Holographic Action for the Self-Dual Field

TL;DR

This work develops a holographic framework for the quantum self-dual field in $4\ell+2$ dimensions by encoding its dynamics in a Chern-Simons theory living in one higher dimension. A careful quantization using differential cohomology and a generalized spin structure (QRIF) yields a partition function that is a holomorphic section of a line bundle, expressed in terms of a theta-function with a metric-dependent period matrix and a chosen Lagrangian decomposition. The authors construct the spin abelian Chern-Simons term, analyze the resulting Gauss law, and derive an action principle for the self-dual field whose metric dependence is tied to a determinant- or Quillen-type norm, thereby clarifying topological and geometric subtleties previously obscured. The framework unifies known special cases (e.g., chiral scalars and Henneaux-Teitelboim actions) and clarifies the role of background torsion, tadpole constraints, and boundary degrees of freedom, with implications for RR-field actions in string theory and moduli stabilization.

Abstract

We revisit the construction of self-dual field theory in 4l+2 dimensions using Chern-Simons theory in 4l+3 dimensions, building on the work of Witten. Careful quantization of the Chern-Simons theory reveals all the topological subtleties associated with the self-dual partition function, including the generalization of the choice of spin structure needed to define the theory. We write the partition function for arbitrary torsion background charge, and in the presence of sources. We show how this approach leads to the formulation of an action principle for the self-dual field.

Figures (16)

• Figure 1: The quantum Hilbert space $\mathcal{H}^{qu}$ is defined as a subbundle inside the trivial bundle $\mathcal{H}^{cl}\times \mathcal{T}$. The fiber $\mathcal{H}^{qu}_J$ over the complex structure $J$ is defined by $L^2$-normalizable $J$-holomorphic sections of $\boldsymbol{\mathcal{L}}\to P_{q\Sigma}$.
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Theorems & Definitions (36)

• Lemma 6.1
• proof
• Theorem 6.1
• Remark 6.1
• proof : Proof of the theorem.
• Corollary 6.1
• Remark 6.2
• Theorem 6.2
• Theorem 6.3: Quantum equation of motion
• proof