[Paper Review] Nahm's, Basu-Harvey-Terashima's equations and Lie superalgebras

Roger Bielawski|arXiv (Cornell University)|Mar 12, 2015

Advanced Topics in Algebra3 references3 citations

TL;DR

This paper establishes a deep correspondence between the Basu-Harvey-Terashima (BHT) equations—governing the dynamics of BPS states in ABJM theory—and Lie superalgebras, particularly via the double superbracket equation on the odd part of $\mathfrak{gl}_{n|m}(\mathbb{C})$. It shows that solutions to the BHT equations yield solutions to Nahm's equations through a hyperkähler gradient flow, and provides a Lax pair and spectral curve interpretation using line bundles on a double cover of a spectral curve in $\mathbb{P}^2$, linking the dynamics to linear flows on Jacobians with $\tau$-equivariance.

ABSTRACT

We discuss the correspondence between Nahm's equations, the Basu-Harvey-Terashima equations, and Lie superalgebras.

Motivation & Objective

• To clarify the mathematical structure underlying the BHT equations in the context of BPS states in ABJM theory.
• To establish a correspondence between solutions of the BHT equations and solutions of Nahm's equations via a hyperkähler gradient flow on $\operatorname{Mat}_{n,m}(\mathbb{C}) \oplus \operatorname{Mat}_{m,n}(\mathbb{C})$.
• To interpret the BHT equations as a double superbracket equation on the odd part of $\mathfrak{gl}_{n|m}(\mathbb{C})$ equipped with a quaternionic automorphism.
• To provide a Lax pair and spectral curve formulation of the BHT equations, identifying them with linear flows on the Jacobian of a spectral curve with $\tau$-equivariance.

Proposed method

• The BHT equations are derived as the gradient flow of the quartic function $F = \frac{1}{2}\|\mu_1\|^2 - \frac{1}{2}\|\nu_1\|^2$ on $W_{n,m} = \operatorname{Mat}_{n,m}(\mathbb{C}) \oplus \operatorname{Mat}_{m,n}(\mathbb{C})$, with respect to the $U(n) \times U(m)$-invariant hyperkähler structure.
• Solutions to the BHT equations are shown to satisfy Nahm's equations via the identity $I_1X_{\mu_1} - I_1X_{\nu_1} = I_2X_{\mu_2} - I_2X_{\nu_2} = I_3X_{\mu_3} - I_3X_{\nu_3}$, linking the dynamics to moment maps.
• The BHT equations are reformulated as a double superbracket equation $\dot{C} = \frac{1}{2}[[J(C), C], C]$ on the odd part of $\mathfrak{gl}_{n|m}(\mathbb{C})$, where $C = \begin{pmatrix} 0 & A \\ B & 0 \end{pmatrix}$ and $J(A,B) = (-B^*, A^*)$.
• A Lax pair formulation is established: if $Z = XY$ satisfies $\dot{Z} = [M, Z]$, then there exists $N$ such that $\dot{X} = XN - M X$, $\dot{Y} = Y M - N Y$, linking the flow to linear dynamics on a spectral curve.
• The spectral curve is defined as the zero locus of $\hat{P}(\zeta, \lambda) = \det(\lambda - C(\zeta))$ in $\mathbb{P}^2$, and the flow corresponds to a linear flow on the moduli space of $\tau$-equivariant line bundles on the double cover $\hat{S}$.

Experimental results

Research questions

• RQ1 How are the Basu-Harvey-Terashima equations related to Nahm's equations through hyperkähler geometry?
• RQ2 Can the BHT equations be interpreted as a double superbracket equation on a Lie superalgebra?
• RQ3 What is the spectral curve and Lax pair structure underlying the BHT equations?
• RQ4 How does the $\tau$-equivariance of line bundles on the spectral curve relate to the BHT flow?

Key findings

• Solutions to the BHT equations on $W_{n,m}$ generate solutions to Nahm's equations via the hyperkähler gradient flow of $F = \frac{1}{2}\|\mu_1\|^2 - \frac{1}{2}\|\nu_1\|^2$, with $\mu_i$ and $\nu_i$ being moment maps for $U(n)$ and $U(m)$, respectively.
• The BHT equations are equivalent to the double superbracket equation $\dot{C} = \frac{1}{2}[[J(C), C], C]$ on the odd part of $\mathfrak{gl}_{n|m}(\mathbb{C})$, providing a Lie superalgebraic interpretation.
• The spectral curve $\hat{S} \subset \mathbb{P}^2$ is defined by $\hat{P}(\zeta, \lambda) = \det(\lambda - C(\zeta))$, and the BHT flow corresponds to a linear flow on the Jacobian of $\hat{S}$ restricted to $\tau$-equivariant line bundles.
• For $n = m$, the spectral curve $S$ in $T\mathbb{P}^1$ satisfies $\hat{P}(\zeta, \lambda) = P(\zeta, \lambda^2)$, so $\hat{S}$ is a double cover of $S$ ramified over $\eta = 0$, with genus $g_{\hat{S}} = (n-1)(2n-1)$ and $g_S = (n-1)^2$.
• Acyclic $\tau$-line bundles $\mathcal{L}$ on $\hat{S}$ satisfy $\mathcal{L} \simeq \pi^*\mathcal{F} \otimes [\Delta_B] \simeq \pi^*\mathcal{G} \otimes [\Delta_A]$, where $\mathcal{F}, \mathcal{G}$ are line bundles on $S$ defined by $A(\zeta)B(\zeta)$ and $B(\zeta)A(\zeta)$.