JHEP12(2017)050 Springer December 4, 2017 December 12, 2017 : September 12, 2017 : : Accepted Received Published Published for SISSA by https://doi.org/10.1007/JHEP12(2017)050 [email protected] , . 3 1708.07840 The Authors. Conformal and W Symmetry, M-Theory, Supersymmetric Gauge Theory, c

Recently, a physical derivation of the correspondence Alday-Gaiotto-Tachikawa , [email protected] Institute for Theoretical Physics, UniversityScience Park of 904, Amsterdam, 1098 XH Amsterdam, TheE-mail: Netherlands Open Access Article funded by SCOAP Supersymmetry and Duality ArXiv ePrint: description of these defects andpole find region that using the generalized generalizedclarifies conifolds. AGT several Furthermore, setup features we can of argue the be that derivation modeled the of in ordinary the a conifold originalKeywords: AGT correspondence. has been put forward. Ain crucial the role is 3d-3d played correspondence, bysurface. the whose We complex explore boundary Chern-Simons several modes theory features arising ofextended lead this to to derivation and a Toda subsequently generalization argue theory of thattwo on it defects the can in a AGT be six correspondence. Riemann dimensions The that latter wrap involves the codimension Riemann surface. We use a purely geometrical Abstract: Sam van Leuven and Gerben Oling Generalized Toda theory from sixconifold dimensions and the JHEP12(2017)050 ], ]. 5 21 22 5 8 – = 4 4 f N partition 4 S ]. In this construction, 4 12 = 2 Yang-Mills theories [ theories provided great insight N S 9 12 14 = 2 SU(2) Yang-Mills with 18 N 6 17 ] can be expressed as correlation functions 4 [ ]. The correspondence states that 1 − 2 – 1 – 4 , N 1 A 22 with C´ordova-Jafferis partition function of 1 4 , 1 S K and Drinfeld-Sokolov reduction λ m theories of type N S and 1 25 k and AGT ] of class ,m theory from generalized conifolds 3 1 λ Toda theory. In particular, the conformal blocks of the Toda theory were shown K 1 − N The correspondence can arguably be viewed as the culmination of a long effort towards 4.3 General 3.2 Intersecting D6s from the generalized conifold 4.1 Compatibility of 4.2 2.1 Principal Toda2.2 theory from six Partitions dimensions of 3.1 Codimension two defects and their geometric realization 1.1 Overview and summary of results A and strong-weak dualities arethe interpreted as Riemann a surface. change The offour-dimensional AGT ‘pairs Yang-Mills correspondence of into then pants’-decomposition the explicitlyfruitful of realm brings since of the (non-perturbative) latter two-dimensional class CFT.S-duality of theories invariance This is of in the connection general is much better understood. For example, whereas the three-point functions reproduce the one-loop determinants. the understanding of theIn non-perturbative particular, structure the of systematicinto the construction strong of coupling the limitsgauge of couplings class these are identified super with Yang-Mills the theories complex [ structure parameters of a Riemann surface BPS sectors of four-dimensional supersymmetricsupersymmetric gauge conformal field theories theories and [ two-dimensionalfunctions non- [ in to be equivalent to the instanton partition functions, computed in the Ω background [ 1 Introduction The Alday-Gaiotto-Tachikawa (AGT) correspondence is a remarkable relation between 5 Conclusions and outlook A The conifold 4 Toda 2 Review 3 Orbifold defects and the generalized conifold Contents 1 Introduction JHEP12(2017)050 4 Σ S × 4 ]. Us- ]. It is S 4 21 and Toda on Indeed, the S 1 ]. Moreover, 1 − theory of type N 11 , S . These features A T 10 of type M5 branes on Σ [ (Σ) T ]. A constructive derivation N Toda 20 – 0 Z → 12 4 S 0) theory ,
Σ 0) theory on a circle and five-dimensional , × 4 – 2 – S ⇐⇒ 6d (2 ], one performs a Kaluza-Klein reduction on T 1 24 − Z – N ) Chern-Simons theory on a manifold with asymptot- 22 A C 0 blocks a straightforward implementation of this strategy. → N, Σ T
4 S ]. S 9 theories already hints at this since it assigns a class Z S Toda theory on a Riemann surface Σ. The Toda fields are understood 1 ) Yang-Mills theory [ − N N A to a punctured Riemann surface Σ, by compactifying 1 In this paper we will build on a recent derivation by and C´ordova Jafferis [ Over the past few years, many different approaches have been taken to overcome To be precise, the six-dimensional interpretation of the AGT correspondence is that A physical interpretation of the AGT correspondence and its generalizations to higher Increasing the rank, however, the AGT correspondence maps unsolved problems in the Relevant references will be given in the main body of the paper. − = 2 SU( 1 N N to obtain as boundary fluctuations of SL( ically hyperbolic boundary. This is understood in the following way. Near the boundary, correspondences between supersymmetric sectorsmodels. of Moreover, due gauge to theorieson its the six-dimensional and worldvolume origin, exactly theory such of solvable a multiple derivation M5 may branes. also sheding light the relation between the type partition function, as longof as a we Lagrangian restrict description to of the supersymmetric sector. However,this the difficulty. lack See forof an the incomplete correspondence list is of desirable references as [ it could provide an idea of the scope of AGT-like The arrows denote a supersymmetrictheory zero-mode reduction respectively. to the gauge Thethrough theory a equivalence topological of twist performedenable the on us lower Σ to two and the send partition Weyl the functions invariance of size is of explained either manifold to zero without affecting the value of the following diagram. precisely this Riemann surfacedenotes on the which number the of Toda primary theory insertions lives. in The the number Todathe of correlation supersymmetric punctures function. partition functionhas of a the four- 6d and (2 two-dimensional incarnation, which are equal. This is illustrated by the phrased in very distinct settings,a leading complete solution to to new either insights problem and would progress kill [ tworank birds and with inclusion one of stone. construction defects of seems class to relyA on a six-dimensional perspective. proven some time ago [ gauge theory to otherof unsolved partition problems functions in of Todageneral non-Lagrangian theory. three-point theories For function. is example, mapped the However, computation onto the the correspondence determination allows of these a problems to be corresponds to crossing symmetry of the Liouville four-point function, which was rigorously JHEP12(2017)050 ] , 2 + N N N C C T 36 are d sl sl × = (1.1) N m sl N Z / ) theories = 2 SU(2) C N ], such con- N ]. 21 SU( WZW theories. . This technique 35 ∗ , λ N d 34 sl = 2 ]. N embeddings into . An explicit check was 26 , 2 N WZW theories. This was d sl sl 25 2 c sl .
0 . These generalized Toda theories σ = 3, the reduced boundary theory λ embedding found in [ , whose conformal blocks were shown N W 2 O , would be equivalent to the conformal (2) sl 3 + N ) results were obtained as well, but could W ) gauge theories and + N of T N algebra, which is also labeled by a partition λ ) instanton partition functions with more gen- σ partition functions of du λ , which sits together with a raising operator – 3 – N 4 + N W S . Each choice leads to a reduced boundary theory sl H ) are well known to provide a reduction of the N σ . The corresponding generalized AGT correspondence dσ 1.1 of λ . For example, for λ correspondence. N λ A → ]. For the principal sl ]. Consequently, one of the building blocks in establishing the ] that the SU( 29 symmetry. The standard AGT and full surface operator setup by Toda – 30 33 λ into λ 27 W 2 ] to compute the sl 37 ] to a relation between SU( + 1 respectively. The subalgebra. In a type IIA frame, these boundary conditions arise from a 32 ... N sl WZW correlation function. For SU( ] a relation was proposed between instanton partition functions of 2 ⊂ corresponds to the maximum number of parts of the partition is an element of the Cartan of symmetry if the embedding is the principal one, but it has Polyakov-Bershadsky c sl 2 31 = 1 + m 3 sl H symmetries for the diagonal embedding. More generally, , is obtained by quantum Drinfeld-Sokolov reduction of N In the following, we will denote the generalized Toda theory resulting from an Then, based on mathematical results in instanton moduli spaces, it was realized in [ It was conjectured in [ In [ Boundary conditions such as ( However, the residual symmetries of the constrained WZW theory strongly depend W N (2) 3 the full not be compared due to lack of results onreduction the with WZW partition side. will be referred to as the AGT that the instanton partitioncould function be in conveniently computed the as presencewhere an of ordinary a instanton partition general functionwas surface on further operator used on inin [ the presence of a full surface operator, and was shown in the case of SU(2) to reproduce and of performed for the Polyakov-Bershadsky algebra to agree with instanton partitionpartition functions 3 in = the 2 + presence 1. of a Further checks simple of surface the defect, proposal with have appeared in [ These cases dealt with the so-called full surface operators. eral surface operators, labeled byblocks a of partition theories with are now special cases of this more general setup, corresponding to the partitions play a role in extensions of the AGT correspondence. quiver gauge theories with ansion insertion of two a defect surface in operator,generalized which the in arises 6d from [ a theory, codimen- and conformal blocks of on the embedding of has W labeled by the integer partitions with different symmetries, which we will denote by WZW theory inducedspace, by see Chern-Simons for on examplestraints the [ give Toda boundary theory [ ofAGT asymptotically correspondence is hyperbolic obtained. Here, in an Nahm pole on the scalars of D4 branes ending on D6 branes [ the Chern-Simons connection satisfies the boundary conditions JHEP12(2017)050 ] for ) for U(1) 38 × 1.1 U(1) isome- × can be understood ] for the 3d-3d back- N correspondences using d sl 38 λ , connects the 4d-2d correspon- 2.1 ] after Weyl rescaling and an R-gauge 39 , 3 – 4 – ]. ] are related to a squashed sphere with U(1) 21 39 ] that the Killing spinors as obtained in [ 21 embedding. Therefore, we wish to show that upon including the N sl ⊂ on which the resulting Chern-Simons theory lives has nontrivial boundary. 2 3 sl M The Weyl rescaling of the full supergravity background should also allow one to further A crucial element in the original derivation is that the Nahm pole on the scalars These boundary conditions manifest themselves in a IIA frame in the form of a Nahm The original derivation, which we review in section In the present paper, we propose a setup to derive these AGT ground become the usual 4dtransformation. Killing This spinors argument is of not [ frame completely satisfactory, are since the related spinors to intry, the a whereas 3d-3d the squashed Killing sphere spinorsisometry. geometry in that We [ note preserves that an this SU(2) slight discrepancy may in fact be immaterial at the level of Nahm pole arising in3d-3d the frame. 