The Geometry of Time in Topological Quantum Gravity
of the Ricci Flow

At the center of our investigations in this paper will be the symmetries of time, and their interplay with the geometry and symmetries of the $D$-dimensional spatial slices of the $D+1$ dimensional spacetime. Perhaps the more appropriate label for the topics studied here would be the “chronometry of time and supertime”. Indeed, already more than six decades ago synge , one of the pioneers of the modern geometric approach to general relativity J.L. Synge advocated convincingly that measurements of space in general relativity are all secondary to the measurements of time, suggesting that spacetime “geometry”¹¹1Geometry: From the Greek $\gamma\varepsilon\omega\mu\varepsilon\tau\!\rho\acute{\iota}\alpha$, “measurement of earth or land”. should be more properly referred to as spacetime “chronometry”²²2Chronometry: From the Greek $\chi\rho\acute{o}\nu o\varsigma$, “time”; thus “measurement of time”. (see also jammer ). This suggestion appears particularly suitable in the context of nonrelativistic gravity mqc ; lif , where time indeed plays a privileged role, quite different from that of spatial dimensions. Moreover, in the context of topological quantum gravity grf associated with the nonrelativistic Ricci flow on Riemannian manifolds, time is far from being simply a real parameter $t$ taking values in ${\bf R}$. In this nonrelativistic gravity, the “geometry” associated with time can be quite sophisticated, including a supersymmetrization of time – but not of space – to a supermanifold ${\mathscr{M}}_{0}$ of dimension $(1|2)$, which we will naturally refer to as “supertime.” In this paper, we will find further geometric structure associated with time in nonrelativistic topological gravity, in particular a rather unusual form of local time reparametrization gauge symmetry that turns out to underlie the theory presented in grf . One can thus say that the present paper represents an investigation into this geometry of time, or “chronometry,” in topological quantum gravity of the Ricci flow.

We begin by reviewing some salient features of the nonrelativistic gravity theory presented in grf . The spacetime manifold ${\cal M}$ is a $D+1$ dimensional manifold, on which we choose coordinates $t,x^{i}$, with $i=1,\ldots D$. Even though we are primarily interested in the case of $3+1$ dimensions, it is not difficult to keep $D$ more general during our arguments. We equip ${\cal M}$ with the additional structure of a codimension-one foliation ${\cal M}_{\cal F}$ by spatial slices $\Sigma$ of constant time $t$. Locally, this foliation defines a natural projection ${\cal M}\to{\cal M}_{0}$, where ${\cal M}_{0}$ is the one-dimensional time, parametrized by $t$. Typically, ${\cal M}_{0}$ can be an open interval $I=(t_{\textrm{init}},t_{\textrm{fin}})$, and with the initial time $t_{\textrm{init}}$ and the final time $t_{\textrm{fin}}$ allowed to be finite or infinite; or it can be chosen to be a compact interval, when one considers either the initial-value problem, or transition amplitudes between two instants in time.

In the process of constructing a topological theory of gravity on ${\cal M}$, one anticipates the existence of at least one supercharge $Q$, which will play the role of the BRST charge of the underlying gauge symmetry. In grf , we required in addition the existence of another supercharge $\overline{Q}$, the “anti-BRST charge”. They form a closed algebra with the generator of rigid time translations,

$$Q^{2}=0,\qquad\overline{Q}^{2}=0,\qquad\{Q,\overline{Q}\}=\partial_{t}.$$

(1)

This algebra of symmetries can be most efficiently implemented in the appropriate ${\cal N}=2$ superspace. Thus, the bosonic time manifold ${\cal M}_{0}$ is promoted to an ${\cal N}=2$ supersymmetric superspace ${\mathscr{M}}_{0}$, the “supertime”; ${\mathscr{M}}_{0}$ is parametrized by coordinates $(t,\theta,\overline{\theta})$. The supersymmetry algebra (1) is represented on ${\cal M}_{0}$ by differential operators

$$Q=\frac{\partial}{\partial\theta},\qquad\overline{Q}=\frac{\partial}{\partial\overline{\theta}}+\theta\frac{\partial}{\partial t}.$$

(2)

The spatial leaves $\Sigma$, on the other hand, play a much less prominent, almost a spectator role, and in particular they do not undergo any additional promotion to a supermanifold. Thus, our full supersymmetrized spacetime ${\mathscr{M}}$ is of dimension $(D+1|2)$, with a natural projection ${\mathscr{M}}\to{\mathscr{M}}_{0}$ to supertime, which in coordinates simply maps $(t,\theta,\overline{\theta};x^{i})\mapsto(t,\theta,\overline{\theta})$.

The simplest topological quantum theory of gravity with (1) symmetry (referred to as the “primitive” theory in grf ) describes the dynamics of the spatial metric $g_{ij}(t,x^{k})$. In the superspace formulation, $g_{ij}$ is the lowest component of a superfield $G_{ij}$,

$$G_{ij}=g_{ij}+\theta\psi_{ij}+\overline{\theta}\chi_{ij}+\theta\overline{\theta}B_{ij}.$$

(3)

The component fields have a very natural interpretation: $\psi_{ij}$ is the topological ghost, $\chi_{ij}$ the topological antighost, and $B_{ij}$ is its associated auxiliary field. These are the standard BRST multiplets associated with the BRST gauge fixing of the topological gauge symmetry consisting of all local deformations of the metric,

$$\delta g_{ij}=f_{ij}(t,x^{k}),$$

(4)

and with $Q$ playing the role of the BRST charge. The antighost-auxiliary multiplet has been chosen such that the gauge-fixing condition that fixes the topological symmetry will also be given by a symmetric 2-tensor ${\mathscr{F}}_{ij}$, which can depend on several coupling constants. As we showed in grf , for specific values of the couplings, this gauge fixing is equivalent to Hamilton’s Ricci flow collrf ; hrf

$${\mathscr{F}}_{ij}=\dot{g}_{ij}+2R_{ij}=0.$$

(5)

Thus, the primitive topological quantum gravity contains Hamilton’s Ricci flow, but does not yet contain Perelman’s generalization to the coupled flow of $g_{ij}$ and Perelman’s “dilaton” field $\phi$.

