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I have been reviewing some string theory for a project I'm working on and I have some questions regarding string compactifications on a circle and the precise definition/origin of the moduli space that arises, given that the radion/scalar field $\phi$ parameterizing the metric only gains a kinetic term (i.e. has no potential) after the Kaluza-Klein reduction.

The setup is to consider a $D = d+1$ dimensional line element $$ds^{2} = G_{MN}dX^{M}dX^{N} = e^{2\alpha\phi}g_{\mu\nu}dX^{\mu}dX^{\nu}+e^{2\beta\phi}(dX^{d})^{2}$$

such that the $D$'th spatial direction is taken to be a circle $S^{1}$ of radius $R$, $$X^{d} \cong X^{d}+2\pi R$$ The constants $\alpha,\beta, $ and $R$ are all related but those relationships aren't important to consider here.

Due to the compactness of the $D$'th spatial direction, one can show using Kaluza-Klein reduction that the $D$-dimensional Einstein-Hilbert action with metric as above can be written as a $d$-dimensional effective theory as $$\int dX^{D}\sqrt{-G}R^{D} = \int dX^{d}\sqrt{-g}\Big(R^{d}-\frac{1}{2}(\partial\phi)^{2}\Big)$$ where $R^{D}$ and $R^{d}$ are the scalar curvatures of the $D$-dimensional theory and $d$-dimensional theory respectively (the superscript is notation to indicate this, not an exponent).

My questions are as follows:

  1. I've seen authors refer to the $d$-dimensional theory above as "having a moduli space parameterized by $\phi$". What does this mean since $\phi$ has no potential? I'm used to seeing the term "moduli space" to refer to the space of solutions/field configurations which minimize a potential, or even more generally as to just mean "parameter space", but I'm confused about how/if either of those notions of "moduli space" apply here.
  2. Authors will also say that the coefficient in front of the scalar $\phi$ kinetic term can be interpreted as a metric on this resulting moduli space. How is this the case? For example, if we had more than one scalar field, the kinetic term might be written as something like $h_{ij}\partial\phi^{i}\partial\phi^{j}$ where one would say $h_{ij}$ is the induced metric on the moduli space. This reminds me of how one discusses a non-linear sigma model, but I'm not sure of the exact connection between what is done here and the formalism of non-linear sigma models.

The closest related post I could find was here: precise definition of "moduli space", but this answer again insists on a correspondence between states and solutions to the minima of a potential (and again to reiterate, as far as I can tell $\phi$ has no potential here).

Thanks!

EDIT: In Tongs String Theory Notes, bottom of page 169 he says

Notice that there is no potential term for the dilaton and therefore nothing that dynamically sets its expectation value in the bosonic string. However, there do exist backgrounds of the superstring in which a potential for the dilaton develops, fixing the string coupling constant.

Maybe this is related?

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  • $\begingroup$ Yes, moduli space is the space parametrized by vacuum values of scalars. When there is no potential, any value is vacuum $\endgroup$
    – Kosm
    Mar 29 at 12:11
  • $\begingroup$ @Kosm Is this because when there is no potential, any constant value for the fields will minimize the total energy (to 0)? I.e. constant values for the fields would just be killed by the derivatives in the corresponding fields kinetic terms. Does this explain at all the interpretation of the coefficients of the kinetic terms as being components of a metric? $\endgroup$
    – cpollack
    Mar 29 at 12:19
  • $\begingroup$ Yes and yes. Any value will minimize the potential. $\endgroup$
    – Kosm
    Mar 29 at 12:25
  • $\begingroup$ @Kosm for this particular case then, does that mean there are an infinite number (on the order of $\mathbb{R}$) of equally valid vacua for the theory, which are physically distinct and not dynamically determined? $\endgroup$
    – cpollack
    Mar 29 at 16:12
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    $\begingroup$ sure, if these interactions are not forbidden by any symmetry, then they could give physically distinct vacua. $\endgroup$
    – Kosm
    Mar 29 at 16:34

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The physics notion of a "modulus" or it taking values in a "moduli space" is very much a vague notion, not something for which you could write down a formal (let alone a mathematically rigorous) definition that applies to all instances where this terminology is used.

Generally, a "modulus" is some object whose values "control" an aspect of a theory that may (more or less broadly) be interpreted as the "shape" of some geometrical object.

For your particular case, the scalar $\phi$ is related to the "shape", i.e. the circumference, of the circular compact dimension: The circumference of the circular compact dimension is $$ \mathrm{Vol}(S^1) = \int_{S^1}\sqrt{g_{dd}}\mathrm{d}x^d = 2\pi R \sqrt{g_{dd}}$$ and since you have $g_{dd} = \mathrm{e}^{2\beta\phi}$, your $\phi$ is therefore directly related to the physical size of the compactified dimension.

Hence, the possible values of $\phi$ form a "moduli space" for this circular dimension, and your interpretation of the coefficient of the kinetic term as a metric is actually spot-on - it really just is the one-dimensional version of $h^{ij}\partial\phi_i \partial_j\phi_j$ - you have just a single $h = h_{00}$ as the sole coefficient.

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  • $\begingroup$ As a follow up, I think I need more clarity as to why and when we can interpret the coefficients in front of a kinetic term as being a metric on some geometry parameterized by the field of which the kinetic term is related. I do not see an obvious reason as to why this would always be the case. $\endgroup$
    – cpollack
    Mar 30 at 16:22

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