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Symmetries and Higher-Form Connections in Derived Differential Geometry

Severin Bunk, Lukas Müller, Joost Nuiten, Richard J. Szabo

TL;DR

Symmetries and Higher-Form Connections in Derived Differential Geometry provides a comprehensive derived-geometric framework for higher-form connections on principal $\infty$-bundles. It constructs the Atiyah $L_\infty$-algebroid $T(M/X)$ associated to maps $x:M\to X$ (where $X$ has deformation theory), and defines $p$-form connections as order-$p$ splittings of this algebroid, with curvature as the obstruction to lifting to higher forms. It proves that, for higher $U(1)$-bundles, the derived spaces of connections coincide with the Čech–Deligne descriptions, giving an intrinsic model of differential cohomology via $L_\infty$-algebroid mapping spaces. The work further relates derived higher symmetries to higher Courant algebroids, showing canonical projections from Courant-level symmetry algebras to Atiyah-type structures and exploring weak/enhanced connection data in the supergravity context. Altogether, the paper provides a unifying, deformation-theoretic approach to higher gauge theory, extends classical Atiyah theory to the derived setting, and connects to differential cohomology and generalized geometry.

Abstract

We introduce a general definition of higher-form connections on principal $\infty$-bundles in differential geometry. This is achieved by developing the formal differentiation and integration of maps from smooth manifolds to derived stacks with sufficient deformation theory. That allows us to introduce the Atiyah $L_\infty$-algebroid of a principal $\infty$-bundle and establish its global sections as the $L_\infty$-algebra of the derived higher symmetry group of the bundle. We define the space of $p$-form connections on an $\infty$-bundle as the space of order $p$ splittings of its Atiyah $L_\infty$-algebroid. We demonstrate that our new concept of derived geometric $p$-form connections recovers the known notion of connections on higher U(1)-bundles defined via Čech-Deligne differential cocycles. We further relate the $L_\infty$-algebras of derived higher symmetries of higher U(1)-bundles and higher Courant algebroids. Some applications in higher gauge theory and in supergravity are mentioned.

Symmetries and Higher-Form Connections in Derived Differential Geometry

TL;DR

Symmetries and Higher-Form Connections in Derived Differential Geometry provides a comprehensive derived-geometric framework for higher-form connections on principal -bundles. It constructs the Atiyah -algebroid associated to maps (where has deformation theory), and defines -form connections as order- splittings of this algebroid, with curvature as the obstruction to lifting to higher forms. It proves that, for higher -bundles, the derived spaces of connections coincide with the Čech–Deligne descriptions, giving an intrinsic model of differential cohomology via -algebroid mapping spaces. The work further relates derived higher symmetries to higher Courant algebroids, showing canonical projections from Courant-level symmetry algebras to Atiyah-type structures and exploring weak/enhanced connection data in the supergravity context. Altogether, the paper provides a unifying, deformation-theoretic approach to higher gauge theory, extends classical Atiyah theory to the derived setting, and connects to differential cohomology and generalized geometry.

Abstract

We introduce a general definition of higher-form connections on principal -bundles in differential geometry. This is achieved by developing the formal differentiation and integration of maps from smooth manifolds to derived stacks with sufficient deformation theory. That allows us to introduce the Atiyah -algebroid of a principal -bundle and establish its global sections as the -algebra of the derived higher symmetry group of the bundle. We define the space of -form connections on an -bundle as the space of order splittings of its Atiyah -algebroid. We demonstrate that our new concept of derived geometric -form connections recovers the known notion of connections on higher U(1)-bundles defined via Čech-Deligne differential cocycles. We further relate the -algebras of derived higher symmetries of higher U(1)-bundles and higher Courant algebroids. Some applications in higher gauge theory and in supergravity are mentioned.
Paper Structure (39 sections, 62 theorems, 443 equations, 1 figure)

This paper contains 39 sections, 62 theorems, 443 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Towards higher-form connections
  3. Symmetries and connections from Atiyah algebroids.
  4. Summary of main results
  5. Derived differential geometry and Atiyah $\boldsymbol{L_\infty}$-algebroids.
  6. Moduli of geometric structures on manifolds, globally.
  7. Higher-form connections and their curvatures.
  8. Computing spaces of connections on higher ${\mathrm{U}}(1)$-bundles.
  9. Higher Courant algebroids and weak connections.
  10. Outline of the paper
  11. Acknowledgements
  12. Notation and conventions
  13. Deformation theory
  14. Homotopy theory of $L_\infty$-algebroids
  15. Homotopy theory of dg $C^\infty$-rings
  16. ...and 24 more sections

Key Result

Theorem 1.5

(Theorem thm:formal stack vs lie) Let $M$ be a smooth manifold and let $x \colon M \longrightarrow X$ be a map to a stack with deformation theory. The relative tangent complex which is a priori a dg $C^\infty(M)$-module, has a natural lift to a complete $L_\infty$-algebroid on $M$. This determines a right adjoint functor of $\infty$-categories from stacks with deformation theory under $M$ to com

Figures (1)

  • Figure 1: Maps from $\overline{{\mathrm{Sym}}}{}_\mathbbm{R}^{\leqslant p}(TM[1])$ to the cochain complex underlying the dg Lie algebroid $\widehat{{\mathfrak{g}}}_l$ of the fibrant simplicial resolution $\widehat{{\mathfrak{g}}}$ of ${\check{C}}{\mathrm{At}}(g)$.

Theorems & Definitions (230)

  • Theorem 1.5
  • Definition 1.6
  • Theorem 1.7
  • Definition 1.8
  • Theorem 1.10
  • Definition 2.1
  • Example 2.2
  • Proposition 2.3: Nuiten:HoAlg_for_Lie_Algds
  • Definition 2.4
  • Remark 2.6
  • ...and 220 more