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date: 26 October 2020

Supersymmetric QFT in Six Dimensions

Abstract and Keywords

Quantum field theory (QFT) in six dimensions is more challenging than its four-dimensional counterpart: most models tend to become ill-defined at high energies. A combination of supersymmetry and string theory has yielded many QFTs that evade this problem and are low-energy effective manifestations of conformal field theories (CFTs). Besides the usual vector, spinor and scalar fields, the new ingredients are self-dual tensor fields, analogs of the electromagnetic field with an additional spacetime index, sometimes with an additional non-Abelian structure. A recent wave of interest in this field has produced several classification results, notably of models that have a holographic dual in string theory and of models that can be realized in F-theory. Several precise quantitative checks of the overall picture are now available, and give confidence that a full classification of all six-dimensional CFTs may be at hand.conformal field theories, supersymmetry, extra dimensions, holography, string theory, D-branes, F-theory

Keywords: conformal field theories, supersymmetry, extra dimensions, holography, string theory, D-branes, F-theory

1. Introduction

Defining interacting quantum field theories (QFTs) gets harder as the spacetime dimension d gets higher. An operator with scaling dimension Δ>d is called irrelevant; this means that it gets less important at low energies, but conversely its coefficient grows at high energies, rendering the QFT non-renormalizable, or in other words cutoff-dependent—an effective field theory in need of a more fundamental definition.

For example, a scalar ϕ has classical dimension d21; an interaction ϕk in the Lagrangian becomes irrelevant for d2kk2. For a cubic interaction, k=3, this gives d6; for larger k, the critical dimension is even lower. A Yang–Mills (YM) coupling 12TrFμvFμvTr|F|2 has dimension four, and thus is non-renormalizable in d=5 and above. The situation is even worse for gravity, whose Lagrangian density reads gR (with R the scalar curvature), and hence has dimension two. In a sense, in d>4 all QFTs become as bad as general relativity.

In spite of these difficulties, string theory arguments have led to several proposals for interacting conformal field theories (CFTs) in d>4, which have been checked in many ways. CFTs are in particular scale invariant, and so are defined at all energy scales. Moreover, a non-conformally-invariant QFT can become a CFT at high energies; one says that the CFT gives an ultraviolet (UV) completion of the QFT. Such a breaking of scale invariance could come in general by adding an additional operator to the QFT, or from giving a vacuum expectation value (vev) to an operator. In the latter case, one can also say that the QFT gives an effective description of the CFT.

This article focuses on supersymmetric theories in d=6, which is the highest dimension where a supersymmetric CFT (SCFT) can exist (Nahm, 1978). Historically, the first SCFT suggested by string theory was the theory governing the dynamics of a stack of M5-branes, which has N=(2,0) supersymmetry (Strominger, 1996; Witten, 1995, 1996). Other suggestions soon followed for theories with N=(1,0) supersymmetry, involving for example orbifolds (Intriligator, 1997, 1998), IIA brane intersections (Brunner & Karch, 1998; Hanany & Zaffaroni, 1998), and M-theory boundaries (Ganor & Hanany, 1996; Seiberg & Witten, 1996). At a later stage, the subject was revitalized by several related classification results for N=(1,0) theories: of theories with a holographic dual in string theory (Apruzzi, Fazzi, Rosa, & Tomasiello, 2014), of theories that can be engineered in F-theory (Heckman, Morrison, Rudelius, & Vafa, 2015; Heckman, Morrison, & Vafa, 2014), and of effective descriptions allowed by anomaly cancellation constraints (Bhardwaj, 2020a). (In d=6, an SCFT can flow at low energies to a supersymmetric QFT only by vev breaking (Córdova, Dumitrescu, & Intriligator, 2016b; Louis and Lüst, 2015).)

In the meantime it also became clear that many CFTs in d=4 can be obtained by compactifying a CFT in d=6 on a Riemann surface. Evidence for this came from brane intersections (Witten, 1997) and from holography (Maldacena & Núñez, 2001), and was later greatly expanded in both directions (Bah, Beem, Bobev, & Wecht, 2012; Gaiotto, 2012; Gaiotto & Maldacena, 2012) for N=(2,0) theories and by several other authors for N=(1,0) (e.g., Del Zotto, Vafa, & Xie, 2015; Gaiotto & Razamat, 2015; Ohmori, Shimizu, Tachikawa, & Yonekura, 2015a, 2015b; Razamat, Vafa, & Zafrir, 2017). While those developments are not covered here, they have fueled speculation that perhaps all CFTs in d=4 have a d=6 origin, and have given yet another powerful piece of motivation for studying d=6 CFTs.

