Quadratic functions in geometry, topology, and M-theory

By M. Hopkins, I. Singer. 2002

428 citations recorded by the source corpus.

We describe an interpretation of the Kervaire invariant of a Riemannian manifold of dimension $4k+2$ in terms of a holomorphic line bundle on the abelian variety $H^{2k+1}(M)\otimes R/Z$. Our results are inspired by work of Witten on the fivebrane partition function in $M$-theory (hep-th/9610234, hep-th/9609122). Our construction requires a refinement of the algebraic topology of smooth manifolds better suited to the needs of mathematical physics, and is based on our theory of differential functions. These differential functions generalize the differential characters of Cheeger-Simons, and the bulk of this paper is devoted to their study.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/20b621888f68616ed362949389abeea189c32f60

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. Clay Córdova
2. Ho Tat Lam
3. D. Gaiotto
4. Z. Komargodski
5. N. Seiberg
6. D. Freed
7. U. Bunke
8. G. Moore

In Mathematical Physics

1. U. Bunke
2. G. Moore
3. N. Seiberg
4. D. Freed
5. Clay Córdova
6. D. Gaiotto
7. Z. Komargodski
8. Thomas Schick

Citations in the corpus

• Holographic Action for the Self-Dual Field (2006, background): “In particular, we hope that our paper will make clear the physical relevance of the main theorem of Hopkins and Singer in [29].”
• Relative Quantum Field Theory (2012, background): “similarly expect that the evenness in Data 5.1(ii) can be omitted to define a larger class of theories. 15It will certainly include an orientation and, at least in many cases, an integral Wu structure [HS]. 16 D. S. FREED AND C. TELEMAN A useful formal picture is to imagine X g as constructed from cl”
• Axiomatic characterization of ordinary differential cohomology (2007, background): “In [9], an analog of differential characters, called differential cohomology, is constructed for any generalized cohomology theory (meaning that the Eilenberg–Steenrod axioms are satisfied except that the groups of a point may not be concentrated in degree zero).”
• Heisenberg groups and noncommutative fluxes (2006, methodology): “The generalization to gaugetheorieswith ‘-form eldstrengthsisnow im m ediate: 7 D e nition TheCheeger-Sim onsgroup ofdi erentialcharacters,denoted H ‘(M ),isthe subgroup ofallhom om orphism sfrom closed (‘ 1)-cyclesto U (1): H ‘(M ) Hom (Z‘ 1 (M );U (1)) (2:4) satisfying the following property: For…”
• The Uncertainty of Fluxes (2006, methodology): “7 To define gauge fields and gauge transformations one needs to introduce groupoids which represent the differential cohomology groups. Several models are considered in [ HS ].”
• Anomalies in the space of coupling constants and their dynamical applications II (2019, background): “rs by Cheeger-Simons [71] in the early 1970s. Dierential cohomology groups may be viewed as an analog of Deligne cohomology [72] for smooth manifolds; see also [73, x6.3]. The work of Hopkins-Singer [46] extends the dierential theory to generalized cohomology theories and also develops a version wit”
• Four-Dimensional Wall-Crossing via Three-Dimensional Field Theory (2008, background): “(7)Such quadratic refinements frequently appear in the precise formulations of self-dual gauge theories [33, 34, 35, 36].”
• Framed BPS states (2010, neutral): “In the path integral of the abelian six-dimensional theory the sum over topological sectors is weighted by a quadratic refinement of the mod-two intersection form on H(3)(M6;Z) [39, 40, 41, 42].”
• Comments on abelian Higgs models and persistent order (2017, methodology): “One way to describe the U(1) 2-form gauge fieldK in a way that mathematically separates small and large gauge transformations is using the theory of differential cocycles [36].”
• Structured Vector Bundles Define Differential K-Theory (2008, background): “Following their approach, Freed, as well as Hopkins and Singer and perhaps others like ourselves, were aware that a model for differential K-theory could be constructed based on pairs (E,O), where E is a bundle with connection and O is a total odd form with an equivalence relation generalizing that ”