# Infinitesimal moduli for the Strominger system and Killing spinors in generalized geometry

###### Abstract.

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 and solutions of the Strominger system.

###### 2010 Mathematics Subject Classification:

58D27, 53D18cm12

###### Contents

## 1. Introduction

The Strominger system couples a pair of Hermite–Yang–Mills
connections with a conformally balanced hermitian metric on a
Calabi–Yau threefold , by means of an equation for
-forms—known as *the Bianchi identity*. Although originated
in string theory [39, 63], its mathematical study was proposed by
Yau [70] as a natural generalization of the Calabi problem
[11, 69], in relation to moduli spaces of Calabi-Yau
threefolds which are not necessarily Kählerian.

Pioneered by Fu, Li and Yau [27, 50], the existence problem for the Strominger system has been an active area of research in mathematics in the last ten years (see [4, 20, 21, 25, 26, 28, 64] and references therein). There is an important conjecture by Yau [71], which states that any stable holomorphic vector bundle over a homologically balanced Calabi–Yau threefold [58] with admits a solution of the Strominger system. This conjecture is widely open, the main difficulties being its non-Kähler nature and the lack of understanding of the geometry of the equations.

A problem closely related to Yau’s conjecture is the construction of a moduli space of solutions of the Strominger system. This moduli problem remained almost unexplored for a long time, despite its interest in string theory, where it describes the most basic pieces (scalar massless fields) of the four-dimensional theory induced by a heterotic string compactification. Indeed, only a few references that tackle the preliminary question of constructing the tangent space at a given solution can be found in the physics literature [3, 7, 8, 16, 57, 17]. This first step turns out to be rather challenging, and a complete answer has been so far elusive.

The prime motivations for this work are the construction of the moduli space of solutions of the Strominger system and its interrelation with Yau’s conjecture. In this paper we make a contribution to the first problem, constructing the space of infinitesimal variations of a solution and an obstruction space to integrability. We initiate the study of the geometry of the moduli space, describing the infinitesimal structure of a natural foliation, whose leaves are intimately related to generalized geometry [37]. By investigating the tangent to a leaf, we give an interpretation of the leaves as moduli spaces of solutions of suitable Killing spinor equations on a Courant algebroid. This last tangent space arises naturally as a quotient of a bigger finite-dimensional vector space by the second de Rham cohomology group of . Our construction provides a unifying framework for metrics with holonomy and solutions of the Strominger system, that we expect will have future applications to Yau’s conjecture. To explain our results, let us first recall the definition of the equations.

### Background

Let be a Calabi-Yau threefold, that is, a complex manifold of dimension three endowed with a nowhere vanishing holomorphic section of the canonical bundle . We do not assume that is Kählerian. Let be a principal bundle over with compact structure group . The Strominger system is

(1.1) |

with unknowns given by a hermitian metric on with fundamental form , a connection on and a metric connection on the (smooth) tangent bundle of . Here, is a positive real constant and and denote, respectively, the curvature -forms of and . The notation refers to the Killing form on the Lie algebra of . In this paper we impose that is unitary with respect to the hermitian structure .

The Strominger system comprises, essentially, three conditions—the
first two well understood in the literature. First, the equation in
the third line, often known as the *dilatino equation*, is
strongly reminiscent of the complex Monge-Ampère equation on a
Kähler manifold (see e.g. [29]). It restricts the holonomy of the Bismut connection

to , where denotes the Levi-Civita connection of the metric . Furthermore, as observed by Li and Yau [50], the dilatino equation is equivalent to the condition

(1.2) |

which implies that is the
fundamental form of a balanced metric, namely , and
hence is *conformally balanced*. A classical result of
Michelsohn [58] characterizes the existence of balanced
metrics on a complex manifold using a condition on the
homology—formulated in terms of currents and known as the
*homologically balanced condition*.

