Advances in Theoretical and Mathematical Physics

Volume 17 (2013)

Number 2

Framed BPS states

Pages: 241 – 397



Davide Gaiotto (School of Natural Sciences, Institute for Advanced Study, Princeton, New Jersey, U.S.A.)

Gregory W. Moore (NHETC and Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey, U.S.A.)

Andrew Neitzke (Department of Mathematics, University of Texas at Austin, Texas, U.S.A.)


We consider a class of line operators in $d = 4, \mathcal{N} = 2$ supersymmetric field theories, which leave four supersymmetries unbroken. Such line operators support a new class of BPS states which we call “framed BPS states.” These include halo bound states similar to those of $d = 4, \mathcal{N} = 2$ supergravity, where (ordinary) BPS particles are loosely bound to the line operator. Using this construction, we give a new proof of the Kontsevich-Soibelman wall-crossing formula (WCF) for the ordinary BPS particles, by reducing it to the semiprimitive WCF. After reducing on $S^1$, the expansion of the vevs of the line operators in the IR provides a new physical interpretation of the “Darboux coordinates” on the moduli space M of the theory. Moreover, we introduce a “protected spin character” (PSC) that keeps track of the spin degrees of freedom of the framed BPS states. We show that the generating functions of PSCs admit a multiplication, which defines a deformation of the algebra of holomorphic functions on $\mathcal{M}$. As an illustration of these ideas, we consider the sixdimensional (2, 0) field theory of $A_1$ type compactified on a Riemann surface $\mathcal{C}$. Here, we show (extending previous results) that line operators are classified by certain laminations on a suitably decorated version of $\mathcal{C}$, and we compute the spectrum of framed BPS states in several explicit examples. Finally, we indicate some interesting connections to the theory of cluster algebras.

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