4d-2d However, frame a toof direct the a implementation Drinfeld-Sokolov Lagrangian of boundary description this of condition procedure multiple in is M5 the ruled branes. outexplain by the the lack claim in [ the Chern-Simons connection. Itto is combine argued Nahm that data thisunique boundary into combination that condition a achieves is flat this. aunderstanding connection, However, of natural we this but way have point the not andthat been our Drinfeld-Sokolov able construction carefully form still to examining is relies obtain the on a notcorresponding this Weyl better the assumption. worldvolume rescaling supersymmetry We of expect the equations full should supergravity allow background one and to the translate the level. We point outbranes, that which the we Nahm refer pole toD6 as branes, should D6’ and instead branes. will be still The attributed be original to related branes a to will distinct thetransforms always set Chern-Simons be under coupling. of referred Weyl to rescaling as to the relevant Drinfeld-Sokolov boundary condition on manifold With specific boundary conditions, its boundary excitations lead topole Toda on the theory. worldvolume scalars of aattributes D4 the brane Nahm ending pole on to a the D6 D6 brane. branes The that original are derivation also related to a non-zero Chern-Simons briefly sketch the main logic and possible pitfalls ofdence our to arguments here. the 3d-3dof this correspondence connection through is athe that 3d-3d Weyl a correspondence, rescaling. full which can supergravity One then background of be was the put already main to derived use virtues in in [ the 4d-2d setting. The three- general boundary conditions. Alongof the the way, analysis we in will the also original be paper [ able to1.1 clarify some aspects Overview and summarySince of the results story is rather intricate and hinges on some important assumptions, we will the path laid outat by and C´ordova hand, Jafferis. since This the approachfrom general is a quantum very Chern-Simons Drinfeld-Sokolov natural reduction perspective fora of as the general well, problem by imposingappropriate the codimension boundary two conditions defects ( in the six-dimensional setup, one finds these more JHEP12(2017)050 ]. 42 This 2 ]. ) Chern- R 36 , N, 35 U(1) isometry. × theories. In the λ ) and SL( U(1), but still only two C . Here, we crucially use × N, 3 , we believe that this claim 3 M ] in the context of 3d partition 40 = 1 the reduction to generalized real Toda k – 5 – , one obtains complex Toda ]. k 41 theories. = 1 the Hilbert spaces of SL( λ k theories similarly treat such codimension two defects geometrically [ ]. Therefore, at S 45 ]. Subsequently, we give an overview of the relation between Chern- 38 , Although this approximation suffices for our purposes, it comes with does not break any additional isometries, hence the number of preserved 21 ] for a like-minded approach to the 3d-3d correspondence. 4 3 S 44 , 43 theories. λ Finally, it is known that at An important assumption in our derivation is that the connection to the 3d-3d cor- The radial slices of the divisor of the generalized conifold have a U(1) We propose to use the conifold geometry as an approximation to the pole region of a The uplift to M-theory of the setup we propose leads to M5 branes on a holomorphic background. The gravity duals ofSee class also [ 2 3 4 correspondence [ Simons theory and Wess-Zumino-WittenToda models and their Drinfeld-Sokolov reduction to correspondence for complex Toda 2 Review In this section we review the derivation and by Jafferis C´ordova of both the 3d-3d and AGT Simons theories agree [ theories proceeds asprincipal usual. case, the For original higher real derivation paraToda puts forward with a a duality decoupled between complex coset. Toda and It would be interesting to formulate a similar respondence still stands. Evenonly though manifest additional themselves defects inthese are the defects present boundary we are conditions claim locatedis of that justified. at the these the Chern-Simons asymptotic theory. boundary Since of to a trivial surfacesupercharges operator, are present. the isometryoperator This enhances to may to break seem part SU(2) strange,fully of since squashed the one supersymmetries. expects However,supercharges a on placing a nontrivial a fully surface squashed surface background operator is the on same a with or without a surface operator. worldvolume theory. It would therefore bebackgrounds very for interesting arbitrary to parameter obtain a values class where of this supergravity singularity canFurthermore, be avoided. it supportsbackground. two supercharges, In the in special agreement case with where the only a four-dimensional single Ω D6’ brane is present, corresponding coupling the worldvolume theory to additional degrees of freedomfull on supergravity the background defect. thatS would be needed toa account particular for value a of defectresponding in the to a squashing the squashed parameter conifold point. that leads In to principle, a such a curvature singularity singularity could cor- couple to the M5 functions and properly understood in [ divisor in a generalizedthe orbifold conifold, description which of weenables codimension us discuss two to defects in treat that the section defects was purely advocated geometrically, in so [ that we do not have to worry about partition functions, as was indeed originally found in [ JHEP12(2017)050 4 1. (2.1) (2.2) − 2 ` √ i = 2 off-shell + Σ 2 k Σ N = q ]. 49 – × 47 is a warped product of a the 3d-3d frame. The Hopf 3 . 2 M 0) theory on a circle and 5d (b) . R , 3 3 = 2 SYM to 5d (2 × with coupling M M 3 ] 3 1 N × − M 38 M ∗ k N fibered over an interval, shrinking to 3 Z T A Σ / M k 3 ` × × Z S × k / for an illustration. k k 3 Z Z 1 Z / S / ⇒ 3 / ` 4 ` = 3 on two geometries which are related by a Weyl ` – 6 – S S Weyl ⇐ S . We will refer to these geometries as the 4d-2d the 4d-2d frame and 1 1 ], these general results allow one to preserve four 3 ⊂ − Σ S Σ 24 (a) 3 N (a) (b) – × A M 22 k ) Chern-Simons on × 0) theory is reduced on the Hopf fiber. This translates in Z C is a squashing parameter which controls the ratio between , k / 4 ` ` Z N, S / 3 ` × S and 0) theory on the geometry [ as the Lens space , R k ] it was shown how to couple 5d Z / 0) CFT of type 46 4 , ] S 21 is indicated in blue. ) Yang-Mills theory [ 3 N The geometries associated to S which gives rise to SL( 3 In older work [ S This provides a derivation of the 3d-3d correspondence as formulated in [ = 2 SU( 4 5d to a flux forFor the general graviphoton, squashing, which it is is compatible requiredmultiplet. with to the turn The on 5d resulting all supergravity bosonic background background. the fields allows in for the a off-shell supergravity supersymmetric zero mode reduction on N supercharges from the (2 In the original derivation, the (2 zero size at theRiemann endpoints. surface Σ The and three-dimensionalthe manifold Hopf fiber andand base 3d-3d radius geometries of respectively. the See figure supergravity. Using the equivalence between the Consider the 6d (2 transformation [ We think of the Figure 1. fiber of the 2.1 Principal Toda theory from six dimensions JHEP12(2017)050 ) 5 2 Σ. C R ∪ (2.3) N, .  ) becomes a A N ∧ A is the full SL( 3 ∧ 0) theory come to be , M A 2 2 3 and a continuous param- R . Equivalently, thinking of , so that in the far IR the k + k 3 gauge group. The ghost and S Z / R dA 4 . Σ. In the 3d-3d frame the entire S ∧ ) ∗ gauge group SU( due to the topological twist. We 3 N A ⊂ T R ( 3  k M Z i su Tr / . 3 3 ⊕ S iX non-compact M ) compact Z 2.2 Σ + N π ( k A 4 – 7 – . su that arises from the Weyl rescaling of the 4d-2d = 3 3 ⇒ ∼ = M A ) k M C Z ) = . They correspond to movement in the remaining / a F 4 N, Y S ( ∧ sl F 0 1 2 3 4 5 6 7 8 9 10 Yang-Mills with arises from the squashing parameter of the three-sphere. ∧ ` C M5 x x x x x x M-theory background relevant for the AGT correspondence. Tr ( N Chern-Simons theory with 3 M × ). 2 -centered Taub-NUTs glued along their asymptotic boundary, one reduces S k Z 2.2 Table 1. provides the fibers of its cotangent bundle 2 supersymmetric 1 π 3 8 . The latter combine into a one-form on i R X as two , which summarizes the 4d-2d setup. The theory is topologically twisted along Σ. ⊂ . The former arises from the graviphoton flux that couples to the D4 gauge fields 4 2 1 ` The setup is reduced on the Hopf fiber of the To understand what type of boundary conditions have to be imposed, let us look at We now return to the particular A salient detail of the reduction is that the fermions of the (2 The complex Chern-Simons coupling consists of an integer S R is used for the topological twist on 3 R the on the Taub-NUT circle fiber.fiber of It is a well multi-Taub-NUT known yields that D6 the branes M-theory at reduction the on Taub-NUT the centers. circle The IIA setup background. Note that itSo has we a need nontrivial to boundaryToda specify consisting theory of boundary on two Σ, conditions, components as which Σ we ultimately will lead review to intable section non-chiral complex An is built out of the originalscalars Yang-Mills connection together with three of thedenote five the worldvolume other twodirections scalars of by ( gauge fixing is undone and theChern-Simons final theory. result for The the complex effective connection theory on This provides ation concrete of explanation 5d non-supersymmetric for the puzzlegauge fixing that terms in the the supersymmetric effective action reduc- are subleading in The continuous parameter interpreted as Faddeev-Popov ghosts forgauge algebra the gauge fixing of the non-compact part of the eter through the 5d Chern-Simons coupling JHEP12(2017)050 D4 (2.4) N triplet of i X denotes the = 1. ` ]. k ] for D3 branes 25 parametrizes the 26 are instead similar σ 2 . Here, the 2 2 3 R (2) and R su 3 3 R R would translate to in the 3d-3d picture. . i i X σ T Σ Σ → – 8 – i satisfy the Nahm pole boundary conditions ]. The latter deal with (the T-dual of) a D4-D6 i X D6 branes sit at the north and south pole of the ]). The corresponding orbifold singularities in the X 26 2 , 51 ). However, they cannot give rise to a Nahm pole. In 3 25 x x x x monopole on the D6 worldvolume. Indeed, the Nahm S 2.3 N 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 ` RR is fibered. This can be understood by thinking of the dimensional representation of k correct codimensions between D4 and D6 branes for a Nahm , transforming the supersymmetry equations that lead to a Reduction of M-theory background to type IIA. Z N / ] (see also [ D4 x x x x x D4 3 D6 x x x x x x x D6 x x x x x x x D6 x x x x x x x . The D6 and 1.1 and 50 S N k k N k 2 Table 2. constitute an i T Type IIA setup in which Nahm poles arise as boundary conditions on the , we propose an alternative perspective that simultaneously allows for a non-zero 4 Leaving these comments aside for the moment, we must understand precisely what a We want to pause here for a moment to note that it is not quite clear why the present respectively and the D4 branes end on them. The boundary conditions on the D4 Note that the resulting three-sphere has curvature singularities at the poles even for 5 3 Nahm pole in the topologicallyAs twisted remarked scalars inNahm section pole under the Weyl transformation is a dificult problem. However, we know that 4d-2d frame reduce toChern-Simons a coupling graviphoton through flux ( insection the 3d-3d frame, whichChern-Simons is coupling responsible for the pole to arise. setup is related tobrane the system analyses with of different [ codimensions,fact as that described the D4 in branes table to end on the the ones D6 studied branes. in The [ D6 branes in table Table 3. D4 scalars. is then given by table Here, the interval over which the branes as comprising apole charge boundary conditions were originally discovered in a similar context [ S worldvolume fields are thenending claimed on D5 to branes. be Thatconditions, would similar imply while to that the the those D4 triplet gauge studied of field in scalars satisfies Dirichlet [ boundary JHEP12(2017)050 . 2 3 as sl ]. It iT (2.8) (2.7) (2.5) (2.6) ∓ 52 , 2 ) denote A ∧ A T 29 ¯ z + = z, A  d L If ( = 6 F ) to the connection and . 1 ] 2.4 A iT δ Toda theory. It would be vanishes. Thus, a natural = . Boundary conditions on 3 ¯ z 0 A ∧ L A ∂M . Tr[
