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All references I have consulted have been extremely sketchy about this point. The (2 dimensional) nonlinear sigma model in some Riemannian manifold $M$ with metric $g_{\mu\nu}$ has action $$S = \frac{1}{2} \int_M d^2 \sigma~ g_{\mu \nu} \partial_\alpha X^\mu \partial_\alpha X^\nu$$ Here, $\alpha$ is the "world sheet" index describing the embedding, and $\mu$ is the target space index. When we define the path integral, it should be $$\int \mathcal{D} X e^{-S}$$ Standard techniques to compute the renormalization of the sigma model use the background field expansion, suitably adapted to preserve manifest target space coordinate invariance. BUT, a crucial assumption going into the background field expansion and the whole apparatus of perturbative quantum field theory is that the path integrals are over linear field spaces; otherwise the Gaussian integral formulas which give us the determinants and Feynman diagrams break down even at the formal level. However, the NLSM is defined as an integral over maps to a curved manifold. Putting it differently, the usual discretized measure is not generally covariant. How does the "textbook approach" account for this point?

The quickest fix would be to discretize and define $$\mathcal{D}X = \sqrt{\det g(X)} \prod_i dX_i,$$ but even this is not very satisfactory since it renders the integrals non-Gaussian. Any insight would be appreciated.

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  • $\begingroup$ Indeed, the measure typically includes a factor of $\det(A)^{1/2}$ implicit, where $A$ is the Vilkovisky metric (cf. this PSE post). If the space of field configurations is not flat, the determinant will depend on $X$, and it cannot be reabsorbed as a normalisation constant. $\endgroup$ – AccidentalFourierTransform Jun 21 '18 at 3:06
  • $\begingroup$ Thanks for your comment. My question was more oriented toward what justifies the apparent neglect of this factor in the background field expansion. The measure is always taken to be translation invariant to define such expansions, but this seems obviously untrue here. How is there no inconsistency? $\endgroup$ – Spencer Tamagni Jun 21 '18 at 3:11
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This is an active research area and the main topic of e.g. Ref. 1. The first step is to exponentiate the measure factor $$\sqrt{\det G_{\mu\nu}(X)}~=~\int \! {\cal D}a~ {\cal D}b~{\cal D}c \exp\left(\frac{i}{\hbar}\int d^2\sigma \left(b^{\mu}G_{\mu\nu}(X)c^{\nu}+\frac{1}{2}a^{\mu}G_{\mu\nu}(X)a^{\nu} \right)\right) $$ with a Gaussian action of auxiliary fields. Here the field $a^{\mu}$ is Grassmann-even, while $b^{\mu}$ and $c^{\mu}$ are Grassmann-odd.

References:

  1. F. Bastianelli and P. van Nieuwenhuizen, Path Integrals and Anomalies in Curved Space, 2006.
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For a lot of target spaces, you can avoid these questions by realizing the nonlinear sigma model as the low energy description of a gauged linear sigma model.

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  • $\begingroup$ Thanks for the answer. I am aware of this in cases like O(N), but in most string theory textbooks, the beta functions are calculated for general metrics and other fields. It is unclear to me what formalism/approximation/argument could be used to circumvent these difficulties in the general case. $\endgroup$ – Spencer Tamagni Jun 21 '18 at 3:14

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