Semiclassical framed BPS states

Abstract

We provide a semiclassical description of framed BPS states in fourdimensional N = 2 super Yang-Mills theories probed by 't Hooft defects, in terms of a supersymmetric quantum mechanics on the moduli space of singular monopoles. Framed BPS states, like their ordinary counterparts in the theory without defects, are associated with the L 2 kernel of certain Dirac operators on moduli space, or equivalently with the L 2 cohomology of related Dolbeault operators. The Dirac/Dolbeault operators depend on two Cartan-valued Higgs vevs. We conjecture a map between these vevs and the SeibergWitten special coordinates, consistent with a one-loop analysis and checked in examples. The map incorporates all perturbative and nonperturbative corrections that are relevant for the semiclassical construction of BPS states, over a suitably defined weak coupling regime of the Coulomb branch. We use this map to translate wall crossing formulae and the no-exotics theorem to statements about the Dirac/Dolbeault operators. The no-exotics theorem, concerning the absence of nontrivial SU(2) R representations in the BPS spectrum, implies that the kernel of the Dirac operator is chiral, and further translates into a statement that all L 2 cohomology of the Dolbeault operator is concentrated in the middle degree. Wall crossing formulae lead to detailed predictions for where the Dirac operators fail to be Fredholm and how their kernels jump. We explore these predictions in nontrivial examples. This paper explains the background and arguments behind the results announced in the short note [1].

Index-like theorems from line defect vevs

“…In this chamber, the coordinate transformation is of the form a = e a = f 1 − f 2 1 − 1 , f 1 = in terms of the spectral network coordinates in the c 1 chamber. Note that this matches the direct computation from spectral networks (6.7) [40]. where P = 1 2 diag(2, −2) and ζ ∈ c 1 .…”

“…As shown in [17,40], the expectation value of the 't Hooft defect of minimal charge in terms of Darboux coordinates in the chamber c n is given by 25…”

Quantum line defects and refined BPS spectra

“…UV line defects. Theories of class S admit BPS line operators [22,31]. Quantum field theories contain far more information than what is captured by local correlators.…”

“…When we consider the theory on R 3 × S 1 R with the defect wrapping the S 1 , the vev of the line operator can be expanded [22,31]…”

“…String theory associates to ℘ a UV line operator in a four dimensional quantum field theory with N = 2 on S 1 R × R 3 , for example via the low energy description of a system of M5-M2 branes. In the IR such a line operator acquires an expectation value, which is a function on the moduli space of vacua of the theory [22,31] L ℘ = γ∈Γ Ω(γ; ℘) X γ (1.1)…”