References

An M-Theoretic Derivation of a 5d and 6d AGT

Correspondence, and Relativistic and Elliptized

Integrable Systems





Meng-Chwan Tan

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

[0 mm] mctan@nus.edu.sg

Abstract

We generalize our analysis in [arXiv:1301.1977], and show that a 5d and 6d AGT correspondence for – which essentially relates the relevant 5d and 6d Nekrasov instanton partition functions to the integrable representations of a -deformed and elliptic affine -algebra – can be derived, purely physically, from the principle that the spacetime BPS spectra of string-dual M-theory compactifications ought to be equivalent. Via an appropriate defect, we also derive a “fully-ramified” version of the 5d and 6d AGT correspondence where integrable representations of a quantum and elliptic affine -algebra at the critical level appear on the 2d side, and argue that the relevant “fully-ramified” 5d and 6d Nekrasov instanton partition functions are simultaneous eigenfunctions of commuting operators which define relativistic and elliptized integrable systems. As an offshoot, we also obtain various mathematically novel and interesting relations involving the double loop algebra of , elliptic Macdonald operators, equivariant elliptic genus of instanton moduli space, and more.

1. Introduction, Summary and Acknowledgements

In 2009, Alday-Gaiotto-Tachikawa verified in [1] 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 soon proposed to also hold for a 4d asymptotically-free theory [2], and a 4d conformal quiver theory where the corresponding 2d CFT is an conformal Toda field theory which has -algebra symmetry [3].

The basis for the AGT correspondence for – as first pointed out in [4] – 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 proved mathematically in [5, 6], and physically derived in [7] via the principle that the spacetime BPS spectra of string-dual M-theory compactifications ought to be equivalent.

“Ramified” generalizations of the AGT correspondence for pure involving arbitrary surface operators were later proposed in [8, 9], where they were also physically derived in [7] via the principle that the spacetime BPS spectra of string-dual M-theory compactifications ought to be equivalent. For a full surface operator, the correspondence was also proved purely mathematically in [10].

The AGT correspondence also implies certain relations between the Nekrasov instanton partition function and 2d integrable systems. An example would be the conjecture in [4], which asserts that the “fully-ramified” Nekrasov instanton partition function should be related to Hitchin’s integrable system. This was again physically derived using M-theory in [7].

Our main aim is to furnish in a pedagogical manner, a fundamental M-theoretic derivation of a 5d and 6d analog of the above 4d-2d relations, by generalizing our analysis in [7]. Let us now give a brief plan and summary of the paper.


A Brief Plan and Summary of the Paper

In 2, we will first review the M-theoretic derivation of a 4d pure AGT correspondence in [7]. Then, we will generalize the analysis in loc. cit. to furnish an M-theoretic derivation of a 5d AGT correspondence for pure in the topological string limit. In particular, we find that the corresponding 5d Nekrasov instanton partition function can be expressed as the inner product of a coherent state in an integrable module over a universal central extension of the double loop algebra of . Our result therefore sheds further light on an earlier field-theoretic result by Nekrasov-Okounkov in [11] regarding the aforementioned 5d Nekrasov instanton partition function.

In 3, we will first review the M-theoretic derivation of a 4d AGT correspondence for a conformal linear quiver theory in [7]. Then, we will generalize the analysis in loc. cit. to furnish an M-theoretic derivation of a 5d AGT correspondence for with fundamental matter. In starting with the case, we will first make contact with what is known as a Ding-Iohara algebra, whence for the case, we find that the corresponding 5d Nekrasov instanton partition function can be expressed as a four-point correlation function on a sphere of vertex operators of an integrable module over a -deformed affine -algebra. Our result therefore serves as a purely physical M-theoretic proof of a mathematical conjecture by Awata-Feigin-Hoshino-Kanai-Shiraishi-Yanagida in [13], of a 5d analog of the AGT correspondence for with fundamental matter. Last but not least, we will proceed in the same manner to furnish an M-theoretic derivation of a 5d AGT correspondence for pure , where we find that the corresponding 5d Nekrasov instanton partition function – which is also given by an equivariant index of the Dirac operator on the moduli space of -instantons – can be expressed as the inner product of a coherent state in an integrable module over a -deformed affine -algebra – which is also given by a Whittaker function on the -deformed affine -algebra. Our result therefore serves as an generalization of a 5d analog of the AGT correspondence for pure first proposed and checked by Awata-Yamada in [14]. Furthermore, as an offshoot of our physical analysis in the topological string limit, we also have the novel representation-theoretic result that a Whittaker vector in a level module over a (certain specialization of the) Ding-Iohara algebra is also a Whittaker vector in a module over a universal central extension of the double loop algebra of .

