Torsion forms and particle physics

[itssilva]

Hi; I've been doing some research for my master's on gauge theory in the language of fiber bundles, and something occurred to me. Both GR and the Standard Model (SM) can be described in terms of connections (potentials) and curvatures (field strengths), but there's this generalization of GR, Einstein-Cartan (EC) theory, which incorporates a new object, the torsion 2-form, as something that might have physical importance, as it seems to be necessary to deal with couplings with fields with spin (e.g., Dirac fermions). Regardless of that being correct or not, torsion is a geometrical concept that (theoretically) can also be adopted in the principal bundle formalism of SM and Yang-Mills theories in general, and, carrying on the analogy with EC, we could talk about the spin of, say, SU(N) (not the usual "Lorentzian" spin, but the more general definition of Spin as double cover of a rotation group or something like that), introduce torsion, couple terms à la EC, and then derive new, potentially observable coupling terms involving the SU(N)-spin fields, right? However, I haven't found anything in the literature that deals with torsion in a field-theoretical context without involving EC or another theory of gravitation; does anyone know of such work, or an argument why torsion (in the generalized, purely geometrical sense referred above) isn't important to particle physics research?

 

[dextercioby]

I am not an expert on this (@haushofer, @samalkhaiat , @fzero ), but particle physics, at least when it comes to QFT of the SM, is governed by the Coleman-Mandula no-go theorem, which prevents a non-trivial coupling of background (flat Minkowski) geometry to matter content (gauge fields). So, yes, Minkowski space-time is torsionless, the Yang-Mills fiber bundle is torsionless (?), but they are unrelated under the SM. A study of supersymmetric QFT/SuGra is not a domain where I can formulate an opinion.

 

[itssilva]

I am not an expert on this (@haushofer, @samalkhaiat , @fzero ), but particle physics, at least when it comes to QFT of the SM, is governed by the Coleman-Mandula no-go theorem, which prevents a non-trivial coupling of background (flat Minkowski) geometry to matter content (gauge fields). So, yes, Minkowski space-time is torsionless, the Yang-Mills fiber bundle is torsionless (?), but they are unrelated under the SM. A study of supersymmetric QFT/SuGra is not a domain where I can formulate an opinion.

Thanks for the reply; but, in your own words, I don't think the no-go theorem would apply here because I asked about torsion in the G-bundle, not the frame bundle of GR (spacetime is still Minkowski). Also, SuSyQFT is not necessary, as this question is more fundamental. Another way of asking: if the Riemann tensor is to the Faraday tensor as the Levi-Civita connection is to the EM potential, what is the Yang-Mills equivalent of torsion? By making a (generalized ) Yang-Mills theory with torsion, what kinds of prediction can one make?

 

[samalkhaiat]

It has been done long time ago. And in the late 80's identical set up was formulated in string theory and other M theories. Look up the work of T.L. Curtright, C.K. Zachos and L. Mezinescu. In particular
T.L. Curtright, C.K. Zachos, Phys. Rev. Lett. 53 (1984) 1799.
E. Braaten, T.L. Curtright, C.K. Zachos, Nucl. Phys. B (?) (1985).

 

[itssilva]

It has been done long time ago. And in the late 80's identical set up was formulated in string theory and other M theories. Look up the work of T.L. Curtright, C.K. Zachos and L. Mezinescu. In particular
T.L. Curtright, C.K. Zachos, Phys. Rev. Lett. 53 (1984) 1799.
E. Braaten, T.L. Curtright, C.K. Zachos, Nucl. Phys. B (?) (1985).

Thanks, but I'd also be interested to see a study done within old-school QFT - I'm not much of a string guy, nor do I worry about stuff like SuSy.

 

[Igael]

Try Correspondence between Einstein-Yang-Mills-Lorentz systems and dynamical torsion models

In the framework of Einstein-Yang-Mills theories, we study the gauge Lorentz group and establish a particular equivalence between this case and a certain class of theories with torsion within Riemann-Cartan space-times. This relation is specially useful in order to simplify the problem of finding exact solutions to the Einstein-Yang-Mills equations. Solutions for non-vanishing torsion with rotation and reflection symmetries are presented by the explicit use of this correspondence. Although these solutions were found in previous literature by a different approach, our method provides an alternative way to obtain them and it may be used in future research to find other exact solutions within this theory.