3 0 complex . σ b ∂M + . Z a 3 O ) translate to constraints in the WZW π L k generators ) 4 + b 2.5 ∂M 2 σ dz − sl ] + a + L ) would be – 9 – to be chiral on the boundary. ] for reviews. We simply wish to point out how + ] = ( correspondence. A ∧ F = 0 on 2.4 b 56 with boundary, Chern-Simons theory with gauge A δ , the relation between Chern-Simons theory and σ λ – ¯ L z dσ will have to be flat. Furthermore, we expect that the 3 , 0 A Tr[ a 53 M L 3 2.2 L iX [ algebra. The antiholomorphic connection of the complex M = + Z N A π A k W 2 Wess-Zumino-Witten model on = = N ) under the Weyl transformation, as we already pointed out in A and Drinfeld-Sokolov reduction d sl ) on Σ, one can set towards the boundary should still be fixed. CS ¯ 2.5 z N then gives rise to a full (non-chiral) δS A z, 3 ∂M algebra to the induces an N ]. Here we restrict to the holomorphic sector of the complex Chern-Simons . d sl N 59 – sl 1.1 57 We first recall how one obtains a (Brown-Henneaux) Virasoro algebra in the As we will review in section Nonchiral boundary conditions lead to reduced theories with nonzero chemical potentials [ 6 Using coordinates ( would be interesting to see if they have a role to play in further generalizations of the AGT correspondence. The bulk term would leadthe us equations to of identify motion. the vanishingIt However, of this is the is most curvature not commonly justified dealt unless with the boundary by term requiring vanishes. one of the boundary components to vanish. they can be used in deriving the AGT case [ theory. The following holdsChern-Simons similarly action for is the antiholomorphic sector. The variation of the On a three-dimensionalalgebra manifold the connection such as thosemodel. we encountered Many in of ( theand results three-dimensional we gravity, see discuss [ are well known in the literature on WZW models interesting to directly verify theboundary transformation condition of ( thesection Nahm pole ( 2.2 Partitions of These boundary conditions areboundary precisely the ones thatChern-Simons correspond theory to the behaves reduction incomponents of the of the same way. Adding the contributions from the two This is a flat connection.They satisfy We have the defined standard the commutation relations leading behavior of Wess-Zumino-Witten models requires (anti)holomorphic coordinates on Σ, weequivalent should of demand the that boundary conditions ( the resulting connection JHEP12(2017)050 2 0, sl ≡ (2.9) leads 0 (2.12) (2.13) (2.10) (2.11) ρ J ]. Again, 61 , ) and central z ( 60 , − J dz 52 , − . The reduced symmetry 3 L 29 ) describing the WZW ) = ) z z dz. dz. ( ( AdS + + . − 2.10 T 0 L L J L ρ ρ ρ ρ . e e − . e 0 e ) up to leading order in L
+ + N ) is then ρ is empty + e sl 0 dz dρ dρ ) ) 2.9 2.10 A 0 0 ∈ z dz ( dz L 0 L 0 . a − , the leading order of the transformed L L 2 T = ) = J ) sl z 0 /σ 0 = . ) and gauge fix the radial component as z ( ) and ( + L ( 0 L 0 ∈ ρ ρ ρ a L J e 2.8 ) are the currents of the chiral Wess-Zumino- e ρ 0 2.8 A J dz = 1 z e
) L ( + + ) ρ + a – 10 – ] and higher spin gravity [ L dz dz , = − d J dz dz + + e ( ρ a + + 53 0 L L , = 0 L A d L L ) ) can be written as ) ρ 3 + + . The chiral connection of ( z z 0 − 30 a ( d d , ( L 2.5 e a ∂M ( T ρ | + , the reference connection J 0 0 ¯ z − = 1. We can use the resulting gauge symmetry to fix J 28 L N L , e ρ A ρ + ρ , where the corresponding situation has been studied in the G e ≡ A − − 4 27 d = N e e ( + + l/ 0 sl A J = = L = dρ ρ 0 0 k 0 − A A L e WZW model. generators by = = 2 N c A sl On the level of the action, the reduction outlined above produces Liouville sl 7 . . In particular, we can use the bulk gauge freedom to fix the radial component k ∂M = 6 c ), geometries. Now let us return to 3 2.9 From the perspective of three-dimensional Einstein gravity, which can be described by two chiral 7 Chern-Simons actions with algebra is a chiral halforder of radial the falloff Brown-Henneaux corresponds asymptoticAdS to Virasoro imposing symmetries. Dirichlet Constraining constraints the on leading- the boundary metric of asymptotically Its asymptotic behavior isformed constrained Nahm pole, by the leading order behavior of the Weyl trans- we can impose chiralin ( boundary conditions ( We denote the model becomes Its residual symmetries formcharge a Virasoro algebratheory with from current the context of current algebras [ to a first class constraint leading to a second class set of constraints. The reduced on-shell phase space consists of particular, using the radialNahm pole coordinate boundary conditions ( Comparing this to the WZW current components in ( The remaining chiral degrees ofWitten freedom model. One can consider the reduction of this model using certain constraints. In model on to be The most general flat connection satisfying ( With these boundary conditions, Chern-Simons theory describes a Wess-Zumino-Witten JHEP12(2017)050 , ], λ 3 sl 63 (2.14) (2.15) (2.17) (2.18) (2.19) (2.20) (2.16) in 2 . We are sl multiplet. M subalgebra. 2 sl 2 . a . For example, sl sl , T , N 0 ) for the first two 2 sl A . To be precise, if 1 0 . ⊕ 2.16 L basis in multiplets of a . 2 , k T 2 N 2 ) n 3 2 sl z in the connection is then ( . ), the choice of partition 2 ⊕ ⊕ a a under X 1 2 2 J J a 5 3 ρ = , ) 2.17 T a ( = = = 8 N w 2 2 2 Polyakov-Bershadsky algebra [ is in ( − embedding through the generators 1 1 1 → ∞ 0 generators, e z : k ρ − − − (2) 3 n N < A = λ 3 3 3 ) sl 0 W ¯ 3 ¯ 3 ¯ 3 a P ( L ρ ⊂ ⊗ ⊗ ⊗ w e = 3 3 3 a 2 of 3 3 3 under the adjoint action of its T N ] and sl (1) as a 0 – 11 – ←→ N L J fundamental representation are labeled by parti- ρ 62 sl , WZW model constrained by ( . − 2 and a spin one current. In the final decomposition, ⇒ ⇒ ⇒ N a ≡ O e 30 component of / N = = = ) L sl 0 2 d a sl z ( k T embeddings and the corresponding multiplet structure. a k 2 J n N 1 0 for all other negative weight generators A − A sl multiplets in the adjoint representation of = =1 N ⊕ algebra [ k M 2 a ≡ It is therefore useful to organize the 2 2 ⊂ T 3 sl 1 1 a = 8 2 a T W T ⊕ ⊕ sl | J N z 2 2 2 N 3 2 1 A = = = to denote a k-dimensional fundamental representation of current algebra. 3 3 3 , we see that the 2 a 3 3 3 M T c sl 1 and k ) a , ( ≡ N w appearing in it. + L } of subalgebra corresponding to J ] = + 2 λ a = 3 we can choose 3 = 3, 3 = 2 + 1 or 3 = 1 + 1 + 1, corresponding to L The residual symmetries of the We shortly review In particular, the radial falloff of a current component sl , From the point of view of three-dimensional gravity, this choice of embedding corresponds to choosing ,T 0 8 0 N L while the latter comes with twono spin positive 3 radial weightsthe appear full so affine no constraints are imposed and wean are Einstein still sector left within higher with spin gravity. These partitions correspond torespectively. the principal, diagonal and trivial embedding of decompositions are the respectively. In addition to the Virasoro current, the former contains a spin three current, if Here, we use interested in the multiplet structureThrough of the corresponding decomposition determines the number of as highest-weight or Drinfeld-Sokolov gauge, with a single currentThe for possible each decompositions oftions the we constrain all the current components These constraints generate additional gaugehighest freedom, weight currents which of can each be multiplet used to to zero. fix This all brings but us the to what is usually known Thus, if we fix the non-normalizable part of the connection in terms of {L the determined by the weight[ of the corresponding generator The latter dictates a particular choice of JHEP12(2017)050 λ ]. by 35 . In ) or λ – 2 R C 33 , Toda N, 3 embedding, are coupling N s . ]. sl theories [ 1 N − λ 37 ) ⊂ k and k multiplet appearing ( 2 correspondence. This (1) k 2 b sl su u b λ sl describes the conformal ] ]. Both real and complex k 21 ) + 64 , whose geometry we review in N, k, b k,m K ]. As we will see in section models and their reductions, whereas . Under the correspondence, the rami- summarizes the M-theory background . λ 4 53 N , is is sl 28 − + , , where the geometry locally looks like k k 4 27 [ S real paraToda( r – 12 – N = sl b ⇔ that we are interested in. For the moment, we zoom = 1 are identified. ) λ k 0) theory. These defects wrap Σ and lie along a two- = 1. After reduction, the complex and real Toda theory , k algebra contains a current for each N, k, s λ ] for W theory at 45 of the parafermions in paraToda. The real Toda coupling is λ /k 1 − complex Toda( we briefly summarize the setup pertaining to the AGT 1 gives the rank of the Toda theory. The parameters . 