This article assumes textbook knowledge of QFT and working knowledge of supersymmetry, both in four dimensions. While much of the intuition behind the development of this field came from string theory, most of the article should be accessible to readers who are not too familiar with it, emphasizing instead field theory methods. The string theory comments are often packaged in paragraphs that can be skipped without excessive damage; for readers who wish to follow those as well, some familiarity with any introductory textbook on the subject should be enough.

The section “Motivating Examples” reviews some simple examples that historically motivated the development of the field; then the section “Classification” surveys the more recent classification results.

2. Motivating Examples

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 α and α˙ 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 Qα and Q¯α˙, 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 QαQα1 one can give a name to its conjugate, Qα2(Qα1)c; then by construction Qα1(Qα2)c. This is called a symplectic Majorana pair. In this article, N=(p,q) supersymmetry is defined to have p independent Qα of positive chirality and q of negative chirality, thus counting the symplectic Majorana pairs as one single supercharge. So, for example, for N=(1,0) supersymmetry there is a single pair Qαa. 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: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 thenN=4, which is rather constrained. For example, for N=(1,1) one can build a super-Yang–Mills theory rather reminiscent of N=4 in four dimensions. This theory is not conformal and will not be considered further in this article. N=(1,0) gives eight supercharges; upon dimensional reduction to d=4 this becomes then N=2 supersymmetry. This case will be the focus of this article. 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 μjμ=I41, with I41 a four-form. The so-called descent formalism gives a way to summarize this conveniently by a formal six-form I6 with cubic dependence on the gauge field-strength two-form F12Fμvdxμdxv; the relation is via I6=dI7, δI7=dI61, where δ denotes a gauge transformation. In d=6, these statements are very similar; the anomaly is summarized by a formal eight-form I8, now quartic in F. The descent equations now read I8=dI7, δI7=dI61.

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:

Tμμ=aE+iciIi,
(2.1)

up to total derivatives. Here E is the Euler density, and Ii 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 Fp 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 G4.

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 H. M-theory has M5-branes, which couple to the potential for G4. 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 N=(2,0) theory in d=6.

2.2 Multiplets

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

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

  • • Vector multiplet: vector field Aμ, two spinors λaα˙.

  • • Tensor multiplet: a real scalar ϕ, two spinors χaα, and a potential bμv with a field-strength hμvρ=[μbvρ] which is self-dual: hμvρ=16ϵμvραβγhαβγ. In the short-hand “form” notation one writes h=db, h=*h.

  • • Hypermultiplet: two complex scalars ha, one complex spinor ψα.

  • • Linear multiplet: four real scalars lab, two spinors ξaα.

The linear multiplet is a variant of the hypermultiplet, and plays a more limited role. The difference between the two lies in the SU(2)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 13; 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, N=(1,0) reduces toN=2. The vector multiplet becomes the vector multiplet in d=4, with two components of Aμ becoming two scalars. The tensor multiplet also becomes a vector multiplet, upon taking the dual of the four-dimensional components of bμv. The hypermultiplet reduces with no further difficulty.

2.3 N=(2,0) Theories

In the N=(2,0) superconformal algebra, the R-symmetry is Sp(2)RSO(5)R/2. One option to obtain such an enhancement is to combine a N=(1,0) tensor multiplet with a linear multiplet; the 1 and 13 of Sp(1)R come together to form a 5 of Sp(2)R.

This 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 N=(2,0) tensor multiplet; it is denoted by TN A different string realization exists in IIB, involving 2/N singularities. This suggests (Witten, 1995) a generalization TΓ where N is replaced with other discrete subgroups Γ of SU(2).

In spite of many attempts at writing a Lagrangian for them, the theories TN and TΓ remain quite mysterious. Various arguments indicate that the degrees of freedom of this non-Abelian tensor TN should go like N3. In string theory this manifests itself via an anomaly-inflow computation (Harvey, Minasian, & Moore, 1998), or by using holographic duality to the AdS7×S4 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 TN reads

a(TN)=167(N3N)+N,
(2.2)

in a normalization where a=1 for the single N=(2,0) tensor multiplet,T1. There are three more Weyl anomalies ci in d=6, but N=(2,0) supersymmetry fixes them to be proportional to a.