In order to extend the theory to accommodate the Perelman-Ricci flow perel1 ; perel2 ; perel3 , in grf we gauged the foliation-preserving diffeomorphism symmetry of spacetime. In a bosonic theory, gauging spatial diffeomorphisms requires the introduction of a shift vector $n^{i}$, which plays the role of the gauge field for ${\rm Diff}(\Sigma)$. In order to make this gauging compatible with our ${\cal N}=2$ supersymmetry, $n^{i}$ is also viewed as the lowest component of a superfield,³³3Note that we choose $N^{i}$ to be nonchiral; thus, we focus throughout this paper on the balanced, nonchiral theory referred to as “Type B” in grf .

$$N^{i}=n^{i}+\theta\psi^{i}+\overline{\theta}\chi^{i}+\theta\overline{\theta}B^{i}.$$

(6)

The $N^{i}$ superfield covariantizes the time derivative with respect to the gauge symmetry ${\rm Diff}(\Sigma)$ or, more precisely, its supersymmetric extension generated by the superfield $\Xi^{i}(t,x^{j},\theta,\overline{\theta})$ whose lowest component is $\xi^{i}(t,x^{j})$. For instance, the covariant derivative $\nabla_{t}$ on the metric superfield is given by

$$\nabla_{t}G_{ij}\equiv\dot{G}_{ij}-\nabla_{i}N_{j}-\nabla_{j}N_{i},$$

(7)

where $N_{i}\equiv G_{ij}N^{j}$, and $\nabla_{i}$ is the spatial covariant derivative that uses the Levi-Cività connection built from $G_{jk}$. The higher superpartners of $\xi^{i}$ in $\Xi^{i}$ can eventually be gauge-fixed by going to Wess-Zumino gauge grf , reducing the gauge symmetries to the desired bosonic ${\rm Diff}(\Sigma)$.

In the next step, the ${\cal N}=2$ supersymmetry requires that the odd derivatives ${{\mathrm{D}}}$ and ${\overline{\mathrm{D}}}$

$${{\mathrm{D}}}=\frac{\partial}{\partial\theta}-\overline{\theta}\frac{\partial}{\partial t},\qquad{\overline{\mathrm{D}}}=\frac{\partial}{\partial\overline{\theta}}$$

(8)

also be covariantized, which in turn required in grf the introduction of gauge superfields $S^{i}$ and $\overline{S}^{i}$,

	$\displaystyle S^{i}$	$\displaystyle=$	$\displaystyle\sigma^{i}+\theta\overline{X}^{i}+\overline{\theta}Y^{i}+\theta\overline{\theta}\rho^{i},$		(9)
	$\displaystyle\overline{S}^{i}$	$\displaystyle=$	$\displaystyle\overline{\sigma}^{i}+\theta X^{i}+\overline{\theta}\overline{Y}^{i}+\theta\overline{\theta}\overline{\rho}^{i}.$		(10)

Since $S^{i}$ and $\overline{S}^{i}$ are Grassmann odd, the components $\sigma^{i}$, $\overline{\sigma}^{i}$, $\rho^{i}$ and $\overline{\rho}^{i}$ are fermions, and $X^{i},\overline{X}^{i},Y^{i}$ and $\overline{Y}^{i}$ are bosons. This is a large and unwanted proliferation of component fields, and we expect them to be reduced by a number of covariant constraints. Such constraints were identified in grf :

	$\displaystyle{{\mathrm{D}}}S^{i}$	$\displaystyle=$	$\displaystyle S^{k}\partial_{k}S^{i},$		(11)
	$\displaystyle{\overline{\mathrm{D}}}\overline{S}^{i}$	$\displaystyle=$	$\displaystyle\overline{S}^{k}\partial_{k}\overline{S}^{i},$		(12)
	$\displaystyle N^{i}$	$\displaystyle=$	$\displaystyle-{\overline{\mathrm{D}}}S^{i}-{{\mathrm{D}}}\overline{S}^{i}+S^{k}\partial_{k}\overline{S}^{i}+\overline{S}^{k}\partial_{k}S^{i}.$		(13)

With these shift-sector superfields $N^{i}$, $S^{i}$ and $\overline{S}^{i}$, one then has all the ingredients to covariantize the action of the primitive theory, to make it gauge invariant under (the ${\cal N}=2$ superfield extension of) the spatial diffeomorphism gauge symmetry.

When we also gauge time reparametrizations, more superfields need to be introduced, leading to a rather complex set of constraints grf . We postpone discussing those until Section 4 of this paper.

In the standard process of constructing a gauge theory, we first specify the fields, and then the gauge symmetries acting on those fields. Then we apply the algorithm of BRST gauge fixing, which requires additional choices of gauge-fixing conditions. In this section, we choose our fields to be the spatial metric $g_{ij}$ and the shift vector $n^{i}$. We postpone the addition of the lapse function $n$ and time-reparametrization symmetries until Section 4 below, and focus first on clarifying the gauging of the spatial diffeomorphisms ${\rm Diff}(\Sigma)$.