3.1 = 1 the right hand side reduces to real Toda theory [ − ) Chern-Simons theories. Geometric quantization of the latter two theories yields k 3.2 C N . Its corresponding λ ], as we will discuss in more detail below. Table There exists a natural class of codimension two defects that are labeled by partitions of A six-dimensional perspective on this correspondence is provided by including codi- At So far, we have been working with complex More generally, we denote the reduced theory obtained from a general partition N, 4 [ gauge theory, and change the theory on the RiemannN surface. for the particular instancein of on AGT the region near the north pole of the fied instanton partition functions are mappedSimilarly, to it conformal is blocks expected of Toda that one-loopfunctions. determinants This in has the gauge been theory checked map for to the three-point case ofmension a two full defects surface in operatordimensional the [ surface 6d in the (2 gauge theory. Therefore, they represent a surface defect in the section 3.1 Codimension twoWe defects now and consider their the geometrictheory generalization realization partition of function, AGT which we that refer includes to as surface AGT operators in the gauge 3 Orbifold defects andIn the section generalized conifold leads us to consider generalized conifolds, denoted by generalized Toda theories canSL( be obtained as Drinfeld-Sokolovidentical reductions Hilbert of spaces [ SL( therefore agree at thiscomplex particular and real level. Toda Likewise, using a general dimension ∆ = 1 the following relation is suggested for the principal embedding [ Here, constants in the complex Toda theory. On the right hand side, Toda in the decomposition ofcan the be adjoint obtained of from six dimensions using the generalizedthe conifold. original AGT correspondence involves a real version of Toda theory. To mediate this, JHEP12(2017)050 , an with (3.1) ] and k 4 Z 36 / S 2 C D6 branes. m ) is mapped to 2.1 ]. This generaliza- in ( with the 910 directions 67 ] that was already cov- , k 2 orbifold singularity that Z w 31 R 66 / m , intersecting along a two- 3 [ ]. More recently, it has also Z m S / 4 Z -centered Taub-NUT space, it 2 69 / , C m ]). Note that these codimension 2 centers. The M5 branes wrap the . 3 68 C setup described above. This will 65 m is then naturally associated to the R , m n λ m λ and + w k k z with the 23 and ... Z . Upon reduction on the Taub-NUT circle z / = Σ m + 2 1 ] (see also [ C xy . n – 13 – 36 = , : 4.3 D4 branes are distributed among the Generalized AGT setup. . A partition 35 N [ C N k,m reflects the AGT 4 4 ] concerns instanton partition functions on S K : × ,m λ 21 m 1 Z K in section . 0 1 2 3 4 5 6 7 8 9 10 / Table 4. k is thought of as a limit of an C m Z / m 2 with the 01 directions, Z M5 x x x x x x C / and y 2 k Defect x x x x x x x N C or x ]. Note that 70 with defect included. This is a considerable simplification to the full su- . For earlier occurrences of this space, see [ 4 4 S after the Weyl transformation. The geometry near the north pole of this quotiented k We will use the generalized conifold as an approximation in the pole region of a We now observe that there exists a simple (local) Calabi-Yau threefold that provides Another generalization of the original AGT correspondence [ This means that from the gauge theory perspective, i.e. the 0123 directions, that the Z / 4 therefore be the geometry wecomment focus on on general in the following, although we willsquashed also make a brief pergravity background that would be needed to preserve supersymmetry on the where we identify in the table appeared in [ S four-sphere is precisely a particular realization ofdimensional two subspace. ALE This spaces is the (partially resolved) generalized singular conifold ‘cigars’ in the second relativefiber, homology the of partition TN specifies how the ered in the originalorbifold derivation singularity [ that spanstion also the naturally 0123 arises directions from of a 3d-3d table perspective since the It specifies the numberAlternatively, when of M5 branesspecifies with how a the M5 particular branes charge are distributed under among the the orbifold group. only the orbifold singularity. Thison interpretation the is also equivalence supported between byreferences ramified mathematical therein. instantons results and instantons on orbifolds. See [ geometry locally looks like M5 branes this region, we considerspans the the realization 01910 of directions thetwo of defect defects table as are usually a described usingwith an an additional orbifold set singularity. of We intersecting want M5 to branes emphasize together here that we describe the defects using JHEP12(2017)050 3 × in S (3.4) (3.5) (3.6) (3.7) (3.2) (3.3) ' k,m K 2. / ) 2 ϕ to be K¨ahler, − 1 , 1 1 ϕ K − . 0 orbifold with ψ D6 branes located at
 6= 0, respectively. We k 2 2 0 Z k . For = ( . z 2 3 / ) dϕ
correspondence. This will ψ 2 y S D6 branes upon reduction. 2 . 2 1 θ C λ , the standard conifold is a × 2 S m 1 dϕ , πi/k 2 can be described by SU(2) 1 2 S 2) ' − K / T and

2 6= 0 and + sin θ . k 0 x, e ( 2 2 2 2 T D6 branes in IIA. We then introduce w dθ U(1) k ds described by πi/k , so that its metric is 2 2 ∈ . 3 e ρ + T . + cos ( C k iα
A 1 + 2 1 e z gives rise in IIA to zw. | → 2 dϕ ) = dϕ 2 = C dρ y 1 2) C – 14 – θ iα / xy = xy 2 1 ∈ − 1 θ ALE for fixed is homeomorphic to , ) . In other words, ( 1 2 ψ 2 K x, e T ]. However, we can still obtain a better understanding m 9 ALE space and ] + sin x, y Z iα ds 38 2 k 1 e 71 ( Z + cos ALE along dθ  :( and ) that k  and show how it effectively glues two such ALEs together into k dψ k := Z 1 6 Z 3.5 Z C k,m 9 4 + ∈ K 1 = ALE space as a surface in 2 T given by k ds Z 4 C is a cone. We denote its base by 1 , ) describes a 1 -fold angular deficit at the origin in the circle K 3.1 k U(1). The quotient by U(1) serves to identify the Hopf fibers of the SU(2) / In its most common form, which we will denote by Consider the Note that this reproduces the standard conifold metric upon redefining 9 This describes two two-spheres, eachshared with Hopf one unit fiber of parametrizedSU(2) magnetic charge by with respect tofactors. the More details can be found in appendix the base has to have the metric [ The space It can easily be seen from ( the origin and stretched alongequation the ( transverse directions.will The make generalized these conifold considerations more precise in the following. hypersurface in The latter makes itthere clear is that a the resulting space is singular. In particular, weReducing see M-theory that on the Equivalently, we can think of this four-dimensional space as a In this section, we provide morerelation detail between on M-theory the on geometry a ofthe our generalized proposal. conifold First, wea recall single the six-dimensional manifold, leading to two sets of tailed and rigorous analysis asof in the [ emergence of abe general the Nahm main pole result and the of ensuing this AGT paper. 3.2 Intersecting D6s from the generalized conifold defect. One might worry that much information is lost by refraining from a similarly de- JHEP12(2017)050 U(1) (3.9) (3.8) (3.11) (3.10) × , which y = 0 in the and coordinates, z. . w 4 x 2
C ) describing the 2 , together with a 3.7 dϕ T . 2) ) . / x = 0 along ) . Setting 2 α ). This is illustrated in π θ w ) leads to two D6-branes ( w 2 − ) restricts to 2 3.8 , D : = 3.8 [0 3 3.6 y α, ψ ∈ ) + cos = + x A 2 dψ , ) 2 α, ϕ , α 2 ρ 9 ) ϕ 4 + + 1 . ψ + ( i D6’ from = 0. These are two branches meeting along iψ
, ϕ e w e 2 2 y, z, w 2 y 2 2 2 2 θ iα θ , θ dϕ 1 − – 15 – 2 = 0, as we can see in ( w, θ these Hopf coordinates are no longer valid and the θ ( 2 w π x, e = 0 or = cos = sin 7→ iα z orbits on the base two-spheres of x e ) x z ( i + sin θ , ψ , in terms of the Hopf coordinates in ( 2 2 2 = 0 or 7→ ) in the ALE case, we are led to consider the action i = 0 along = 0 divisor, which we denote by axes. A ) dθ = 0 and θ , ϕ directions, as illustrated in figure z 3.4 1 z w w 2 with a particular squashing. Note that it preserves SU(2) 6 ρ = w , ϕ 3 2 y S and + ) can shrink to a point. In terms of the embedding , θ 2 z = x, y, z, w 1 and ) implies that ( θ y 3.9 x dρ ( z 3.5 = = 1 it is parametrized by (see appendix w ρ 2 D ds D6 from Each axis represents the modulus of a complex number, the surrounding circles denote . Reduction of M-theory along the circle generated by ( axis. We will choose the latter one, so that the metric ( 2 Now let us look at the Now we want to choose a circle which leads to intersecting D6 branes upon reduction z This is a radially fibered isometries. At conifold equation ( the these loci are hypersurfaces figure stretched along the Thus the circle consistsrotation of in the equal Hopf fiber.circle At described by ( to IIA. Following the circle ( As can be seenbulk from of appendix the base of the conifold, the circle is the orbit of Figure 2. its phase. We willshrink reduce to on a the point red at the (solid) circles, corresponding to the phase of JHEP12(2017)050 at = 0 = 0. -fold w y (3.13) (3.14) (3.15) (3.12) z m = = x U(1). We = 0 divisor w at ⊂ w = z m y Z = ) shrinks to a point x ) by 3.8 3.8 x = 0. α. axis. Reducing to IIA leads w δ. z + M5 branes on the β = . → ) N circle. circles of the second SU(2) factor -fold angular defect along ) that are sometimes used to describe x 2 → m θ xy p, q ) shifts to the Hopf fiber, α, ψ It is given by , δ D4 ) + 3.8 . iδ 10 , β of this space. From the point of view of . 2 m , e 0) w N , , ϕ , intersecting only at w D 2 m,m w iβ m θ e K z ( D = 0 of the first SU(2) factor. = 0. ) = (0 – 16 – = 1 z → θ ) ) = ( = x, z 6= 0 leads to a xy and are located at D6 y , ψ = 0. In contrast, the orbit of ( 2 x, z ) whose orbit defines the M-theory circle is w z ρ = = 0 divisor :( D6’ k = 0, the action ( , ϕ after reduction on the 3.8 , x 2 2 π w ρ θ θ ( = k,m = 0, m are unrelated to the labels ( 2 K z = 0 : ( θ m,n 2 for all θ K π y -fold angular deficit stretches along the = m 2 D6 and D6’ branes, the M-theory circle should shrink with an θ m , an w = 0. This defect is pointlike in D The resulting IIA brane content corresponding to z = D6’ branes that stretch along y A similar analysis for To obtain Where does this circle shrink? Again, we have to be careful about the range of our of the generalized conifold Note that the labels on m = 10 w to x possible base spaces of the conifold. denote the resulting generalized conifold by We are mainly interested inthe the divisor Therefore, from the perspective ofand the the divisor, the D6 D6’ brane brane is stretches pointlike along at angular deficit. We can achieve this by quotienting the action ( That is a circle ofat finite the size south unless pole We see that the corresponding circlesof are the just the conifold, equal sitting at the north pole coordinates. At the north pole D In these coordinates, the action ( Figure 3. JHEP12(2017)050 D6’ (3.17) (3.18) (3.16) m ). In a . 3.15 background 2 axis needs to  C 2 2 x × dϕ m axis of ( 2) Z / / , except that it has a 2 w θ C ( 2 m,m ), K cos ) D6 branes to infinity along 3.1 k = 0 defect, and we can place 1 m − w + -fold one. Consequently, reducing m U(1). The M-theory circle shrinks D6 branes gives rise to the Chern- = , m ) k dψ × y  /m 2 correspondence in the spirit of C´ordova- 2 = . ϕ ρ 9 λ ), it is described by m + x 4 . ψ w ( k i is equivalent to the + iψ 3.11 e z e 2 w  2 2 2 θ 2 2 = θ – 17 – D axis. Note that the three-spheres at fixed radius dϕ xy 2 z θ D6’ branes. = cos 2 = sin = 1, ] for a similar setup in IIB. D6 branes. The conifold point at the origin corresponds is described by equation ( z x axis to be able to move off the defect. Upon reduction, m k 72 sin m y 2 k,m 1 m K + 2 2 dθ  2 . 