We consider $G_2$-structures with torsion coupled with $G_2$-instantons, on a compact $7$-dimensional manifold. The coupling is via an equation for $4$-forms which appears in supergravity and generalized geometry, known as the Bianchi identity. First studied by Friedrich and Ivanov, the resulting system of partial differential equations describes compactifications of the heterotic string to three dimensions, and is often referred to as the $G_2$-Strominger system. We study the moduli space of solutions and prove that the space of infinitesimal deformations, modulo automorphisms, is finite dimensional. We also provide a new family of solutions to this system, on $T^3$-bundles over $K3$ surfaces and for infinitely many different instanton bundles, adapting a construction of Fu-Yau and the second named author. In particular, we exhibit the first examples of $T$-dual solutions for this system of equations.
We construct the space of infinitesimal variations for the Strominger system and an obstruction space to integrability, using elliptic operator theory. We initiate the study of the geometry of the moduli space, describing the infinitesimal structure of a natural foliation on this space. The associated leaves are related to generalized geometry and correspond to moduli spaces of solutions of suitable Killing spinor equations on a Courant algebroid. As an application, we propose a unifying framework for metrics with holonomy $\SU(3)$ and solutions of the Strominger system.
We consider $G_2$ structures with torsion coupled with $G_2$-instantons, on a compact $7$-dimensional manifold. The coupling is via an equation for $4$-forms which appears in supergravity and generalized geometry, known as the Bianchi identity. The resulting system of partial differential equations can be regarded as an analogue of the Strominger system in $7$-dimensions. We initiate the study of the moduli space of solutions and show that it is finite dimensional using elliptic operator theory. We also relate the associated geometric structures to generalized geometry.
We construct new examples of solutions of the Hull-Strominger system on non-Kähler torus bundles over K3 surfaces, with the property that the connection $\nabla$ on the tangent bundle is Hermite-Yang-Mills. With this ansatz for the connection $\nabla$, we show that the existence of solutions reduces to known results about moduli spaces of slope-stable sheaves on a K3 surface, combined with elementary analytical methods. We apply our construction to find the first examples of T-dual solutions of the Hull-Strominger system on compact non-Kähler manifolds with different topology.
These notes give an introduction to the Strominger system of partial differential equations, and are based on lectures given in September 2015 at the GEOQUANT School, held at the Institute of Mathematical Sciences (ICMAT) in Madrid. We describe the links with the theory of balanced metrics in hermitian geometry, the Hermite-Yang-Mills equations, and its origins in physics, that we illustrate with many examples. We also cover some recent developments in the moduli problem and the interrelation of the Strominger system with generalized geometry, via the cohomological notion of string class.
We describe the infinitesimal moduli space of pairs $(Y, V)$ where $Y$ is a manifold with $G_2$ holonomy, and $V$ is a vector bundle on $Y$ with an instanton connection. These structures arise in connection to the moduli space of heterotic string compactifications on compact and non-compact seven dimensional spaces, e.g. domain walls. Employing the canonical $G_2$ cohomology developed by Reyes-Carrión and Fernández and Ugarte, we show that the moduli space decomposes into the sum of the bundle moduli $H^1_{\check{d}_A}(Y,\mathrm{End}(V))$ plus the moduli of the $G_2$ structure preserving the i…
Heterotic string compactifications on integrable $G_2$ structure manifolds $Y$ with instanton bundles $(V,A), (TY,\tildeθ)$ yield supersymmetric three-dimensional vacua that are of interest in physics. In this paper, we define a covariant exterior derivative $\cal D$ and show that it is equivalent to a heterotic $G_2$ system encoding the geometry of the heterotic string compactifications. This operator $\cal D$ acts on a bundle ${\cal Q}=T^*Y\oplus{\rm End}(V)\oplus{\rm End}(TY)$ and satisfies a nilpotency condition $\check{\cal D}^2=0$, for an appropriate projection of $\cal D$. Furthermore, …
This note revisits recent results regarding the geometry and moduli of solutions of the heterotic string on manifolds $Y$ with a $G_2$ structure. In particular, such heterotic $G_2$ systems can be rephrased in terms of a differential $\check {\cal D}$ acting on a complex $\checkΩ^*(Y , {\cal Q})$, where ${\cal Q}=T^*Y\oplus{\rm End}(TY)\oplus{\rm End}(V)$ and $\check {\cal D}$ is an appropriate projection of an exterior covariant derivative ${\cal D}$ which satisfies an instanton condition. The infinitesimal moduli are further parametrised by the first cohomology $H^1_{\check {\cal D}}(Y,{\cal…
