Adjusted connections on non-abelian bundle gerbes

Abstract: Higher gauge theory for non-abelian structure 2-groups faces significant challenges when extending beyond the fake-flat sector, which suffers from limited applicability in physical models. A promising resolution involves equipping 2-groups with additional structure, known as adjustments. We present a comprehensive theory of adjusted connections on non-abelian bundle gerbes, classified by Saemann's adjusted version of non-abelian differential cohomology. This theory enables, in particular, a new coordinate-independent formulation of Tellez-Dominguez' lifting theorem, establishing a correspondence between adjusted connections on non-abelian bundle gerbes and connections on abelian bundle 2-gerbes.

• The paper introduces a novel framework for non-abelian bundle gerbes by employing adjusted connections that overcome the limitations of fake flatness.
• It systematically constructs a sheafified bicategory of local adjusted connections, enabling effective gluing of non-fake-flat curvature data.
• The work establishes a lift from abelian gerbe theory, proving the existence and classification of non-abelian adjusted connections via differential cohomology.

Adjusted Connections on Non-Abelian Bundle Gerbes

Introduction and Motivation

The theory presented in "Adjusted connections on non-abelian bundle gerbes" (2604.24513) addresses the longstanding challenge of formulating a robust, physically viable differential geometry for bundle gerbes associated with non-abelian structure 2-groups. While abelian bundle gerbes—associated to 2-groups of the form $BA$ for $A$ abelian—are well-understood and have been deployed in string theory for modeling the B-field, the extension to non-abelian Lie 2-groups has been stymied by technical and conceptual obstacles. Chief among these is the issue that non-abelian higher gauge theory, without extra structure, cannot consistently impose connections with arbitrary (i.e., non-flat or non-fake-flat) curvature in a way compatible with bicategorical descent and gluing on higher intersections. The "fake flatness" restriction, while mathematically convenient, severely limits physical applicability.

This work synthesizes and extends insights from categorical differential cohomology and adjusted gauge theory, providing the most comprehensive to date coordinate-free description of connections on non-abelian bundle gerbes framed in the language of adjusted non-abelian differential cohomology. It introduces both the necessary algebraic structures ("adjustments" and "splittings") on Lie 2-groups and a systematic procedure for sheafifying the category of local adjusted connections, yielding a bicategory of non-abelian gerbes suitable for gluing via arbitrary (non-fake-flat) local data.

Theoretical Framework and Key Innovations

Lie 2-Groups, Connections, and Fake Flatness

Bundle gerbes for a 2-group $\Gamma$ (modeled as a Lie crossed module $(H \to G)$) locally involve 1-forms $A$ ($\mathfrak{g}$-valued) and 2-forms $B$ ($\mathfrak{h}$-valued), with "fake curvature" defined as

$$\mathrm{fcurv}(A,B) = dA + \frac{1}{2}[A \wedge A] - t_{*}(B).$$

For abelian $\Gamma = BA$, $A$0 and the theory is tractable. In the non-abelian setting, enforcing $A$1 ("fake flatness") allows a theory admitting parallel transport along surfaces but is too constraining for many physical contexts, as it excludes nontrivial curvature in stringy sigma models and obstructs local-to-global gluing beyond the abelian sector.

The core observation is that unrestricted non-fake-flat local data cannot be consistently glued using the standard formulation, as the bicategorical structure breaks down: nontrivial 2-morphisms (gauge 2-transformations) fail to exist generically unless fake flatness is imposed. This pathological behavior signals that the classical model for connections and gauge transformations is insufficient for higher gauge theory with non-abelian 2-groups.

Adjustments and Adjusted Connections

This paper adopts and refines the notion of "adjustments" introduced by Saemann, Kim, Rist, et al. The necessary additional algebraic datum is a map

$A$2

that is linear in the $A$3-argument and satisfies a set of coherence conditions aligning it with the structure maps of $A$4. Adjustments specify how the curvature of a gauge transformation

$A$5

is coupled to the underlying fake curvature:

$A$6

formula (2.16), generalizing the vanishing condition for the abelian or fake-flat case. Without an appropriate $A$7, descent fails for non-abelian non-fake-flat connections. For abelian 2-groups, the adjustment is necessarily trivial ($A$8), recovering standard Deligne/Čech-Dolbeault theory.

Further, the introduction of a "splitting" $A$9 (essentially a lift of $\Gamma$0 modulo the image of $\Gamma$1) refines the types of connections ("adapted" vs. general adjusted ones) and partitions the adjusted connections into subcategories with desirable descent properties, as formalized in Tellez-Dominguez's adapted connection concept.

Sheafification and Non-Abelian Differential Cohomology

The main technical contribution is a systematic sheafification (plus-construction) of the (pre-)2-stack of local adjusted $\Gamma$2-connections, explicitly constructed in full generality for non-abelian and non-fake-flat data. The resulting sheaf of bicategories $\Gamma$3 over a smooth base $\Gamma$4 assembles local coordinate data into globally defined bundle gerbes with adjusted connections, manifesting all higher descent and bicategorical gluing required for applications.