In order to combine the topological gauge symmetries with the spatial diffeomorphisms, one could naturally start with a redundant system of symmetries,

	$\displaystyle\delta n^{i}$	$\displaystyle=$	$\displaystyle f^{i}(t,x^{j})+\dot{\xi}^{i}+\xi^{k}\partial_{k}n^{i}-\partial_{k}\xi^{i}n^{k},$		(14)
	$\displaystyle\delta g_{ij}$	$\displaystyle=$	$\displaystyle f_{ij}(t,x^{k})+\xi^{k}\partial_{k}g_{ij}+\partial_{i}\xi^{k}g_{kj}+\partial_{j}\xi^{k}g_{ik}.$		(15)

Here $f_{ij}$ is the topological gauge symmetry acting by arbitrary local deformations of $g_{ij}$, $f^{i}$ generates a similar topological symmetry on the shift vector $n^{i}$, and $\xi^{i}$ is the standard generator of ${\rm Diff}(\Sigma)$. These symmetries are indeed redundant: A given diffeomorphism transformation $\xi$ can be compensated by the choice of $f_{ij}$ and $f^{i}$ so that the combined transformation is zero. In such cases, it is well-known that the proper BRST gauge fixing requires not only the fermionic ghosts $\Psi_{ij}$, $\Psi^{i}$ and $c^{i}$ associated with the gauge symmetries generated by $f_{ij},f^{i}$ and $\xi^{i}$ (all of ghost number one), but also a bosonic ghost-for-ghost $\phi^{i}$ of ghost number two ewcoho ; htbook , which accounts for the redundancy of the original gauge symmetries (14) and (15). These fields form a BRST multiplet of the BRST charge ${Q^{\ }_{\textrm{BRST}}}$ (with ${Q^{\ }_{\textrm{BRST}}}^{2}=0$),

	$\displaystyle{Q^{\ }_{\textrm{BRST}}}\,n^{i}$	$\displaystyle=$	$\displaystyle\Psi^{i}+\dot{c}^{i}+c^{k}\partial_{k}n^{i}-\partial_{k}c^{i}n^{k},$
	$\displaystyle{Q^{\ }_{\textrm{BRST}}}\,g_{ij}$	$\displaystyle=$	$\displaystyle\Psi_{ij}+c^{k}\partial_{k}g_{ij}+\partial_{i}c^{k}g_{kj}+\partial_{j}c^{k}g_{ik},$
	$\displaystyle{Q^{\ }_{\textrm{BRST}}}\,c^{i}$	$\displaystyle=$	$\displaystyle c^{k}\partial_{k}c^{i}+\phi^{i},$		(16)
	$\displaystyle{Q^{\ }_{\textrm{BRST}}}\,\Psi^{i}$	$\displaystyle=$	$\displaystyle-\dot{\phi}^{i}+n^{k}\partial_{k}\phi^{i}-\phi^{k}\partial_{k}n^{i}+c^{k}\partial_{k}\Psi^{i}+\Psi^{k}\partial_{k}c^{i},$
	$\displaystyle{Q^{\ }_{\textrm{BRST}}}\,\Psi_{ij}$	$\displaystyle=$	$\displaystyle-\phi^{k}\partial_{k}g_{ij}-\partial_{i}\phi^{k}g_{jk}-\partial_{j}\phi^{k}g_{ik}+c^{k}\partial_{k}\Psi_{ij}+\partial_{i}c^{k}\Psi_{jk}+\partial_{j}c^{k}\Psi_{ik},$
	$\displaystyle{Q^{\ }_{\textrm{BRST}}}\,\phi^{i}$	$\displaystyle=$	$\displaystyle\phi^{k}\partial_{k}c^{i}-c^{k}\partial_{k}\phi^{i}.$

In contrast, in the hybrid two-step construction of grf , no such ghost-for-ghost fields $\phi^{i}$ of ghost number two appeared to be necessary, and we wish to understand why.

To clarify this, we note first that this redundancy between topological and spacetime symmetries is almost inevitable in relativistic theories, especially in higher than the lowest few dimensions, if one wishes to maintain their relativistic symmetries. In the case of our nonrelativistic gravity, however, there is a more elementary theory, which does not require ghost-for-ghosts. The reasoning is reminiscent of the effects observed in the context of relativistic topological gravity in two dimensions in htcomm . Our non-redundant gauge symmetries will be

	$\displaystyle\delta n^{i}$	$\displaystyle=$	$\displaystyle\dot{\xi}^{i}+\xi^{k}\partial_{k}n^{i}-\partial_{k}\xi^{i}n^{k},$		(17)
	$\displaystyle\delta g_{ij}$	$\displaystyle=$	$\displaystyle f_{ij}+\xi^{k}\partial_{k}g_{ij}+\partial_{i}\xi^{k}g_{kj}+\partial_{j}\xi^{k}g_{ik}.$		(18)

We have simply left out from (14) the topological gauge symmetry generated by $f^{i}$ and acting on $n^{i}$. Indeed, these symmetries are now non-redundant: While the action of the spatial diffeomorphism $\xi^{i}$ on the spatial metric can be compensated for by a shift in the topological symmetry $f_{ij}$, no such compensation is possible in the action on $n^{i}$, and the two symmetries generated by $f_{ij}$ and $\xi^{i}$ are mutually independent. Yet, clearly, these symmetries are powerful enough to eliminate all local propagating degrees of freedom, and the resulting theory will therefore still be “topological” in this sense. The BRST charge acts as in (16), but with the fields $\Psi^{i}$ and $\phi^{i}$ omitted:

	$\displaystyle{Q^{\ }_{\textrm{BRST}}}\,n^{i}$	$\displaystyle=$	$\displaystyle\dot{c}^{i}+c^{k}\partial_{k}n^{i}-\partial_{k}c^{i}n^{k},$
	$\displaystyle{Q^{\ }_{\textrm{BRST}}}\,g_{ij}$	$\displaystyle=$	$\displaystyle\Psi_{ij}+c^{k}\partial_{k}g_{ij}+\partial_{i}c^{k}g_{kj}+\partial_{j}c^{k}g_{ik},$
	$\displaystyle{Q^{\ }_{\textrm{BRST}}}\,c^{i}$	$\displaystyle=$	$\displaystyle c^{k}\partial_{k}c^{i},$		(19)
	$\displaystyle{Q^{\ }_{\textrm{BRST}}}\,\Psi_{ij}$	$\displaystyle=$	$\displaystyle c^{k}\partial_{k}\Psi_{ij}+\partial_{i}c^{k}\Psi_{jk}+\partial_{j}c^{k}\Psi_{ik}.$

