R^{3} index for four-dimensional (N)=2 field theories

Area of Science:

• Theoretical High Energy Physics
• Supersymmetry and String Theory
• Quantum Field Theory

Background:

• N=2 supersymmetric theories on R^{3,1} exhibit BPS indices (Ω(γ,u)) that are discontinuous across walls of marginal stability.
• These discontinuities arise from the decay of supersymmetric bound states in the Coulomb branch parameter space.
• The familiar Witten index (Tr(-1)^{F}e^{-βH}) is a key quantity in such theories.

Purpose of the Study:

• To introduce and analyze a generalized supersymmetric index (I) that includes contributions from 1/2-BPS states.
• To propose a formula for this index (I) that is continuous across walls of marginal stability.
• To provide a universal prediction for the contributions of multiparticle states to the index (I).

Main Methods:

• Consideration of a generalized supersymmetric index (I) incorporating 1/2-BPS states.
• Conjecturing a formula for (I) based on BPS indices (Ω(γ,u)), drawing inspiration from N=2 string vacua.
• Analysis of the index (I) on the moduli space after dimensional reduction on a circle.

Main Results:

• A conjectured formula for the supersymmetric index (I) is proposed, which remains continuous across walls of marginal stability.
• This formula correctly accounts for contributions from single-particle states at large β.
• The index (I) is shown to be a function on the moduli space, related to hyperkähler geometry.

Conclusions:

• The proposed index (I) offers a smooth generalization of the Witten index across walls of marginal stability.
• This provides a universal framework for understanding multiparticle state contributions in N=2 supersymmetric theories.
• The index (I) has deep connections to the hyperkähler metric and hyperholomorphic connection of the moduli space.