Supersymmetric QFT in Six Dimensions

2.1 Background

Since this article focuses on supersymmetric theories, some relevant background is needed about supersymmetry in six dimensions.

Spinors in six and four dimensions have slightly different properties. One introduces indices $\alpha$ and $\dot{\alpha }$ for positive- and negative-chirality spinors respectively, but in $d=6$ the number of complex independent components of each is four rather than two. A more important difference is that in $d=6$ Majorana conjugation preserves chirality, and that it squares to minus the identity; so there are no Majorana spinors.

This has important consequences for the supersymmetry algebra. In $d=4$, supercharges are expressed as a Majorana spinor, with components $Q_{\alpha }$ and ${\overline{Q}}_{\dot{\alpha }}$, the two chiralities related to each other by Majorana conjugation. In $d=6$ one can have an independent number of supercharges of either chirality. There are no Majorana spinors, but given a supercharge $Q_{\alpha }\equiv Q_{\alpha }^1$ one can give a name to its conjugate, $Q_{\alpha }^2\equiv {(Q_{\alpha }^1)}^c$; then by construction $Q_{\alpha }^1\equiv -{\left(Q_{\alpha }^2\right)}^c$. This is called a symplectic Majorana pair. In this article, $\mathcal{N}=(p,q)$ supersymmetry is defined to have $p$ independent $Q_{\alpha }$ of positive chirality and $q$ of negative chirality, thus counting the symplectic Majorana pairs as one single supercharge. So, for example, for $\mathcal{N}=(1,0)$ supersymmetry there is a single pair $Q_{\alpha }^a$. Since each chiral spinor has four complex components, the number of supercharges in this terminology is $8(p+q)$.

For 16 or fewer supercharges, there are only three possibilities:$\mathcal{N}=(1,0)$, $(2,0)$, or $(1,1)$. (Higher numbers lead to theories that include gravity.) The latter two possibilities have 16 supercharges; upon dimensional reduction to $d=4$ this becomes then$\mathcal{N}=4$, which is rather constrained. For example, for $\mathcal{N}=(1,1)$ one can build a super-Yang–Mills theory rather reminiscent of $\mathcal{N}=4$ in four dimensions. This theory is not conformal and will not be considered further in this article. $\mathcal{N}=(1,0)$ gives eight supercharges; upon dimensional reduction to $d=4$ this becomes then $\mathcal{N}=2$ supersymmetry. This case will be the focus of this article. $\mathcal{N}=(2,0)$ will also appear, as anticipated in the introduction.

Some background about anomalies is also needed, namely about how classical symmetries are broken quantum-mechanically. In $d=4$, this can be parameterized by ${\partial }_{\mu }{j}^{\mu }{=}_{\ast }{I}_4^1$, with $I_4^1$ a four-form. The so-called descent formalism gives a way to summarize this conveniently by a formal six-form $I_6$ with cubic dependence on the gauge field-strength two-form $F\equiv \frac{1}{2}{F}_{\mu v}d{x}^{\mu }\wedge d{x}^v$; the relation is via $I_6=d{I}_7$, $\delta I_7=d{I}_6^1$, where $\delta$ denotes a gauge transformation. In $d=6$, these statements are very similar; the anomaly is summarized by a formal eight-form $I_8$, now quartic in $F$. The descent equations now read $I_8=d{I}_7$, $\delta I_7=d{I}_6^1$.

A peculiar case is Weyl anomalies, namely the quantum non-invariance under rescaling of a classically scale-invariant theory. This manifests itself as a non-zero trace of the stress-energy tensor:

up to total derivatives. Here $E$ is the Euler density, and $I_i$ are built from the Weyl tensor. In $d=4$ there is only one such $c$; in $d=6$ there are three of them (Bonora, Pasti, & Bregola, 1986; Deser & Schwimmer, 1993).

Finally some minimal background about string theory, even though its use will be minimal and its reading optional. The most relevant theories are type IIA and IIB in $d=10$, and M-theory in $d=11$. The bosonic fields of type II theories include analogs of the electromagnetic field-strength $F_p$ called Ramond–Ramond (RR) fields, that are $p$-forms with $p=$ even in IIA and $p=$ odd in IIB, as well as a three-form $H$. M-theory contains a four-form field-strength $G_4$.

All string theories have extended objects called branes. They are denoted by a letter and a number, the latter indicating their space dimensions. For example, type II theories have Dp-branes, which extend along $p$ space dimensions, plus time; $p$ is even in IIA and odd in IIB. These objects couple to the potentials of the RR field-strengths. Type II theories also have NS5-branes, which couple to the potential for the dual field-strength ${}_{\ast }H$. M-theory has M5-branes, which couple to the potential for ${}_{\ast }{G}_4$. Besides this coupling to field-strengths, branes have a dynamics governed by supersymmetric field theories. For example, the effective theory of an M5-brane is believed to be governed by an $\mathcal{N}=(2,0)$ theory in $d=6$.

2.2 Multiplets

As anticipated, this article focuses on the minimal amount of supersymmetry,$\mathcal{N}=(1,0)$. The corresponding superconformal algebra also contains a global symmetry traditionally called R-symmetry, $\text{Sp}{(1)}_{\text{R}}\cong \text{SU}{(2)}_{\text{R}}$, that acts on this index in the fundamental representation.

The multiplets for $\mathcal{N}=(1,0)$ supersymmetric field theories (excluding gravity) are (Howe, Sierra, & Townsend, 1983):

The linear multiplet is a variant of the hypermultiplet, and plays a more limited role. The difference between the two lies in the $\text{SU}{(2)}_{\text{R}}$ representations. As indicated by the index structure, the scalars of the hypermultiplet transform as a complex doublet, while those of the linear multiplet transform in the $\mathbf{1}\oplus \mathbf{3}$; the spinors respectively as a complex singlet and as a doublet. (For more details, see, e.g., Park and Taylor (2012).)

Under dimensional reduction to $d=4$, $\mathcal{N}=(1,0)$ reduces to$\mathcal{N}=2$. The vector multiplet becomes the vector multiplet in $d=4$, with two components of $A^{\mu }$ becoming two scalars. The tensor multiplet also becomes a vector multiplet, upon taking the dual of the four-dimensional components of $b_{\mu v}$. The hypermultiplet reduces with no further difficulty.

2.3 $\mathcal{N}=(2,0)$ Theories

In the $\mathcal{N}=(2,0)$ superconformal algebra, the R-symmetry is $\text{Sp}{(2)}_{\text{R}}\cong \text{SO}{(5)}_{\text{R}}/{\mathbb{Z}}_2$. One option to obtain such an enhancement is to combine a $\mathcal{N}=(1,0)$ tensor multiplet with a linear multiplet; the $\mathbf{1}$ and $\mathbf{1}\oplus \mathbf{3}$ of $\text{Sp}{(1)}_{\text{R}}$ come together to form a $\mathbf{5}$ of $\text{Sp}{(2)}_{\text{R}}$.

