arXiv:submit/1624497 [hep-th] 27 Jul 2016Heterotic Supergravity with Internal Almost-Kähler
Configurations and Gauge SO(32), or E8 × E8, Instantons
Laurenţiu Bubuianu
TVR Iaşi, 33 Lascaˇr Catargi street, 700107 Iaşi, Romania
email: laurentiu.bubuianu@tvr.ro
Klee Irwin
Quantum Gravity Research; 101 S. Topanga Canyon Blvd # 1159. Topanga, CA 90290, USA
email: klee@quantumgravityresearch.org
Sergiu I. Vacaru
Quantum Gravity Research; 101 S. Topanga Canyon Blvd # 1159. Topanga, CA 90290, USA
and
University "Al. I. Cuza" Iaşi, Project IDEI
18 Piaţa Voevozilor bloc A 16, Sc. A, ap. 43, 700587 Iaşi, Romania
email: sergiu.vacaru@gmail.com
July 28, 2016
Abstract
Heterotic supergravity with (1+3)–dimensional domain wall configurations and (warped) internal, six
dimensional, almost-Kähler manifolds 6X are studied. Considering on ten dimensional spacetime, nonholo-
nomic distributions with conventional double fibrations, 2+2+...=2+2+3+3, and associated SU(3) structures
on internal space, we generalize for real, internal, almost symplectic gravitational structures the construc-
tions with gravitational and gauge instantons of tanh-kink type [1, 2]. They include the first α′ corrections
to the heterotic supergravity action, parameterized in a form to imply nonholonomic deformations of the
Yang-Mills sector and corresponding Bianchi identities. We show how it is possible to construct a variety of
solutions, depending on the type of nonholonomic distributions and deformations of ’prime’ instanton configu-
rations characterized by two real supercharges. This corresponds to N = 1/2 supersymmetric, nonholonomic
manifolds from the four dimensional point of view. Our method provides a unified description of embedding
nonholonomically deformed tanh-kink-type instantons into half-BPS solutions of heterotic supergravity. This
allows us to elaborate new geometric methods of constructing exact solutions of motion equations, with first
order α′ corrections to the heterotic supergravity. Such a formalism is applied for general and/or warped
almost-Kähler configurations, which allows us to generate nontrivial (1+3)-d domain walls. This formalism
is utilized in our associated publication [3] in order to construct and study generic off-diagonal nonholonomic
deformations of the Kerr metric, encoding contributions from heterotic supergravity.
Keywords: heterotic supergravity, almost Kähler geometry, nonholonomic (super) manifolds, nonlinear
connections, domain walls.
MSC2010: 8C15, 8D99, 83E99
PACS2008: 04.20.Jb, 04.50.-h, 04.20.Cv
1
Contents
1 Introduction
2
2 Nonholonomic Manifolds with 2+2+... Splitting
4
2.1 N-adapted frames and coordinates
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
2.2 d–torsions and d–curvatures of d–connections . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.3 d–metrics and generic off–diagonal metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.4 The canonical d–connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.5 The Riemann and Ricci d–tensors of the canonical d–connection . . . . . . . . . . . . . . . . . .
9
3 Nonholonomic Domain–Walls in Heterotic Supergravity
9
3.1 The canonical d–connection and BPS equations . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
3.2 Conventions for 10-d nonholonomic manifolds, d–tensor and d–spinor indices . . . . . . . . . . .
10
3.3 Nonholonomic domain–wall backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
4 Almost-Kähler Internal Configurations in Heterotic Supergravity
14
4.1 Almost symplectic structures induced by effective Lagrange distributions . . . . . . . . . . . . .
14
4.2 Almost symplectic connections for N-anhlonomic internal spaces . . . . . . . . . . . . . . . . . .
17
4.3 N-adapted G2 structures on almost-Kähler internal spaces . . . . . . . . . . . . . . . . . . . . .
18
4.4 Nonholonomic instanton d–connections nearly almost Kähler manifolds . . . . . . . . . . . . . .
20
5 The YM Sector and Nonholonomic Heterotic Supergravity
20
5.1 N–adapted YM and instanton configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
5.2 Static and/or dynamic SU(3) nonholonomic structures on almost Kähler configurations
. . . .
22
5.3 Equations of motion of heterotic supergravity in nonholonomic variables . . . . . . . . . . . . .
23
6 Conclusions
24
1 Introduction
The majority of different vacua in string gravity theories, including four dimensional spacetime domains,
are elaborated with 6-d internal manifolds adapted to certain toroidal compactification or warping of extra
dimensions. With the aim of obtaining interesting and realistic models of lower-dimensional physics, elaborations
of 10-d theories with special Calabi–Yau (and/or more general SU(3) structure) manifolds were used. Such
constructions are related to pseduo-Euclidean 4-d domain configurations and warped almost-Kähler internal
spaces. Recent results and reviews related to superstrings, flux compactifications, D-branes, instantons etc., are
cited respectively [1, 2, 4, 5, 6, 7, 8].
Further generalizations with nontrivial solutions in the 4-d domain, such as reproductions of 4-d black
hole solutions and cosmological scenarios related to modified gravity theories (MGTs) encoding information
from extra dimension internal spaces, are possible if richer geometric structures are involved. Nonholonomic
distributions with splitting on 4-d, 6-d and 10-d manifolds as well as almost-Kähler internal manifolds are
considered when bimetric-connection structures, possible nontrivial mass terms for graviton, locally anisotropic
effects etc. can be reproduced in the framework of heterotic supergravity theory.We cite [9, 10].For MGTs and
their applications, we refer to [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]as well as
references therein.
In a series of works [29, 30, 31, 32, 33, 34, 35, 36, 37, 38], the so-called anholonomic frame deformation
method, AFDM, of constructing exact solutions in commutative and noncommutative (super) gravity and
geometric flow theories has been elaborated. By straightforward analytic computations, it was proven that it is
2
possible to decouple the gravitational field equations and generate general classes of solutions in various theories
of gravity with metric, nonlinear, N-, and linear connection structures. The geometric formalism was based on
spacetime fibrations determined by nonholonomic distributions with splitting of dimensions, 2 (or 3) + 2 + 2 +
.... In explicit form, certain classes of N-elongated frames of reference, considered formal extensions/embeddings
of 4-d spacetimes into higher dimensional spacetimes were introduced and necessary types of adapted linear
connections were defined. Such connections are called distinguished, d-connections, and defined in some form
that preserves the N-connection splitting. In Einstein gravity, a d-connection is considered as an auxiliary
one which is additional to the Levi-Civita, LC, connection. For certain well defined conditions, the canonical
d-connection can be uniquely defined by the metric structure following the conditions of metric compatibility
and the conditions of zero values for "pure" horizontal and vertical components but nonzero, nonholonomically-
induced, mixed vertical-horizontal torsion components. Surprisingly, such a canonical d-connection allows us to
decouple the motion equations into general form. As a result, we can generate various classes of exact solutions
in generalized/modified string and gravity theories. Having constructed a class of generalized solutions in
explicit form (depending on generating and integration functions, generalized effective sources and integration
constants), we then impose some additional constraints at the end, resulting in zero induced torsion fields.
