Punctures and Dynamical Systems

TL;DR

This work reframes N=1 punctures of class S_Gamma theories, arising from M5 branes probing ADE singularities, as a discrete dynamical system on an affine A-type quiver. By analyzing the N=1 case, it shows puncture classification reduces to selecting initial data that yield periodic orbits, with momentum values constrained to rational fractions p = a/b and b not divisible by 4. The study reveals a continuous zero mode for eternal punctures, intricate dependence on initial conditions, and a recursive construction for higher order poles. It also outlines how these structures generalize to higher rank and to D and E type orbifolds, and discusses implications for 4D N=1 vacua and potential holographic duals. The results provide a concrete, arithmetic-driven framework for cataloging punctures in 6D to 4D compactifications with broad applicability to related theories.

Abstract

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 N = 1 supersymmetric punctures for class S_Gamma theories, namely the 6D SCFTs obtained from M5-branes probing an ADE singularity. For M5-branes probing a C^2 / 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 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.

Figures (7)

• Figure 1: A-type quiver with the direction along the quiver interpreted as the time of the dynamical system.
• Figure 2: Approximated maxima and minima of $x(i)$ arise for all $i$ where $p(i)=p$ holds. In the plotted example, we have $x(1) = 1/3$, and $p(1) = x(1) + 5$, with $p = 1/3$. For this case, we realize an eternal puncture where no $x(i)$ vanishes, and the total length of the orbit is $k = 60$.
• Figure 3: A generic feature of initial conditions with $p=a/b$ and $b\text{ mod }4=0$ is that the enveloping curves of $x(i)$ and $p(i)$ grow over time. Therefore all examples we have studied numerically do not seem to close into a periodic orbit. Here, this feature is depicted for the initial conditions $p(1)=5/4$, $x(1)=1/4$ and the first 1000 $x(i)$/$p(i)$ of the time evolution.
• Figure 4: Three iterations of the initial line segment $[0 , \infty )$ which give rise to bifurcating behavior in the initial condition $l$.
• Figure 5: Periodic orbits which represent the 1/2 BPS punctures of the $A_k$ theory for $k\le 10$ if one link between the quiver nodes is turned off. Whether a node is filled or not distinguishes between short and long orbits. Each link between two nodes carries the sequence of $\mathop{\rm sgn}\nolimits x(i)$ which has to be appended to the shorter orbit in order to obtain the longer one.

Theorems & Definitions (17)

• Lemma 1
• proof : Proof of Lemma \ref{['lemma1']}
• Lemma 2
• proof : Proof of Lemma \ref{['lemma2']}
• Lemma 3
• proof : Proof of Lemma \ref{['lemma3']}
• Definition 1
• Lemma 4
• proof : Proof of Lemma \ref{['lemma4']}
• Definition 2