In 4, we will first review the M-theoretic derivation of a 4d “fully-ramified” pure AGT correspondence in [7]. Then, we will generalize the analysis in loc. cit. to furnish an M-theoretic derivation of a 5d “fully-ramified” AGT correspondence for (i) pure and (ii) with single adjoint matter, in the Nekrasov-Shatashvilli (NS) limit. In particular, we find that the corresponding 5d Nekrasov instanton partition function can be expressed as (i) the inner product of a coherent state in an integrable module over a quantum affine -algebra at the critical level and (ii) a one-point correlation function on a torus of a vertex operator of an integrable module over a quantum affine -algebra at the critical level. Along the way, we will also argue that the corresponding 5d Nekrasov instanton partition function ought to be a simultaneous eigenfunction of commuting operators which define a (i) relativistic periodic Toda integrable system and (ii) relativistic elliptic Calogero-Moser system. As an offshoot, we also have the interesting result that the simultaneous eigenfunctions of an elliptic generalization of the celebrated Macdonald operators can be understood as one-point correlation functions of a 2d QFT on a torus whose underlying symmetry is generated by a quantum affine -algebra at the critical level.

In 5, we will generalize our analysis in 3 to furnish an M-theoretic derivation of a 6d AGT correspondence for the non-anomalous case of an theory with fundamental matter. In starting with the case, we will first make contact with what is known as an elliptic Ding-Iohara algebra, whence for the case, we find that the corresponding 6d Nekrasov instanton partition function – which is also given by an equivariant elliptic genus on the moduli space of -instantons – can be expressed as a two-point correlation function on a torus of vertex operators of an integrable module over an elliptic affine -algebra.

In 6, we will generalize our analysis in 4 to furnish an M-theoretic derivation of a 6d “fully-ramified” AGT correspondence for with fundamental matter in the NS limit. In particular, we find that the corresponding 6d Nekrasov instanton partition function can be expressed as a two-point correlation function on a torus of vertex operators of an integrable module over an elliptic affine -algebra at the critical level. Along the way, we will also argue that the corresponding 6d Nekrasov instanton partition function ought to be a simultaneous eigenfunction of commuting operators which define an elliptized integrable system.


Acknowledgements

I would like to thank H. Awata, H. Konno, Y. Saito, P. Sulkowski and K. Takemura for very helpful exchanges. This work is supported in part by the NUS Startup Grant.

2. An M-Theoretic Derivation of a 5d Pure AGT Correspondence in the Topological String Limit

2.1. An M-Theoretic Derivation of a 4d Pure AGT Correspondence: A Review

We shall now review how an expected equivalence of the 6d spacetime BPS spectra of physically dual M-theory compactifications would allow us to derive a 4d AGT correspondence for pure  [7].

To this end, recall from [7, (5.9) and (5.11)] that we have the following physically dual M-theory compactifications

(2.1)

Here, we have a common half-BPS boundary condition at the tips of the interval (that is realized by a pair of M9-branes which span all directions normal to them); the radius of is ; ; is a priori the same as ; and the ’s are parameters of the Omega-deformation along the indicated planes,1 where the directions spanned by the M5-branes are:

(2.2)

with and being the coordinates on and , respectively. (Note that if and , can be viewed as a complex surface whose coordinates are . Likewise, if and , can be viewed as a complex surface whose singularity at the origin would be modeled by , where are the coordinates on .)


The Spectrum of Spacetime BPS States on the LHS of (2.1)

Let us first discuss the spectrum of spacetime BPS states along on the LHS of (2.1). In the absence of Omega-deformation whence , according to [7, 5.1], the spacetime BPS states would be captured by the topological sector of the sigma-model on with target the moduli space of -instantons on . However, in the presence of Omega-deformation, as explained in [7, 5.1], as one traverses a closed loop in , there would be a -automorphism of , where , and is the Cartan subgroup. Consequently, the spacetime BPS states of interest would, in the presence of Omega-deformation, be captured by the topological sector of a non-dynamically -gauged version of the aforementioned sigma-model.2 Hence, according to [16] and our arguments in [7, 3.1] which led us to [7, eqn. (3.5)], we can express the Hilbert space of spacetime BPS states on the LHS of (2.1) as

(2.3)

where is the -equivariant intersection cohomology of the Uhlenbeck compactification of the (singular) moduli space of -instantons on with instanton number .

