From Large to Small $\mathcal{N}=(4,4)$ Superconformal Surface Defects in Holographic 6d SCFTs

Six dimensional (6d) superconformal field theories (SCFTs) hold a special place among quantum field theories (QFTs). Owing to the classification discovered in the seminal work by Nahm Nahm:1977tg , superconformal symmetry is only possible in six and fewer spacetime dimensions. Moreover, $\mathcal{N}=(2,0)$ is the maximal amount of supersymmetry (SUSY) that a 6d theory can have. Combining this amount of SUSY with conformal symmetry constrains a 6d $\mathcal{N}=(2,0)$ SCFT to such a degree that the only additional information that is necessary to completely determine the theory is the choice of a gauge algebra.

The study of the 6d $\mathcal{N}=(2,0)$ theory is thus of fundamental importance in QFT, for many reasons. For example, the 6d $\mathcal{N}=(2,0)$ SCFT determines the physics of many other QFTs in 6d, via SUSY-breaking deformations Antoniadis:1998ki ; Antoniadis:1998ep ; Cordova:2016xhm such as orbifolds Blum:1997fw ; Blum:1997mm ; Brunner:1997gf ; Intriligator:1997dh . By suitable (partial) topological twisting, the 6d theory compactified on, e.g., a Riemann surface Gaiotto:2009we or a 3-manifold Dimofte:2011ju can also determine the physics of infinite families of QFTs in $d<6$.

The 6d $\mathcal{N}=(2,0)$ SCFT is also of fundamental importance in quantum gravity. Currently, the leading candidate for an ultra-violet (UV)-complete theory of quantum gravity is M-theory. M-theory’s fundamental objects are M2-branes Duff:1990xz and M5-branes Gueven:1992hh , and the low-energy worldvolume theory on $M$ coincident M5-branes is the 6d $\mathcal{N}=(2,0)$ SCFT with gauge algebra $A_{M-1}$ Strominger:1995ac . Understanding the 6d $\mathcal{N}=(2,0)$ SCFT is thus essential to understanding M-theory in general. In particular, via the Anti-de Sitter/CFT (AdS/CFT) correspondence, the 6d $\mathcal{N}=(2,0)$ SCFT can provide a fully non-perturbative definition of M-theory on an asymptotically 7d AdS spacetime, AdS₇, times a four-sphere, $\mathds{S}^{4}$ Witten:1996hc ; Maldacena:1997re ; Witten:1998wy .

Strongly interacting SCFTs constructed in string- and M-theory, including the non-Abelian 6d $\mathcal{N}=(2,0)$ SCFT, are prohibitively difficult to study, for many reasons, of which we will mention only three. First, the $\mathcal{N}=(1,0)$ and $\mathcal{N}=(2,0)$ SUSY multiplets include a chiral two-form gauge field, and writing a local, gauge-invariant Lagrangian for a non-Abelian higher-form gauge field remains a major open problem. These SCFTs thus have no known Lagrangian descriptions. Second, in the space of renormalization group (RG) flows, these SCFTs are isolated fixed points, and in particular they cannot be reached as infra-red (IR) fixed points of RG flows from free ultra-violet (UV) fixed points. Third, these SCFTs are intrinsically strongly interacting. For example, the 6d $\mathcal{N}=(2,0)$ SCFT has no dimensionless parameter besides $M$ that can be tuned to allow a perturbative expansion.

As a result, practically all of our direct knowledge¹¹1Indirect methods such as dimensional reduction to 5d $\mathcal{N}\leq 2$ SUSY QFTs have also been used to great effect to study these theories, e.g. by using the resulting lower dimensional Lagrangian description together with supersymmetric localization techniques Bullimore:2014upa . of interacting 6d SCFTs comes from non-perturbative methods, such as the superconformal bootstrap Beem:2015aoa , F-theory Heckman:2015bfa , and especially AdS/CFT Apruzzi:2013yva , where holographic computations of quantities like Weyl anomalies Henningson:1998gx and entanglement entropy (EE) Hung:2011xb are used to great effect to characterize 6d SCFTs at large $M$.