This $\mathcal{N}=(2,0)$ tensor multiplet arises in string theory as the theory on a single M5; its five scalars represent the possible motions of the M5 in the directions transverse to it. The theory is free, but the presence of a self-dual tensor makes it challenging to write down a Lagrangian; a popular proposal is found in Pasti, Sorokin, and Tonin (1997).

M5-branes in string theory can be superimposed. Moreover, string dualities relate M5s to D-branes, which when superimposed exhibit non-Abelian gauge symmetries. Thus a stack of $N>1$ superimposed M5-branes is expected to lead to a non-Abelianization of the $\mathcal{N}=(2,0)$ tensor multiplet; it is denoted by ${\mathcal{T}}_N$ A different string realization exists in IIB, involving ${ℂ}^2/{\mathbb{Z}}_N$ singularities. This suggests (Witten, 1995) a generalization ${\mathcal{\text{T}}}_{\Gamma }$ where ${\mathbb{Z}}_N$ is replaced with other discrete subgroups $\Gamma$ of $\text{SU}(2)$.

In spite of many attempts at writing a Lagrangian for them, the theories ${\mathcal{T}}_N$ and ${\mathcal{T}}_{\Gamma }$ remain quite mysterious. Various arguments indicate that the degrees of freedom of this non-Abelian tensor ${\mathcal{T}}_N$ should go like $N^3$. In string theory this manifests itself via an anomaly-inflow computation (Harvey, Minasian, & Moore, 1998), or by using holographic duality to the ${\text{AdS}}_7\times S^4$ solution (Henningson & Skenderis, 1998). In pure field theory terms, an effective description on its moduli space and anomaly matching methods (Córdova, Dumitrescu, & Yin, 2019; Intriligator, 2000; Maxfield & Sethi, 2012) yield the same result. For example, the $a$ Weyl anomaly (defined in (2.1)) of ${\mathcal{T}}_N$ reads

in a normalization where $a=1$ for the single $\mathcal{N}=(2,0)$ tensor multiplet,${\mathcal{T}}_1$. There are three more Weyl anomalies $c_i$ in $d=6$, but $\mathcal{N}=(2,0)$ supersymmetry fixes them to be proportional to $a$.

It would be interesting to establish whether ${\mathcal{T}}_N$ and ${\mathcal{T}}_{\Gamma }$ are the only $\mathcal{N}=(2,0)$ theories. One way to investigate general properties of CFTs is to use the so-called bootstrap, a program that uses conformal invariance to generate a large number of inequalities on what CFTs can possibly exist. For $\mathcal{N}=(2,0)$ theories, strong constraints are already generated from the four-point function of a certain scalar, which is related to the stress-energy tensor by supersymmetry. The numerical evidence indicates that for a $\mathcal{N}=(2,0)$ theory the smallest possible $a$ is $\frac{110}{7}$; this corresponds to (2.2) for $N=2$. So in a sense the smallest possible theory is${\mathcal{T}}_2$. (The free tensor multiplet ${\mathcal{T}}_1$ is ignored in this approach.) This can also be read as independent verification that ${\mathcal{T}}_N$ exist.

2.4 Single Gauge Group

Given the mystery surrounding the $\mathcal{N}=(2,0)$ case, it may look safer to stay within$\mathcal{N}=(2,0)$. Instead of tensor multiplets, using a vector multiplet may look easier.

Two issues immediately appear, however. First of all, as mentioned in the introduction, the usual YM action $\frac{1}{g_{\text{YM}}^2}\text{Tr}|F{|}^2$ is relevant in $d=6$; this is not promising for a CFT. Even more worrisome, the partner of the vector (the “gaugino”) ${\lambda }_{a\dot{\alpha }}$ is chiral (see the section “Multiplets”); this means that it can generate an anomaly for the gauge symmetry, which would make the theory ill-defined.

To cancel the anomaly, some additional matter fields must be introduced, charged under the gauge symmetry. The spinor in a hypermultiplet (the “hyperino”) ${\psi }_{\alpha }$ has chirality opposite to that of ${\lambda }_{a\dot{\alpha }}$; so the two give a contribution to the gauge anomaly with opposite sign. The detailed expression will, however, depend on the gauge group $G$ and on the representation $\rho$ under which the hypermultiplet transforms. Perhaps the first possibility that springs to mind is to take $\rho$ to be the adjoint; in this case the gauge anomaly is indeed canceled, and the theory becomes super-YM with$\mathcal{N}=(1,1)$. In other words, the $\mathcal{N}=(1,0)$ vector and hypermultiplet assemble into a single $\mathcal{N}=(1,1)$ vector multiplet. This theory is, however, non-conformal, because as just mentioned the kinetic term is relevant.

Fortunately there are other possibilities. For example, for $G=\text{SU}(N_c)$ with $N_{\text{f}}$ hypermultiplets in the fundamental representation, the anomaly polynomial (see the section “Background”) $I_8={\text{Tr}}_{\text{Ad}}(F^4)-N_{\text{f}}{\text{Tr}}_{\text{fund}}(F^4)$; expressing the generators of the adjoint in terms of those in the fundamental gives ${\text{Tr}}_{\text{Ad}}\left(F^4\right)=2N_{\text{c}}{\text{Tr}}_{\text{fund}}\left(F^4\right)-6{\left({\text{Tr}}_{\text{fund}}{F}^2\right)}^2$. Thus the term ${\text{Tr}}_{\text{fund}}(F^4)$ can be canceled by taking

The residual

can be canceled by introducing a tensor multiplet. Indeed $I_6^1=I_2^1\wedge I_4$ (with $I_4=d{I}_3$, $\delta I_3=d{I}_2^1$ as in the section “Background”); if the two-form $b$ transforms as $\delta b=I_2^1$ and has a coupling (Green, Schwarz, & West, 1985; Sagnotti, 1992)

the transformation of the new term cancels the anomaly.

To put the theory on a curved background, it is also prudent to cancel the mixed gauge-gravitational anomaly, which depends on the Riemann tensor, viewed as a two-form $R^{ab}\equiv \frac{1}{2}{R}^{ab}{}_{\mu \nu }d{x}^{\mu }\wedge d{x}^{\nu }$. It turns out, however, that this is automatic given (2.3).

