I A Geometric Langlands Duality for Surfaces

M-Theoretic Derivations of 4d–2d Dualities: From a

Geometric Langlands Duality for Surfaces, to the

AGT Correspondence, to Integrable Systems





Meng-Chwan Tan

[0mm] Department of Physics, National University of Singapore

[0 mm] mctan@nus.edu.sg

Abstract

In Part I, we extend our analysis in [arXiv:0807.1107], and show that a mathematically conjectured geometric Langlands duality for complex surfaces in [1], and its generalizations – which relate some cohomology of the moduli space of certain (“ramified”) -instantons to the integrable representations of the Langlands dual of certain affine (sub) -algebras, where is any compact Lie group – can be derived, purely physically, from the principle that the spacetime BPS spectra of string-dual M-theory compactifications ought to be equivalent.

In Part II, to the setup in Part I, we introduce Omega-deformation via fluxbranes and add half-BPS boundary defects via M9-branes, and show that the celebrated AGT correspondence in [2, 3], and its generalizations – which essentially relate, among other things, some equivariant cohomology of the moduli space of certain (“ramified”) -instantons to the integrable representations of the Langlands dual of certain affine -algebras – can likewise be derived from the principle that the spacetime BPS spectra of string-dual M-theory compactifications ought to be equivalent.

In Part III, we consider various limits of our setup in Part II, and connect our story to chiral fermions and integrable systems. Among other things, we derive the Nekrasov-Okounkov conjecture in [4] – which relates the topological string limit of the dual Nekrasov partition function for pure to the integrable representations of the Langlands dual of an affine -algebra – and also demonstrate that the Nekrasov-Shatashvili limit of the “fully-ramified” Nekrasov instanton partition function for pure is a simultaneous eigenfunction of the quantum Toda Hamiltonians associated with the Langlands dual of an affine -algebra. Via the case with matter, we also make contact with Hitchin systems and the “ramified” geometric Langlands correspondence for curves.

1. Introduction, Summary and Acknowledgements

The correspondence between 4d gauge theories and 2d CFT’s have long been observed in the physical and mathematical literature. In a mathematical work [5] that dates back as early as 1994, Nakajima showed that the middle-dimensional cohomology of the moduli space of -instantons on a resolved ALE space of -type can be related to the integrable representations of an affine -algebra of level . Physicists then attempted to seek a physical derivation of this beautiful 4d-2d relation; in particular, Vafa and Witten quickly realized [6] that one needs string theory to “see” Nakajima’s result, whence in 1995, Vafa presented evidence [7] that the correct framework to derive Nakajima’s result is in the context of heterotic-type IIA string duality, following which in 1996, Harvey and Moore argued [8] that it is the equivalence of the algebra of BPS states in heterotic/IIA dual pairs which is relevant. That said, a direct physical derivation – in the sense of an equivalence between generating functions of the middle-dimensional cohomology of the moduli space of -instantons on a resolved ALE space of -type and the integrable representations of an affine -algebra of level – was still lacking.

Six years later in 2002, a similar development took place in the physical literature, where it was conjectured by Nekrasov in [9] that the equivariant cohomology of the moduli space of -instantons on a (resolved) ALE space of -type should be related to WZW models on the SW curve underlying the associated 4d pure theory. Shortly thereafter in 2003, the seminal result in [9] – regarding the exact evaluation of the SW prepotential via the Nekrasov partition function – was made mathematically rigorously by Nekrasov and Okounkov in [4], where a more refined and far-reaching 4d-2d conjecture was also proposed; they asserted that the topological string limit of the dual Nekrasov partition function of a 4d pure theory should be related to the integrable representations of the Langlands dual of an affine -algebra, where is any Lie group.

Then in 2007, Dijkgraaf, Hollands, Sulkowski and Vafa finally gave a direct physical derivation in [10] of Nakajima’s result; the aforementioned generating functions were partition functions of BPS states in two different but dual frames in string/M-theory which could then be equated to each other. Right at about the same time, in an attempt to generalize the geometric Langlands duality for Lie groups [11] to affine Kac-Moody groups, mathematicians Braverman and Finkelberg were also led to formulate a conjecture in [1], which asserts that the intersection cohomology of the moduli space of -instantons on should be related to the integrable representations of the Langlands dual of an affine -algebra. This conjecture was henceforth known as a geometric Langlands duality for surfaces, since it involves -bundles over a complex surface as opposed to a complex curve (which is the underlying ingredient in Beilinson and Drinfeld’s formulation in [12] of a geometric Langlands duality for Lie groups). Witten, in a series of lectures delivered at the IAS in 2008 [13], argued, somewhat abstractly, that a geometric Langlands duality for surfaces can be understood as an invariance of the BPS spectrum of the mysterious 6d SCFT under different compactifications down to 5d.1 Combining the insights from Witten’s lectures and the work of Dijkgraaf, Hollands, Sulkowski and Vafa, the author was able to give a concrete M-theoretic derivation in [15] of this geometric Langlands duality for surfaces; he showed that for the groups, the duality can be derived from the principle that the spacetime BPS spectra of string-dual M-theory compactifications ought to be equivalent.