An aspect of 6d SCFTs, and generally of QFTs in three and higher dimensions, that requires particularly careful treatment to characterize is the spectrum of 2d, string-like or surface, defects. In the co-dimension one case, 2d defects in 3d QFTs arise as boundaries or interfaces, and so are more easily studied and, thus, more familiar than their higher co-dimension realizations. Despite being somewhat more exotic in standard treatments of QFTs, 2d defects of co-dimension two and greater show up in a number of settings²²2This is by no means an exhaustive list of the work done on 2d defects. For a recent review of boundaries and defects in QFTs and further references on the topic, see Andrei:2018die .: from free field theories Soderberg:2017oaa ; Lauria:2020emq ; Giombi:2021uae ; Bianchi:2021snj to strongly interacting and non-Lagrangian 4d QFTs, e.g. Gukov:2006jk ; Gukov:2008sn ; Alday:2009fs , to being fundamental objects in 6d SCFTs Ganor:1996nf and in the study of EE and Renyi entropies Hung:2014npa ; Lewkowycz:2014jia . As such, the last few decades have seen tremendous advancements in characterizing Billo:2016cpy and constraining Jensen:2015swa ; Kobayashi:2018lil the properties of 2d defects, and it is vitally important in the study of QFTs, generally, to continue this effort by finding novel constructions of surface defects and examining their unique physics.

Of interest to us in the current work are the holographic descriptions, afforded by AdS/CFT, of 6d SCFTs and the defects that they support. In particular, we will primarily focus our attention on solutions to 11d supergravity (SUGRA) that are contained in a one-parameter family of solutions with superisometry given by the exceptional Lie superalgebra $\mathfrak{d}(2,1;\gamma)\oplus\mathfrak{d}(2,1;\gamma)$ Bachas:2013vza . Crucially, an asymptotically AdS${}_{7}\times\mathds{S}^{4}$ solution is possible only at certain values of $\gamma$. Indeed, as a historical note, prior to the full classification given in Bachas:2013vza , evidence of $\mathfrak{d}(2,1;\gamma)\oplus\mathfrak{d}(2,1;\gamma)$ invariant SUGRA solutions that exist for general $\gamma$ were found in superconformal Janus solutions in $4d$ gauged $\mathcal{N}=8$ SUGRA Bobev:2013yra , which extended beyond the known $\gamma=1$ AdS${}_{4}\times\mathds{S}^{7}$ Janus solution of 11d SUGRADHoker:2009lky .

More pertitent to the cases that we will study below, the most well studied case among the values of $\gamma$ that admit asymptotically locally AdS${}_{7}\times\mathds{S}^{4}$ solutions is $\gamma=-1/2$, which holographically describes 1/2-BPS Wilson surface operators in the 6d $A_{M-1}$ $\mathcal{N}=(2,0)$ SCFT that preserve a large $\mathcal{N}=(4,4)$ 2d SUSY. These BPS Wilson surface operators have a long history in M-theory descriptions of 6d SCFTs Strominger:1995ac ; Ganor:1996nf ; Howe:1997ue ; Maldacena:1998im ; Berenstein:1998ij , and there has been a recent resurgence of interest in these defects where holographic Gentle:2015jma ; Estes:2018tnu ; Jensen:2018rxu and field theoretic Chalabi:2020iie ; Drukker:2020atp ; Drukker:2020dcz computations have characterized these defect CFTs through their EE and Weyl anomalies.

Recently, new solutions in 11d SUGRA have been constructed that are proposed to be holographically dual to 2d BPS surface defects in 6d $\mathcal{N}=(1,0)$ SCFTs preserving “small” $\mathcal{N}=(4,4)$ or $\mathcal{N}=(0,4)$ 2d SUSY Faedo:2020nol . In the sections below, we will clearly demonstrate that these new solutions also fit into the one-parameter family of solutions in Bachas:2013vza in the limit where $\gamma\to-\infty$. It will turn out that the solutions in Faedo:2020nol are, in fact, a particular case within a broader class of solutions that can be realized in the $\gamma\to-\infty$ limit, which we will characterize by computing their contributions to the EE of a spherical region.

