Higher Automorphism Bundles

• Higher automorphism bundles are categorical and geometric structures that extend classical symmetry groups of fiber bundles using higher groupoids.
• They enable a systematic treatment of moduli rigidity, invariant connections, and the interplay between base and fiber symmetries in diverse geometric settings.
• Applications span algebraic geometry, gauge theory, and quantum field theory, providing a unified framework for symmetry gauging and computation of invariants.

Higher automorphism bundles are categorical and geometric structures that generalize the classical notion of automorphism groups of fiber bundles and moduli spaces, encoding symmetries at higher (typically groupoid or n-group) levels that act on associated bundles, moduli spaces, or even stacks of geometric or physical fields. This concept has deep roots in algebraic geometry, gauge theory, moduli theory, and modern mathematical physics, where their structure controls the interactions between underlying geometric spaces and symmetry operations, notably in the context of higher gauge theory, moduli of decorated bundles, and higher-form symmetries. The formal paper of higher automorphism bundles clarifies how extended symmetries systematically arise, how they are computed or classified, and how they impact phenomena such as moduli rigidity, symmetry gauging, fiberwise structures, and quantum field theoretic constructions.

1. Automorphism Groups of Bundles and Moduli Spaces

In the context of fiber bundles over a scheme or complex variety, the classical automorphism group is generalized to the automorphism group scheme, which may be equipped with additional structure in the presence of group actions, decorations, or added geometric data. For principal $G$-bundles $E_G$ over a manifold $M$, the internal automorphism group is encoded as the adjoint bundle $\mathrm{Ad}(E_G)$, with higher automorphism structures corresponding to torus (or more generally, n-group) subbundles of $\mathrm{Ad}(E_G)$ that control the existence of compatible reductions or invariant connections (Biswas et al., 2019).

In moduli theory, the automorphism group of moduli spaces of vector bundles, symplectic bundles, or principal $G$-bundles is typically generated by explicit geometric operations:

• For moduli spaces $M(r,\Lambda)$ of rank $r$ vector bundles of fixed determinant $\Lambda$ over a curve $X$, the automorphism group is generated by pullbacks via automorphisms of $X$, tensorizations with suitable line bundles, and dualization (when $r|2\deg(\Lambda)$) (Biswas et al., 2012).
• For moduli spaces $M_{Sp}(L)$ of symplectic bundles of fixed determinant, the automorphisms are generated by tensoring with 2-torsion line bundles and automorphisms of the base curve, forming an exact sequence

$$0 \to J(X)_2 \to \mathrm{Aut}(M_{Sp}(L)) \to \mathrm{Aut}(X) \to 0$$

with $J(X)_2$ the 2-torsion of the Jacobian (Biswas et al., 2010).

• For moduli of parabolic bundles, the generators extend to Hecke transforms and actions from the automorphisms of the marked curve, with constraints imposed by the chamber structure of the parabolic weights (Alfaya et al., 2019, Alfaya, 2021).

Such groups can often be systematically encoded by higher groupoids or group schemes, especially when considering moduli stacks with nontrivial automorphism structure at geometric and stack-theoretic levels (Herrero, 2022, Fringuelli, 2021).

2. Higher Automorphism Groups, n-Groups, and Symmetries

The notion of a higher automorphism bundle naturally extends to higher categories (n-groupoids, stacky group schemes) in contexts such as higher gauge theory, where the symmetries themselves are encoded as higher group bundles (e.g., 2-groups, 3-groups). In higher gauge theory, for any moduli stack $\mathcal{X}$, the group of global symmetries is the automorphism $\infty$-group $\mathrm{Aut}(\mathcal{X})$ in the sense of smooth higher stacks. Symmetry actions on fields, modeled as $H(M, \mathcal{X})$ for a spacetime $M$, are naturally induced by composition with flat principal $\mathrm{Aut}(\mathcal{X})$-bundles, leading to global higher-form symmetries whose parameters are determined by the flatness condition imposed by the shape modality $\Pi$ (Perez-Lona, 18 Sep 2025).