The complete supersymmetric Lagrangian for this simple model can be found by specializing the formalism in Samtleben, Sezgin, and Wimmer (2011), which does indeed allow for a term of the form $L_{\text{GSWS}}$. (To be more precise, it is a pseudo-Lagrangian, where the self-duality constraint $b{=}_{*}b$ has to be imposed by hand.) Schematically it reads

$Dh$ is the gauge covariant derivative, ${\sigma }^I$ are the Pauli matrices, and $D_I$ is a triplet of auxiliary fields; again $|{|}^2$ is the norm of a form by index contraction. In particular (2.6) contains $L_{\text{GSW}}$ and a partner term

where $\varphi$ is the scalar in the tensor multiplet, following the notation in the section “Multiplets.” Since a scalar in $d=6$ has dimension 2, (2.7) has now dimension 6 and is marginal, not changing with scale. This is an improvement over the usual YM coupling $\text{Tr}|F{|}^2$, which has dimension 4 and is relevant.

This model, however, still encounters problems from a quantum-mechanical point of view. The standard quantization approach is to choose a vev for $\varphi$, and to treat its fluctuations perturbatively. Notice that

If $〈\varphi 〉\ne 0$, the action becomes the usual YM one; the theory is non-renormalizable with an energy cutoff of order ${〈\varphi 〉}^{1/2}$. If on the other hand $〈\varphi 〉=0$, then $g_{\text{YM}}\to \infty$ and the theory is strongly coupled at all scales.

On the one hand, this may sound discouraging: the $〈\varphi 〉=0$ theory is strongly coupled, and it may seem that not much can be done with it. On the other hand, it is scale invariant. If it actually makes sense, it is a CFT. In fact, even the theories for $〈\varphi 〉\ne 0$ approach this CFT when they become strongly coupled at energies far above the cutoff ${〈\varphi 〉}^{1/2}$. So the theory (2.6) may be an effective description of a CFT. The space ${ℝ}_{\ge 0}$ of possible vev’s for $\varphi$ is called a tensor branch, since $\varphi$ is the scalar in a tensor multiplet.

As an additional motivation to take this possible CFT seriously, it turns out that it can also be engineered in string theory, for example using two M5-branes on the singularity ${ℝ}^4/{\mathbb{Z}}_2\times ℝ$. A stack of M5s at the origin of ${ℝ}^4/{\mathbb{Z}}_2$ but separated by $\delta$ in $ℝ$ are described by (2.6) with $〈\varphi 〉\propto \delta$. To be more precise, this engineers a version of (2.6) with gauge group $\text{U}(N_{\text{c}})$ rather than $\text{SU}(N_{\text{c}})$. The $\text{U}(1)$ gives rise to further anomalies besides those already discussed (involving for example $\text{Tr}F$); these are canceled by a similar mechanism to the one in (2.5), but now involving the linear multiplet mentioned in the sections “Multiplets” and “$\mathcal{N}=(2,0)$ Theories.”

2.3. The full modification to (2.6) involving supersymmetry has not been worked out yet.

A similar analysis can be performed for any simple gauge group (Bhardwaj, 2020a; Danielsson, Ferretti, Kalkkinen, & Stjernberg, 1997). In fact, Bhardwaj (2020a) also takes into account some additional constraints, not mentioned so far. One is the vanishing of certain global anomalies: certain gauge groups $G$ have subtle global features that generate additional potential anomalies, not captured by the descent formalism, and which have to be imposed separately. Another constraint comes from the so-called tensionless strings. These are defects in the gauge field similar to an instanton in $d=4$ (indeed, both have codimension four). They have a coupling ${\int }^b$ to the tensor multiplet two-form, and their tension is proportional to $〈\varphi 〉$, which hence goes to zero in the CFT limit. Flux quantization applied to these strings results in an integrality constraint on the coefficient of (2.5). This constraint is summarized in (3.13) of Bhardwaj (2020a). This turns out, however, to be redundant.

As an outcome of this classification, it turns out that all simple gauge groups are allowed, but there are severe restrictions on the representations. The list can be found, for example, in Bhardwaj, Jefferson, Kim, Tarazi, and Vafa (2019; table 1) and Bhardwaj (2020a; tables 2 and 3).

2.5 Chains of Unitary Gauge Groups

The example in (2.6) can be generalized straightforwardly to multiple unitary gauge groups (Brunner & Karch, 1998; Hanany & Zaffaroni, 1998).

Consider a chain of $N-1$ vector multiplets $(A_i,{\lambda }_i)$, $i=1,\dots ,N-1$ with gauge group $\text{U}(r_i)$; hypermultiplets $h_i$, $i=1,\dots ,N-2$ in the bifundamental $\left({\mathbf{\text{r}}}_i,{\overline{\mathbf{\text{r}}}}_{i+1}\right)$ of $\text{U}(r_i)\times \text{U}(r_{i+1})$ and $f_i$ hypermultiplets ${\tilde{h}}_i$, $i=1,\dots ,N-1$ in the fundamental ${\mathbf{\text{r}}}_i$ of $\text{U}(r_i)$; and tensor multiplets $b_i$, $i=1,\dots ,N$. (Linear multiplets are also needed for each gauge groups, along the lines explained in the section “Single Gauge Group,” but this fact will be ignored here, for simplicity.) One can check anomaly cancellation for each gauge group $\text{U}(r_i)$ separately. The $\left({\mathbf{\text{r}}}_i,{\overline{\mathbf{\text{r}}}}_{i+1}\right)$ bifundamental is seen by it as $r_{i+1}$ fundamentals; so in total (2.3) gives

This cancels the ${\text{Tr}}_{\text{fund}}(F_i^4)$ term in the anomaly polynomial $I_8$. What remains is of the form

where $C_{ij}=2{\delta }_{ij}-{\delta }_{i-1,j}-{\delta }_{i+1,j}$ is the Cartan matrix for the Lie algebra ${\text{A}}_N$. This generalizes $I_4$ in (2.4). As in that case, this term in the anomaly polynomial can be cancelled by introducing a coupling $3C_{ij}{b}_i{\text{Tr}}_{\text{fund}}{F}_j^2$ in the (pseudo-)Lagrangian, and letting $\delta b_i=I_2^i$ be related to $I_4^i=\text{Tr}{(F_i^2)}^2$ by descent. Supersymmetry relates this to a coupling of the type $({\varphi }_{i+1}-{\varphi }_i){\text{Tr}}_{\text{fund}}|F_i{|}^2$.

Once again this theory is non-renormalizable, and has a hope of being a CFT at the point where the YM couplings all diverge: $〈{\varphi }_i〉=0$, namely the origin of what was called tensor branch in the section “Single Gauge Group.” Again it turns out that all these theories can be engineered in string theory, and thus perhaps one should take these potential CFTs seriously, since string theory can be defined at arbitrarily high energies.

A first simple example is to take all gauge group ranks to be equal, $r_i=k$. This satisfies (2.9) with all $f_i=0$ except for $f_1=k$ and $f_{N-1}=k$. A second example is obtained by taking $r_i=ri$, a linear growth in ranks. Again (2.9) is satisfied by taking all $f_i=0$ except $f_1=r$ and $f_{N-1}=rN$.

