

One type of supersymmetric deformations one can add to an SCFT is the following superpotential term,

$\displaystyle W=\lambda_{\cal O}\,\int d^{4}xd^{2}\theta\,{\cal O}+c.c.\,,$

(22)

where ${\cal O}$ is a chiral operator and $\lambda_{\cal O}$ is a complex number. Relevant supersymmetric deformations are given by chiral operators ${\cal O}$ with scaling dimension smaller than $3$. For such deformations $\lambda_{\cal O}$ has a positive mass dimension and thus grows with the RG-flow to the IR. Remember that $d\theta$ has mass dimension $+\frac{1}{2}$. Note also, that for chiral operators the superconformal algebra relates the dimension to the superconformal R-charge, $\Delta=\frac{3}{2}\,R$; thus, the R-charge of relevant deformations ${\cal O}$ is smaller than $2$.

The marginal superpotential deformations are given by chiral operators ${\cal O}$ of dimension exactly $3$, or equivalently superconformal R-charge of exactly $2$, and as we will soon discuss these can be either marginally irrelevant or exactly marginal.

Given an SCFT with a subgroup $G$ of the global symmetry group with vanishing ’t Hooft anomaly we can couple it to dynamical gauge fields in the standard way. In particular we introduce the superpotential term,

$\displaystyle W=\tau_{G}\int d^{4}xd^{2}\theta\,W_{\alpha}W^{\alpha}+c.c.\,,$

(23)

with $\tau$ being the complexified gauge coupling and $W_{\alpha}$ the chiral superfield with the field strength $F_{\mu\nu}$ as one of its components. As with other superpotential couplings $\tau$ can be either a relevant deformation, irrelevant, or marginal. To determine which of the three possibilities holds one needs to perform a one loop computation of the gauge beta-function: classically gauge-couplings are marginal in 4d. In supersymmetric theories the result of this computation can be expressed in an elegant form Benini:2009mz ,⁸⁸8See also Meade:2008wd for an important discussion.

$\displaystyle\beta_{G}\propto-\mathop{\mathrm{Tr}}\nolimits R_{{\cal P}}\,G^{2}\,.$

(24)

That is the beta function is proportional to minus the ’t Hooft anomaly of the superconformal R-symmetry, denoted here by $R_{{\cal P}}$, and two currents of the gauged symmetry. The proportionality coefficient is a positive number which will not play a role for us (and can be easily fixed by computing it for simple Lagrangians). The reason this expression can be derived is because the superconformal R-symmetry is in the same multiplet as the stress-energy tensor. In particular, if $\mathop{\mathrm{Tr}}\nolimits R_{{\cal P}}\,G^{2}$ is positive the gauging is UV free, if $\mathop{\mathrm{Tr}}\nolimits R_{{\cal P}}\,G^{2}$ is negative the gauging is IR free, and if $\mathop{\mathrm{Tr}}\nolimits R_{{\cal P}}\,G^{2}=0$ the gauging is marginal. Note that this way of determining relevance of a gauging does not rely on a description of the theory in terms of weakly coupled fields and thus will turn out to be useful also for strongly coupled SCFTs. Note also that the marginality of gauging is determined by whether the superconformal R-symmetry at the fixed point is anomalous or not: that is whether it is consistent with the gauging. This is very analogous to marginality of superpotentials: marginal superpotentials do not break R-symmetry.

Finally, we need to determine whether the marginal deformations we discussed are exactly marginal, marginally irrelevant, or marginally relevant. In fact for supersymmetric theories with supersymmetry preserving deformations the last option is not possible. The reason is as follows Green:2010da . When deforming a theory by a supersymmetric marginal operator, in order for the deformation to cease being marginal its dimension needs to change after the RG-flow. However, such deformations are chiral and as such form shortened, protected, representations of the superconformal symmetry group. The only way for such a deformation to cease being marginal is for it to recombine during the flow with another shortened multiplet to form a long multiplet. Studying the representation theory of the superconformal group it is possible to show that the only multiplet with which the marginal deformations can recombine are conserved currents Beem:2012yn , see Appendix A. The primaries of the resulting long multiplet have dimensions which must be larger than three. As such, after such a recombination the dimension of the marginal deformation, which was $3$ since it was the chiral primary of the short multiplet, must increase and thus it is marginally irrelevant. This implies in particular that there are no supersymmetric marginally relevant deformations. Another simple implication of this logic is that if a deformation does not break any symmetry it is exactly marginal. See Figure 7 for a depiction of exactly marginal, marginally irrelevant, and relevant deformations. Also, see Figure 8 for interesting cases of RG-flows.

