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Duality and Moduli Spaces for Time-Like Reductions

We consider the dimensional reduction/compactification of supergravity, string and M-theories on tori with one time-like circle. We find the coset spaces in which the massless scalars take their values, and identify the discrete duality groups.


D=11-d
In three dimensions, the bosonic sector consists of gravity coupled to an E 8 /SO (16) coset space, and reduction to D = 2 gives scalars in E 8 /SO(16) × R, coupled to gravity and a vector field. This has an infinite dimensional symmetry [2] E 9 (9) (which is an affine E 8 (8) ), and the quantum U-duality symmetry is conjectured to be a discrete subgroup of this [10]. The natural subgroup H 2 to consider in this context is an infinite dimensional subgroup of E 9 (9) containing SO (16), which can be characterised as a fixed set of a Cartan involution of E 9 (9) , as in the case of symmetric spaces. It is also the subspace of the adjoint representation on which the Killing form, or rather the invariant bilinear form, is negative definite [4] and deserves the name of maximal compact subalgebra by analogy with the finite dimensional semi-simple situation with the Cartan compactness criterion. H 2 , as acting on solutions in the Geroch [11] group fashion, does not contain the central charge of E 9 . A different subalgebra arises in the dressing method, see [8].
Instead of reducing on a space-like torus to obtain a theory with Lorentzian signature in 11 [12,13].
Timelike dimensional reductions of supergravity and string theories were investigated in [14,9,15,16], and can be useful for constructing stationary solutions in 12 − d dimensions. For example, in [15] the case d = 8 was considered and the resulting Euclidean 3-dimensional theory was used to construct stationary solutions of N = 8 supergravity in 3 + 1 dimensions. In [15], it was found that the Euclidean maximal supergravity in 3 dimensions still had G d = E 8(+8) but that H d , which had been SO(16) in 2 + 1 dimensions, had become the non-compact group SO * (16); the noncompact SO * (2k) had been discussed already in [9]. In [16], the bosonic string was compactified to zero dimensions on T 25,1 , allowing an algebraic study of the theory.
The remaining possibility is to consider compactification on a torus which includes a light-like circle; such null reductions were studied in [17]. This could be viewed as the result of compactifying on a spacelike torus, boosting in one of the toroidal directions and taking the limit of infinite boost. M-theory compactifications on such null tori have recently been the focus of considerable interest, as taking a suitable limit defines the matrix theory for toroidal compactifications of M-theory [18][19][20][21][22]. It is of particular interest to study the duality symmetries for such null reductions and compactifications, and their implications for the U-duality of M-theory. Such reductions would also be useful in constructing solutions with null Killing vectors.
Our aim here is to study the duality symmetries and scalar coset spaces of gravity and supergravity theories compactified on tori including a time-like circle, and to return to the null case in a future publication. values. There will be massless scalars taking values in N X independent of whether one keeps the full Kaluza-Klein theory (compactification) or whether one performs a consistent truncation to the massless sector (dimensional reduction). For compactification of string or M-theories, the massless fields can be studied using the appropriate (super)-gravity theory, and they again take values in the appropriate N X .
In this paper we will discuss compactifications in which X is a torus and the moduli space takes the form For compactification on Euclidean tori, H is the maximal compact subgroup of G so that G/H is a Riemannian manifold, a symmetric space of the noncompact type, and M G is a Hausdorff space with cusps in which the massless scalars take their values. For compactification on a torus with a time-like direction, the situation is more complicated [16]. In this case, H is non-compact and G/H has indefinite signature. The action of H on G(Z)\G is typically ergodic, with generic orbits being space-filling. This leads to non-commutative geometry [16,23]. (See also [24].) In the following we will not discuss such points further, but restrict ourselves to finding the groups G, H, G(Z) that arise for various compactifications.

Toroidal Compactifications of Gravity
Consider the dimensional reduction of For reductions to D = 3 dimensions, the d vector fields g iµ can be dualised to give additional scalars, and the GL(d, R) symmetry is enlarged by Ehlers-type transformations. For example, reduction from 3 + 1 Lorentzian dimensions on a circle gives a scalar and a vector, which can be dualised to give a second scalar.
For reductions on a spacelike circle, the two scalars take values in SL(2, R)/SO (2) and the Ehlers symmetry is SL(2, R) [25,11]. For reduction on a timelike circle, there are two minus signs which conspire to leave the structure intact, so that the scalar coset is again SL(2, R)/SO (2), but 4-dimensional euclidean gravity reduced to 3 dimensions leads to the coset space SL(2, R)/SO(1, 1) [14].