Both of these simple examples are shown in Figure 1, along with a third, more complicated one. The convention here is as follows: a circle with a number $r$ denotes a gauge group $\text{U}(r)$; a horizontal link denotes a bifundamental hypermultiplet, and a tensor multiplet that couples to both gauge groups in the way detailed in the comment following (2.10). A vertical link to a square with a number $f$ denotes the presence of $f$ fundamental hypermultiplets (but no tensor multiplets).

Open in new tab

Figure 1. Examples of chains of unitary gauge groups.

This gives rise to a large number of possibilities. The choice of the $r_i$ is, however, not completely arbitrary. The combination of $r_{i+1}+r_{i-1}-2r_i=(r_{i+1}-r_i)-(r_i-r_{i-1})$ in (2.9) is a discrete double derivative; since $f_i$ is by definition non-negative, it follows that the discrete function $i\mapsto r_i$ is convex. The function $i\to s_i\equiv r_i-r_{i-1}$ is then monotonic: in general, it will start positive, go through zero, and then arrive at a negative number. Collecting the positive and negative number gives rise to two Young diagrams ${\rho }_{\text{L}}$, ${\rho }_{\text{R}}$ with the same number of boxes $k$, or equivalently two partitions of $k$. The theories described in this section can be labeled by the number $N$ of gauge groups (plus one), and the two partitions:

In this notation, the three examples of theories in Figure 1 are ${\mathcal{T}}_{N,[1^k],[1^k]}$, ${\mathcal{T}}_{k,[k],[1^k]}$, and ${\mathcal{T}}_{N,[k],[k]}$, respectively.

It has been conjectured (Gaiotto & Tomasiello, 2014) that the theories (2.11) are related by RG-flows—induced by giving vev’s to operators, which is the only possibility for SCFT flows in six dimensions. There is a natural partial ordering among Young diagrams, obtained by moving blocks down one at a time; the conjecture is that ${\mathcal{T}}_{N,{\mu }_{\text{L}},{\mu }_{\text{R}}}$ can flow from ${\mathcal{T}}_{N,{{\mu }^{\prime }}_{\text{L}},{\mu }_{\text{R}}}$ if and only if ${\mu }_{\text{L}}>{{\mu }^{\prime }}_{\text{L}}$, and similarly for the partition on the right.

The chains of gauge groups considered in this subsection may appear to be only a very special choice of theories. However, the next section shows that in fact all six-dimensional SCFTs have the structure of a linear chain, in a sense that will become clear.

2.6 Brane Engineering of the Chains

The theories (2.11) can be engineered in string theory using $N$ NS5-branes (Brunner & Karch, 1998; Hanany & Zaffaroni, 1998), and variable numbers of D6- and D8-branes. The NS5s are parallel; they have four common transverse directions ${ℝ}^3\times ℝ$; they are all at the origin of the ${ℝ}^3$; and they are separated along the $ℝ$ direction, their positions corresponding to the $〈{\varphi }_i〉$, the vev’s of the tensor multiplet scalars. There are $r_i$ D6-branes parallel to the NS5 and stretched along $ℝ$ between the i-th and $(i+1)$-th NS5. Finally, there are D8-branes parallel to the NS5 and stretched along the ${ℝ}^3$. They are arranged in stacks, each with $f_i$ D8s in it. The D6s end on the D8s in a pattern related to the partitions ${\mu }_{\text{L}}$, ${\mu }_{\text{R}}$. If the NS5s are in generic positions, the D8-branes can be moved around without changing the field theory, while respecting the Hanany–Witten rules (Hanany & Witten, 1997). The fact that the D6-branes on the left and right of each NS5 is not the same is due to the Bianchi identity for the flux $F_2$ sourced by the D6s, which dictates that $n_{\text{D}6,\text{R}}-n_{\text{D}6,\text{R}}=2\pi F_0$, which is in turn non-zero because of the D8-branes.

For more details, an explanation of the D6–D8 pattern, and of the algorithm connecting the brane diagram to the theories (2.11), see, for example, Cremonesi and Tomasiello (2016, section 2).

For example, for the theory ${\mathcal{T}}_{N,[1^k],[1^k]}$, the IIA realization can be simplified: the D8-branes can be moved to infinity without any effect on the field theory. This results in a stack of $k$ D6-branes with $N$ NS5-branes on top. This can also be lifted to M-theory, where it becomes an ${ℝ}^4/{\mathbb{Z}}_k\times ℝ$ singularity with $N$ M5s on top—similar to the discussion for the single gauge group in the section “Single Gauge Group.” The separations among the M5s in the $ℝ$ direction corresponds to the $〈{\varphi }_i〉$. The candidate CFT point is at the origin of the tensor branch parameterized by the $〈{\varphi }_i〉$, namely at $〈{\varphi }_i〉=0$; this corresponds to placing the M5s (or NS5s) on top of each other. This theory is also called ${\mathcal{T}}_{N,k}$; for $k=1$, it reduces to the $\mathcal{N}=(2,0)$ theory ${\mathcal{T}}_N$ of the section “$\mathcal{N}=(2,0)$ Theories.”

Moreover, for ${\mathcal{T}}_{k,[k],[1^k]}$, the IIA realization can be simplified by sending the D8-branes to infinity. The result is $N$ NS5-branes, and $r_i=ri$ D6-branes suspended between the i-th and $(i+1)$-th NS5. Even though the D8-branes are gone, there is a non-zero $F_0=\frac{r}{2\pi }$ everywhere. Again the CFT corresponds to placing the NS5s together: there are no D6-branes on the left and $rN$ on the right of the NS5 stack.

2.7 E-string

A new ingredient will now be introduced: an SCFT ${\epsilon }_N$ known as “E-string” (perhaps confusingly, since it is not a string theory).

This was originally proposed (Ganor & Hanany, 1996; Seiberg & Witten, 1996) as the SCFT describing $N$ M5-branes which are brought to an M-theory boundary. The boundary breaks supersymmetry by a factor of one-half, so this SCFT has only $\mathcal{N}=(1,0)$. The R-symmetry is correspondingly broken to $\text{SO}(4)$, of which an $\text{SU}{(2)}_{\text{R}}$ is R-symmetry and another is a “flavor” symmetry. An M-theory boundary carries an $E_8$ gauge symmetry, which is related to the $E_8\times E_8$ gauge symmetry of the heterotic string. This manifests itself in the E-string as an $E_8$ flavor symmetry.

The positions of the $N$ M5s in the direction transverse to the M-theory boundary correspond to tensor branch parameters, just as for the $\mathcal{N}=(2,0)$ theory ${\mathcal{T}}_N$. In that case, however, one “center of mass” tensor was actually decoupled, while for ${\epsilon }_N$ this does not happen (since the boundary breaks translation invariance). Related to this, even the $N=1$ theory ${\epsilon }_1$ is interacting, unlike the ${\mathcal{T}}_1$ theory which is just a free $\mathcal{N}=(2,0)$ tensor multiplet.