Notably, as explained in [7, 5.1], the partition function of these spacetime BPS states would be given by the following 5d (brane) worldvolume expression

(2.4)

where are the generators of the Cartan subgroup of ; are the corresponding purely imaginary Coulomb moduli of the gauge theory on ; are the rotation generators of the - and - planes, respectively, corrected with an appropriate amount of the -symmetry to commute with the two surviving worldvolume supercharges; and is the space of holomorphic functions on the moduli space of -instantons on with instanton number .


The Spectrum of Spacetime BPS States on the RHS of (2.1)

Let us next discuss the corresponding spectrum of spacetime BPS states along on the RHS of (2.1). As explained in [7, 5.2], the spacetime BPS states would be furnished by the I-brane theory in the following type IIA configuration:

(2.5)

Here, we have a stack of coincident D6-branes whose worldvolume is given by , and a single D4-brane whose worldvolume is given by .

Let us for a moment turn off Omega-deformation in (2.5), i.e., let . Then, as explained in [7, 5.2], the spacetime BPS states would be furnished by complex chiral fermions on which effectively realize a chiral WZW model at level 1 on , , where is the affine -algebra.

Now turn Omega-deformation back on. As indicated in (2.5), as one traverses around a closed loop in , the - plane in would be rotated by an angle of together with an -symmetry rotation of the supersymmetric gauge theory along . As such, Omega-deformation in this instance would effect a -automorphism of as we traverse around a closed loop in , where is the moduli space of -instantons on with instanton number ; ; is the rotation generator of the - plane corrected with an appropriate amount of -symmetry to commute with the D6-brane worldvolume supercharges; are the generators of the Cartan subgroup ; and are the corresponding purely imaginary Coulomb moduli of the gauge theory on . In fact, since is also the space of self-dual connections of an -bundle on , and since these self-dual connections correspond to differential one-forms valued in the Lie algebra , Omega-deformation also means that there is a -automorphism of the space of elements of and thus , as we traverse a closed loop in .

Note at this point that can be regarded as a (chiral half of a) WZW model at level 1 on . Since a WZW model on is a bosonic sigma-model on with target the -manifold, according to the last paragraph, it would mean that Omega-deformation would effect a -automorphism of the target space of as we traverse a closed loop in , where . In turn, according to footnote id1, it would mean that in the presence of Omega-deformation, we would have to non-dynamically gauge by .

That being said, notice also from (2.5) that as one traverses around a closed loop in , the - plane in would be rotated by an angle of together with an -symmetry rotation of the supersymmetric gauge theory living on the single D4-brane, i.e., Omega-deformation is also being turned on along the D4-brane. As explained in [7, 5.2], this means that we would in fact have to non-dynamically gauge not by but by .

At any rate, because , where is a Borel subgroup, it would mean that . Also, is never bigger than the Cartan subgroup , where is the subgroup of strictly upper triangular matrices which are nilpotent and traceless whose Lie algebra is . Altogether, this means that our gauged WZW model which corresponds to the coset model , can also be studied as an -gauged WZW model which corresponds to the coset model , where . As physically consistent -gauged WZW models are such that is necessarily a connected subgroup of , it will mean that . Therefore, what we ought to ultimately consider is an -gauged WZW model.

Before we proceed any further, let us make a slight deviation to highlight an important point regarding the effective geometry of . As the simple roots of form a subset of the simple roots of , the level of the affine -algebra ought to be the equal to the level of the affine -algebra [17] which is 1. However, it is clear from our discussion hitherto that the affine -algebra, in particular its level, will depend nontrivially on the Omega-deformation parameters which may or may not take integral values; in other words, its level will not be equal to 1. A resolution to this conundrum is as follows. A deviation of the level of the affine -algebra from 1 would translate into a corresponding deviation of its central charge; since a central charge arises due to an introduction of a macroscopic scale in the 2d system which results from a curvature along  [18], it would mean that Omega-deformation ought to deform the a priori flat into a curved Riemann surface with the same topology – that is, a Riemann sphere with two punctures – such that the anomalous deviation in the central charge and therefore level, can be consistently “absorbed” in the process. Thus, we effectively have , so can be viewed as an fibration of whose fiber has zero radius at the two end points and , where ’’ is a holomorphic coordinate on .

