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We consider (2,0) theory on a manifold M_6 that is a fibration of a spatial S^1 over some five-dimensional base manifold M_5. Initially, we study the free (2,0) tensor multiplet which can be described in terms of classical equations of motion in six dimensions. Given a metric on M_6 the low energy effective theory obtained through dimensional reduction on the circle is a Maxwell theory on M_5. The parameters describing the local geometry of the fibration are interpreted respectively as the metric on M_5, a non-dynamical U(1) gauge field and the coupling strength of the resulting low energy Maxwell theory. We derive the general form of the action of the Maxwell theory by integrating the reduced equations of motion, and consider the symmetries of this theory originating from the superconformal symmetry in six dimensions. Subsequently, we consider a non-abelian generalization of the Maxwell theory on M_5. Completing the theory with Yukawa and phi^4 terms, and suitably modifying the supersymmetry transformations, we obtain a supersymmetric Yang-Mills theory which includes terms related to the geometry of the fibration.
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In this note we consider the gauge field equation of motion for the dimensional reduction of the (2,0) tensor multiplet on singular circle fibrations. The fibrations are characterized by the corresponding U(1) action having a codimension four fixed point locus W. Along W, the dimensional reduction of the (2,0) receives a modification described by a WZW model. We consider the emergence of the additional degrees of freedom through the topological term in the action, which in addition to the gauge field strength involves the U(1) connection of the space-time fibration. We also consider the Taub-NUT space as a simple example of a singular fibration, and in particular consider spherically symmetric solutions for the field strength.
We study twisted circle compactification of 6d $(2,0)$ SCFTs to 5d $mathcal{N} = 2$ supersymmetric gauge theories with non-simply-laced gauge groups. We provide two complementary approaches towards the BPS partition functions, reflecting the 5d and 6d point of view respectively. The first is based on the blowup equations for the instanton partition function, from which in particular we determine explicitly the one-instanton contribution for all simple Lie groups. The second is based on the modular bootstrap program, and we propose a novel modular ansatz for the twisted elliptic genera that transform under the congruence subgroups $Gamma_0(N)$ of $text{SL}(2,mathbb{Z})$. We conjecture a vanishing bound for the refined Gopakumar-Vafa invariants of the genus one fibered Calabi-Yau threefolds, upon which one can determine the twisted elliptic genera recursively. We use our results to obtain the 6d Cardy formulas and find universal behaviour for all simple Lie groups. In addition, the Cardy formulas remain invariant under the twist once the normalization of the compact circle is taken into account.
We discuss how D=5 maximally supersymmetric Yang-Mills theory (MSYM) might be used to study or even to define the (2,0) theory in six dimensions. It is known that the compactification of (2,0) theory on a circle leads to D=5 MSYM. A variety of arguments suggest that the relation can be reversed, and that all of the degrees of freedom of (2,0) theory are already present in D=5 MSYM. If so, this relation should have consequences for D=5 SYM perturbation theory. We explore whether it might imply all orders finiteness, or else an unusual relation between the cutoff and the gauge coupling. S-duality of the reduction to D=4 may provide nonperturbative constraints or tests of these options.
We consider a twisted version of the abelian $(2,0)$ theory placed upon a Lorenzian six-manifold with a product structure, $M_6=C times M_4 $. 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 $M_4$ 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 $M_4$. 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 $M_4$ 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.
A $Q$-exact off-shell action is constructed for twisted abelian (2,0) theory on a Lorentzian six-manifold of the form $M_{1,5} = Ctimes M_4$, where $C$ is a flat two-manifold and $M_4$ is a general Euclidean four-manifold. The properties of this formulation, which is obtained by introducing two auxiliary fields, can be summarised by a commutative diagram where the Lagrangian and its stress-tensor arise from the $Q$-variation of two fermionic quantities $V$ and $lambda^{mu u}$. This completes and extends the analysis in [arXiv:1311.3300].
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