Constraining Spacetime Dimensions in Quantum Gravity by Scale Invariance and Electric-Magnetic Duality

We consider a low energy effective theory of $p$-branes in a $D$-dimensional spacetime, and impose two conditions: 1) the theory is scale invariant, and 2) the electric-magnetic dual $(D-p-4)$-branes exist and they obey the same type of interactions to the $p$-branes. (We also assume other natural conditions such as Lorentz invariance but not string theory, supersymmetry, supergravity and so on.) We then ask what $p$ and $D$ are consistent with these conditions. Using simple dimensional analysis, we find that only two solutions are possible: $(p,D)=(2,11)$ and $(p,D)=(2n-1,4n+2)$, ($n=1,2,3,\cdots$). The first solution corresponds to M-theory, and the second solutions at $n=1$ and $n=2$ correspond to self-dual strings in little string theory and D3-branes in type IIB superstring theory, respectively, while the second solutions for $n \ge 3$ are unknown but would be higher spin theories. Thus, quantum gravity (massless spin two theory) satisfying our two conditions would only be superstring theories, and the conditions would be strong enough to characterize superstring theories in quantum gravity.


Introduction
The construction of the underlying quantum gravity theory in our universe is an important goal in theoretical physics, and the most promising candidate is superstring theories [1][2][3].
One remarkable property of superstring theories is that spacetime is constrained to ten dimensions 1 .Furthermore, there may be a theory of two-dimensional membranes [4] in an eleven-dimensional spacetime (M-theory) that unifies the five superstring theories [5][6][7].
Therefore, the ten and eleven dimensions have a special significance in quantum gravity.
There are several derivations of these dimensions such as the analysis of the maximum supersymmetry [8], supersymmetric extended objects [4], supergravity-like dilaton gravities [9,10], and the anomaly cancellation on the world-sheets of superstrings [1][2][3].However, these derivations are mathematically sophisticated, and it is not that obvious why ten and eleven are important.Moreover, it is not clear whether other candidates for quantum gravity are possible or not.
In this article, we suggest that scale invariance and electric-magnetic duality may provide clues to answer these questions.We assume that the underlying theory of quantum gravity is defined in a D-dimensional spacetime and elementary objects are a p-brane and its electricmagnetic dual (D − p − 4)-brane so that the branes are quantized by the Dirac quantization condition.By imposing several natural assumptions of quantum gravity, including scale invariance, simple dimensional analysis tells us that only two-brane and five-brane in D = 11 and a self-dual three-brane in D = 10 would be consistent with conventional gravity.These solutions correspond to superstring theories, and it means that scale invariance and electricmagnetic duality may be strong enough to characterize superstring theories.This derivation of the ten and eleven dimensions is not only simple, but also gives us a novel understanding of superstring theories in quantum gravity.