It would be interesting to establish whether TN and TΓ are the only 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 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 N=(2,0) theory the smallest possible a is 1107; this corresponds to (2.2) for N=2. So in a sense the smallest possible theory isT2. (The free tensor multiplet T1 is ignored in this approach.) This can also be read as independent verification that TN exist.

2.4 Single Gauge Group

Given the mystery surrounding the N=(2,0) case, it may look safer to stay withinN=(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 1gYM2Tr|F|2 is relevant in d=6; this is not promising for a CFT. Even more worrisome, the partner of the vector (the “gaugino”) λaα˙ 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”) ψα has chirality opposite to that of λaα˙; 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 ρ under which the hypermultiplet transforms. Perhaps the first possibility that springs to mind is to take ρ to be the adjoint; in this case the gauge anomaly is indeed canceled, and the theory becomes super-YM withN=(1,1). In other words, the N=(1,0) vector and hypermultiplet assemble into a single 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=SU(Nc) with Nf hypermultiplets in the fundamental representation, the anomaly polynomial (see the section “Background”) I8=TrAd(F4)NfTrfund(F4); expressing the generators of the adjoint in terms of those in the fundamental gives TrAd(F4)=2NcTrfund(F4)6(TrfundF2)2. Thus the term Trfund(F4) can be canceled by taking

Nf=2Nc.
(2.3)

The residual

6(TrfundF2)26I42
(2.4)

can be canceled by introducing a tensor multiplet. Indeed I61=I21I4 (with I4=dI3, δI3=dI21 as in the section “Background”); if the two-form b transforms as δb=I21 and has a coupling (Green, Schwarz, & West, 1985; Sagnotti, 1992)

LGSWS=6b2I4,
(2.5)

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 Rab12Rabμνdxμdxν. 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 LGSWS. (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

LϕTr(F|2D2)+ϕ2+|h|2+(bTrF2)+|Dh|2+h¯ahσabIDIhb.
(2.6)

Dh is the gauge covariant derivative, σI are the Pauli matrices, and DI is a triplet of auxiliary fields; again ||2 is the norm of a form by index contraction. In particular (2.6) contains LGSW and a partner term

ϕTr|F|2
(2.7)

where ϕ 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 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 ϕ, and to treat its fluctuations perturbatively. Notice that

1gYM2=ϕ
(2.8)

If ϕ0, the action becomes the usual YM one; the theory is non-renormalizable with an energy cutoff of order ϕ1/2. If on the other hand ϕ=0, then gYM and the theory is strongly coupled at all scales.

On the one hand, this may sound discouraging: the ϕ=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 ϕ0 approach this CFT when they become strongly coupled at energies far above the cutoff ϕ1/2. So the theory (2.6) may be an effective description of a CFT. The space 0 of possible vev’s for ϕ is called a tensor branch, since ϕ 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 4/2×. A stack of M5s at the origin of 4/2 but separated by δ in are described by (2.6) with ϕδ. To be more precise, this engineers a version of (2.6) with gauge group U(Nc) rather than SU(Nc). The U(1) gives rise to further anomalies besides those already discussed (involving for example TrF); 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 “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 b to the tensor multiplet two-form, and their tension is proportional to ϕ, 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 N1 vector multiplets (Ai,λi), i=1,,N1 with gauge group U(ri); hypermultiplets hi, i=1,,N2 in the bifundamental (ri,r¯i+1) of U(ri)×U(ri+1) and fi hypermultiplets h˜i, i=1,,N1 in the fundamental ri of U(ri); and tensor multiplets bi, i=1,,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 U(ri) separately. The (ri,r¯i+1) bifundamental is seen by it as ri+1 fundamentals; so in total (2.3) gives

ri+1+ri12ri=fi.
(2.9)

This cancels the Trfund(Fi4) term in the anomaly polynomial I8. What remains is of the form

3CijTrfundFi2TrfundFj2
(2.10)

where Cij=2δijδi1,jδi+1,j is the Cartan matrix for the Lie algebra AN. This generalizes I4 in (2.4). As in that case, this term in the anomaly polynomial can be cancelled by introducing a coupling 3CijbiTrfundFj2 in the (pseudo-)Lagrangian, and letting δbi=I2i be related to I4i=Tr(Fi2)2 by descent. Supersymmetry relates this to a coupling of the type (ϕi+1ϕi)Trfund|Fi|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: ϕi=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, ri=k. This satisfies (2.9) with all fi=0 except for f1=k and fN1=k. A second example is obtained by taking ri=ri, a linear growth in ranks. Again (2.9) is satisfied by taking all fi=0 except f1=r and fN1=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 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).