A careful analysis of the above logic Green:2010da (see also Kol:2002zt ) leads to the following conclusion. Given an SCFT ${\cal P}$ with a space of marginal operators ${\cal O}^{i}_{\cal P}$ and couplings $\lambda^{i}_{\cal P}$, the set of exactly marginal deformations is given by the following Kahler quotient,

$\displaystyle\{\lambda^{i}_{\cal P}\}/({G_{\cal P}})_{\mathbb{C}}\,,$

(25)

where ${G_{\cal P}}$ is the global symmetry of the SCFT ${\cal P}$ and $({G_{\cal P}})_{\mathbb{C}}$ is its complexification. When the marginal deformations involve also gauge couplings of some gauge group, we need to include in $G_{\cal P}$ also the symmetries which exist before the gauging but are anomalous once the gauging is introduced. The coupling which transforms well under the anomalous symmetry is $\exp(2\pi i\tau)$ with $\tau$ being the complexified gauge coupling. Note that in the exponentiation we took $\tau$ with the positive sign.⁹⁹9Usually in computations of Kahler quotients both signs are allowed. However, here taking the negative sign might lead to solutions with imaginary YM coupling. Formally in perturbation theory this would give a line of exactly marginal non-unitary SCFTs (See for similar effects e.g. Gorbenko:2018ncu ). However it is not clear whether this would make sense non perturbatively. In some cases, e.g AGT correspondence Alday:2009aq ; Gaiotto:2014bja or 5d gauge theories Aharony:1997ju ; Bergman:2013aca (where gauge coupling is related to real mass for instanton symmetry), going to lower half plane for $\tau$ is allowed but it is interpreted as going to infinite coupling (and “beyond”) where the gauge theory description breaks down. In our analysis we are dealing with weakly coupled gauge fields and thus will restrict to upper half-plane for $\tau$. We thank Z. Komargodski for discussions of this subtlety. This procedure is very abstract and has again the advantage that it does not rely on having a description of a theory in terms of weakly coupled fields. One way to compute this quotient is to list all the $({G_{\cal P}})_{\mathbb{C}}$ invariant independent combinations of the couplings: though often this is practically hard to do. We refer the reader to Razamat:2020pra for a detailed analysis of such quotients in many examples as well as a discussion of how to approach the problem in a general case.

First, a free R-symmetry assignment to all the matter fields is non anomalous,

$\displaystyle\mathop{\mathrm{Tr}}\nolimits\,R_{\cal P}SU(N)^{2}={\color[rgb]{1,0,0}N}+{\color[rgb]{.75,.5,.25}\frac{1}{2}(\frac{2}{3}-1)\,3N}+{\color[rgb]{0,0,1}\frac{1}{2}(\frac{2}{3}-1)\,3N}=0\,.$

(26)

Thus, according to the previous exercise it is also the superconformal one. Moreover, since this is also the superconformal R-symmetry of the free collection of chiral fields before gauging, the gauging is marginal. Since the superconformal R-charges at the free UV fixed point is $\frac{2}{3}$ we can turn on marginal superpotentials using chiral operators with R-charge $2$ which are built from cubic combinations of the basic fields. However, for $N\neq 3$ no such operators exist. For $N=3$ the operators $B=\epsilon_{ijk}Q^{i}_{(a}Q^{j}_{b}Q^{k}_{c)}$ and $\widetilde{B}=\epsilon^{ijk}\widetilde{Q}_{i}^{(a}\widetilde{Q}_{j}^{b}Q_{k}^{c)}$ are singlets under the gauged $SU(3)$ symmetry: these are the baryons and the anti-baryons. The analysis thus is different for $N\neq 3$ and $N=3$. We start with the former.

