The global formulation of generalized Einstein-Scalar-Maxwell theories
We summarize the global geometric formulation of Einstein-Scalar-Maxwell theories twisted by flat symplectic vector bundle which encodes the duality structure of the theory. We describe the scalar-electromagnetic symmetry group of such models, which consists of flat unbased symplectic automorphisms of the flat symplectic vector bundle lifting those isometries of the scalar manifold which preserve the scalar potential. The Dirac quantization condition for such models involves a local system of integral symplectic spaces, giving rise to a bundle of polarized Abelian varieties equipped with a symplectic flat connection, which is defined over the scalar manifold of the theory. Generalized Einstein-Scalar-Maxwell models arise as the bosonic sector of the effective theory of string/M-theory compactifications to four-dimensions, and they are characterized by having non-trivial solutions of “U-fold” type.
Key words and phrases:Supergravity, Lorentzian geometry, symplectic geometry
Supergravity theories [1, 2] are supersymmetric theories of gravity which extend general relativity and gauge theory and arise in the low energy limit of string/M-theory and of their compactifications. It is known that the construction of such theories involves interesting structures, such as Kähler-Hodge and special Kähler manifolds, symmetric spaces, exceptional Lie groups, generalized complex structures, differential cohomology and differential K-theory etc. However, the global formulation of these theories is not yet fully understood. This note is part of a larger project (see also [4, 5, 6]) aimed at obtaining the full mathematical formulation of supergravity theories (in the generality required by their relation to string theory) and at studying the global geometry of their solutions.
Supergravity theories are classical theories of gravity coupled to matter, formulated using systems of “fields” defined on a manifold of the appropriate dimension and subject to certain partial differential equations, known as the “equations of motion”. An unambiguous formulation of such theories requires that one specifies the global nature of the fields and of the partial differential operators arising in the equations of motion. Currently, however, the supergravity literature gives only local descriptions111Descriptions which are valid only if one restricts all fields to sufficiently small open subsets of . of the fields and of these differential operators. The globalization problem is the problem of giving globally-unambiguous mathematical definitions of such theories which reduce locally to the local description found in the supergravity literature. The solution of this problem is non-unique since there can be many global definitions of “fields” subject to globally-defined partial differential equations which reduce to a given local description.
Since supergravity theories are supersymmetric, they require spinors for their formulation. In this note, we simplify the globalization problem by ignoring the spinor field content and the supersymmetry conditions, thus considering only the so-called universal bosonic sector. This sector arises in any supergravity theory, though it is subject to increasingly stringent supplementary constraints (not discussed in this paper) as the number of supersymmetries present in the theory increases. In addition, we focus exclusively on the case when is a four-manifold.
In four dimensions, the universal bosonic sector is the so-called Einstein-Scalar-Maxwell (ESM) model defined on a four-manifold , which involves gravity (modeled globally by a Lorentzian metric on ), a finite number of real scalar fields (modeled globally by a smooth map from to a manifold of arbitrary dimension) and a finite number of Abelian gauge fields, whose field strengths can be modeled locally as 2-forms defined on . While the local form of ESM theories is well-known, their precise global formulation was systematically addressed only recently . It turns out that the naive globalization of the local formulation fails to capture the classical limit of certain string theory backgrounds known as “U-folds” and hence is insufficient for the application of such models to string theory. The geometric description of the classical limit of U-fold backgrounds  requires that one globalizes ESM models by including a “twist” of the Abelian gauge field sector through the (pull-back of) a flat symplectic vector bundle defined on . This produces so-called generalized ESM models, which are locally indistinguishable from the naive globalization but have a rather different global behavior. The naive globalization corresponds to using a trivial flat symplectic vector bundle on .
The global mathematical formulation of generalized ESM models given in  is summarized in this note. We follow the notations and conventions of loc. cit.; in particular, all manifolds considered are smooth and connected and all bundles and maps considered are smooth. In this note, a Lorentzian metric is a smooth metric of signature defined on a four-manifold.
2. Generalized Einstein-Scalar-Maxwell theories
2.1. Scalar structures and related notions
A scalar structure is a triplet , where is a Riemannian manifold (called the scalar manifold) and is a smooth real-valued function defined on (called the scalar potential).
Let be a scalar structure. Let be an oriented four-dimensional smooth manifold (which need not be compact).
The modified density of a smooth map relative to a Lorentzian metric and to the scalar structure is the following smooth real-valued map defined on :
where and denotes trace taken with respect to .