In this way, we can always "extract" solutions for LC-configurations and/or Einstein gravity. It should be
emphasized that it is important to impose the zero-torsion conditions at the end, i.e. after we found a class of
generalized solutions. We can not decouple and solve in general forms the corresponding systems of PDEs if we
use the LC-connection from the very beginning. Here it should be noted that to work with nontrivial torsion
configurations is important in order to find exact solutions in string gravity and gauge gravity models.
Using the AFDM, a series of exact and/or small parameter depending solutions were constructed, which
for small deformations mimic rotoid Kerr - de Sitter like black holes/ellipsoids self–consistently embedded into
generic off-diagonal backgrounds of arbitrary finite dimensions. A number of examples for 5,6 and 8 dimensional
(non) commutative and/or supersymmetric spacetimes are provided, see examples in [30, 34, 35, 37, 38] and
references therein. Such backgrounds can be of solitonic/ vertex / instanton type. In this paper, we develop
and apply these nonholonomic geometric constructions to heterotic string theory. The motion equations are
re-written in certain nonholonomic variables as generalized (effective) Einstein equations for 4-d spacetimes,
encoding nontrivial geometric constructions on extra dimension internal spaces. Here we note that by using
nonholonomic distributions and corresponding classes of solutions for heterotic string gravity, it is possible to
mimic physically important effects in modified gravity. In a series of works [23, 39]), we studied the acceleration
of the universe, certain dark energy and dark matter locally anisotropic interactions and effective renormalization
of quantum gravity models via nonlinear generic off-diagonal interactions on effective Einstein spaces. In our
associated publication [3] we shall elaborate in more detail the AFDM for constructing general classes of exact
solutions with generic off-diagonal metrics in heterotic supergravity and generalize connections depending on all
4-d and 10-d coordinates via corresponding classes of generating and integration functions. The main results of
this and the associated publications are based on the idea that we can generate physically interesting domain wall
configurations (for instance, 4-d deformed black holes) by considering richer geometric structures on the internal
space. We shall prove that we can parametrize and generalize all possible off-diagonal solutions in heterotic
supergravity in terms of such variables with internal geometric objects that are determined by an almost-Kähler
geometry. This is possible for nonholonomic distributions with conventional splitting 2 + 2 + 2 = 3 + 3 (such
constructions are based on former results in [9, 10, 40]).
In this paper, we apply methods used in the geometry of nonhlonomic and almost-Kähler manifolds in order
to study heterotic supergravity derived in the low-energy limit of heterotic string theory [41, 42, 43]. We cite
also section 4.4 in [1] for a summary of previous results and certain similar conventions on warped configurations
and modified gravitational equations.1 The main goal of this work is to formulate a geometric formalism which
is applied in [3] for integrating in generic off-diagonal forms, and for generalized connections, the equations of
1Nevertheless, we shall elaborate a different system of notation with N-connections and auxiliary d–connections which allows us
to define geometric objects on higher order shells of nonholonomically decomposed 10-d spacetimes.
3
additional parametrizations
µs, αs, ... = 1, 2, ...10, on a 10-d nonholonomic manifold with shell coordinates u
µs = (xi, ya, ya1 , ya2 , ya3) on M;
µˇ, αˇ, ... = 1, 2, ...10, equivalently, with general indices and coordinates uµˇ on M;
iˇ, jˇ, ... = 1, 2, 3, for a 3-d pseudo-Euclidean signature (+ +−) and coordinates uiˇ = xiˇ = (xi, y3 = t) = (x1, x2, t);
aˇ, bˇ, ... = 5, 6, 7, 8, 9, 10, with uaˇ = yaˇ on N-anholonomic 6X with Euclidian signature ;
a˜, b˜, ... = 4, 5, 6, 7, 8, 9, 10, with ua˜ = ya˜ on N-anholonomic 7X with Euclidian signature ;
We shall underline such indices in order to emphasize that we work in a coordinate base ∂aˇ: For instance,
Aaˇeaˇ = A
aˇ∂aˇ if we consider decompositions of a d–vector.
Anti–symmetrization is performed with a factorial factor, for instance, A[µˇBαˇ] :=
1
2 (AµˇBαˇ − BαˇAµˇ). A
p–form ω is expressed ω = 1
p!ωµˇ1...µˇpe
µˇ1 ∧ ...∧ eµˇp , see (4). The Clifford action of such a form on a spinor ǫ can
be defined in N–adapted form as
ω · ǫ :=
1
p!
ωµˇ1...µˇpγ
µˇ1...µˇpǫ,
for γµˇ1...µˇp := γ[µˇ1 ...γµˇp ] if the Clifford d–algebra is introduced following convention
{γµˇ, γνˇ} := γµˇγνˇ + γνˇγµˇ = 2gµˇνˇ
for 10-d gamma matrices γµˇ with gµˇνˇ = {gµsνs} admitting a shell decomposition (10). A theory of N–adapted
spinors and Dirac operators is elaborated upon [44, 47, 48, 49], see also references therein. We omit such
considerations in this work.
It is proven in [2] that at first order in α′, the BPS equations are solved by A∇ˇ = c∇, Hˇ = 0, φˇ = const,
where
c∇ is the LC–connection on cˇ( 6X). A series of less trivial solutions with Hˇ
6= 0 have been studied
in [2, 56, 57, 58, 59, 1] under assumptions that the gauge field is chosen to be an instanton and within the
framework of dynamic SU(3) structures. In this work, we shall extend those results in string theory by proving
that the equations of motion of heterotic string supergravity can be decoupled and solved in very general off–
diagonal forms with dependence, in principle, of all 10-d spacetime cooridinates (using the AFDM for higher
dimensions [37, 38, 34]). In order to preserve certain relations to former "holonomic" solutons, we shall work
with N–anholonomic manifolds M and 6X determined by d–metric (10). We shall prove that for any solution
of certain generalized/modified 10-d Einstein equations for
sD̂|T̂ =0 →
s∇ and additional extensions with
gauge and scalar fields, the submanifold
6X can be endowed with almost-Kähler variables [9, 54, 40]. In
result, the curvature R˜ of d–connection D˜ can be determined for arbitrary solutions of the equations of motion
of heterotic supergravity with generic off–diagonal interactions, nonholonomically deformed connections and
various parametrizations of effective sources.