Global symmetry anomalies of ${\epsilon }_N$ have been computed using anomaly-inflow methods (Ohmori, Shimizu, & Tachikawa, 2014). Using results to be discussed in the section “Field Theory Developments,” one obtains, for example, for the $a$ Weyl anomaly the value (Córdova, Dumitrescu, & Intriligator, 2016a)

This theory has also been targeted by the bootstrap approach. This time it proves more fruitful to use a scalar related by supersymmetry to the $E_8$ flavor current. Several central charges are tested (including the one for flavor symmetry), and they all agree rather nicely with those of the smallest E-string theory ${\epsilon }_1$.

3.1 Field Theory Developments

This section considers more recent developments. It starts by reviewing some recent developments that were obtained by purely field-theoretical methods, and that are in a sense a continuation of results obtained in the first wave of work on the subject in the mid-1990s.

The first development is progress in computing anomalies for a CFT, given an effective theory. All gauge anomalies should of course cancel, as discussed in the section “Chains of Unitary Gauge Groups,” but anomalies of global symmetries can still be non-zero. These represent an interesting invariant of the theory that can tell us, for example, if two SCFTs can possibly be dual. The method to compute them is “anomaly matching,” the idea that the anomaly should not change under an RG flow. (Imagine that this wasn’t true and $I_8^{\text{UV}}\ne I_8^{\text{IR}}$. Now add some free matter so that $I_8^{\text{UV total}}=0$ at high energies, and then gauge the global symmetry. This would result in $I_8^{\text{IR total}}\ne 0$ at low energies, and a consistent theory would have become inconsistent under an RG flow.)

Thanks to anomaly matching, the anomalies can be computed directly in an effective description of an SCFT (Intriligator, 2014; Ohmori, Shimizu, Tachikawa, & Yonekura, 2014). When doing this, attention should be paid to the contributions from the tensors that have been used to cancel gauge anomalies. For example, to compute the anomaly of the global symmetry $\text{SU}{(2)}_{\text{R}}$, switch on a background gauge field for it; $I_8$ includes polynomials in its field strength $F_{\text{R}}$. Already the index structure of the fermions in the section “Multiplets” shows that both gauginos and tensorinos give a contribution. For a chain theory of the type we had in the section “Chains of Unitary Gauge Groups,” a computation similar to that for the gauge anomalies gives contributions $(2(N-1)-{\displaystyle {\sum }_i{r}_i^2}){(c_2)}^2-12{\displaystyle {\sum }_i{r}_i{c}_2\text{Tr}{F}_i^2}$, where now $c_2\equiv \text{Tr}{F}_R^2$, in addition to (2.10) (all traces now being in the fundamental). However, the transformation of the $b_i$ should now be modified because of the mixed gauge-R anomaly $c_2\text{Tr}{F}_i^2$; the anomaly can still be canceled by completing the square, but at the cost of modifying $I_4^i$ to $\text{Tr}{F}_i^2+2C_{ij}^{-1}{r}_j$ and of introducing a term

in $I_8$. This gives another contribution to the R-symmetry anomaly. Recall from the comment following (2.10) that $C_{ij}$ is the Cartan matrix for ${\text{A}}_N$.

The anomalies for global symmetries, such as the one just discussed, have also been related to the Weyl anomaly, extending previous work in lower dimensions. Including the gravitation anomalies $p_1$ and $p_2$ (polynomials in the $R^{ab}$ two-form introduced below (2.5)) in the computation of the previous paragraph results in

for some coefficients $\alpha ,\dots ,\delta$, again after canceling the gauge anomalies. The Weyl anomaly $a$ is then (Córdova et al., 2016a):

This result is quite general, and not limited to the chains of the section “Chains of Unitary Gauge Groups.” Similar formulas for the three $c_i$ Weyl anomalies have been given in Beccaria and Tseytlin (2016).

For chain theories, when the number $N-1$ of gauge groups is large, (3.1) is in fact the dominant contribution both to the coefficient $\alpha$ and to $a$. For the case where $r_i=k$, (3.1) can be evaluated using the so-called “strange formula” of Freudenthal and de Vries (2011):

displaying again an $N^3$ behavior. The other contributions to $\alpha$, and the coefficients $\beta$, $\gamma$ and $\delta$ are all subleading in $N$. Inserting (3.4) in (3.1) and (3.3),

at large $N$. Taking the formal limit $r\to 1$, this reproduces the leading coefficient of (2.2). In other chain theories, the coefficients change, but it is still true that (3.1) gives the dominant contribution both to $\alpha$ and to $a$.

The section “Single Gauge Group” reviewed the classification of theories with a single gauge group. The extension to theories with several gauge groups, such as the chain theories of the section “Chains of Unitary Gauge Groups,” was done in Bhardwaj (2020a), again by imposing supersymmetry and anomaly cancellation. While the results are quite involved, some features emerge:

So in fact one of the outcomes of this classification is that the chains of the section “Chains of Unitary Gauge Groups ” are rather close to the general story. The gauge groups are not limited to be unitary, however; this will be explored in the section “F-theory.”

3.2 Holography

Holography (see, e.g., Aharony, Gubser, Maldacena, Ooguri, & Oz, 2000) relates a ${\text{SCFT}}_6$ to an ${\text{AdS}}_7$ spacetime with at least 16 supercharges. This section is about the classification of such solutions in string theory.

There is only one solution with 32 supercharges: the ${\text{AdS}}_7\times S^4$ solution of 11-dimensional supergravity. This is dual to the $\mathcal{N}=(2,0)$ theory ${\mathcal{T}}_N$. Indeed, a near-horizon limit around the solution describing a stack of $N$ M5-branes reproduces ${\text{AdS}}_7\times S^4$.

So we will focus on 16 supercharges, corresponding to $\mathcal{N}=(1,0)$ supersymmetry. Eleven-dimensional supergravity doesn’t produce many more solutions. The supersymmetry equations require the internal space $M_4$ to have Killing spinors, namely $\eta$ such that ${\nabla }_m\eta =\gamma {\gamma }_m\eta$. Such spinors are known to exist if and only if the cone $C(M_4)$ has special holonomy. However, such a space in five dimensions can only be an orbifold ${ℝ}^4/\Gamma \times ℝ$, where $\Gamma$ is a discrete subgroup of $\text{SU}(2)$; this is a cone over a quotient $S^4/\Gamma$. The $\Gamma$ action has two fixed points at the two poles of the $S^4$.