A crucial point that we emphasize in our construction of the $\gamma\to-\infty$ solutions is that the superisometry algebra of the near-horizon limit of a stack of M5-branes, $\mathfrak{osp}(8^{*}|4)$, does not contain $\mathfrak{d}\left(2,1;\gamma\right)\oplus\mathfrak{d}\left(2,1;\gamma\right)$ as a sub-superalgebra frappat1996dictionary ; D_Hoker_2008 ; D_Hoker_2008b . In the supergravity solution, this is manifested by the appearance of additional terms in the four-form flux as compared to that due to pure M5-branes. Nevertheless, the geometry is still locally asymptotically AdS${}_{7}\times\mathds{S}^{4}$, which is in line with the fact that the bosonic subalgebra of $\mathfrak{d}\left(2,1;\gamma\right)\oplus\mathfrak{d}\left(2,1;\gamma\right)$ is a subalgebra of $\mathfrak{osp}(8^{*}|4)$.

One upshot of our analysis that follows from the identification of the global symmetries in the $\gamma\to-\infty$ limit is that it allows for a suitable regulation scheme, which we will employ when we compute the defect sphere EE. Lacking a generalized program of holographic renormalization for SUGRA solutions dual to defects on the field theory side, we will use a background subtraction scheme in order to remove the ambient degrees of freedom and isolate contributions to physical quantities coming from the defect. The key step in this background subtraction scheme is the correct identification of the vacuum solution, and as will be made clear, the ambient theory with a “trivial defect” is a deformed 6d $\mathcal{N}=(2,0)$ SCFT preserving the bosonic superconformal subalgebra ${\mathfrak{so}}(2,2)\oplus{\mathfrak{so}}(4)\oplus{\mathfrak{so}}(5)\subset{\mathfrak{osp}}(8*|4)$. Since the solutions in Faedo:2020nol belong to the class of solutions that we study below, in finding a vacuum solution that manifests the corresponding isometries and carefully carrying out background subtraction, we will also resolve a puzzle in Faedo:2020nol , where physical quantities like the “defect central charge” were divergent.

Ultimately, we will see that that universal part of the defect sphere EE, $S_{\rm EE}^{\tiny\rm(univ)}$, i.e. the coefficient of its log-divergent part, is determined in terms of the highest weight vector, $\varpi$, of an $A_{M-1}$ irreducible representation determined by a Young diagram that encodes the partition of M5-branes that specifies the defect. Explicitly, we will show that

where $(\cdot,\cdot)$ is the scalar product on the weight space. This result is similar to the Wilson surface sphere EE Estes:2018tnu in that both are expressible in terms of scalar quantities derived from representation data but differ in that eq. (1) is negative definite³³3Unlike in an ordinary 2d CFT, eq. (1) being strictly negative does not necessarily signal that the theory may be non-unitary. Indeed, the 2d defect sphere EE is expressible as a signed linear combination of defect Weyl anomaly coefficients Jensen:2018rxu , which is not bounded from below. and is completely determined by the highest weight vector.