Next came a mini revolution in 2009, when Alday, Gaiotto and Tachikawa, motivated by the insights from Gaiotto’s work in [16], verified in [2] that the Nekrasov instanton partition function of a 4d conformal quiver theory is equivalent to a conformal block of a 2d CFT with -algebra symmetry that is Liouville theory. This celebrated 4d-2d correspondence, better known since as the AGT correspondence, was anticipated to hold for other gauge theories as well. In particular, it was soon proposed and checked to some extent in [3], that the correspondence should hold for 4d asymptotically-free theories; it was also proposed and checked to some extent in [17], that the correspondence should hold for a 4d conformal quiver theory whereby the corresponding 2d CFT is an conformal Toda field theory which has -algebra symmetry; and last but not least, the correspondence for a 4d pure arbitrary theory was also proposed and checked to hold up to the first instanton level in [18]. The basis for the AGT correspondence for – as first pointed out by Alday and Tachikawa in [19] – is a conjectured relation between the equivariant cohomology of the moduli space of -instantons and the integrable representations of an affine -algebra. This conjectured relation was first proved mathematically for finite -algebras in [20], and later proved mathematically for affine -algebras in [21, 22]. An original effort to furnish a fundamental physical derivation of the AGT correspondence from the viewpoint of 6d SCFT was also undertaken by Yagi in [23, 24], although certain assumptions made in loc.cit. require further investigation. Also, in the Nekrasov-Shatashvili limit, the AGT correspondence in [2] has also been derived via a certain bispectral duality between two integrable systems in [25].

“Ramified” generalizations of the AGT correspondence for pure to include surface operators were also proposed and checked to some extent in [26, 27], although the correspondence for pure arbitrary with a full surface operator had already been proved mathematically in 2004 by Braverman in [28] (as made known to physicists in [19]). Nonetheless, based on peripheral physical evidence, it was later conjectured by Chacaltana, Distler and Tachikawa in [29], that the AGT correspondence should hold for pure arbitrary with not just a full but with any surface operator, where on the 2d CFT side, one has a most general affine -algebra.

The AGT correspondence for was further proposed in [30, 31] to hold on , where on the 2d CFT side, one has an -th para--algebra; this proposal was checked to be true for in [30, 32]. Ideas for this proposal were based on physical evidence presented in [33], where it was also conjectured that the AGT correspondence on should hold not just for but for any group, where on the 2d CFT side, one has an -th para--algebra derived from the affine -algebra.

As 2d CFT’s can often be associated with integrable systems, the AGT correspondence also implies certain relations between the Nekrasov instanton partition function and integrable systems. An example which actually predates the AGT correspondence would be Nekrasov’s conjecture in [9], which asserts that the Nekrasov instanton partition function should be related to a tau-function of Toda lattice hierarchy. A more recent example that arose from the AGT correspondence would be Alday and Tachikawa’s conjecture in [19], which asserts that the “fully-ramified” Nekrasov instanton partition function should be related to Hitchin’s integrable system.

Our main aim is to furnish in a pedagogical manner, a fundamental M-theoretic derivation of all the above 4d-2d relations, and more. Let us now give a brief plan and summary of the paper.


A Brief Plan and Summary of the Paper

In 2, we will employ a chain of string dualities to physically relate distinct compactifications of M-theory down to six-dimensions, where around the five compactified directions, there can be (i) coincident M5-branes; (ii) coincident M5-branes and an orientifold fiveplane; (iii) coincident M5-branes, an orientifold fiveplane, and a worldvolume defect of the kind studied in [29] which can be realized in M-theory by an orbifold in the transverse directions. The relation under string dualities between multi-Taub-NUT space and D6-branes and NS5-branes, and the relation under string dualities between Sen’s four-manifold and D6-branes/O6-planes and NS5-branes/ON5-planes, play a central role in our arguments; they are described in detail in Appendix A.

