A categorified excision principle for elliptic symbol families

Upmeier, Markus

doi:10.48550/arxiv.1901.10818

Cited by 4 publications

(11 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This is the third of four papers: Upmeier [27], Joyce-Tanaka-Upmeier [21], this paper, and Cao-Gross-Joyce [7], on orientability and canonical orientations for gauge-theoretic moduli spaces. The first [27] proves the Excision Theorem (see Theorem 2.15 below), which relates orientations on different moduli spaces.…”

mentioning

confidence: 89%

“…Then our main object of study, the orientation torsor Or E of an SU(m)-bundle E → X, is introduced along with its basic properties. We recall from [27] the excision technique from index theory in the context of orientations. It can be regarded as extending the functoriality of orientation torsors from globally defined isomorphisms to local ones.…”

Section: Outline Of the Papermentioning

confidence: 99%

“…Donaldson first applied excision to gauge theory in [9], see also [10, §7]. In [27], the second author observes that on the level of orientations these ideas can be formalized into a 'categorification' of the classical calculus for the numerical index. Here is [27, Th.…”

Section: Excisionmentioning

confidence: 99%

“…This paper studies orientations of moduli spaces in dimension 7. It uses results from [21,27], but is self-contained and can be read independently. The sequel [7] concerns dimension 8.…”

mentioning

confidence: 99%

See 3 more Smart Citations

Canonical orientations for moduli spaces of $G_2$-instantons with gauge group SU(m) or U(m)

Joyce,

Upmeier

2018

Preprint

Self Cite

View full text Add to dashboard Cite

Suppose (X, g) is a compact, spin Riemannian 7-manifold, with Dirac operator /Let G be SU(m) or U(m), and E → X be a rank m complex bundle with G-structure. Write BE for the infinite-dimensional moduli space of connections on E, modulo gauge. There is a natural principal Z2-bundle O is the Z2-bundle of orientations on M G 2 E . Thus, our theorem induces canonical orientations on all such G2-instanton moduli spaces M G 2 E . This contributes to the Donaldson-Segal programme [11], which proposes defining enumerative invariants of G2-manifolds (X, ϕ, g) by counting moduli spaces M G 2 E , with signs depending on a choice of orientation.

show abstract

mentioning

confidence: 89%

Section: Outline Of the Papermentioning

confidence: 99%

Section: Excisionmentioning

confidence: 99%

“…This paper studies orientations of moduli spaces in dimension 7. It uses results from [21,27], but is self-contained and can be read independently. The sequel [7] concerns dimension 8.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Canonical orientations for moduli spaces of $G_2$-instantons with gauge group SU(m) or U(m)

Joyce,

Upmeier

2018

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Equations (2.3) and (2.5) are examples of the kind of explicit formula relating orientations referred to in Problem 1.3(c). See also [62,Prop. 3.5(ii)].…”

Section: Orientations On Products Of Moduli Spacesmentioning

confidence: 99%

On orientations for gauge-theoretic moduli spaces

Joyce¹,

Tanaka²,

Upmeier³

2020

Advances in Mathematics

Self Cite

View full text Add to dashboard Cite

Let X be a compact manifold, G a Lie group, P → X a principal G-bundle, and BP the infinite-dimensional moduli space of connections on P modulo gauge, as a topological stack. For a real elliptic operator E• we previously studied orientations on the real determinant line bundle over BP , twisting E• by connections ∇ Ad(P ) . These are used to construct orientations in the usual sense on smooth gauge theory moduli spaces, and have been extensively studied since the work of Donaldson [14-16].Here we consider complex elliptic operators F• and introduce the idea of spin structures, square roots of the complex determinant line bundle of F• twisted over BP . These may be used to construct spin structures in the usual sense on smooth complex gauge theory moduli spaces. We study the existence and classification of such spin structures.Our main result identifies spin structures on X with orientations on X × S 1 . Thus, if P → X and Q → X × S 1 are principal G-bundles with Q| X×{1} ∼ = P , we relate spin structures on (BP , F•) to orientations on (BQ, E•) for a certain class of operators F• on X and E• on X × S 1 .Combined with [24], we obtain canonical spin structures for positive Diracians on spin 6-manifolds and gauge groups G = U(m), SU(m). In a sequel [25] we will apply this to define canonical orientation data for all Calabi-Yau 3-folds X over C, as in Kontsevich and Soibelman [27, §5.2], solving a long-standing problem in Donaldson-Thomas theory.

show abstract

Orientability of moduli spaces of Spin(7)-instantons and coherent sheaves on Calabi–Yau 4-folds

Cao¹,

Gross²,

Joyce³

2020

Advances in Mathematics

View full text Add to dashboard Cite

Suppose (X, Ω, g) is a compact Spin(7)-manifold, e.g. a Riemannian 8-manifold with holonomy Spin (7), or a Calabi-Yau 4-fold. Let G be U(m) or SU(m), and P → X be a principal G-bundle. We show that the infinite-dimensional moduli space BP of all connections on P modulo gauge is orientable, in a certain sense. We deduce that the moduli space M Spin(7) P ⊂ BP of irreducible Spin(7)-instanton connections on P modulo gauge, as a manifold or derived manifold, is orientable. This improves theorems of Cao and Leung [9] and Muñoz and Shahbazi [42].If X is a Calabi-Yau 4-fold, the derived moduli stack M of (complexes of) coherent sheaves on X is a −2-shifted symplectic derived stack (M, ω) by Pantev-Toën-Vaquié-Vezzosi [46], and so has a notion of orientation by Borisov-Joyce [7]. We prove that (M, ω) is orientable, by relating algebro-geometric orientations on (M, ω) to differential-geometric orientations on BP for U(m)-bundles P → X, and using orientability of BP .This has applications to defining Donaldson-Thomas type invariants counting semistable coherent sheaves on a Calabi-Yau 4-fold, as in Donaldson and Thomas [15], Cao and Leung [8], and Borisov and Joyce [7].

show abstract

A categorified excision principle for elliptic symbol families

Cited by 4 publications

References 16 publications

Canonical orientations for moduli spaces of $G_2$-instantons with gauge group SU(m) or U(m)

Canonical orientations for moduli spaces of $G_2$-instantons with gauge group SU(m) or U(m)

On orientations for gauge-theoretic moduli spaces

Orientability of moduli spaces of Spin(7)-instantons and coherent sheaves on Calabi–Yau 4-folds

Contact Info

Product

Resources

About