Twisted Spectral Triples without the First-Order Condition

Abstract

We extend twisted inner fluctuations to twisted spectral triples that do not meet the twisted first order condition, following what has been done in [6] for the non twisted case. We find a similar non-linear term in the fluctuation, and work out the twisted version of the semi-group of inner perturbations.

Minimal twist for the Standard Model in noncommutative geometry: The field content

“…(3.1), replacing the commutator for a twisted one [20]. In addition, if the twisted first-order condition does not hold, one should add a nonlinear term [13,22]. We thus consider the twisted-covariant Dirac operator…”

“…To guarantee that the twisted covariant operator (3.28) is self-adjoint, one assumes that the twisted 1-form A ð1Þ is self-adjoint (Proposition 3.8 in Ref. [22]) (actually, this is not a necessary condition, but requiring A ð1Þ to be selfadjoint makes sense viewing the fluctuation D → D A as a three-step process…”

“…By Proposition 4.2 in Ref. [22], the operator D A , viewed as a function of the components a i and b i of the twisted 1-form A ¼ A ð1Þ ¼ P i a i ½D; b i , transforms under a gauge transformation in the operator D A u , where…”

A critical survey of twisted spectral triples beyond the Standard Model

“…Therefore it becomes relevant to extend to the twisted case the results of [14] regarding inner fluctuations without the first-order condition. This has been done in [49], where it was shown that the removal of the twisted first-order condition yields a second order term in the twisted fluctuation (38), which is a straightforward adaptation of the term worked out in the non-twisted case.…”

“…Hence the necessity to adapt to the twisted case the fluctuations without first-order condition introduced in [14]. This has been done in [49] and is summarised in section 4.3.…”