Parallel spinors and connections with skew-symmetric torsion in string theory

By T. Friedrich, Stefan Ivanov. 2001

396 citations recorded by the source corpus.

We describe all almost contact metric, almost hermitian and G2-structures admitting a connection with totally skew-symmetric torsion tensor, and prove that there exists at most one such connection. We investigate its torsion form, its Ricci tensor, the Dirac operator and the ∇-parallel spinors. In particular, we obtain solutions of the type II string equations in dimension n = 5,6 and 7.

Subfields: Astronomy and Astrophysics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/baa0a1eea262367db42069407eabf82e9eb131e8

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Astronomy and Astrophysics

1. J. Gauntlett
2. D. Martelli
3. D. Waldram
4. M. Graña
5. I. Agricola
6. J. Sparks
7. Nakwoo Kim
8. Stathis Pakis

In Nuclear and High Energy Physics

1. J. Gauntlett
2. D. Martelli
3. D. Waldram
4. J. Sparks
5. M. Graña
6. Nakwoo Kim
7. I. Agricola
8. Stathis Pakis

Citations in the corpus

• Prolongations of Lie algebras and applications (2007, background): “(iii) if R T = 0 its Ricci contraction must vanish and the claim follows from Lemma 2.1, (ii) by a positivity argument.”
• Six-Dimensional Nearly Kähler Manifolds of Cohomogeneity One (II) (2009, background): “Recall also that the NK manifolds constitute one of the sixteen classes of Gray and Hervella’ s classification of almost Hermitian manifolds and their canonical Hermitian connection D has totally skew and D-parallel torsion ([1,10,15,16,23]).”
• Special Hermitian metrics on compact solvmanifolds (2015, background): “1 in [22] implies that in the strictly almost-Kähler case such connections cannot exist.”
• On the holonomy of connections with skew-symmetric torsion (2003, methodology): “Moreover, we know many non integrable geometric structures on Riemannian manifolds admitting a unique metric connection preserving the structure and with non vanishing skew-symmetric torsion (see [ 15 ], [14]).”
• All supersymmetric solutions of minimal supergravity in five- dimensions (2002, neutral): “1 The utility of using G-structures in analysing supersymmetric solutions of supergravity was discussed earlier in [ 4 , 5].”
• Killing spinor equations in dimension 7 and geometry of integrable G2-manifolds (2001, methodology): “he number of parallel spinors. In 10-dimensional string theory, the Killing spinor equations with non-constant dilation Φ and the 3-form field strength H can be written in the following way [39], (see [25, 24, 16]) (∗) ∇Ψ = 0, (dΦ − 1 2 H) ·Ψ = 0, where Ψ is a spinor field and ∇ is a metric connection”
• Nearly Kaehler geometry and Riemannian foliations (2002, background): “From this point of view they are interesting in theoretical physics [10].”
• The Srní lectures on non-integrable geometries with torsion (2006, background): “roved that many non-integrable geometric structures (almost contact metric structures, nearly Kahler and weak G 2-structures) admit a unique invariant connection ∇ with totally skew-symmetric torsion [FI02], thus being a natural replacement for the Levi-Civita connection. Non-integrable geometries c”
• Maximally homogeneous para-CR manifolds of semisimple type (2006, neutral): “enoted simply by T. The following result of Friedrich and Ivanov clarifies in which circumstances an almostHermitian structure admitsaHermitianconnexion withtotallyskew-symmetric torsion. Theorem 2.2. [30] Let (M2m,g,J) be almost Hermitian. There exists an almost Hermitian connexion with torsion in Λ”
• Some remarks on G_2-structures (2003, background): “30) were found in [7, 8], where those authors considered what they called ‘integrable G2-structures’, which, in the terminology of this article, means G2-structures σ satisfying τ2 = 0.”
• Mirror symmetry in generalized Calabi–Yau compactifications ☆ (2002, neutral): “2On leave from IFIN-HH Bucharest”
• Superstrings with intrinsic torsion (2003, background): “The necessary and sufficient conditions for the G2 in d = 7 [8, 9, 10] and Spin(7) in d = 8 [11] cases have also been analysed from this point of view.”
• Fivebranes wrapped on SLAG three-cycles and related geometry (2001, background): “The general theory of a single G2 connection with totally anti-symmetric torsion has been recently developed in [ 35 ].”
• The classification of naturally reductive homogeneous spaces in dimensions $n \leq 6$ (2014, background): “√ r2+1 E35, Λg(ξ) = r(a−2) 2 √ r2+1 E12 + 2−b 2 √ r2+1 E34, and we see that ξis a Killing field. Consequently, the quasi-Sasaki structure admits a characteristic connection ∇ whose torsion is given by [FI02] T= η∧ dη= ar √ r2 +1 η∧ α1 ∧ α2 − b √ r2 +1 η∧ β1 ∧β2, and it is described by the linear map ”
• Flux compactifications in string theory: A Comprehensive review (2005, background): “ly possible when its structure group is reduced. Let us start by briefly reviewing the main facts about G-structures. For detailed explanations, we refer the reader to the mathematical references Ref. [70, 71, 72, 73]. For a review of G-structures in the context of compactifications with fluxes, see Re”
• G structures, fluxes and calibrations in M theory (2003, background): “Thus, by the results of [46, 47], we indeed have a solution.”