The symmetry of the free fixed point is $G_{\cal P}=U(6N^{2})$. We choose $G=SU(N)$ subgroup which we gauge. The commutant of $G$ in $G_{\cal P}$ is $G_{\cal P|G}=U(3N)\times U(3N)$. The Kahler quotient we need to compute is thus,

$\displaystyle\{\exp(2\pi i\tau_{SU(N)})\}/(U(3N)\times U(3N))_{\mathbb{C}}\,.$

(27)

The gauge coupling is not charged under the $SU(3N)\times SU(3N)\times U(1)_{B}$ non-anomalous symmetry, but transforms under the $U(1)_{A}$ anomalous symmetry. Thus this quotient is empty and there are no exactly marginal deformations. The gauge coupling is marginally irrelevant and the theory is free.

Next we analyze the case of $N=3$ where we have additional marginal operators $B$ and $\widetilde{B}$. These transform in $({\bf 84},{\bf 1})_{+3,+3}$ and $({\bf 1},{\bf 84})_{+3,-3}$ under $G_{\cal P|G}$ which we parametrize as $(SU(9),SU(9))_{U(1)_{A},U(1)_{B}}$, where $U(1)_{A}$ is the symmetry under which both quarks and anti-quarks have charge $+1$ while $U(1)_{B}$ is the baryonic symmetry under which they are oppositely charged (that is with charges $\pm{1}$). The representation ${\bf 84}$ is the three index completely antisymmetric irrep of $SU(9)$. The gauge coupling transforms as $({\bf 1},{\bf 1})_{9,0}$. The charge of the gauge coupling $\tau$, or rather of $\exp(2\pi i\tau)$, is given by the ’t Hooft anomaly $\mathop{\mathrm{Tr}}\nolimits\,U(1)_{A}G^{2}$. In this case it is $+1\times\frac{1}{2}\times 18=9$. Note that the baryonic couplings have charge $-3$ as all the matter fields have charge $+1$. In particular this implies that we always can solve for the quotient with the anomalous $U(1)_{A}$. Thus the quotient we need to compute is,

$\displaystyle\{({\bf 84},{\bf 1})_{+3},\,({\bf 1},{\bf 84})_{-3}\}/(SU(9)\times SU(9)\times U(1)_{B})_{\mathbb{C}}\,.$

(28)

One way to construct the quotient is by finding all the independent monomial holomorphic combinations of the couplings invariant under the symmetries. In general this is a well defined group theoretic question which is nevertheless tricky to solve.

A way to proceed is to find a deformation which is exactly marginal. Understand what symmetry it breaks. Deform the theory by this deformation and repeat the process. In the given case a simple deformation which was found by Leigh and Strassler Leigh:1995ep (See Appendix B.) in the seminal paper on conformal manifolds is,

$\displaystyle W=\lambda\left(Q_{1}Q_{2}Q_{3}+Q_{4}Q_{5}Q_{6}+Q_{7}Q_{8}Q_{9}+\widetilde{Q}_{1}\widetilde{Q}_{2}\widetilde{Q}_{3}+\widetilde{Q}_{4}\widetilde{Q}_{5}\widetilde{Q}_{6}+\widetilde{Q}_{7}\widetilde{Q}_{8}\widetilde{Q}_{9}\right)\,.$

(29)

This deformation breaks $U(1)_{B}$ and each $SU(9)$ is broken to $SU(3)^{3}$. Note that under this breaking,

	$\displaystyle{\bf 84}\,\to\,1+1+1+{\bf 3}_{1}{\bf 3}_{2}{\bf 3}_{3}+\sum_{i,j=1;\,i\neq j}^{3}{\bf 3}_{i}\overline{\bf 3}_{j}\,,$		(30)
	$\displaystyle{\bf 80}\,\to\,1+1+{\bf 8}_{1}+{\bf 8}_{2}+{\bf 8}_{3}+\sum_{i,j=1;\,i\neq j}^{3}{\bf 3}_{i}\overline{\bf 3}_{j}\,.$

