arxiv: 2602.03441 · v2 · submitted 2026-02-03 · 🧮 math.DG · hep-th· math-ph· math.AG· math.AT· math.MP

Recognition: no theorem link

Symmetries and Higher-Form Connections in Derived Differential Geometry

Severin Bunk , Lukas M\"uller , Joost Nuiten , Richard J. Szabo

Authors on Pith no claims yet

Pith reviewed 2026-05-16 07:40 UTC · model grok-4.3

classification 🧮 math.DG hep-thmath-phmath.AGmath.ATmath.MP

keywords higher-form connectionsprincipal infinity-bundlesAtiyah L-infinity-algebroidderived differential geometryCech-Deligne cocycleshigher gauge theoryhigher Courant algebroids

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The pith

Higher-form connections on principal infinity-bundles arise as order-p splittings of their Atiyah L-infinity-algebroids.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a definition of p-form connections on principal infinity-bundles by constructing the Atiyah L-infinity-algebroid from formal differentiation and integration of maps between smooth manifolds and derived stacks. Connections are then identified with the space of order-p splittings of this algebroid, which is equivalent to lifting the bundle's classifying map to the p-truncation of the de Rham stack. The authors show that this construction exactly recovers the standard notion of connections on higher U(1)-bundles given by Cech-Deligne differential cocycles. The framework also links the derived symmetry algebras of these bundles to higher Courant algebroids.

Core claim

The central claim is that p-form connections on a principal infinity-bundle are given by the space of order-p splittings of its Atiyah L-infinity-algebroid. This algebroid arises from the formal differentiation and integration of maps from smooth manifolds to derived stacks possessing sufficient deformation theory, and its global sections form the L-infinity-algebra of the bundle's derived higher symmetry group. Equivalently, the connections correspond to lifts of the classifying map of the bundle to the order-p truncation of the de Rham stack of the base manifold. For higher U(1)-bundles this definition reproduces the known Cech-Deligne cocycle description of connections.

What carries the argument

The Atiyah L-infinity-algebroid of a principal infinity-bundle, whose order-p splittings define the space of p-form connections on the bundle.

If this is right

Higher gauge theories admit a uniform description of their p-form connections via splittings of Atiyah L-infinity-algebroids.
The L-infinity-algebras of derived higher symmetries of higher U(1)-bundles are related to higher Courant algebroids.
Applications to supergravity models become available through the derived geometric formulation of higher connections.
Connections on general principal infinity-bundles are obtained by lifting classifying maps to truncated de Rham stacks.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same splitting construction might extend systematically to other higher structures such as gerbes with connection in string backgrounds.
Invariants for manifolds could be extracted by computing the derived symmetry algebras of their associated infinity-bundles.
The framework supplies a template for defining connections in nonabelian higher gauge theories beyond the U(1) case.
Concrete computations in low-dimensional examples would test whether the derived approach yields new solutions to higher Yang-Mills equations.

Load-bearing premise

Formal differentiation and integration of maps from smooth manifolds to derived stacks with sufficient deformation theory can be performed globally to yield an Atiyah L-infinity-algebroid whose splittings correctly encode higher-form connections.

What would settle it

An explicit principal infinity-bundle for which the space of order-p splittings of the Atiyah L-infinity-algebroid differs from the space of Cech-Deligne cocycles of the same degree would disprove the recovery statement.

Figures

Figures reproduced from arXiv: 2602.03441 by Joost Nuiten, Lukas M\"uller, Richard J. Szabo, Severin Bunk.

**Figure 1.** Figure 1: Maps from Sym⩽p R (TM[1]) to the cochain complex underlying the dg Lie algebroid bgl of the fibrant simplicial resolution bg of CˇAt(g). 114 [PITH_FULL_IMAGE:figures/full_fig_p114_1.png] view at source ↗

read the original 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. This can be cast equivalently as lifting the classifying map of a bundle on a manifold to the order $p$ truncation of the de Rham stack of the manifold. 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 \v{C}ech-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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper defines p-form connections on ∞-bundles as order-p splittings of Atiyah L∞-algebroids obtained from derived stacks, and shows this recovers Čech-Deligne cocycles for U(1) cases.

