[Paper Review] N=2 Quantum Field Theories and Their BPS Quivers

Murad Alim, Sergio Cecotti+4 more|arXiv (Cornell University)|Dec 16, 2011

Algebraic structures and combinatorial modelsMathematics71 references132 citations

TL;DR

This paper establishes a systematic correspondence between four-dimensional $χ=2$ quantum field theories and their BPS quivers, demonstrating that the BPS spectrum can be computed via quiver quantum mechanics and quiver mutations. The key contribution is a novel 'mutation method' that uses quantum mechanical dualities—encoded as quiver mutations—to fully determine the BPS spectrum across different regions of moduli space, validated in examples including $SU(N)$ SYM and Gaiotto-type theories.

ABSTRACT

We explore the relationship between four-dimensional N=2 quantum field theories and their associated BPS quivers. For a wide class of theories including super-Yang-Mills theories, Argyres-Douglas models, and theories defined by M5-branes on punctured Riemann surfaces, there exists a quiver which implicitly characterizes the field theory. We study various aspects of this correspondence including the quiver interpretation of flavor symmetries, gauging, decoupling limits, and field theory dualities. In general a given quiver describes only a patch of the moduli space of the field theory, and a key role is played by quantum mechanical dualities, encoded by quiver mutations, which relate distinct quivers valid in different patches. Analyzing the consistency conditions imposed on the spectrum by these dualities results in a powerful and novel mutation method for determining the BPS states. We apply our method to determine the BPS spectrum in a wide class of examples, including the strong coupling spectrum of super-Yang-Mills with an ADE gauge group and fundamental matter, and trinion theories defined by M5-branes on spheres with three punctures.

Motivation & Objective

• To establish a one-to-one correspondence between $\mathcal{N}=2$ quantum field theories and their associated BPS quivers.
• To understand how quiver structures encode physical properties such as flavor symmetries, gauging, decoupling limits, and dualities.
• To develop a systematic method for computing the BPS spectrum across different patches of moduli space.
• To demonstrate that quantum mechanical dualities, realized as quiver mutations, impose strong consistency conditions that fully determine the BPS spectrum.

Proposed method

• Construct BPS quivers from the BPS spectrum using quiver quantum mechanics, where nodes represent primitive BPS states and arrows represent interactions.
• Use quiver representations and holomorphic descriptions to analyze wall-crossing phenomena and stability conditions.
• Apply quiver mutation as a one-dimensional analog of Seiberg duality to relate distinct quivers valid in different regions of moduli space.
• Utilize the mutation method: consistency of BPS spectra under quiver mutations acts as a powerful constraint to determine the full spectrum.
• Apply the method to concrete examples, including $SU(N)$ gauge theories with matter and Gaiotto-type theories from M5-branes on punctured Riemann surfaces.
• Use glueing and splitting rules for trinion theories to build and analyze higher-genus theories.

Experimental results

Research questions

• RQ1 How can BPS quivers be systematically constructed for a wide class of $\mathcal{N}=2$ quantum field theories?
• RQ2 What is the role of quiver mutations in encoding quantum mechanical dualities and relating different BPS spectra across moduli space patches?
• RQ3 How do flavor symmetries, gauging, and decoupling limits manifest in the quiver formalism?
• RQ4 To what extent can the BPS spectrum be uniquely determined by consistency conditions arising from quiver mutations?
• RQ5 Can the mutation method be applied to compute the strong-coupling BPS spectrum of $SU(N)$ gauge theories with fundamental matter?

Key findings

• The BPS spectrum of $SU(3)$ pure super-Yang-Mills theory at strong coupling is fully determined via the mutation method, confirming known results through a new algebraic framework.
• For $SU(N)$ gauge theories with fundamental matter, the mutation method successfully computes the BPS spectrum, including the strong-coupling regime.
• The $E_6$ Minahan-Nemeschansky theory coupled to $SU(2)$ with a fundamental flavor is shown to be dual to $SU(3)$ with six fundamental flavors via quiver mutation, confirming a strong-coupling duality.
• The method reproduces the BPS spectrum of trinion theories (e.g., $\mathcal{T}_3$) by constructing quivers from glueing rules and verifying consistency under mutations.
• The analysis shows that quiver mutations encode quantum monodromy, and that the spectrum is invariant under mutation when consistency conditions are satisfied.
• The paper demonstrates that the BPS quiver provides a unique and complete characterization of the theory on a patch of moduli space, with mutations enabling global spectrum reconstruction.