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A String Theory and Einstein-Cartan Gravity

  1. Sep 4, 2017 #1


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    I know two essential points where General Relativity plays a central role in String Theory:
    i) definition of the theory using a target spacetime with some Riemannian background metric in the Polyakov action and
    ii) recovery of the Einstein field equations as conditions regarding conformal invariance on the world-sheet, i.e. vanishing beta functions.

    My question related to (i) is if anybody has ever thought about a construction which is not based on using a Riemann but a Riemann-Cartan manifold?

    The question related to (ii) would be if that may result in different consistency conditions, i.e. beta functions, effective / low-energy equations, anomaly cancellation, dimensions and geometry of the background spacetime etc.?
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  3. Sep 4, 2017 #2


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    I've never entirely understood this, but as far as I can tell EC gravity at the effective field theory level is equivalent to GR modulo an additional tensor field (eg the torsion tensor is just the usual affine connection plus some new tensor).

    Since various solutions of string theory have a lot of those extra fields floating around (contractions of various 3 form fields etc), and almost anyone of them could be reinterpreted in such a way, i'm not entirely sure what physical difference it would make (at least perturbatively)...

    Already at the formal level, you need something like EC to define fermions and spin in a gravitational theory, and it's taught like that in almost every textbook i've ever read, so the only difference i can see relative to the classical level is whether or not the additional tensor comes from a dynamic principle (eg whether it arises from the use of variational principle of some other variables or whether it is some sort of fixed component in the Lagrangian).

    I'd be happy to learn otherwise though.

    Said another way. String theory has stringy symmetry principles. The theory outputs all possible terms consistent with those principles including a Dilaton, an Axion, a B field and its exterior derivative (etc etc). Some of those terms can (and have) been identified with a torsion tensor. However you aren't allowed to simply add another field on top of the theory (it would no longer be string theory), and its really up to you whether you want to group certain terms together and call it something else, the physical predictions ought to be the same. Or not?
    Last edited: Sep 4, 2017
  4. Sep 4, 2017 #3
    I have not the intellectual level to give you a precise answer but you may eventually enjoy the lecture of arXiv: 0904 1738 v2 [math.DG] 03 August 2009 and discover in that way numerous references inside this article. This will also gives you an overview on E. Cartan's work.
  5. Sep 5, 2017 #4


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    In ECT there's an additional torsion field. It is non-dynamical, that means it can be expressed algebraically in terms of the matter fields (spin density). It does not propagate, that means it is identically zero in the vacuum. Therefore ECT is identical with GR in vacuum and deviates only in non-vacuum areas with non-vanishing spin density.

    However there are some interesting effects. Minimal coupling of spin 1/2 spinors to geometry and integrating out torsion results in four-spinor self-interaction terms (studied in LQG). ECT seems to behave diffently w.r.t. the renormalization group flow when G, Lambda and the Immirzi parameter are treated as coupling constants (studied in the asymptotic safety approach).

    Anyway - these are the effects; I am asking if and how one could formulate strong theory taking into account torsion.
  6. Sep 5, 2017 #5
    Thesis of Felix Rennecke says the target space of the strings can already be viewed as a Riemann-Cartan manifold.

    For the basics of how to think about this, I would start with Ben Moshe vs Maimon. Ben Moshe talks about dynamical torsion, while Maimon explains how Einstein-Cartan gravity is the natural way to add a spin density in general relativity.
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