Letters in Mathematical Physics

, Volume 109, Issue 3, pp 449–495 | Cite as

Punctures and dynamical systems

  • Falk HasslerEmail author
  • Jonathan J. Heckman


With the aim of better understanding the class of 4D theories generated by compactifications of 6D superconformal field theories (SCFTs), we study the structure of \(\mathcal {N}=1\) supersymmetric punctures for class \(\mathcal {S}_{\varGamma }\) theories, namely the 6D SCFTs obtained from M5-branes probing an ADE singularity. For M5-branes probing a \(\mathbb {C}^2 / \mathbb {Z}_{k}\) singularity, the punctures are governed by a dynamical system in which evolution in time corresponds to motion to a neighboring node in an affine A-type quiver. Classification of punctures reduces to determining consistent initial conditions which produce periodic orbits. The study of this system is particularly tractable in the case of a single M5-brane. Even in this “simple” case, the solutions exhibit a remarkable level of complexity: Only specific rational values for the initial momenta lead to periodic orbits and small perturbations in these values lead to vastly different late-time behavior. Another difference from half BPS punctures of class \(\mathcal {S}\) theories includes the appearance of a continuous complex “zero mode” modulus in some puncture solutions. The construction of punctures with higher-order poles involves a related set of recursion relations. The resulting structures also generalize to systems with multiple M5-branes as well as probes of D- and E-type orbifold singularities.


6D SCFTs Punctures Dynamical systems 4D SCFTs 

Mathematics Subject Classification

37N20 17B08 



We thank F. Apruzzi, J. Marincel, N. Miller and T. Rudelius for helpful discussions. JJH thanks the 2017 Summer Workshop at the Simons Center for Geometry and Physics as well as the Aspen Center for Physics Winter Conference in 2017 on Superconformal Field Theories in \(d\ge 4\), NSF Grant PHY-1066293, for hospitality during part of this work. The work of FH and JJH is supported by NSF CAREER Grant PHY-1756996. The work of FH is also supported by NSF Grant PHY-1620311.


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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of Physics and AstronomyUniversity of PennsylvaniaPhiladelphiaUSA
  2. 2.Department of PhysicsUniversity of North CarolinaChapel HillUSA

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