The modified tension field of a smooth map relative to the Lorentzian metric and to the scalar structure is the section of the pulled-back bundle defined through:
Here is the gradient vector field of with respect to and is the tension field of relative to and :
where denotes the -valued one-form associated to the differential and is the connection induced on by the Levi-Civita connections of and .
2.2. Duality structures
Let be a manifold.
A duality structure on is a flat symplectic vector bundle defined over , where denotes the symplectic pairing on the vector bundle and denotes the -compatible flat connection on .
Let with be two duality structures defined on . A morphism of duality structures from to is a based morphism of vector bundles such that and such that .
With this notion of morphism, duality structures on form a category which we denote by . Let be a duality structure defined on such that . Let denote the category of finite-dimensional symplectic vector spaces over and linear symplectic morphisms. Let denote the unit groupoid of this category and denote the fundamental groupoid of . Let denote the parallel transport of along a path .
The parallel transport functor of is the functor which associates to any point the symplectic vector space and to any homotopy class with fixed initial point and fixed final point the invertible symplectic morphism , where is any path which represents the class .
Notice that can be viewed as a -valued local system defined on . The map which takes into is an equivalence between the category and the functor category . This implies that duality structures on are classified up to isomorphism by the symplectic character variety:
A duality frame of is a -flat symplectic frame of defined on an open subset .
The duality structure is called trivial if it is trivial as a flat symplectic vector bundle.
A duality structure is trivial iff it admits a globally-defined duality frame. If is simply connected, then any duality structure on is trivial.
2.3. Electromagnetic structures
Let be a manifold.
An electromagnetic structure defined on is a quadruplet , where is a duality structure defined on and is a taming of the symplectic vector bundle .
Notice that we do not require to be compatible with . Together with , defines an Euclidean scalar product on given by .
Let and be two electromagnetic structures defined on . A morphism of electromagnetic structures from to is a morphism of duality structures such that .
With this definition of morphism, electromagnetic structures defined on form a category which we denote by . This fibers over the category of duality structures ; the fiber at a duality structure can be identified with the set of tamings of , which is a contractible topological space. Accordingly, the set of isomorphism classes of fibers over the disjoint union of character varieties . Let be an electromagnetic structure defined on and be the Hermitian scalar product defined by and on .
The fundamental form of is the -valued one-form on defined through:
The electromagnetic structure is called unitary if , i.e. if is parallel with respect to .
If is unitary, then is a unitary connection on the Hermitian vector bundle . In this case, we have for all . The category of unitary electromagnetic structures defined on is the full sub-category of whose objects are the unitary electromagnetic structures. This is equivalent with the category of Hermitian vector bundles defined on and endowed with a flat -linear Hermitian connection. In particular, isomorphism classes of unitary electromagnetic structures are in bijection with the points of the character variety:
where acts by conjugation.
2.4. Scalar-duality and scalar-electromagnetic structures
A scalar-duality structure is an ordered system , where:
is a scalar structure
is a duality structure defined on .
A scalar-electromagnetic structure is an ordered system , where:
is a scalar structure
is an electromagnetic structure defined on .
In this case, the system is called the underlying scalar-duality structure, where is the duality structure underlying .
Let be a scalar-electromagnetic structure as in the definition.
The fundamental field of the scalar-electromagnetic structure is defined through:
2.5. Pulled-back electromagnetic structures
Let be a scalar-electromagnetic structure with underlying scalar structure and underlying electromagnetic structure . Let be a four-manifold and be a smooth map from to .
The -pullback of the electromagnetic structure defined on is the electromagnetic structure defined on .
The Hodge operator of induces the endomorphism of the bundle .
The twisted Hodge operator of is the bundle endomorphism defined through:
Let be the main automorphism of . We have:
The operator preserves the sub-bundle , on which it squares to plus the identity. Accordingly, we have a direct sum decomposition:
where are the sub-bundles of eigenvectors of corresponding to the eigenvalues .
An -valued two-form defined on is called positively polarized with respect to and if it is a section of the vector bundle , which amounts to the requirement that it satisfies the positive polarization condition:
For any open subset of , let , and let be the sheaf of smooth sections of the bundle . Globally-defined and positively-polarized -valued forms are the global sections of this sheaf. Notice that is positively polarized iff is.
2.6. The mathematical formulation of generalized ESM theories
Let be a four-manifold and be a scalar-electromagnetic structure with underlying scalar structure and underlying electromagnetic structure . The -pullback of the Euclidean scalar product induced by and on is a Euclidean scalar product on . Let be the bundle morphism given by -contraction of the two middle indices. This is uniquely determined by the condition:
where is the pseudo-Euclidean metric induced by on . Viewing as the sub-bundle of antisymmetric 2-tensors inside , this restricts to a morphism of vector bundles , which we call the inner -contraction of 2-forms.