Note that the components of the currents (the ${\bf 80}$) in representation $\sum_{i,j=1;\,i\neq j}^{3}{\bf 3}_{i}\overline{\bf 3}_{j}$ recombine with the same components of the marginal operators (the ${\bf 84}$). Moreover the currents of the four $U(1)$s in the decomposition of $SU(9)^{2}$ to $SU(3)^{6}\times U(1)^{4}$ as well as the current of $U(1)_{B}$ recombine with five out of the six singlets in the decomposition of the two ${\bf 84}$s. We are thus left with marginal deformations in,

$\displaystyle{\color[rgb]{1,0,0}1}+{\bf 3}_{1}{\bf 3}_{2}{\bf 3}_{3}+{\bf 3}_{4}{\bf 3}_{5}{\bf 3}_{6}\,,$

(31)

with the singlet being the exactly marginal deformation. We thus deduce that there is one direction on the conformal manifold on which the symmetry is $SU(3)^{6}$. Next we can turn on the marginal deformation ${\bf 3}_{1}{\bf 3}_{2}{\bf 3}_{3}$. This will break $SU(3)^{3}$ down to diagonal $SU(3)$. Note that under this decomposition,

$\displaystyle{\bf 3}_{1}{\bf 3}_{2}{\bf 3}_{3}\,\to\,1+{\bf 10}+{\bf 8}+{\bf 8}\,.$

(32)

In particular the two ${\bf 8}$ recombine with two of the three $SU(3)$ currents leaving only $SU(3)$ symmetry and marginal operators in,

$\displaystyle{\color[rgb]{1,0,0}1}+{\color[rgb]{.75,.5,.25}1}+{\bf 10}_{1}+{\bf 3}_{4}{\bf 3}_{5}{\bf 3}_{6}\,.$

(33)

Thus we have a two dimensional conformal manifold on which the symmetry is $SU(3)^{4}$. Next we can break in the same manner using ${\bf 3}_{4}{\bf 3}_{5}{\bf 3}_{6}$ another triplet of $SU(3)$ symmetries down to diagonal $SU(3)$. We will obtain a three dimensional conformal manifold with symmetry $SU(3)^{2}$ and marginal operators in,

$\displaystyle{\color[rgb]{1,0,0}1}+{\color[rgb]{.75,.5,.25}1}+{\color[rgb]{0,0,1}1}+{\bf 10}_{1}+{\bf 10}_{2}\,.$

(34)

In lecture III we will have a geometric interpretation of this conformal manifold as corresponding to complex structure moduli of a sphere with six marked points. We can next continue exploring the conformal manifold by turning on operators in the ${\bf 10}$. As from ${\bf 10}$ one can build two independent invariant (the fourth and the sixth symmetric powers contain singlets), each gives two exactly marginal directions on which the relevant $SU(3)$ is completely broken. All in all we obtain a seven dimensional conformal manifold on which all the symmetry is broken.

Instead of starting with (29) we can start with the following,

$\displaystyle W=\lambda\,\left(Q_{[i}^{1}Q_{j}^{2}Q_{k]}^{3}+\widetilde{Q}_{[i}^{1}\widetilde{Q}_{j}^{2}\widetilde{Q}_{k]}^{3}\right)\,.$

(35)

Here we think of the nine quarks as forming ${\bf 3}\times{\bf 3}$ representation of the $SU(3)\times SU(3)$ subgroup of $SU(9)$. The lower flavor indices in (35) are antisymmetrized as well as the gauge indices are. This superptential is exactly marginal. This can be understood either performing the LS analysis (all the fields have same anomalous dimensions but we have two couplings: gauge coupling and $\lambda$), or performing the Kahler quotient. The two terms are charged oppositely under the $U(1)_{B}$ symmetry and we note that,

	$\displaystyle{\bf 84}\,\to\,{\bf 10}_{1}+{\bf 10}_{2}+{\bf 8}_{1}\times{\bf 8}_{2}\,,$		(36)
	$\displaystyle{\bf 80}\,\to\,{\bf 8}_{1}+{\bf 8}_{2}+{\bf 8}_{1}\times{\bf 8}_{2}\,.$