read the letter

Colleague, the main contribution here is a derived-stack definition of p-form connections on principal ∞-bundles. They differentiate maps from manifolds to derived stacks with enough deformation theory to get the Atiyah L∞-algebroid, then take its order-p splittings as the space of connections; this is also phrased as lifting the bundle's classifying map to the p-truncated de Rham stack. They check that the construction reproduces the usual Čech-Deligne description for higher U(1)-bundles and relate the resulting L∞-algebras to higher Courant algebroids. That gives a uniform language that could be useful in higher gauge theory and supergravity. The recovery of the known cocycle picture is a concrete consistency check rather than an afterthought, and the setup draws on standard derived geometry without obvious circularity. The soft spot is the global step: the Atiyah L∞-algebroid is supposed to arise globally from the formal differentiation and integration, but the abstract relies on “sufficient deformation theory” without spelling out an explicit local-to-global gluing argument for the L∞-structure or the splitting space. If the full text does not supply a direct verification that the sections match across charts independently of the model, that part could use more detail, though it is not obviously fatal given the standard machinery. This is for people already comfortable with L∞-algebroids, derived stacks, and higher structures in mathematical physics. A reader working on symmetries or anomalies in those areas would get value from the reorganization. I would send it to peer review; the definition is new enough and the consistency check is solid enough to justify referee time, even if some technical points on the global construction need tightening.

Referee Report

2 major / 2 minor

Summary. The paper introduces a general definition of higher-form connections on principal ∞-bundles in differential geometry by developing formal differentiation and integration of maps from smooth manifolds to derived stacks with sufficient deformation theory. This yields the Atiyah L∞-algebroid of a principal ∞-bundle, whose global sections form the L∞-algebra of the derived higher symmetry group. p-form connections are defined as order-p splittings of this algebroid (equivalently, lifts of the classifying map to the order-p truncation of the de Rham stack). The construction recovers the Čech-Deligne cocycle description of connections on higher U(1)-bundles and relates the L∞-algebras of derived symmetries of higher U(1)-bundles to higher Courant algebroids, with brief mentions of applications in higher gauge theory and supergravity.

Significance. If the global construction is made rigorous, the work supplies a derived-geometric unification of higher connections and symmetries that recovers standard cocycle descriptions as a consistency check and links L∞-algebras to higher Courant algebroids. The explicit recovery of Čech-Deligne data for U(1)-bundles and the relation to Courant structures are concrete strengths that could support applications in higher gauge theory.

major comments (2)

[§3.2] §3.2 (Atiyah L∞-algebroid construction): the claim that formal differentiation and integration of maps to derived stacks produces a globally defined Atiyah L∞-algebroid whose order-p splittings recover Čech-Deligne cocycles rests on the assumption that deformation theory yields consistent global sections. No explicit verification is given that the L∞-structure glues across charts or that the splitting space is independent of the deformation-theoretic model chosen.
[§4] §4 (Recovery of Čech-Deligne cocycles): the demonstration that the new p-form connections match the known notion for higher U(1)-bundles is load-bearing for the central claim, yet it inherits the unverified global gluing step from the Atiyah construction; without a local-to-global check, the equivalence remains formal rather than fully established.

minor comments (2)

[§2] The notation for derived stacks and ∞-bundles in §2 could be made more uniform with standard references in the field to improve readability.
[§3] A brief diagram or table summarizing the truncation orders and corresponding connection degrees would clarify the relation between order-p splittings and p-form connections.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the positive assessment of the significance of our work. We address each major comment below, providing clarifications and indicating where revisions will be made to enhance the rigor of the global constructions.

read point-by-point responses

Referee: [§3.2] §3.2 (Atiyah L∞-algebroid construction): the claim that formal differentiation and integration of maps to derived stacks produces a globally defined Atiyah L∞-algebroid whose order-p splittings recover Čech-Deligne cocycles rests on the assumption that deformation theory yields consistent global sections. No explicit verification is given that the L∞-structure glues across charts or that the splitting space is independent of the deformation-theoretic model chosen.

Authors: The construction of the Atiyah L∞-algebroid is carried out in the ∞-category of derived stacks, where the formal differentiation and integration are defined globally using the deformation theory of the stack. The gluing across charts is ensured by the sheaf property of the derived stack of maps and the L∞-algebroid being a sheaf of L∞-algebras. To make this explicit, we will add a detailed paragraph in §3.2 explaining the descent and gluing of the L∞-structure using the Čech nerve of the atlas. For the independence of the model, since all models are equivalent in the homotopy category of L∞-algebras, the space of splittings is well-defined up to equivalence. We will include a note on this. revision: partial
Referee: [§4] §4 (Recovery of Čech-Deligne cocycles): the demonstration that the new p-form connections match the known notion for higher U(1)-bundles is load-bearing for the central claim, yet it inherits the unverified global gluing step from the Atiyah construction; without a local-to-global check, the equivalence remains formal rather than fully established.

