Holographic Duals of Argyres-Douglas Theories

We present new $AdS_5$ solutions in 11d supergravity and identify them as the gravity duals of a class of 4d $\mathcal N = 2$ SCFTs of Argyres-Douglas type, engineered by a stack of M5-branes wrapped on a sphere with one irregular puncture and one regular puncture. The gravity solutions feature an internal M5-brane source, which is dual to the irregular puncture. Thanks to a novel St\"uckelberg mechanism involving an axion originating from the expansion of the M-theory 3-form, one of the isometry generators on the gravity side is not mapped to a continuous symmetry on the SCFT side.


INTRODUCTION
Strong-coupling phenomena in quantum field theory (QFT) are of crucial importance, both conceptually and phenomenologically, but their study poses considerable theoretical challenges. In the endeavor of exploring the vast and largely uncharted landscape of strongly coupled phases in QFT, valuable lessons can be learned from theories with a higher degree of symmetry. Superconformal field theories (SCFTs) of Argyres-Douglas (AD) type in four dimensions constitute a prominent example. These theories are intrinsically strongly-coupled and describe interactions among mutually non-local massless dyons [1]. Their spectrum contains relevant Coulomb branch operators of fractional dimension. Establishing the existence and surprising properties of these QFTs has been complicated by their lack of an N = 2 weakcoupling Lagrangian description, and hence, exploring less conventional windows into their physics is especially valuable.
A vast class of SCFTs of AD type is expected to admit holographic duals, but their identification has remained an open problem for years. In this letter, we present a new class of fully explicit AdS 5 solutions in 11d supergravity and we propose them as holographic duals to SCFTs of AD type. Our results give the opportunity to analyze these QFTs from a new angle, providing novel insights on their properties. Furthermore, a subclass of SCFTs of AD type can be realized as N = 2 supersymmetric IR fixed points of renormalization group (RG) flows preserving N = 1 supersymmetry [2,3]. Our solutions pave the way to the exciting possibility of studying the gravity dual of supersymmetry enhancing RG flows, which could shed new light on holography in general.
A crucial feature of our AdS 5 solutions is the presence of suitable singularities, which we interpret as the low-energy approximation to well-defined brane sources in M-theory. Localized sources in the internal space constitute an important ingredient in the holographic dictionary that allows for arbitrary flavor symmetries (see e.g. [4][5][6][7][8][9][10][11]). This letter describes novel controlled examples allowing a better understanding of these sources, pivotal for enlarging the scope of the AdS/CFT correspondence.

SUPERGRAVITY SOLUTIONS
Our AdS 5 solutions in 11d supergravity preserve 4d N = 2 superconformal symmetry. They were obtained in 7d gauged supergravity and uplifted on S 4 , as will be reported in [12]. The 7d solutions are a warped product of AdS 5 and a 2d space Σ, consisting of a circle fibered over an interval. Σ is supported by a U (1) gauge flux, does not have a constant curvature metric, and admits a non-constant Killing spinor. Thus, as in [13][14][15], supersymmetry is not realized in the standard topological twist paradigm.