Second, the first two lines in (1.1) correspond to the Hermite–Yang–Mills condition for the connections and with respect to the conformally balanced metric . There is a well-known theory for Hermite–Yang–Mills connections on a hermitian manifold [53], which ranges from existence results to the construction of the moduli space (which turns out to be Kähler when is conformally balanced). The main result of the theory is Li–Yau’s theorem [49], which characterizes the existence of solutions in terms of (slope) stability of the bundle, generalizing the Donaldson–Uhlenbeck–Yau theorem in Kähler geometry [18, 65].

Finally, the most demanding and less understood condition of the
system is the *Bianchi identity*

(1.3) |

which is ultimately responsible for the non-Kähler nature of the
problem. The non-vanishing of the *Pontryagin term*
prevents
the hermitian form to be closed and hence allows the complex
manifold to be non-Kählerian. This subtle condition, which
arises in the quantization of the physical theory, was studied by Freed [24] in the context
of index theory for Dirac operators and more recently by Sati–Schreiber–Stasheff from the point of view of *differential string structures* [62]. Despite these important topological
insights, we have an almost total lack of
understanding of this last equation from an analytical point of view.

### Main results

In this work we add to the understanding of the moduli problem for the Strominger system. The first contribution of this work is a complete and direct construction of the vector space of infinitesimal variations of a given solution—the infinitesimal moduli space— using an elliptic complex .

###### Theorem 1.1.

The space of infinitesimal deformations of solutions of the Strominger system is given by the first cohomology group of an elliptic complex of multi-degree differential operators . This complex admits a natural extension , and the space of obstructions is defined as .

To clarify the exposition, we first undertake the construction of the complex for a toy model in Section 2. For this, we introduce an abelian version of the equations (1.1) depending on a real parameter. The analysis in the abelian case will show that the combination of the Bianchi identity with the conformally balanced equation (1.2) is well-behaved at the level of symbols.

In Section 3 we construct the elliptic complex of differential operators and identify its first cohomology

with the infinitesimal moduli of solutions of the Strominger system. Some of the difficulties that arise in the construction of come from the symmetries of the system, which turn out to have a Lie groupoid structure due to the compatibility of the connection with the metric .

In Section 4 we investigate the geometry on the moduli space of solutions of the Strominger system derived from the Bianchi identity. This moduli space is endowed with a canonical -valued closed -form

which is constructed via the variation of (1.3). The kernel of defines an integrable distribution on the tangent bundle of and hence a foliation on the moduli space. A striking fact about this foliation is that its leaves can be understood by using Hitchin’s theory of generalized geometry [37]. The aim of Section 4 is to give a rigorous account of the infinitesimal version of this picture. The construction of a differentiable structure on and the local structure of the foliation will be addressed in future work.

Neglecting obstructions to integrability, the tangent to a leaf at a point is defined by an exact sequence

As the notation suggests, is the cohomology of a complex which, surprisingly, needs generalized geometry for its rigorous definition. In this new framework has a natural interpretation, as variations of a suitable generalized metric modulo generalized diffeomorphisms. A special feature of is that it cannot be constructed by standard elliptic operator theory, as the space of generalized vector fields cannot be identified with the space of global sections of a vector bundle (similarly as the space of symplectic vector fields on a symplectic manifold). Motivated by this problem, we construct a refinement of , which fits into the following exact diagram

(1.4) |

Unlike , the refined vector space
is constructed by considering *inner
symmetries* of a smooth, transitive, Courant algebroid, and is
defined as the first cohomology of an elliptic complex of degree
differential operators. Motivation for the previous construction comes
from two basic principles in the physics of the heterotic string,
given by the *Green-Schwarz mechanism* [34], and
the *flux quantization condition*. We should stress that the
space is the one that comes closer to the physics
of the heterotic string.

Section 5 gives a geometric interpretation of the leaves of the foliation in the moduli space, showing the strong connection between the Strominger system and generalized geometry. For this, we define suitable Killing spinor equations

(1.5) |

for an admissible metric on a smooth transitive Courant algebroid and prove the following result.

###### Theorem 1.2.

This result builds on previous work of the first author in the relation between generalized geometry and heterotic supergravity [30]. Theorem 1.2 gives a precise interpretation of the vector space , as infinitesimal deformations for solutions to the Killing spinor equations (1.5) modulo infinitesimal symmetries of the Courant algebroid. As a consequence, a leaf of the foliation determined by can be interpreted as a moduli space of solutions of these equations.