For $\Gamma ={\mathbb{Z}}_k$, these

solutions are dual to the ${\mathcal{T}}_{N,k}={\mathcal{T}}_{N[1^k],[1^k]}$ solutions of the sections “Chains of Unitary Gauge Groups ” and “Brane Engineering of the Chains.” Indeed (see the section “Brane Engineering of the Chains”), one of the string theory realizations of this theory is in terms of $N$ M5-branes on top of an ${ℝ}^4/{\mathbb{Z}}_k\times ℝ$ singularity. A near-horizon limit on these M5s gives (3.6).

From the way (3.6) was obtained as an orbifold, it is clear that it also has a generalization where ${\mathbb{Z}}_k$ is replaced by another subgroup $\Gamma$ of $\text{SU}(2)$. Their field theory duals will be considered in the section “F-theory.”

Returning to ${\mathcal{T}}_{N,k}$, it also has another realization, in terms of NS5-branes on D6-branes (see the section “Brane Engineering of the Chains”). It is possible to reduce (3.6) along a $\text{U}(1)$-isometry that contains the ${\mathbb{Z}}_k$ action. This can be understood by writing the metric of $S^4/{\mathbb{Z}}_k$ as an $S^3$-fibration over an interval: $d{\zeta }^2+{\sin }^2\zeta d{s}_{S^3/{\mathbb{Z}}_k}^2$. The $\text{U}(1)$ isometry is now the one that realizes $S^3/{\mathbb{Z}}_k$ as a Hopf fibration over an $S^2$. So in IIA the metric is an $S^2$-fibration over an interval. The topology is that of an $S^3$, but the metric is not the round one. It has in fact two singularities at the two poles, where $\zeta =0$ and $\pi$; a cartoon is shown in Figure 2(a). These have in fact a physical meaning. When one reduces on a circle from M-theory to IIA, a locus where the circle shrinks is a stack of D6-branes. The singularities just mentioned are then those expected for a stack of $k$ D6-branes at one pole, and of a stack of $k$ anti-D6-branes at the other pole. As a cross-check of the overall picture, one can also obtain this solution as a near-horizon limit near a system of $N$ NS5-branes on $k$ D6-branes.

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Figure 2. Cartoons of the internal $M_3$ dual to the theories of Figure 1.

One can try to obtain more ${\text{AdS}}_7$ solutions by working directly in type II, without getting there from an M-theory reduction. This was initiated in Apruzzi, Fazzi, Passias, Rota, and Tomasiell (2014) using the pure spinor formalism, without prejudice as to the metric on the internal space $M_3$. One result was that there are no solutions in IIB supergravity. In IIA supergravity, on the other hand, it was shown that $M_3$ had to be a fibration of a round $S^2$ over an interval $I$. Both features were already present in the M-theory reduction described in the previous paragraph. The presence of the $S^2$ was in fact guaranteed: the $\mathcal{N}=(1,0)$ superalgebra has an $\text{SU}{(2)}_{\text{R}}$ R-symmetry (see the section “Multiplets”). Global symmetries are realized in holography as isometries of the internal space. Thus, the internal space $M_3$ needs to have an $\text{SU}(2)$ isometry. This guarantees the presence of a round $S^2$ fibration. The presence of an interval was a priori not as clear; it may have been replaced by a circle. However, a posteriori this is also very reasonable, as will be seen shortly.

In Apruzzi et al. (2014) the explicit form of this $S^2$-fibration over an interval $I$ was reduced to solving a system of ordinary differential equations, which at the time could only be solved numerically. The system was later solved analytically by relating it to systems relevant for ${\text{AdS}}_5$ and ${\text{AdS}}_4$ solutions (Apruzzi et al., 2015). Further simplification was achieved with a better coordinate system (Cremonesi & Tomasiello, 2016). All this results in an infinite set of ${\text{AdS}}_7$ solutions. The existence of these solutions is not in contradiction with the scarcity of M-theory solutions, since most of them have non-zero value of $F_0$, which prevents the uplift to M-theory.

The most general ${\text{AdS}}_7$ solution of type IIA supergravity has the (string-frame) metric, dilaton $\varphi$, $B$ field and $F_2$ flux:

Here, $z$ denotes a coordinate on the interval $I$; dots denote derivatives with respect to $z$; and $\alpha =\alpha (z)$ is a function such that

where $F_0$ is the Romans mass.

A few remarks are in order. First some boundary conditions are needed on $\alpha$ at the endpoints of the interval $I$. For example, demanding that the $S^2$ shrinks at the endpoints so that the space $M_3$ is smooth results in

at the endpoint. However, if one requires $M_3$ to be smooth everywhere, it turns out that no solutions exist. This is because (3.8) would require $\alpha$ to be a cubic function, and (3.9) cannot be satisfied at two points for such a function; there are four conditions for four parameters, and the only solution would be $\alpha =0$.

Introducing D8-branes, namely loci where $F_0$ can jump, gives more freedom. Now $\alpha$ is a piecewise-cubic function. Demanding that the metric is continuous requires continuity of $\alpha$, $\dot{\alpha }$, and $\ddot{\alpha }$. The metric will still be non-differentiable at a D8, but this is to be interpreted as its back-reaction, and can be seen already on a D8 in flat space. Flux quantization should also be imposed: $n_0\equiv 2\pi F_0\in \mathbb{Z}$ and $n_2\equiv \frac{1}{2\pi }{\displaystyle {\int }_{S^2}{\tilde{F}}_2}$, where ${\tilde{F}}_2\equiv F_2-B_2{F}_0$. The second requirement leads to placing the D8-branes at integer values of $z$. Moreover, these integer values are related to the D6 charge of the D8-branes, which are thus more properly to be regarded as being D8/D6 bound states. Flux quantization for $H=dB$ is automatically satisfied if the interval $I$ has integer length: $z\in [0,N]$.

With these rules in place, it is very easy to find solutions. $\ddot{\alpha }$ can be any function that gives integers when restricted to integers:

As an example, the following solution

describes a solution with a stack of $n_0$ D8-branes at $z=k$, each with D6-charge $k$, and another stack of $n_0$ D8-branes at $z=N-k$, each with D6-charge $-k$. The Romans mass $F_0$ vanishes in the central region.

More general boundary conditions can also be contemplated. $\alpha \to 0$, for example, leads to a singular geometry, but it is the same singularity generated by a D6-brane in flat space. Allowing for this gives more solutions. For example, if $\ddot{\alpha }$ is constant:

From (3.8) it follows that $F_0=0$. This solution has $k$ D6-branes at $z=0$ and $k$ anti-D6-branes at $z=5$. When plugged in (3.7) and lifted to M-theory, this reproduces (3.6). A simple solution with $F_0\ne 0$ is

which describes a solution with $k$ D6-branes at $z=0$. In Figure 2 cartoons are shown for $M_3$ in the three solutions (3.11), (3.12), and (3.13).