Moreover the characters of ${\bf 10}$ and ${\bf 8}$ are given by

$\displaystyle\chi_{\bf 10}=\chi_{\bf 8}-1+z_{1}^{3}+z_{2}^{3}+\frac{1}{z_{1}^{3}z_{2}^{3}}\,,\qquad\chi_{\bf 8}=1+1+\sum_{i\neq j}z_{i}/z_{j}\,.$

(37)

Using these observations we can conclude that (35) breaks the symmetry to $SU(3)\times SU(3)\times U(1)^{4}$ where the abelian symmetries are the Cartan generators of the $SU(3)$s broken by (35),¹⁰¹⁰10This is very reminiscent of the ${\cal N}=1$ $\beta$ deformation of ${\cal N}=4$ SYM, see e.g. Lunin:2005jy .

	$\displaystyle 1+{\bf 84}+\widetilde{\bf 84}-1-1-{\bf 80}-\widetilde{\bf 80}\to$		(38)
	$\displaystyle{\color[rgb]{1,0,0}1}+{\bf 10}_{1}+\chi_{\bf 3}((z^{2}_{1})^{3},(z^{2}_{2})^{3})+\widetilde{\bf 10}_{1}+\chi_{\bf 3}((\widetilde{z}^{2}_{1})^{3},(\widetilde{z}^{2}_{2})^{3})-{\bf 8}_{1}-\widetilde{\bf 8}_{1}-1-1-1-1\,.$

Here $z^{2}_{i}$ are fugacities for the Cartan of $SU(3)_{2}$ and $\widetilde{z}^{2}_{i}$ are fugacities for the Cartan of $\widetilde{SU(3)}_{2}$. We can further break the rest of the symmetries. For example, the two $z^{2}_{i}$ can be broken together by turning on the three operators in $\chi_{\bf 3}((z^{2}_{1})^{3},(z^{2}_{2})^{3})$ and the $SU(3)_{1}$ can be broken first to the Cartan and then completely by turning on operators in ${\bf 10}_{1}$.

$\square$

The matter content is conformal, i.e. the one loop beta function vanishes, and thus all the fields have free R-charges. Let us start by discussing the symmetries of the theory. We have $n$ anomalous symmetries $U(1)_{A_{i}}$ under which the adjoint fields $\Phi_{i}$ have charge $+1$ and all the other fields are not charged. We have $n$ $U(1)_{\alpha_{i}}$ symmetries under which $Q_{i}$ has charge $+1$ and $\widetilde{Q}_{i}$ charge $-1$ with all the rest of the fields not charged. Finally we have $n$ $U(1)_{t_{i}}$ symmetries under which fields $Q_{i}$ and $\widetilde{Q}_{i}$ have charge $+1$, $\Phi_{i}$ and $\Phi_{i+1}$ charge $-1$ while the other fields are not charged. The $U(1)_{\alpha}$ and $U(1)_{t}$ symmetries are not anomalous. Let us consider the following superpotentials,

$\displaystyle W=\sum_{i=1}\lambda_{i}\Phi_{i}Q_{i}\widetilde{Q}_{i}+\lambda^{\prime}_{i}\,\Phi_{i}Q_{i-1}\widetilde{Q}_{i-1}\,.$

(39)

Note that the couplings $\lambda_{i}$ have charges encoded in fugacities as $A_{i}^{-1}t^{-1}_{i}t_{i-1}$ while $\lambda^{\prime}_{i}$ have charges $A_{i}^{-1}t_{i}t_{i-1}^{-1}$. Then $\lambda_{i}\lambda_{i}^{\prime}$ dressed with appropriate power of $\exp(2\pi i\tau_{i})$ is a singlet for each $i$. These thus give us $n$ independent exactly marginal parameters preserving in fact ${\cal N}=2$ supersymmetry. Along these directions the $U(1)_{t_{i}}$ symmetries are broken to a diagonal combination. Note that as all of the couplings are charged under the anomalous symmetries with negative charge these can be taken care of by exponents of gauge couplings.