The proof of Theorem 1.2 reveals a strong parallelism with the theory of metrics with holonomy , once generalized geometry enters into the game. The same equations, formulated instead on an exact Courant algebroid, pin down precisely Riemannian metrics with holonomy on a six dimensional manifold. Generalized geometry provides a unifying framework for the theory of the Strominger system and the well-established theory of Calabi–Yau metrics, which we expect will have interesting applications in the former. We will explore further this analogy in future work.

In the physics literature, de la Ossa and Svanes [17] and Anderson, Gray and Sharpe [3] have recently proposed a close approximation to the infinitesimal moduli for the Strominger system in terms of the Dolbeault cohomology of a holomorphic double extension—based on previous ideas by Melnikov and Sharpe [57]. In a sequel to the present paper we will show that their proposal admits a natural interpretation in generalized geometry, and relates in a certain way to the infinitesimal moduli of the Strominger system. After Section 5 was completed, we were informed by A. Coimbra that an alternative formulation of the Strominger system using generalized geometry was provided recently in the physics literature [15]. Our approach has the benefit of making evident that the Strominger system is invariant under generalized diffeomorphisms.

Acknowledgments: We thank Luis Álvarez-Cónsul, Bjorn Andreas, Vestislav Apostolov, Henrique Bursztyn, Ryushi Goto, Marco Gualtieri, Nigel Hitchin, Laurent Meersseman, Xenia de la Ossa, Dan Popovici, Brent Pym and Eirik Svanes for useful discussions. Part of this work was undertaken while CT was visiting IMPA, UFRJ, CRM, during visits of MGF and CT to CIRGET, and of RR to EPFL and ICMAT. We would like to thank these very welcoming institutions for providing a nice and stimulating working environment.

## 2. Infinitesimal moduli: abelian case

The aim of this section is to study a toy model for the construction of the infinitesimal moduli space of the Strominger system, which avoids the difficulties arising from the treatment of the unitary connection on the tangent bundle and non-abelian groups. In particular, we will consider deformations of a Calabi-Yau structure on a compact, six dimensional, smooth manifold , endowed with a holomorphic line bundle. We do not require our complex manifolds to be Kählerian.

### 2.1. The abelian equations

Let be a compact, oriented, six dimensional smooth manifold. Let be a hermitian line bundle over . We fix a non-zero real constant . We denote by the smooth tangent bundle of and its complexification by . Consider triples where is a complex -form such that

(2.1) |

determines an almost complex structure on , is a unitary connection on , and is a -compatible -form, that is,

(2.2) |

is a Riemannian metric on . We aim to construct a space of infinitesimal deformations for solutions of the equations

(2.3) |

where is a constant that depends on the unitary structure determined by and the first Chern class of , as follows from the identity

(2.4) |

Our convention for the point-wise norm of a -form with respect to is

(2.5) |

Recall that the integrability of is equivalent to the condition

and hence, by the first two equations in (2.3), any solution determines a Calabi-Yau threefold structure on endowed with a holomorphic line bundle.

When and the system (2.3)
corresponds to the field equations of *abelian heterotic
supergravity* considered in [55] (note that in our notation
is a purely imaginary -form).
The reason why we do not work directly with this case, which is the
situation that comes closer to the Strominger system, is the following
observation, that shall be compared with [12, p. 55].

###### Proposition 2.1.

Let be a solution of (2.3). If and is a -manifold, then is flat and has holonomy (in particular, is Kähler Ricci-flat).

