Surface Defect Indices and 2d-4d BPS States

Abstract

We conjecture a formula for the Schur index of four-dimensional $\mathcal{N}=2$ theories coupled to $(2,2)$ surface defects in terms of the $2d$-$4d$ BPS spectrum in the Coulomb phase of the theory. The key ingredient in our conjecture is a refined $2d$-$4d$ wall-crossing invariant, which we also formulate. Our result intertwines recent conjectures expressing the four-dimensional Schur index in terms of infrared BPS particles, with the Cecotti-Vafa formula for limits of the elliptic genus in terms of two-dimensional BPS solitons. We extend our discussion to framed $2d$-$4d$ BPS states, and use this to demonstrate a general relationship between surface defect indices and line defect indices. We illustrate our results in the example of $SU(2)$ super Yang-Mills coupled to the ${\mathbb{CP}}^1$ sigma model defect.

50 pages, 1 figure

References in corpus (7)

• The 4d Superconformal Index from q-deformed 2d Yang-Mills
• Infrared Computations of Defect Schur Indices
• Chiral Algebras for Trinion Theories
• Bubbling Surface Operators And S-Duality
• Framed BPS States, Moduli Dynamics, and Wall-Crossing
• The BPS Spectrum Generator In 2d-4d Systems
• ADE Spectral Networks and Decoupling Limits of Surface Defects

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• Schur sector of Argyres-Douglas theory and $W$-algebra
• On the Correspondence between Surface Operators in Argyres-Douglas Theories and Modules of Chiral Algebra
• Four-dimensional Lens Space Index from Two-dimensional Chiral Algebra
• The Coulomb and Higgs Branches of $\mathcal{N}=1$ Theories of Class ${\mathcal{S}}_k$
• Testing Macdonald Index as a Refined Character of Chiral Algebra
• Closed form fermionic expressions for the Macdonald index
• Abelian mirror symmetry of $\mathcal{N}=(2,2)$ boundary conditions
• An infrared bootstrap of the Schur index with surface defects
• Deconstructing Defects
• A Family of Vertex Algebras from Argyres-Douglas Theory
• Orbifold Schur Index and IR formula
• Surface Operators in Superspace