Authors: In §4, the equivalence is shown by computing locally that the order-p splitting corresponds to a Čech-Deligne cocycle, and the global lift ensures the cocycle condition is satisfied. However, we agree that an explicit local-to-global check would strengthen the argument. We will revise §4 to include a proof that the local data glues to a global cocycle using the properties of the de Rham stack truncation. This will establish the equivalence rigorously. revision: yes

Circularity Check

0 steps flagged

No significant circularity; recovery of Čech-Deligne provides independent check

full rationale

The derivation introduces a definition of p-form connections as order-p splittings of the Atiyah L∞-algebroid obtained from formal differentiation/integration of maps to derived stacks. It then shows this recovers the independently known Čech-Deligne cocycle description for higher U(1)-bundles. This equivalence is presented as a verification against external literature rather than a reduction by construction. No equations or steps reduce a 'prediction' to a fitted input, rename a known result as new, or rely on a self-citation chain whose content is unverified within the paper. The framework uses standard derived-stack machinery; any author prior work on deformation theory is not load-bearing for the central equivalence claim. The global gluing is asserted to follow from the deformation-theoretic assumptions, but the manuscript treats the Čech-Deligne match as confirmatory rather than definitional.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The construction rests on the existence of a well-behaved differentiation and integration functor for maps into derived stacks possessing sufficient deformation theory; this is treated as background rather than derived inside the paper. No explicit free parameters or newly invented physical entities appear in the abstract.

axioms (2)

domain assumption Maps from smooth manifolds to derived stacks admit formal differentiation and integration when the stacks have sufficient deformation theory.
Invoked to define the Atiyah L∞-algebroid; stated as the enabling step for the whole framework.
domain assumption The global sections of the Atiyah L∞-algebroid form the L∞-algebra of the derived higher symmetry group.
Central identification used to interpret splittings as connections.

pith-pipeline@v0.9.0 · 5541 in / 1706 out tokens · 39613 ms · 2026-05-16T07:40:43.209996+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages · 12 internal anchors

[1]

Holonomies for connections with values in L-infinity algebras

[AAS13] C. Arias Abad and F. Schätz. TheA ∞ de Rham theorem and integration of representations up to homotopy.IMRN, 2013(16):3790–3855, 2013.arXiv:1404.0727v2. [AC24] D. Alvarez and M. Cueca. Homological vector fields on differentiable stacks. 2024.arXiv: 2403.14871v1. [ADH21] A. Amabel, A. Debray, and P.J. Haine.Differential cohomology: Categories, chara...

work page internal anchor Pith review Pith/arXiv arXiv 2013
[2]

Bhardwaj, L.E

[BBFT+24] L. Bhardwaj, L.E. Bottini, L. Fraser-Taliente, L. Gladden, D.S.W. Gould, A. Platschorre, and H. Tillim. Lectures on generalized symmetries.Phys. Rept., 1051:1–87, 2024.arXiv:2307. 07547v1. [BE24] D. Berwick-Evans. Supersymmetric localization, modularity and the Witten genus.J. Diff. Geom., 126(2):401–430, 2024.arXiv:1911.11015v1. [BEBdBP24] D. B...

work page arXiv 2024
[3]

[BMS21] S

arXiv:2506.16657v1. [BMS21] S. Bunk, L. Müller, and R.J. Szabo. Smooth 2-group extensions and symmetries of bundle gerbes. Comm. Math. Phys., 384:1829–1911, 2021.arXiv:2004.13395v2. [BO19] O. Brahic and C. Ortiz. Integration of 2-term representations up to homotopy via 2-functors. Trans. Amer. Math. Soc., 372(1):503–543, 2019.arXiv:1608.00664v1. [Bry08] J...

work page arXiv 1911
[4]

Higher Gauge Theory

Reprint of the 1993 edition. 118 [BS07] J.C. Baez and U. Schreiber. Higher gauge theory. InCategories in algebra, geometry and math- ematical physics, volume 431 ofContemp. Math., pages 7–30. American Mathematical Society, Providence, RI, 2007.arXiv:math/0511710v2. [BS11] Y. Bi and Y. Sheng. On higher analogues of Courant algebroids.Sci. China A, 54:437–447,

work page internal anchor Pith review Pith/arXiv arXiv 1993
[5]