Derivation of spacetime dimensions
We assume that the underlying theory of quantum gravity is defined in a D-dimensional spacetime and an elementary object is a p-brane, where p is the number of the spatial dimensions of the brane (D ≥ 2 and p ≥ 0).We discuss what p and D are consistent by considering a low-energy effective theory of the branes under certain assumptions of quantum We put two p-branes nearly parallel and far enough separated (Fig. 1).Then, this system at low energy would be described by the following effective theory, where we have taken natural units c = ℏ = 1.Here, x µ (µ = 0, • • • , p) are the coordinates on the world volume of the p-branes and ϕ ) are the target space coordinates of the i-th brane in the D-dimensional spacetime.Since we will consider scale invariance later, ϕ I i are canonically normalized for convenience and their mass dimension is given by Note that we are interested in the low energy physics and the non-relativistic limit has been taken in the effective action (2.1) 2 .Also, since we will study dimensional analysis, we have omitted the numerical factors of the coefficients and indexes of the derivatives.Besides, generally other fields such as gauge fields or fermions may exist on the branes, but we omit them and focus on the ϕ I i fields.Let us consider possible interaction terms L int between the two nearly parallel p-branes in the low energy effective theory (2.1).For simplicity, we define ϕ := |ϕ 1 − ϕ 2 | and focus Figure 2: Scale invariant interaction between the two p-branes.We cannot distinguish between "small" and "large", if the system is scale invariant.on the radial mode of the relative motion of the two p-branes by considering the situation where other modes and fields excite much more weakly.Also, we impose the following assumptions on the effective theory, Assumption 1 The system obeys the D-dimensional Poincaré symmetry.
These are reasonable assumptions, but they are not enough to restrict the theory.Hence, as a strong constraint, we assume the condition on scale invariance, Assumption 3 (scale invariance) The system is scale invariant and no dimensionful coupling exists in the Lagrangian (2.1).
Since scale invariance is one of beautiful symmetries in quantum theory (Fig. 2), it may be natural to expect that the underlying theory of our universe respects it.Now, the leading interaction terms of the effective action (2.1) satisfying these three assumptions are given by Here, the power n is restricted to be a non-negative integer by Assumption 2, and the derivative expansion has been considered and "• • • " denotes higher order terms.The power X is determined by dimensional analysis as (2.4) (X is not fixed when p = 1, since [ϕ] = 0, and this case is discussed separately in Appendix A.) If X is an integer, the factor of 1/ϕ X in the interaction (2.3) can be regarded as a conventional long range interaction between two separated parallel p-branes in the D = X + p + 3 dimensional spacetime induced by bulk massless modes 3 .Hence, we impose, Assumption 4 X defined in Eq. (2.4) should be an integer, and the spacetime dimension This assumption constrains n and p, and we obtain non-trivial relations between p and D 4 .
However, we still have many possible solutions (n, p) and we impose an additional assumption, Assumption 5 (electric-magnetic duality) If a p-brane exists in the theory, its electricmagnetic dual (D − p − 4)-brane also exists.The interactions between two dual branes are also described by Eq. ( 2.3) with the same power of n.
This assumption is also reasonable.If a p-brane exists, we naturally expect that it couples to a (p + 1)-form field similar to string theories, and, to quantize the charge through the Dirac quantization, its dual (D − p − 4)-brane that couples to the dual (D − p − 3)-form field should be involved in the theory [13].(The existence of the dual brane is discussed in the context of the swampland conjecture, too [14].)Then, if the p-branes and the dual branes obey the same dynamics, we expect that the common power of n appears in the interaction (2.3), although this point is more speculative 5 .
Assumption 5 is expressed by the following equations, Assumption 5' (n, p) satisfies the relations: 3 When X = 0, the factor of 1/ϕ X may be replaced by log ϕ.However, it is possible only when p = 1 and [ϕ] = 0 from Assumptions 2 and 3. Hence, we consider this case in Appendix A. 4 In the n = 2 case, we obtain three solutions [11]: (p, D) = (2, 11), (5,11) and (3,10).They correspond to M2-branes and M5-branes in M-theory and D3-branes in type IIB superstring theory.Actually, the interaction (2.3) at n = 2 naturally arises in maximally supersymmetric conformal field theories on the branes [12].Thus, if we assume a similar symmetry and restrict n = 2, these three solutions are our final answer.Therefore, the condition n = 2 is powerful enough to extract M-theory and type IIB superstring theory.A related analysis by using a dilaton gravity theory has been done in Refs.[9,10]. 5If we relax the condition on n and allow different n for the dual branes, Assumption 5 is still restrictive, but weakened.
Here, p and X are the quantities for the dual brane corresponding to p and X of the original p-brane.We can solve these equations easily as shown in Appendix A, and there are only (2.9) The first solution corresponds to M-theory (p = 2 and 5 are M2-branes and M5-branes, respectively).The n = 2 interaction (2.3) is also known to be consistent with the supergravity [12,15].
The second solution (2.9) is characterized by p = p.In the obtained dimension D = 4n + 2, the self-dual (p + 2)-form field strength is possible, where p + 2 = 2n + 1 = D/2 by Eq. (2.9), and we presume that the branes are self-dual.Actually, when n = 2, we obtain (p, D) = (3, 10) corresponding to D3-branes in type IIB superstring theory that are self-dual.Also, when n = 1, we obtain (p, D) = (1, 6) corresponding to self-dual strings [16,17] in little string theory [18][19][20][21].The n = 1 [16,17] and 2 [3,15] interactions are also consistent with string theories.(Notice that little string theory is a non-gravitational theory.In our analysis, we have not used genuine gravity, and it is not surprising that the non-gravitational theory is obtained.)The theories corresponding to the solutions (2.9) for n ≥ 3 (D ≥ 14) are unknown 7 .However, the self-dual field that couples to the selfdual brane causes a gravitational anomaly [22], and it can be cancelled only for D ≤ 10 in conventional gravitational theories.Thus, the solutions for n ≥ 3 would not correspond to conventional gravitational theories.(If the solutions for n ≥ 3 are not self-dual, they can be gravitational theories, although it is unnatural, since the solutions n = 1 and 2 would be self-dual.)Therefore, if the solution (2.9) is self-dual, only (p, D) = (3, 10) corresponds to a conventional gravitational theory.
In this way, we have obtained the scale invariant branes and the spacetime dimensions in superstring theories {(p, D) = (3, 10), (2,11) and (5,11)} as the solutions satisfying our five assumptions of quantum gravity and corresponding to conventional gravitational theories.
Although the obtained properties of our brane systems are just a part of the properties 6 In our derivation, quantum mechanics is used only in two places.One is that the Planck's constant ℏ is used to relate the energy scale to the length scale in dimensional analysis.The other is in the Dirac quantization.In this sense, our analysis is semi-classical. 7Although (p, D) = (11, 26) appears at n = 6, the interaction between two D11-branes in 26 dimensional bosonic string theory does not agree with the effective Lagrangian (2.3) at n = 6.So the solution (2.9) at n = 6 does not correspond to bosonic string theory.
of the corresponding branes in superstring theories, our systems should be identified with superstring theories, because the knowledge of string theory tells us that it is highly unlikely that there are consistent theories of these branes other than superstring theories.Therefore, the assumptions proposed in our study are strong enough to characterize superstring theories.
Since we have obtained the scale invariant branes in superstring theories, we can derive other branes and fundamental strings by using string dualities.It would be instructive to review the derivation of type IIA superstrings from the M2-branes (p, D) = (2, 11) [5,23].
It is useful to rewrite the scalars ϕ I = y I /l 3/2 , where l is a constant and y I are coordinates of the target spacetime, and both l and y I have a dimension of length.Then, the effective action (2.1) at p = 2 becomes where y := l 3/2 ϕ.In our dimensional analysis, it is possible that an additional dimensionless constant appears as a coefficient of the interaction term, but it is known that no such constant exists in M-theory.From this expression, we see that 1/l 3 and l 9 represent the tension of the proper coefficient for the interaction of two p-branes.)Now, we compactify one of the brane direction x ∼ x + R where R is the period which breaks the scale invariance.By introducing constants l s and g s through the relations R = g s l s and l 2 s = l 3 /R, where [l s ] = −1 and [g s ] = 0, we obtain the effective action after the Kaluza-Klein reduction as, This action shows p = 1 and X = 6, and it describes a (p, D) = (1, 10) system.The tension and the Newton's constant G 10 are given by 1/l 2 s and g 2 s l 8 s , respectively.This system indeed corresponds to the fundamental strings in type IIA superstring theory, where l s and g s represent the string length and the string coupling.Similarly, we can extract other strings and branes through the string dualities.(See, for example, appendix of Ref. [15] for concrete derivations of the brane effective actions.)

