[Paper Review] Partial supersymmetry breaking in Multidimensional N=4 SUSY QM

E. E. Donets, A. Pashnev+2 more|ArXiv.org|Jan 28, 2000

Particle physics theoretical and experimental studies3 citations

TL;DR

This paper constructs multidimensional N=4 supersymmetric quantum mechanics (SUSY QM) with central charges, demonstrating that partial supersymmetry breaking can yield models with 1/4, 1/2, or 3/4 of supersymmetries unbroken. By tuning superpotential-dependent parameters and boundary conditions, the authors show that all such partial breaking patterns are realizable in D ≥ 3 dimensions, generalizing earlier one-dimensional results and providing a framework for studying extended SUSY breaking in quantum systems.

ABSTRACT

The multidimensional N=4 supersymmetric quantum mechanics (SUSY QM) is constructed and the various possibilities for partial supersymmetry breaking are discussed. It is shown that quantum mechanical models with one quarter, one half and three quarters of unbroken(broken) supersymmetries can exist in the framework of the multidimensional N=4 SUSY QM.

Motivation & Objective

• To generalize one-dimensional N=4 SUSY QM with partial supersymmetry breaking to higher dimensions (D ≥ 2).
• To investigate the role of central charges in enabling partial supersymmetry breaking in multidimensional N=4 SUSY QM.
• To classify and construct models with one-quarter, one-half, and three-quarters of unbroken supersymmetries in D ≥ 3.
• To identify the conditions under which specific supersymmetries remain exact while others are broken, based on superpotential form and boundary conditions.
• To lay the groundwork for applications to quantum mechanics on AdS spaces and to Calogero-type models related to black holes and SYM theories.

Proposed method

• Constructing a D-dimensional N=4 SUSY QM with bosonic fields φⁱ, fermionic fields ψᵃⁱ and ȳᵃⁱ, and an arbitrary superpotential A(φⁱ).
• Defining the Hamiltonian and supercharges using covariant momentum operators Lᵢ and Rᵢ, incorporating derivatives of the superpotential and constant matrices λᵃᵢᵇ.
• Introducing central charges via real constants mⁱ, complex conjugate pairs nⁱ, ȳⁱ, and SU(2)-valued matrices Λᵃᵢᵇ to generate nontrivial algebraic structures.
• Deriving the superalgebra relations {Qᵃ, Qᵇ} and {Q̄ᵃ, Q̄ᵇ} to determine the conditions under which supersymmetries are unbroken or broken.
• Analyzing the algebraic structure of the superalgebra under different parameter choices (e.g., m¹Λ³₁, Λ¹₂, N², M³) to identify unbroken supersymmetry sectors.
• Using the Bogomol’ny bound and ground state energy conditions to determine which supercharges remain exact, depending on the sign and magnitude of central charge products.

Experimental results

Research questions

• RQ1 Can partial supersymmetry breaking with 1/4, 1/2, or 3/4 of unbroken supersymmetries be realized in multidimensional N=4 SUSY QM?
• RQ2 What role do central charges play in enabling partial supersymmetry breaking in higher-dimensional N=4 SUSY QM?
• RQ3 How does the dimensionality (D ≥ 1) affect the possible patterns of supersymmetry breaking in N=4 SUSY QM?
• RQ4 What specific conditions on the superpotential A(φⁱ) and its derivatives are required to realize one-quarter or three-quarters of unbroken supersymmetries?
• RQ5 Can the same algebraic structure used in one-dimensional N=4 SUSY QM be generalized to D ≥ 3 dimensions to preserve partial supersymmetry?

Key findings

• Models with one-quarter of unbroken supersymmetry (i.e., three out of four supersymmetries exact) are realizable in three-dimensional N=4 SUSY QM when m¹Λ³₁ = Λ¹₂N² = –Λ²₃M³ and m¹Λ³₁ < 0.
• Models with one-half of unbroken supersymmetries (two exact) are realizable in one-dimensional N=4 SUSY QM when m¹Λ³₁ > 0 and the ground state energy is |m¹Λ³₁|, with S² and T² supersymmetries exact.
• Models with three-quarters of unbroken supersymmetries (three exact) are realizable in two-dimensional N=4 SUSY QM when m¹Λ³₁ = Λ¹₂N² and m¹Λ³₁ > 0, making only T² supersymmetry exact.
• The number of unbroken supersymmetries depends critically on the sign and magnitude of products of central charge parameters (e.g., m¹Λ³₁, Λ¹₂N²), not on the dimensionality alone.
• All patterns of partial supersymmetry breaking (1/4, 1/2, 3/4, fully broken, fully unbroken) are possible in D ≥ 3 N=4 SUSY QM, depending on the superpotential and boundary conditions.
• The existence of central charges in the superalgebra is essential for partial breaking, as Witten’s theorem forbids such patterns in the absence of central charges.