nLab adjusted Weil algebra

Idea

The traditional notion of connection on a principal bundle with given structure Lie algebra $\mathfrak{g}$ has a slick dg-algebraic-formulation in terms of the Chevalley-Eilenberg algebra $CE(\mathfrak{g})$ and the Weil algebra $W(\mathfrak{g})$ of $\mathfrak{g}$ (due to Henri Cartan 1950).

These concepts of Cartan connection may be generalized to $L_\infty$-algebras (originally so in SSS09, SSS12, FSS12, which was the starting point for the more comprehensive development in dcct12) – see at connection on a smooth principal $\infty$-bundle. In fact, noticing (see here) that the Chevalley-Eilenberg algebra-construction $CE(-)$ constitutes a full subcategory-embedding of the 1-category of finite type $L_\infty$-algebras into the opposite 1-category of dgc-algebras, there is an immediate $L_\infty$-analog of the notion of Weil algebra (SSS09, Def. 16).

However, this is a 1-category theoretic analog, hence is a particularly strict model, defined up to isomorphism, of what the would-be $\infty$-Weil algebra of an $L_\infty$-algebra should be. The latter will only be well-defined up to compatible quasi-isomorphism.

The need to pass to compatible deformations of $L_\infty$-Weil algebras for a satisfactory definition of string 2-connections (1-brane connections) and analogous higher notions (5-brane-connection, 9-brane, etc.) was first discussed around SSS09, Prop. 21 (and used to exhibit the higher gauge-theoretic nature of the Green-Schwarz mechanism in SSS12, §3.2), and was justified there by the condition that the canonical invariant polynomial in that situation does lift as expected.

A more general discussion of the necessary adjustments of $L_\infty$-Weil algebras was then given in Saemann & Schmidt 20, who observe that a good general condition to impose is that the induced BRST complexes of $L_\infty$-connections (SSS09, §9.3) remain well-defined when the antifields and auxiliary fields are required to strictly vanish. These authors introduce the terminology adjusted Weil algebras (Saemann & Schmidt 20, Def. 4.2) and the resulting adjusted $L_\infty$-connections and adjusted higher parallel transport (Kim & Saemann 2020).

Concretely, the choice of compatible deformation of the Weil algebra determines the Bianchi identities on the curvature characteristic forms of corresponding $L_\infty$-algebra valued differential forms (these Bianchi identities are embodied by the differential in the Weil algebra restricted to shifted generators) and without adjustment the Bianchi identities are stronger than (i.e. just special cases of what) they ought to be.

Notice that, while the Weil algebra by itself is contractible homotopy type, its adjustments along quasi-isomorphisms must satisfy horizontality constraints (such as the vanishing of those antifields) which makes this a non-trivial procedure.

While the characterization of adjusted Weil algebras for $L_\infty$-algebra in Saemann & Schmidt 20, Def. 4.2 clearly (generalizes and) conceptually improves on SSS09, Prop. 21, and while in applications it clearly gives the right answers (the correct higher Bianchi identities), what is still missing is a purely homotopy theoretic justification. Since in practice these adjusted Weil algebras behave and are used much like resolutions by minimal fibrations in model category-theory, it is natural to wonder if there is the structure of a homotopical fibration category on the Weil algebra choices for $L_\infty$-algebras, such that this is indeed the case. This question is open.

On the other hand, Borsten, Kim & Saemann 2021 argue that adjustment is naturally understood after embedding $L_\infty$-algebras within $E L_\infty$-algebras (and that this is what exhibits tensor hierarchies as a higher gauge theory-phenomenon).

Related entries

• principal infinity-connection

• nonabelian differential cohomology

• supergravity Lie 6-algebra

References

The original discussion for the special case of string 2-connections and their higher analogs (such as fivebrane 6-connections, Ninebrane 10-connections etc.):

• Urs Schreiber, Obstructions to $n$-Bundle Lifts Part II (Oct 2007) [bottom right corner in this hand-drawn diagram: pdf]

• Hisham Sati, Urs Schreiber, Jim Stasheff, Def. 23 on p. 47 with Prop. 21 on p. 48 in: $L_{\infty}$ algebra connections and applications to String- and Chern-Simons $n$-transport, in Quantum Field Theory, Birkhäuser (2009) 303-424 [arXiv:0801.3480, doi:10.1007/978-3-7643-8736-5_17]

• Hisham Sati, Urs Schreiber, Jim Stasheff: middle boxes on p. 28 and p. 32 of: Twisted Differential String and Fivebrane Structures, Communications in Mathematical Physics 315 1 (2012) 169-213 [arXiv:0910.4001, doi:10.1007/s00220-012-1510-3))]

• Domenico Fiorenza, Urs Schreiber, Jim Stasheff, p. 77 and p. 80 of: Čech cocycles for differential characteristic classes, Advances in Theoretical and Mathematical Physics, 16 1 (2012) 149-250 [arXiv:1011.4735, doi:10.1007/BF02104916]

• Urs Schreiber, Def. 5.2.91 on p. 645 in: differential cohomology in a cohesive topos (2013-)

With hindsight, an early example of such adjustment (namely for the “supergravity Lie 6-algebra”, see there), is given in:

• Leonardo Castellani, Riccardo D'Auria, Pietro Fré, vol 2, III (8.24) of: Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991) [doi:10.1142/0224, epdf]

(recognized as a “modified Weil algebra” in revision 8, 2011, long before the “adjusted”-terminology was proposed).

Characterization of adjusted $\mathfrak{g}$-Weil algebras by requiring well-behaved BRST-complexes for $L_\infty$-algebra valued differential forms:

• Christian Saemann, Lennart Schmidt, Towards an M5-Brane Model II: Metric String Structures, Fortschr. Phys. 68 (2020) 2000051 [arXiv:1908.08086, doi:10.1002/prop.202000051]

Revisiting the special case of the string Lie 2-algebra:

• Lennart Schmidt, Twisted Weil Algebras for the String Lie 2-Algebra, in Christian Saemann, Urs Schreiber, Martin Wolf (eds.) Higher Structures in M-Theory Durham Symposium 2018, Fortschritte der Physik 2019 [arXiv:1903.02873, doi:10.1002/prop.201910016]

Application to higher parallel transport:

• Hyungrok Kim, Christian Saemann, Adjusted Parallel Transport for Higher Gauge Theories, J. Phys. A 52 (2020) 445206 [arXiv:1911.06390, doi:10.1088/1751-8121/ab8ef2]

Relation to $E L_\infty$-algebras and tensor hierarchies:

• Leron Borsten, Hyungrok Kim, Christian Saemann. $E L_\infty$-algebras, Generalized Geometry, and Tensor Hierarchies [arXiv:2106.00108]

Survey and review:

• Christian Saemann, Adjusted Higher Gauge Theory: Connections and Parallel Transport Lisbon (2021) [pdf, pdf]

• Christian Saemann, $E L_\infty$-algebras, Generalized Geometry, and Tensor Hierarchies, talk at SFT@Cloud 2021 (2021) [pdf, pdf]

• Christian Saemann, Atiyah Algebroids for Higher and Groupoid Gauge Theories, talk at M-Theory and Mathematics, CQTS (2024) [pdf]

Application to geometric refinement of topological T-duality:

• Hyungrok Kim, Christian Saemann, T-duality as Correspondences of Categorified Principal Bundles with Adjusted Connections [arXiv:2303.16162]