###### Proof.

As , is exact. By the -lemma, for some smooth function on . After conformal re-scaling of the hermitian metric on , we obtain a flat Chern connection on the holomorphic line bundle . Then, since is the Chern connection of a hermitian-Einstein metric on , has to be flat by uniqueness, and hence it follows that is strong Kähler with torsion. By the conformally balanced condition, the Bismut connection of has holonomy (see [63, Section II]). Applying now [41, Corollary 4.7], is Kähler and the result follows. ∎

In this work we are mainly interested in non-Kähler solutions of (2.3), and therefore we will assume that . Non-Kähler solutions of (2.3) can be obtained using the perturbative method in [4], from holomorphic line bundles over a projective Calabi-Yau threefold with non-torsion satisfying

###### Remark 2.1.

It is perhaps more natural to consider the Hermite-Yang-Mills equations in (2.3) with respect to the conformally balanced metric . The linearization of these alternative abelian equations is, however, more involved and does not add to the understanding of the Strominger system.

### 2.2. Notation, parameter space and symmetries

Let be a complex -form on such that (2.1) determines an almost complex structure on with anti-holomorphic tangent bundle . Denote by (resp. ) the space of real (resp. complex) smooth -forms on . We denote by the space of -forms on and by the space of -forms taking values in a vector bundle . Given a -form on (that may take values in a vector bundle) and , we define a -form by

(2.6) |

where denotes the skew-symmetric part of the tensor satisfying :

As is nowhere vanishing, it induces an isomorphism on forms

(2.7) |

We will also denote by the induced isomorphism in cohomology [59].

To study the infinitesimal moduli of (2.3) we define the following parameter space. Let be the space of unitary connections on and the non-linear subspace of complex -forms such that (2.1) determines an almost complex structure on . We set

(2.8) |

where the compatibility condition is as in (2.2).

Let be the identity component of the group of diffeomorphisms of . Consider the group of automorphisms of that preserve the unitary bundle structure and cover an element in . This group preserves , exchanging solutions of (2.3). Denote by the gauge group of the hermitian bundle . Then we have an exact sequence [1]

(2.9) |

Given a connection on , we have a lift and at the level of vector spaces the corresponding Lie algebra sequence splits

(2.10) |

### 2.3. Linearization and ellipticity

In the sequel, we fix a solution of (2.3). The integrable almost complex structure determined by will be denoted by and the curvature of will be denoted by . The complex encoding infinitesimal deformations of (2.3) is built from an elliptic complex parameterizing infinitesimal deformations of the complex structure that preserve the Calabi-Yau condition, that is, with trivial canonical bundle, [33]

(2.11) |

Here, the first non-trivial arrow is defined by the infinitesimal action of , given by the Lie derivative of and we use the following characterization of the tangent space of at :

The variations in are just rescaling of the holomorphic -form, while elements in correspond via to deformations of the complex structure. Given , we will denote by the associated variation of almost complex structure given by

(2.12) |

with

Let be the tangent space of at the initial solution:

Note that the equations that define can be equivalently written as

(2.13) |

and therefore there is a canonical isomorphism

(2.14) |

where denotes the space of real -forms on , given explicitly by

(2.15) |

Consider the linearization of the equations (2.3)

Using the vector space splitting (2.10) given by the fixed connection , the infinitesimal action

reads explicitly

(2.16) |

with . We construct a complex of differential operators

(2.17) |

combining the operators and with the isomorphism (2.14). Our aim is to prove that this complex is elliptic. Note that an arbitrary unitary connection on is of the form where , with corresponding curvature , and also that is a function of the hermitian structure given by (2.4). Using this, we obtain the following expression for the differential , regarded as an operator with domain .

###### Lemma 2.1.

The differential is given by

(2.18) |

where and are, respectively, the infinitesimal variations of almost-complex structure and constant defined by and , and

(2.19) |

###### Proof.

We note that the differential operator is of first order in the components and , but has order two. We shall use the generalized notion of ellipticity provided by Douglis and Nirenberg [19]. For the general theory of linear multi-degree elliptic differential operators we refer to [51, 52]. Here we recall the basic definition. Let and be smooth real vector bundles over the compact manifold with a direct sum decomposition

and a linear differential operator with corresponding decomposition .

###### Definition 2.1.

Two tuples, and of non-negative integers form a system of orders for if for each , we have order (if then ). The -principal part of is obtained by replacing each by its terms which are exactly of order , and the -principal symbol of is obtained by replacing each with its principal symbol.

We apply now this definition to our setup.