For targets $\mathcal{X} = \mathrm{B}G$ or higher analogues, the "center" phenomena manifest categorically as central extensions:

$$B\mathcal{Z}(G) \to \mathrm{AUT}(G) \to \mathrm{OUT}(G)$$

where $B\mathcal{Z}(G)$ is the delooping of the center of $G$. The higher automorphism group structure encodes both the conventional (0-form) and higher-form symmetries, and describes their interplay in tensoring, Wilson operators, and more generally in the symmetry structure of the underlying gauge theory (Perez-Lona, 18 Sep 2025).

In concrete terms, for $U(1)$ gauge theory or bundle gerbes, the higher automorphism bundle acts by tensoring, inducing higher-form center symmetries. For the string 2-group $String(G_k)$ associated to a compact Lie group $G$, the center of $\mathrm{Aut}(B\,String(G_k))$ is a braided 2-group with $\pi_0 \simeq Z(G)$ and $\pi_1 \simeq U(1)$, encapsulating both discrete and continuous higher-form symmetries (Perez-Lona, 18 Sep 2025).

3. Lifting, Descent, and Structure of Automorphism Bundles

For principal bundles and fiber bundles, the structure of higher automorphism bundles is reflected both in lifting and descent properties:

• For torsors under group schemes, especially abelian varieties, every automorphism of the base lifts (up to finite ambiguity) to an equivariant automorphism of the total space. This lifting produces, for a $G$-torsor $X \to Y$, exact sequences

$$1 \to \mathrm{Aut}_Y(X) \to \mathrm{Aut}_G(X) \to \mathrm{Aut}(Y)$$

and in the abelian case, a splitting $\mathrm{Aut}(X) = G \cdot H$ where $H$ maps isogenously to $\mathrm{Aut}^0(Y)$, encoding how higher automorphism data decomposes into base and fiber symmetries (Brion, 2010).

• In the context of Levi reduction, principal $G$-bundles over $M$ with torus subbundles of $\mathrm{Ad}(E_G)$ have higher automorphism structure directly related to reductions of structure group and invariant connections. The key criterion for a connection to descend or be compatible with a reduction is its preservation of the torus subbundle. There is a bijective correspondence between such torus subbundles and quadruple data including principal $W$-bundles, holomorphic reductions, and compatible actions, clarifying the modular parameters of the higher automorphism bundle (Biswas et al., 2019).
• For moduli stacks with adequate moduli spaces (notably for semistable $G$-bundles with decorations), automorphism groups of points are rigid: the kernel of the restriction morphism to the fiber is unipotent and finite (and trivial in characteristic 0). This rigidity underlies both local-to-global principles in moduli theory and allows the stack to be presented as a $G$-linearized global quotient (a GIT problem) (Herrero, 2022).

4. Applications: Moduli Rigidity, Quantum Cohomology, and Physics

A central application of higher automorphism bundles is in formulating Torelli-type theorems, rigidity results, and explicit classifications of automorphism groups of moduli spaces:

• For the moduli space of symplectic bundles $M_{Sp}(L)$ or principal $G$-bundles, automorphisms are entirely determined by concrete operations—tensorizations, pullbacks by curve automorphisms, and appropriate outer automorphisms or center twistings. There are no "exotic" symmetries unaccounted for by these operations, and the automorphism group fits into explicit exact sequences (Biswas et al., 2010, Fringuelli, 2021).
• In moduli of parabolic bundles, only those basic transformations (pullbacks, tensorizations, Hecke transformations, dualization) that preserve weight chamber data (numerical invariants $M(r,a,d,n)$) can extend to honest automorphisms, and Torelli-type rigidity is obtained: the 3-birational class of the moduli space determines the underlying curve (Alfaya et al., 2019, Alfaya, 2021).
• For moduli of Higgs bundles, higher automorphism bundles (incorporating $\mathbb{C}^*$-actions, automorphisms of the underlying SU-type moduli space, and Hamiltonian flows) lead to infinite-dimensional vertical groups, with restrictions for symplectic or Kähler isomorphisms (Baraglia, 2014). Refined indices such as the automorphism equivariant Hitchin index are computable by localization to fixed loci and are connected to quantum topology via the Verlinde formula and Chern–Simons theory (Andersen et al., 24 Jan 2024).