Coming back to our main discussion, it is clear that in the schematic notation of [7, 3.1], our -gauged WZW model can be expressed as the partially gauged chiral CFT

(2.6)

on , where the level would, according to our discussions thus far, necessarily depend on the Omega-deformation parameters and . (, being a purely real number, should not depend on the purely imaginary parameter ).

In sum, the sought-after spacetime BPS states ought to be given by the states of the partially gauged chiral CFT in (2.6), and via [7, B] and [19, eqn. (6.67)], we find that this chiral CFT realizes an affine -algebra obtained from via a quantum Drinfeld-Sokolov reduction. In other words, the states of the chiral CFT would be furnished by a Verma module over the affine -algebra, and the Hilbert space of spacetime BPS states on the RHS of (2.1) can be expressed as

(2.7)

An AGT Correspondence for Pure

Clearly, the physical duality of the compactifications in (2.1) will mean that in (2.3) is equivalent to in (2.7), i.e.,

(2.8)

Moreover, as explained in [7, 5.2], the central charge and level of the affine -algebra are given by

(2.9)

respectively.

In the limit that , it is well-known [20] that

(2.10)

of (2.4) behaves as , whence the 4d Nekrasov instanton partition function can be written as

(2.11)

where ; is some constant; and can be interpreted as the inverse of the observed scale of the space on the LHS of (2.1).

Note that equivariant localization [21] implies that must be endowed with an orthogonal basis , where the ’s denote the fixed points of a -action on . Thus, since is a weighted count of the states in , it would mean that one can write

(2.12)

where , and the dependence on , and arises because the energy level of each state – given by the eigenvalue of the operator which generates translation along in (2.1) whence there is an Omega-deformation twist of the theory along the orthogonal spaces indicated therein – ought to depend on these Omega-deformation parameters.

Notice that (2.12) also means that

(2.13)

where

(2.14)

Here, the state , and is a Poincaré pairing in the sense of [10, 2.6].

Now consider the state

(2.15)

By substituting (2.13) in the RHS of (2.11), and by noting that , one can immediately see that

(2.16)

where . In turn, the duality relation (2.8) would mean that

(2.17)

whence

(2.18)

where . Since the RHS of (2.18) is defined at (see the RHS of (2.11)), and since we have in a common boundary condition at and , and ought to be a state and its dual associated with the puncture at and , respectively (as are the points in where the fiber has zero radius). Furthermore, as the RHS of (2.15) is a sum over states of all possible energy levels, it would mean from (2.17) that is actually a coherent state. This is depicted in fig. 1.

Figure 1: and its -fold cover with the states and at and

At any rate, since we have D6-branes and D4-brane wrapping (see (2.5)), we effectively have an -fold cover of . This is also depicted in fig. 1. Incidentally, is also the Seiberg-Witten curve which underlies ! Moreover, it is by now well-established (see [22] and references therein) that can be described in terms of the algebraic relation

(2.19)

where (for some complex variable ) is a section of ; the ’s are -holomorphic differentials on given by

(2.20)

where ; while for weights of the -dimensional representation of , and for , . This is consistent with our results that we have, on , the following -holomorphic differentials

(2.21)

whence we can naturally identify, up to some constant factor, with . (In fact, a -symmetry of the 4d theory along on the LHS of (2.1) which underlies and , can be identified with the rotational symmetry of ; the duality relation (2.1) then means that the corresponding -charge of the operators that define , ought to match, up to a constant, the conformal dimension of the operators on , which is indeed the case.)

In arriving at the relations (2.8), (2.9), (2.17) and (2.18), and the identification between the ’s and the ’s, we have just derived an AGT correspondence for pure .

