Actions for twisted spectral triple and the transition from the Euclidean to the Lorentzian

Abstract

This is a review of recent results regarding the application of Connes' noncommutative geometry to the Standard Model, and beyond. By twisting (in the sense of Connes-Moscovici) the spectral triple of the Standard Model, one does not only get an extra scalar field which stabilises the electroweak vacuum, but also an unexpected 1-form field. By computing the fermionic action, we show how this field induces a transition from the Euclidean to the Lorentzian signature. Hints on a twisted version of the spectral action are also briefly mentioned. IntroductionNoncommutative geometry "a la Connes" [10] allows to obtain the Lagrangian of the Standard Model of elementary particles -including the Higgs sector -minimally coupled with Einstein-Hilbert action (in Euclidean signature) from geometrical principles. In addition, it offers some guidelines to go beyond the Standard Model by playing with the mathematical rules of the game (for a recent review, see [9]).Early attempts "beyond the SM" were considering new fermions (see e.g.[28] and other papers of the same author). One may also relax one of the axioms of noncommutative geometry, the first-order condition discussed below [8]; or modify another axiom regarding the real structure (also discussed below) [3,4]. Other proposals are based on some Clifford bundle structure [16], or non-associativity [2]. Here we focus on a model consisting in twisting the original noncommutative geometry [19,23,22].From the examples listed above, all but the first are minimal extensions of the Standard Model: they allow to produce the kind of extra scalar field σ suggested by particle physicists to stabilise the electroweak-vacuum (which also permits to make the calculation of the Higgs mass in noncommutative geometry compatible with its experimental value), without adding new fermions.By using twisted noncommutative geometry, one gets in addition an supplementary piece, namely a 1-form field, which surprisingly turns out to be related to the transition from Euclidean to Lorentzian signature.We give an overview of these results below, beginning in §2 with a recalling on the Standard Model in noncommutative geometry. Then we summarise in §3 how to apply a twist to the geometry, and why this is related to a transition from the Euclidean to the Lorentzian. In §4 we show how this transition is actually realised at the level of the fermionic action. We also stress some projects regarding the spectral (i.e. bosonic) action.1

Twisted Spectral Triples without the First-Order Condition

“…Whether this happens in other examples of minimal twist (as for the U (1) × U (2) model above should be investigated in a systematic way; but this requires first to stabilise a definition for the spectral action in a twisted context. Some ideas have been proposed in [16] but deserve more study, especially in the light of a possible signature transition towards the lorentzian [15] [27].…”

“…This is the same law 4.6 as when the twisted first-order condition holds. As a consequence, the gauge invariance of the fermionic action defined in [15,16] still holds, even if the twisted first-order condition is violated.…”

A critical survey of twisted spectral triples beyond the Standard Model

“…And more importantly, does the identification between twisted fluctuation of / ∂ and the 4-momentum still makes sense for the bosonic part of the action, given by the spectral action? Not to mention that the definition of the latter in a twisted context has not been established yet [31].…”