Supersymmetric QFT in Six Dimensions

Figure 1. Examples of chains of unitary gauge groups.

This gives rise to a large number of possibilities. The choice of the ri is, however, not completely arbitrary. The combination of ri+1+ri12ri=(ri+1ri)(riri1) in (2.9) is a discrete double derivative; since fi is by definition non-negative, it follows that the discrete function iri is convex. The function isiriri1 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 ρL, ρR 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:

TN,μL,μR.
(2.11)

In this notation, the three examples of theories in Figure 1 are TN,[1k],[1k], Tk,[k],[1k], and TN,[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 TN,μL,μR can flow from TN,μL,μR if and only if μL>μL, 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 3×; they are all at the origin of the 3; and they are separated along the direction, their positions corresponding to the ϕi, the vev’s of the tensor multiplet scalars. There are ri D6-branes parallel to the NS5 and stretched along between the i-th and (i+1)-th NS5. Finally, there are D8-branes parallel to the NS5 and stretched along the 3. They are arranged in stacks, each with fi D8s in it. The D6s end on the D8s in a pattern related to the partitions μL, μR. 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 F2 sourced by the D6s, which dictates that nD6,RnD6,R=2πF0, 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 TN,[1k],[1k], 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 4/k× 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 direction corresponds to the ϕi. The candidate CFT point is at the origin of the tensor branch parameterized by the ϕi, namely at ϕi=0; this corresponds to placing the M5s (or NS5s) on top of each other. This theory is also called TN,k; for k=1, it reduces to the N=(2,0) theory TN of the section “N=(2,0) Theories.”

Moreover, for Tk,[k],[1k], the IIA realization can be simplified by sending the D8-branes to infinity. The result is N NS5-branes, and ri=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 F0=r2π 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 εN 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 N=(1,0). The R-symmetry is correspondingly broken to SO(4), of which an SU(2)R is R-symmetry and another is a “flavor” symmetry. AnM-theory boundary carries an E8 gauge symmetry, which is related to the E8×E8 gauge symmetry of the heterotic string. This manifests itself in the E-string as an E8 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 N=(2,0) theory TN. In that case, however, one “center of mass” tensor was actually decoupled, while for εN this does not happen (since the boundary breaks translation invariance). Related to this, even the N=1 theory ε1 is interacting, unlike the T1 theory which is just a free N=(2,0) tensor multiplet.

Global symmetry anomalies of εN 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)

a=17(64N3+144N2+99N).
(2.12)

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 E8 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 ε1.

3. Classification

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 I8UVI8IR. Now add some free matter so that I8UV total=0 at high energies, and then gauge the global symmetry. This would result in I8IR total0 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 SU(2)R, switch on a background gauge field for it; I8 includes polynomials in its field strength FR. 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(N1)iri2)(c2)212iric2TrFi2, where now c2TrFR2, in addition to (2.10) (all traces now being in the fundamental). However, the transformation of the bi should now be modified because of the mixed gauge-R anomaly c2TrFi2; the anomaly can still be canceled by completing the square, but at the cost of modifying I4i to TrFi2+2Cij1rj and of introducing a term

12i,jCij1rirjc22
(3.1)

in I8. This gives another contribution to the R-symmetry anomaly. Recall from the comment following (2.10) that Cij is the Cartan matrix for AN.