The metric of the uplifted 11d solution is where m is a mass scale, ds 2 AdS5 is the metric on the unitradius AdS 5 , and ds 2 S 2 is the metric on the unit-radius S 2 . The functions h(w), H(w, µ) are defined as where 0 < B < 1 is a constant parameter. The coordinates µ, w have ranges 0 ≤ µ ≤ 1 and 0 ≤ w ≤ w 1 , with w 2 . The angular coordinates φ, z have period 2π, and C is a constant. The 1-form Dφ and the warp factor are given by < l a t e x i t s h a 1 _ b a s e 6 4 = " X v O V 5 a N M k 9 / h 3 j t 2 V H Q 8 4 R d b t 3 4 = " > A A A B 6 3 i c b Z C 7 T g J B F I b P 4 g 3 x h l r a T A Q T C 0 N 2 S Y y U J D a W G O W S w E p m h w N M m J l d Z 2 Z J y I a n s D A x F l r 4 M D 6 C b + O C N I B / 9 e X 8 5 5 L / B J H g x r r u j 5 P Z 2 N z a 3 s n u 5 v b 2 D w 6 P 8 s c n D R P G m m G d h S L U r Y A a F F x h 3 X I r s B V p p D I Q 2 A x G t z O / O U Z t e K g e 7 S R C X 9 K B 4 n 3 O q E 1 L f v H h q V w k Z q i 5 G p l u v u C W 3 L n I O n g L K M B C t W 7 + u 9 M L W S x R W S a o M W 3 P j a y f U G 0 5 E z j N d W K D E W U j O s C E S m M m M p i S C 0 n t 0 K x 6 s + J / X j u 2 / Y q f c B X F F h V L W 1 K v H w t i Q z L L Q 3 p c I 7 N i k g J l m q e X C R t S T Z l N U y 9 v M q i o R H N F e m M e m T n 7 y f y F 0 1 y a 3 V t N u g 6 N c s m 7 L r n 3 5 U K 1 s v h C F s 7 g H C 7 B g x u o w h 3 U o A 4 M n u E V P u D T k c 6 L 8 + a 8 / 7 V m n M X M K S z J + f o F m S 6 O H g = = < / l a t e x i t > S 2 shrinks < l a t e x i t s h a 1 _ b a s e 6 4 = " b r l 2 G u A 9 0 5 H F 4 5 H 0 6 W u t d 4 j b J o 4 = " > A A A B / X i c b Z D N S s N A F I U n / t b 6 F 3 U p Q r A I L r Q k h W K X B T d u h A r 2 B 5 p Q J p P b d u j M J M x M C i U U F z 6 L C 0 F c 6 M J n 8 B F 8 G 6 c 1 m 7 a e 1 c c 9 d + 5 w T p g w q r T r / l h r 6 x u b W 9 u F n e L u 3 v 7 B o X 1 0 3 F J x K g k 0 S c x i 2 Q m x A k Y F N D X V D D q J B M x D B u 1 w d D v z 2 2 O Q i s b i U U 8 S C D g e C N q n B G s z 6 t l n v o i p i E B o R 3 H A E i L f L 9 5 X r 0 O J B a i e X X L L 7 l z O K n g 5 l F C u R s / + 9 q O Y p N z c I w w r 1 f X c R A c Z l p o S B t O i n y p I M B n h A W S Y K z X h 4 d S 5 4 F g P 1 b I 3 G / 7 n d V P d r w U Z F U m q Q R C z Y r x + y h w d O 7 O E T k Q l E M 0 m B j C R 1 P z s k C G W m G j T w + I l B Q J z U F d O N K a J m n O Q z U u d F k 1 2 b z n p K r Q q Z a 9 a d h 8 q p X o t b 6 G A T t E 5 u k Q e u k F 1 d I c a q I k I e k a v 6 A N 9 W k / W i / V m v f + t r l n 5 m x O 0 I O v r F 0 9 M l U o = < / l a t e x i t > smeared M5-branes < l a t e x i t s h a 1 _ b a s e 6 4 = " + 3 h 0 p U m 5 4 t o c h 2 0 P X h l C K F v I 1 C Q = " > A