6 ρ D6 branes and 3 ) then becomes k + 2 ]. In terms of the coordinates ( 3.10 theory from generalized conifolds dρ angular deficit along the 36 axis. The resulting geometry is similar to that of ]. We will see that our proposal also sheds some light on the derivation of the λ = y m 21 w Z 2 D or ds As we will show in the next section, three of the scalars on the D4 branes will obtain Note that the M5 branes are in a sense fractional: a brane along the Finally, we should comment on how the M5 branes are placed in this geometry. The By resolving all the way to The generalized conifold divisor has two components ending on the x w -fold angular deficit intersecting at the origin with an roles in our construction. 4 Toda In this section, we outline aJafferis derivation of [ the AGT illustrate in figure a Nahm pole boundarybranes. condition On dictated the by other howSimons hand, they coupling the are in flux partitioned the coming 3d-3d among from frame. the the Thus, both sets of D6 branes play distinct but crucial the M5s together alongis either holomorphic. one. See This for setup instance preserves [ supersymmetry sincepair the up divisor with athese brane fractional along the M5 branes correspond to D4 branes that end on the D6’ branes, as we with a are in fact also squashed.corresponding to This the leads to conifold an point. additional ‘squashing’ singularity atD the origin, The metric ( In this case, the sphere isometries are broken to U(1) studied in [ IIA frame, such a resolutionthe corresponds to moving out ( k to IIA produces to the location where the two orbifold singularities intersect. It can be obtained by partially resolving the singularity along the Upon reduction to IIA, it leads to JHEP12(2017)050 can 4 S ], where . 1 , 21 1 K ], the D4-D6’ 26 , 25 This feature is not correspondence. This 11 λ produces a D6 and D6’ 2 1 , which we take as further , 1 R ,m ] where only the D6 brane is K 1 ] we note that the D6 brane K 21 38 3 R . By the analysis of [ -plane in the pole region. z . In this limit, the chirality and amount 5 ,m 1 K Σ – 18 – = 1 there is no orbifold singularity. Thus, we can 2 correspondence, a crucial role is played by the gener- m λ x x x x is decoupled from the field theory. In our description, N with C´ordova-Jafferis 1 2 3 4 5 6 7 8 9 10 ` RR 1 = , . This provides a reinterpretation of the results in [ 1 N K D4 2.1 D6 x x x x x x x D6’ x x x x x x x N . In the presence of our defect, the pole region of a squashed ,m 1 Type IIA setup in the pole region after reduction on the circle of with defect included. However, since the supersymmetry analysis is partic- ] for related discussions. K in section k 51 i , Z / X 50 4 Killing spinors agree with those of the divisor in S , our circle degenerates along the entire 4 4 Table 5. S S ). In contrast to the Hopf fiber of the three-sphere, which only shrinks at the poles ] translates to a flux for the graviphoton field in the 3d-3d frame, which is ultimately Our brane configuration is illustrated in table To check that we can still build on the results of [ It should be noted that we use the (generalized) conifold merely as a technical sim- Moving on to the general AGT 3.8 See also [ 21 11 system leads to adenoted (principal) by Nahm poleit boundary is condition claimed on that three the D6 of brane the is D4 responsible scalars, for the Nahm pole. in [ responsible for a non-zerolost Chern-Simons in level our in construction,located three since at dimensions. reduction the on north the pole. conifold still produces a similar D6 brane brane if webrane. reduce The to latter type isconsidered. IIA. not present The In in main our the reasonin original setup, for ( derivation this the [ discrepancy D4 isof that branes the we end choose properly a different on circle the fiber D6’ ularly easy for the conifold,tleties that we arise specialize are to expected the to parameter be resolved values in it the dictates. general4.1 set Several of sub- supergravity backgrounds. Compatibility of As explained in the previous section, the standard conifold of the evidence for our proposal. plification. We expect asquashed general set of supergravity backgrounds exists that allows for a view the original AGT correspondencewill as be a special discussed case first. of the AGT alized conifolds be identified with an appropriate divisor in original AGT correspondence. Thewith reason the for trivial this partition isthis that is the reflected surface by operator the associated fact that for JHEP12(2017)050 ]. 74 (4.1) , 73 ). .
σ 2 ( σ f . . As we can .  2 O 2 ),
, π
2 + dϕ σ 3.10 dϕ = 0 2 3 2) 2 / σ θ/ 2 ( θ ) = 2 2 σ ( f + cos + cos and dψ dψ 6 . as above, the Ricci scalar singular- 3 2 √ Since the Killing spinors are chiral ` 0) theory cannot couple to curvature ) ] then goes through. Namely, after 2 . , . In principle, it could couple to the σ ρ 9 d = ( chiral 4 2 21 4 . We will not be too concerned about 12 ξ µν ` ` f + R ) and ⊗ +
σ = 0 divisor is given by ( µν
2 2 ( ], the (2 2 R f w d ) and dϕ chiral 2 21 dϕ 2 σ ξ – 19 – ) ( θ θ f 2 ( = 2 ) that vanishes linearly near σ ( d + sin chiral 6 f + sin ξ 2 2 2 dθ 4) supersymmetry in the remaining two dimensions [ dθ , 2 ] of the AGT correspondence considers the class of geome- 6 ρ 2 ` 21 2 and any + ) 4 2 ` Here, we will show that the amount and chirality of supercharges σ ( dρ f  = w + 2 D 2 Recall that the metric on the ds dσ SU(2) subgroup is used to perform the twist. ⊂ = ) for general 2 ) for these particular choices of 4.1 ds 4.1 It is well known that an M5 brane wrapped on a holomorphic divisor inside a Calabi- For the singular conifold, which dictates Note that the conifold singularity translates to a squashing singularity of the met- We will now turn to consistency checks of our identification of the conifold as an This follows from the fact that the four supercharges form two doublets under the SU(2) R-symmetry 0) theory in the AGT setup. Subsequently, we will relate them to the supercharges in , 12 dimensions as well, whose U(1) Yau three-fold has at mostHowever, (0 since the latterlogically lie twisted, along only the two Riemannfrom Killing both surface a spinors Σ, six-dimensional survive. and on a which two-dimensional perspective, the they theory must be is chiral topo- in four Supersymmetries. preserved by the M5 brane(2 on the conifold divisor matchthe with 3d-3d what frame. one expects for the 6d scalars of dimension fourRicci or scalar, higher but this such can as be resolved by anity appropriate is choice present. of the However, wesolutions function expect that the exclude general singularities setbe in of merely the supergravity an backgrounds Ricci artifact to of scalar. contain the parameter The values curvature imposed singularity by should the conifold. tries ( see, the divisor of the conifold imposes the values ric ( these curvature singularities. As argued in [ The original derivation [ Geometry. This is the pole region of the metric of a squashed four-sphere, approximation of the supergravity backgroundthe relevant to Nahm the AGT pole correspondence. onwards,Weyl From rescaling the original to analysis theWZW of model 3d-3d to [ frame, Toda the theory as Nahm reviewed pole in section translates into the reduction of the JHEP12(2017)050 m v with (4.2) (4.3) ]). ] and 4 77 40 ] makes S .  38 )  isometry of 2 σ ) 2 2 σ which preserves cos( ]. cos( 4 2 θ ], i 2 θ S 74 i , ie 75 ie , gauge transformation, 73 − , − ) R 39 , 2 σ ) ] which are independent of 2 σ . ˙ . This is required to make α A 38 ¯ σ ξ cos( ˙ α cos( 2 θ α i 2 θ ) i − m = 2 theory on a squashed − ie σ SO(5) ( − N , ie , ⊂ m ) ) In terms of groups, the R-symmetry is v 2 σ 2 σ 1 2 13 isometry of the squashed sphere. Since − ]. sin( sin( 2 SO(4) = 2 θ 2 θ 76 = 2 theory on a squashed i i e ⊂ – 20 – αA , e N − ) , U(1) isometries. As was first observed in [ 2 σ ) doublet of Killing spinors [ 2 σ U(2) × . R 3 sin( ⊂ , ξ ) become independent of sin( M 2 θ ˙ α A i 2 θ δ ], the three-dimensional supersymmetric partition function − i 4.2 e − U(1) = 41  e become holomorphic two-forms on the divisor [ ) ˙ U(2) holonomy on the divisor (see e.g. appendix A in [  α A 2 1 ¯ ) ) precisely reflect the ordinary (Donaldson-Witten) topological ξ , φ 2 1 ⊂ one sees that the Killing spinors are indeed chiral to zeroth order + φ K 1 4.3 + φ σ 1 ( i φ ( 1 2 ], after Weyl rescaling to the 3d-3d frame, in a suitable R-symmetry i 1 2 − index is identified with the dotted spinors index. The twist implemented 21 e e R ) = ) = 2 1 ¯ ¯ ξ ξ , , is a Killing vector that generates a linear combination of the U(1) 2 1 Another perspective on the equivalence of the preserved supersymmetries in the ξ ξ U(1) isometries has an SU(2) m U(1) isometries. However, the derivation of the 3d-3d correspondence in [ v = ( = ( × × This feature is also reflected in the standard AGT setup. Indeed, at zeroth order in A final subtlety is to be mentioned here. In the above, we have made contact with As noted in [ Now let us turn to the usual AGT setup. A 4d . Hence, the amount and chirality of the supercharges preserved by the divisor of the 1 2 The normal bundle to the divisor is its canonical bundle. Then, the two scalars corresponding to σ ε ε due to the topological twist on 13 , the Killing spinors ( σ twist: the SU(2) by the conifold is a special version of this twist whentransverse movement the inside holonomy is reduced to U(2). is used to twist the U(1) Twist. conifold case and thevolume AGT theory setup on lies theK¨ahlercycle in M5 inside their branes a relation Calabi-Yau is threefold tobroken automatically [ by topological topologically the twists. twisted, setup since to The U(2). it world- wraps The U(1) a R-symmetry given by the embedding U(1) use of a squashedthen sphere properly with understood SU(2) in [ is in fact insensitive toproposed these relation extra between the symmetries. partition This functions should evaluated provide in a the 4d-2d justification and for 3d-3d the frame. gauge, the Killing spinorscontact with in the Killing ( spinorsσ in the 3d-3d correspondence [ the Killing spinors corresponding to the the Ω background. Itvanishes descends linearly from with the U(1) in conifold are consistent with the ordinary AGT setup. Near the north pole theseto reduce, the up Ω to background a local Lorentz and SU(2) Here, U(1) JHEP12(2017)050 (4.5) (4.6) (4.4) corre- ]. λ ) WZW . 38 N ,m 1 K . The half-line 6 of D4 branes ending i n . . D4 worldvolume scalars 2    a    0 R ∗ X D6’ branes translates in the 0 1 0 0 ∗ ∗ ∗ ∗ m . ], which obtains complex Chern-       ], this leads to a derivation of the = 1 Chern-Simons level [ m 38 = 3 n = 45 . Recalling the equivalence between k R λ + + . In the 3d-3d frame, the D6 brane is L λ = 1 [ ... k + in the Weyl rescaled frame. , 1 Σ 3 n D4 branes on ←→ A ··· M – 21 – = N + N 2 dz + , such a constraint precisely reduces a SL( x x x x L : ρ e 2.2 1 2 3 4 5 6 7 8 9 10 ` λ RR corresponding to the directions 145 in table + 3 dρ M embedding associated to D4 D6’ x x x x x x x : 3 = 2 + 1 = ) Chern-Simons at level λ N λ D6 x x x x x x x m N A sl N is therefore unaffected by the presence of the additional D6’ branes 3 ⊂ 2 M Type IIA setup in the pole region after reduction on the circle of sl . We thus claim that only the constraint on the boundary behavior of the 3 and AGT M D6’ brane. This imposes a Nahm pole on the three ,m 1 th Table 6. i correspondence. K λ To be precise, the partition of the The bulk of In the M-theory frame, Weyl rescaling from 4d-2d to 3d-3d results in an asymptotically As we have outlinedmodel to in the section Toda theorycomplex associated and to real the SL( partition AGT The Nahm pole then maps to a constraint in this block diagonal form, Chern-Simons theory is different. 