Note one additional useful fact: The symmetries are not only non-redundant, their Lie algebra decomposes into a direct sum of a symmetry acting solely on the shift $n^{i}$, and a topological symmetry acting only on $g_{ij}$. This follows from a simple change of basis in the symmetry algebra, from $\xi^{i}$ and $f_{ij}$ to $\xi^{i}$ and $\hat{f}_{ij}$, with

$$\hat{f}_{ij}\equiv f_{ij}+\xi^{k}\partial_{k}g_{ij}+\partial_{i}\xi^{k}g_{kj}+\partial_{j}\xi^{k}g_{ik}.$$

(20)

This is a change of coordinates on the symmetry algebra, whose Jacobian is equal to one. Calling by $\psi_{ij}$ the ghost associated with the shifted topological symmetry $\hat{f}_{ij}$, the BRST transformations now simplify to

	$\displaystyle{Q^{\ }_{\textrm{BRST}}}\,n^{i}$	$\displaystyle=$	$\displaystyle\dot{c}^{i}+c^{k}\partial_{k}n^{i}-\partial_{k}c^{i}n^{k},$
	$\displaystyle{Q^{\ }_{\textrm{BRST}}}\,c^{i}$	$\displaystyle=$	$\displaystyle c^{k}\partial_{k}c^{i},$
	$\displaystyle{Q^{\ }_{\textrm{BRST}}}\,g_{ij}$	$\displaystyle=$	$\displaystyle\psi_{ij},$		(21)
	$\displaystyle{Q^{\ }_{\textrm{BRST}}}\,\psi_{ij}$	$\displaystyle=$	$\displaystyle 0.$

The fact that such a shift exists, and that in the new basis the ${\rm Diff}(\Sigma)$ symmetry acts trivially on $g_{ij}$, is important: The supermultiplets of $Q$ introduced in grf in the process of gauging ${\rm Diff}(\Sigma)$ exhibit the same type of decoupling between the two symmetries. It is this form (21) of the BRST transformations that will be best suited for the comparison against our topological gravity constructed in Section 3 of grf .

We claim that the ${\cal N}=2$ supersymmetry multiplets found in Section 3 of grf are indeed equivalent to the standard multiplets associated with the BRST gauge fixing of (17) and (18), and that the supercharge $Q$ of grf is simply the standard BRST charge of this one-step construction. This will be best seen when we rewrite the theory developed in grf in the component formalism. In order to determine the independent component fields and their properties, we must first solve the constraints relating $N^{i},S^{i}$ and $\overline{S}^{i}$. Since the constraints (11) and (12) involve only $S^{i}$ and $\overline{S}^{i}$ respectively, they can be solved first, yielding

	$\displaystyle S^{i}$	$\displaystyle=$	$\displaystyle\sigma^{i}+\theta\sigma^{k}\partial_{k}\sigma^{i}+\overline{\theta}Y^{i}+\theta\overline{\theta}(\dot{\sigma}^{i}+Y^{k}\partial_{k}\sigma^{i}-\sigma^{k}\partial_{k}Y^{i}),$		(22)
	$\displaystyle\overline{S}^{i}$	$\displaystyle=$	$\displaystyle\overline{\sigma}^{i}+\theta X^{i}+\overline{\theta}\overline{\sigma}^{k}\partial_{k}\overline{\sigma}^{i}+\theta\overline{\theta}(\overline{\sigma}^{k}\partial_{k}X^{i}-X^{k}\partial_{k}\overline{\sigma}^{i}).$		(23)

Thus, the independent components of $S^{i}$ and $\overline{S}^{i}$ have been reduced to $\sigma^{i},\overline{\sigma}^{i},X^{i}$ and $Y^{i}$, transforming as follows under $Q$:

	$\displaystyle Q\,\sigma^{i}$	$\displaystyle=$	$\displaystyle\sigma^{k}\partial_{k}\sigma^{i},$		(24)
	$\displaystyle Q\,Y^{i}$	$\displaystyle=$	$\displaystyle-\dot{\sigma}^{i}-Y^{k}\partial_{k}\sigma^{i}+\sigma^{k}\partial_{k}Y^{i},$		(25)
	$\displaystyle Q\,\overline{\sigma}^{i}$	$\displaystyle=$	$\displaystyle X^{i},$		(26)
	$\displaystyle Q\,X^{i}$	$\displaystyle=$	$\displaystyle 0.$		(27)

These are not yet the appropriate component fields of our desired supersymmetry multiplets, as we still need to solve the constraint (13) that expresses the components of $N^{i}$ in terms of the independent components in $S^{i}$ and $\overline{S}^{i}$. First, solving (13) at the lowest component gives

$$n^{i}=-Y^{i}-X^{i}+\sigma^{k}\partial_{k}\overline{\sigma}^{i}+\overline{\sigma}^{k}\partial_{k}\sigma^{i}.$$

(28)

Note that because $X^{i}$ is $Q$ invariant, it is our candidate for the auxiliary field in a trivial BRST multiplet, paired up with $\overline{\sigma}^{i}$ which will play the role of the antighost. This means that the correct way of reading (28) is to interpret $Y^{i}$ as a composite field, determined by this equation in terms of the four independent component fields $n^{i},X^{i},\sigma^{i}$ and $\overline{\sigma}^{i}$,

$$Y^{i}=-n^{i}-X^{i}+\sigma^{k}\partial_{k}\overline{\sigma}^{i}+\overline{\sigma}^{k}\partial_{k}\sigma^{i}.$$

(29)

This is the sense in which $Y^{i}$ should be understood as a composite field when it appears in expressions such as (22).

