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Defects in Supersymmetric Gauge Theory and Integrable Lattice Models

太田, 敏博 大阪大学 DOI:10.18910/82001

2021.03.24

概要

本研究は超対称ゲージ理論に含まれる欠陥演算子と可解格子模型の転送行列の間の新たな対応関係を見出す理論的研究である.ゲージ理論は電磁気学や素粒子標準模型の基礎となっている理論であり,理論物理学の普遍的言語の一つである.ゲージ理論の運動方程式の特解を与える自己双対方程式やそこから次元還元で得られる方程式は非線型であるにもかかわらず厳密に解けることが知られており,一般に可積分系となっている.ゲージ理論に内在する可積分性の役割は超対称性のあるときにはより顕著になり,特にN=2の超対称ゲージ理論では場の量子論としての低エネルギー有効理論を厳密に決定するほどである.その際に現れる可積分構造は2次元の可積分場の理論のスペクトル曲線として出現し,Seiberg-Witten曲線と呼ばれる.Seiberg-Witten曲線は当初は発見法的に構成され,実際ゲージ理論そのものだけを見ている限りは天下り的に与えられるように見えるが,考えている物理系を弦理論に埋め込んで考えると余剰次元から極めて自然に現れることが知られている.この結果に代表されるように,超対称ゲージ理論の背後には一般に可積分構造が存在することが示唆され,その可積分性によってゲージ理論の非摂動的な性質を理解できることが期待される.さらにそのような可積分構造は弦理論のセットアップを考慮することでより統一的に理解することができ,弦理論の双対性を駆使することでさらなる応用も可能となるはずである.

これらを踏まえて,本研究では可解格子模型の転送行列,N=2超対称ゲージ理論の欠陥演算子であるWilson-‘t Hooft演算子,2次元共形場理論のVerlinde演算子,という3つの量の間の等価性を理論的に示した.さらにこれらの等価性は弦理論のセットアップに埋め込むことにより統一的に理解できることもわかった.
具体的には,超対称ゲージ理論としてN=2周期的箙ゲージ理論と呼ばれるクラスのゲージ理論を考え,その理論に含まれるWilson-‘t Hooft演算子を考える.Wilson-‘t Hooft演算子は電荷と磁荷の両方を持つダイオンと呼ばれる粒子の世界線に対応しており,一次元的な広がりを持つ欠陥演算子である.この演算子の期待値は超対称局所化の方法を用いて厳密に計算することができ,Coulombモジュライ空間上の関数として得られる.ここで,Wilson-‘t Hooft演算子の期待値が非可換な積であるMoyal積を満たしていることを考慮してさらにWeyl量子化(変形量子化)を行うことによって,ある状態空間に作用する演算子として表し直すことができる.この演算子が三角関数型の可解格子模型の転送行列となっていること,および作用している状態空間がその格子模型のスピン鎖の状態空間となっていることを示した.加えて,AGT対応(の変種)を用いることにより転送行列を戸田共形場理論のVerlinde演算子と同定することもでき,このときスピン鎖の状態空間は戸田共形場理論の共形ブロックのなす空間とみなすこともできるということも示した.最後に,考えていたN=2周期的箙ゲージ理論を弦理論のセットアップに埋め込
み,弦理論の双対性を用いることで4次元Chern-Simons理論と関係づけられることもわかった.4次元Chern-Simons理論は可解格子模型や2次元可積分場の理論の統一理論として期待されており,Wilson-‘t Hooft演算子が転送行列とみなせることの自然な説明を与えるとともに,さらなる応用をも示唆する.
本研究は,今まで知られていなかった超対称ゲージ理論と可解格子模型の新たな対応関係を構築する足がかりとなり,さらなる発展によってゲージ理論,可積分系の双方に応用が考えられる.異なる時空や異なるゲージ群の理論を考えることで全く新しい可積分系を提案できることが予想されると同時に,可積分系の知見を用いたゲージ理論の新たな非摂動的性質の理解に有用であることが期待される.

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