3d-3d frame to a block diagonal form of the connection at the boundary. For example, constraint at the asymptotic boundary of introduced by the generalized conifold.Simons The theory analysis of from [ reductionthe bulk of of the M5 worldvolume theory, should then still hold in in terms of the solely reflected as a graviphoton flux which leads tohyperbolic a three-manifold along the 1 directionbranes is are stretched located to at a line the with edge an of asymptotic this boundary. half-line, Since the the Nahm D6’ pole they induce becomes a fractional M5 branes in the orbifold background After reduction on the circle fiber,on this the partition encodes the number 4.2 We now want tospondence. explain First, the a relevance partition of is the associated to generalized the conifold divisor for that the specifies the AGT charges of the JHEP12(2017)050 in the 4 ] can be S coincident 21 m . Here, we ex- . 1 U(1) isometries, n − ) k,m k × k , as we showed in K ( C (1) b su u b × reproduces the setup in m ) + Z ,m / 1 with U(1) C K 4 , where the excitations of the n, k, b S 4 ( λ S . together with a boundary condition m k n = 1 + 14 2 ]. | 2 ˜ U(1) isometries since the base of the Hopf ` ... x | 36 × + + = 1 case, is also visible there. The squashed 1 is equivalent to real paraToda 2 n – 22 – | 2 = 0 does not break these isometries any further. m ` z ,m | = 1 ? x ⇔ = 0) theory in the presence of these orbifold singularities was K + , N k 2 ) : t λ n, k, s ( λ m The generalized conifolds preserve the same amount of supersym- ) Chern-Simons theory at level = 0 divisor of C w and N, correspondence was studied [ = 1 conifold. This is particularly clear from the IIA perspective, where k λ The m ]. Complex Toda 65 . The non-trivial Ω background that is manifested by the squashing of a radially 3.2 ], one could naively ask if Furthermore, the divisor of the generalized conifold is still K¨ahler,so only chiral su- Likewise, the general AGT setup concerns a squashed The superconformal index of the 6d (2 21 14 5 Conclusions and outlook We have argued thatunderstood the derivation by of replacing the the AGT correspondence north proposed pole incomputed region [ in of [ the However, such a statementeralized requires Toda one theories. to Weobvious understand are first how step not would parafermions aware be couple of to figure the to out existence how gen- parafermions of could any couple to such affine constructions. subsectors. An According to this partitioncomplex one Toda theory. should obtainof To [ a arrive at quantum a Drinfeld-Sokolov reduction duality with of a real paraToda theory, in the spirit Finally, we comment on apect conjecture to obtain arising SL( from thedetermined general by conifold the partition setup does not break any additional supersymmetry. persymmetries survive. This agrees withpole the region. chirality of the Killing spinors of an 4.3 General It is clear thatTherefore, including just our as defect in at our conifold construction, including such defects in the general AGT Supersymmetries. metry as the instead of a single D6D6’ and branes. D6’ brane, we now have one D6 intersecting with section fibered three-sphere, just asthree-sphere in in this the geometryfibration preserves U(1) is now orbifolded.which the This AGT shows that the divisor in Geometry. JHEP12(2017)050 ) ]. N 36 , allow ]. For 35 38 ,m 1 K theory. For λ ]. shrinks at the with boundary, 21 3 3 ] following similar S M 21 . One can understand 3 and vacuum characters ] is in a sense only con- 1 − 38 N EAdS A size. On an × 3 k S Z / 3 ` S on the two-dimensional side. We used an λ 0) theory of type theory corresponds to a real Toda – 23 – , geometry. λ 3 fibration over a disc, where the 3 S ]. This relation was also derived in [ EAdS 20 , can be Weyl rescaled to 1 as an S 5 × S 5 S , involving the inclusion of surface operators on the gauge theory side λ = 1, using the equality between the Hilbert spaces of complex and real SL( . This interpretation has two main virtues. Firstly, it provides a clear per- k 1 , 1 K , it would be interesting to understand the equivalent of the parafermions that algebras discovered in [ k theory is unknown. In other words, one cannot directly transform the supersymmetry N 1 In the main body of this paper, we have not touched upon the relation between (a limit At level Furthermore, the derivation of the 3d-3d correspondence [ An obstruction to performing this transformation is that these equations can only Let us now turn to the possible pitfalls of our analysis and their potential resolutions. − W N arguments to their derivationsuperconformal of index, the AGT correspondence.this The by geometry thinking relevant of toboundary the the of theinfinite disc. length and The produces Weyl the rescaling stretches the radial direction of the disk to Chern-Simons theory, the complex Toda higher were necessary to make contact with real Toda inof) the the original superconformal derivation [ indexof of the 6d (2 the boundary constraints ofif the such Chern-Simons ghost connection. termsDrinfeld-Sokolov It can constraints would be in the be obtained setting interesting in weNahm to propose, the pole and see 3d-3d setting. how frame, they translate if to they the 4d-2d correspond to the proper correct Chern-Simons boundary conditions. cerned with the bulk offix the the Chern-Simons noncompact theory. part of Evenone the there, gauge ghosts should group appear, similarly at which finite gauge impose boundary conditions on these ghosts, which are related to Chern-Simons connection. be written down inA a five-dimensional setting, sinceequations the leading Lagrangian formulation to of a the Nahm 6d pole in 4d-2d to the 3d-3d equivalent which should give the a precise understanding ofthe the supergravity origin background of andpresence the supersymmetry of Nahm equations our pole, of additionalresembles one a the defects, conifold should 4d-2d such near furthermore a frame.the its obtain background 3d-3d pole frame In should regions. should the include Transforming then a the give supersymmetry rise geometry to equations which the to Drinfeld-Sokolov boundary conditions on the equivalent description of these surface operators as orbifold defects, as advocated inFirst [ of all, wesupergravity make background a and number supersymmetry ofpresence equations assumptions our of and additional the simplifications defects. worldvolume dueand theory Even supersymmetry to in in equations a the the have lack original only of case, been a a written full full down supergravity for background the 3d-3d frame [ conifold spective on the originus of to outline the a Nahm generalizationwe pole. of denote the by Secondly, derivation AGT the ofand the generalized a original conifolds AGT generalization correspondence, of which Toda theory to Toda four-dimensional gauge theory are localized, with a holomorphic divisor inside the singular JHEP12(2017)050 5 ], S 20 = 2 N should 3 that maps M 4 S . One can then couple such a 3 ] by equivariantly integrating M 12 algebra is equal to the 6d superconformal index. . – 24 – 1 λ Ω background. Reproducing the generalized Toda S W m , which we can associate to the choice of a (possibly Z Ω background. Following the geometric description of / N 4 C R × times the radial direction of C 3 S ]. It would be interesting to reproduce this central charge from six 28 ]. This leads to a derivation of the conjecture, appearing before in [ 65 ]. It would be interesting to interpret the role these defects play on this side 3 M [ T SvL acknowledges support from the ERC Advanced Grant 268088-EMERGRAV and Finally, the central charge of generalized Toda theories is known for any embedding, We can also include other types of defects. Codimension two defects that are pointlike Further possibly interesting directions of research include the following. In the 3d-3d As in the derivation of the AGT correspondence, this D6 brane does not have the The boundary conditions on the Chern-Simons connection are again argued to be of the Spinoza Grant ofFoundation for the Fundamental Dutch Research on Scienceconsortium, Matter Organisation a (FOM). (NWO). program This of GO workand is the is Science part NWO supported (OCW). of funded the by by ∆-ITP the the Dutch Ministry of Education, Culture Acknowledgments It is a pleasure to thankand Jan encouragement, de and Boer, and Clay Yuji Erik C´ordova of Tachikawa Verlinde for codimension for useful correspondence two discussions defects. on the geometric realization polynomial on the orbifolded central charge from such aof computation a would geometric provide description a of convincing the check codimension on two the defects. validity see for example [ dimensions. This hasthe been anomaly eight-form done over the forthe principal codimension Toda two defects, in one [ could integrate a suitable generalization of the anomaly on the Riemann surfacesimilarly labeled translate by to a operatorsemi-degenerate) partition Toda of insertions primary. in Inunder the Weyl six rescaling Toda to dimensions, theory. an thesecodimension They defects two defect are wrap to an theproduce five-dimensional a Yang-Mills Wilson theory. line Reducing in to complex Chern-Simons theory. frame, the additional defects we introducedtheory. affect the On boundary the other conditions hand, of they Chern-Simons theory also have two directions along theof three-dimensional the correspondence. reproduces the D6 brane and additionallyconstruction produces generalizes the to D6’ the inclusion on of whichas codimension the two D4s studied defects can as in end. orbifold [ singularities, This that the vacuum character of a general Drinfeld-Sokolov type. The D6the brane boundary that circle arises of from the reduction disc and on the the Hopfcorrect fiber codimensions wraps for apole. D4 Again, brane we claim to that end the analogous on identification it, of the and divisor for of a its conifold scalars in the to acquire a Nahm JHEP12(2017)050 (A.8) (A.4) (A.5) (A.6) (A.7) (A.1) (A.2) (A.3) is topologically . , . , T ) = 1, 2 with a seven-sphere ϕ . = 1 = 1 r + SU(2) 2 2 1 . , 2 , | | l 1 2 . b ψ = 1 | | ( ) ⊂ i l K iψ defined by e Z e + + iθ † 4 bk. 2 SU(2) 2 2) − 2) | ] but choose slightly different | Z C / / k U(1) a = ∈ 2 . The base | 2 | 71 ). Setting θ k, e θ Tr ∈ T iθ SU(2) parametrization. The U(1) ! A.2 − , × e , , iθ ( 2 2 , 0 − = cos( = sin( al, w 3 C C e , L, R = 0 → S − ∈ ∈ ! iθ ) Z 0 ) of SU(2) matrices, ' e ) ) = xy.