The remaining components of $N^{i}$ can now be determined in terms of the independent fields $n^{i},X^{i},\sigma^{i}$ and $\overline{\sigma}^{i}$ by evaluating the higher-order terms in (28). They are found to be

	$\displaystyle\psi^{i}$	$\displaystyle=$	$\displaystyle\dot{\sigma}^{i}+\sigma^{k}\partial_{k}n^{i}-\partial_{k}\sigma^{i}n^{k},$		(30)
	$\displaystyle\chi^{i}$	$\displaystyle=$	$\displaystyle\dot{\overline{\sigma}}^{i}+\overline{\sigma}^{k}\partial_{k}n^{i}-\partial_{k}\overline{\sigma}^{i}n^{k},$		(31)
	$\displaystyle B^{i}$	$\displaystyle=$	$\displaystyle-\dot{X}^{i}+n^{k}\partial_{k}X^{i}-X^{k}\partial_{k}n^{i}+\dot{\sigma}^{k}\partial_{k}\overline{\sigma}^{i}+\sigma^{j}\partial_{j}n^{k}\partial_{k}\overline{\sigma}^{i}-n^{j}\partial_{j}\sigma^{k}\partial_{k}\overline{\sigma}^{i}$		(32)
			$\displaystyle\ {}+\overline{\sigma}^{k}\partial_{k}\dot{\sigma}^{i}+\overline{\sigma}^{k}\partial_{k}\sigma^{j}\partial_{j}n^{i}+\overline{\sigma}^{k}\sigma^{j}\partial_{k}\partial_{j}n^{i}-\overline{\sigma}^{k}\partial_{k}n^{j}\partial_{j}\sigma^{i}-\overline{\sigma}^{k}n^{j}\partial_{k}\partial_{j}\sigma^{i}.$

Thus, the entire sector of superfields $N^{i}$, $S^{i}$ and $\overline{S}^{i}$ reduces in components to four independent fields, belonging to two BRST multiplets of the BRST charge $Q$, with BRST transformations

	$\displaystyle Q\,n^{i}$	$\displaystyle=$	$\displaystyle\dot{\sigma}^{i}+\sigma^{k}\partial_{k}n^{i}-n^{k}\partial_{k}\sigma^{i},$		(33)
	$\displaystyle Q\,\sigma^{i}$	$\displaystyle=$	$\displaystyle\sigma^{k}\partial_{k}\sigma^{i},$		(34)

and

	$\displaystyle Q\,\overline{\sigma}^{i}$	$\displaystyle=$	$\displaystyle X^{i},$		(35)
	$\displaystyle Q\,X^{i}$	$\displaystyle=$	$\displaystyle 0.$		(36)

Clearly, $\sigma^{i}$ is the ghost of the spatial diffeomorphism symmetry, and should be identified with $c^{i}$ in (16), while $\overline{\sigma}^{i}$ and $X^{i}$ form the trivial BRST multiplet consisting of an antighost and an auxiliary field. In addition to these two BRST multiplets, the theory that was constructed in Section 3 of grf by gauging spatial diffeomorphisms of the primitive topological gravity contains also the two BRST multiplets in the metric superfield of (3),

	$\displaystyle Q\,g_{ij}=\psi_{ij},\qquad Q\,\psi_{ij}=0,$		(37)
	$\displaystyle Q\,\chi_{ij}=-B_{ij},\qquad Q\,B_{ij}=0.$		(38)

We see that these component supermultiplets (33), (34) and (37) of $Q$ match exactly the multiplets (21) of ${Q^{\ }_{\textrm{BRST}}}$, which we obtained by the one-step BRST gauge fixing of the non-redundant gauge symmetries (17) and (18). And the remaining multiplets of $Q$, listed in (35), (36) and (38), are the standard antighost-auxiliary BRST multiplets ready for the implementation of our gauge fixing choice. In this way, the two-step construction in Section 3 of grf can indeed be consistently interpreted as the standard one-step gauge fixing of a gauge theory with non-redundant gauge symmetries, with $Q$ identified as the BRST charge.

In Section 4 of grf , the theory was further extended by gauging foliation-preserving time reparametrizations, and adding the appropriate ${\cal N}=2$ supersymmetrization of the lapse function $n$. In order to clarify whether this extended theory can also be interpreted as a one-step BRST gauge fixing of appropriate non-redundant gauge symmetries, we need to overcome additional challenges which were absent in the theory with ${\rm Diff}(\Sigma)$ gauge symmetry discussed above. We begin by briefly reviewing the symmetries and fields introduced in grf .

The gauge symmetries of spacetime-dependent spatial diffeomorphisms ${\rm Diff}(\Sigma)$ are extended to all foliation-preserving diffeomorphisms of the $D+1$ dimensional spacetime ${\cal M}_{\cal F}$. Besides the generators $\xi^{i}(t,x^{k})$ that were considered in the previous paragraph, this extended symmetry also contains time-dependent time reparametrizations, generated by $f(t)$:

	$\displaystyle\delta n$	$\displaystyle=$	$\displaystyle\dot{f}n+f\dot{n},$
	$\displaystyle\delta n^{i}$	$\displaystyle=$	$\displaystyle\dot{f}n^{i}+f\dot{n}^{i},$		(39)
	$\displaystyle\delta g_{ij}$	$\displaystyle=$	$\displaystyle f\dot{g}_{ij}.$

Here we have introduced besides the spatial metric $g_{ij}$ and the shift vector $n^{i}$ also the lapse function $n$, which plays the role of the gauge field associated with time reparametrizations: More precisely, it is $\log n$ that transforms as a gauge field: $\delta\,\log n=\dot{f}+\ldots$. While it is mathematically consistent in nonrelativistic gravity to declare the field $n$ to be a projectable field $n(t)$, it is physically more interesting to allow the lapse to be a nonprojectable field $n(t,x^{i})$, i.e., to promoting it to a spacetime-dependent field. Allowing the lapse to be nonprojectable was indeed crucial for us in grf in establishing contact with Perelman’s theory of the Ricci flow, since we found that the role of Perelman’s “dilaton” field is played by the nonprojectable lapse field.

In Section 4 of grf , we constructed a nonrelativistic gravity theory which is gauge invariant under $f(t)$ and consistent with our requirement of rigid ${\cal N}=2$ supersymmetry of supertime, using the standard techniques for gauging symmetries in superspace. We promoted $n(t,x^{i})$ to a nonchirial, nonprojectable superfield, whose lowest component is the nonprojectable lapse, $N=n+\ldots$ grf . In fact, in grf we found it convenient to introduce the inverse lapse superfield, $E=1/N$, whose lowest component is the nonprojectable field $e\equiv 1/n$,

$$E=e+\theta\psi+\overline{\theta}\chi+\theta\overline{\theta}B.$$

(40)

This $E$ superfield then covariantizes the time derivative under (the supersymmetric extensions of) the $f(t)$ symmetries, in the way explained in detail in grf .

