Twelve-dimensional Effective Action and T-duality

We propose a twelve-dimensional supergravity action, which describes low energy dynamics of F-theory. Dimensional reduction leads the theory to eleven-dimensional, IIA, and IIB supergravities. Self-duality of the four-form field in IIB supergravity is understood. It is necessary to abandon twelve-dimensional Poincare symmetry by making one dimension compact, which is to be decompactified in some region of parameter space, such that the physical degrees of freedom are the same as those of eleven-dimensional supergravity. This makes T-duality explicit as a relation between different compactification schemes.

The ideas of Kaluza and Klein (KK) [1], generalized to higher dimensions, are beautiful ones that translate the known field degrees of freedom and their interactions into geometry of extra dimensions. Most of the supergravity theories, which are hoped to have intimate connection to our world, can be obtained by dimensional reduction of eleven-dimensional one [2]. However, it does not directly give type IIB supergravity in ten-dimension, although their relations is well-understood in the context of string theory.
Eleven-dimensional supergravity is a low-energy description of the M-theory [3]. It is also shown that type IIB superstring theory is obtained by reduction of Ftheory on a torus, with its complex structure identified by axion-dilaton, and the latter is shown to be T -dual to M-theory [4]. Thus, the effective field theory of Ftheory should be twelve-dimensional, however it is not easy to write down the action. One crucial difficulty might be that the twelve-dimensional minimal fermion with Lorentzian signature (11,1), which must be the case for F-theory, should have superpartner components with spin higher than two in the four dimensional language, whose interacting theory would be inconsistent [5]. Another obstacle is, if F-theory is dual to M-theory, there should be no surplus field degrees of freedom, although the former is a higher dimensional theory.
An important hint comes from a careful look at the derivation of F-theory [4,6]. Although it is T -dual to Mtheory, F-theory has one more dimension than the latter. Now, this extra dimension is a dual dimension to one of the dimension shared by the two theories. In other words, F-theory has two redundant dimensions which are T -dual to each other. Although we cannot maintain twelve-dimensional Poincaré symmetry fully, each tenand eleven-dimensional theories can be symmetric in its own. There is no contradiction if we cannot see both at once. Therefore, it is natural to keep both the dimensions. In this picture, M-theory looks like a compactification Ftheory on a circle, as schematically shown in Figure 1.
In this Letter, we propose the bosonic part a desired twelve-dimensional effective action, whose dimensional reductions leads to those of all known supergravities in F-theory M-theory IIB string IIA string 12D SUGRA Relation among superstrings and supergravities (SUGRA). In twelve-dimension, we make T -duality explicit in terms of compactification, by taking the other routes. The diagonal direction is the zero size limit. eleven and ten dimensions, found in standard textbooks [7]. Since we follow and make use of the duality relation between M-and F-theory from the eleven-dimensional supergravity, this theory shall provide the effective field theory for F-theory.
Supergravity is powerful enough in the sense that many of new results here, like existence of three-brane and generalized T -duality are obtained without referring to string theory. Of course, the effective field description of F-theory is timely in realistic model building, because we have so far borrowed descriptions, for instance of the gauge fields, from M-theory [8][9][10].

THE BOSONIC ACTION OF TWELVE DIMENSIONAL SUPERGRAVITY
We start with the fundamental bosonic degrees of freedom of eleven-dimensional supergravity: graviton G mn and rank three antisymmetric tensor field C map . The last one is promoted to a four-form field with total antisymmetrization, for instance C mny ′ p ≡ −C mnpy ′ . Here y ′ denotes the twelfth direction. Although this field is twelve-dimensional, we do not introduce any more degrees of freedom if one of the indices is forced to be on y ′ and the others are eleven-dimensional.
The graviton is also regarded as a part of the twelvedimensional one We suggest a formally twelve-dimensional action with the twelve-dimensional Hodge star operator. The Ricci scalar R is made of the twelve-dimensional metric (2). We will define κ 12 shortly. The presence of last term is noticed in Refs. [11,12]. Other definitions and derived relations are in order.
It is important to note that the indices assume only eleven-dimensional coordinates. Therefore the action (3) has at best eleven-dimensional Poincaré invariance. Nevertheless this form is useful, since we may also have tendimensional invariance in which we include y ′ and exclude some of the other directions. There is another loop correction term, having the form C 4 ∧I 8 where I 8 is again dependent on eleven-dimensional metric only, given in Ref [13]. The equation of motion and the Bianchi identity of C 4 follow Exchanging the role of the two, we also have a dual field which defines a six-form C 6 . In components, the dual field strength to G 5 is defined as where the indices are raised by the metric (2). Note that we have converted the eleven-dimensional field C 3 to the twelve-dimensional field C 4 = rC 3 using the metric (2). They should not be treated as independent degrees of freedom, otherwise we cannot match the equation of motion with the eleven-dimensional one. The four form structure (1) suggests that there is a coupled three-brane wrapped on y ′ direction, becoming M2brane of M-theory [18]. When a dimension is compact, this wrapping behavior should not be strange, since in the decompactification limit it becomes D3-brane along the y-direction, which we are familiar with.
We consider in this letter only the bosonic degrees of freedom. The fermonic part will be dealt with elsewhere [18].

REDUCTION TO ELEVEN-DIMENSIONAL SUPERGRAVITY
The action (3) is meaningful only if we take the y ′direction as a circle with a radius 2πr, measured in a length unit ℓ. Dimensional reduction gives us the KK tower of the fields C 4 , G mn , r with masses All of them shall play important role later in decompactification, but we keep the zero modes only for the moment. We can show that the kinetic terms of graviton and and three-form field become the standard form of eleven-dimensional supergravity. The last term in (3) is The eleven-dimensional coupling κ 11 may reversely define the coupling κ 12 with the scale r is to be fixed shortly.