It is well-known that every 6-dimensional strictly nearly Kähler manifold $(M,g,J)$ is Einstein with positive scalar curvature $scal>0$. Moreover, one can show that the space $E$ of co-closed primitive (1,1)-forms on $M$ is stable under the Laplace operator $Δ$. Let $E(a)$ denote the $a$-eigenspace of the restriction of $Δ$ to $E$. If $M$ is compact, we prove that the moduli space of infinitesimal Einstein deformations of the nearly Kähler metric $g$ is naturally isomorphic to the direct sum $E(scal/15)\oplus E(scal/5)\oplus E(2scal/5)$. It is known that the last summand is itself isomorphic w…
This paper considers the natural geometric structure on the moduli space of deformations of a compact special Lagrangian submanifold $L^n$ of a Calabi-Yau manifold. From the work of McLean this is a smooth manifold with a natural $L^2$ metric. It is shown that the metric is induced from a local Lagrangian immersion into the product of cohomology groups $H^1(L)\times H^{n-1}(L)$. Using this approach, an interpretation of the mirror symmetry discussed by Strominger, Yau and Zaslow is given in terms of the classical Legendre transform.
We describe the first order moduli space of heterotic string theory compactifications which preserve $N=1$ supersymmetry in four dimensions, that is, the infinitesimal parameter space of the Strominger system. We establish that if we promote a connection on $TX$ to a field, the moduli space corresponds to deformations of a holomorphic structure $\bar D$ on a bundle $\cal Q$. The bundle $\cal Q$ is constructed as an extension by the cotangent bundle $T^*X$ of the bundle $E= {\rm End}(V) \oplus {\rm End}(TX) \oplus TX$ with an extension class $\cal H$ which precisely enforces the anomaly cancela…
We study the four-dimensional effective theory arising from ten-dimensional heterotic supergravity compactified on manifolds with torsion. In particular, given the heterotic superpotential appropriately corrected at $\mathcal{O}(α')$ to account for the Green-Schwarz anomaly cancellation mechanism, we investigate properties of four-dimensional Minkowski vacua of this theory. Considering the restrictions arising from F-terms and D-terms we identify the infinitesimal massless moduli space of the theory. We show that it agrees with the results that have recently been obtained from a ten-dimensiona…
Infinitesimal rigidity of Hermitian gravitational instantons
University of Potsdam（波茨坦大学）；University of Edinburgh（爱丁堡大学）；Max-Planck-Institut für Gravitationsphysik (Albert-Einstein-Institut)（马克斯·普朗克引力物理研究所（爱因斯坦研究所））
An analytic approach and description are presented for the moduli cotangent sheaf for suitable stable curve families including noded fibers. For sections of the square of the relative dualizing sheaf, the residue map at a node gives rise to an exact sequence. The residue kernel defines the vanishing residue subsheaf. For suitable stable curve families, the direct image sheaf on the base is locally free and the sequence of direct images is exact. Recent work of Hubbard-Koch and a formal argument provide that the direct image sheaf is naturally identified with the moduli cotangent sheaf. The res…
We describe the geometry of generic heterotic backgrounds preserving minimal supersymmetry in four dimensions using the language of generalised geometry. They are characterised by an $SU(3)\times Spin(6+n)$ structure within $O(6,6+n)\times\mathbb{R}^+$ generalised geometry. Supersymmetry of the background is encoded in the existence of an involutive subbundle of the generalised tangent bundle and the vanishing of a moment map for the action of diffeomorphisms and gauge symmetries. We give both the superpotential and the Kähler potential for a generic background, showing that the latter defines…
This article studies the geometry of moduli spaces of G2-manifolds, associative cycles, coassociative cycles and deformed Donaldson-Thomas bundles. We introduce natural symmetric cubic tensors and differential forms on these moduli spaces. They correspond to Yukawa couplings and correlation functions in M-theory. We expect that the Yukawa coupling characterizes (co-)associative fibrations on these manifolds. We discuss the Fourier transformation along such fibrations and the analog of the Strominger-Yau-Zaslow mirror conjecture for G2-manifolds. We also discuss similar structures and transform…
We study deformations of shrinking Ricci solitons on a compact manifold M, generalising the classical theory of deformations of Einstein metrics. Using appropriate notions of twisted slices S_f inside the space of all Riemannian metrics on M, we define the infinitesimal solitonic deformations and the local solitonic pre-moduli spaces. We prove the existence of a finite dimensional submanifold of S_f x C^infty(M), which contains the pre-moduli space of solitons around a fixed shrinking Ricci soliton as an analytic subset. We define solitonic rigidity and give criteria which imply it.