In section 2, we begin by reviewing the 11d supergravity solutions dual to 2d small $\mathcal{N}=(4,4)$ defects in 6d $\mathcal{N}=(1,0)$ SCFTs found in Faedo:2020nol . In section 3 we briefly review the $\mathfrak{d}(2,1;\gamma)\oplus\mathfrak{d}(2,1;\gamma)$-invariant solutions to 11d supergravity found in Bachas:2013vza , and we show that by orbifolding the solutions in the $\gamma\to-\infty$ limit, we can recover the solutions in Faedo:2020nol . We then use the $\gamma\to-\infty$ limit to construct new 2d small $\mathcal{N}=(4,4)$ defects with finite Ricci scalar in section 4. Further in section 4, we demonstrate that the naïve AdS${}_{7}\times\mathds{S}^{4}$ vacuum is inappropriate to use in a background subtraction scheme for regulating holographic computations in the $\gamma\to-\infty$ limit, and we identify the correct background to use in this scheme. In section 5, we utilize the new $\gamma\to-\infty$ supergravity solutions and correct regulating scheme in a computation of the contribution of a flat 2d small $\mathcal{N}=(4,4)$ superconformal defect to the EE of a spherical region in a 6d SCFT. We then summarize our findings and discuss remaining issues and open questions surrounding these new defect solutions in section 6.

In addition, there are two appendices that detail technical aspects of the computations in the main text. First, in appendix A, we analyze the asymptotic expansion of the supergravity data that specify the new solutions in the $\gamma\to-\infty$ limit and construct the map to Fefferman-Graham (FG) gauge. Lastly, in appendix B, we carefully treat the integral in the area functional of the Ryu-Takayanagi (RT) surface in the computation of the holographic EE of the defect in the dual field theory.

We begin with a brief summary of the results of Faedo:2020nol . The particular 11d supergravity metric constructed therein is the uplift of a 7d charged AdS${}_{3}\times\mathds{S}^{3}$ domain wall initially found in Dibitetto:2017tve , and is given by

for some parameter $Q_{\text{M5}}$ and a function $H_{\text{M5}^{\prime}}$ defined over a 4d space parametrized by the coordinates $\{z,\rho,\varphi^{1},\varphi^{2}\}$. The solution above captures the near-horizon geometry of the brane intersection depicted in table 1. Namely, a “bound state” (in the sense discussed in footnote 4) of M2- and M5-branes, with charges $Q_{\text{M2}}$ and $Q_{\text{M5}}$, intersects an orthogonal stack of M5^′-branes, thus forming a 1/4-BPS brane setup. In 2d notation, this corresponds to $\mathcal{N}=(4,4)$ supersymmetry, with the large R-symmetry realized geometrically as the isometry of the two 3-spheres $\mathds{S}^{3}$ and $\tilde{\mathds{S}}^{3}$ with coordinates $\{\chi,\theta^{1},\theta^{2}\}$ and $\{\phi,\varphi^{1},\varphi^{2}\}$, respectively. In addition, the two stacks of 5-branes can be made to probe ALE singularities by introducing two Kaluza-Klein (KK) monopoles, with charges $k$ and $k^{\prime}$ and Taub-NUT directions $\partial_{\chi}$ and $\partial_{\phi}$, respectively. The inclusion of any one of the two KK monopoles results in a further breaking of the preserved supersymmetries and a degeneration to an 1/8-BPS setup, which in 2d language corresponds to (large) $\mathcal{N}=(0,4)$ supersymmetry. The presence of the second KK monopole does not incur a further loss of supersymmetry, so the final brane configuration is always at least a $1/8$-BPS supergravity solution.

