[Paper Review] Lie 3-Algebra Non-Abelian (2,0) Theory in Loop Space

Kuo-Wei Huang, Wung-Hong Huang|arXiv (Cornell University)|Aug 23, 2010

Advanced Topics in Algebra12 references3 citations

TL;DR

This paper proposes a non-abelian (2,0) theory for multiple M5-branes using Lie 3-algebra structures within loop space formalism, where all fields—including the 2-form potential—are reformulated as loop fields. The approach incorporates non-locality via loop space coordinates and noncommutative relations, and while supersymmetry closure leads to complex constraints, the framework provides a systematic scheme for constructing field equations and derives essential $̳$-matrix identities for further analysis.

ABSTRACT

It is believed that the multiple M5-branes are described by the non-abelian (2,0) theory and have the non-local structure. In this note we investigate the non-abelian (2,0) theory in loop space which incorporates the non-local property. All fields will be formulated as loop fields and the two-form potential becomes a part of connection. We make an ansatz for field supersymmetry transformation with a help of Lie 3-algebra and examine the closure condition of the transformation to find the field equations. However, the closure conditions lead to several complex terms and we have not yet found a simple form for some constrain field equations. In particular, we present the clear scheme and several detailed calculations in each step. Many useful $Γ$ matrix algebras are derived in the appendix.

Motivation & Objective

• To explore a non-abelian (2,0) theory for multiple M5-branes by combining Lie 3-algebra structures with loop space formalism.
• To incorporate the non-local nature of the (2,0) theory through loop field formulations and noncommutative loop coordinates.
• To construct a supersymmetry transformation ansatz using Lie 3-algebra and examine its closure conditions.
• To derive field equations from the closure constraints, despite encountering complex terms in the resulting constraints.
• To provide a detailed computational framework with explicit $̳$-matrix algebra relations for future analysis of the (2,0) theory.

Proposed method

• Formulates all fields—scalar, fermion, and 2-form potential—as loop fields via integration over loop parameter $s$ using $\dot{C}^\mu(s)$ as a weighting measure.
• Introduces a noncommutative loop space structure via $[C^\mu(s), C^\nu(s)] = Q^{\mu\nu\lambda} \dot{C}^\mu(s)$, embedding non-locality at the fundamental level.
• Defines an effective gauge connection $A^b_{\mu a}(C) = \oint ds~{} B_{\mu\nu a}^b(C(s)) \dot{C}^\nu(s)$, incorporating the 2-form potential into the connection.
• Proposes a supersymmetry transformation ansatz using Lie 3-algebra structure constants, with field components redefined via normalization by $L^\mu / \sqrt{|(L^\nu)^2|}$.
• Derives and utilizes extensive $\Gamma$-matrix identities in 11D spacetime, including commutation and anticommutation relations for $\Gamma^{a_1...a_n}$, to analyze supersymmetry closure.
• Applies summation identities such as $\Gamma^{ab}\Gamma^a = (1-n)\Gamma^b$ to simplify expressions during closure checks.

Experimental results

Research questions

• RQ1 Can a non-abelian (2,0) theory for multiple M5-branes be consistently formulated in loop space using Lie 3-algebra structures?
• RQ2 How does the non-local structure of the (2,0) theory emerge naturally in loop space with noncommutative coordinates?
• RQ3 What are the implications of formulating the 2-form potential as part of a connection in loop space?
• RQ4 Can a supersymmetry transformation ansatz based on Lie 3-algebra close, and what constraints arise from this closure?
• RQ5 What role do 11D $\Gamma$-matrix identities play in simplifying the closure conditions of the supersymmetry algebra?

Key findings

• The paper successfully formulates scalar, fermion, and 2-form fields as loop fields using integration over loop parameter $s$, with proper normalization to maintain scale dimensions.
• The 2-form potential $B^{\mu\nu}$ is incorporated into an effective connection $A^b_{\mu a}(C)$, suggesting a gauge-like structure in loop space despite the absence of a standard 2-form gauge theory.
• Noncommutative loop coordinates $[C^\mu(s), C^\nu(s)] = Q^{\mu\nu\lambda} \dot{C}^\mu(s)$ are introduced as a fundamental feature of the loop space, encoding non-locality.
• Supersymmetry closure leads to complex field equations, and no simple form is found for certain constraint equations, indicating the theory's nontrivial structure.
• The paper derives and systematically applies a comprehensive set of $\Gamma$-matrix identities in 11D, including $\Gamma^{ab}\Gamma^a = (1-n)\Gamma^b$ and $\Gamma^{abc}\Gamma^a = (n-2)\Gamma^{bc}$, essential for supersymmetry analysis.
• The appendix provides detailed $\Gamma$-matrix algebra, including general identities like $\Gamma^{b_1...b_n}\Gamma^{a_1...a_n} = \sum_{p=0}^{\min(n,m)} \frac{n!m!}{(n-p)!(m-p)!p!} \Gamma^{[b_1...b_{n-p}}{}^{[a_{p+1}...a_m}} g^{b_{n-p+1}...b_n]}{}_{a_1...a_p]}$, crucial for higher-rank tensor computations.