0 1 0 0 k, l ( k, l a, b

( = ( L, R ) is overcomplete. Two pairs of SU(2) matrices – 25 – = : det , Θ = bl, z ) is a hypersurface in zw 0 b T A.3 − , k 1 , , , iθ ) 1 1 = ϕ in ( , K a, e ) + † 1 , l T iθ ,Z SU(2) ! SU(2) ψ 1 e Θ † ( ( i iψ ∈ ∈ R e e ,R 0 ak, y z x y w → U(1) quotient is the conifold. Indeed, the invariant coordinates ! 2) ! 2) Θ l ¯ b / ¯

/ ) = / ¯ a LZ ¯ k 1 L 1 − − r 1 ( θ x θ = a, b l b a k ( and can be conveniently parametrized in the following way, 7→ :=

Z SU(2) ) 3 Z × S = cos( := := = sin( a b × L L, R R ( 2 coordinates by S 3 and conjugating it with a pair ( S , we can study its base, which we denote by 0 r Z ) describe the same solution if and only if they are related by the U(1) action L, R The resulting SU(2) under this U(1) action correspond to the ones used in ( This degeneracy should be quotientedacts out of on the the SU(2) Note that the parametrization of ( Now introduce Hopf coordinates on each SU(2) Each SU(2) factor can be described using two complex coordinates, We can write downsolution the most general solution to these equations by taking a particular This equation defines aof six-dimensional radius cone.equivalent to By intersecting Let us review some facts oncoordinates the in conifold. places. We mainly The follow conifold [ A The conifold JHEP12(2017)050 ) = 2 A.9 ] N (A.9c) (A.12) (A.9a) (A.10) (A.11) (A.9b) (A.9d) ]. ] 71 SPIRE IN . . Thus we are at the ][
 π arXiv:0906.3219 2 2 [ 2 = dϕ
= 0 divisor of the conifold. 2 2 2 θ θ w is given by [ 2 dϕ T , 2) ) / , 2 (2010) 167 2 ) = 0, the parametrization in ( 2 . The remaining sphere, together + sin ϕ

1 θ = 0 or . 1 T ( + 2 2 ϕ )  θ 1 , 91 2 1 2 + ) vanishes. In terms of the coordinates ϕ arXiv:0907.2189 θ ϕ dθ , 2 ψ dZ [ + ( + ϕ y † i iψ ψ ψ e + ( e ( + Z ) of the conifold. In terms of the Hopf i . i ψ + cos 2 or 2 2 e (  2. 2
e θ i θ 1 iψ / 2 1 2 2 x e 2 is parametrized by e ) 2 A.1 θ θ Tr 2 2 . Setting 2 dϕ 2

Liouville Correlation Functions from sin dϕ 2 θ sin 3 T ) joins the two Hopf fiber coordinates in the θ ϕ 1 2 9 1 2) ). It describes two three-spheres with a shared – 26 – 1 S θ (2009) 002 cos 2 cos 2 θ / θ − 2 1 1 − 1 A.7 2 base factors of 1 2 3.7 θ θ θ 11 (  ϕ = cos 2 sin cos = sin 2 Lett. Math. Phys. S , . Then x z − − − + sin dZ 2 0 † 2 1 ψ ψ = cos = = = sin JHEP ), which permits any use, distribution and reproduction in + cos dZ dθ z y + x , w  = ( 1  dψ ψ 1 6 Tr ψ 4 9 2 3 ) implies that either + := = = ψ is K¨ahlerimplies that the metric on A.1 CC-BY 4.0 2 T 1 , 1 ds This article is distributed under the terms of the Creative Commons K ), the U(1) quotient ( A(N-1) conformal Toda field theory correlation functions from conformal In the main text, we make extensive use of the A.5 ]. quiver gauge theories to zero in ( ) N w ), these choices corresponds to setting either SPIRE IN Four-dimensional Gauge Theories [ SU( A.9 L.F. Alday, D. Gaiotto and Y. Tachikawa, N. Wyllard, [1] [2] Open Access. Attribution License ( any medium, provided the original author(s) and source areReferences credited. reduces to Hopf coordinates The divisor. Setting in ( north (or south) polewith of the one fiber, of now the describes an ordinary This is the metricHopf we fiber. wrote If down we infeel one think ( unit of of this magneticthe fibration charge. coordinate as Note redefinition that an this electromagnetic metric U(1) is equivalent bundle, to both the usual spheres one under Demanding that coordinates ( invariant combination They are related by the defining equation ( JHEP12(2017)050 , ]. , (2015) and JHEP = 2 09 (2015) (2013) SPIRE 06 flat ]. = 2 N IN ]. B 500 = 1, 05 07 M ][ ]. N β (1994) 19 JHEP JHEP , SPIRE SPIRE , IN JHEP JHEP [ IN SPIRE , , ][ IN B 426 (1995) 65 ][ Adv. Theor. Math. Nucl. Phys. ]. , , ]. arXiv:1605.03997 , ]. B 363 SPIRE ]. arXiv:1302.3778 SPIRE IN [ IN ]. SPIRE ][ Nucl. Phys. ][ IN , SPIRE ][ arXiv:1403.3657 hep-th/9408099 IN ]. , [ SPIRE (2015) 1 ][ arXiv:0904.2715 Phys. Lett. IN [ , ][ 19 SPIRE ]. Dualities: From a Geometric Langlands IN -Triality symmetry in six dimensions d ]. n ]. ][ 2 (1994) 484 A direct proof of AGT conjecture at - A Toda 3-Point Functions From Topological Strings hep-th/9608111 Liouville/Toda central charges from M5-branes W d SPIRE – 27 – [ ]. 4 (2012) 034 IN arXiv:0712.2824 SPIRE [ SPIRE ][ arXiv:0909.4776 ]. 08 IN [ IN [ B 431 hep-th/9407087 -background and the AGT correspondence SPIRE ][ ]. ]. ]. Ω Supersymmetric gauge theories, quantization of IN arXiv:1002.0888 SPIRE Toda Theory From Six Dimensions ][ (1996) 753 [ JHEP The Omega Deformation, Branes, Integrability and Liouville IN (2012) 71 ]. , Electric-magnetic duality, monopole condensation and Monopoles, duality and chiral symmetry breaking in SPIRE SPIRE SPIRE Toda Theories, Matrix Models, Topological Strings and ][ Toda 3-Point Functions From Topological Strings arXiv:1412.3395 IN IN IN [ 313 Adv. Theor. Math. Phys. Nucl. Phys. ][ ][ ][ supersymmetric Yang-Mills theory (1994) 485] [ B 388 , (2010) 141601 , SPIRE hep-th/0206161 IN dualities [ (2010) 092 = 2 ][ 105 arXiv:0909.2453 Seiberg-Witten prepotential from instanton counting N 09 arXiv:1012.3137 , B 430 = 2 [ (2016) 066 On the Liouville three point function N arXiv:1205.6820 Five-dimensional SUSY field theories, nontrivial fixed points and string M-Theoretic Derivations of Solutions of four-dimensional field theories via M-theory Localization of gauge theory on a four-sphere and supersymmetric Wilson loops Phys. Lett. [ 08 hep-th/9703166 , ]. Compactification on the JHEP (2003) 831 [ , 7 arXiv:1404.1079 arXiv:1301.1977 arXiv:1409.6313 [ [ [ (2011) 067 JHEP SPIRE , IN hep-th/9507109 Erratum ibid. 017 [ dynamics Duality for Surfaces, to171 the AGT Correspondence, to Integrable Systems conformal field theory Gauge Systems (2012) 101 Theory 02 II Phys. Rev. Lett. (1997) 3 [ 049 confinement in [ supersymmetric QCD Commun. Math. Phys. Phys. J. Teschner, N. Seiberg and E. Witten, E. Witten, D. Gaiotto, N.A. Nekrasov, N. Seiberg and E. Witten, V. Pestun, C. and C´ordova D.L. Jafferis, N. Seiberg, J. Teschner and G.S. Vartanov, C. Beem, L. Rastelli and B.C. van Rees, M. Aganagic, N. Haouzi and S.J. Shakirov, Yagi, M.-C. Tan, N. Nekrasov and E. Witten, A. Mironov, A. Morozov and S. Shakirov, R. Dijkgraaf and C. Vafa, M. Isachenkov, V. Mitev and E. Pomoni, L.F. Alday, F. Benini and Y. Tachikawa, V. Mitev and E. Pomoni, [9] [7] [8] [4] [5] [6] [3] [21] [22] [19] [20] [16] [17] [18] [13] [14] [15] [11] [12] [10] JHEP12(2017)050 ]. 