[Paper Review] BPS Quivers and Spectra of Complete N=2 Quantum Field Theories

Murad Alim, Sergio Cecotti+4 more|arXiv (Cornell University)|Sep 22, 2011

Black Holes and Theoretical PhysicsPhysics and Astronomy41 references78 citations

TL;DR

This paper establishes a systematic method to compute BPS spectra in complete $χ=2$ quantum field theories using BPS quivers derived from triangulations of Riemann surfaces. By encoding the BPS spectrum as a quiver quantum mechanics problem, the authors prove that asymptotically free theories, Argyres-Douglas models, and conformal theories on punctured spheres and tori all possess a chamber with finitely many BPS states, and they provide an algorithmic procedure to compute the full spectrum in these cases.

ABSTRACT

We study the BPS spectra of N=2 complete quantum field theories in four dimensions. For examples that can be described by a pair of M5 branes on a punctured Riemann surface we explain how triangulations of the surface fix a BPS quiver and superpotential for the theory. The BPS spectrum can then be determined by solving the quantum mechanics problem encoded by the quiver. By analyzing the structure of this quantum mechanics we show that all asymptotically free examples, Argyres-Douglas models, and theories defined by punctured spheres and tori have a chamber with finitely many BPS states. In all such cases we determine the spectrum.

Motivation & Objective

• To develop a systematic framework for computing BPS spectra in complete $χ=2$ quantum field theories.
• To establish a direct correspondence between triangulations of Gaiotto's Riemann surface and the BPS quiver and superpotential of the theory.
• To identify and characterize chambers with finitely many BPS states across a broad class of $χ=2$ theories.
• To provide an algorithmic method for computing the BPS spectrum in such finite chambers using quiver quantum mechanics.
• To extend the quiver formalism to include self-folded triangles and exceptional theories beyond the Gaiotto construction.

Proposed method

• Construct BPS quivers from ideal triangulations of the Riemann surface via D3-brane worldvolume quantum mechanics on special Lagrangian cycles.
• Derive the superpotential from the triangulation by assigning cyclic terms to internal triangles, including special rules for self-folded triangles and punctures.
• Use quiver mutation to relate different chambers and track BPS spectrum changes across moduli space.
• Apply quiver representation theory and quantum mechanics to solve for BPS states in the finite chamber.
• Use the triangulation-glueing rule to combine quivers for composite surfaces, enabling recursive construction of spectra.
• Extend the formalism to include self-folded triangles via auxiliary nodes and modified arrow rules, ensuring consistency with flip operations.

Experimental results

Research questions

• RQ1 Can the BPS spectrum of complete $χ=2$ theories be computed algorithmically using quiver data derived from Riemann surface triangulations?
• RQ2 Which classes of $χ=2$ theories admit a chamber with a finite number of BPS states?
• RQ3 How do quiver mutations correspond to physical chamber transitions in moduli space?
• RQ4 What is the role of self-folded triangles in the quiver construction, and how can they be consistently handled?
• RQ5 Can the BPS quiver and superpotential be systematically derived for exceptional complete theories outside the Gaiotto construction?

Key findings

• All asymptotically free $χ=2$ theories possess a chamber with finitely many BPS states, and their spectra can be computed via the quiver method.
• Argyres-Douglas theories on punctured spheres and tori also admit finite chambers, and their BPS spectra are fully determined by the algorithm.
• For theories defined by punctured spheres with $n \geq 4$ punctures and tori with $n \geq 2$ punctures, the BPS spectrum is explicitly computed and found to be finite in a specific chamber.
• The quiver construction is extended to include self-folded triangles via auxiliary nodes and modified arrow rules, preserving consistency with quiver mutations.
• All eleven exceptional complete theories (excluding one) are shown to admit a finite chamber, and their superpotentials are constructed explicitly.
• The method provides a complete, algorithmic procedure to compute the BPS spectrum in finite chambers for an infinite class of $χ=2$ theories.