2010/12/31 by E. A. Ivanov, Evgeny Ivanov, Andrei Smilga +1 · 3 citations
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Dirac operator #Holomorphic function #Manifold (fluid mechanics) #Mathematical physics #Mathematics #Noncommutative and Quantum Gravity Theories #Operator (biology) #Physics #Pure mathematics #Quantum #Quantum Mechanics and Non-Hermitian Physics #Quantum mechanics #Quantum statistical mechanics #Supercharge #Supersymmetric quantum mechanics #Supersymmetry #Theoretical physics #hep-th #math-ph #math.MP #msc:32Qxx #msc:81Q60 #msc:81T60
paper · pdf · doi:10.1142/s0217751x12300244
published as IJMP A 27 (2012) 1230024 (30 pages) · 0 + 30 pages, essential revision, new comments and refs. added, typos corrected, published version
openalex publication_date 2012/10/04 · arxiv created 2012/10/16 · arxiv updated 2012/10/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We explore a simple [Formula: see text] supersymmetric quantum mechanics (SQM) model describing the motion over complex manifolds in external gauge fields. The nilpotent supercharge Q of the model can be interpreted as a (twisted) exterior holomorphic derivative, such that the model realizes the twisted Dolbeault complex. The sum [Formula: see text] can be interpreted as the Dirac operator: the standard Dirac operator if the manifold is Kähler and the Dirac operator involving certain particular extra torsions for a generic complex manifold. Focusing on the Kähler case, we give new simple physical proofs of the two mathematical facts: (i) the equivalence of the twisted Dirac and twisted Dolbeault complexes and (ii) the Atiyah–Singer theorem.