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This is a very specific question about a paper by P. Glorioso and H. Liu that can be found here https://arxiv.org/pdf/1805.09331.pdf. In particular I want to understand how the authors get from the Lagrangian (6.71) to the constrained symmetry (6.72).

BACKGROUND:

The set up is as follows: We have four scalar fields $\sigma^A(x)$, ($A=0,1,2,3$) living in a 4D spacetime. The fields $\sigma^i$ for $i=1,2,3$ correspond to fluid elements and the field $\sigma^0$ can be thought of as co-moving, internal clocks for each fluid element. The action has the following symmetries

  1. Fluid element re-labeling: $$ \sigma^i\to\sigma'^i(\sigma^j)$$ for $i,j=1,2,3$ and $\sigma'^i$ is any diffeomorphism of $\sigma^i$.

  2. Re-parameterizations of the 'internal clocks' $$ \sigma^0\to \sigma^0+\alpha(\sigma^i)$$ for generic infinitely differentiable function $\alpha$.

At lowest order in derivatives, the action expressed in physical space-time takes the form $$S=\int d^4 x \sqrt{-g}~P(b), $$ where $b\equiv-g_{\mu\nu}(K^{-1})^\mu_0 (K^{-1})^\nu_0$ and $K^A_\mu\equiv \partial_\mu\sigma^A$ and $P$ is an arbitrary function (that can be specified by experiments). [Note: in the paper they have a different form of the action given in equation (6.71) defined on the fluid corrdinates $\sigma$. The action I have written down above is the same action but expressed as an integral over physical space-time coordinates $x$.]

The authors argue starting from the above action, it is possible to integrate out the $\sigma^0$ fields to obtain an action for just the $\sigma^i$ fields. They then claim that "When solving $\sigma^0$ in terms of other variables, one finds an arbitrary integration function, which can be fixed to be unity and in turn breaks the [fluid element re-labeling] symmetry down to volume-preserving spatial diffeomorphisms" $$ \sigma^i\to f^i(\sigma_j)~~~~~\det\bigg(\frac{\partial f^i}{\partial \sigma^j}\bigg)=1.$$

QUESTION: I am having trouble carrying out the computation of integrating out the $\sigma^0$ fields to obtain the volume-preserving spatial diffeomorphism symmetry. Can someone please show me how to do this computation explicitly and in detail?

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