More general boundary conditions are also possible: $\ddot{\alpha }\to 0$ describes the local behavior of an O6, and $\dot{\alpha }\to 0$, $\ddot{\alpha }\to 0$ describes an O8. This would significantly enrich the discussion, but for simplicity these ingredients are not discussed further in this section.

What are the ${\text{SCFT}}_6$ dual to all these ${\text{AdS}}_7$ solutions? They include D6-branes and D8-branes; the fluxes present are $F_0$, $F_2$, and $H=dB$. These are more or less the same ingredients considered in the section “Brane Engineering of the Chains ” for the string realization of the chain theories of the section “Chains of Unitary Gauge Groups.” NS5-branes are absent in the ${\text{AdS}}_7$ solutions, but this can be explained as a result of a near-horizon procedure; it would be similar to the way M5-branes disappear in the near-horizon limit to ${\text{AdS}}_7\times S^4$ (or in the even better-known example of D3-branes producing ${\text{AdS}}_5\times S^5$). The fact that the internal $M_3$ has the topology of an $S^3$ (an unexpected outcome of solution classification) would also be explained as the result of a near-horizon: this sphere would simply be the set of angular directions surrounding the coincident NS5s in the brane configurations of the section “Brane Engineering of the Chains.”

All this leads to conjecture (Gaiotto & Tomasiello, 2014) that the ${\text{AdS}}_7$ solutions (without O-planes) are dual to the chain theories of the section “Chains of Unitary Gauge Groups.” The precise version of this statement is naturally suggested from the integrality requirement: the gauge group ranks should be simply given by

Showing directly that this holographic duality works would need the gravity solution describing the brane intersection in the section “Brane Engineering of the Chains,” which is far beyond current capabilities. However, some indirect tests are possible. Global symmetries match. An $\text{SU}(f_i)$ global symmetry group acts on each set of $f_i$ fundamental hypermultiplets (gauged by the $i$-th gauge group). The gravity solution has precisely $f_i$ D8-branes at $z=i$. These branes have an $\text{SU}(f_i)$ gauge group on them. This is in agreement with the general principle that in holography, global symmetries in the CFT become gauge symmetries in AdS.

A more quantitative test is offered by the Weyl anomaly $a$ (Cremonesi & Tomasiello, 2016). The section “Field Theory Developments” has already shown how to compute this in field theory. In gravity, the methods of Henningson and Skenderis (1998) reduce to computing the integral $a_{\text{hol}}=\frac{3}{56{\pi }^4}{\displaystyle \int {\text{vol}}_{M_3}{e}^{5A-2\varphi }}$, where $e^{2A}$ is the function multiplying $d{s}_{{\text{AdS}}_7}^2$ in (3.7). This results in

at large $N$.

For example, on the $F_0=0$ solution (3.12) this reproduces the field theory result (3.5). This particular match may be regarded as coming from the ${\text{AdS}}_7\times S^4$ computation in Henningson and Skenderis (1998), but it still works for more complicated theories. For example, in the solution (3.11), $\ddot{\alpha }$ is a function that goes up linearly, then has a plateau, then goes down linearly again. The correspondence (3.14) then relates this solution to the theory in Figure 1(c). Both the field theory and the holographic computation agree on the result $a\sim \frac{16}{7}{k}^2(N^3-4N{k}^2+\frac{16}{5}{k}^3)$. (It is possible to keep $k$ large and of order $N$ to make all these terms equally relevant at large $N$.) To see more generally why this kind of check works, observe that $C_{ij}$ in (3.1) is a discrete double derivative, so $C_{ij}^{-1}$ is a discrete double integral. Since the ranks $r_i$ correspond to $\ddot{\alpha }$ under (3.14), (3.15) is a continuous version of (3.1). A more precise analysis (Cremonesi & Tomasiello, 2016) confirms this intuition, and proves that the $a$ anomaly always matches at large $N$ for this class of theories.

All this gives quite nontrivial evidence that the proposed holographic identification (3.14) is valid, and hence that the chain theories of the section “Chains of Unitary Gauge Groups” are indeed SCFTs, as originally proposed (Brunner & Karch, 1998; Hanany & Zaffaroni, 1998).

3.3 F-theory

F-theory is a way to access non-perturbative solutions of IIB string theory. The idea is to interpret the axio-dilaton $\tau =C_0+i{e}^{-\varphi }$ as the modular parameter of a torus fibered over spacetime. Since in many classes of solutions $\tau$ is holomorphic, the total space of the fibration can be studied with algebraic-geometric methods. Degenerations of the fiber correspond to D7-branes or to its images under $\text{SL}(2,\mathbb{Z})$ duality, called $(p,q)$-seven-branes.

To study six-dimensional SCFTs, spacetime is taken to be of the form ${ℝ}^{1,5}\times M_4$; the SCFT will live on the first factor. The torus is now fibered over $M_4$ only. One way to get a solution is then to take the total space $M_6$ of the fibration to be a Calabi–Yau three-fold, which is said to be “elliptically fibered” because of the torus. It is common to describe such an $M_6$ as the locus $\{y^2=x^3+fx+g\}$, where $f$ and $g$ are functions on $M_4$, and $(x,y)$ are coordinates on the torus. The loci where the fiber degenerates and the type of degeneration can be read off from the local behavior of $f$ and $g$. Recall that these loci are interpreted as $(p,q)$-seven-branes. It turns out that stacks of branes with different $(p_i,q_i)$ realize non-Abelian gauge groups of non-unitary type. This goes beyond what is possible with D-branes and perturbative methods, and is one of the reasons F-theory is useful in the context of this article. For example, the equation

describes a seven-brane with gauge group $E_8$ located at $\{u=0\}$.

A sequence of string dualities relates (3.16) to M-theory on a singularity ${ℝ}^4/{\Gamma }_{E_8}\times ℝ$, where ${\Gamma }_{E_8}$ is the so-called “binary icosahedral group.” The notation comes from the McKay correspondence, which associates to a discrete subgroup $\Gamma \subset \text{SU}(2)$ a Lie group of A, D, or E type. Indeed, other equations like (3.16) exist that describe seven-branes with the other ADE Lie groups $G$, and they are dual to M-theory on ${ℝ}^4/\Gamma \times ℝ$.

In the section “Brane Engineering of the Chains” such a singularity was used to obtain an SCFT in the $\Gamma ={\mathbb{Z}}_k$ case, which is associated by the McKay correspondence (McKay, 1980) to $\text{SU}(k)$. In that case, one puts M5s on top of the singularity. It turns out that this corresponds in F-theory to a chain of several seven-branes of ${\text{A}}_k$-type. This suggests how to realize more generally an F-theory dual of $N$ M5-branes on top of ${ℝ}^4/\Gamma \times ℝ$: a chain of seven-branes, each intersecting the next, of type $G_{\Gamma }$ (the Lie group associated to $\Gamma$ by the McKay correspondence), as in Figure 3. Each seven-brane wraps an $S^2$ in the geometry of the base $M_4$; the description in this section shows that they are holomorphic and hence non-trivial in homology. In algebraic geometry these are called “curves,” because they have complex dimension one. For example, an intersection of two $E_8$-branes can be described by the equation

Indeed, near any point with $\upsilon \ne 0$, $\upsilon$ becomes a constant and gives an equation such as (3.16); the same happens near any point $\upsilon \ne 0$. There are two $E_8$-branes, one at $\{u=0\}$ and one at $\{\upsilon =0\}$. More complicated geometries, such as the ones in Figure 3, can be similarly arranged.