The moduli space NK of infinitesimal deformations of a nearly Kähler structure on a compact 6-dimensional manifold is described by a certain eigenspace of the Laplace operator acting on co-closed primitive (1,1) forms. Using the Hermitian Laplace operator and some representation theory, we compute the space NK on all 6-dimensional homogeneous nearly Kähler manifolds. It turns out that the nearly Kähler structure is rigid except for the flag manifold F(1,2)=SU_3/T^2, which carries an 8-dimensional moduli space of infinitesimal nearly Kähler deformations, modeled on the Lie algebra su_3 of the i…
Apart from global topological problems an affine homogeneous space is locally described by its curvature, its torsion and a slightly less tangible object called its connection in a given base point. Using this description of the local geometry of an affine homogeneous space we construct an algebraic variety $\mathfrak{M}(\mathfrak{gl}\,V)$, which serves as a coarse moduli space for the local isometry classes of affine homogeneous spaces of dimension dim V. Moreover we associate a $\mathrm{Sym}V^*$-comodule to a point in $\mathfrak{M}(\mathfrak{gl}\,V\,)$ and use its Spencer cohomology in order…
An \emph{$ω$-admissible almost complex structure} on a $2n$-dimensional symplectic manifold $(M,ω)$ is a $ω$-calibrated almost complex structure $J$ admitting a nowhere vanishing $\bar{\partial}_J$-closed $(n,0)$-form $ψ$. After giving some examples we consider the moduli space of admissible almost complex structures and we study infinitesimal deformations. As special case, we write down explicit computations for the complex torus.
We introduce a symplectic structure on the space of connections in a G-principal bundle over a four-manifold and the Hamiltonian action on it of the group of gauge transformations which are trivial on the boundary. The symplectic reduction becomes the moduli space of flat connections over the manifold. On the moduli space of flat connections we shall construct a hermitian line bundle with connection whose curvature is given by the symplectic form. This is the Chern-Simons prequantum line bundle. The group of gauge transformations on the boundary of the base manifold acts on the moduli space of…
In this thesis, we study moduli in compactifications of ten-dimensional heterotic supergravity. We consider supersymmetric compactifications to four-dimensional maximally symmetric space, commonly referred to as the Strominger system. The compact part of space-time $X$ is a six-dimensional manifold of what we refer to as a heterotic $SU(3)$-structure. We show that this system can be put in terms of a holomorphic operator $\bar D$ on a bundle $\mathcal{Q}=T^*X\oplus\mathrm{End}(TX)\oplus\mathrm{End}(V)\oplus TX$, defined by a series of extensions. We proceed to compute the infinitesimal deforma…
We study heterotic supergravity at $\mathcal{O}(α')$, first described in detail in 1989 by Bergshoeff and de Roo. In particular, we discuss an ambiguity of a connection choice on the tangent bundle. It is well known that at $\mathcal{O}(α')$ the Hull connection gives a consistent supergravity theory with supersymmetry transformations given in the usual way. We consider deformations of this connection corresponding to field redefinitions, and the necessary corrections to the supersymmetry transformations. We are interested in the moduli space of such field redefinitions which allow for supersym…
In this article we discuss the geometry of moduli spaces of (1) flat bundles over special Lagrangian submanifolds and (2) deformed Hermitian-Yang-Mills bundles over complex submanifolds in Calabi-Yau manifolds. These moduli spaces reflect the geometry of the Calabi-Yau itself like a mirror. Strominger, Yau and Zaslow conjecture that the mirror Calabi-Yau manifold is such a moduli space and they argue that the mirror symmetry duality is a Fourier-Mukai transformation. We review various aspects of the mirror symmetry conjecture and discuss a geometric approach in proving it. The existence of rig…