Discussions
We have studied the low energy effective theory of two p-branes in a D-dimensional spacetime, and asked what p and D satisfy the five assumptions proposed for quantum gravity.
We have shown that the assumptions of the scale invariance and the electric-magnetic duality strongly constrain (p, D), and only the two solutions (2.8) and (2.9) are possible.If the second solution (2.9) is self-dual, just the D = 10 case can be a gravitational theory (massless spin two theory), and both (2.8) and (2.9) at D = 10 correspond to superstring theories.It means that quantum gravities satisfying our assumptions would be superstring theories only.As we know, superstring theories exhibit several important properties, and the scale invariance and the electric-magnetic duality might be thought to be only two of them.
However, our study shows that these two may be essential and they are strong enough to extract superstring theories.Also, M-theory is derived as an exceptional solution (2.8), and it may emphasize the uniqueness of M-theory.
Assumptions 1, 3 and 5 we have made in this paper are based on the philosophy that the quantum gravity describing our universe is as simple a theory as possible.Assumptions 1 and 3 are made from the point of view that the system will have as much symmetry as possible.
Assumption 5 requires that the fundamental objects are subject to the Dirac quantization, otherwise there could be many objects with unrestricted charges in the system, which may complicate the theory.(Note that Assumption 2 and 4 are the conditions that the system is described by the low energy effective theory through the derivative expansions, which is different from the simplicity of the theory.) The ten and eleven dimensions in superstring theories and supergravities are related to supersymmetries [1,3,4,8] 8 .However, there seems to be no clear answer to the question of why the supersymmetries are essential in our universe.On the other hand, our derivation of the ten and eleven dimensions is mainly based on the simplicity of the theory.Thus, simplicity and supersymmetries may be related, and it would be interesting to explore this relationship further.
We should emphasize that, although the ten and eleven dimensions in superstring theories are obtained in our analysis, they are derived as the dimensions of the scale invariant brane systems rather than those of string theories.We have to use the knowledge of string theory and string dualities to reproduce all superstring theories from our scale invariant brane systems.Consequently, we cannot obtain bosonic string theory in a 26 dimensional spacetime from our analysis.On the other hand, our assumptions of quantum gravity, including the one about the scale invariance, sound natural because of the simplicity.Thus, if superstring theory is the underlying theory of our nature, one might claim that superstring theory is chosen but not bosonic string theory, because our nature prefers the natural assumptions.
Finally, the theories corresponding to the solution (2.9) with D ≥ 14 (n ≥ 3) are unknown.One possibility is that these solutions are simply unphysical.Another possibility is that they correspond to self-dual branes in some higher spin theories [24][25][26][27][28].If this is the case, since the systems are scale invariant, it may suggest the existence of a correspondence between the higher spin theory on the AdS 2n+1 × S 2n+1 spacetime and the 2n dimensional CFT on the (2n−1)-branes similar to the AdS/CFT correspondence [12].It may be valuable to pursue this possibility.

Figure 1 :
Figure 1: Two p-branes nearly parallel in a D-dimensional spacetime.

M2-brane and
the D-dimensional Newton's constant G D at D = 11, respectively.(Recall that [G D ] = 2 − D, which is consistent with G 11 ∼ l 9 at D = 11, and G D ×(tension) 2 is a