3.3 Nonholonomic domain–wall backgrounds
In Einstein–Cartan and gauge field theories, torsion fields have certain sources subjected to algebraic equa-
tions. If a background with vanishing fermionic vacuum expectations does not have prescribed nonholonomic
distributions, the supersymmetry transformations of the corresponding fermionic fields are zero. Such condi-
tions for the holonomic backgrounds are known as BPS equations. Up to and including terms of order α′ for
N–anholonomic backgrouns, the BPS equations are formulated for a Majorana–Weyl spinor ǫ,
(D̂− Ĥ) · ǫ = 0,
(20)
(d̂φ̂−
1
2
Ĥ) · ǫ = 0,
F̂ · ǫ = 0.
11
We note that in this work hatted boldface objects denote 10–d geometric/physical objects on N–anholonomic
manifolds enabled with d–connection structure D̂.
In heterotic string gravity, (see details and references in [1]) backgrounds given by 4-d domain walls with
6 internal directions stating a compact manifolds
6X with SU(3) structure for a 10-d metric ansatz with
quadratic element
ds2[˚g] =
g˚αˇβˇ(y
4, yaˇ)duαˇduβˇ = g˚αsβs(y
4, yaˇ)duαsduβs =
(21)
= e2A(y
4,yaˇ)
[(
dx1
)2
+
(
dx2
)2
−
(
dy3
)2
+ e2B(y
aˇ)
(
dy4
)2
+ g˚bˇcˇ(y
4, yaˇ)dybˇdycˇ
]
,
are considered, where y3 = t and y4 is chosen to be transverse to the domain wall enabled with coordi-
nates (xi, t). We can consider orthonormal frames eaˇ = eaˇ(y4, ycˇ)eaˇ
′
for a prescribed N–connection structure
N˚ = {N˚as
is
(y4, ycs) → N˚ aˇ
is
(y4, ycˇ)} defined for a local system of coordinates in the internal 6-d manifold 6X
embedded via warped coordinate y4 into higher dimensional ones and transform it into N-anholonomic manifold
6X ⊂ 7X ⊂M, endowed with a d–metric structure of type (10),
ds2[˚g] = e2A˚(y
4,yaˇ)[
(
dx1
)2
+
(
dx2
)2
−
(˚
e3
)2
+ e2B˚(y
aˇ)
(˚
e4
)2
+
(22)
g˚a1(y
4, ya1) (˚ea1)2 + g˚a2(y
4, ya2) (˚ea2)2 + g˚a3(y
4, ya3) (˚ea3)2],
where
a0 = 3 : e˚
3 = dt+ n˚i0dx
i0 , for N˚3
i0
= n˚i0(x
k, y4), for k, i0 = 1, 2;
(23)
a0 = 4 : e˚
4 = dy4 + w˚i0dx
i0 , for N˚4
i0
= w˚i0(x
k, y4);
a1 = 5 : e˚
5 = dy5 + n˚i1dx
i1 , for N˚5
i1
= n˚i1(x
k, y4, y6), for i1 = 1, 2, 3, 4;
a1 = 6 : e˚
6 = dy6 + w˚i1dx
i1 , for N˚6
i1
= w˚i1(x
k, y4, y6);
a2 = 7 : e˚
7 = dy7 + n˚i2dx
i2 , for N˚7
i2
= n˚i2(x
k, y4, y6, y8), for i2 = 1, 2, 3, 4, 5, 6;
a2 = 8 : e˚
8 = dy8 + w˚i2dx
i2 , for N˚8
i2
= w˚i2(x
k, y4, y5, y6, y8);
a3 = 9 : e˚
9 = dy9 + n˚i3dx
i3 , for N˚9
i3
= n˚i3(x
k, y4, y5, y6, y7, y8, y10), for i3 = 1, 2, 3, 4, 5, 6, 7, 8;
a3 = 10 :
e˚10 = dy10 + w˚i3dx
i3 , for N˚10
i3
= w˚i3(x
k, y4, y5, y6, y7, y8, y10).
In this paper, we shall study stationary configurations with Killing symmetry on ∂t when the metric/d–metric
ansatz do not depend on coordinate y3 = t. We call an ansatz g˚ (21) as a prime off–diagonal metric and an
ansatz g˚ (22) (prime metrics will be labeled by a circle ◦).7
The overall goal of this and associated [3] articles is to study nonholonomic deformations of a primary metric
g˚ into a target metric sg,
g˚ = [˚gαs , N˚
as
is
]→ sg = [gαs = ηαs g˚αs , N
as
is
= ηas
is
N˚as
is
],
(24)
→
s
εg = [gαs = (1 + εχαs )˚gαs , N
as
is
= (1 + εχas
is
)N˚as
is
],
for ηαs ≃ 1 + εχαs , η
as
is
, ηas
is
≃ 1 + εχas
is
, where 0 ≤ ε≪ 1.
In these formulas, we do not consider summations on repeating indices. Any target metric sg will be subjected
to these conditions to define new classes of solutions for certain systems of nonlinear PDEs in heterotic string
7It should be emphasized that indices a1 = (5, 6), a2 = (7, 8), a3 = (9, 10) are shell adapted but indices aˇ, cˇ, aˇ
′ may take, in
general, values 6, 7, ...10 with shell mixing of indices. In order to apply the AFDM, we shall always consider certain frame/coordinate
transforms with N–adapted shell redefinitions of interior indices and coordinates. Indices with "inverse" hats are convenient for
parametrization of almost-Kähler structures but shell indices are important for constructing exact off-diagonal solutions in 10-d
gravity.
12
gravity (see next section) or in a MGT. It is always possible to model the internal 6-d space as an almost-Kähler
manifold. For certain subclasses of solutions with limε→0 sg→ g˚, the η–polarization functions (ηαs , η
as
is
) → 1.
In general, such limits with a small parameter ε may not exist, or can behave singularly.
In our next constructions, we show how G structures [58, 59] can be adapted to N–connections. The Killing
spinor ǫ is adapted to above prime and target d-metric ansatz as
ǫ(xi, t, y4, yaˇ) = ρ(xi, t)⊗ η(y4, yaˇ)⊗ θˇ,
where the domain wall spinor ρ has two real components corresponding to the two real supercharges which the
holonomic background R2,1 preserves; η is a covariantly constant Majorana spinor on 7X = (y4, 6X); and
θˇ
is an eigenvector of the respective Pauli matrix.
It is possible to preserve, in the tangent bundle TM, the (1 + 2)–dimensional Lorentz invariance together
with N–connection splitting if restrictions on φ̂ and Ĥαsβsµs :
eiˇφ̂ = 0, Ĥ
iˇa˜b˜
= 0, Ĥ
iˇjˇb˜
= 0.