In 3, we will show that the Braverman-Finkelberg (BF) conjecture [1] of a geometric Langlands duality for surfaces, can, for the , , , and groups, be derived, purely physically, from the principle that the spacetime BPS spectra of the string-dual M-theory compactifications obtained in 2 ought to be equivalent. As an offshoot, we would be able to also demonstrate (i) an identity of the dimension of the intersection cohomology of the moduli space of -, - and -instantons on singular ALE spaces; (ii) a Langlands duality of the dimension of the intersection cohomology of the moduli space of - and -instantons on singular ALE spaces. Likewise for the and groups, we will show that the Langlands duality can be derived, purely physically, from the principle that the spacetime BPS spectra of string-dual compactifications of M-theory and type IIB theory on singular K3 manifolds ought to be equivalent. Furthermore, for the simply-laced and groups, we would be able to also derive (1) a McKay-type correspondence of the intersection cohomologies of the moduli spaces of instantons, which serves as a generalization of Proudfoot’s conjecture in [34] to completely blown-down ALE spaces; (2) a level-rank duality of chiral WZW models; (3) a 4d-2d Nakajima-type duality involving completely blown-down ALE spaces. In particular, for the groups, (2), (3), and our main derivation of a geometric Langlands duality for surfaces, physically realize the commutative diagram in [35, 1]; and for the groups, (1), (2), (3), and our main derivation of a geometric Langlands duality for surfaces, physically realize a -type ALE space generalization thereof.

In 4, we will derive a non-singular and quasi-singular generalization of the geometric Langlands duality for surfaces for the and groups. In turn, this would allow us to make contact with and generalize a closely-related field-theoretic result obtained earlier by Witten [13], and reproduce, purely physically, Nakajima’s celebrated result in [5]. Via the string-dual M-theory compactifications with worldvolume defects obtained in 2, we will also derive a “ramified” version of the geometric Langlands duality for surfaces for the , , , and groups.

In 5, to the setup in 3, we will introduce Omega-deformation via the fluxbrane background studied in [36, 37], add half-BPS boundary defects realized by M9-branes [38], and go on to show that the pure AGT correspondence for the , , , and groups, can likewise be derived from the principle that the spacetime BPS spectra of string-dual M-theory compactifications ought to be equivalent. Our derivation physically reproduces the mathematical conjecture by Braverman et al. in [20], that the Nekrasov instanton partition function for pure is given by the norm of a coherent state in the Verma module over the Langlands dual affine -algebra. Furthermore, the underlying Seiberg-Witten curve – interpreted as an - or -fold cover of the two-punctured Gaiotto curve  [16, 3] – also arises naturally in our picture. A crucial ingredient in our derivation is the realization by a gauged WZW model of affine -algebras obtained from a quantum Drinfeld-Sokolov reduction procedure; this realization is described in detail in Appendix B.

In 6, we will first add worldvolume defects to our setup in 3, and derive a “ramified” generalization of the pure AGT correspondence for the , , , and groups. Our derivation reproduces the conjecture by Chacaltana-Distler-Tachikawa in [29], that the “ramified” Nekrasov instanton partition function for pure is given by the norm of a coherent state in the Verma module over the Langlands dual affine -algebra associated with an arbitrary embedding of in the underlying Lie algebra. In anticipation of a connection to integrable systems, we then specialize our formulas to the case of a full worldvolume defect. In so doing, we would be able to reproduce exactly the mathematical result by Braverman in [28], that relates the “fully-ramified” Nekrasov instanton partition function for pure to the norm of a coherent state in the Verma module over the Langlands dual of an affine -algebra. Second, based on our setup in 4.1 which underlies our earlier derivation of a non-singular generalization of the geometric Langlands duality for surfaces, we will derive a smooth -type ALE generalization of the pure AGT correspondence for the , , , and groups. Our derivation reproduces and generalizes to nonsimply-laced gauge groups the conjecture by Nishioka-Tachikawa in [33], that the Nekrasov instanton partition function for pure simply-laced on an -type ALE space is given by the norm of a coherent state in a Verma module over the sum of a parafermionic coset affine algebra and the -th para--algebra derived from the affine -algebra. In particular, our derivation furnishes us with a concrete definition of even when – see eqns. (6.62)–(6.63) and eqns. (6.72)–(6.73). Last but not least, via building blocks defined by M-theory compactifications with M9-brane boundaries that are in one-to-one correspondence with the three-punctured sphere and cylinder of Gaiotto’s construction in [16], we will derive the AGT correspondence with matter. For brevity, we will consider just conformal linear and necklace quiver theories with gauge groups, although our arguments can be straightforwardly generalized to other Gaiotto-type theories as well. Once again, the underlying Seiberg-Witten curve – this time interpreted as an -fold cover of the generically multi-punctured Gaiotto curve that is a sphere and a torus, respectively [16] – arises naturally in our picture.

