Effective gravitational action for 2D massive Majorana fermions on arbitrary genus Riemann surfaces

• Published in: The journal of high energy physics Vol. 2023; no. 11; pp. 194 - 100
• Main Authors: Namuduri, Manojna, Bilal, Adel
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 27.11.2023; Springer Nature B.V; Springer; SpringerOpen
• Subjects: 2D Gravity; Classical and Quantum Gravitation; Effective Field Theories; Elementary Particles; Fermions; Green's functions; High energy physics; High Energy Physics - Theory; Mathematical Physics; Models of Quantum Gravity; Operators (mathematics); Physics; Physics and Astronomy; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Regular Article - Theoretical Physics; Relativity Theory; Riemann surfaces; Scalars; String Theory; 2D Gravity; Effective Field Theories; Models of Quantum Gravity; gravitation: action; zero mode; expansion: mass; dimension: 2; operator: Dirac; fermion: Majorana; Green function; Riemann surface; background: metric; heat kernel; conformal; Liouville; Euclidean; zeta function; higher-order
• ISSN: 1029-8479; 1126-6708; 1029-8479
• DOI: 10.1007/JHEP11(2023)194

A bstract We explore the effective gravitational action for two-dimensional massive Euclidean Majorana fermions in a small mass expansion, continuing and completing the study initiated in a previous paper [ 1 ]. We perform a detailed analysis of local zeta functions, heat kernels, and Green’s functions of the Dirac operator on arbitrary Riemann surfaces. We obtain the full expansion of the effective gravitational action to all orders in m 2 . For genus one and larger, this requires the understanding of the role of the zero-modes of the (massless) Dirac operator which is worked out. Besides the Liouville action, at order m 0 , which only involves the background metric and the conformal factor σ , the various contributions to the effective gravitational action at higher orders in m 2 can be expressed in terms of integrals of the renormalized Green’s function at coinciding points of the squared (massless) Dirac operator, as well as of higher Green’s functions. In particular, at order m 2 , these contributions can be re-written as a term ∫ e 2 σ σ characteristic of the Mabuchi action, much as for 2D massive scalars, as well as several other terms that are multi-local in the conformal factor σ and involve the Green’s functions of the massless Dirac operator and the renormalized Green’s function, but for the background metric only, and certain area-like parameters related to the zero-modes.