A A C B 3 i c b V D L T s J A F J 3 i C / F V d e l m I h h d G G y J R p Y k b l y i k U e k l U y H C 0 y Y d p q Z K Q k h f I A L v 8 W F i X G h C 5 d + g n 9 j i 3 U B e F b n 3 n M f O c c L O V P a s r 6 N z N L y y u p a d j 2 3 s b m 1 v W P u 7 t W V i C S F G h V c y K Z H F H A W Q E 0 z z a E Z S i C + x 6 H h D a 4 S v T E E q Z g I 7 v Q o B N c n v Y B 1 G S U 6 b r X N 4 4 L j E 9 3 3 P H z 7 c I 7 P 8 F 9 1 3 3 a A 8 w I W 0 m N d w T t t M 2 8 V r S n w I r F T k k c p q m 3 z y + k I G v k Q a M q J U i 3 b C r U 7 J l I z y m G S c y I F I a E D n P x f + 4 8 V S a G N Z P 6 S 0 s b m 1 v V P e r e z t H x w e V Y 9 P 2 j p K F M c W j 2 S k u h 7 T K E W I L S O M x G 6 s k A W e x I 4 3 e V j 4 n S k q L a L w 2 c x i d A M 2 C o U v O D N Z q d c P m B l r n z q D m 0 G 1 Z t W t X H Q d 7 A J q U M g Z V L / 7 w 4 g n A Y a G S 6 Z 1 z 7 Z i 4 6 Z M G c E l z i v 9 R G P M + I S N M G W B 1 r P A m 9 O L / O C q t y j < l a t e x i t s h a 1 _ b a s e 6 4 = " j v J g e h C i e g F Y f G b 6 T H N j F W n J 5 0 8 = " > A A A B 6 X i c b Z D L S s N A F I Z P 6 q 3 W W 9 W l m 8 E i u J C S S M U u C 2 5 c V r A X a E O Z T E / a o Z N J m J k U S u h D u B D E h S 5 8 G h / B t z G J 2 b T 1 X 3 2 c / 1 z 4 j x c J r o 1 t / 1 i l r e 2 d 3 b 3 y f u X g 8 O j 4 p H p 6 1 t V h r B h 2 W C h C 1 f e o R s E l d g w 3 A v u R Q h p 4 A n v e 7 C H z e 3 N U m o f y 2 S w i d A M 6 k d z n j J q 0 N B g G 1 E y 1 T 9 q j x q h a s + t 2 L r I J T g E 1 K N Q e V b + H 4 5 D F A U r D B N V 6 4 N i R c R O q D G c C l 5 V h r D G i b E Y n m N B A 6 0 X g L c l V f n D d y 4 r / e Y P Y + E 0 3 4 T K K D U q W t q S e H w t i Q p K l I W O u k B m x S I E y x d P L h E 2 p o s y k m V c 3 a Z Q 0 Q H 1 D x n M e 6 Z z d J H / g s p J m d 9 a T b k L 3 t u 7 c 1 e 2 n R q 3 V L L 5 Q h g u 4 h G t w 4 B 5 a 8 A h t 6 A C D E F 7 h A z 6 t m f V i v V n v f 6 0 l q 5 g 5 h x V Z X 7 + x C o 2 h < / l a t e x i t > P 4 < l a t e x i t s h a 1 _ b a s e 6 4 = " A / t o S W I q F n 5 Y t y c j 8 k X I V v 2 v h 7 U = " > A A A B 6 X i c b Z D L T s J A F I Z P 8 Y Z 4 Q 1 2 6 m U h M X B j S E o w s S d y 4 h E Q u C T R k O p z C h O m 0 m Z m S k I a H c G F i X O j C p / E R f B t L 7 Q b w X 3 0 5 / 7 n k P 1 4 k u D a 2 / W M V d n b 3 9 g + K h 6 W j 4 5 P T s / L 5 R V e H s W L Y Y a E I V d + j G g W X 2 D H c C O x H C m n g C e x 5 s 8 e V 3 5 u j 0 j y U z 2 Y R o R v Q i e Q + Z 9 S k p c E w o G a q f d I e 1 U b l i l 2 1 M 5 F t c H K o Q K 7 W q P w 9 H I c s D l A a J q j W A 8 e O j J t Q Z T g T u C w N Y 4 0 R Z T M 6 w Y Q G W i 8 C b 0 l u s o O b 3 q r 4 n z e I j d 9 w E y 6 j 