Noncompact symmetries in string theory

Abstract

Noncompact groups, similar to those that appeared in various supergravity theories in the 1970's, have been turning up in recent studies of string theory. First it was discovered that moduli spaces of toroidal compactification are given by noncompact groups modded out by their maximal compact subgroups and discrete duality groups.Then it was found that many other moduli spaces have analogous descriptions. More recently, noncompact group symmetries have turned up in effective actions used to study string cosmology and other classical configurations. This paper explores these noncompact groups in the case of toroidal compactification both from the viewpoint of low-energy effective field theory, using the method of dimensional reduction, and from the viewpoint of the string theory world sheet. The conclusion is that all these symmetries are intimately related. In particular, we find that Chern-Simons terms in the three-form field strength H µνρ play a crucial role.SU(1, 1) invariance in an appropriate formulation of N = 4, D = 4 supergravity [7].The qualification "appropriate formulation" refers to the fact that duality transformations allow n-forms to be recast as (D − n − 2)-forms in D dimensions (dà = * dA), interchanging the role of Bianchi identities and equations of motion. Only after appropriate transformations is the full noncompact symmetry exhibited. In the SU(1, 1) theory there are two scalar fields, which parametrize the coset space SU(1, 1)/U(1).A year later, Cremmer and Julia showed that N = 8, D = 4 supergravity could be formulated with E 7,7 symmetry [8]. In this case the 70 scalars parametrize the coset E 7,7 /SU(8). In analogous manner the N = 8, D = 5 theory was found to have E 6,6 symmetry, with the 42 scalars parametrizing E 6,6 /USp (8) [9]. The largest known symmetry of this type occurs in the N = 16, D = 3 theory, which has E 8,8 symmetry, with 128 scalars parametrizing E 8,8 /O(16) [10]. It appears to be a general feature that the scalars parametrize G/H, where H is the maximal compact subgroup of G. Two-dimensional examples, in which G and H are both infinite, have also been considered [11].The examples listed above (except for the SU(1, 1) case) all refer to maximally supersymmetric theories. If they have any connection to string theory, it is with the type II superstring. Although it may be worthwhile to do so, that line of inquiry will not be pursued here. Rather, we shall focus on theories with half as much supersymmetry (N = 1 in D = 10 or N = 4 in D = 4). These should be relevant to heterotic string theories. In [12] it was shown that the N = 1, D + d = 10 supergravity theory, dimensionally reduced to D dimensions (by dropping the dependence of the fields on d dimensions), has global O(d, d) symmetry. One exception occurs for D = 3, where duality transformations allow the symmetry to be extended to O(8, 8) [10]. Moreover, when the original N = 1 D = 10 theory has n Abelian vector supermultiplets in addition to the supergravity multiplet, the global symmetry of the dimensionallyreduced...

Scale factor duality and the energy condition inequalities

“…Subsequent generalisations to background fields depending on more than one coordinate have been carried out in [4]. Such noncompact symmetries of string theory [5] have been exploited to a great extent to construct inequivalent string vacua and has yielded interesting background geometries representing black holes [6] and black p-branes [4], [7] as well as cosmologies [8], [9].…”

Transformation of black-hole hair under duality and supersymmetry

“…This we will do in Section 1 getting manifest O(2, 4) due to the presence of the two Abelian vector fields in four dimensions [10]. We will also find the S duality transformations in their four-dimensional form.…”

“…As announced in the Introduction, it will be assumed that the metric has a timelike and a spacelike rotational isometry 10 . The former is physically associated to the stationary (but not static, in general) character of the spacetime and the other to the axial symmetry 11 .…”

“…This action can be obtained from the ten-dimensional heterotic string effective action by first considering only the lowest order in α ′ terms (so the Yang-Mills fields and R 2 terms are consistently excluded), then compactifying the theory on T 6 to four dimensions following essentially Ref. [10] and afterwards setting to zero all the scalars and identifying the six Kaluza-Klein vector fields with the six vector fields that come from the ten-dimensional axion. This last truncation is done in the equations of motion and it is perfectly consistent with them and with the supersymmetry transformation rules.…”

Integrability and generalized monodromy matrix

“…which is derived by dimensional reduction of the effective action of string theory [12,13]. In this expression, g is the determinant of the metric g αβ where α, β = 0, 1 are the twodimensional space time indices and R is the associated Ricci scalar curvature.…”