Above we turned on $\lambda$ and $\lambda^{\prime}$ couplings in pairs. Now consider turning on only $\lambda$ couplings. Then the combination $\prod_{i=1}^{n}\lambda_{i}$ (dressed by appropriate power of all $\exp(2\pi i\tau_{i})$) is also a singlet of all the symmetries. and thus turning on only $\lambda_{i}$ gives us an exactly marginal deformation. Here also the $U(1)_{t_{i}}$ symmetries are broken to a diagonal combination. This deformation is only ${\cal N}=1$ supersymmetric. Note that turning on only $\lambda^{\prime}$ couplings also is exactly marginal, but it is a combination of the above and the ${\cal N}=2$ exactly marginal couplings.

Note that (39) is not the most general cubic superpotential we can write. Specifically, for $N>2$, we can also add the terms $\sum_{i=1}\hat{\lambda}_{i}\Phi^{3}_{i}$, where we use the fully symmetric cubic invariant polynomial of $SU(N)$ to contract the gauge indices and form a gauge invariant. The couplings $\hat{\lambda}_{i}$ then have the charges $A_{i}^{-3}t^{3}_{i}t^{3}_{i-1}$. We can cancel the $U(1)_{A_{i}}$ charges by dressing with appropriate powers of the gauge couplings, but we cannot cancel all the $U(1)_{t_{i}}$ charges. As such, for generic $N$, these operators are marginally irrelevant. However, for $N=3$, we further have the baryonic superpotential $\sum_{i=1}\lambda^{B}_{i}Q^{3}_{i}+\lambda^{\tilde{B}}_{i}\widetilde{Q}^{3}_{i}$, where we use the epsilon tensor to contract the $SU(3)$ indices to a singlet. The $\lambda^{B}_{i}$ have the charges $t^{-3}_{i}\alpha^{3}_{i}$ while $\lambda^{\tilde{B}}_{i}$ have the charges $t^{-3}_{i}\alpha^{-3}_{i}$. We then see that the $\lambda^{B}_{i}\lambda^{\tilde{B}}_{i}$ combination of couplings is uncharged under $U(1)_{\alpha_{i}}$ and together with $\hat{\lambda}_{i}$ and the gauge couplings, can be used to form flavor symmetry singlets. Thus, we see that for $N=3$ there are additional exactly marginal deformations that preserve only ${\cal N}=1$ supersymmetry.

We start with the $SU(3)$ SQCD with six flavors. This theory is asymptotically free,

$\displaystyle\mathop{\mathrm{Tr}}\nolimits R\,SU(3)^{2}=3+\frac{1}{2}(\frac{2}{3}-1)\times 2\times 6=1>0\,,$

(40)

where we used the free assignment of R-charges. Assigning R-charge $\frac{1}{2}$ to all the chiral fields is anomaly free,

$\displaystyle\mathop{\mathrm{Tr}}\nolimits R\,SU(3)^{2}=3+\frac{1}{2}(\frac{1}{2}-1)\times 2\times 6=0\,,$

(41)

and in fact this is the superconformal R-symmetry at the fixed point. We have one abelian $U(1)$ symmetry which is non-anomalous under which the fundamentals and the antifundamentals have opposite charges. We thus define a trial $a$-anomaly,

$\displaystyle a(\alpha)=a_{v}\times 8+a_{R}(\frac{1}{2}+\alpha)\times 18+a_{R}(\frac{1}{2}-\alpha)\times 18=\frac{3}{64}\left(41-324\alpha^{2}\right)\,,$

(42)

and the maximum is at $\alpha=0$. Next we add the $12$ fields $\Phi$ and couple them with superpotential as on the left of Figure 11. This interaction breaks some of the nonabelian symmetry. The R-charge of the superpotential at the UV fixed point is $\frac{1}{2}+\frac{1}{2}+\frac{2}{3}<2$ and thus it is relevant and the theory flows in the IR to a fixed point where there is a non-anomalous R-symmetry under which the R-charges of $q_{L}$, $S_{L}$ and $Q_{L}$ are still $\frac{1}{2}$ but that of $\Phi$ is 1 (such that the R-charge of the superpotential is 2). We are still left with finding the superconformal R-symmetry at that IR fixed point. The theory has two non anomalous abelian symmetries $(U(1)^{L}_{a},U(1)_{b})$ so that the charges of the various fields are,