2 K B k a U v q + b E g J i S r N G T M F T I j F i l Q p n h 6 m b A p V Z S Z N P P 6 J o 2 S B q j v y H j O I 5 2 x m 2 Q P X J b S 7 M 5 m 0 m 3 o 1 q r O f d V u 1 y v N R v 6 F I l z B N d y C A w / Q h C d o Q Q c Y h P A K H / B p z a w X 6 8 1 6 / 2 s t W P n M J a z J + v o F r 5 G N o A = = < / l a t e x i t > Q 2 < l a t e x i t s h a 1 _ b a s e 6 4 = " Y G 0 i h R E b i / R b M q Q H 6 y e E j 1 g g J o U = " > A A A B 4 X i c b Z D L S s N A F I b P 1 F u t t 6 h L N 8 E i u J C S i G K X B T c u K 9 o L t K F M p q f t 0 J k k z K V Q Q h / A h S A u d O E T + Q i + j W n N p q 3 / 6 u P 8 5 8 J / w k R w b T z v h x Q 2 N r e 2 d 4 q 7 p b 3 9 g 8 M j 5 / i k q W O r G D Z Y L G L V D q l G w S N s G G 4 E t h O F V I Y C W + H 4 f u 6 3 J q g 0 j 6 N n M 0 0 w k H Q Y 8 Q F n 1 G S l p 6 6 0 P a f s V b y F 3 H X w c y h D r n r P + e 7 2 Y 2 Y l R o Y J q n X H 9 x I T p F Q Z z g T O S l 2 r M a F s T I e Y U q n 1 V I Y z 9 0 J S M 9 K r 3 r z 4 n 9 e x Z l A N U h 4 l 1 m D E s p b M G 1 j h m t i d p 3 D 7 X C E z Y p o B Z Y p n l 1 0 2 o o o y k 2 V d 3 q Q x o h L 1 l d u f 8 E Q v O E g X j 5 u V s u z + a t J 1 a F 5 X / N u K 9 3 h T r l X z L x T h D M 7 h E n y 4 g x o 8 Q B 0 a w G A I r / A B n 4 S R F / J G 3 v 9 a C y S f O Y U l k a 9 f j K G K r Q = = < / l a t e x i t > µ < l a t e The G 4 flux supporting the solution reads

x i t s h a 1 _ b a s e 6 4 = " 7 o D A q y L 4 s H B U h T 2 5 c w 5 Z G P U 8 y Q Y = " > A A A B 3 3 i c b Z C 7 S g N B F I b P x F u M t 6 i l z W I Q L C T s i m L K g I 1 l A u Y C y R L O T k 6 S I b M X Z m Y D Y U l t I Y i F F j 6 S j + D b u B u 3 S e J f f Z z / X P i P F 0 m h j W 3 / s M L W 9 s 7 u X n G / d H B 4 d H x S P j 1 r 6 z B W n F o 8 l K H q e q h J i o B a R h h J 3 U g R + p 6 k j j d 9 z P z O j J Q W Y f B s 5 h G 5 P o 4 D M R I c T V p q O o N y x a 7 a S 1 m b 4 O R Q g V y N Q f m 7 P w x 5 7 F N g u E S t e 4 4 d G T d B Z Q S X t C j 1 Y 0 0 R 8 i m O K U F f 6 7 n v L a w r H 8 1 E r 3 t Z 8 T + v F 5 t R z U 1 E E M W G A p 6 2 p N 4 o l p Y J r S y D N R S K u J H z F J A r k V 6 2 + A Q V c p M m X d 2 k K U C f 9 I 0 1 n I l I L 9 l N l m 9 b l N L s z n r S T W j f V p 3 7 q t 2 8 q 9 R r + R e K c A G X c A 0 O P E A d n q A B L e B A 8 A o f 8 M m Q v b A 3 9 v 7 X W m D 5 z D m s i H 3 9 A r N n i Y w = < / l a t e x i t >
where vol S 2 is the volume form of the S 2 . The space M 6 is an S 1 z × S 1 φ × S 2 fibration over the rectangle [0, w 1 ] × [0, 1] in the (w, µ) plane, see Figure 1. The directions w, S 1 z in (2) are identified with Σ in the 7d solution, while µ, S 1 φ , S 2 span the S 4 used in the uplift.

