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K. Dasgupta et al.
henceforth express as g
θ k
s , with θ k is as given by eq. (2.10) in [14]. This is expressed
in terms of polynomial powers of the curvature tensor and G-flux components
(which are denoted in eq (2.10) in [14] by l i ∈ Z); parameters k i ∈
Z
2 that denote
how the g s dependences of the various G-flux components are expressed in eq.
(2.5) of [14]; as well as on n 0 and n 1,2,3 , the former being related to the number
of temporal derivatives and the latter to the number of internal spatial derivatives
in the quantum series given as eq. (2.6) of [14]. If the G-flux components are timeindependent (i.e. when k i = 0), then θ k , which we write as θ 0 , is as in eq. (2.11) of
[14].
The important thing to note now is the difference between eq. (2.10) and eq.
(2.11) in [14]. If k 1 0, k 2
3
2 and n 3 = 0, then eq. (2.10) has no relative minus
signs, whereas eq. (2.11) does have relative minus signs. Interestingly switching on
n 3 , i.e. derivatives along the toroidal directions, both the expressions have relative
minus signs. For the time being let us assume that we have switched off n 3 (we
will come back to non-zero n 3 just a bit later). In that case, the g s scalings of the
energy-momentum tensors along the various directions in M-theory (i.e. along the
space-time, internal six-manifold, and the internal toroidal directions) now become
eq. (2.12) of [14]. What does that imply?
Let us take the 2+1 dimensional space-time directions to clarify the implications
of the above g s scalings. The energy-momentum tensors of the quantum terms will
appear on the RHS of the Einstein’s equation. The g s scalings of the Einstein as
well as of the classical G-flux terms can be shown to scale as g 0
s , i.e. they are at
zeroth order in g s (there is some subtlety associated with the scaling as shown in
section 4.1 of [13]). Now comparing the g s scalings of the quantum terms along
the space-time directions, i.e. eq. (2.12) of [14], we see that θ k =
8
3 , implying that
the quantum terms are constructed out of eighth order polynomials in curvature and
G-flux components. This seems like a marvellous thing, until we notice that for the
time-independent case the equations θ 0 =
8
3 do not have a finite number of integer
solutions for (l i , n 1,2 , n 0 ). The loss of finiteness is precisely because of the presence
of relative minus signs in the expression for θ 0 in eq. (2.12) of [14]. On the other
hand, with θ k =
8
3 in eq. (2.11), there are no relative minus signs (we have made
n 3 = 0), so there is only a finite number of operators to zeroth order in g s .
Such a conclusion would appear to immediately rule out the time-independent
compactifications, because allowing an infinite number of operators means that we
have lost g s hierarchy. However the scenario is a bit more subtle than what appears at
face-value. In fact a careful analysis reveals a few caveats. First is of course the case
with n 3 > 0. Second, and this may be more important, all the quantum terms (finite
or infinite) at zeroth order in g s have different M p suppressions. Thus although we
have lost our g s hierarchy for the time-independent case, we seem to still retain an
M p hierarchy in the system. Could we then just make M p → ∞ and get rid of
operators suppressed by higher powers of M p and retain a finite set of operators for
both time-dependent and time-independent cases, respectively? What goes wrong
with such a line of reasoning?
K. Dasgupta et al.
henceforth express as g
θ k
s , with θ k is as given by eq. (2.10) in [14]. This is expressed
in terms of polynomial powers of the curvature tensor and G-flux components
(which are denoted in eq (2.10) in [14] by l i ∈ Z); parameters k i ∈
Z
2 that denote
how the g s dependences of the various G-flux components are expressed in eq.
(2.5) of [14]; as well as on n 0 and n 1,2,3 , the former being related to the number
of temporal derivatives and the latter to the number of internal spatial derivatives
in the quantum series given as eq. (2.6) of [14]. If the G-flux components are timeindependent (i.e. when k i = 0), then θ k , which we write as θ 0 , is as in eq. (2.11) of
[14].
The important thing to note now is the difference between eq. (2.10) and eq.
(2.11) in [14]. If k 1 0, k 2
3
2 and n 3 = 0, then eq. (2.10) has no relative minus
signs, whereas eq. (2.11) does have relative minus signs. Interestingly switching on
n 3 , i.e. derivatives along the toroidal directions, both the expressions have relative
minus signs. For the time being let us assume that we have switched off n 3 (we
will come back to non-zero n 3 just a bit later). In that case, the g s scalings of the
energy-momentum tensors along the various directions in M-theory (i.e. along the
space-time, internal six-manifold, and the internal toroidal directions) now become
eq. (2.12) of [14]. What does that imply?
Let us take the 2+1 dimensional space-time directions to clarify the implications
of the above g s scalings. The energy-momentum tensors of the quantum terms will
appear on the RHS of the Einstein’s equation. The g s scalings of the Einstein as
well as of the classical G-flux terms can be shown to scale as g 0
s , i.e. they are at
zeroth order in g s (there is some subtlety associated with the scaling as shown in
section 4.1 of [13]). Now comparing the g s scalings of the quantum terms along
the space-time directions, i.e. eq. (2.12) of [14], we see that θ k =
8
3 , implying that
the quantum terms are constructed out of eighth order polynomials in curvature and
G-flux components. This seems like a marvellous thing, until we notice that for the
time-independent case the equations θ 0 =
8
3 do not have a finite number of integer
solutions for (l i , n 1,2 , n 0 ). The loss of finiteness is precisely because of the presence
of relative minus signs in the expression for θ 0 in eq. (2.12) of [14]. On the other
hand, with θ k =
8
3 in eq. (2.11), there are no relative minus signs (we have made
n 3 = 0), so there is only a finite number of operators to zeroth order in g s .
Such a conclusion would appear to immediately rule out the time-independent
compactifications, because allowing an infinite number of operators means that we
have lost g s hierarchy. However the scenario is a bit more subtle than what appears at
face-value. In fact a careful analysis reveals a few caveats. First is of course the case
with n 3 > 0. Second, and this may be more important, all the quantum terms (finite
or infinite) at zeroth order in g s have different M p suppressions. Thus although we
have lost our g s hierarchy for the time-independent case, we seem to still retain an
M p hierarchy in the system. Could we then just make M p → ∞ and get rid of
operators suppressed by higher powers of M p and retain a finite set of operators for
both time-dependent and time-independent cases, respectively? What goes wrong
with such a line of reasoning?