Higher Wess-Zumino-Witten term from semistrict higher Chern-Simons theory

Abstract: We show that in any $2n+2$ dimensions, the higher Chern-Simons action built from a semistrict Lie 2-algebra gives a non-trivial higher Wess-Zumino-Witten (WZW) term under a higher gauge transformation. The key point is that the non-zero Jacobiator directly generates the higher WZW term, hereas it vanishes when reduced to the strict case.

• The paper demonstrates that the Jacobiator in semistrict Lie 2-algebras is crucial for yielding nontrivial higher WZW terms under finite gauge transformations.
• It employs a Cartan homotopy framework to generalize the Chern-Simons action to arbitrary 2n+2 dimensions, bridging anomalies with topological invariants.
• The results unify strict and semistrict cases, establishing new quantization conditions and invariants for higher-dimensional quantum field theories.

Semistrict Higher Wess-Zumino-Witten Term from Higher Chern-Simons Theory

Background and Motivation

The classical Wess-Zumino-Witten (WZW) term is central in quantum field theory and topological physics, capturing anomalous behavior and quantization in low-dimensional gauge theories. Its intimate connection with anomalies and boundary effects in three-dimensional Chern-Simons (CS) theory has motivated a search for proper higher-dimensional analogues. Extending this correspondence requires a framework for gauge structures beyond Lie algebras, specifically via semistrict Lie 2-algebras and higher gauge theory. Higher Chern-Simons actions serve as generalized topological field theories, relevant to extended objects like strings and branes.

Previous studies have established two distinct behaviors: the strict Lie 2-algebra case yields no additional higher WZW term under gauge transformations in any dimension, while the semistrict case in four dimensions demonstrates a non-trivial higher WZW term attributable to the Jacobiator. The present work addresses the previously unresolved question of whether nontrivial higher WZW terms persist for arbitrary $2n+2$ dimensions with semistrict Lie 2-algebra structure.

Semistrict Lie 2-Algebra Framework and Higher Gauge Transformations

Semistrict Lie 2-algebras, or 2-term $L_\infty$ algebras, generalize Lie algebras by incorporating a trilinear bracket (Jacobiator) and relaxed Jacobi identities, capturing higher homotopical structure. A $\mathfrak{v}$-connection is specified by a pair $(A, B)$, with $A \in \Omega^1(M, \mathfrak{v}_0)$ and $B \in \Omega^2(M, \mathfrak{v}_1)$, and curvatures

$$\mathcal{F} = dA + \frac{1}{2}[A, A] - \alpha(B),\quad \mathcal{H} = dB + [A, B] - \frac{1}{6}[A, A, A],$$

subject to Bianchi and flatness conditions. Finite higher gauge transformations are detailed as automorphisms of $\mathfrak{v}$ with associated flat connection doublets and derivations, yielding transformed connections $(A^g, B^g)$ with explicit dependence on the Jacobiator and associated structures.

Higher Chern-Simons Action via Cartan Homotopy

The construction leverages the Cartan homotopy formula and homotopy derivation operators to interpolate connections and curvatures, generalizing the Chern-Weil transgression. The invariant polynomial in $(A, B, \mathcal{F}, \mathcal{H})$ of degree $L_\infty$0 and its transgression yield the semistrict higher Chern-Simons action: $L_\infty$1 encoding topological and gauge characteristics in arbitrary dimension.

Gauge Variation and the Emergence of Higher WZW Term

The gauge variation of the higher Chern-Simons action is computed explicitly for finite gauge transformations in arbitrary $L_\infty$2 dimensions, clarifying the dependence of higher WZW term on the semistrict structure. The Jacobiator—not present in strict Lie 2-algebras—emerges as the necessary and sufficient condition for the appearance of a non-zero higher WZW term. The result unifies prior observations: the strict case yields vanishing contributions, while the semistrict case produces a topological term directly tied to the flat connection associated with the gauge parameter.

For $L_\infty$3 (four dimensions), the higher WZW term is recovered as

$L_\infty$4

For general $L_\infty$5, the explicit formula shows that the Jacobiator's presence ensures the persistence and quantization of the higher WZW term, generalizing the traditional boundary anomaly relation and demonstrating that the topological quantization remains robust under higher gauge symmetry.

Implications and Future Directions

The identification of the higher WZW term with the Jacobiator in semistrict higher gauge theory not only generalizes the anomaly structure of ordinary CS/WZW correspondence but also establishes new topological invariants for field theories with extended gauge symmetry. This result has theoretical significance for anomaly cancellation, quantization of coupling constants, and boundary effects in higher-dimensional topological field theories. Practically, the formalism supports model-building for six-, eight-, and higher-dimensional theories relevant in string theory and higher brane dynamics.

Future research directions include extending the finite gauge transformation framework to general $L_\infty$6 algebras, systematic exploration of associated anomalies, and classification of invariant polynomial structures. The explicit topological actions offered by this generalization hold promise for new quantum field theory models, holographic dualities, and mathematical understanding of categorified symmetry.

Conclusion

The paper provides a rigorous construction of higher Wess-Zumino-Witten terms arising from semistrict higher Chern-Simons theories in arbitrary $L_\infty$7 dimensions. By leveraging the Cartan homotopy formalism and semistrict Lie 2-algebra structure, the authors demonstrate that the Jacobiator is the key ingredient for non-vanishing higher WZW terms under finite higher gauge transformations. The results unify earlier strict and semistrict cases, broaden the scope of CS/WZW correspondence, and open avenues for new topological and anomaly-related phenomena in higher gauge theories (2605.29282).