And finally in 7, via our results in 5 and 6, we will make contact with chiral fermions, integrable systems, and the “ramified” geometric Langlands correspondence for curves. First, by considering the topological string limit in our derivation of the AGT correspondence for a conformal necklace quiver with gauge groups, we will reproduce and generalize a purely field-theoretic result by Nekrasov-Okounkov in [4], that relates the corresponding Nekrasov instanton partition function of the theory to the theory of chiral fermions on a torus. Second, by considering the topological string limit in our derivation of the pure AGT correspondence for , we will reproduce the conjecture by Nekrasov-Okounkov in [4], which implies that the corresponding Nekrasov instanton partition function for pure is equal to the norm of a coherent state in the integrable highest weight module over the Langlands dual of an affine -algebra of level 1. Moreover, if , we find that the corresponding Nekrasov instanton partition function for pure is a tau-function of Toda lattice hierarchy; this also coincides with Nekrasov’s conjecture in [9]. Third, by considering the Nekrasov-Shatashvilli limit in our derivation of the “fully-ramified” pure AGT correspondence for , we will show that the corresponding “fully-ramified” Nekrasov instanton partition function for pure is a simultaneous eigenfunction of the quantum Toda Hamiltonians associated with the Langlands dual of an affine -algebra. And last, guided by the relation between the elliptic Calogero-Moser system and the “tamely-ramified” Hitchin system on a single-punctured torus, we will show that in the Nekrasov-Shatashvili limit, the corresponding “fully-ramified” Nekrasov instanton partition function of a conformal linear and necklace quiver theory of gauge groups is also a -module in the “tamely-ramified” geometric Langlands correspondence for at genus zero and one, respectively. In turn, this confirms the conjecture by Alday-Tachikawa in [19], that the aforementioned Nekrasov instanton partition function is a simultaneous eigenfunction of the quantum Hitchin Hamiltonians for .


Shorter Routes Through This Paper

As indicated in the contents page, this paper can actually be broken up into four parts. Part I, or 2–4, discusses the geometric Langlands duality for surfaces and its various generalizations. Part II, or 5–6, discusses the AGT correspondence and its various generalizations. Part III, or 7, discusses the relation of the AGT correspondence to chiral fermions, integrable systems and the “ramified” geometric Langlands correspondence for curves. Part IV, or the Appendix, contains materials in support of our discussions in 2 and 5.

Readers who are interested in the physical derivation of a geometric Langlands duality for surfaces, should read 2.1–2.2 and 3.1–3.3. Readers who are interested in the physical derivation of a non-singular or quasi-singular generalization of the geometric Langlands duality for surfaces, should read 2.1, 3.1, and 4.1 or 4.2, respectively. Readers who are interested in the physical derivation of the “ramified” geometric Langlands duality for surfaces, should read 2.3, 3.1–3.2, and 4.3. Readers who are interested in the physical derivation of (i) a McKay-type correspondence of the intersection cohomologies of the moduli spaces of instantons, (ii) a level-rank duality of chiral WZW models, and (iii) a 4d-2d Nakajima-type duality involving singular ALE spaces, should read 2.1–2.2, 3.1–3.2, and 3.4.

Readers who are interested in the physical derivation of the pure AGT correspondence, should read 2.1–2.2, 3.1–3.2, and 5.1–5.3. Readers who are interested in the physical derivation of a “ramified” generalization of the pure AGT correspondence, should read 2.3, 4.3, 5.1–5.3, and 6.1. Readers who are interested in the physical derivation of an -type ALE generalization of the pure AGT correspondence, should read 2.1–2.2, 3.1–3.2, 4.1, and 6.2. Readers who are interested in the physical derivation of the AGT correspondence with matter, should read 2.1, 3.1, 5.1–5.2, and 6.3.

Readers who are interested in the relation of the AGT correspondence to chiral fermions, should read 2.1, 3.1, 5.1–5.2, 6.3, and 7.1. Readers who are interested in the relation of the AGT correspondence to the Nekrasov-Okounkov conjecture in [4] and the tau-function of Toda lattice hierarchy, should read 2.1–2.2, 3.1–3.2, 5.1–5.3, and 7.2. Readers who are interested in the relation of the AGT correspondence to quantum affine Toda systems, should read 2.1–2.2, 3.1–3.2, 5.1–5.3, 6.1, and 7.3. Readers who are interested in the relation of the AGT correspondence to the “ramified” geometric Langlands correspondence for curves and the Alday-Tachikawa conjecture in [19], should read 2.1, 3.1, 5.1–5.2, 6.1, 6.3, 7.3, and 7.4.


