The trouble with twisting (2,0) theory

Louise Anderson 0 Hampus Linander 0 0 Department of Fundamental Physics, Chalmers University of Technology , S-412 96 Goteborg, Sweden We consider a twisted version of the abelian (2, 0) theory placed upon a Lorentzian six-manifold with a product structure, M6 = C M4. This is done by an investigation of the free tensor multiplet on the level of equations of motion, where the problem of its formulation in Euclidean signature is circumvented by letting the time-like direction lie in the two-manifold C and performing a topological twist along M4 alone. A compactification on C is shown to be necessary to enable the possibility of finding a topological field theory. The hypothetical twist along a Euclidean C is argued to amount to the correct choice of linear combination of the two supercharges scalar on M4. This procedure is expected and conjectured to result in a topological field theory, but we arrive at the surprising conclusion that this twisted theory contains no Q-exact and covariantly conserved stress tensor unless M4 has vanishing curvature. This is to our knowledge a phenomenon which has not been observed before in topological field theories. In the literature, the setup of the twisting used here has been suggested as the origin of the conjectured AGT-correspondence, and our hope is that this work may somehow contribute to the understanding of it. 1 Introduction 2.1 Details of reinterpreting the fields 2.2 Some useful relations 2.3 Compactifying on C The theory after twisting 3.1 Equations of motion 3.2 Supersymmetry 4.1 Actions 4.2 Ansatz and modifications 4.3 Q-exactness The case when M4 is curved 5.1 Covariant conservation of T in the curved case Conclusion and outlook Introduction This work is an investigation of the topological twisting of the (2, 0) theory which has been suggested to be relevant in the explanation of the origin of the AGT-conjecture. Herein, the simpler model of the free tensor multiplet is considered, and we find that the resulting twisted theory exhibits some curious, undesirable properties. The most severe of these is the lack of any satisfactory formulation of a stress tensor. This surprising result will be clear eventually, but let us first start at the very beginning. The theory known as (2, 0) theory [1, 2] is a six-dimensional superconformal theory that continue to resist attempts at unraveling its mysteries. One way to obtain information about the theory is to look at its different compactifications. For example, when compactified on a circle it gives rise to five-dimensional maximally supersymmetric Yang-Mills theory [3]. Recently a whole class of four-dimensional gauge theories have been constructed in this way by compactifying (2, 0) theory on a two-dimensional Riemann surface with possible defects [46]. This class of theories is sometimes referred to as class S in the literature [7, 8]. The way these theories are obtained through compactification has led to a conjecture about the relation of certain objects in four-dimensional- and two-dimensional theories, the so-called AGT correspondence [9]. More specifically, this correspondence states that the correlation functions in twodimensional Liouville theory are related to the Nekrasov partition function [10, 11] of certain N = 2 superconformal gauge theories in four dimensions. One natural way to derive it [9, 12, 13] would be to link it to a certain geometric setup in (2, 0) theory, where the spacetime is taken to be a product of a two-dimensional- and a four-dimensional manifold. In such a setting, compactifications could either be carried out on the two- or on the fourmanifold, after which one could search for protected quantities which have survived the compactification. A relation should then exist between the protected quantities of both compactifications. However, one is here faced with the great challenge of a lack of any satisfactory definition of (2, 0) theory that would permit such detailed calculations. While this is indeed true for the full, interacting (2, 0) theory, this is not the whole story for the abelian version. Here, a classical formulation in terms of equations of motion exists. Moreover, it is important to notice that for a general background all supersymmetry will be broken and such a situation cannot be expected to shed any light on the AGTcorrespondence. In order to preserve some supersymmetry, one must first perform a topological twisting [14]. In a case where the six-manifold has the product structure mentioned previously, i.e. M6 = C M4, and M6 is of Euclidean signature, (thus the holonomy groups of both C and M4 are compact), such a theory admits a unique twisting, which has been claimed [12] to be analogous to the Donaldson-Witten twist of four-dimensional N = 2 Yang-Mills theory [14]. In the literature (see for example [12, 15]), it has been stated that the six-dimensional twist of would result in a theory which would be topological along M4 and holomorphic along C [16]. Herein, the behaviour of the Lorentzian theory (especially along the four-manifold), is investigated explicitly by computing a stress tensor. Our result seems to indicate that this Lorentzian twist may as conjectured coincide with the Donaldson-Witten twist on a flat background, however it does not seem to be true in the general case. A more detailed discussion of this shall be presented towards the end of this work. However, the elusive side of (2, 0) theory once again comes back to bite us here, since not even the abelian version of this theory has a satisfactory description on a Euclidean sixmanifold, but rather only on a six-manifold with Minkowski signature. In such a situation, the holonomy group would be non-compact, and a topological twisting that results in a scalar supercharge cannot be performed. If the light-like direction is taken to lie in C, one may still obtain supercharges that are scalars on M4 by a twisting procedure. One of these charges has properties that would make it scalar along C as well, were we in the Euclidean scenario. In this work, this is the supercharge we will consider, and the behaviour of the theory under it is the subject of investigation. The final conclusion is that, on a general M4, the stress tensor of the theory cannot be both Q-exact and conserved, and the theory is thus not topological in the traditional sense. The outline of this work is as follows: in section 2 we describe the twisting procedure giving rise to the supercharge that is scalar on M4 and give a detailed description of the field content in this new, twisted theory. Section 3 deals with the equations of motion as well as the supersymmetry transformations of the twisted theory. In section 4, a stress tensor is computed in the flat case which is shown to have all desired properties. An attempt at generalising this to a general M4 is made, and any Q-exact stress tensor is shown to not be covariantly conserved. It is also shown that no modifications to (...truncated)