Since the odd derivatives also need to be covariantized, two more superfields $\Theta$ and $\overline{\Theta}$ had to be introduced in addition to $E$; they are also nonprojectable, nonchiral, but odd, and we introduce the following notation for their components:⁴⁴4The minus signs introduced here in the process of naming the component fields will simplify some future component formulas below, and they also make the parallel between the odd superfields $\Theta,\overline{\Theta}$ of the lapse sector and the odd superfields $S^{i},\overline{S}^{i}$ of the shift sector more explicit.

	$\displaystyle\Theta$	$\displaystyle=$	$\displaystyle-\nu-\theta\overline{w}-\overline{\theta}z-\theta\overline{\theta}\rho,$		(41)
	$\displaystyle\overline{\Theta}$	$\displaystyle=$	$\displaystyle-\overline{\nu}-\theta w-\overline{\theta}\overline{z}-\theta\overline{\theta}\overline{\rho}.$		(42)

As in the case of spatial diffeomorphisms discussed in the previous section of the present paper, this proliferation of component fields is reduced to the minimal independent set by suitable constraints, which now involve both the shift sector and the lapse sector superfields grf :

	$\displaystyle{{\mathrm{D}}}\Theta-S^{k}\partial_{k}\Theta$	$\displaystyle=$	$\displaystyle-\Theta(\dot{\Theta}-N^{k}\partial_{k}\Theta),$		(43)
	$\displaystyle{\overline{\mathrm{D}}}\overline{\Theta}-\overline{S}^{k}\partial_{k}\overline{\Theta}$	$\displaystyle=$	$\displaystyle-\overline{\Theta}(\dot{\overline{\Theta}}-N^{k}\partial_{k}\overline{\Theta}),$		(44)

and

$$E=1-{{\mathrm{D}}}\overline{\Theta}+S^{k}\partial_{k}\overline{\Theta}-{\overline{\mathrm{D}}}\Theta+\overline{S}^{k}\partial_{k}\Theta-\Theta(\dot{\overline{\Theta}}-N^{k}\partial_{k}\overline{\Theta})-\overline{\Theta}(\dot{\Theta}-N^{k}\partial_{k}\Theta).$$

(45)

In addition to these mixed constraints, the shift sector superfields $S^{i},\overline{S}^{i}$ and $N^{i}$ still satisfy constraints (11), (12) and (13). These constraints appear somewhat more involved than in the case of the shift sector alone, and we postpone solving them until Section 4.1.

Once the nonprojectable lapse sector has been introduced, the pattern that we uncovered in the case with ${\rm Diff}(\Sigma)$ gauge symmetry will continue to be valid if the BRST multiplets in the lapse sector can be interpreted as having originated in BRST gauge fixing of a hidden symmetry associated with time. Note that since the lapse sector contains only nonprojectable fields, this hidden symmetry cannot be simply the projectable time reparametrizations generated by $f(t)$. This hidden symmetry will have to be nonprojectable, generated by some space and time dependent parameter $\zeta(t,x^{i})$. Is there such a hidden symmetry, and if so, what is its geometric interpretation?

The first guess might be that $\zeta(t,x^{i})$ should perhaps be some kind of nonprojectable time reparametrization symmetry, which would extend the symmetry of spatial diffeomorphisms generated by $\xi^{i}(t,x^{k})$. This symmetry would have to satisfy several stringent requirements. For example, it would have to act on $n^{i}$ (and also on $g_{ij}$) trivially, or more accurately, in a way which can be absorbed into a redefinition of the spatial diffeomorphisms $\xi^{i}$ (and the topological gauge transformations $\hat{f}_{ij}$). This phenomenon is analogous to what happened above, when we found the basis of symmetry generators in which $\xi^{i}$ acts trivially on $g_{ij}$, and therefore ${Q^{\ }_{\textrm{BRST}}}\,g_{ij}$ is independent of $c^{i}$ (see (21)). Is it possible to construct such a symmetry $\zeta$? We could try to read off the rules directly from the transformation properties of the component fields under $Q$, assuming that they will conform to the interpretation as having come from the BRST fixing of a symmetry, with $Q$ being the BRST charge. However, we find it more instructive to look first for available symmetries in the context of previously studied symmetries of spacetime, which we will do in the next section. In the process, we will learn a few surprising facts which might be of more general interest for quantum gravity, beyond the applications to the main subject of this paper. The reader who is solely interested in our topological nonrelativistic gravity construction can proceed directly to Section 4.

In order to discuss the nonrelativistic decomposition of a relativistic spacetime, we wish to re-introduce the measurement of time and space in unrelated units: $L$ and $T$. First, we recall a few elementary facts about scaling and dimensions in quantum field theory, and propose a simple refinement to the case when the system contains a dynamical spacetime metric.

In quantum field theory in a Minkowski spacetime, scaling dimensions of various objects (quantum fields, spacetime derivatives, composite operators built out of them) play a central role, controlled by the concept of the renormalization group (RG). In relativistic field theories, it is natural to set the speed of light $c=1$, and all dimensions are then expressed in terms of one dimensionful scale. By convention, largely for historical reasons, in particle physics this scale is typically selected to be the momentum scale, or equivalently the inverse length scale when one sets $\hbar=1$. In momentum units, Cartesian spacetime coordinates $x^{\mu}$ of the Minkowski spacetime are of dimension $-1$, and the derivatives $\partial_{\mu}$ are of dimension $1$. Dimensions of quantum fields and various composite operators then depend on the type of theory in question, typically controlled by an RG fixed point. Sometimes, however, there is a strong geometric reason to assign a given field a particular scaling dimension. For example, in general relativity, the components $g_{ij}$ of the spacetime metric relate the proper length element $ds$ to the coordinate length elements $dx^{i}$ via $ds^{2}=g_{ij}dx^{i}dx^{j}$. It is only sensible to consider $ds^{2}$ to be of dimension $-2$ in momentum units, hence implying the often-quoted fact that for the purposes of power counting, the components $g_{ij}$ of the metric are dimensionless. This picture is of course well-known, and almost universally accepted, but as we explain below, in need of a slight conceptual modification in theories with dynamical inhomogeneous geometries.