Acknowledgements

I would like to thank V. Balaji, P. Bouwknegt, A. Braverman, S. Cherkis, M. Douglas, K. Maruyoshi, N. Nekrasov, S. Reffert, A. Sen and S. Wu, for illuminating exchanges.

I would especially like to thank H. Nakajima for his patient explanation of various related mathematical works and concepts; J.-J. Ma and C.-B. Zhu for their generous expertise on nilpotent orbits; D. Orlando for his assistance with an important formula; and Y. Tachikawa for his ever prompt and detailed reply to all my queries on the AGT correspondence, and more.

Last but not least, I would like to thank O. Foda for spotting some imprecisions in the previous version of this paper.

This work is supported in part by the NUS Startup Grant.

Part I A Geometric Langlands Duality for Surfaces

2. Dual Compactifications of M-theory with M5-Branes, OM5-Planes and 4d Worldvolume Defects

2.1. Dual Compactifications of M-theory with M5-Branes

Consider a six-dimensional compactification of M-theory on the five-manifold . Here, is a singular ALE manifold of type ; is a circle of radius ; and the subscript ’’ means that we perform, in the sense of [39], a “-twist” of the theory as we go around the circle – that is, we evoke a -outer-automorphism of (and of the geometrically-trivial six-dimensional spacetime) as we go around the circle and identify the circle under an order translation. Wrap on this five-manifold a stack of coincident M5-branes, such that its worldvolume, in Euclidean signature,2 will be given by , where is the “time” direction. In other words, let us consider the following M-theory configuration:

(2.1)

Taking the “eleventh circle” to be one of the decompactified directions along the subspace, we see that (2.1) actually corresponds to the following ten-dimensional type IIA background with coincident NS5-branes wrapping , where the IIA string coupling and string length are such that :

(2.2)

Let us now T-dualize along the direction of the worldvolume of the stack of NS5-branes. From A.3, we learn that T-dualizing along any one of the worldvolume directions of an NS5-brane (where the background solution is trivial), will bring us back to an NS5-brane. This means that we will arrive at the following type IIB configuration with IIB string coupling (since , and , where is the radius of ):

(2.3)

Next, let us T-dualize along a direction that is transverse to the stack of NS5-branes. As explained in A.3, since the NS5-branes are coincident, one will end up having a multi-Taub-NUT manifold with an singularity at the origin, with no NS5-branes. Thus, as one can view one of the ’s in to be a circle of infinite radius, in doing a T-duality along this circle, we will arrive at the following type IIA background:

(2.4)

where is a multi-Taub-NUT manifold with an singularity at the origin and asymptotic radius . (As explained in A.3, because we are T-dualizing along a trivially-fibered circle of infinite radius.) At this stage, one also finds that . Consequently, this can be interpreted as the following M-theory background with a very small “eleventh circle” :

(2.5)

From A.1, we learn that the singular ALE space is simply with an singularity at the origin whose asymptotic radius . Note also from A.2 that M-theory on such a space is equivalent upon compactification along its circle fiber to type IIA string theory with coincident D6-branes filling out the directions transverse to the space. In other words, starting from (2.5), one can descend back to the following type IIA background:3

(2.6)

Note however, that we now have a type IIA theory that is strongly-coupled, since the effective type IIA string coupling from a compactification along the circle fiber is proportional to the asymptotic radius which is large. (See A.2, again.)

Let us proceed to do a T-duality along , which will serve to decompactify the circle, as well as convert the D6-branes to D5-branes in a type IIB theory. By coupling this step with a type IIB S-duality that will convert the D5-branes into NS5-branes, we will arrive at the following type IIB configuration at weak-coupling:

(2.7)

Finally, let us do a T-duality along , which will bring us back to a type IIA background with NS5-branes and .4 Lifting this IIA background back up to M-theory, we will arrive at the following configuration:

(2.8)

where there is a nontrivial -outer-automorphism of as we go around the circle.

Hence, from the chain of dualities described above, we conclude that the six-dimensional M-theory compactifications with and coincident M5-branes wrapping the five compactified directions along the manifolds and as shown in (2.1) and (2.8), respectively, ought to be physically dual to each other.