01 , , (2011) Lett. JHEP B 357 ] , , , ]. SPIRE (1997) 220 ]. A 44 (1998) 158 JHEP IN ]. , (2011) 011 (2012) 033 3 ][ theory 67 SPIRE Super Yang-Mills 02 09 0) SPIRE IN B 503 , AdS SPIRE [ Nucl. Phys. IN J. Phys. IN = 4 , ][ , ]. gauge theories ][ = (2 N JHEP JHEP conformal blocks from d ]. , 4 , arXiv:1012.1355 ) ]. [ The Geometry of d N N SPIRE 6 ( Liouville and Toda Theories (1989) 214 Nucl. Phys. IN sl ]. theory SPIRE , arXiv:1309.5876 gauge theories with a general ][ IN [ SPIRE superconformal field theories ) 0) ]. , ][ IN gauge theories N ]. (2 SPIRE B 227 ][ Affine (2011) 114 = 2 IN arXiv:1008.1412 = 2 Nucl. Phys. Proc. Suppl. arXiv:1408.4132 SPIRE N [ ][ 02 ]. , [ IN N SPIRE (2014) 124 = 2 SU( ][ IN N ][ 01 conformal blocks from SPIRE JHEP ]. Phys. Lett. , IN , arXiv:0804.2902 – 28 – (2011) 045 [ ][ ]. hep-th/9302006 [ (2015) 1277 SL(2) JHEP 01 SPIRE Covariantly coupled chiral algebras arXiv:1305.2891 ]. , IN [ 19 SPIRE ][ arXiv:1109.4734 N=2 supersymmetric theories on squashed three-sphere IN [ Affine The gravity duals of Complex Chern-Simons from M5-branes on the Squashed JHEP SPIRE ][ (2009) 789 Instanton counting with a surface operator and the chain-saw arXiv:1105.0357 Seiberg-Witten Theories on Ellipsoids ]. ]. , [ IN (1994) 317 ]. ]. Supersymmetric Boundary Conditions in Thermodynamics of higher spin black holes in arXiv:1005.4469 ][ super Yang-Mills theory and [ 135 (2017) 119 The relation between quantum W algebras and Lie algebras 160 SPIRE SPIRE arXiv:0904.4466 = 5 11 SPIRE SPIRE [ IN IN ]. IN IN D D-branes, monopoles and Nahm equations ][ ][ (2011) 119 (2012) 025015 ][ ][ (2010) 87 arXiv:1102.0076 On On W-algebras and the symmetries of defects of gauge theories 06 JHEP [ SPIRE ) , 94 Instanton partition functions in W-algebras and surface operators in IN arXiv:1302.0816 Notes on theories with 16 supercharges Givental J-functions, Quantum integrable systems, AGT relation with surface D 85 N [ [ (2012) 189 Adv. Theor. Math. Phys. ]. J. Statist. Phys. arXiv:1011.0289 , JHEP 10 , [ , (2011) 043 = 2 SU( SPIRE arXiv:1206.6359 IN hep-th/9705117 arXiv:1012.2880 hep-th/9608163 Phys. Rev. Supersymmetric Partition Functions JHEP Three-Sphere [ 03 quiver operator 155401 surface operator and their W-algebra[ duals Math. Phys. N Commun. Math. Phys. (2014) 023 as Conformally Reduced WZNW Theories Theory (1991) 632 [ [ [ C. Closset, T.T. Dumitrescu, G. Festuccia and Z. Komargodski, D. Gaiotto and J. Maldacena, C. and C´ordova D.L. Jafferis, N. Hama and K. Hosomichi, Y. Imamura and D. Yokoyama, H. Kanno and Y. Tachikawa, S. Nawata, N. Wyllard, Y. Tachikawa, L.F. Alday and Y. Tachikawa, C. Kozcaz, S. Pasquetti, F. Passerini and N. Wyllard, N. Wyllard, J. de Boer and J.I. Jottar, P. Forg´acs,A. Wipf, J. Balog, L. Feh´erand L. O’Raifeartaigh, D. Gaiotto and E. Witten, F.A. Bais, T. Tjin and P. van Driel, J. de Boer and T. Tjin, M.R. Douglas, D.-E. Diaconescu, N. Seiberg, [41] [42] [38] [39] [40] [36] [37] [34] [35] [31] [32] [33] [29] [30] [26] [27] [28] [24] [25] [23] JHEP12(2017)050 ]. (2015) Lett. ] 223 , , Phys. , , SPIRE 892 Commun. IN , (1995) 2961 ][ ]. 12 Phys. Rept. ]. , SPIRE IN Correspondence (2006) 119 ]. ]. hep-th/9901148 arXiv:1107.0290 Adv. Theor. Math. Phys. ][ d , SPIRE [ , 3 01 (2010) 007 - IN On the general structure of [ Lect. Notes Phys. ]. d Asymptotic symmetries of ]. PoS(Modave2015)001 , 3 SPIRE SPIRE 11 , arXiv:0709.4446 IN IN [ Indices JHEP ][ ][ SPIRE d , SPIRE ]. ]. 3 (2011) 113 IN IN Class. Quant. Grav. JHEP [ ][ , Supersymmetric gauge theories, , 09 Asymptotic W-symmetries in ]. SPIRE SPIRE arXiv:1305.2886 (2008) 106 IN IN [ hep-th/9112068 ][ The asymptotic dynamics of three-dimensional ][ JHEP 02 , SPIRE , ]. IN ][ – 29 – arXiv:1409.0857 arXiv:1108.4389 I-brane dynamics Vortex Counting and Lagrangian 3-manifolds Gauge Theories Labelled by Three-Manifolds 3-Manifolds and ]. [ [ Higher Spin Black Holes JHEP Equivariant Verlinde algebra from superconformal (2017) 003 SPIRE , arXiv:1407.3844 IN arXiv:1605.06528 [ 10 W symmetry in conformal field theory ][ , SPIRE IN Five-Dimensional Maximally Supersymmetric Yang-Mills in ]. ]. ][ arXiv:1006.0977 (2015) 619 (2014) 367 ]. arXiv:1501.01310 Boundary conditions and partition functions in higher spin JHEP [ ]. [ , hep-th/9405171 (2016) 107 SPIRE 339 325 [ ]. SPIRE Equivariant Verlinde formula from fivebranes and vortices IN SPIRE IN 04 SPIRE ][ IN ][ IN (2017) 1 ][ (2011) 225 ][ arXiv:1112.5179 [ JHEP Three-dimensional quantum geometry and black holes Global charges in Chern-Simons field theory and the (2+1) black hole 98 355 Complex Chern-Simons Theory at Level k via the , hep-th/9210010 Asymptotic dynamics of three-dimensional gravity (1996) 5816 2 [ ]. ]. D 52 /CFT arXiv:1402.1465 3 [ (2013) 975 SPIRE SPIRE arXiv:1008.4744 IN gr-qc/9506019 IN arXiv:1602.09021 hep-th/0508025 three-dimensional gravity coupled to higher-spin[ fields three-dimensional higher-spin gauge theories [ Einstein gravity with a[ negative cosmological constant [ [ Rev. Hamiltonian reductions of the WZNW theory (1993) 183 265 [ AdS Commun. Math. Phys. 17 intersecting branes and free fermions Supergravity Backgrounds Math. Phys. Math. Phys. index and Argyres-Seiberg duality Commun. Math. Phys. A. Campoleoni, S. Fredenhagen, S. Pfenninger and S. Theisen, A. Campoleoni, S. Fredenhagen and S. Pfenninger, O. Coussaert, M. Henneaux and P. van Driel, M. Ba˜nados, L. Donnay, M. Ba˜nados, P. Bouwknegt and K. Schoutens, A. Perez, D. Tempo and R. Troncoso, N. Itzhaki, D. Kutasov and N. Seiberg, J. de Boer and J.I. Jottar, L. L. Feh´er, O’Raifeartaigh, P. Ruelle, I. Tsutsui and A. Wipf, T. Dimofte, D. Gaiotto and S. Gukov, R. Dijkgraaf, L. Hollands, P. Sulkowski and C. Vafa, C. and C´ordova D.L. Jafferis, T. Dimofte, S. Gukov and L. Hollands, T. Dimofte, D. Gaiotto and S. Gukov, S. Gukov, D. Pei, W. Yan and K. Ye, T. Dimofte, S. Gukov and D. Pei, [60] [61] [58] [59] [56] [57] [54] [55] [51] [52] [53] [49] [50] [46] [47] [48] [44] [45] [43] JHEP12(2017)050 B ]. , B , 12 B 551 , , SPIRE B 699 IN [ JHEP (1990) 246 , -models Nucl. Phys. σ , Nucl. Phys. = 2 , Nucl. Phys. ]. N , B 342 Nucl. Phys. , (1993) 653 (1999) 022 Theory with Defects Commun. Math. Phys. SPIRE ]. , 0) 01 , IN arXiv:1306.4320 , (2 ][ arXiv:1109.4264 B 390 , SPIRE JHEP Nucl. Phys. IN , , ][ gauge theories = 1 ]. Nucl. Phys. N , S matrices for perturbed Black hole entropy in M-theory gauge theories ]. Symmetry in Conformal Field Theories arXiv:1106.1172 ]. ) [ SPIRE Calabi-Yau black holes and (0,4) ]. = 2 n IN SPIRE – 30 – N ][ hep-th/9904217 SL( D-branes and topological field theories ]. ]. ]. IN [ SPIRE [ SPIRE IN Fivebranes and 4-manifolds IN Supersymmetric Gauge Theories in Four Dimensions ][ Comments on Conifolds ][ SPIRE SPIRE SPIRE Central charges of para-Liouville and Toda theories from Hidden Relating Conifold Geometries to NS5-branes IN IN IN = 2 The Superconformal Index of the (2011) 046009 ][ ][ ][ Brane constructions, conifolds and M-theory (2000) 325 (1989) 49 ]. ]. N D 84 ]. 209 126 arXiv:1412.3872 SPIRE SPIRE [ IN IN . ][ ][ SPIRE hep-th/9511222 arXiv:1101.3552 Conformal field theories via Hamiltonian reduction IN [ [ Brane configurations for branes at conifolds [ Coset conformal blocks and Phys. Rev. hep-th/9811139 hep-th/0311084 hep-th/9711053 Holomorphic D7 branes and flavored Localization for , [ [ [ (2015) 048 ]. ]. 05 (1996) 420 (2011) 573 (1991) 71 SPIRE SPIRE IN IN hep-th/9811004 hep-th/9206041 [ arXiv:1412.7134 463 (2004) 207 (1997) 002 Commun. Math. Phys. 849 [ [ (1999) 204 JHEP M-5-branes Commun. Math. Phys. 139 superconformal field theory from quantum[ groups V. Pestun, M. Bershadsky, C. Vafa and V. Sadov, A. Gadde, S. Gukov and P. Putrov, J.M. Maldacena, A. Strominger and E. Witten, R. Minasian, G.W. Moore and D. Tsimpis, J. McOrist and A.B. Royston, P. Candelas and X.C. de la Ossa, P. Ouyang, N. Wyllard, A.M. Uranga, K. Dasgupta and S. Mukhi, M. Bullimore and H.-C. Kim, T. Nishioka and Y. Tachikawa, M. Bershadsky, A. LeClair, D. Nemeschansky and N.P. Warner, M. Bershadsky and H. Ooguri, [75] [76] [77] [73] [74] [70] [71] [72] [67] [68] [69] [65] [66] [63] [64] [62]