2.2. An M-Theoretic Derivation of a 5d Pure AGT Correspondence in the Topological String Limit

Let us now consider the topological string limit in our derivation of the 4d pure AGT correspondence in the last subsection. In this limit, Omega-deformation on the RHS of (2.1) effectively vanishes. According to our discussions in the last subsection, (i) in the I-brane configuration (2.5) would become a flat finite cylinder again; (ii) the partially gauged chiral CFT behind (2.6) would be ungauged. This means that instead of (2.18), we now have

(2.22)

Here, is the energy scale; ; is the Coulomb moduli of the underlying 4d pure theory; , where is an integrable highest weight module over an affine Lie algebra of level 1; is a coherent state generated from the primary state of conformal dimension ; and is the generator of time translations along which propagates the state at one end by a distance to the other end whence it is annihilated by the state , where is the underlying gauge coupling.3


The 4d Nekrasov Instanton Partition Function and Complex Chiral Fermions

Note that if are the Chevalley generators of , where the ’s and ’s correspond to lowering and raising operators, respectively, we can write

(2.23)

where is some normalization constant, and is the state corresponding to the highest weight vector .

As mentioned in the last subsection, we have complex chiral fermions

(2.24)

which realize on . Consequently, by choosing , we can also express (2.23) as (c.f. [11])

(2.25)

whence from (2.22), the 4d Nekrasov instanton partition function would be given by

(2.26)

where is a vacuum state in a standard fermionic Fock space whose energy level is , and are creation and annihilation operators in , respectively, which are constructed out of the chiral fermions. In particular, these operators obey the Heisenberg algebra

(2.27)

because they span the modes of a free chiral boson

(2.28)

where

(2.29)

is a single chiral fermion and its conjugate such that

(2.30)

We also have

(2.31)

Note that by using the commutator , we can, up to a constant factor of , write

(2.32)

where . Moreover, by comparing this with (2.18), we find that we can interpret and as a coherent state and its dual at the and point on .


A 5d AGT Correspondence for Pure in the Topological String Limit

We are now ready to derive a 5d version of (2.32). To this end, first note that the 5d version of (2.18) is just (2.18) itself but at . As such, if is the 5d Nekrasov instanton partition function at , then, according to our discussion following (2.18), we can write

(2.33)

where is a coherent state that has a projection onto a circle of radius in .

The explicit form of can be determined as follows. Firstly, recall that when a quantum system is confined to a space of infinitesimal size, its higher excited states would be decoupled; nevertheless, as we gradually increase the size of the space, we would start to observe these states. Now notice that ; the preceding statement and the last two paragraphs then mean that , where the ’s are constants depending on the indicated parameters.

Secondly, notice that (i) in (2.4) which underlies in (2.33) is invariant under the simultaneous rescalings , where is some real constant; (ii) in (2.32) (which is just the limit of ) is invariant under the simultaneous rescalings ; and (iii) since the underlying worldvolume theory of the M5-branes on the LHS of (2.1) is scale-invariant, it would mean that the physics ought to be invariant under the simultaneous rescalings . Altogether, this means that the ’s should be invariant under the simultaneous rescalings . Thus, the ’s should depend on through the rescaling-invariant combinations and .

Thirdly, notice that in (2.32), we can write the exponent as , where and is a constant depending on the indicated parameters. The preceding two paragraphs then mean that in order to arrive at , we ought to replace this exponent by , where the ’s are distinct constants depending on and .

Fourthly, note that when whence only the mode acts on the vacuum state , must reduce to . On the other hand when , would be a state that has a projection onto a circle of infinite radius whence it would only have zero energy; in other words, must reduce to the vacuum state when .

Last but not least, notice that (2.15) and (2.17) mean that the power of ought to accompany the state of energy level in (which is indeed the case as can be seen from the expansion of four paragraphs earlier). In turn, this means that ought to depend on .

The above five points and a little thought then lead one to conclude that

(2.34)

where

(2.35)

Here, , and

(2.36)

At any rate, note that by using the commutator , we can, up to a constant factor, also write (2.34) as

(2.37)

where

(2.38)

The relations (2.37) and (2.38) agrees with the results of Nekrasov-Okounkov in [11, 7.2.3], as expected.

Comparing (2.34)–(2.35) with (2.32), it is clear that one can interpret as a state in an integrable module over some -dependent affine algebra. One can ascertain this -dependent affine algebra as follows. Firstly, recall that the (chiral) WZW model which underlies on , can be regarded as a bosonic sigma model with worldsheet and target an group manifold. Thus, , which is defined over a point in , would be associated with a point in the space of all points into the target that is the group itself. Similarly, , whose projection is onto a loop in , would be associated with a point in the space of all loops into the target that is the loop group of . Via (2.34), this implies that the 5d theory underlying