Furthermore, the defect M2- and M5-branes are fully localized within a 2d submanifold of the worldvolume of the M5^′-branes. This is to be expected from the holographic realization of a surface defect in a 6d SCFT; this interpretation was first attached to the 7d domain wall in Dibitetto:2017klx and to the full 11d SUGRA background in Faedo:2020nol . On the other hand, the defect branes are smeared in the directions transverse to the worldvolume of the M5^′-branes⁴⁴4It is this property – that the M2- and M5-branes do not share transverse directions with the M5^′-branes, other than those along which the former are smeared – which we are implicitly referring to when we describe the M2- and M5-branes as forming a “bound state”. This aligns with the terminology used in Faedo:2020nol , and should not to be confused with the dyonic supermembrane Izquierdo:1995ms , which is a rather different multimembrane solution of 11d supergravity. Indeed, in the setup described by table 1, there is no M2-brane charge dissolved within the M5-brane worldvolume, nor do the M2-branes polarize into a fuzzy 3-sphere via the Myers effect., so that their charge is localized within $\mathds{S}^{3}/\mathbb{Z}_{k}$, but not $\tilde{\mathds{S}}^{3}/\mathbb{Z}_{k^{\prime}}$. Therefore, while the metric in eq. (2) manifests the isometry groups of both (orbifolded) 3-spheres, the R-symmetry is partially broken, giving rise to small $\mathcal{N}=(0,4)$ supersymmetry. For $k^{\prime}=1$, the solution above fits into the classification of $\mathcal{N}=(0,4)$ AdS${}_{3}\times\mathds{S}^{3}/\mathbb{Z}_{k}\times\operatorname{CY}_{2}$ backgrounds foliated over an interval performed in Lozano:2020bxo , where $\operatorname{CY}_{2}=\mathbb{R}^{4}$ contains the (round) $\tilde{\mathds{S}}^{3}$. In particular, the solution above corresponds to taking the M5^′-branes to be completely localized in their transverse space.

On shell, the defect brane charges are constrained to be equal, $Q_{\text{M2}}=Q_{\text{M5}}$, while the function $H_{\text{M5}^{\prime}}$ satisfies Faedo:2020nol

where we rescaled $\hat{\rho}=\rho^{2}/(4k^{\prime})$, and denoted by $\mathbb{R}^{3}_{\hat{\rho}}$ the three-dimensional subspace which is transverse to the M5-branes, parallel to the M5^′-branes, and along which the M2-branes are smeared. Following the parametrization adopted in table 1, $\mathbb{R}^{3}_{\hat{\rho}}$ is then the space spanned by $\{\partial_{\hat{\rho}},\partial_{\varphi^{1}},\partial_{\varphi^{2}}\}$, with $\hat{\rho}$ being the radial coordinate and $\{\varphi^{1},\varphi^{2}\}$ parametrizing a 2-sphere.

In terms of the brane setup described above, then, the spacetime in eq. (2) is reached by approaching the brane intersection locus from within the worldvolume of the M5^′-branes in a radial fashion, i.e. by taking $r\to 0$. In this limit, the $ISO(1,1)$ isometry group gets promoted to $SO(2,2)$, and the M5^′-brane worldvolume becomes AdS${}_{3}\times\mathds{S}^{3}/\mathbb{Z}_{k}$.

A particular solution to eq. (3) is, for any $\alpha\in\mathbb{R}$,

where $P_{\pm}=\sqrt{z^{2}+(\rho\pm\alpha)^{2}}$. By redefining

and setting $\alpha=(2^{7/4}g^{3/2}kQ_{\text{M5}})^{-1}$, the particular solution in eq. (4) can be matched to the one found in eq. (2.17) of Faedo:2020nol , which we now reproduce for clarity⁵⁵5To see this, it is convenient to first rationalize the product $P_{+}P_{-}$ as $P_{+}P_{-}=\alpha^{2}\big{(}1+\mu^{5}-(1-\mu^{5})\cos(2\xi)\big{)}/2(1-\mu^{5})$.:

Furthermore, in Faedo:2020nol it was argued that, as $\mu\to 1$ (which corresponds to a non-linear limit in the original coordinates $\hat{\rho}$ and $z$), the near-horizon geometry in eq. (2) locally recovers the AdS${}_{7}/\mathbb{Z}_{k}\times\mathds{S}^{4}/\mathbb{Z}_{k^{\prime}}$ vacuum of M-theory. This was shown by realizing the 11d line element as the uplift of the domain wall solution to $\mathcal{N}=1$, $d=7$ supergravity (whose gauge coupling constant $g$ appears in eq. (6) above) found in Dibitetto:2017tve , which interpolates between AdS₇ (as $\mu\to 1$) and an infrared singularity (at $\mu=0$).

