Vanishing theorems and string backgrounds

By Stefan Ivanov, Georgios Papadopoulos. 2000

219 citations recorded by the source corpus.

We show various vanishing theorems for the cohomology groups of compact Hermitian manifolds for which the Bismut connection has a (restricted) holonomy contained in SU(n) and classify all such manifolds of dimension four. In this way we provide necessary conditions for the existence of such structures on compact Hermitian manifolds with vanishing first Chern class of non-Kahler type. Then we apply our results to solutions of the string equations and show that such solutions admit various cohomological restrictions such as, for example, that under certain natural assumptions the plurigenera vanish. We also find that under some assumptions the string equations are equivalent to the condition that a certain vector is parallel with respect to the Bismut connection.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/083e862c62777da0613cc9796dce00f298bbbc4a

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Citations in the corpus

Canonical bundles of complex nilmanifolds, with applications to hypercomplex geometry (2007, background): “The Lee form can be expressed (locally) in terms of the torsion tensor c as follows (see [ IP ]):”
Geometric Model for Complex Non-Kähler Manifolds with SU (3) Structure (2002, background): “pactiﬁcations (see e.g. [11] and references therein). In many diﬀerent contexts, the “manifolds with intrinsic torsion” or “manifolds with SU(3) structure” have been studied for a long time (see e.g. [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27]). Remarkably, all these manifolds a”
Exploring curved superspace (2012, background): “7 See also the discussion in [35], where these manifolds are identified with compact Hermitian surfaces for which the restricted holonomy of the Bismut connection is contained in SU(2).”
Geometry of Hermitian manifolds (2010, background): “For more vanishing theorems on special Hermitian manifolds, one can see [1], [25], [16], [17] and references therein.”
A survey on strong KT structures (2009, methodology): “In [34], it was shown that a conformally balanced strong KT structure (J, g) on a compact manifold M of real dimension 2n with Hol(∇) ⊆ SU(n) is necessarily Kähler and therefore it gives rise to a Calabi-Yau structure.”
Regularity results for pluriclosed flow (2010, methodology): “We start by recording some basic calculations which appear in Ivanov and Papadopoulos [16].”
Anomaly cancellation and smooth non-Kähler solutions in heterotic string theory (2006, background): “Not being Calabi–Yau, many theorems of Kähler geometry do not apply which makes their analysis more arduous.”
Complex geometry of conifolds and a 5-brane wrapped on a 2-sphere (2000, methodology): “In particular, it was shown in [13] that if one assumes that M(6) is compact and the dilaton is a globally defined function on it, then the only such spaces that preserve at least four supersymmetries are the Calabi-Yau spaces with H = 0 and Φ = const.”
BPS Action and Superpotential for Heterotic String Compactifications with Fluxes (2003, methodology): “The Ricci two–form of the Bismut connection is the same as the Ricci form of the Chern connection up to an exact form [70, 13],”
Families of strong KT structures in six dimensions (2002, methodology): “This connection was used by Bismut [3] to prove a local index formula for the Dolbeault operator when the manifold is non-Kähler, and is the subject of a number of other interesting results [12, 16].”
The geometry of D = 11 Killing spinors (2002, unknown): “Keywords: M-Theory, Supergravity Models, Differential and Algebraic Geometry. c© SISSA/ISAS 2003 http://jhep.sissa.it/archive/papers/jhep042003039/jhep042003039.pdf J H E P 0 4 ( 2 0 0 3 ) 0 3 9 Contents”
Ricci Curvatures on Hermitian manifolds (2014, background): “See also [1, 3, 4, 11, 12, 23, 33, 34, 42, 58, 59] for some curvature relations on complex surfaces.”
Non-Kähler Calabi-Yau manifolds (2014, background): “much studied in the mathematical physics literature [2, 21, 24, 38, 42, 45, 61], and Conjecture 4.”
Mirror symmetry in generalized Calabi–Yau compactifications ☆ (2002, background): “irror symmetric to NS-ﬂux backgrounds. As a consequence, one ﬁnds rather diﬀerent conditions. Since in both cases one wants N = 2 supersymmetry in four dimensions, the class of manifolds discussed in [19,20,21,30,37,40,41] are also manifolds with SU(3) structure. However, in these cases the torsion ”
Non-K\"ahler Calabi-Yau manifolds (2014, background): “much studied in the mathematical physics literature [2, 21, 24, 38, 42, 45, 61], and Conjecture 4.”
Calabi-Yau connections with torsion on toric bundles (2003, background): “†Address: Department of Mathematics, Florida International University, Miami, FL 33199, U.S.A..”
Strominger connection and pluriclosed metrics (2019, background): “, we refer the readers to [1], [7], [11], [17], [20], [23], [24], [27], [28], [29], [37], [38], [40], [44], [49] and the references therein for more discussions and backgrounds.”