Bosonic and Heterotic String Compactifications
Consider first the bosonic string compactified on T d (with D + d = 26), or a theory of (D + d)-dimensional gravity coupled to a 2-form gauge field and dilaton with action We will first consider the case of reduction to more than 4 dimensions, with D > 4.
For a Euclidean torus, this has a moduli space The simplest coset space with these properties is G d = O(d, d) and and this is indeed the correct moduli space [9,26,16].
with n = 16, while similar arguments to the above give the moduli spaces for compactification on T d−1,1 . The same spaces emerge for toroidal compactification of half-maximal supergravity coupled to n vector multiplets, for any n [30].

Compactifications to 4 Dimensions and Less
For reduction of the bosonic or heterotic string, or of the related supergravity theories to 4 dimensions or less gives scalars in the moduli spaces (2) or ( In both cases, the moduli of the torus become the coupling constants on reduction and the SL(2, Z) torus symmetry becomes the S-duality of the 4-dimensional theory. This Euclidean theory can be twisted to give a topological field theory, as in [12,13].
In both cases, the expected discrete symmetry is SL(d + 1, Z).
For the bosonic string, the dimensional reduction on a d torus gives moduli in or together with 2d vectors from the reduction of the metric and 2-form gauge field.
In four dimensions S duality appears due to the 2-form, and for compactifications to 3 dimensions, the vectors are dual to scalars, and can be combined with the other scalars to take values in the coset spaces or Ehlers-type symmetries, and the SL(2, Z) S-dualities inherited from 4-dimensions, as in [31]. For compactifications to 3 dimensions of the heterotic string (with d = 7, n = 16), or for any theory of half-maximal supergravity coupled to n vector multiplets, a similar analysis gives the coset spaces for compactification on a spacelike torus [31], or for compactification on a Lorentzian torus [31]. For the heterotic string, the coset space is and there is an O(8, 24; Z) U-duality symmetry [31,10].
For reductions to 2 dimensions, the situation is more complicated [4,5,7,8,6]. In the reduction of the heterotic string to 1+1 dimensions, one obtains scalars × R coupled to 2-dimensional gravity (which includes a coupling d 2 x √ gRe −φ where φ is the dilaton) [6]. The classical supergravity theory has an affine O(8, 24) symmetry [6] in which the central charge acts through scale transformations, and a discrete subgroup of this is conjectured to be the U-duality symmetry of the full string theory [10,6].
For the reduction on a Lorentzian torus the situation is similar, with scalars taking values in coupled to gravity, with an affine O (8,24) symmetry. However, if one first reduces on a Lorentzian torus to three dimensions, dualises the vectors to scalars so that the scalar target space is (13), then reduces to two dimensions, one obtains a coset which is different from (14). However, in two Euclidean dimensions, the dual of a scalar φ with kinetic term (Dφ) 2 is a scalarφ with kinetic term −(Dφ) 2 , and some of the scalars in the coset space (15) can be dualised so that the resulting coset space is (14). Thus in two Euclidean dimensions, duality can be used to analytically continue the coset space to another real form, as T-dualities for Euclidean worldsheets can change the target space signature.
One can also consider reduction to 0 dimensions. For the bosonic string, this was considered in [16].
and other real forms of this can be obtained by going via 2 or 3 dimensions and performing duality transformations.

Compactifications of 11-Dimensional Supergravity and M-Theory
We now turn to the toroidal compactification of M-theory.
and h, k can be taken to be anti-hermitian. The algebraH d is obtained from this by substituting h →h = ih, so that the generatorsh inH d \ K d are hermitian and so are non-compact generators.
The bosonic part of the supergravity action in 3 dimensions is gravity coupled to an E 8(8) /SO * (16) sigma-model and reducing to 2 dimensions gives scalars in E 8(8) /SO * (16) ×R coupled to gravity, with an E 9(9) symmetry. In two dimensions, H * 2 is an infinite dimensional subgroup of E 9(9) containing SO * (16), which is a fixed set of an involution of E 9 (9) .