While the singular nature of the solution is not immediately obvious, it can be made manifest by studying the Ricci scalar, $\mathcal{R}$, in the $z\to 0$ limit. For metrics generally of the form of eq. (2), and following D_Hoker_2008b , we can write the expression for the Ricci scalar for an arbitrary harmonic function $H_{\text{M5}^{\prime}}$ as

The function $H_{\rm M5^{\prime}}$ has a branch point located at $z=0$ and $\rho=\alpha$ and correspondingly admits two different expansions as $z\rightarrow 0$, depending on whether $\rho$ is larger or smaller than $\alpha$. The choice of sign for the branch cut can be determined by the requirement $P_{+}P_{-}\geq 0$. For $\rho>\alpha$, this leads to

which has a pole as $\rho\rightarrow\alpha$. This corresponds to setting $\xi=0$, with the pole appearing as $\mu\rightarrow 0$. For $\rho<\alpha$, we find

which corresponds to taking $\mu\rightarrow 0$ with $\xi\neq 0$. Thus for all values of $\xi$, we find that the Ricci scalar diverges as $\mu\rightarrow 0$.

To begin, we will briefly review the classification due to D_Hoker_2008b ; Bachas:2013vza of the $d=11$ supergravity solutions with superisometry given by two copies of the exceptional Lie superalgebra ${\mathfrak{d}}(2,1;\gamma)$. This classification represents the foundation for the new defect solutions we present below.

In particular, we will be interested in the real form $\mathfrak{d}(2,1;\gamma;0)$ which arises as a real subsuperalgebra of the fixed points of an involutive semimorphism of $\mathfrak{d}(2,1;\gamma)$ frappat1996dictionary . Recall that the 9-dimensional maximal bosonic subalgebra of the real form ${\mathfrak{d}}(2,1;\gamma;0)$ is ${\mathfrak{so}}(2,1)\oplus{\mathfrak{so}}(3)\oplus{\mathfrak{so}}(3)$. Labelling each factor in this subalgebra by an index $a\in\{1,2,3\}$, the generators $T^{(a)}$ satisfy

where $\varepsilon_{ijk}$ and $\eta^{kl}_{a}$ are the totally anti-symmetric tensor (with $\varepsilon_{123}=1$) and the canonical metric induced by the Killing form, respectively. In addition to the bosonic sector, ${\mathfrak{d}}(2,1;\gamma)$ contains an 8-dimensional fermionic generator with components $F_{A_{1}A_{2}A_{3}}$, where the indices $A_{a}\in\{\pm\}$ transform in the spinorial representation $\boldsymbol{2}$ of the $a^{\rm th}$ factor in the even subalgebra. Furthermore, the components $F_{A_{1}A_{2}A_{3}}$ obey the following anti-commutation relation

where $C\equiv i\sigma^{2}$ is the charge conjugation matrix, $\{\sigma^{i}\}_{i=1}^{3}$ are the Pauli matrices, and $\{\beta_{i}\}_{i=1}^{3}$ are real parameters satisfying $\sum_{a=1}^{3}\beta_{a}=0$, which follows from the (generalized) Jacobi identity. This last constraint, together with the possibility of absorbing any rescaling $\{\lambda\beta_{1},\lambda\beta_{2},\lambda\beta_{3}\}$ for $\lambda\in\mathbb{C}$ into a redefinition of the normalization of the fermionic generator, implies that ${\mathfrak{d}}(2,1;\gamma)$ is entirely specified by the choice of a ratio of any two of the three $\beta$ parameters; here, we take $\gamma\equiv\beta_{2}/\beta_{3}$. Note that ${\mathfrak{d}}(2,1;\gamma)$ is the only (finite-dimensional) Lie superalgebra admitting a continuous parametrization.

