[Paper Review] From Loop Space Mechanics to Nonabelian Strings

Urs Schreiber|ArXiv.org|Sep 21, 2005

Black Holes and Theoretical PhysicsPhysics and Astronomy170 references17 citations

TL;DR

This paper establishes a categorified framework for nonabelian strings by lifting supersymmetric quantum mechanics on loop space to higher gauge theories via 2-bundles with 2-holonomy. It introduces nonabelian Deligne hypercohomology to describe global surface holonomy, generalizing gerbes and connecting to M-theory branes, spinning strings, and derived categories. The key contribution is a consistent action functional for nonabelian strings using Lie 2-algebras and the String-group structure.

ABSTRACT

Lifting supersymmetric quantum mechanics to loop space yields the superstring. A particle charged under a fiber bundle thereby turns into a string charged under a 2-bundle, or gerbe. This stringification is nothing but categorification. We look at supersymmetric quantum mechanics on loop space and demonstrate how deformations here give rise to superstring background fields and boundary states, and, when generalized, to local nonabelian connections on loop space. In order to get a global description of these connections we introduce and study categorified global holonomy in the form of 2-bundles with 2-holonomy. We show how these relate to nonabelian gerbes and go beyond by obtaining global nonabelian surface holonomy, thus providing a class of action functionals for nonabelian strings. The examination of the differential formulation, which is adapted to the study of nonabelian p-form gauge theories, gives rise to generalized nonabelian Deligne hypercohomology. The (possible) relation of this to strings in Kalb-Ramond backgrounds, to M2/M5-brane systems, to spinning strings and to the derived category description of D-branes is discussed. In particular, there is a 2-group related to the String-group which should be the right structure 2-group for the global description of spinning strings.

Motivation & Objective

• To develop a global, categorified description of nonabelian string theories beyond abelian gerbes.
• To generalize loop space mechanics to include nonabelian 2-form gauge fields and their holonomy.
• To establish a connection between the String-group and the global structure of spinning strings via 2-group gauge theory.
• To formulate a differential cohomology theory—nonabelian Deligne hypercohomology—for classifying local and global 2-connections.
• To relate the formalism to physical systems such as M2/M5-branes, Kalb-Ramond backgrounds, and D-branes in derived categories.

Proposed method

• Lifts supersymmetric quantum mechanics on loop space to derive worldsheet invariants and boundary states, identifying them with string background fields.
• Introduces 2-bundles with 2-connections and 2-holonomy using strict 2-groups and categorified holonomy functors on p-path groupoids.
• Constructs the differential picture via Lie 2-algebras and L∞-algebras, particularly the gk algebra associated with the String-group.
• Derives the 2-curvature and consistency conditions for 2-connections using chain maps on differential graded algebras.
• Applies the formalism to infinitesimal 2-bundles, showing that gauge transformations and cocycle conditions reduce to abelian gerbe structure up to a k-dependent Chern-Simons shift.
• Uses Čech-extended p-path groupoids to define global p-functors for p=2, generalizing holonomy to higher categories.

Experimental results

Research questions

• RQ1 How can loop space mechanics be extended to describe nonabelian strings via categorification?
• RQ2 What is the global structure of nonabelian surface holonomy, and how does it generalize gerbes and 1-holonomy?
• RQ3 How does the String-group arise as the structure 2-group for spinning strings in this formalism?
• RQ4 What is the role of nonabelian Deligne hypercohomology in classifying 2-connections and their gauge transformations?
• RQ5 How do the differential and global formulations of 2-connections relate, especially in the infinitesimal limit?

Key findings

• The formalism realizes nonabelian surface holonomy via 2-bundles with 2-connection, providing a global action functional for nonabelian strings.
• The 2-curvature satisfies a 2-Bianchi identity due to the Jacobi identity in the underlying Lie 2-algebra, ensuring consistency.
• In the infinitesimal limit, the gauge transformation of the 2-form connection Bi acquires a k-dependent shift involving the Chern-Simons form of the 1-connection A.
• The cocycle conditions for 2-connections reduce to those of an abelian gerbe when the 1-connection is flat and a suitable gauge is chosen, with the k-parameter encoding nonabelian corrections.
• The 2-group PkG is shown to be equivalent to the Lie 2-algebra gk, and this equivalence underlies the global description of spinning strings.
• The differential formalism correctly reproduces known consistency conditions for 3-connections, validating the approach at higher categorical levels.