Regularity and Flux Quantization
As we approach a point in the interior of the P 1 P 2 segment in the (w, µ) plane (see Figure 1), the S 2 shrinks smoothly. The Killing vector ∂ φ shrinks smoothly in the interior of P 3 P 4 . The linear combination ∂ φ + ∂ z shrinks smoothly along P 2 P 3 , where is given as The quantization of stems from analyzing the local geometry of the 4d space spanned by w, µ, φ, z near P 3 , and requiring it to be locally an orbifold R 4 /Z . The internal space M 6 admits non-trivial 4-cycles which lead to flux quantization conditions for G 4 . The 4-cycle C 4 is obtained by combining the segment Q 1 Q 2 , S 1 φ , and S 2 . C 4 has the topology of a 4-sphere because the S 2 shrinks at Q 1 and the S 1 φ shrinks at Q 2 . The flux of G 4 through C 4 with suitable orientation defines where p is the 11d Planck length. Next, we define the 4-cycle B 4 by combining S 2 , the segment P 2 P 3 , and the linear combination of S 1 φ and S 1 z that does not shrink in the interior of P 2 P 3 . B 4 is topologically a 4-sphere, because the S 2 shrinks at P 2 and both S 1 φ and S 1 z shrink at the orbifold point P 3 . For = 1 we have B 4 ∼ = C 4 , but B 4 is an independent 4-cycle for > 1, with Finally, we construct the 4-cycle D 4 by combining P 3 P 4 with S 2 -which shrinks at P 4 -and the combination of S 1 φ and S 1 z that does not shrink in the interior of P 3 P 4 .
In the vicinity of P 1 P 4 , the geometry is singular and e 2λ vanishes. We interpret this in terms of a smeared M5-brane source, as inferred from G 4 near w = 0, This term yields a finite flux equal to N (7) when integrated along S 2 , µ, S 1 φ , signaling a source of the form Comparing the 11d metric near w = 0 with the standard M5-brane solution, we see that the M5-branes are extended along AdS 5 and the S 1 z , smeared along the µ, S 1 φ directions, and sitting at the origin w = 0 of the local R 3 space dw 2 + w 2 ds 2 S 2 .

Solutions in Canonical N = 2 Form
The general form of an AdS 5 M-theory solution preserving 4d N = 2 superconformal symmetry was determined in [16] by Lin, Lunin, and Maldacena (LLM). The 11d metric and flux are summarized in [17]. In LLM form, the internal space M 6 is an S 1 χ × S 2 fibration over a 3d space with local coordinates (x 1 , x 2 , y). The Killing vector ∂ χ is associated with the U (1) r R-symmetry of the dual SCFT, while the isometries of the S 2 are mapped to the SU (2) R R-symmetry. The solution is determined by a function D(x 1 , x 2 , y) satisfying the Toda equation Our solutions can be cast in canonical LLM form, with the S 2 in (2) identified with the S 2 in LLM. Defining x 1 + i x 2 = r e iβ , the map between χ, β and φ, z is With reference to the uplift from 7d, the isometry ∂ χ mixes the Σ and S 4 directions. This is in contrast to the solutions of [17,18], in which ∂ χ = ∂ φ .
The function D and the map between the LLM coordinates y, r and the coordinates µ, w, are This determines a class of exact solutions D to the Toda equation (11) which are separable in the variables µ, w. Crucially, in our setup D does not describe a constant curvature Riemann surface, in contrast to the 4d N = 2 Maldacena-Nuñez solutions [18].
