Formalism
Definition
Spacetime configurations
Properties
Spacetimes
black hole spacetimes | vanishing angular momentum | positive angular momentum |
---|---|---|
vanishing charge | Schwarzschild spacetime | Kerr spacetime |
positive charge | Reissner-Nordstrom spacetime | Kerr-Newman spacetime |
Quantum theory
superalgebra and (synthetic ) supergeometry
The equations of motion of supergravity typically imply – or are even equivalent to (Candiello-Lechner 93, Howe 97), that the super-torsion of the super-vielbein fields vanishes. At least in some cases these supergravity torsion constraints may naturally be understood as saying that supergravity solutions are (higher) super-Cartan geometry modeled on extended super Minkowski spacetime with its canonical torsion of a G-structure, due to the fact that the left invariant 1-forms on super-Minkowski space are not closed.
The torsion constraint is naturally understood by regarding supergravity as Cartan geometry for the inclusion of the orthogonal group into a super Poincare group and by noticing that the corresponding local model space, which is super-Minkowski spacetime , canonically has non-vanishing torsion.
Let be the canonical coordinates on the supermanifold underlying the super translation group. Then the left-invariant 1-forms are
.
.
Here the extra summand in the equation for (necessary to make it left-invariant) causes it to be non-closed:
Taking the spin connection on to vanish, as usual, this means that there is non-vanishing torsion:
Depending on perspective one might say that it is the supertorsion that vanishes (see at super-Minkowski spacetime and at D'Auria-Fre formulation of supergravity for this perspective), or, alternatively, that one is dealing with Cartan geometry/G-structure whose local model space carries non-vanishing torsion, see below.
Notice that the torison-full but left-invariant forms are of course obtained from the torsion-free but non-left-invartiant forms by a -valued function:
This shows that regarding
as a super-vielbein is consistent: this is indeed a homotopy in
but not the tautological one given by
where the left triangle is that which exhibits the canonical trivialization of the frame bundle of .
Given a subgroup of the general linear group of a linear model space (e.g. super-Minkowski spacetime ), then a G-structure is first-order integrable if on the first-order infinitesimal neighbourhoods of any point it is equal to the canonical (trivial) -structure on . Ordinarily the standard torsion on vanishes, and if so then so does that of any first-order integrable -structure, which is the reason why for these the torsion of a G-structure vanishes.
But in the situation of being super-Minkowski spacetime as above, the torsion of the local model space does not vanish, and so accordingly neither does that of a first-order integrable -structure in this case.
This perspective on the torsion constraints in supergravity is adopted in (Lott 01), see there around (38) of the original article or section 4 of the review on the arXiv.
The supergravity equations of motion typically imply the torsion constraints. See at super p-brane – On curved spacetimes for more.
With enough supersymmetry, the torsion constraints (always together with the Bianchi identities on the superfields, see at D'Auria-Fre formulation of supergravity) may even become equivalent to the supergravity equations of motion. This is so for 11-dimensional supergravity (Candiello-Lechner 93, Howe 97, see Cederwall-Gran-Nilsson-Tsimpis 04, section 2.4) and maybe its maximally supersymmetric KK-compactifications. See at Examples – 11d SuGra.
A close analogy between CR geometry and supergravity superspacetimes (as both being torsion-ful integrable G-structures) is pointed out in (Lott 01 exposition (4.2)).
In accord with the above, typically the equations of motion of a supergravity theory constrain the spinorial part of the torsion to have components .
The torsion constraint for 11-dimensional supergravity is discussed for instance by Bergshoeff, Sezgin & Townsend 1987, (14).
Here something special happens:
The authors Candiello & Lechner 1993 (5.6) and Howe 1997 (see Cederwall, Gran, Nilsson, Tsimpis 2004, section 2.4) show that imposing the torsion constraint (on any chart) as well as implies the equations of motion of 11d supergravity.
For heterotic supergravity in 10d the equations of motion are equivalent to the condition that
the super-torsion of the bosonic part of the super vielbein is a bosonic form
the super-torsion of the odd part of the super vielbein is of the form
for
proportional to the bispinor formed by tracing the square of the gaugino field
the curvature 2-form of the gauge field has vanishing bispinorial component:
(this is the 10d super Yang-Mills theory sector)
This is due to (Witten 86 (5)+(27)), see also (Atick-Dhar-Ratra 86 (4.1)). These authors do not state explicitly that . (Among authors using a similar but different parameterization this statement is made explicit in Candiello-Lechner 93 (2.5) with (2.29)). But this follows by taking the differential of the bispinorial part of the 3-form field (which is the cocycle term for the heterotic Green-Schwarz superstring)
where we used the relation (Witten 86 (8)) (recalled for instance in Bonora-Bregola-Lechner-Pasti-Tonin 87 (2.28), Lechner-Tonin 08 (2.13)).