2.2. Dual Compactifications of M-theory with M5-Branes and OM5-Planes

To the stack of coincident M5-branes in (2.1), one can add a fiveplane that is intrinsic to M-theory known as the OM5-plane [40]. Then, we would have the following M-theory configuration:

(2.9)

where as before, we evoke a -outer-automorphism of (and of the geometrically-trivial spacetime) as we go around the circle and identify the circle under an order translation.

Unlike the usual Op-planes, the OM5-plane has no (discrete torsion) variants and is thus unique. Its presence will serve to identify opposite points in the spatial directions transverse to its worldvolume. Consequently, the gauge symmetries associated with the stack of M5-branes will be modified, much in the same way how Op-planes modify the effective worldvolume gauge symmetry on a stack of Dp-branes by identifying open-string states with exchanged Chan-Paton indices that connect between the Dp-branes. An essential fact to note at this point is that the OM5-plane can be interpreted as a ON-plane in type IIA string theory under a compactification along an “eleventh circle” that is transverse to its worldvolume [40]; here, the ‘-’ superscript just indicates that its presence will result in an orthogonal gauge symmetry in the type IIA theory, while the ‘N’ just denotes that it can only be associated with NS5-branes. This means that the presence of an OM5-plane will serve to convert an existing gauge symmetry (in a certain regime) of the worldvolume theory on the stack of coincident M5-branes to that of an orthogonal (and not symplectic) type. This fact will be important later.

Let us now take the “eleventh circle” to be one of the decompactified directions along the subspace. We then see that (2.9) actually corresponds to the following ten-dimensional type IIA background with coincident NS5-branes wrapping on top of an ON-plane, where :

(2.10)

Let us next T-dualize along the direction of the NS5-branes/ON-plane configuration. From A.6, we learn that T-dualizing along any one of the worldvolume directions of an NS5-brane/ON-plane configuration (where the background solution is trivial), will bring us back to an NS5-brane/ON-plane configuration. This means that we will arrive at the following type IIB configuration where :

(2.11)

Here, the ON-plane is the T-dual counterpart of the ON-plane. It is also the S-dual counterpart of the usual O-plane in type IIB theory [40].

Now, let us T-dualize along a direction that is transverse to the stack of NS5-branes/ON-plane. As explained in A.6, one will end up with Sen’s four-manifold with a singularity at the origin [41] (which one can roughly regard as with a -identification of its fiber and base), with no NS5-branes and no ON-plane. Thus, as one can view one of the ’s in to be a circle of infinite radius, in doing a T-duality along this circle, we will arrive at the following type IIA background:

(2.12)

where is Sen’s four-manifold with a singularity at the origin and asymptotic radius . (As explained in A.6, because we are T-dualizing along a trivially-fibered circle of infinite radius.) This is consistent with the fact that a T-duality along a direction transverse to the ON-plane gives rise to a solution that can be identified with a unique OM6-plane in M-theory [40], which, in turn, implies the -symmetry that is inherent in Sen’s four-manifold [41]. At this stage, one also finds that . In other words, (2.12) can also be interpreted as the following M-theory background with a very small “eleventh circle” :

(2.13)

From A.1, we learn that the singular ALE space is simply with an singularity at the origin whose asymptotic radius . Also from A.2, we learn that M-theory on such a space is equivalent upon compactification along its circle fiber to type IIA string theory with coincident D6-branes filling out the directions transverse to this space. In other words, starting from (2.13), one can descend back to the following type IIA background:5

(2.14)

Note however, that we now have a type IIA theory that is strongly-coupled, since the effective type IIA string coupling from a compactification along the circle fiber is proportional to the asymptotic radius which is large. (See A.2, again.)

Let us proceed to do a T-duality along , which will serve to decompactify the circle, as well as convert the D6-branes to D5-branes in a type IIB theory. By coupling this step with a type IIB S-duality that will convert the D5-branes into NS5-branes, we will arrive at the following type IIB configuration at weak-coupling:

(2.15)

Finally, let us do a T-duality along , which will bring us back to a type IIA background with NS5-branes and .6 Lifting this IIA background back up to M-theory, we will arrive at the following configuration:

(2.16)

where there is a nontrivial -outer-automorphism of as we go around the circle.