Holographic Central Charge, Flavor Central Charge, and Probe M2-Branes The holographic central charge is extracted from the warped volume of the internal space [19], where vol M6 is the volume form of ds 2 M6 in (1). Expanding the M-theory 3-form C 3 onto the resolution cycles of the R 4 /Z orbifold singularity at P 3 , one obtains − 1 Abelian gauge fields. The gauge group enhances to SU ( ) by virtue of states from M2-branes wrapping the resolution cycles [17]. We compute the associated flavor central charge k SU ( ) using the 't Hooft anomaly inflow methods of [20], yielding M2-brane probes wrapping calibrated 2-cycles in the internal space are dual to BPS operators in the SCFT. The calibration condition was written in [19] for a generic solution preserving 4d N = 1 superconformal symmetry and can be adapted to the N = 2 solutions at hand. The conformal dimension ∆ of the operator dual to an M2-brane wrapping the calibrated 2-cycle C 2 is [19] where vol C2 is the volume form on C 2 induced from ds 2 M6 . We identify two supersymmetric M2-brane probes in our setup. Firstly, we can wrap an M2-brane on the S 2 on top of the orbifold point P 3 in the (w, µ) plane. We denote the corresponding operator as O 1 . Secondly, we can wrap an M2-brane on the 2d subspace consisting of the segment P 3 P 4 and the combination of The U (1) r × SU (2) R charges of O 1 , O 2 can be computed from the M2-brane coupling to the 11d 3-form C 3 [19], with R the Cartan generator of SU (2) R . Thus O 1 and O 2 have the R-charges of N = 2 Coulomb branch and Higgs branch operators, respectively.

A NOVEL STÜCKELBERG MECHANISM
The Killing vector ∂ β in (12) is a symmetry of the 11d metric and flux, but it does not correspond to a continuous flavor symmetry of the dual SCFT. This is due to a Stückelberg mechanism in the 5d low-energy effective action of M-theory on M 6 . The components of the 11d metric with one external leg and one leg along ∂ β yield a U (1) gauge field A β . When A β is turned on, the 1-form dβ must be replaced by the gauge invariant combination dβ + A β . This replacement affects the closure of G 4 , which is restored by adding suitable terms, including The improved G tot 4 is built with the closed but not exact 3-form ω 3 ∝ ι ∂ β G 4 , whose non-exactness hinges on the M5-brane source at w = 0. The 1-form Da 0 is the field strength of an external axion a 0 .
Closure of G 4 requires dDa 0 ∝ dA β , signaling a nontrivial Stückelberg coupling between A β and a 0 . As a result, A β is massive and is dual to a spontaneously broken U (1) symmetry in the SCFT. As discussed in detail in [21], this mechanism provides a non-trivial physical realization of a mathematical obstruction to promoting G 4 to an equivariant cohomology class [22]. In contrast, ι ∂χ G 4 is exact, and the U (1) gauge field A χ (originating from the components of the 11d metric with one external leg and one leg along ∂ χ ) does not participate in any Stückelberg coupling to a 0 and remains massless. This is expected since ∂ χ is dual to the U (1) r R-symmetry of the SCFT. Similar versions of the Stückelberg mechanism for isometries in flux backgrounds are known for flat internal spaces (see e.g. [23]). The internal geometry discussed in this letter is richer, and this Stückelberg mechanism is novel in the context of holographic M-theory solutions.

FIELD THEORY DUALS
We claim that the 11d supergravity solutions presented above are holographically dual to 4d N = 2 SCFTs that arise from the low energy limit of N M5-branes wrapping a sphere with an irregular puncture of type A (N ) N −1 [k], labeled by the integer k > −N . For = 1 the irregular puncture is the only puncture on the sphere, and the 4d field theories coincide with the Type I theories with b = N and J = A N −1 in the classification of [24,25] (also called I N,k in [26]). These are the AD theories of type (A N −1 , A k−1 ), obtained in Type IIB in [27] (generalizing the N = 2 cases obtained in [1,28,29]). For > 1 there is an additional regular puncture at the opposite pole of the sphere that is labeled by a box Young diagram with columns and N/ rows, contributing an SU ( ) non-Abelian flavor symmetry [30]. We label the resulting 4d theories by (A , which belong to the class labeled Type IV in [24,25]. For = N the regular puncture is of maximal type and these are the D N k+N (SU (N )) theories studied in [31][32][33]. The case = 1 is the "nonpuncture", equivalent to the (A N −1 , A k−1 ) theories.
