Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2312.10516

Bordism categories and orientations of gauge theory moduli spaces

Dominic Joyce·Markus Upmeier
View original source ↗Read source document ↗Export metadata ↓

Abstract

This is the second paper of a series that develops a bordism-theoretic point of view on orientations in enumerative geometry. The first paper is arXiv:2312.06818. This paper focuses on those applications to gauge theory that can be established purely using formal arguments and calculations from algebraic topology. We prove that the orientability of moduli spaces of connections in gauge theory for all principal $G$-bundles $P\to X$ over compact spin $n$-manifolds at once is equivalent to the vanishing of a certain morphism $Ω_n^{\rm Spin}(\mathcal L BG)\to{\mathbb Z}_2$ on the $n$-dimensional spin bordism group of the free loop space of the classifying space of $G,$ and we give a complete list of all compact, connected Lie groups $G$ for which this holds. Moreover, we apply bordism techniques to prove that mod-$8$ Floer gradings exist for moduli spaces of $G_2$-instantons for all principal SU(2)-bundles. We also prove that there are canonical orientations for all principal U$(m)$-bundles $P\to X$ over compact spin $8$-manifolds satisfying $c_2(P)-c_1(P)^2=0.$ The proof is based on an interesting relationship to principal $E_8$-bundles. These canonical orientations play an important role in many conjectures about Donaldson-Thomas type invariants on Calabi-Yau $4$-folds, and resolve an apparent paradox in these conjectures.

Explore related subjects

math.ATmath.AGmath.DG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Dominic Joyce, Markus Upmeier. 2025-03-27. Bordism categories and orientations of gauge theory moduli spaces. https://arxiv.org/abs/2312.10516

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The fifth algebraic transfer in generic degrees and validation of a localized Kameko's conjecture

This paper develops our previous works concerning the classical Peterson hit problem for the polynomial algebra on five variables over the mod--2 Steenrod algebra $\mathscr A$ in a generic family of degrees, together with applications to the fifth Singer algebraic transfer and a localized variation of Kameko's conjecture. As a topological illustration of the usefulness of the Steenrod algebra, we prove that $\mathbb{C}P^4/\mathbb{C}P^2$ and $\mathbb{S}^6\vee \mathbb{S}^8$ are not homotopy equivalent by showing that their mod--2 cohomologies are not isomorphic as $\mathscr A$-modules, and we further determine the homotopy type of the quotient $\mathbb{C}P^n/\mathbb{C}P^{\,n-2}$ for all $n\ge 3$. For the generic degrees under consideration, we determine the relevant cohit spaces and describe the associated $GL(5,\mathbb F_2)$-module structure. As a consequence, the fifth algebraic transfer is an isomorphism in an explicit infinite family of internal degrees. These results were independently verified by implementations in \texttt{SageMath} and \texttt{OSCAR}. We also study a localized form of Kameko's conjecture concerning the dimensions of the indecomposables $\mathbb F_2\otimes_{\mathscr A}\mathbb F_2[x_1,\ldots,x_m]$ relative to parameter vectors, and prove that this conjecture holds for all $m\ge 1$ in certain degrees.

math.AT↗
arXiv2026

A rigidity theorem for $π_*$-étale $\mathbb E_k$-algebras

We prove a rigidity theorem for $π_*$-étale $\mathbb E_k$-algebras over an $\mathbb E_{k+1}$-ring spectrum: the category of $π_*$-étale extensions of an $\mathbb E_k$-algebra is identified with the ordinary category of étale Dirac algebras over its graded homotopy Dirac ring. The proof develops a relative Goerss-Hopkins type obstruction theory in synthetic spectra, including an $I$-complete version. As an application, the completed obstruction theory constructs the $I_n$-complete $\mathbb E_3$-$MU_{(p)}$-algebra realization of the Lubin-Tate theory, hence an $\mathbb E_4$-orientation $MU_{(p)}\to E_n$.

math.AT↗
arXiv2026

Automated proofs of unstable Adams differentials

We present a computer-based approach to computing differentials in the unstable Adams spectral sequence by systematically applying the unstable Leibniz rule and naturality with respect to maps in the EHP sequence. We record our results in tables of upper and lower bounds on the orders of 2-primary unstable homotopy groups of spheres through the unstable 50-stem. We provide examples of proofs for several differentials and give a guide to interpreting the associated unstable Adams charts and flow chart diagrams for differential proofs.

math.AT↗