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Figure 3. F-theory realization of $N$ M5-branes on a ${\Gamma }_G$ singularity.

The advantage of the F-theory setup is that the physics at the intersections can be investigated. The usual F-theory rules don’t apply near $u=\upsilon =0$ in (3.17); but a geometrical procedure called “blow-up” allows replacing this singularity with a different space, equivalent from the point of view of algebraic geometry but with a milder singularity. These blow-ups reveal a new seven-brane; in the particular example (3.17) the procedure has to be performed several times to obtain degenerations that make sense in F-theory. This allows reading off the gauge groups on each of these new branes.

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Figure 4. $E_8$ conformal matter theory.

The result for two $E_8$-branes is shown in Figure 4. Each brane is depicted as a round node, in analogy with the quiver chains of the section “Chains of Unitary Gauge Groups.” The numbers below each curve represent a geometrical feature called self-intersection, which in field theory corresponds to the string charge mentioned in the section “Single Gauge Group”; they will also be important later. (More precisely, the numbers in the figure are minus the self-intersection, which in this context is always negative.) Another popular way of representing this result is via a diagram where gauge groups are displayed on top of (minus) the self-intersection. With this notation, the chains obtained by blowing up an intersection of two $G$-branes, for $G$ an ADE group, are:

The square brackets denote non-compact curves. The last entry in this list is the same as Figure 4. These theories were dubbed “conformal matter” in Del Zotto, Heckman, Tomasiello, and Vafa (2015). They can be thought of as the analogue of a hypermultiplet, connecting two gauge groups that are more general than the unitary ones in the section “Chains of Unitary Gauge Groups.” The physical interpretation is as follows. The theory describing the intersection of two $E_8$-branes is really an SCFT, which has a tensor branch of dimension 11, corresponding to the size of each of the curves in Figure 4. The blow-ups correspond to making these curves of non-zero size, and to going on a generic point $〈{\varphi }_i〉\ne 0$ of the tensor branch. For other ADE groups, the tensor branch has a smaller dimension, but the same comments apply.

So a diagram such as the one in Figure 4 gives an effective description of sorts to the SCFT that lives at the intersection of two $E_8$-branes. The gauge groups are now more general and include some exceptional groups such as $G_2$ and $F_4$. There are also some “empty” curves without any gauge group. Their presence is not immaterial, since they contribute a tensor multiplet (whose scalar regulates their size). A crucial observation is that the curves with self-intersection $-1$ (corresponding to a $1$ in Figure 4 and in (3.18)) are always connected to groups that are subgroups of $E_8$. For example, in Figure 4 it is connected directly to an $E_8$, or in another case to $G_2\times F_4$, which also turns out to be a subgroup of $E_8$. It is natural to surmise (Heckman et al., 2014) that such curves are just the E-string theory of the section “E-string,” whose $E_8$ flavor symmetry is sometimes partially gauged. The empty curves with self-intersection are interpreted as copies of ${\mathcal{T}}_2$.

The conformal matter theories (3.18) represent one M5 on a $({ℝ}^4/\Gamma )\times ℝ$ singularity. $N$ M5s are obtained by concatenating several copies of (3.18). For example, for $N=2$ and $\Gamma$ associated to $E_6$, the tensor-branch effective description is

This is called a “rank 2” conformal matter theory.

Thus F-theory produces interesting SCFTs, which go beyond the ones of previous sections. Classifying the most general theory that can be produced in F-theory may then be expected to give a handle on the most general SCFT in six dimensions.

Such a classification is in fact feasible and was done in two stages. A first “coarse” classification looked at a particular tensor branch locus, namely the one obtained by shrinking to zero size all curves with self-intersection $-1$. After doing so the self-intersection of neighboring curves changes by $+1$; this may create new curves of self-intersection $-1$, which should then also shrink to zero size. The “endpoint” of this process is a locus in the tensor branch that is not the SCFT point and also not the generic locus where all curves have non-zero size. In Heckman et al. (2014), the possible endpoints were classified. There is a finite list of outliers, and an infinite set of the type

where $\alpha$ and $\beta$ belong to a finite set of “tails,” and ${}^t$ denotes inverting the order. Each of these numbers represents a self-intersection, of the type appearing in (3.18). (This list of endpoints was interpreted in Mekareeya, Ohmori, Shimizu, and Tomasiello (2017) in terms of M-theory.) Another interesting outcome was that at the conformal point all $M_4$ are orbifolds ${ℂ}^2/\Gamma$.

This endpoint has a lot of information about the SCFT, but a lot is also missing. The gauge groups are not shown on any curves. Indeed, the same endpoint can correspond to several different SCFTs. For example, all the chain theories of the section “Chains of Unitary Gauge Groups” have an endpoint where $\alpha =\beta =\cancel{0}$. This is because there are no $-1$-curves to begin with: all gauge groups come from a $-2$-curve. For example, in the notation of (3.18) the theory in Figure 1(b) can be written as

More generally, the endpoint does not change for two theories related by an RG-flow obtained by giving a vev to a Higgs-branch operator.

The classification of endpoints was a first important step, but not the full story. In Heckman et al. (2015) a finer classification was given, which gives a list of individual theories. Each SCFT is again represented by nodes and links, but the nodes are gauge groups of D and E types, while the links are themselves conformal theories such as (3.18). Unfortunately the full list of theories is quite complicated, but a few interesting features emerge:

Thus the “linear chain” structure encountered in the section “Chains of Unitary Gauge Groups” and in the classification of the section “Field Theory Developments” is still present in this more general F-theory classification. Moreover, (3.22) can also be seen as a generalization of convexity condition for the $r_i$ for the unitary chains, which was derived from (2.9).

Actually, while the F-theory classification is clearly broader in scope (because, for example, of the inclusion of E-strings), a comparison with the field theory classification of the section “Field Theory Developments” reveals that some possibilities are missing. For example, field theory would seem to allow symmetric representations for $\text{SU}(N)$ gauge groups. This particular discrepancy was resolved by the study of “frozen” F-theory seven-branes in Bhardwaj, Morrison, Tachikawa, and Tomasiello (2018), but a few more seem to remain; see Bhardwaj (2020b) for a detailed comparison.