This is the second in a series of five papers math.DG/0211294, math.DG/0302355, math.DG/0302356, math.DG/0303272 studying special Lagrangian submanifolds (SL m-folds) X in (almost) Calabi-Yau m-folds M with singularities x_1,...,x_n locally modelled on special Lagrangian cones C_1,...,C_n in C^m with isolated singularities at 0. Readers are advised to begin with the final paper math.DG/0303272 which surveys the series, gives examples, and proves some conjectures. In this paper we study the deformation theory of compact SL m-folds X in M with conical singularities. We define the moduli space M_…
This paper is dedicated to the study of deformations of coassociative 4-folds in a G_2 manifold which have conical singularities. We stratify the types of deformations allowed into three problems. The main result for each problem states that the moduli space is locally homeomorphic to the kernel of a smooth map between smooth manifolds. In each case, the map in question can be considered as a projection from the infinitesimal deformation space onto the obstruction space. Thus, when there are no obstructions the moduli space is a smooth manifold. Furthermore, we calculate a lower bound on the e…
It is shown that the space of infinitesimal deformations of 2k-Einstein structures is finite dimensional at compact non-flat space forms. Moreover, spherical space forms are shown to be rigid in the sense that they are isolated in the corresponding moduli space.
The proper action functional of (4k+3)-dimensional U(1)-Chern-Simons theory including the instanton sectors has a well known description: it is given on the moduli space of fields by the fiber integration of the cup product square of classes in degree-(2k+2) differential cohomology. We first refine this statement from the moduli space to the full higher smooth moduli stack of fields, to which the higher order-ghost BRST complex is the infinitesimal approximation. Then we generalize the refined formulation to cup product Chern-Simons theories of nonabelian and higher nonabelian gauge fields, su…
Let $Σ$ be a closed surface, $G$ a compact Lie group, with Lie algebra $g$, $ξ\colon P \to Σ$ a principal $G$-bundle, let $N(ξ)$ denote the moduli space of central Yang-Mills connections on $ξ$, for suitably chosen additional data, and let $\roman{Rep}_ξ(Γ,G)$ be the space of representations of the universal central extension $Γ$ of the fundamental group of $Σ$ in $G$ that corresponds to $ξ$. We construct smooth structures on $N(ξ)$ and $\roman{Rep}_ξ(Γ,G)$ and show that the assignment to a smooth connection $A$ of its holonomies with reference to suitable closed paths yields a diffeomorphism …
We prove each embedded, constant mean curvature (CMC) surface in Euclidean space with genus zero and finitely many coplanar ends is nondegenerate: there is no nontrivial square-integrable solution to the Jacobi equation, the linearization of the CMC condition. This implies that the moduli space of such coplanar surfaces is a real-analytic manifold and that a neighborhood of these in the full CMC moduli space is itself a manifold. Nondegeneracy further implies (infinitesimal and local) rigidity in the sense that the asymptotes map is an analytic immersion on these spaces, and also that the copl…
We study the Bonnet problem for surfaces in 4-dimensional space forms, where two isometric surfaces have the same mean curvature if there exists a parallel vector bundle isometry between their normal bundles that preserves the mean curvature vector fields. We deal with the moduli space of congruence classes of isometric surfaces with the same mean curvature, and with properties inherited on a surface by its structure. The study of this problem led us to a new conformally invariant property, called isotropic isothermicity, that coincides with the usual concept of isothermicity for surfaces lyin…
The space $\mathbf{H}^{4,2}$ of vectors of norm -1 in $\mathbb{R}^{4,3}$ has a natural pseudo-Riemannian metric and a compatible almost complex structure. The group of automorphisms of both of these structures is the split real form $G_2'$. In this paper we consider a class of holomorphic curves in $\mathbf{H}^{4,2}$ which we call alternating. We show that such curves admit a so called Frenet framing. Using this framing, we show that the space of alternating holomorphic curves which are equivariant with respect to a surface group are naturally parameterized by certain $G_2'$-Higgs bundles. Thi…