Amongst the possible values that $\gamma$ can take, there are clearly three special values corresponding to the vanishing of any one of the $\beta$ parameters. Specifically, choosing $\beta_{1}=0$ fixes $\gamma=-1$. The more interesting case, and the one relevant to our analysis, is $\beta_{3}=0$, which corresponds to $\gamma\to\pm\infty$. The case $\beta_{2}=0$ corresponds to $\gamma=0$ and is equivalent under a group involution as discussed at the end of this section. In the limit $\beta_{3}=0$, the anticommutator in eq. (11) degenerates to

Consequently, the $\mathfrak{so}(4)_{R}\cong\mathfrak{so}(3)\oplus\mathfrak{so}(3)$ R-symmetry of the large superalgebra is broken into the single $\mathfrak{so}(3)_{R}$ factor which constitutes the R-symmetry of a small superalgebra. Note that the other $\mathfrak{so}(3)$ factor remains a bosonic symmetry of the supergravity solution; however, it is now realized as a flavor symmetry, rather than as an outer automorphism of the supersymmetry algebra.

In addition, there are isolated points of interest in the $\gamma$-parameter space where the real form $\mathfrak{d}(2,1;\gamma;0)$ reduces to classical Lie superalgebras:

In particular, $\gamma=-1/2$ is the only value (up to algebra involutility, as we will discuss shortly) for which $\mathfrak{d}(2,1;\gamma;0)\oplus\mathfrak{d}(2,1;\gamma;0)$ admits a canonical inclusion into the superisometry algebra $\mathfrak{osp}(8^{*}|4)$ of AdS${}_{7}\times\mathds{S}^{4}$. This case was studied extensively in Estes:2018tnu ; it is the holographic realization of Wilson surfaces, 1/2-BPS codimension-4 superconformal solitons, within the 6d $\mathcal{N}=(2,0)$ SCFT. In this case, the ambient 6d SCFT is undeformed, in the sense that the supergroup of symmetries preserved by the defect is a subgroup of the 6d superconformal symmetry group. This is a common feature throughout the study of defects embedded in higher-dimensional theories.

For generic values of $\gamma$, including the limit $\gamma\to\pm\infty$, the superisometry algebra $\mathfrak{d}(2,1;\gamma;0)\oplus\mathfrak{d}(2,1;\gamma;0)$ is not a subalgebra of the $\mathfrak{osp}(8^{*}|4)$ superconformal Lie superalgebra Estes:2018tnu . As a result, we expect the 6d ambient theory to be deformed as we tune $\gamma$ away from the special value $\gamma=-1/2$. In the supergravity solutions we construct below, this deformation will appear at leading order in the asymptotic expansion of the four-form flux. Additionally, the warp factors of the symmetric spaces corresponding to $T^{(1)}$ and $T^{(2)}$ have the correct normalization to create an AdS₇ space only when $|\beta_{1}|=|\beta_{2}|$, which can happen only when $\gamma=-1/2$ or $\gamma\to\pm\infty$.

Finally, we note that the complex Lie superalgebra $\mathfrak{d}(2,1;\gamma)$ enjoys a triality symmetry generated by $\gamma\mapsto\gamma^{-1}$, $\gamma\mapsto-(\gamma+1)$, and $\gamma\mapsto-\gamma/(\gamma+1)$, any two of which are linearly independent. However, only the $\gamma\mapsto\gamma^{-1}$ involutility survives at the level of the real form $\mathfrak{d}(2,1;\gamma;0)$, due to the distinguished nature of the $T^{(1)}$ generator. Therefore, our analysis below can be equivalently carried out in the $\gamma\to\pm\infty$ limits, or even in the $\gamma\to 0$ limit (albeit with a permutation of the two ${\mathfrak{so}}(3)\oplus{\mathfrak{so}}(3)$ factors contained in two copies of ${\mathfrak{d}}(2,1;\gamma;0)$)⁶⁶6In spite of the involution relating the $\gamma\to-\infty$ and $\gamma\to 0$ limits, these two descriptions cannot be smoothly deformed into one another by varying $\gamma$. Indeed, the two regimes are separated by the decompactification point $\gamma=-1$.. The physical interpretation of the choice of either limit corresponds to fixing the orientation of the M5-branes that engineer the ambient 6d SCFT.