Real Spectral Triples and Charge Conjugation

We construct a noncommutative geometry with generalised 'tangent bundle' from Fell bundle C*-categories (E) beginning by replacing pair groupoid objects (points) with objects in E. This provides a categorification of a certain class of real spectral triples where the Dirac operator D is constructed from morphisms in a category. Applications for physics include quantization via the tangent groupoid and new constraints on D finite (the fermion mass matrix).

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PRODUCT OF REAL SPECTRAL TRIPLES

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s021988781100597x ◽

2011 ◽

Vol 08 (08) ◽

pp. 1833-1848 ◽

Cited By ~ 18

Author(s):

LUDWIK DĄBROWSKI ◽

GIACOMO DOSSENA

Keyword(s):

Dirac Operator ◽

Clifford Algebra ◽

Finite Dimension ◽

Spectral Triples ◽

Inequivalent Representations ◽

Real Spectral Triples

We construct the product of real spectral triples of arbitrary finite dimension (and arbitrary parity) taking into account the fact that in the even case there are two possible real structures, in the odd case there are two inequivalent representations of the gamma matrices (Clifford algebra), and in the even-even case there are two natural candidates for the Dirac operator of the product triple.

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The Geometry of Quantum Lens Spaces: Real Spectral Triples and Bundle Structure

Mathematical Physics Analysis and Geometry ◽

10.1007/s11040-015-9179-4 ◽

2015 ◽

Vol 18 (1) ◽

Cited By ~ 2

Author(s):

Andrzej Sitarz ◽

Jan Jitse Venselaar

Keyword(s):

Lens Spaces ◽

Spectral Triples ◽

Real Spectral Triples ◽

Bundle Structure

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The graded product of real spectral triples

Journal of Mathematical Physics ◽

10.1063/1.4975410 ◽

2017 ◽

Vol 58 (2) ◽

pp. 023507 ◽

Cited By ~ 8

Author(s):

Shane Farnsworth

Keyword(s):

Spectral Triples ◽

Real Spectral Triples

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Gauge transformations for twisted spectral triples

Letters in Mathematical Physics ◽

10.1007/s11005-018-1099-3 ◽

2018 ◽

Vol 108 (12) ◽

pp. 2589-2626 ◽

Cited By ~ 5

Author(s):

Giovanni Landi ◽

Pierre Martinetti

Keyword(s):

Spectral Triples ◽

Gauge Transformations ◽

Twisted Spectral Triples

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2, 12, 117, 1959, 45171, 1170086, …: a Hilbert series for the QCD chiral Lagrangian

Journal of High Energy Physics ◽

10.1007/jhep01(2021)142 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Lukáš Gráf ◽

Brian Henning ◽

Xiaochuan Lu ◽

Tom Melia ◽

Hitoshi Murayama

Keyword(s):

Hilbert Series ◽

Charge Conjugation ◽

External Fields ◽

Dynkin Diagrams ◽

Non Linear ◽

Very High ◽

Chiral Lagrangian

Abstract We apply Hilbert series techniques to the enumeration of operators in the mesonic QCD chiral Lagrangian. Existing Hilbert series technologies for non-linear realizations are extended to incorporate the external fields. The action of charge conjugation is addressed by folding the $$ \mathfrak{su}(n) $$ su n Dynkin diagrams, which we detail in an appendix that can be read separately as it has potential broader applications. New results include the enumeration of anomalous operators appearing in the chiral Lagrangian at order p8, as well as enumeration of CP-even, CP-odd, C-odd, and P-odd terms beginning from order p6. The method is extendable to very high orders, and we present results up to order p16.(The title sequence is the number of independent C-even and P-even operators in the mesonic QCD chiral Lagrangian with three light flavors of quarks, at chiral dimensions p2, p4, p6, …)

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