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 p1 and p2 (polynomials in the Rab two-form introduced below (2.5)) in the computation of the previous paragraph results in

I8=124(αc22+βc2p1+γp12+δp2)
(3.2)

for some coefficients α,,δ, again after canceling the gauge anomalies. The Weyl anomaly a is then (Córdova et al., 2016a):

a=167(αβ+γ)+67δ,
(3.3)

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

For chain theories, when the number N1 of gauge groups is large, (3.1) is in fact the dominant contribution both to the coefficient α and to a. For the case where ri=k, (3.1) can be evaluated using the so-called “strange formula” of Freudenthal and de Vries (2011):

12i,jCij1=k2(N3N),
(3.4)

displaying again an N3 behavior. The other contributions to α, and the coefficients β, γ and δ are all subleading in N. Inserting (3.4) in (3.1) and (3.3),

a167N3k2
(3.5)

at large N. Taking the formal limit r1, 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 α 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:

  • • There is a list of allowed representations connecting two simple gauge groups (see Bhardwaj, 2020a, table 4).

  • • The integrality of string charges, mentioned in the section “Single Gauge Group,” is again guaranteed by anomaly cancellation.

  • • No loops are allowed. Roughly speaking, the reason is the following. If there were a loop, the number of tensor multiplets needed to cancel the gauge anomaly would be smaller than that of simple gauge groups. While this may seem good, it would signal a string state (discussed in the section “Single Gauge Group ”) whose tension would remain constant on the tensor branch. But if a CFT point existed, it would have no mass parameter. So theories with loops have no CFT points.

  • • The allowed bifurcations are rather minimal.

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 SCFT6 to an AdS7 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 AdS7×S4 solution of 11-dimensional supergravity. This is dual to the N=(2,0) theory TN. Indeed, a near-horizon limit around the solution describing a stack of N M5-branes reproduces AdS7×S4.

So we will focus on 16 supercharges, corresponding to N=(1,0) supersymmetry. Eleven-dimensional supergravity doesn’t produce many more solutions. The supersymmetry equations require the internal space M4 to have Killing spinors, namely η such that mη=γγmη. Such spinors are known to exist if and only if the cone C(M4) has special holonomy. However, such a space in five dimensions can only be an orbifold 4/Γ×, where Γ is a discrete subgroup of SU(2); this is a cone over a quotient S4/Γ. The Γ action has two fixed points at the two poles of the S4.

For Γ=k, these

AdS7×S4/k
(3.6)

solutions are dual to the TN,k=TN[1k],[1k] 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 4/k× 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 k is replaced by another subgroup Γ of SU(2). Their field theory duals will be considered in the section “F-theory.”

Returning to TN,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 U(1)-isometry that contains the k action. This can be understood by writing the metric of S4/k as an S3-fibration over an interval: dζ2+sin2ζdsS3/k2. The U(1) isometry is now the one that realizes S3/k as a Hopf fibration over an S2. So in IIA the metric is an S2-fibration over an interval. The topology is that of an S3, but the metric is not the round one. It has in fact two singularities at the two poles, where ζ=0 and π; 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.

Supersymmetric QFT in Six Dimensions

Figure 2. Cartoons of the internal M3 dual to the theories of Figure 1.

One can try to obtain more AdS7 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 M3. One result was that there are no solutions in IIB supergravity. In IIA supergravity, on the other hand, it was shown that M3 had to be a fibration of a round S2 over an interval I. Both features were already present in the M-theory reduction described in the previous paragraph. The presence of the S2 was in fact guaranteed: the N=(1,0) superalgebra has an SU(2)R R-symmetry (see the section “Multiplets”). Global symmetries are realized in holography as isometries of the internal space. Thus, the internal space M3 needs to have an SU(2) isometry. This guarantees the presence of a round S2 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 S2-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 AdS5 and AdS4 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 AdS7 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 F0, which prevents the uplift to M-theory.

The most general AdS7 solution of type IIA supergravity has the (string-frame) metric, dilaton ϕ, B field and F2 flux:

1π2ds2=8αα¨dsAdS72+αα¨(dz2+α2α˙22αα¨dsS22);
(3.7a)

eϕ=25/4π5/234(α/α¨)3/4α˙22αα¨;
(3.7b)

B=π(z+αα˙α˙22αα¨)volS2,F2=(α¨162π2+πF0αα˙α˙22αα¨)volS2.
(3.7c)

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

α=162π3F0
(3.8)

where F0 is the Romans mass.