The systematic paper of higher automorphism bundles also clarifies the structure of automorphism groups of bundles over non-uniruled bases, P$^1$-bundles over ruled surfaces, and multiprojective or symmetric powers of projective bundles, often leading to split exact sequences and explicit combinatorial control by base automorphisms and symmetric group actions (Fong, 2023, Bandman et al., 2021, Bansal et al., 22 Aug 2025).

In mathematical physics, higher automorphism bundles provide the universal framework to understand global higher-form symmetries as acting by flat principal bundles for higher automorphism stacks, unifying various phenomena (Wilson operator charge assignments, symmetry gauging, instanton restriction, etc.) under the functorial symmetry group of the moduli stack of fields (Perez-Lona, 18 Sep 2025).

5. Structural Features and Classification Theorems

The explicit structure of higher automorphism bundles is elucidated via:

• Exact sequences relating fiberwise and base automorphisms, often incorporating central or 2-torsion contributions (e.g., $$0 \to J(X)_2 \to \mathrm{Aut}(M_{Sp}(L)) \to \mathrm{Aut}(X) \to 0$$ for symplectic bundles (Biswas et al., 2010)).
• Decomposition results for multiprojective and symmetric bundles, as in

$$1 \to \prod_i (\mathrm{Aut}_{\mathbb{P}^n}(P_i))^{a_i} \rtimes S_{a_i} \to \mathrm{Aut}(P) \to H \to 1$$

where $H$ is the subgroup of base automorphisms preserving the isomorphism types of factors (Bansal et al., 22 Aug 2025).

• Jordan-type finiteness and commutativity conditions for automorphism groups of certain bundles, such as P$^1$-bundles over non-uniruled or "poor" bases, where the connected identity component is abelian and sits in split short exact sequences with complex tori (Bandman et al., 2021).
• Rigidity theorems which reduce the automorphism group of a moduli stack to the automorphism group of the curve plus groups generated by tensorizations and torsion data, imposing strong restrictions and thereby limiting possible moduli isomorphisms or deformations (Herrero, 2022, Fringuelli, 2021).

6. Impact, Open Directions, and Interplay with Other Theories

Higher automorphism bundles form the organizing principle behind the extended symmetry structure in both algebraic geometry and quantum field theory. They clarify when and how automorphisms can act at higher levels (e.g., via non-trivial 2-groupoids or n-groups), what constraints are imposed by moduli-theoretic "chambers," and how symmetries are transferred between base and fiber in geometric structures. Their rigorous formulation enables:

• Systematic understanding of global and higher-form symmetries in gauge theory, including center symmetries, higher-form anomaly cancellation, and the structure of the TQFT mapping class group representations.
• Computation of geometric and quantum invariants (e.g. Hitchin indices, Verlinde formulas) via localization over higher automorphism bundles.
• Clarification of the functorial relationships between field theories with higher gauge symmetry and the associated moduli of connections and torsors.

Continuing work leverages these structures in the paper of derived categories, modular invariants, extended TQFT, and in the explicit geometric construction of moduli stacks and their equivariant refinements for varieties and stacks with additional structures, such as parabolic, symplectic, or decorated bundles (Perez-Lona, 18 Sep 2025, Andersen et al., 24 Jan 2024). A plausible implication is that as the categorical technology advances, higher automorphism bundles will underlie a unified symmetry theory for moduli problems and quantum field theories alike.