The Atiyah–Singer index theorem

Abstract

The Atiyah-Singer index theorem, a landmark achievement of the early 1960s, brings together ideas in analysis, geometry, and topology. We recount some antecedents and motivations, various forms of the theorem, and some of its implications, which extend to the present. Contents 1. Introduction 1 2. Antecedents and motivations from algebraic geometry and topology 2 3. Antecedents in analysis 9 4. The index theorem and proofs 12 5. Variations on the theme 17 6. Heat equation proof 24 7. Geometric invariants of Dirac operators 29 8. Anomalies and index theory 38 About the author 45 References 45

QCD topology with electromagnetic fields and the axion-photon coupling

“…Hence, for small fields we expect a linear relation between the topological charge and ì 𝐸 • ì 𝐵. We can understand this behaviour further through the Atiyah-Singer index theorem [20]. The electromagnetic topological charge induced by a nonzero ì 𝐸 • ì 𝐵 contributes to the zero modes of the Dirac operator as…”

KK Theory and K Theory for Type II Strings Formalism

“…Now, for the linkage of 𝐾 − 𝑇ℎ𝑒𝑜𝑟𝑦 𝑡𝑜 𝐾 − Homology and c*-algebras for the locally compact Hausdorff spaces, there can be a relatable definition of the c*-algebra through noncommutative topology where there exists a detailed constructions to be discussed below [13,17,18,25] .…”

“…Towards the establishment of noncommutative topology as described above in the paper the relation between noncommutative topology with noncommutative geometry over a non-trivial prescriptions of 𝑐 *− Hilbert modules and Hausdorff space that gets channelized further to establish the 𝐾𝐾 − 𝑇ℎ𝑒𝑜𝑟𝑦 and Morita equivalence; a considerable fact is that for the proper extensions of 𝑐 *− algebras there is a defined category of the operator formalisms in the algebraic notions of 𝐾 − 𝑇ℎ𝑒𝑜𝑟𝑦 where it can be shown that for the parent group (taken before) ^ with the 𝑐 *− algebra, any reduced category for the completion of 𝑐 𝑟𝑒𝑑 * (^) formalism through a locally compact Topological group (denoting with a trivial notation just for the formulations) as ^' for a translation invariant norm through bounded functions; this 𝑐 𝑟𝑒𝑑 * (^) has an isomorphism for 𝑐 * (^) where any defined 𝑐 *− algebra can be expressed taking the 𝑐 𝑟𝑒𝑑 * (^) as a quotient of 𝑐 * (^) for the Hilbert space 𝐻 having the defined norm 𝐻 𝑙 2 there exists 5 − 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟𝑠 that connects the 𝐾 − 𝑇ℎ𝑒𝑜𝑟𝑦 𝑡𝑜 𝐾 − Homology for making 𝐾𝐾 − 𝑇ℎ𝑒𝑜𝑟𝑦 which provides a relation to Gromov hyperbolic groups [26] along with the groups that defined ^' for a translation invariant norm through bounded functions in 𝑆𝐿 3 (ℤ)along with other 𝑟𝑎𝑛𝑘 − 1 Lie Groups and other discrete Lie Groups 𝑆𝑂(𝑛, 1) and 𝑆𝑈(𝑛, 1) with Gromov's 𝑎 − 𝑇 − 𝑚𝑒𝑛𝑎𝑏𝑙𝑒 property for the assembly mapping parameter ℸ (which will be extremely useful later in the paper) for isomorphism having the representation of [16,18,25] ,…”

A Panorama Of Physical Mathematics c. 2022

“…These are related to parity anomalies [475][476][477][478] and are known as Dai-Freed theories. For two recent reviews summarizing the status of these theories see [479,480]. The general theory of nontopological invertible theories is under rapid development; a small sample of very recent papers is [481,482].…”