According to (Bonora-Bregola-Lechner-Pasti-Tonin 90) in fact all these constraints follow from just .
The formulation of supergravity equations of motion in terms of constraints on the torsion tensor goes back to
A mathematical formulation in terms of torsion-full first-order integrable G-structures on supermanifolds (for low dimensional supergravity theories) is given in
John Lott, The Geometry of Supergravity Torsion Constraints [arXiv:0108125]
following:
John Lott, Torsion constraints in supergeometry, Comm. Math. Phys. 133 (1990) 563-615 [doi:10.1007/BF02097010]
which is followed up in
Discussion of torsion constrains for 11-dimensional supergravity from the point of view of consistency of the membrane Green-Schwarz action functional is in
The claim that this torsion constraint in 11-dimensional supergravity is already equivalent to all of the equations of motion is due to
A. Candiello, Kurt Lechner, Duality in Supergravity Theories, Nucl.Phys. B412 (1994) 479-501 (arXiv:hep-th/9309143)
Paul Howe: Weyl Superspace, Physics Letters B, 415 2 (1997) 149-155 [arXiv:hep-th/9707184, doi:10.1016/S0370-2693(97)01261-6]
concisely reviewed in
For commentary see also (Nilsson 00, section 2) and
Martin Cederwall, Ulf Gran, Mikkel Nielsen, Bengt Nilsson, Manifestly supersymmetric M-theory, JHEP 0010 (2000) 041 (arXiv:hep-th/0007035)
Paul Howe, Ergin Sezgin, The supermembrane revisited, Class.Quant.Grav. 22 (2005) 2167-2200 (arXiv:hep-th/0412245)
also
Discussion of possible deformations of the torsion constraint (M-theory corrections) includes
Martin Cederwall, Ulf Gran, Mikkel Nielsen, Bengt Nilsson, Generalised 11-dimensional supergravity, in A. Semikhatov, M. Vasiliev and V. Zaikin (eds.) Proceedings of “Quantization, Gauge Theory & Strings”, Moscow 2000 (arXiv:hep-th/0010042)
Paul Howe, Dimitrios Tsimpis, On higher-order corrections in M theory, JHEP 0309 (2003) 038 (arXiv:hep-th/0305129)
Discussion of torsion constraints for heterotic supergravity goes back to (Nilsson 81) and includes
Paul Howe, A. Umerski, On superspace supergravity in ten dimensions, Phys. Lett. B 177 (1986) 163.
Joseph Atick, Avinash Dhar, and Bharat Ratra, Superspace formulation of ten-dimensional N=1 supergravity coupled to N=1 super Yang-Mills theory, Phys. Rev. D 33, 2824, 1986 (doi.org/10.1103/PhysRevD.33.2824)
Edward Witten, Twistor-like transform in ten dimensions, Nuclear Physics B Volume 266, Issue 2, 17 March 1986
Loriano Bonora, M. Bregola; Kurt Lechner, Paolo Pasti, Mario Tonin, Anomaly-free supergravity and super-Yang-Mills theories in ten dimensions, Nuclear Physics B
Volume 296, Issue 4, 25 January 1988 (doi:10.1016/0550-3213(88)90402-6)
Loriano Bonora, M. Bregola; Kurt Lechner, Paolo Pasti, Mario Tonin, A discussion of the constraints in SUGRA-SYM in 10-D, International Journal of Modern Physics A, February 1990, Vol. 05, No. 03 : pp. 461-477 (doi:10.1142/S0217751X90000222)
Paul Howe, Heterotic supergeometry revisited (arXiv:0805.2893)
Bengt Nilsson, A superspace approach to branes and supergravity (arXiv:hep-th/0007017)
Kurt Lechner, Mario Tonin, Superspace formulations of ten-dimensional supergravity, JHEP 0806:021,2008 (arXiv:0802.3869)
For d=4 N=1 supergravity the torsion is again constrained to be equal to the left-invariant torsion of super-Minkowski spacetime, see for instance
Leonardo Castellani, Riccardo D'Auria, Pietro Fré, volume 2, (III.2.28a), (III.3.66a) of Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991)
Daniel Patrick Butter, section 2.2.5 of On conformal superspace and the One-Loop Effective Action in Supergravity, 2010 (pdf)
Last revised on October 6, 2024 at 10:38:09. See the history of this page for a list of all contributions to it.