The twisted inner contraction of -valued 2-forms is the unique morphism of vector bundles which satisfies:
for all and all .
Let be the fundamental field of and let be its pullback through . Let be the pseudo-Euclidean scalar product induced by and on the vector bundle . For any vector bundle defined on , we extend this trivially to a -valued pairing (denoted by the same symbol) between the bundles and . Similarly, we trivially extend the twisted wedge product defined in Appendix C of reference  to a -valued pairing (denoted by the same symbol) between the bundles and .
The sheaf of ESM configurations determined by is the sheaf of sets defined on through:
for all open subsets , with the obvious restriction maps. An element is called a local ESM configuration of type defined on . The set of global configurations of type is the set:
of global sections of this sheaf. An element is called a global ESM configuration of type .
The generalized ESM theory associated to is defined by the following set of partial differential equations on with unknowns :
The Einstein equation:
with energy-momentum tensor given by:
The scalar equations:
The twisted electromagnetic equations:
where is the de Rham differential of twisted by the pulled-back flat connection .
A local ESM solution of type defined on is a smooth solution of these equations which is defined on . A global ESM solution of type is a smooth solution of these equations which is defined on . The sheaf of local ESM solutions of type is the sheaf of sets defined on whose sections on an open subset is the set of all local solutions defined on .
It is shown in  that a generalized ESM model is locally indistinguishable from an ordinary ESM model, in the sense that the global partial differential equations (7), (9) and (10) reduce locally to those used in the supergravity literature (see for example reference 222Notice however that we use different conventions.) upon choosing a local flat symplectic frame of the duality structure . The supergravity literature tacitly assumes that the local formulas globalize trivially, which amounts to working with a trivial duality structure; this assumption implies existence of a globally-defined duality frame. Generalized ESM models with a non-trivial duality structure are globally quite different from the models used in the supergravity literature, since a non-trivial duality structure does not admit global duality frames. Due to this fact, global solutions of generalized ESM models afford a geometric description of a certain type of classical U-folds, thereby realizing the proposal of .
2.7. Sheaves of scalar-electromagnetic configurations and solutions
Let and be as above and fix a metric .
The sheaf of local scalar-electromagnetic configurations relative to is the sheaf of sets defined on whose set of sections on an open subset is defined through:
The set of global scalar-electromagnetic configurations relative to is the set of global sections of this sheaf.
The sheaf of local scalar-electromagnetic solutions relative to is the sheaf of sets defined on whose set of sections on a an open subset is defined as the set of all solutions of the scalar and twisted electromagnetic equations (9) and (10) defined on . The set of global scalar-electromagnetic solutions relative to is the set of global sections of .
Since it will be of use later, we define:
where is a scalar-duality structure.
2.8. Electromagnetic field strengths
An electromagnetic field strength on with respect to and relative to and to the map is an -valued 2-form which satisfies the following two conditions:
is positively polarized with respect to , i.e. we have .
is -closed, i.e.:
The second condition is called the electromagnetic equation.
For any open subset of , let:
denote the set of electromagnetic field strengths defined on , which is an (infinite-dimensional) subspace of the -vector space . This defines a sheaf of electromagnetic field strengths relative to and , which is a locally-constant sheaf of -vector spaces defined on .
3. Scalar-electromagnetic dualities and symmetries
Let be a duality structure on and be a taming of . Let be the corresponding electromagnetic structure with underlying duality structure .
An unbased automorphism is called:
A symmetry of the duality structure , if is symplectic with respect to and covariantly constant with respect to .
A symmetry of the electromagnetic structure , if is complex with respect to and is a symmetry of the duality structure .
Let and denote the groups of symmetries of and . We have:
Given a symplectic automorphism , the endomorphism is again a taming of , where denotes the adjoint action of on ordinary sections of the bundle (see ). Hence for any electromagnetic structure having as its underlying duality structure, the quadruplet:
is again an electromagnetic structure having as its underlying duality structure. This defines a left action of the group on the set of all electromagnetic structures whose underlying duality structure equals .
Let be a four-manifold and be a scalar-electromagnetic structure with underlying scalar structure and underlying electromagnetic structure . Let be the scalar-duality structure underlying , where . Let be a Lorentzian metric on . Let:
where denote the isometry group of .
