Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Mechanics on flag manifolds

Andrew Kuzovchikov

TL;DR

This paper builds a link between SU(n) spin chains and one-dimensional sigma models on complete flag manifolds, enabling computation of geometric spectra and geodesics on CP^1 and F_3 for a class of metrics. The authors formalize flag manifolds as F_n = SU(n)/S(U(1)^n) and describe a Lagrangian embedding into (CP^{n-1})^n, focusing on F_2 ≈ CP^1 and F_3. For CP^1, the quantum spectrum matches SU(2) representations with eigenvalues Lambda_l = l(l+1), and in the large p limit, the spin-chain dynamics reproduce geodesic motion on S^2; the twisted partition function also converges to the particle-on-a-sphere result. For F_3, the SU(3) spin chain with three sites yields a metric on F_3 and an explicit spectrum when beta equals gamma, plus a geodesic solution; these results extend to supersymmetric generalizations.

Abstract

We study the connection between $\mathrm{SU}(n)$ spin chains and one-dimensional sigma models on flag manifolds. Using this connection, we calculate the spectrum of the Laplace-Beltrami operator and geodesics for a particular class of metrics on $\mathbb{CP}^1$ and $\mathcal{F}_3$, which is a manifold of complete flags in $\mathbb{C}^3$.

Mechanics on flag manifolds

TL;DR

This paper builds a link between SU(n) spin chains and one-dimensional sigma models on complete flag manifolds, enabling computation of geometric spectra and geodesics on CP^1 and F_3 for a class of metrics. The authors formalize flag manifolds as F_n = SU(n)/S(U(1)^n) and describe a Lagrangian embedding into (CP^{n-1})^n, focusing on F_2 ≈ CP^1 and F_3. For CP^1, the quantum spectrum matches SU(2) representations with eigenvalues Lambda_l = l(l+1), and in the large p limit, the spin-chain dynamics reproduce geodesic motion on S^2; the twisted partition function also converges to the particle-on-a-sphere result. For F_3, the SU(3) spin chain with three sites yields a metric on F_3 and an explicit spectrum when beta equals gamma, plus a geodesic solution; these results extend to supersymmetric generalizations.

Abstract

We study the connection between spin chains and one-dimensional sigma models on flag manifolds. Using this connection, we calculate the spectrum of the Laplace-Beltrami operator and geodesics for a particular class of metrics on and , which is a manifold of complete flags in .

Paper Structure

This paper contains 4 sections, 2 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Case of $\mathbb{CP}^1$
  3. Case of $\mathcal{F}_{3}$
  4. Acknowledgement

Key Result

Proposition 1

The eigenvalues of $\mathrm{H}_{\text{spin}}$, for a fixed value of $p$, take the form where $p_1^A+p_2^A=2p,\, p_2^A\leq p\leq p_1^A$, $p_1^B+p_2^B+p_3^B=3p,\, p_2^B\leq p_1^A\leq p_1^B,\, p_3^B\leq p_2^A \leq p_2^B$, $C_2(p_1,p_2,p_3)=\sum_{j=1}^3 s_j(s_j - 2j)$ and $s_i = p_i - \frac{1}{3} \sum_{j=1}^3 p_j$.

Theorems & Definitions (2)

  • Proposition 1: Bykov2024
  • Proposition 2: Bykov2024