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Moduli Space Quantum Mechanics

Luis Anchordoqui, Muldrow Etheredge, Dieter Lust

Abstract

In this paper, continuing the discussion about Species Quantum Mechanics, we investigate quantum mechanics in moduli spaces using a mini-superspace approach. From this perspective, moduli-dependent functions can be viewed as operators, and we explore how the taxonomic relations from the Emergent String Conjecture can constrain the non-commutativity between these operators. Next, we study wave functions on moduli spaces, and we find that the geometry of moduli space plays an important role and leads to excited wave functions localised in the bulks of moduli spaces, and with positive energy eigenvalues. For cases when potentials are present, these effects result in moduli localised away from classical minima, and often result in excited, positive energy states.

Moduli Space Quantum Mechanics

Abstract

In this paper, continuing the discussion about Species Quantum Mechanics, we investigate quantum mechanics in moduli spaces using a mini-superspace approach. From this perspective, moduli-dependent functions can be viewed as operators, and we explore how the taxonomic relations from the Emergent String Conjecture can constrain the non-commutativity between these operators. Next, we study wave functions on moduli spaces, and we find that the geometry of moduli space plays an important role and leads to excited wave functions localised in the bulks of moduli spaces, and with positive energy eigenvalues. For cases when potentials are present, these effects result in moduli localised away from classical minima, and often result in excited, positive energy states.
Paper Structure (15 sections, 114 equations, 2 figures)

This paper contains 15 sections, 114 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Commutation relations, patterns, and taxonomy relations
  3. Asymptotic constraints
  4. Assumptions and limitations
  5. Commutation relations and patterns with potential
  6. AdS vacua and negative potentials
  7. Positive potentials
  8. Moduli space wave functions
  9. The free case without potential
  10. One-dimensional moduli space
  11. Higher-dimensional moduli space with finite volume
  12. Modular invariant wave functions
  13. Comparison with the species scale
  14. Non-vanishing potentials
  15. Conclusions

Figures (2)

  • Figure 1: A saxionic limit of moduli space. The circular periodicity (highlighted in purple), has radius controlled by the axion decay constant, and this becomes exponentially small as the saxion increases. This exponential decay squeezes the normalizable wavefunctions into the bulk.
  • Figure 2: The potential $V$, together with the geometric contribution $V_\text{geo}$ together produce an effective potential $V_\text{eff}=V+V_\text{geo}$ with a minimum at a finite region of moduli space.