Thus, from the chain of dualities described above, we conclude that the six-dimensional M-theory compactifications with and coincident M5-branes wrapping the five compactified directions along the manifolds (in the presence of an OM5-plane) and as shown in (2.9) and (2.16), respectively, ought to be physically dual to each other:

2.3. Dual Compactifications of M-theory with M5-Branes, OM5-Planes and 4d Worldvolume Defects

To the stack of coincident M5-branes in (2.1), one can add a 4d worldvolume defect of the kind studied in [29] which can be realized in M-theory by a -orbifold in the transverse directions (see [27, 2.2]). For definiteness, let us consider the following M-theory configuration:

(2.17)

Here, the ’’ sign in the column labeled by means that the particular brane or worldvolume defect extends along the direction with coordinate ; similarly, the ’’ sign in the column labeled by means that the -orbifold realizing the worldvolume defect extends along the direction with coordinate . We take and to be the coordinates on and , respectively, so that would be the coordinates on . Then, if and , the singular ALE manifold can be viewed as a complex surface whose coordinates are identified under the -action , where . According to (2.17), the 4d worldvolume defect then wraps and the -plane. Consequently, the presence of the 4d worldvolume defect (i) modifies the theory living on ; (ii) introduces – when observation scales are much larger than the radius of – a surface worldvolume defect (which we will describe below) in the 4d SYM theory living on the “constant-time” hypersurface , at . Such a 4d worldvolume defect was first considered in [19].


Characterization of the 4d Worldvolume Defect by a Partition of

This 4d worldvolume defect can be labeled by a partition of when , as follows. First, set for ease of illustration. (The same arguments will apply for , except that one must further take into account the above-mentioned identification under the -action.) As usual, freeze the center-of-mass degrees of freedom of the stack of coincident M5-branes; then, along the 2345-directions, we have an , theory on with a Gukov-Witten surface operator [42] along the -plane.

Second, note that this surface operator introduces a singularity in the gauge field : if are the polar coordinates of the transverse -plane in , i.e., , the gauge field diverges as

(2.18)

near the surface operator. By a gauge transformation, one can assume that .

Third, note that the commutant of is a subgroup which is called the Levi subgroup; in other words, the gauge group reduces to along the surface defined by the -plane. The structure of can take the general form

(2.19)

where .7 This means that the Levi subgroup is

(2.20)

where . It is in this sense that the underlying 4d worldvolume defect can be characterized by the partition of , and be called one of type .


Reduction of Gauge Group and Parabolic Subgroups

It will be useful for later to also discuss the connection between (i) the reduction, along the surface, of the gauge group to its Levi subgroup , and (ii) parabolic subgroups of (the complexification of ).

To this end, let be a subalgebra of (the Lie algebra of ) spanned by elements satisfying

(2.21)

Then, is called a parabolic subalgebra, and the corresponding subgroup is called a parabolic subgroup.

Note that since is the commutant of , (2.21) means that there ought to exist a correspondence between and . For example, consider and ; according to our above discussion, is associated to the partition and ; in this case, the corresponding parabolic subgroup is , and its elements take the form

(2.22)

where the sign ’’ denotes some complex number such that the determinant of the matrix is one.

As a second example, consider and ; according to our above discussion, is associated to the partition and ; in this case, the corresponding parabolic subgroup is , and its elements can be any complex semi-lower triangular matrix of determinant one. In general, when the Levi subgroup is , the corresponding parabolic subgroup is just the Borel subgroup .

As a final example, consider and ; according to our above discussion, is associated to the partition and ;8 in other words, there is no defect. In this case, the corresponding parabolic subgroup would be spanned by all complex matrices of determinant one, i.e., .

Note that one can also understand the above correspondence between and to be a consequence of the fact that as Riemannian manifolds. This isomorphism also means that we can describe the reduction of the gauge group along the surface in terms of parabolic subgroups: the gauge group is reduced along the surface by an amount , and from the preceding observations, this is the same as .


Dual Compactifications with M5-Branes and 4d Worldvolume Defects

Now consider the M-theory configuration given in (2.17):

(2.23)

where the coordinates are , with . Here, (i) we evoke a -outer-automorphism of the transverse ten-dimensional space as we go around the circle and identify the circle under an order translation; (ii) the 4d worldvolume defect wraps and the -plane in ; (iii) can be regarded as the -plane identified under the -action , where ; (iv) can be regarded as the -plane identified under the -action , where ; and (v) can be regarded as the -plane identified under the -action . The -action, in addition to acting geometrically, also acts representation-theoretically: at a low-energy scale much larger than the radius of with , the -dimensional representation of the gauge group of the 4d theory living on the “constant-time” hypersurface gets multiplied, under the -action, by (c.f. [43, 44])

(2.24)

Taking the “eleventh circle” to be the decompactified -direction along the subspace, we see that (2.23) actually corresponds to the following ten-dimensional type IIA background with coincident NS5-branes wrapping , where the IIA string coupling and string length are such that :

(2.25)