(25)
are considered. For a flat 3-d Minkowski spacetime, the only non-zero components of the NS 3-form flux are
Ĥ4aˇbˇ, Ĥaˇbˇcˇ and
Ĥiˇjˇkˇ = ℓ
√
|giˇjˇ |ǫˇijˇkˇ
for ℓ = const and the totally antisymmetric tensor on R2,1 with normalization ǫ123 = 1.
Let us define G2 d–structure adapted to the N–connection splitting in
7X = R× 6X enabled with an
arbitrary d–metric (of type included in (21) and (22)), respectively,
ds2[ 7g˚] = e2B(y
aˇ)
(
dy4
)2
+ g˚bˇcˇ(y
4, yaˇ)dybˇdycˇ and
(26)
ds2[7g˚] = e2B˚(y
aˇ)
(˚
e4
)2
+ g˚a1(y
4, ya1) (˚ea1)2 + g˚a2(y
4, ya2) (˚ea2)2 + g˚a3(y
4, ya2 , ya3) (˚ea3)2 .
Such holonomic structures were studied in [58, 59] for certain special parametrizations of functions B, B˚ and
g˚as . For nonholonomic deformations (24), we generated d–metrics on
7X with g˚4 = e
2B˚(yaˇ) → g4 = e
2B(yaˇ)
(via coordinate transforms, we can consider parametrizations with B = 0 but we keep a nontrivial value of
B in order to compare our results with those for holonomic Kähler configurations outlined in [1], where ∆ is
considered for B),
ds2[7g] = [e2B(y
aˇ)
(
e4
)2
+ ga1(y
4, ya1) (ea1)2 + ga2(y
4, ya2) (ea2)2 + ga3(y
4, yaˇ) (ea3)2].
In this formula and (26), the internal spaces, the indices and coordinates can be written in any form we need
for the definition of almost-Kähler or diadic shell structures as we explained above in footnote 7. The d–metric
7g defines the Hodge operator ∗7. The G2 d–structure is given by a 3-form ̟ ∈ ∧
3( 7X) and its 7-d Hodge
dual W := 7 ∗̟ ∈ ∧4( 7X) (in a similar form, d-structures can be introduced for d–spinors and 7-d gamma
matrices). In N–adapted differential form with absolute differential operator 7d on 7X, the BPS equations
(20) imply
7d̟ = 2 7dφ̂ ∧̟ − 7 ∗ Ĥ− ℓW, 7dW = 2 7dφ̂ ∧W,
7 ∗ 7dφ̂ = −
1
2
Ĥ ∧̟, 7 ∗ ℓ = 2Ĥ ∧W.
(27)
Such relations can be written in N-adapted form with respect to frames (4) and for the canonical d–connection
sD̂.
13
4 Almost-Kähler Internal Configurations in Heterotic Supergravity
We analyze important geometric structures which can be defined for the decomposition 7X = R× 6X. For
holonomic distributions, it is always possible to rewrite the equation (27) in terms of an SU(3) structure defined
on 6X and the domain wall direction. Such constructions are related to the complex structure and Kähler
geometry [58, 59, 1]. In order to elaborate a heterotic theory with generic off–diagonal metrics g = ( 4g, 6g) (10),
we need a richer, real geometric structure for internal space 6X. Up to certain classes of frame transforms, any
6g of Euclidean signature can be uniquely related to an almost-Kähler geometry. The approach was considered
for elaborating certain methods of deformation of the Einstein and Finsler modified gravity theories and for
formulating models of almost-Kähler geometric flows and Lie algebroid structures, see [9, 10, 54, 40]. The goal
of this subsection is to apply those methods in the study of heterotic string gravity models.
4.1 Almost symplectic structures induced by effective Lagrange distributions
In order to enable the internal space with a complex Kähler structure, one considers a decomposition of
the Majorana spinor η into two 6-d spinors of definite chirality, η = 1√
2
(η+ + η−). We can specifiy any SU(3)
structure on 6X via a couple of geometric objects (J, θ) a real 2-form J and a complex 3-form θ = θ+ + iθ−,
where i2 = −1. Such values can be defined for any fixed values y4 and by using the chiral spinors η±. The
relation between holonomic G2 structure (̟,W) and (J, θ) is studied in [1, 58, 59, 60].
The real nonholonomic almost-Kähler geometry is also determined by a couple (J˜ , θ˜) which in our work is
constructed to be uniquely determined by a N–connection structure N˜ and a 6-d metric 6g→θ˜ defined by a
Lagrange type distribution L˜(y4, yaˇ). In this case, we also have a nontrivial nonolonomically induced torsion
with conventional splitting of internal coordinates and indices in the form yaˇ = (yı´, ya`), where ı´, j´, ... = 5, 6, 7
and (for conventional "vertical", v, indices) a`, b`, ... = 8, 9, 10.
We can re-parametrize a general Riemannian metric 6g ⊂ g on 6X, when possible dependencies on 4-d
spacetime coordinates 0u = (xi, ya) are considered as parameters [for simplicity, we shall omit writting of 4-d
coordinates if it will not result in ambiguities],
ds2[ 6g] = gbˇcˇ(x
i, ya, yaˇ)dybˇdycˇ and/or
ds2[6g˚] = ga1(
0u, ya1) (ea1)2 + ga2(
1u, ya1) (ea2)2 + ga3(
2u, ya2) (ea3)2 .
in the form:
6g = gı´j´(
0u, yı´, ya`)dyı´ ⊗ dyj´ + g
a`b`
( 0u, yı´, ya`)ea` ⊗ eb`,
(28)
ea` = dya` −N a`
ı´ (u)dy
ı´.
In formluas (28), the vierbein coefficients eaˇaˇ of the dual basis e
aˇ = (eı´, ea`) = eaˇaˇ(u)dy
aˇ, are parametrized to
define a formal 3 + 3 splitting with N–connection structure 6N = {N a`
ı´ }.
It is possible to prescribe any generating function L(u) = L(xi, ya, yaˇ) on 6X with nondegenerate Hessian
det | g˜
a`b`
| 6= 0 for
g˜
a`b`
:=
1
2
∂2L
∂ya`∂yb`
.
(29)
We define a canonical N–connection structure
N˜ a`
ı´ =
∂Ga`
∂y7+ı´
,
(30)
Ga` =
1
4
g˜a` 7+ı´(
∂2L
∂y7+ı´∂yk´
y7+k´ −
∂L
∂yı´
).
14
In these formulas,
g˜a`b` is inverse to
g˜
a`b`
and respective contractions of h– and v–indices,
ı´, j´, ... = 5, 6, 7 and
a`, b`, ... = 8, 9, 10, are performed following this rule: For example, we take an up v–index a` = 3+ ı´ and contract
it with a low index ı´ = 1, 2, 3. Using (29) and (30), we construct an internal 6-d d–metric
6g˜ =
g˜ı´j´dy
ı´ ⊗ dyj´ +
g˜
a`b`
e˜a` ⊗ e˜b`,
(31)
e˜a` = dya` + N˜ a`
ı´ dy
ı´, { g˜
a`b`
} = {g˜7+ı´ 7+j´}.