$\displaystyle q_{L}:\,(1,2),\quad S_{L}:\,(1,-1),\quad Q_{L}:\,(-1,0),\quad\Phi:\,(0,-2)\,.$

(43)

We thus compute the trial anomaly depending on two parameters $\alpha_{a}$ and $\alpha_{b}$ corresponding to the two symmetries,

	$\displaystyle a(\alpha^{L}_{a},\alpha_{b})=a_{v}\times 8+a_{R}(1-2\alpha_{b})\times 12+$		(44)
	$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;a_{R}(\frac{1}{2}+\alpha^{L}_{a}-\alpha_{b})\times 12+a_{R}(\frac{1}{2}+\alpha^{L}_{a}+2\alpha_{b})\times 6+a_{R}(\frac{1}{2}-\alpha^{L}_{a})\times 18\,.$

Here in the first line we have the contribution of the vectors and $\Phi$ and on the second line the charged matter fields. The charges of $\Phi$ are fixed by the superpotential as stated above. We compute the maximum of $a(\alpha^{L}_{a},\alpha_{b})$ to be at

$$(\alpha^{L}_{a},\alpha_{b})=(0.0045,0.0672)\,,$$

approximately. Now we couple the second copy of the $SU(3)$ SQCD to the theory through a superpotential as on the right of Figure 11. The additional term in the superpotential at the new fixed point has R-charge

$$\frac{1}{2}+\frac{1}{2}+R_{\Phi}=1+(1-2\times 0.0672)=1.8657<2\,,$$

and thus it is relevant. The second copy of the SQCD adds another $U(1)$ symmetry and we compute the new trial $a$-anomaly to be,

	$\displaystyle a(\alpha^{L}_{a},\alpha_{b},\alpha^{R}_{a})=a_{v}\times 8\times 2+a_{R}(1-2\alpha_{b})\times 12+$		(45)
	$\displaystyle\;\;\;\;\;\;a_{R}(\frac{1}{2}+\alpha^{L}_{a}-\alpha_{b})\times 12+a_{R}(\frac{1}{2}+\alpha^{L}_{a}+2\alpha_{b})\times 6+a_{R}(\frac{1}{2}-\alpha^{L}_{a})\times 18+$
	$\displaystyle\;\;\;\;\;\;\;\;\;a_{R}(\frac{1}{2}+\alpha^{R}_{a}-\alpha_{b})\times 12+a_{R}(\frac{1}{2}+\alpha^{R}_{a}+2\alpha_{b})\times 6+a_{R}(\frac{1}{2}-\alpha^{R}_{a})\times 18\,.$

Here we used the fact that the superpotential identifies $U(1)_{b}^{L}$ with $U(1)_{b}^{R}$. We find the maximum of $a(\alpha^{L}_{a},\alpha_{b},\alpha^{R}_{a})$ to be at,

$$\alpha_{a}^{L}=\alpha_{a}^{R}=0.00135\,,\qquad\alpha_{b}=0.03669\,.$$

Finally we compute the beta function of the $SU(2)$ gauging at the new fixed point,

$\displaystyle\mathop{\mathrm{Tr}}\nolimits\,R\,SU(2)^{2}=2+\frac{1}{2}(R_{\phi}-1)\times 2+\frac{1}{2}(R_{\Phi}-1)\times 6+\frac{1}{2}(R_{q_{L/R}}-1)\times 6\,.$

(46)

Now $R_{\phi}=\frac{2}{3}$ as these are free fields we add at this point and

$$R_{q_{L/R}}=\frac{1}{2}+\alpha^{L/R}_{a}+2\alpha_{b}=0.5747\,,\qquad R_{\Phi}=1-2\alpha_{b}=0.9266\,.$$

We thus deduce that the anomaly in (46) is

$$\mathop{\mathrm{Tr}}\nolimits\,R\,SU(2)^{2}=0.1707>0\,,$$

and thus at the new fixed point the $SU(2)$ gauging is asymptotically free although it has $14$ fundamental fields.