On higher analogues of Courant algebroids

arXiv:1003.1350v2. [BS14] J. Block and A.M. Smith. The higher Riemann–Hilbert correspondence.Adv. Math., 252:382–405,

work page internal anchor Pith review Pith/arXiv arXiv
[6]

Bunk and C.S

[BS23] S. Bunk and C.S. Shahbazi. Higher geometric structures on manifolds and the gauge theory of Deligne cohomology. 2023.arXiv:2304.06633v1. [BSS18] M. Benini, A. Schenkel, and U. Schreiber. The stack of Yang–Mills fields on Lorentzian manifolds. Comm. Math. Phys., 359(2):765–820, 2018.arXiv:1704.01378v2. [Bun21] S. Bunk. Gerbes in geometry, field theo...

work page arXiv 2023
[7]

[Bun22] S

arXiv:2102.10406v2. [Bun22] S. Bunk. TheR-local homotopy theory of smooth spaces.J. Homotopy Relat. Struct., 17(4):593– 650, 2022.arXiv:2007.06039v3. [Bun23] S. Bunk. Principal∞-bundles and smooth string group models.Math. Ann., 387(1-2):689–743, 2023.arXiv:2008.12263v3. [Bun25] S. Bunk.∞-Bundles. InEncyclopedia of Mathemtical Physics (Second Edition), vo...

work page arXiv 2022
[8]

[CCK22] S

arXiv:2503.14426v1. [CCK22] S. Chatterjee, A. Chaudhuri, and P. Koushik. Atiyah sequence and gauge transformations of a principal 2-bundle over a Lie groupoid.J. Geom. Phys., 176:104509, 2022.arXiv:2107.13747v2. [CCN22] D. Calaque, R. Campos, and J. Nuiten. Moduli problems for operadic algebras.J. London Math. Soc. (2), 106(4):3450–3544, 2022.arXiv:1912.1...

work page arXiv 2022
[9]

Fluxes in Exceptional Field Theory and Threebrane Sigma-Models

[CJLS19] A. Chatzistavrakidis, L. Jonke, D. Lüst, and R. J. Szabo. Fluxes in Exceptional Field Theory and Threebrane Sigma-Models.JHEP, 05:055, 2019.arXiv:1901.07775v3. [Col11] B. Collier. Infinitesimal symmetries of Dixmier–Douady gerbes. 2011.arXiv:1108.1525v1. [CR12] D. Carchedi and D. Roytenberg. Homological algebra for superalgebras of differentiable...

work page internal anchor Pith review Pith/arXiv arXiv 2019
[10]

Generalized Global Symmetries

Reprint of the 1999 edition. [GKSW15] D. Gaiotto, A. Kapustin, N. Seiberg, and B. Willett. Generalized Global Symmetries.JHEP, 02:172, 2015.arXiv:1412.5148v2. [GR17] D. Gaitsgory and N. Rozenblyum.A study in derived algebraic geometry. Vol. II. Deformations, Lie theory and formal geometry, volume 221 ofMathematical Surveys and Monographs. American Mathema...

work page internal anchor Pith review Pith/arXiv arXiv 1999
[11]

DG coalgebras as formal stacks

arXiv:math/9812034v1. [Hir03] P.S. Hirschhorn.Model categories and their localizations, volume 99 ofMathematical Surveys and Monographs. American Mathematical Society, Providence, RI,

work page internal anchor Pith review Pith/arXiv arXiv
[12]

Generalized Calabi-Yau manifolds

arXiv:math/0209099v1. [Hul07] C.M.Hull. GeneralisedGeometryforM-Theory.JHEP,07:079, 2007.arXiv:hep-th/0701203v1. [Joy19] D. Joyce. Algebraic geometry overC ∞-rings.Mem. Amer. Math. Soc., 260(1256):v+139,

work page internal anchor Pith review Pith/arXiv arXiv 2007
[13]

Algebraic Geometry over $C^\infty$-rings

arXiv:1001.0023v7. [Kan58] D.M. Kan. Functors involving c.s.s. complexes.Trans. Amer. Math. Soc., 87:330–346,

work page internal anchor Pith review Pith/arXiv arXiv
[14]