REDUCTION TO IIB SUPERGRAVITY
Next, we compactify two more dimensions on a torus. It has a complex structure τ = τ 1 + iτ 2 , and we take the coordinate x, y such that we identify x + τ y ∼ x + τ y + 2πℓ ∼ x + τ y + 2πτ ℓ. Still we keep the y-direction orthogonal to the other directions, as in (2). The most general metric is From now on, fields and their Greek indices are ninedimensional.
We identify the fields of IIB supergravity as in Table I. They have either all indices nine dimensional or one component fixed to be y ′ . Consider the reduction from G αβγyy ′ to H αβγ = 3∂ [α B βγ] , given in (31) in the appendix. Neglecting the normalization, there are two possible expressions up to a total derivative which is gauge transformation. The left-hand side is the result of dimensional reduction of the ten-dimensional IIA field {H (10) µνρ , H µν } coupled to the KK field b µ , whereas the right-hand side looks 10D field type (9+1)D components 12D components  (4). After decompactifying y ′ or y directions ten-dimensional Poincaré covariance is recovered.
as dimensional reduction of the IIB field {H (10) µνρ , (db) (10) µν } coupled to the KK field K µ = r −1 C µxyy ′ under the metric [14] In the latter picture, the vectors a µ and b µ become components A µy ′ and B µy ′ , respectively, of rank two Neveu-Schwarz Neveu-Schwarz (NSNS) and Ramond-Ramond (RR) tensors. We already have the KK tower of the fields (9) completing the fields B µν , b µ , K µ , g ′ µν , r to be ten-dimensional. This is crucial necessary condition to recover ten-dimensional Poincaré symmetry and fully covariant interactions. In low-energy theory, this is a possible way to see the presence of extra dimensions, if we admit that the gravitational interactions are not observable.
The RR four-form is obtained as with one of the indices fixed to be y ′ . We can perform dimensional reduction, as in (30) in the appendix (with a different nine-dimensional metric), and decompactification in the y ′ direction with the help of one-form K 1 as above. The corresponding part of the second term in (3) gives the kinetic term for Remember that one of the indices is fixed to be y ′ for every term in (15). To avoid confusion later, we name this asF w(10) 5 . Due to the fixing of the component in (15) we do not have complete ten-dimensional four-form. The other part may come from another twelve-dimensional field (7). The only possible nine-dimensionally covariant four-form can be 1 Due to the twelve-dimensional structure in (8), the fields in (16) cannot have any index on y ′ . The left-hand side of the duality relation (8) becomes the dC 6 − 1 2 C 3 ∧G 4 as in (7), with two components fixed to be x and y. By expansion and decompactification, we obtain precisely the same expression as (15), up to normalization, with all the indices to be nine-dimensional. Hence we may call the result as L −2 τ −1 2F wo(10) 5 .
Expressing the duality relation (8) in a local Lorentz frame by ten-dimensional fields, we havẽ F wo(10) 5 with the ten-dimensional Hodge operation * 10 . The different components of F 5 have the different origins, therefore the Lorentz symmetry is not trivial. For the covariance we need the same coefficient This means that the three radii in the x, y, y ′ directions are inverse among themselves, so there is no point in the moduli space where we can have all the twelve dimensions nonimpact. Therefore we have arrived at the self-duality condition for the fully covariant ten-dimensional four-form field via its modified field strengthF (10) 5 . This is mere reexpressing the relation (17) We emphasize that this self-duality condition (19) is the defining relation of half the components of the four-form field in (16). The ten-dimensional Einstein-Hilbert term is obtained as where G ′ = g ′ r 2 is the determinant of ten dimensional metric (13), with which the Ricci scalar R (10) is calculated. Noting that τ 2 = g −1 IIB , if we require L should be absent from the the IIB supergravity action. Careful investigation shows that rescaling g ′ µν ≡ L −1 g µν and g y ′ y ′ ≡ Lg ′ y ′ y ′ = L 3 τ 2 can pull out the overall factor r, which should be absorbed by the coupling The rescaling should also rescale the coordinate periodicity as Finally, dimensional reduction of the last term in (3)  can be exchanged by the covariant one (19). The remaining expansions give the kinetic terms for the IIB supergravity action in the standard form [7,18].

REDUCTION TO IIA SUPERGRAVITY
We may decompactify the y-direction in (11) using the KK field b µ . Decompactification takes place in the same way. For example, the relation (32) in the appendix, after the decompactification, gives the reduction of G mnpq to ten-dimensional fields with one of the indices fixed on y, whereas Eq. (30) provides the remaining components. The A 1 is again the KK gauge field decompactifying x-direction. This gives IIA supergravity. We identify ten-dimensional couplings L 3 = g 2 IIA ≡ e 2Φ . It is straightforward to have the type IIA supergravity action, because we know it is also obtained by further compactification of elevendimensional supergravity action along the x-circle.
In the unit (22) we can naturally convert between IIA and IIB theories in ten dimensions. The relation between the two radii from (11) now becomes the familiar T -duality relation Without referring to string theory, we can perform Tduality by two different compactifications, as in Figure  1. In particular, the relation (18) also allows us to interpret the KK tower of fields above Eq. (9) as ones arisen by wrapping M2-branes on the torus, whose mass is proportional to the dimensionless volume of the torus L 2 τ 2 [14][15][16] . This will also be useful in describing physics around the self-dual radius where the two theories are not so much distinct, or in a strong coupling regime of one theory.