Nonrelativistic gravity of the Lifshitz type mqc ; lif , and its topological counterpart studied here, belong to the class of quantum field theories with two distinct, a priori unrelated scales: a length scale $L$, and a time scale $T$, or their inverses: an energy scale $E=1/T$ and a momentum scale $M=1/L$ (having again set $\hbar=1$). The relativistic strategy of assigning dimensions needs to be refined: We can declare that the time coordinate carries dimension $T\equiv E^{-1}$, and the spatial coordinates $x^{i}$ are of dimension $L\equiv M^{-1}$. (This was also the convention we used for assigning classical scaling dimensions to various objects in topological gravity of the Ricci flow in Section 2 of grf .) In addition, one simplifying convention seems appropriate: Instead of writing the dimension $[{\cal O}]$ of an object ${\cal O}$ multiplicatively in terms of powers of $T$ and $L$ (or $E$ and $M$), we will simply write $[{\cal O}]=(n,m)$ when ${\cal O}$ is of dimension $E^{n}M^{m}$ in energy and momentum units. With this convention we have, for instance,

$$[\partial/\partial t]=(1,0),\qquad[\partial_{i}]=(0,1).$$

(49)

Applying this analysis to the fields of nonrelativistic gravity, we thus obtain the “standard” dimension assignments:

$$[g_{ij}]=(0,0),\qquad[n]=(0,0),\qquad[n^{i}]=(1,-1).$$

(50)

They simply follow from the fact that in nonrelativistic gravity, the geometry of spacetime is characterized by an invariant line element $d\sigma$ of spatial distance, defined via

$$d\sigma^{2}=g_{ij}(dx^{i}-n^{i}dt)(dx^{j}-n^{j}dt),$$

(51)

which should naturally be of dimension $[d\sigma]=(0,-1)$, and an independent invariant element of time,

$$d\tau=n\,dt,$$

(52)

naturally of dimension $(-1,0)$.

In theories with two (or more) a priori unrelated scales, the “one-scale” physics of the renormalization group scaling emerges in the vicinity of an RG fixed point, where the two scales become locked together by a scaling relation

$$E=M^{z},$$

(53)

involving the so-called dynamical critical exponent $z$. This exponent is characteristic of the specific RG fixed point, not of the theory itself – it is indeed possible to have a theory with more than one RG fixed points, with distinct values of $z$. In the vicinity of a given fixed point, one is then free to use (53) to express the RG scaling properties of the system in terms of just one independent scale (say $E$).

In the present context of topological gravity of the Ricci flow, and more generally in nonrelativistic gravity with a dynamical metric, it has become increasingly apparent that a slightly modified strategy for assigning dimensions would be more logical. There are several reasons for that. In the flat Minkowski spacetime, or in a theory with a preferred and highly symmetric background, it may be natural to continue with the picture developed in quantum field theory, and see the spacetime coordinates $x^{\mu}$ or the derivatives $\partial_{\mu}$ as the carriers of a scaling dimension: After all, the idea of scaling is often phrased as a study of the properties of the system under the rescalings $x^{\mu}\mapsto bx^{\mu}$ for constant $b$. However, this intuition is increasingly problematic in a theory whose background geometries are often highly inhomogeneous, such as our topological theory of the Ricci flow, whose path integral localizes to the solutions of generalized Ricci flow equations with arbitrarily inhomogeneous initial conditions. On such backgrounds, there is no analog of a “preferred coordinate system” $x^{\mu}=(t,x^{i})$. Instead, we prefer to treat all coordinate systems equally; this will make sense only if we assign the scaling dimension $(0,0)$ to both $t,x^{i}$ and the derivatives $\partial_{t},\partial_{i}$. Only then the elementary covariance of our rules will be restored, and the concept of scaling dimensions will not have to rely on the existence of a preferred system of coordinates associated with a maximally symmetric ground-state geometry.

In this modified picture, the dimensions are now carried by the physical spatial length element $d\sigma^{2}$ and time element $d\tau$,

$$[d\sigma^{2}]^{\prime}=(0,-2),\qquad[d\tau]^{\prime}=(-1,0),$$

(54)

just as we declared above. However, since now $dx^{i}$ and $dt$ are declared dimensionless, this implies that the old-fashioned rules (49) and (50) are modified to

$$[g_{ij}]^{\prime}=(0,-2),\qquad[n]^{\prime}=(-1,0),\qquad[n^{i}]^{\prime}=(0,0),$$

(55)

and

$$[\partial/\partial t]^{\prime}=(0,0),\qquad[\partial_{i}]^{\prime}=(0,0).$$

(56)

We have denoted the dimension in the modified counting system by $[\ ]^{\prime}$, to distinguish it from the dimension $[\ ]$ in the old-fashioned system.

One can see that both systems of assigning dimensions are for all physical purposes equivalent to each other, for instance they lead to the same dimension counting rules for the terms that can appear in the action. However, the modified system in which the underlying spacetime coordinates are naturally dimensionless is mathematically more accurate and conceptually cleaner that the old-fashioned one, since it is manifestly consistent with nonlinear changes of coordinates, and does not lead to apparent violations of the additivity properties of the classical scaling dimensions during such nonlinear coordinate transformations.

