On the $A_{\infty}$-Category of a Holomorphic Moment Map
Ahsan Z. Khan

TL;DR
This paper investigates the structure of the $A_{ abla}$-category of A-branes in a Landau-Ginzburg model with a hyperK"ahler target, revealing semi-simplicity for generic complex structures and stability of the category at exceptional points.
Contribution
It characterizes the $A_{ abla}$-category for hyperK"ahler manifolds with a $U(1)$ isometry, showing semi-simplicity in generic cases and no instanton corrections at exceptional structures.
Findings
The $A_{ abla}$-category is semi-simple for generic complex structures.
At exceptional structures, the category remains free of instanton corrections.
Explicit examples are provided for the cotangent bundle of the projective line.
Abstract
Let be a hyperK\"{a}hler manifold equipped with a hyperK\"{a}hler isometry, and let be a complex structure on . In this note, we study the -category of A-branes for the Landau-Ginzburg model with target space , and superpotential being the -holomorphic moment map. We show that if is a generic complex structure, the -category is semi-simple. For exceptional complex structures, though typically not semi-simple, the category still has no instanton corrections. We illustrate the -category at both generic and exceptional loci when is the cotangent bundle of the projective line.
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Taxonomy
TopicsBlack Holes and Theoretical Physics · Homotopy and Cohomology in Algebraic Topology · Advanced Algebra and Geometry