It should be emphasized that any d-metric 6g (28) can be parametrized by coefficients 6gaˇbˇ = [gı´j´, ga`b`, N
a`
ı´ ]
computed with respect to a N-adapted basis eaˇ = (eı´ = dyı´, ea`) which is related to the metric
6g˜aˇbˇ =
[g˜ı´j´, g˜a`b`, N˜
a`
ı´ ] (31) with coefficients defined with respect to a N–adapted dual basis
e˜aˇ = (eı´ = dyı´, e˜a`) if the
conditions
6gaˇ′ bˇ′e
aˇ′
aˇe
bˇ′
bˇ
=
6g˜aˇbˇ related to corresponding frame transfoms are satisfied. Fixing any values
6gaˇ′ bˇ′ and
6g˜aˇbˇ, we have to solve a system of quadratic algebraic equations with unknown variables e
aˇ′
aˇ. A
nonholonomic 2+2+2 = 3+3 splitting of 6X with 6gaˇbˇ = [gı´j´ , ga`b`, N
a`
ı´ ] is convenient for constructing generic
off-diagonal solutions but similar N–connection splitting with equivalent 6g˜aˇbˇ = [g˜ı´j´ , g˜a`b`, N˜
a`
ı´ ] will allow us to
define real solutions for effective EYMH systems.
A set of coefficients N˜ = {N˜ a`
ı´ } defines an N–connection splitting as a Whitney sum,
T 6X = h 6X⊕ v 6X
(32)
into conventional internal horizontal (h) and vertical (v) subspaces. In local form, this can be written as
N˜ = N˜ a`
ı´ (u)dy
ı´ ⊗
∂
∂ya`
,
(33)
with 6X = 3+3X. As a result, there are N–adapted frame (vielbein) structures,
e˜aˇ =
(
e˜ı´ =
∂
∂yı´
− N˜ a`
ı´
∂
∂ya`
, ea` =
∂
∂ya`
)
,
(34)
with dual frame (coframe) structures
e˜aˇ, see (31). These vielbein structures define the nonholonomy relations
[e˜aˇ, e˜bˇ] = e˜aˇe˜bˇ − e˜bˇe˜aˇ = w˜
cˇ
aˇbˇ
e˜cˇ
(35)
with (antisymmetric) anholonomy coefficients w˜b`
ı´a` = ∂a`N˜
b`
ı´ and w˜
a`
j´ ı´
= Ω˜a`
ı´j´
, where Ω˜a`
ı´j´
= e˜j´
(
N˜ a`
ı´
)
− e˜ı´
(
N˜ a`
j´
)
are
the coefficients of N–connection curvature (defined as the Neijenhuis tensor).
Using the canonical N–connection splitting, we introduce a linear operator J˜ acting on vectors on
6X
following formulas J˜(eı´) = −e7+ı´ and J˜(e7+ı´) = e˜ı´, where J˜◦J˜= −I, for I being the unity matrix, and construct
a tensor field,
J˜ =
J˜aˇ
bˇ
eaˇ ⊗ e
bˇ = J˜
aˇ
bˇ
∂
∂uaˇ
⊗ dubˇ
(36)
=
J˜aˇ
′
β′ e˜aˇ′ ⊗ e˜
β′ = −e7+ı´ ⊗ e
ı´ + e˜ı´ ⊗ e˜
7+ı´
= −
∂
∂yı´
⊗ dxı´ +
(
∂
∂yı´
− N˜ a`
ı´
∂
∂ya`
)
⊗
(
dy7+ı´ + N˜7+ı´
k´
dyk´
)
.
The corresponding d-tensor field defined globally by an almost complex structure on 6X is completely deter-
mined by a prescribed generating function L˜(y4, yaˇ) ⊂ L(uαs). In this subsection, we consider only structures
J = J˜ induced by a N˜7+ı´
k´
. In general, we can define an almost complex structure J for an arbitrary N–connection
15
N, stating a nonholonomic 3 + 3 splitting by using N–adapted bases (34) which can be included (if necessary)
into respective nonholonomic frames of the 10-d spacetime, see (4).
The Neijenhuis tensor field for J˜ (equivalently, the curvature of N–connection N˜) is
JΩ˜(X,Y) := −[X,Y] + [J˜X, J˜Y]− J˜[J˜X,Y]− J˜[X, J˜Y],
(37)
for any d–vectors X,Y ∈T 6X. With respect to N–adapted bases (34), a subset of the coefficients of the
Neijenhuis tensor defines the N–connection curvature,
Ω˜a`
ı´k´
=
∂N˜ a`
ı´
∂yk´
−
∂N˜ a`
k´
∂yı´
+ N˜ b`
ı´
∂N˜ a`
k´
∂yb`
− N˜ b`
k´
∂N˜ a`
ı´
∂yb`
.
(38)
The nonholonomic structure is integrable if Ω˜a`
ı´k´
= 0. We get a complex structure if and only if both the h– and
v–distributions are integrable, i.e. if and only if
Ω˜a`
ı´k´
= 0 and
∂N˜ a`
ı´
∂yk´
−
∂N˜ a`
k´
∂yı´
= 0.
An almost symplectic structure on a manifold is introduced by a nondegenerate 2–form
θ =
1
2
θaˇbˇ(u)e
aˇ ∧ ebˇ =
1
2
θ
ı´k´
(u)eı´ ∧ ek´ +
1
2
θ
a`b`
(u)ea` ∧ eb`.
An almost Hermitian model of an internal 6-d Riemannian space equipped with a N–connection structure
N is defined by a triple H3+3 = ( 6X, θ,J), where θ(X,Y) + g (JX,Y) for any g (28). A space H3+3 is
almost-Kähler, denoted K3+3, if and only if dθ = 0.
Using g = g˜ (31) and structures N˜ (30) and J˜, we define
θ˜(X,Y) : = g˜
(
J˜X,Y
)
,
for any d–vectors X,Y ∈T 6X. In local N–adapted form, we have
θ˜ =
1
2
θ˜aˇbˇ(u)e
aˇ ∧ ebˇ =
1
2
θ˜aˇbˇ(u)du
aˇ ∧ dubˇ
(39)
=
g˜
ı´k´
(xi, ya, yaˇ)e7+ı´ ∧ dyk´ =
g˜ı´j´(x
i, ya, yaˇ)(dy7+ı´ + N˜7+ı´
k´
dyk´) ∧ dyj´ .