The irregular puncture is identified with the M5-brane source in the gravity dual. Due to the irregular puncture, the U (1) r R-symmetry of the SCFT is given as the combination r = R φ + N k+N R z , where R φ is the generator of the R-symmetry that would be preserved in the absence of the irregular defect and R z is the generator of the global U (1) z isometry of the sphere [24,25]. Comparison with (12) gives the map between k in the SCFT and the flux quantum K, The central charges of the (A  Table I. They are computed in the literature [24,[32][33][34][35] using useful formulae from [36]. For > 1, an especially simple way to compute the central charges as a function of is to apply the results of [33,37] for the partial closure of a maximal puncture, initiated by a nilpotent VEV for the moment map operator of the maximal puncture's flavor symmetry. The third row of Table I gives the central charge in the limit N, k → ∞ with k/N finite. Using (20), we get a perfect match with the holographic central charge (14).
The dimensions of the Coulomb branch operators u i of the theory (A (N ) , Y ) are conveniently captured by a Newton polygon [24] and obey the bounds The upper bound is saturated by exactly one u i , which has the correct dimension and R-charges to be identified with the M2-brane operator O 1 in (17), (18) [38]. Using (20), the k SU ( ) central charge (15) reads For = N it matches the field theory computation of [32]. For generic , it matches the conjecture of [26] that the flavor central charge is equal to twice the maximal Coulomb branch operator dimension-see (21). For = 1, the rank of the global symmetry of the (A N −1 , A k−1 ) theories is GCD(k, N ) − 1 [33]. The maximal rank N − 1 on the SCFT side matches with the maximal rank that can be achieved via the M5-brane source on the gravity side. It would be interesting to establish a precise match with the SCFT formula for generic k, N . When = 1 and k/N is an integer, a Lagrangian description of the SCFT was obtained in [2,3] (see also [39,40] for the case N = 2). Using the dual Lagrangian, a set of 2 N −2 Higgs branch operators can be constructed, with dimension [3] At large N , this exactly matches with the dimension of the wrapped M2-brane operators O 2 in (17), (18). We expect that the field-theory degeneracy factor 2 N − 2 could be understood on the gravity side by studying the possible ways in which the M2-brane can end on the M5brane source. Heuristically, we can picture the M2-brane worldvolume, which has a disk topology, as the collapsed version of a multi-pronged configuration that can have a boundary component on each of the N M5-branes independently. Since the M2-brane must end on at least one of them, the degeneracy is 2 N − 1. Notice the mismatch by one between the degeneracy in field theory and in gravity. It would be interesting to sharpen this argument and to understand the origin of the additional decoupled mode, which we expect is associated to the center-of-mass mode of the M5-brane source stack.

DISCUSSION
We have proposed gravity duals for the 4d N = 2 SCFTs (A (N ) N −1 [k], Y ) of AD type, performing checks on the central charge, the SU ( ) flavor central charge, and the dimensions of suitable Coulomb branch and Higgs branch operators. Our AdS 5 solutions contain internal M5-brane sources. They admit an isometry algebra su(2) R ⊕ u(1) r ⊕ u(1) β . The su(2) R ⊕ u(1) r is dual to the SCFT R-symmetry, while u(1) β does not yield a continuous flavor symmetry thanks to a Stückelberg mechanism in which the u(1) β vector eats an axion originating from the expansion of the M-theory 3-form. There could be still a discrete symmetry remnant of u(1) β , which we plan to study elsewhere.
We expect our 11d solutions to admit generalizations corresponding to a regular puncture labeled by an arbitrary Young diagram. Constructing Lagrangian descriptions for these cases would yield further insights into SCFTs of AD type and allow for precision tests of the holographic duality.
It would be interesting to investigate whether the classification of irregular punctures in field theory can be recovered by a systematic study of exact solutions to the Toda equation of the class we discovered.
Our results set the stage for a broader study of holographic duals of AD theories. The supergravity constructions can be generalized to obtain N = 1 systems. More interestingly, our solutions can be used to study the holographic dual of the supersymmetry enhancing flows observed in the Lagrangian realizations of AD theories.