A few remarks are in order. First some boundary conditions are needed on α at the endpoints of the interval I. For example, demanding that the S2 shrinks at the endpoints so that the space M3 is smooth results in

α0,α¨0
(3.9)

at the endpoint. However, if one requires M3 to be smooth everywhere, it turns out that no solutions exist. This is because (3.8) would require α 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 α=0.

Introducing D8-branes, namely loci where F0 can jump, gives more freedom. Now α is a piecewise-cubic function. Demanding that the metric is continuous requires continuity of α, α˙, and α¨. 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: n02πF0 and n212πS2F˜2, where F˜2F2B2F0. 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[0,N].

With these rules in place, it is very easy to find solutions. α¨ can be any function that gives integers when restricted to integers:

iα(k).
(3.10)

As an example, the following solution

α=816π2n0{z(z2+3k(kN))z[0,k],k(3z23Nz+k2)z[k,Nk],(Nz)((Nz)2+3k(kN))z[Nk,N],
(3.11)

describes a solution with a stack of n0 D8-branes at z=k, each with D6-charge k, and another stack of n0 D8-branes at z=Nk, each with D6-charge k. The Romans mass F0 vanishes in the central region.

More general boundary conditions can also be contemplated. α0, 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 α¨ is constant:

α=z(Nz).
(3.12)

From (3.8) it follows that F0=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 F00 is

α=816π2n0z(Nz)(z2N),
(3.13)

which describes a solution with k D6-branes at z=0. In Figure 2 cartoons are shown for M3 in the three solutions (3.11), (3.12), and (3.13).

More general boundary conditions are also possible: α¨0 describes the local behavior of an O6, and α˙0, α¨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 SCFT6 dual to all these AdS7 solutions? They include D6-branes and D8-branes; the fluxes present are F0, F2, 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 AdS7 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 AdS7×S4 (or in the even better-known example of D3-branes producing AdS5×S5). The fact that the internal M3 has the topology of an S3 (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 AdS7 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

ri=α¨(i).
(3.14)

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 SU(fi) global symmetry group acts on each set of fi fundamental hypermultiplets (gauged by the i-th gauge group). The gravity solution has precisely fi D8-branes at z=i. These branes have an SU(fi) 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 ahol=356π4volM3e5A2ϕ, where e2A is the function multiplying dsAdS72 in (3.7). This results in

ahol1927dzαα¨
(3.15)

at large N.

For example, on the F0=0 solution (3.12) this reproduces the field theory result (3.5). This particular match may be regarded as coming from the AdS7×S4 computation in Henningson and Skenderis (1998), but it still works for more complicated theories. For example, in the solution (3.11), α¨ 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 a167k2(N34Nk2+165k3). (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 Cij in (3.1) is a discrete double derivative, so Cij1 is a discrete double integral. Since the ranks ri correspond to α¨ 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 τ=C0+ieϕ as the modular parameter of a torus fibered over spacetime. Since in many classes of solutions τ 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 SL(2,) duality, called (p,q)-seven-branes.

To study six-dimensional SCFTs, spacetime is taken to be of the form 1,5×M4; the SCFT will live on the first factor. The torus is now fibered over M4 only. One way to get a solution is then to take the total space M6 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 M6 as the locus {y2=x3+fx+g}, where f and g are functions on M4, 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 (pi,qi) 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

y2=x3+u5
(3.16)

describes a seven-brane with gauge group E8 located at {u=0}.

A sequence of string dualities relates (3.16) to M-theory on a singularity 4/ΓE8×, where ΓE8 is the so-called “binary icosahedral group.” The notation comes from the McKay correspondence, which associates to a discrete subgroup ΓSU(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 4/Γ×.

In the section “Brane Engineering of the Chains” such a singularity was used to obtain an SCFT in the Γ=k case, which is associated by the McKay correspondence (McKay, 1980) to 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 Ak-type. This suggests how to realize more generally an F-theory dual of N M5-branes on top of 4/Γ×: a chain of seven-branes, each intersecting the next, of type GΓ (the Lie group associated to Γ by the McKay correspondence), as in Figure 3. Each seven-brane wraps an S2 in the geometry of the base M4; 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 E8-branes can be described by the equation

y2=x3+(uυ)5.
(3.17)

Indeed, near any point with υ0, υ becomes a constant and gives an equation such as (3.16); the same happens near any point υ0. There are two E8-branes, one at {u=0} and one at {υ=0}. More complicated geometries, such as the ones in Figure 3, can be similarly arranged.