Considering the form
ω˜ =
1
2
∂L˜
∂y7+ı´
dyı´,
we prove by a straightforward computation that
θ˜ = d ω˜, i.e. d θ˜ = dd ω˜ = 0. As a result, any canonical
effective Lagrange structure
(
g = g˜, N˜,J˜
)
induces an almost-Kähler geometry. The 2–form (39) can be written
θ =
θ˜ =
1
2
θ˜ı´j´(u)e
ı´ ∧ ej´ +
1
2
θ˜
a`b`
(u)e˜a` ∧ e˜b`
(40)
= g
ı´j´
(u)
[
dyı´ + N˜7+ı´
k´
(u)dyk´
]
∧ dyj´,
where the nontrivial coefficients
θ˜
a`b`
= θ˜7+ı´ 7+j´ are equal to the N-adapted coefficients θ˜ı´
j´ respectively.
16
4.2 Almost symplectic connections for N-anhlonomic internal spaces
Taking a general 2–form θ constructed for any metric g and almost complex J structures on
6X, one
obtains dθ
6= 0. Nevertheless, we can always define a 3+3 splitting induced by an effective Lagrange generating
function when d θ˜ = 0. Considering frame transforms, θaˇ′ bˇ′e
aˇ′
aˇe
bˇ′
bˇ
= θ˜aˇbˇ, we can write d θ = 0 for any set of
2–form coefficients related via frame transforms to a canonical symplectic structure.
There is a unique normal d–connection
D˜ =
{
hD˜ = (D˜
k´
,v D˜
k´
= D˜
k´
); vD˜ = (D˜c`,
vD˜c` = D˜c`)
}
(41)
= {Γ˜aˇ
bˇcˇ
= (L˜ı´
j´k´
, vL˜7+ı´
7+j´ 7+k´
= L˜ı´
j´k´
; C˜ ı´
j´c`
= vC˜7+ı´
7+j´
c`
, vC˜ a`
b`c`
= C˜ a`
b`c`
)},
which is metric compatible, D˜
k´
g˜ı´j´ = 0 and D˜b` g˜ı´j´ = 0, and completely defined by a couple of h– and v–
components D˜aˇ = (D˜k´, D˜b`). The corresponding N–adapted coefficients Γ˜
aˇ
bˇγ
= (L˜ı´
j´k´
, vC˜ a`
b`c`
) are given by
L˜ı´
j´k´
=
1
2
g˜ı´h´
(
e˜
k´
g˜
j´h´
+ e˜j´ g˜h´k´ − e˜h´g˜j´k´
)
, C˜ ı´
j´k´
=
1
2
g˜ı´h´
(
∂g˜
j´h´
∂yk´
+
∂g˜
h´k´
∂yj´
−
∂g˜
j´k´
∂yh´
)
.
(42)
To elaborate a differential form calculus on 6X which is adapted to the canonical N–connection N˜, we
introduce the normal d–connection 1–form
��˜ı´
j´
= L˜ı´
j´k´
ek´ + C˜ ı´
j´k´
ek´.
(43)
Using this linear connection, we prove that the Cartan structure equations are satisfied,
dek´ − ej´ ∧ Γ˜ı´
j´
= −T˜ k´, dek´ − ej´ ∧ Γ˜ı´
j´
= − vT˜ ı´,
(44)
and
dΓ˜ı´
j´
− Γ˜h´
j´
∧ Γ˜ı´
h´
= −R˜ı´
j´
.
(45)
The h– and v–components of the torsion 2–form T˜ aˇ =
(
T˜ k´, vT˜ ı´
)
= T˜aˇ
cˇbˇ
e˜cˇ ∧ e˜bˇ from (44) are computed
T˜ ı´ = C˜ ı´
j´k´
ej´ ∧ ek´, vT˜ ı´ =
1
2
Ω˜ı´
k´j´
ek´ ∧ ej´ + (
∂ N˜ ı´
k´
∂yj´
− L˜ı´
j´k´
)ek´ ∧ ej´ .
(46)
In these formulas, Ω˜ı´
k´j´
are coefficients of the curvature of N˜ ı´
k´
defined by formulas similar to (38). The formulas
(46) parametrize the h– and v–components of torsion T˜aˇ
cˇbˇ
in the form
T˜ ı´
j´k´
= 0, T˜ ı´
j´ a`
= C˜ ı´
j´a`
, T˜ a`
k´j´
= Ω˜a`
k´j´
, T˜ a`
ı´b`
= e
b`
(
N˜ a`
ı´
)
− L˜a`
b`ı´
, T˜ a`
b`c`
= 0.
(47)
We emphasize that T˜ vanishes on h- and v–subspaces, i.e.
T˜ ı´
j´k´
= 0 and T˜ a`
b`c`
= 0, but other nontrivial h–v–
components are induced by the nonholonomic structure determined canonically by g = g˜ (31) and L˜.
An explicit calculus of the curvature 2–form from (45) results in
R˜ı´
j´
= R˜ı´
j´cˇbˇ
ecˇ ∧ ebˇ =
1
2
R˜ı´
j´k´h´
ek´ ∧ eh´ + P˜ ı´
j´k´a`
ek´ ∧ e˜a` +
1
2
S˜ ı´
j´ c`d`
e˜c` ∧ e˜d`.
(48)
The corresponding nontrivial N–adapted coefficients of curvature R˜aˇ
cˇbˇeˇ
of D˜ are
R˜ı´
j´k´h´
=
e˜
k´
L˜ı´
h´j´
− e˜j´L˜
ı´
h´k´
+ L˜m´
h´j´
L˜ı´
m´k´
− L˜m´
h´k´
L˜ı´
m´j´
− C˜ ı´
h´a`
Ω˜a`
k´j´
P˜ ı´
j´k´a`
= ea`L˜
ı´
j´k´
− D˜
k´
C˜ ı´
j´a`
, S˜a`
b`c`d`
= e
d`
C˜ a`
b`c`
− ec`C˜
a`
b`d`
+ C˜ e`
b`c`
C˜ a`
e`d`
− C˜ e`
b`d`
C˜ a`e`c`.
17
By definition, the Ricci d–tensor R˜bˇcˇ = R˜
aˇ
bˇcˇaˇ
is computed
R˜bˇcˇ=
(
R˜ı´j´ , R˜ı´a`, R˜a`ı´, R˜a`b`
)
.
(49)
The scalar curvature R˜ of D˜ is given by two h- and v–terms,
R˜ = g˜bˇcˇR˜bˇcˇ = g˜
ı´j´R˜ı´j´ + g˜
a`b`R˜
a`b`
.