The scalar-electromagnetic duality group of is the following subgroup of :
An element of this group is called a scalar-electromagnetic duality. The duality action is the action of on the set given by:
where is the projection of to and is the based isomorphism of vector bundles induced by .
For any , we have:
The scalar-electromagnetic symmetry group of is the following subgroup of :
An element of this group is called a scalar-electromagnetic symmetry.
We have short exact sequences:
where and are the groups of based automorphisms of and and the groups and consist of those automorphisms of the scalar structure which admit lifts to scalar-electromagnetic duality transformations and scalar-electromagnetic symmetries, respectively. Let be the holonomy group of at a point . Then we can identify with the commutant of inside the group .
4. Twisted Dirac quantization
Let be a manifold.
4.1. Integral duality structures and integral electromagnetic structures
Let be a duality structure of rank defined on . A Dirac system for is a fiber sub-bundle which satisfies the following conditions:
For any , the triple is an integral symplectic space, i.e. is a full lattice in and .
is invariant under the parallel transport of , i.e. the following condition is satisfied for any path :
For every , the lattice is called the Dirac lattice defined by at the point .
An integral duality structure defined on is a pair , where is is a duality structure defined on and is a Dirac system for .
Relation (16) implies that the type (the ordered list of elementary divisors) of the integral symplectic space does not depend on the point . This quantity is denoted and called the type of .
Let and be two integral duality structures defined on . An morphism of of integral duality structures from to is a morphism of duality structures such that .
The set of isomorphism classes of integral duality structures of type defined on is in bijection with the character variety:
where is the modified Siegel modular group of type .
Let be an integral duality structure or rank and type , defined on . For any , the integral symplectic space defines a symplectic torus . These tori fit into a fiber bundle , endowed with a complete flat Ehresmann connection induced by . The Ehresmann transport of this connection is through isomorphisms of symplectic tori so it preserves the group structure and symplectic form of the fibers; in particular, the holonomy group of is contained in .
The pair is called the flat bundle of symplectic tori defined by the integral duality structure .
An integral electromagnetic structure defined on is a pair , where is an electromagnetic structure defined on and is a Dirac system for the underlying duality structure of . The type of the integral duality structure is called the type of :
Let be an integral electromagnetic structure of real rank and type , with underlying duality structure . For every , the fiber is an integral tamed symplectic space which defines a polarized Abelian variety of type , whose underlying symplectic torus is given by . These polarized Abelian varieties fit into a smooth fiber bundle . As above, the connection induces a complete integrable Ehresmann connection on this bundle, whose transport proceeds through isomorphisms of symplectic tori, so it preserves the Abelian group structure and symplectic form of the fibers but not their complex structure.
The pair is called the bundle of polarized Abelian varieties defined by the integral electromagnetic structure .
4.2. The twisted Dirac quantization condition
Let be a Lorentzian four-manifold and be a Riemannian manifold. Let be a fixed smooth map from to . Let be an integral electromagnetic structure defined on , with underlying electromagnetic structure and underlying duality structure . Then the system is an electromagnetic structure on , where is the -pullback of the fiber sub-bundle ; this has underlying duality structure . Let denote the integral duality structure underlying . Let denote the category of finite-dimensional integral symplectic vector spaces. Let denote the total twisted singular cohomology group of with coefficients in the -valued local system and let denote the total twisted singular cohomology space of with coefficients in the -valued local system . The latter can be identified with the total cohomology space of the twisted de Rham complex . Since , the coefficient sequence gives a map , whose image is a graded subgroup of the graded additive group of .
An electromagnetic field is called -integral if its -twisted cohomology class belongs to :
The condition that be -integral is called the twisted Dirac quantization condition defined by the Dirac structure . This should be viewed as a condition constraining semiclassical Abelian gauge field configurations; a mathematical model for such configuration can be given using a certain version of twisted differential cohomology.
4.3. Integral scalar-electromagnetic duality and symmetry groups
An integral scalar-duality structure is a pair , where is a scalar-duality structure and is a Dirac system for . An integral scalar-electromagnetic structure is a pair , where is a scalar-electromagnetic structure and is a Dirac system for the underlying duality structure of the electromagnetic structure .
Let be an integral scalar-electromagnetic structure with underlying scalar-electromagnetic structure , where and . Let be the underlying duality structure and let and be the underlying integral duality structure and integral electromagnetic structure. Let be the underlying scalar-duality structure and be the underlying integral scalar-duality structure.
The integral scalar-electromagnetic duality group defined by the integral scalar-duality structure is the following subgroup of the scalar-electromagnetic duality group :