After this detour, we can now return to the study of the relativistic diffeomorphism symmetry in the ADM decomposition, in a theory with two scales $L$ and $T$. Restoring the dimensions of various objects in the symmetry algebra (46), (47) and (48), we see that the last term on the right-hand side of (47) is the only one whose dimension does not match the rest of the terms, and therefore a dimensionful constant is needed to provide the conversion. This constant is of course the speed of light $c$, of dimension $[c]=[c]^{\prime}=(1,-1)$, which was set equal to one in the relativistic theory. Restoring $c$, the algebra is now

	$\displaystyle\delta n$	$\displaystyle=$	$\displaystyle\xi^{k}\partial_{k}n+\zeta\dot{n}+\dot{\zeta}n,$
	$\displaystyle\delta n^{i}$	$\displaystyle=$	$\displaystyle\dot{\xi}^{i}+\xi^{k}\partial_{k}n^{i}-\partial_{k}\xi^{i}n^{k}+\zeta\dot{n}^{i}+\dot{\zeta}n^{i}-\partial_{k}\zeta n^{k}n^{i}-c^{2}\partial_{k}\zeta g^{ik}n^{2},$		(57)
	$\displaystyle\delta g_{ij}$	$\displaystyle=$	$\displaystyle\xi^{k}\partial_{k}g_{ij}+\partial_{i}\xi^{k}g_{kj}+\partial_{j}\xi^{k}g_{ik}+\zeta\dot{g}_{ij}+(\partial_{i}\zeta g_{jk}+\partial_{j}g_{ik})n^{k}.$

It is an elementary but valuable exercise to verify why this insertion of $c^{2}$ makes the dimensions right in both of our dimension-counting systems.

Now we can return to our quest for a symmetry that would match the ingredients required by the topological nonrelativistic gravity of grf . As we saw at the beginning of Section 2 above, it is precisely this $c^{2}$ term that represents an obstruction against what we need. This term can be simply eliminated by taking the $c\to 0$ limit, well-known in the literature tultra ; hultra (see also gen for the nonrelativistic context) as the “ultralocal” limit of the relativistic diffeomorphism symmetries of general relativity.

Taking $c\to 0$ in (57) will allow us to follow the strategy outlined at the beginning of Section 3, and to make the time transformation generated by $\zeta$ act trivially on the shift vector $n^{i}$. Before taking these steps, let us take a closer look at this “ultralocal” limit of $c\to 0$, especially in the light of our improved prescription for assigning scaling dimensions. According to this prescription, our coordinates $x^{i},t$ are dimensionless, and therefore the generators $\xi^{i},\zeta$ of the corresponding spacetime symmetries are dimensionless as well. Yet, the algebra defined by the transformation rules (57) clearly depends on $c$. Indeed, we are planning to take the contraction of the symmetries by sending $c\to 0$. One might therefore think that the structure constants of this symmetry algebra, which one could obtain from the commutation relations of the transformations in (57), will depend on $c$. However, that clearly represents a puzzle: If the generators $\xi^{i}$ and $\zeta$, and all the derivatives $\partial_{t}$ and $\partial_{i}$ that can appear in the commutation relations are all dimensionless, how could the commutator of two such transformations possibly depend on a dimensionful constant such as $c$?

The resolution is simple, yet perhaps a little surprising. A direct calculation reveals that regardless of the value of $c$, the commutation relations of the transformations in (57) are independent of $c$: The commutator of a transformation $\delta_{1}$ generated by $\xi^{i}_{1}$ and $\zeta_{1}$ with the transformation generated by $\xi^{i}_{2}$ and $\zeta_{2}$,

$$[\delta_{1},\delta_{2}]=\delta_{3},$$

(58)

is the transformation $\delta_{3}$ generated by the following $\xi^{i}_{3}$ and $\zeta_{3}$,

	$\displaystyle\zeta_{3}$	$\displaystyle=$	$\displaystyle\zeta_{1}\dot{\zeta}_{2}-\zeta_{2}\dot{\zeta}_{1}+\xi^{k}_{1}\partial_{k}\zeta_{2}-\xi^{k}_{2}\partial_{k}\zeta_{1},$		(59)
	$\displaystyle\xi^{i}_{3}$	$\displaystyle=$	$\displaystyle\xi^{k}_{1}\partial_{k}\xi^{i}_{2}-\xi^{k}_{1}\partial_{k}\xi^{i}_{2}+\zeta_{1}\dot{\xi}^{i}_{2}-\zeta_{2}\dot{\xi}^{i}_{1}.$		(60)

These are indeed the commutation relations of the general spacetime diffeomorphisms, familiar from the relativistic theory of gravity.

Consider now the theory of bosonic gravity which would be invariant under the $c\to 0$ limit of the symmetries specified in (57). This is of course the theory known in the literature as the “ultralocal” theory of gravity tultra ; hultra , proposed originally as a possible strong-coupling limit of general relativity. Do the relativistic commutation relations of the symmetry generators mean that this theory is somehow relativistic? Not in the standard sense of full relativistic general covariance: While the commutation relations appear relativistic, the realization of the symmetries on the ADM decomposition of the metric does depend on $c$. Consequently, Lagrangians that are invariant under this realization of the symmetries will be contractions of the standard Lagrangians of general relativity. In particular, the lowest-derivative Lagrangian invariant under these symmetries,

$$S=\frac{1}{\kappa^{2}}\int dt\,d^{D}x\left\{\frac{\sqrt{g}}{n}\left(g^{ik}g^{j\ell}-g^{ij}g^{k\ell}\right)\left(\dot{g}_{ij}-\nabla_{i}n_{j}-\nabla_{j}n_{i}\right)\left(\dot{g}_{k\ell}-\nabla_{k}n_{\ell}-\nabla_{\ell}n_{k}\right)-2n\sqrt{g}\Lambda\right\},$$

(61)

contains the standard kinetic term for the spatial metric with two time derivatives, but no spatial-derivative term consistent with it. This of course is the standard result known from tultra ; hultra . It is intriguing, however, that the dependence of this theory on the preferred foliation by constant time slices appears here at the level of the dynamical metric fields, and not at the level of the underlying symmetries of the differentiable structure of spacetime with dimensionless generators $\xi^{i}$ and $\zeta$.