(50)
The normal d–connection D˜ (41) defines a canonical almost symplectic d–connection, D˜ ≡ θD˜, which is
N–adapted to the effective Lagrange and, related to almost symplectic structures, i.e. it preserves the splitting
under parallelism (32),
D˜X g˜= θD˜X θ˜=0,
for any X ∈T 6X and its torsion is constrained to satisfy the conditions T˜ ı´
j´k´
= 0 and T˜ a`
b`c`
= 0.
We conclude that having chosen a regular generating function L(x, y) on a Riemannian internal space V, we
can always model this spacetime equivalently as an-almost Kähler manifold. Using corresponding nonholonomic
frame transforms and deformation of connections, we can work with equivalent geometric data on the internal
space 6X, for convenience
( 6g, 6N, 6D̂) ⇐⇒ ( g˜, N˜,D˜, L˜) ⇐⇒ (
θ˜, J˜, θD��).
The first N-adapted model is convenient for constructing exact solutions in 6-d and 10-d gravity models (this
will be addressed in the associated paper [3], see also examples in [37]). The second nonholonomic model with
"tilde" geometric objects (with so–called Lagrange-Finsler variables, in our case, on a 6–d Riemannian space)
is an example of an internal space with nontrivial nonholonomic 3 + 3 splitting by a canonical N–connection
structure determined by an effective Lagrange function L˜. The (θ˜, J˜, θD˜) defines an almost-Kähler geometric
model on 6X with nontrivial nonholonomically induced d–torsion structure T˜ aˇ. This way, we can mimic a
complex like differential geometry by real values and elaborate on various applications to quantum gravity,
string/brane and geometric flow theories [9, 10, 40]. Introducing the complex imaginary unit i2 = −1, with
J˜ ≈ i... and integrable nonholonomic distributions, we can redefine the geometric constructions for complex
manifolds. Using nonholonomic real 3 + 3 distributions, we can elaborate gravitational and gauge like models
of internal spaces, for instance, with SO(3), or SU(3) symmetries and their tensor products. Two different
approaches can be unified in a geometric language with double nonholonomic fibrations 2 + 2+ 2 = 3+ 3. Any
d–metric with internal 2 + 2 + 2 nonholonomic splitting can be redefined by nonholonomic frame transforms
into an almost symplectic structure with 3 + 3 decomposition. Considering actions of SO(3), or SU(3) on
corresponding tangent spaces, we can reproduce all results with Kähler internal spaces related 4-d, 6-d and 10-d
solutions for holonomic configurations obtained in Refs. [1, 2].
4.3 N-adapted G2 structures on almost-Kähler internal spaces
For any 3–form Θ = Θaˇbˇcˇe˜
aˇ ∧ e˜bˇ ∧ e˜cˇ on 6X endowed with a canonical almost complex structure J˜ (36)
Θ = +Θ+ J˜ −Θ.
(51)
We can fix these conditions such that for J˜→ i, i2 = −1,Θ defines an SU(3) structure defined on 6X and the
tangent space to the domain wall with Θ = +Θ + i −Θ. Defining the gamma matrices γaˇ on 6X from the
relation γ˜aˇγ˜bˇ + γ˜bˇγ˜aˇ = 2
6g˜aˇbˇ, see (31), we can relate the geometric objects in the almost-Kähler model of the
internal space to the models with SU(3) structure on a typical fiber in the tangent bundle T 6X. For integrable
SU(3) structures and Kähler internal spaces, one works with the structure forms (J,Θ), when
Θaˇbˇcˇ = (η+)
†γ
aˇbˇcˇ
η− and Jaˇbˇ = ∓(η±)
†γ
aˇbˇ
η±
18
are considered for a Kähler metric 2 6g
aˇbˇ
= γ
aˇ
γ
bˇ
+ γ
bˇ
γ
aˇ
with a 6-d Hodge star operator ∗. These forms obey
the conditions
J ∧Θ = 0,
i
8
Θ ∧Θ =
1
3!
J ∧ J ∧ J = ∗1, ∗J =
1
2
J ∧ J, ∗Θ± = ±Θ∓,
where Θ means complex conjugation of Θ.
Working with J˜ instead of J, we can define a similar 3-form Θ˜ for an almost-Kähler model (θ˜, J˜, θD˜) and
construct the Hodge star operator ∗˜ corresponding to 6g˜. The relation between 6-d ∗˜ and 7-d 7∗˜ Hodge stars
for an ansatz of type (26) is
7∗˜( 6pω) = e
B(yaˇ) ∗˜( 6pω) ∧ e
4 and 7∗˜(e4 ∧ 6pω) = e
−B(yaˇ) ∗˜( 6pω),
where 6pω is a p–form with legs only in the directions on
6X. The two exterior derivatives 7d and d˜ are related
via
7d( 6pω) = d˜(
6
pω) + dy
4 ∧
∂
∂y4
( 6pω).
Applying these formulas, we decompose the 10-d 3-form Ĥ into three N-adapted parts,
Ĥ = ℓvol[ 3g]+ 6Ĥ+ dy4 ∧ Ĥ4,
where
vol[ 3g] =
1
3
ǫˇijˇkˇ
√
|3giˇjˇ |e
iˇ ∧ ejˇ ∧ ekˇ, 6Ĥ =
1
3!
Ĥaˇbˇcˇe˜
aˇ ∧ e˜bˇ ∧ e˜cˇ, Ĥ4 =
1
2!
Ĥ4bˇcˇe˜
bˇ ∧ e˜cˇ.
The operators (J˜,Θ˜) allow us to generalize, in almost-Kähler form, the original constructions for Kähler
internal spaces provided in [2, 1, 58, 59, 60] for the G2 structure. In our approach, such an N-adapted con-
figuration is adapted by the data for (27), (̟,W) which can be related to the SU(3) almost-Kähler structure
(J˜,Θ˜) by expressions
̟ = eB(y
aˇ)e4 ∧ J˜+Θ˜−,
W = eB(y
aˇ)e4 ∧ Θ˜+ +
1
2
J˜ ∧ J˜.
For any structure group SU(3) and its Lie algebra su(6) = su(3)⊕ su(3)⊥, there is a canonical torsion 0Taˇ
′
bˇ′ cˇ′
∈
∧1⊗ su(3)⊥, where primed indices refer to an orthonormal basis which can be related to any coordinate and/or
N–adapted basis. For instance we write 0T˜aˇ
′
bˇ′cˇ′
if such a basis is for an almost-Kähler structure.
0Taˇ
′
bˇ′ cˇ′
= (3⊕ 3)⊗
(1⊕ 3⊕ 3)
= (1⊕ 1)⊕
(8⊕ 8)⊕
(6⊕ 6)⊕ 2(3⊕ 3)
T1
T2
T3
T4,T5
This classification can be N–adapted if we use derivatives of the structure forms
d˜J˜ = −
3
2
Im(T1Θ˜) + T4 ∧ J˜+T3,
d˜Θ˜ = T1J˜ ∧ J˜+ T2 ∧ J˜+ T 5 ∧ Θ˜,
where Im(T1Θ˜) should be treated as the "vertical" part for almost-Kähler structures (when all values are real)
and as the imaginary part for complex and Kähler structures (J˜, Θ˜)→ (J,Θ).