Isomorphism classes in this bicategory are shown to be classified by the adjusted non-abelian differential cohomology $\Gamma$5. This formulation captures all known cases (abelian, ordinary, and fake-flat), with the adjustment reducing to zero in those subcases.

Reduction to Abelian Theory and Lifting Theorems

A major insight is the geometric relation among principal bundles, abelian gerbes, and general non-abelian $\Gamma$6-gerbes in the presence of adjustment and splitting. There exists a canonical sequence

$\Gamma$7

of bicategories generalizing the extension/reduction paradigm in higher gauge theory. The sequence is exact in the sense of descent obstructions: a non-abelian adjusted gerbe lifts from an abelian gerbe precisely when the associated principal $\Gamma$8-bundle admits a flat (or more generally, arbitrary) section. This extends and globalizes the Chern-Simons 2-gerbe obstruction ("lifting gerbe") theory.

A coordinate-independent, global, and connection-bearing generalization of Tellez-Dominguez's lifting theorem is provided: the data of a non-abelian adjusted gerbe corresponding to a principal $\Gamma$9-bundle with connection is equivalent to a trivialization (with connection) of the associated abelian Chern-Simons 2-gerbe. This produces a reduction of the non-abelian theory to the abelian theory of "one categorification higher," i.e., pairs of principal bundles with 2-gerbe trivializations (main theorems 2 and its corollaries).

Principal Results and Claims

• Adjusted connections on non-abelian bundle gerbes are classified by adjusted non-abelian differential cohomology $(H \to G)$0. This generalizes prior cohomological classifications to the case of arbitrary (non-fake-flat) connections with nontrivial adjustment.
• Sheafification yields a coordinate-free bicategory $(H \to G)$1 with fully consistent 2-categorical descent, gluing, and local-to-global structure, incorporating the adjusted gauge data $(H \to G)$2 and higher gauge 2-transformations.
• The exact sequence $(H \to G)$3 establishes descent/lifting obstructions, which vanish precisely when the relevant principal $(H \to G)$4-bundle possesses a (flat) section.
• Trivializations of the Chern-Simons abelian 2-gerbe (with compatible connection) correspond categorically to non-abelian $(H \to G)$5-gerbes with adjusted/adapted connections projecting to a given principal $(H \to G)$6-bundle. All non-abelian gerbes arise as lifts from this setting.
• Every bundle gerbe for a central, smoothly separable Lie 2-group equipped with splitting and adapted adjustment admits an (adapted and) adjusted connection—the first general existence theorem for such connections in non-abelian higher gauge theory.

Numerical and Structural Highlights

The theory produces a robust formalism that is "fully general": unlike previous work, there are no ad hoc restrictions on curvature, and the bicategorical descent works at all levels. Notably:

• New equivalencies: The bicategory of non-abelian gerbes with connections is shown equivalent to a bicategory of pairs $(H \to G)$7 (principal $(H \to G)$8-bundle with connection and a 2-gerbe trivialization), substantiating claims that all non-abelian higher gauge theory can be recast at the abelian "next level up."
• Sharp classification: Adjusted connections reduce to classic cases exactly when adjustment vanishes, giving a transparent hierarchy among abelian, ordinary, fake-flat, and general non-abelian theories.
• General existence: For the first time, it is proven that, under mild topological assumptions (central, smoothly separable 2-group and adapted adjustment), all such bundle gerbes admit adjusted connections.

Implications and Future Directions

The framework developed here has significant implications for both mathematics and mathematical physics:

Practical applications:

• The results make concrete the integration of non-abelian gerbes into models of string theory, sigma models, and anomaly theory, where non-fake-flatness and higher structure group non-abelianity are essential.
• The reduction to abelian data at the 2-gerbe level suggests that computationally, many non-abelian problems may be made tractable by working in the categorified abelian field and leveraging the equivalence.

Theoretical implications:

• The construction provides a template for higher analogues (possibly for Lie $(H \to G)$9-groups) and suggests a general methodology: the introduction of global "adjustment" structure is necessary to accommodate gluing in higher differential geometry.
• The existence results and the bicategorical formalism pave the way for new investigations into higher parallel transport, categorical characteristic classes, and extensions of index theory.

Speculation and prospects:

• The coordinate-free, globally well-defined formulation of adjusted connections enables systematic study of secondary (categorified) characteristic classes, holonomies, and possibly extends to quantum field-theoretic contexts where non-abelian 2-connections are coupled to topological matter fields.
• Extensions may involve stacks of higher gerbes, non-torsion classes, and non-smooth settings (e.g., diffeological or derived geometries).
• Computational aspects (explicit cocycle representatives, computational algorithms for adjustment data, applications to explicit physical backgrounds) remain a rich area for further work.

Conclusion

This work establishes the definitive local-to-global theory of connections on non-abelian bundle gerbes, validated in full generality by the introduction of adjustment data and a sheafified bicategorical construction. It resolves foundational gluing and existence problems preventing the deployment of non-abelian higher gauge theory in geometric and physical contexts, drawing a precise parallel between non-abelian gerbes with adjusted connections and a categorified abelian 2-gerbe formalism. The approach is technically complete, categorical, and strongly suggests that the methodology of adjustment will be central in any future extension of higher gauge or gerbe theory to higher categorical or derived settings.