Supersymmetric QFT in Six Dimensions

Figure 3. F-theory realization of N M5-branes on a ΓG 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=υ=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.

Supersymmetric QFT in Six Dimensions

Figure 4. E8 conformal matter theory.

The result for two E8-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:

G=SU(k):[SU(k)]2suk2suk[SU(k)]G=SO(2k):[SO(2k)]1usp2k84so2k[SO(2k)]G=E6:[E6]13su316e6[E6]G=E7:[E7]12su23so72su218e7[E7]G=E8:[E8]122su23g215f413g22su2211e82[E8],
(3.18)

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 E8-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 ϕi0 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 E8-branes. The gauge groups are now more general and include some exceptional groups such as G2 and F4. 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 E8. For example, in Figure 4 it is connected directly to an E8, or in another case to G2×F4, which also turns out to be a subgroup of E8. It is natural to surmise (Heckman et al., 2014) that such curves are just the E-string theory of the section “E-string,” whose E8 flavor symmetry is sometimes partially gauged. The empty curves with self-intersection are interpreted as copies of T2.

The conformal matter theories (3.18) represent one M5 on a (4/Γ)× singularity. N M5s are obtained by concatenating several copies of (3.18). For example, for N=2 and Γ associated to E6, the tensor-branch effective description is

[E6]13su316e613su31[E6].
(3.19)

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

α2222βt
(3.20)

where α and β 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 M4 are orbifolds 2/Γ.

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 α=β=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

22su22su32suk[SU(k+1)].
(3.21)

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:

  • • No loops are allowed.

  • • The allowed bifurcations are rather minimal, and can occur only towards the ends of the chain (see Heckman et al., 2015, Figure 1).

  • • In the resulting chain, the DE gauge groups in the nodes grow and then decrease:

G1GmGk.
(3.22)

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 ri 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 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.

4. Conclusions

The recent wave of interest in six-dimensional theories has produced some interesting classifications. The largest is the one of SCFTs with a realization in F-theory (Heckman et al., 2015). An interesting picture has emerged: most SCFTs have an effective field theory description on their tensor branch, which has a description as a linear chain with some minimal ramification. The ingredients of these linear chains are the multiplets discussed in the section “Multiplets,” with the E-string theory of the section “E-string” also appearing quite often.

Several questions remain. The most obvious is whether this list is complete. This is not even known for N=(2,0) theories, where string theory gives a very small list of possibilities (see the section “N=(2,0) Theories”). For N=(1,0) the question is even more pressing. There could in principle be theories that have no realization in string theory. Related to this, in this article all theories were studied by means of an effective realization on their tensor branch, but it is a priori unclear whether all SCFTs have a tensor branch.

Another issue is that the classification in F-theory is still quite messy, and one may perhaps hope for something more memorable and elegant. One possibility was to use the web of Higgs-branch RG flows. As mentioned in the section “Chains of Unitary Gauge Groups,” unitary chain theories are conjectured to be related by a web of such flows; all theories with a given N are connected in such a way. A similar structure may be hoped for in the more general context of theories coming from F-theory. It was in general shown (Heckman, Rudelius, & Tomasiello, 2019) that starting from a single theory with a given endpoint (in the sense of (3.20)), all other theories with the same endpoint can be generated from such RG-flows. This may be used to bring some order to the general classification.

Moreover, the structure of RG-flows is intimately related to the geometry of nilpotent elements in Lie groups. For example, the Young diagrams appearing in (2.11) are in one-to-one correspondence with nilpotent elements in SU(k), up to group conjugation. Maybe the appearance of nilpotent elements is somehow automatic in the fact that the Higgs moduli spaces of N=(1,0) theories is hyper-Kähler, and maybe this fact can be used to rederive the F-theory classification. Techniques such as those in Bourget et al. (2020) may be useful.

Finally, while all this progress in six dimensions is quite exciting, it would be even better to put it to work to generate interesting theories in lower dimensions, such as four. There is already a lot of ongoing work in this direction, and this article cannot do it justice. Compactifying N=(2,0) theories produced many interesting N=2 and N=1 theories, related by duality webs. It is natural to expect that a similarly exciting picture awaits for compactifications of N=(1,0) theories.

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