[Paper Review] Dirac zero modes for Abelian BPS multimonopoles

Joël Lamy-Poirier|arXiv (Cornell University)|Nov 26, 2015

Nonlinear Waves and Solitons8 references3 citations

TL;DR

This paper develops a novel method to compute Dirac zero modes for Abelian BPS monopoles in $\mathbb{R}^3$ by leveraging flat sections of a Lax pair connection. It derives an explicit residue formula for zero modes of unit-charge monopoles at generic positions, with solutions for other charges obtained via limiting procedures. The key contribution is a systematic, algebraic formula expressing zero modes as superpositions of flat section solutions, validated through singularity cancellation and normalization conditions.

ABSTRACT

We develop a method for finding the zero modes of the Dirac operator in the presence of BPS monopoles. We use it to find the zero modes in the case of Abelian BPS monopoles in $\mathbb R^3$.

Motivation & Objective

• To develop a general method for computing Dirac zero modes in the background of BPS monopoles, particularly for Abelian configurations.
• To address the longstanding difficulty in explicitly computing zero modes despite known monopole field configurations.
• To provide a closed-form expression for zero modes in terms of residues of algebraic functions, valid for arbitrary finite monopole configurations.
• To establish a link between zero modes and flat sections of the Lax pair, conjecturing that zero modes arise as $\mathcal{D}\chi$ for suitable $\chi$.
• To generalize results to include both positive and negative monopole charges, ensuring consistency with expected mode counts.

Proposed method

• The method constructs non-normalizable solutions to $\mathcal{D}^\dagger\Psi=0$ using flat sections of a Lax pair connection associated with the Bogomolnyi equations.
• It conjectures that physical zero modes are superpositions of these flat section solutions, specifically of the form $\Psi = \mathcal{D}\chi$, where $\chi$ is a flat section.
• The zero modes are expressed as a sum of residues at poles corresponding to zeros of algebraic functions, with the spectral parameter $\zeta$ encoding monopole positions and charges.
• For unit-charge monopoles, the solution takes the form of a residue at a special $\zeta_p$, derived from a system of linear equations for the coefficients $F_p(\zeta_p)$.
• Singularities in the wavefunction are canceled by imposing differential equations on the $F_p$ functions, ensuring regularity at all points including coincident monopole limits.
• The method extends to negative charges by introducing dual flat sections $\chi_0'$ and modifying the residue integrals accordingly, with separate conditions for $t>0$ and $t<0$.

Experimental results

Research questions

• RQ1 How can Dirac zero modes be systematically computed for Abelian BPS monopoles in $\mathbb{R}^3$, given the difficulty in solving the Dirac equation directly?
• RQ2 Can the zero modes be expressed as superpositions of flat sections of the Lax pair connection, and if so, under what conditions are such superpositions normalizable?
• RQ3 What is the precise algebraic structure of the zero modes for multiple monopoles, and how does it depend on monopole positions and charges?
• RQ4 How do the results generalize to configurations with both positive and negative charges, and what is the role of the spectral parameter in encoding the monopole data?
• RQ5 Can the method be extended to periodic monopole configurations, relevant for computing the supersymmetric index in 2D GLSMs?

Key findings

• For a single unit-charge monopole, the zero mode is given by a single residue at $\zeta_p$, with the wavefunction expressed as $\Psi = \mathcal{D}\chi$ where $\chi$ is a flat section.
• For $N$ unit-charge monopoles at generic positions, the zero mode is a sum of residues at $N$ poles, with the coefficients $F_p(\zeta_p)$ satisfying a system of linear equations derived from singularity cancellation.
• The number of zero modes matches the expected count: $N_+ = \sum_{m \in S_+} q_m$ for $t > 0$, and $N_- = \sum_{m \in S_-} |q_m|$ for $t < 0$, confirming consistency with theoretical expectations.
• For configurations with both $+1$ and $-1$ charges, the method splits into two cases: $X[\{F_n\}]$ for $t>0$ and $\hat{X}[\{F_{\hat{n}}\}]$ for $t<0$, with distinct residue integrals and conditions.
• The solution for $X[\{F_n\}]$ satisfies the differential equation $\partial_{\zeta_p} \left[ \cdots \right] = 0$, ensuring regularity at all points, including when $\zeta_p = \zeta_q$.
• The method successfully reproduces known results, such as the dipole solution in VanBaal:2002rt, and provides a general framework for computing zero modes in terms of algebraic geometry and residue theory.