Clifford Modules

Atiyah, Michael; Bott, Raoul; Shapiro, Alexander

doi:10.1007/978-1-4612-5367-9_1

Cited by 11 publications

(38 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Namely, Remark 7.32 implies that pS ' S˚, ∇ ∇q has a canonical nonflat trivialization over the subset M pSq o Ă M pSq of nondegenerate mass pairings. (This uses [93] and [94, §4].) The nonflat trivialization lifts the deformation class of the anomaly-the cohomology class underlying (7.43)-to a relative class in IZp1q d`2 pX, X o q, where X o " MSpin RiemˆM pSq o .…”

Section: Massive Spinor Fields Partmentioning

confidence: 99%

Anomalies in the space of coupling constants and their dynamical applications I

et al. 2020

View full text Add to dashboard Cite

It is customary to couple a quantum system to external classical fields. One application is to couple the global symmetries of the system (including the Poincaré symmetry) to background gauge fields (and a metric for the Poincaré symmetry). Failure of gauge invariance of the partition function under gauge transformations of these fields reflects 't Hooft anomalies. It is also common to view the ordinary (scalar) coupling constants as background fields, i.e. to study the theory when they are spacetime dependent. We will show that the notion of 't Hooft anomalies can be extended naturally to include these scalar background fields. Just as ordinary 't Hooft anomalies allow us to deduce dynamical consequences about the phases of the theory and its defects, the same is true for these generalized 't Hooft anomalies. Specifically, since the coupling constants vary, we can learn that certain phase transitions must be present. We will demonstrate these anomalies and their applications in simple pedagogical examples in one dimension (quantum mechanics) and in some two, three, and four-dimensional quantum field theories. An anomaly is an example of an invertible field theory, which can be described as an object in (generalized) differential cohomology. We give an introduction to this perspective. Also, we use Quillen's superconnections to derive the anomaly for a free spinor field with variable mass. In a companion paper we will study four-dimensional gauge theories showing how our view unifies and extends many recently obtained results.2 In condensed matter physics, symmetry protected topological orders (SPTs) are also characterized at low energies by such actions. Depending on the precise definitions and context, "SPT" may be synonymous with "invertible field theory", or may instead refer to the deformation class of an invertible field theory, i.e. the equivalence class of invertible theories obtained by continuously varying parameters.3 In certain cases, there is no Y such that BY " X and A on X is extended into Y . Then, one can construct an anomaly free partition function by assuming that X is a component of the boundary of Y and Y has additional boundary components.

show abstract

Section: Massive Spinor Fields Partmentioning

confidence: 99%

Anomalies in the space of coupling constants and their dynamical applications I

et al. 2020

View full text Add to dashboard Cite

show abstract

“…In presence of symmetries that act non-trivially in φ glob this is not the case. Gauge symmetries can be treated as described in [6,5,1], where, essentially, the thimble is defined modulo gauge transformations. But this is not suitable to study the possibility of spontaneous symmetry breaking (SSB).…”

Section: Formulationmentioning

confidence: 99%

The sign problem and the Lefschetz thimble

Cristoforetti

Renzo

Scorzato

2013

J. Phys.: Conf. Ser.

View full text Add to dashboard Cite

show abstract

“…The pullback bundle i * m T r * (j m ) coincides with (3.5). Then BSpin(m) is simply connected for m ≥ 2: this is the consequence of the exact sequence of homotopy groups of the fibration Spin(m) → ESpin(m) → BSpin(m) and the fact that Spin(m) is path connected for m ≥ 2 [1]. So that only the first component of the transfer homomorphism in Lemma 3.4 is relevant.…”

Section: Spin Bundlesmentioning

confidence: 99%

“…Recall from [1] the groups Spin(n) and P in(n) that operate on R n by vector representation. We will use an octonionic representation of Clifford algebra Cl(8, 0) and refer to [10].…”

Section: Preliminariesmentioning

confidence: 99%