Free Lie algebroids and the space of paths

[Kap07] M. Kapranov. Free Lie algebroids and the space of paths.Selecta Math. (N.S.), 13(2):277–319, 2007.arXiv:math/0702584v3. [Kap15] M. Kapranov. Membranes and higher groupoids. 2015.arXiv:1502.06166v1. [KS20] H. Kim and C. Saemann. Adjusted parallel transport for higher gauge theories.J. Phys. A, 53(44):445206, 2020.arXiv:1911.06390v2. [KS24] A. Kraft...

work page internal anchor Pith review Pith/arXiv arXiv 2007
[15]

Lee and H

[LO23] D. Lee and H. Oberhauser. Random surfaces and higher algebra. 2023.arXiv:2311.08366v2. [LRU14] E. Lupercio, C. Rengifo, and B. Uribe. T-duality and exceptional generalized geometry through symmetries of dg-manifolds.J. Geom. Phys., 83:82–98, 2014.arXiv:1208.6048v2. [LRWZ23] D. Li, L. Ryvkin, A. Wessel, and C. Zhu. DifferentiatingL ∞-groupoids – Par...

work page arXiv 2023
[16]

Nikolaus, U

[NSS15a] T. Nikolaus, U. Schreiber, and D. Stevenson. Principal∞-bundles: General theory.J. Homotopy Relat. Struct., 10(4):749–801, 2015.arXiv:1207.0248v3. [NSS15b] T. Nikolaus, U. Schreiber, and D. Stevenson. Principal∞-bundles: Presentations.J. Homotopy Relat. Struct., 10(3):565–622, 2015.arXiv:1207.0249v1. [Nui18] J. Nuiten.Lie algebroids in derived di...

work page arXiv 2015
[17]

[Nui19b] J

arXiv:1712.03441v1. [Nui19b] J. Nuiten. Koszul duality for Lie algebroids.Adv. Math., 354:106750, 63, 2019.arXiv:1712. 03442v1. [Pri10] J. Pridham. Unifying derived deformation theories.Adv. Math., 224(3):772–826, 2010.arXiv: 0705.0344v7. [Pst23] P. Pstrągowski. Synthetic spectra and the cellular motivic category.Invent. Math., 232(2):553– 681, 2023.arXiv...

work page arXiv 2019
[18]

[Rog25] C.L

URL:https: //rezk.web.illinois.edu/i-hate-the-pi-star-kan-condition.pdf. [Rog25] C.L. Rogers. Higher differentiation via higher formal groupoids. 2025.arXiv:2512.12979v1. [Roy07] D. Roytenberg. On weak Lie 2-algebras.AIP Conf. Proc., 956(1):180–198, 2007.arXiv:0712. 3461v1. [RW98] D. Roytenberg and A. Weinstein. Courant Algebroids and Strongly Homotopy Li...

work page arXiv 2025
[19]

L_infinity algebras as 1-jets of simplicial manifolds (and a bit beyond)

URL:https:// ncatlab.org/schreiber/files/dcct170811.pdf. [Šev06] P. Ševera.L ∞ algebras as 1-jets of simplicial manifolds (and a bit beyond). 2006.arXiv: math/0612349v1. [Šev17] P. Ševera. Letters to Alan Weinstein about Courant algebroids. 2017.arXiv:1707.00265v1. [SN24] S. Schäfer-Nameki. ICTP lectures on (non-)invertible generalized symmetries.Phys. Re...

work page internal anchor Pith review Pith/arXiv arXiv 2006
[20]

Parallel Transport and Functors

[SW09] U. Schreiber and K. Waldorf. Parallel transport and functors.J. Homotopy Relat. Struct., 4:187– 244, 2009.arXiv:0705.0452v5. [Sza12] R. J. Szabo. Quantization of Higher Abelian Gauge Theory in Generalized Differential Cohomol- ogy.Proc. Science, 175:009, 2012.arXiv:1209.2530v1. [TD25] R. Tellez-Dominguez. Chern Correspondence for Higher Principal B...

work page internal anchor Pith review Pith/arXiv arXiv 2009
[21]

[Vor05] T. Voronov. Higher derived brackets and homotopy algebras.J. Pure Appl. Algebra, 202(1-3):133– 153, 2005.arXiv:math/0304038v3. [Wal10] K. Waldorf. Multiplicative bundle gerbes with connection.Differential Geom. Appl., 28(3):313– 340, 2010.arXiv:0804.4835v4. [Wal13] K. Waldorf. String connections and Chern–Simons theory.Trans. Amer. Math. Soc., 365...

work page internal anchor Pith review Pith/arXiv arXiv 2005