Toda theory from six dimensions

Journal of High Energy Physics, Dec 2017

We describe a compactification of the six-dimensional (2,0) theory on a foursphere which gives rise to a two-dimensional Toda theory at long distances. This construction realizes chiral Toda fields as edge modes trapped near the poles of the sphere. We relate our setup to compactifications of the (2,0) theory on the five and six-sphere. In this way, we explain a connection between half-BPS operators of the (2,0) theory and twodimensional W-algebras, and derive an equality between their conformal anomalies. As we explain, all such relationships between the six-dimensional (2,0) theory and Toda field theory can be interpreted as statements about the edge modes of complex Chern-Simons on various three-manifolds with boundary.

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Toda theory from six dimensions

HJE Toda theory from six dimensions Clay Cordova 0 2 Daniel L. Ja eris 0 1 Princeton 0 U.S.A. 0 Cambridge 0 U.S.A. 0 0 ric Gauge Theory , Topological Field Theories 1 Je erson Physical Laboratory, Harvard University 2 School of Natural Sciences, Institute for Advanced Study We describe a compacti cation of the six-dimensional (2,0) theory on a foursphere which gives rise to a two-dimensional Toda theory at long distances. This construction realizes chiral Toda elds as edge modes trapped near the poles of the sphere. We relate our setup to compacti cations of the (2,0) theory on the this way, we explain a connection between half-BPS operators of the (2,0) theory and twodimensional W-algebras, and derive an equality between their conformal anomalies. As we explain, all such relationships between the six-dimensional (2,0) theory and Toda eld theory can be interpreted as statements about the edge modes of complex Chern-Simons on various three-manifolds with boundary. E ective Field Theories; Field Theories in Higher Dimensions; Supersymmet- 1 Introduction 1.1 1.2 1.3 Further applications 2 The (2; 0) theory on a squashed S4 2.1 Weyl rescaling and background elds 3 Complex Chern-Simons and Nahm poles 3.1 3.2 Complex Chern-Simons theory from (2,0) on S`3=Zk Boundary conditions 3.3 Fluctuation modes 4 The constrained WZW model and Toda 4.1 The complex WZW model 4.2 Current constraints and complex Toda 5 Duality of complex Toda and ParaToda + coset 5.1 Additional evidence at small k 6 Half-BPS operators and W-algebras 6.1 6.2 The superconfomal index and W-characters W-algebras and conformal anomalies In this paper we construct a compacti cation of the six-dimensional (2; 0) conformal eld theory on S4. We demonstrate that the resulting low-energy e ective action in two dimensions is a complexi ed Toda eld theory. Thus, we obtain a direct derivation of the Alday-Gaiotto-Tachikawa correspondence [1{3]. Previous approaches to this relationship have appeared in [4{12]. We further apply our construction to relate various properties of the Toda theory, such as a its spectrum of chiral operators and associated central charge, to limits of the six-dimensional superconformal index and c-type Weyl anomaly as anticipated by the results of [13, 14]. { 1 { 1.1 The six-dimensional (2; 0) theories [15{17] are maximally supersymmetric and conformally invariant. They are labelled by an ADE Lie algebra g. See for instance [18{20] for a recent survey of their general properties from various perspectives. Starting from these theories, a large class of lower-dimensional supersymmetric theories may be constructed by twisted compacti cation on manifolds of various dimensions [21{26]. A widely studied example results in four-dimensional N = 2 theories. These theories are labelled Tg( ), where is the compacti cation Reimann surface (possibly with punctures). This geometric perspective on four-dimensional eld theories yields insight into many of their physical properties such as non-trivial dualities [22], moduli spaces [21], BPS particles [ 23, 27, 28 ], and spectrum of local operators [29, 30]. The AGT correspondence studied here is a dramatic example in this vein. In this case, the four-dimensional observable of interest is the S4 partition function computed in [31, 32]. This partition function takes the form of an integral over Coulomb branch parameters u. Schematically, for any gauge The parameters k and s are determined by the geometry of our compacti cation. Speci cally, k which is an integer, occurs when we include the orbifold singularity S4 ! S4=Zk, and s (which is either real or pure imaginary) may be activated by deforming the metric { 2 { theory one has of the sphere. takes the form S = q Z 8 Z ZS4 = du e S(u) jF (u)j2 ; where F (u) are generating functions of pointlike instantons [33] localized at the two poles In the context of the theories Tg( ) this partition function is reinterpreted as a twodimensional observable on . The factors F (u) are conformal blocks for a Toda eld theory on , where u plays the role of the exchanged operator dimension, and external vertex operators arise from codimension two defects of the six-dimensional theory which puncture the sphere. The integration over u produces a Toda correlation function. The six-dimensional origin of the theories Tg( ) suggests a natural explanation for the correspondence: the six-dimensional (2; 0) theory reduced on S4 is the Toda eld theory. Then, the correspondence of observables arises from a commuting the order of compacti cation on and S4. Our main result is to carry out the dimensional reduction on S4 and explain how Toda theory emerges. In fact, in our construction we will naturally encounter a complexi ed Toda eld theory. The dynamical variables consist of r complex bosons where r is the rank of g. The action q~ Z 8 i i where Cij is the Cartan matrix of g and the coupling constant q is expressed as q = k + is ; q~ = k is : (1.1) (...truncated)


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Clay Córdova, Daniel L. Jafferis. Toda theory from six dimensions, Journal of High Energy Physics, 2017, pp. 106, Volume 2017, Issue 12, DOI: 10.1007/JHEP12(2017)106