Let us now T-dualize along the direction of the worldvolume of the stack of NS5-branes. From A.3, we learn that T-dualizing along any one of the worldvolume directions of an NS5-brane (where the background solution is trivial), will bring us back to an NS5-brane. Therefore, we will arrive at the following type IIB configuration with IIB string coupling (since , and , where is the radius of ):

(2.26)

Next, let us T-dualize along the -direction transverse to the stack of NS5-branes. As explained in A.3, since the NS5-branes are coincident, one will end up having a multi-Taub-NUT manifold with an singularity at the origin, with no NS5-branes. To this end, note that one can view along the -direction to be a circle of infinite radius. In doing a T-duality along this circle, we arrive at the following type IIA background:

(2.27)

Here, is a multi-Taub-NUT manifold with an singularity at the origin whose -plane (spanning the - directions of its base in the -- directions that supports a nontrivial -fibration in the -direction) is further identified under the -action , and whose asymptotic radius . (As explained in A.3, because we are T-dualizing along a trivially-fibered circle of infinite radius.) At this stage, one also finds that . Consequently, this can be interpreted as the following M-theory background with a very small “eleventh circle” :

(2.28)

To arrive at this configuration, we have noted that from A.1, the singular ALE space is simply with an singularity at the origin whose asymptotic radius . Here, is a multi-Taub-NUT manifold with an singularity at the origin whose -plane (spanning the - directions of its base in the -- directions which supports a nontrivial -fibration in the -direction) is further identified under the -action , and whose asymptotic radius .

From A.2, we learn that M-theory on the space is equivalent upon compactification along its circle fiber to type IIA string theory with coincident D6-branes filling out the directions transverse to the space. In other words, starting from (2.28), one can descend back to the following type IIA background:9

(2.29)

Note however, that we now have a type IIA theory that is strongly-coupled, since the effective type IIA string coupling from a compactification along the circle fiber is proportional to the asymptotic radius which is large. (See A.2, again.)

Let us proceed to do a T-duality along , which will serve to decompactify the circle, as well as convert the D6-branes to D5-branes in a type IIB theory. By coupling this step with a type IIB S-duality that will convert the D5-branes into NS5-branes, we will arrive at the following type IIB configuration at weak-coupling:

(2.30)

Finally, let us do a T-duality along , which will bring us back to a type IIA background with NS5-branes and .10 Lifting this IIA background back up to M-theory, we will arrive at the following configuration:

(2.31)

where the 4d worldvolume defect wraps and the - directions in . (Recall that the -direction is spanned by the -fiber of , while the ---directions are spanned by its base.) Also, there is a nontrivial -outer-automorphism of the ten-dimensional transverse space as we go around the circle.

Note that the -action, in addition to acting geometrically, also acts representation-theoretically: when , the -dimensional representation of the gauge group of the 4d theory along gets multiplied, under the -action, by (c.f. [43, 44])

(2.32)

where . Note that the partition of depends on the partition of , as one would expect. We shall elaborate on this in 4.3.

Assuming that the center-of-mass degrees of freedom of the stack of coincident M5-branes are frozen, the presence of the 4d worldvolume defect means that at a low-energy scale much larger than the radius of with , the gauge group of the 4d theory living on the “constant-time” hypersurface is broken to a Levi subgroup along the - directions that is the commutant of

(2.33)

At any rate, from the chain of dualities described above, we conclude that the six-dimensional M-theory compactifications with and coincident M5-branes wrapping the five compactified directions along the manifolds and in the presence of 4d worldvolume defects as shown in (2.23) and (2.31), respectively, ought to be physically dual to each other.


Dual Compactifications with M5-Branes, OM5-Planes and 4d Worldvolume Defects

To the stack of coincident M5-branes with a 4d worldvolume defect in (2.23), one can, as was done in 2.2, add an OM5-plane [40]. Then, we would have the following six-dimensional M-theory compactification:

(2.34)

where the coordinates are , with . Here, (i) we evoke a -outer-automorphism of the transverse ten-dimensional space as we go around the circle and identify the circle under an order translation; (ii) the 4d worldvolume defect wraps and the -plane in ; (iii) can be regarded as the -plane identified under the -action , where ; (iv) can be regarded as the -plane identified under the -action , where ; and (v) can be regarded as the -plane identified under the -action .

Combining our arguments behind (2.23)–(2.31) with those behind (2.9)–(2.16), we arrive at the following physically dual six-dimensional M-theory compactification:

(2.35)

where the 4d worldvolume defect wraps and the - directions in , Sen’s four-manifold with a singularity at the origin whose asymptotic radius . (Note that the -direction is spanned by the -fiber of , while the -