###### Lemma 2.2.

The leading symbol of is given by the formula

(2.20) |

where . The tuples and form a system of orders for and the associated leading symbol is

(2.21) |

with

(2.22) |

###### Proof.

A linear multi-degree complex of differential operators is elliptic if the induced sequence of symbols is exact, as in the standard case. The usual Fredholm properties of elliptic complexes hold, and therefore given any elliptic complex we have an associated finite dimensional cohomology.

###### Proposition 2.2.

###### Proof.

By -invariance of (2.3), . We prove next that the associated sequence of symbols is exact. Assume that for . From the equations , , we deduce that there is a purely imaginary constant such that . Using the ellipticity of the complex (2.11) and the isomorphism (2.7) there exists a unique such that

In terms of , this translates to

(2.23) |

It remains to show that . From we deduce

(2.24) |

Using now , we obtain

and from this,

(2.25) |

(2.26) |

Define and notice from (2.13) and (2.23) that is a real -form. Furthermore, from (2.26) and (2.25), combined with the vanishing of and , we deduce

(2.27) |

From the last equation

for suitable -forms and . Complete the family into a basis of with forms such that is written

(2.28) |

for some real constant . Now, from the second equation in (2.27) we obtain that is proportional to and therefore

for suitable real constants . We note that a basis of the space of real -forms is

and therefore necessarily

Finally, from the first equation in (2.27) we obtain , which implies that as claimed. ∎

### 2.4. Extension of the complex

We extend the complex (2.17) in order to define a space of obstructions to integrability of infinitesimal deformations of the abelian equations (2.3).

We first define a complex parameterizing joint deformations of a Calabi-Yau structure on endowed with a holomorphic line bundle. For this, we combine an elliptic complex defined by Goto [33] with previous work of Huang [38]. Goto’s complex is an extension of the complex (2.11) given by

(2.29) |

where we use the identification provided by the isomorphism (2.7). Following [38] (cf. [31]) we define an elliptic complex

(2.30) |

where

(by convention ) with differential given by

To include deformations of the metric, we build on the complex for the Hermite–Yang–Mills equations defined by Kim [44] (see also [45, p. 246]). Consider the following commutative diagram

(2.31) |

with

where denotes an element in for . It remains to define the maps and in (2.31), given by suitable modifications of the operator in (2.17) (see Lemma 2.1) and the exterior differential. Firstly, we define

where denotes the part of in (2.18). We note that , but does not vanish in general. In fact, we have the following formula.

###### Lemma 2.3.

The part of is given by

(2.32) |

This motivates the introduction of the map

defined by

We are now ready to prove the main result of this section.

###### Proposition 2.3.

###### Remark 2.2.

The complex has a slightly different flavour from the one of Kim [44] (see also [45]), due to the conformally balanced equation. In Kim’s complex, the linearization of the hermitian-Yang-Mills equation is extended by zero, while in (2.31) we need to introduce an exterior differential to extend the linearization of the conformally balanced equation as part of an elliptic complex.

We start with the proof of (2.32). We need the following.

###### Lemma 2.4.

Let and . Then

(2.33) |

###### Proof.

Write locally so that

Then

The formula

follows from and the expression of in local coordinates. ∎

###### Proof of Lemma 2.3.

###### Proof of Proposition 2.3.

We first prove that (2.31) is a complex. Note that implies and trivially . Hence, using that is a complex, it remains to prove that . Given , using Lemma 2.3 we obtain

which vanishes for type reasons. We next prove that is elliptic. Ellipticity at steps one, two and five follows from ellipticity of the complex and the proof of Proposition 2.2. Ellipticity at step four follows from ellipticity of and of the complex

combined with the formula for the symbol of . We verified the ellipticity of at all steps but one, so it remains to show that an alternated sum of dimensions vanishes. Given a complex as above and , set to be the fiber at of the bundle on whose space of smooth global sections is . With this notation, we need to show that

(2.36) |

By (2.14),

(2.37) |

and note that

(2.38) |

By ellipticity of , the following sum vanishes:

(2.39) |

Then, by (2.39), (2.37) and (2.38), equation (2.36) is equivalent to