19
4.4 Nonholonomic instanton d–connections nearly almost Kähler manifolds
The instanton type connections constructed in [2, 1] can be modelled for almost-Kähler internal spaces if
we work in N–adapted frames for respective nonholonomically deformed connections. For such configurations,
we set
T2 = T3 = T4 = T5 = 0 and T1 =
+T1 + J˜
−T1,
where the last splitting is defined similarly to (51). All further calculations with (J,Θ) in [2, 1, 58, 59, 60] are
similar to those for (J˜,Θ˜) if we work in N–adapted frames on 6X and corresponding model (θ˜, J˜, θD˜). Hereafter,
we shall omit detailed proofs for almost-Kähler structures and send readers to analogous constructions in the
aforementioned references.
Considering an ansatz (26) with possible embedding into a 10-d one of type (22), with A = B = 0 for
simpicity, we can construct solutions on 6X of the first two nonholonomic BPS equations in (20) and (27).
Such almost-Kähler configurations are determined by the system
Ĥ = ℓvol[ 3g]−
1
2
∂4φ+
(
3
2
−T1 +
7
8
ℓ
)
+ dy4 ∧ (2 −T1 + ℓ)J˜ for φ̂ = φ(y4)
(52)
and (J˜,Θ˜) subjected to respective flow and structure equations:
∂4J˜ = (
+T1 + ∂4φ)J˜,
(53)
∂4
−Θ˜ = −(3 −T1 +
15
8
ℓ) +Θ˜ +
3
2
( +T1 + ∂4φ)
−Θ˜,
∂4
+Θ˜ =
3
2
( +T1 + ∂4φ)
+Θ˜ + α˜(y4) −Θ˜, for arbitrary function α˜(y4);
and d˜J˜ = −
3
2
−T1 +Θ˜ +
3
2
+T1
−Θ˜,
(54)
d˜Θ˜ = T1J˜ ∧ J˜.
At the zeroth order in α′ with the Bianchi identity d̂Ĥ = 0, see (19), we can choose F̂ = 0 in order to solve
the third equation in (20). The time–like component of the equations of motion can be solved if ℓ = 0 as in the
pure Kähler case [1]. We obtain a special case of solutions (see [58] for original Kähler ones) when
1.
φ = const.,
+T1 is a free function ,
−T1 = 0,
α˜(y4) is a free function ;
2. φ = 23 log(a0y
4 + b0),
+T1 = 0,
−T1 = 0,
α˜ = 0,
(55)
for integration constants a0 and b0 corresponding to N–adapted frames. Respectively, cases 1 and 2 correspond
to a nearly almost-Kähler geometry, with nonholonomically induced torsion (by off–diagonal N-terms) and
vanishing NS 3-form flux, and a nonholonomic generalized Calabi-Yau with flux.
5 The YM Sector and Nonholonomic Heterotic Supergravity
In this section, we construct a nontrivial gauge d–field F̂ which arises at the first order α′, when the YM sector
can not be ignored. The approach elaborates a noholonomic and almost-Kähler version of YM instantons studied
in [2, 58, 1]. The equations of motion of heterotic supergravity are then re–written in canonical nonholonomic
variables, which allows us to decouple and find general integrals of such systems using methods applied for
nonholonomic EYMH and Einstein–Dirac fields, see [30, 32, 36, 37, 38, 47, 48].
20
5.1 N–adapted YM and instanton configurations
The nonholonomic instanton equations can be formulated on 7X =R× 6X with generalized ’h–cone’ d-metric
7
cg = (e
4)2 + [ ph(y
4)]2 6g(ycˇ) = (e4)2 + [ ph(y
4)]2 6g˜aˇbˇ(y
cˇ),
(56)
e4 = dy4 + wi(x
k, ya),
where 6g (28) can be considered for exact solutions determined in 10-d gravity and 6g˜aˇbˇ = [g˜ı´j´, g˜a`b`, N˜
a`
ı´ ]
(31) is for elaborating respective almost Kähler models.8 We denote by
pe
a˜′ = {e4′,
ph ·
pe˜
aˇ′} ∈ T ∗( 7X) an
orthonormal N–adapted basis on a˜′, b˜′, ... = 4, 5, 6, 7, 8, 9, 10, with
pe˜aˇ′ = eaˇ′
aˇ e˜
aˇ, when 6g˜aˇbˇ = δaˇ′ˇb′e
aˇ′
aˇ e
bˇ′
bˇ
.
We can relate the N–adapted configuration to the orthonormal frame eaˇ′ by introducing certain Kähler
operators in standard form instead of (J˜,Θ˜),
pJ˜
:=
pe˜
5 ∧
pe˜
6 +
pe˜
7 ∧
pe˜
8 +
pe˜
9 ∧
pe˜
10,
pΘ˜
:= ( pe˜
5 + i pe˜
6) ∧ ( pe˜
7 + i pe˜
8) ∧ ( pe˜
9 +
pe˜
10),
where i2 = −1 is used for SU(3). Using properties of such orthonormalized N-adapted bases, we can verify that
standard conditions for the nearly Kähler internal spaces are satisfied [2, 58, 1], but also mimic almost-Kähler
manifolds with
d( +
p Θ˜) = 2
pJ˜ ∧
pJ˜ and d pJ˜ = 3(
−
p Θ˜).
In result, we can consider the same reduction of the instanton equations (third formula in (20)) as for
holonomic Kähler structures with nontrivial torsion structure encoded in N–adapted bases for differential forms
on nonholonomic internal manifolds 7X. Using two types of warping variables,
dy4 = ef(τ)dτ, for ef(τ) =
ph(y
4(τ))
(57)
and two equivalent d–metrics,
7
cg = e
2f 7
zg with
7
zg = dτ
2 + 6g˜aˇbˇ,
we obtain nonholonomic instanton equations
∗z F˜ = −(∗zQz) ∧ F˜,
(58)
where Qz = dτ ∧
+
p Θ˜ +
1
2
pJ˜∧
pJ˜ and ∗zis the Hodge star with respect to the cylinder metric
7
zg. The almost-
Kähler structure of 7X is encoded into boldface operators Qz,
pJ˜ and canonical tilde like for
+
p Θ˜. This does
not allow us to solve such equations with an ansatz for the canonical connection on 6X determined by the
LC–connection as in [2, 1] but imposes the necessity to involve the normal (almost symplectic) d–connection
D˜aˇ = (D˜k´, D˜b`) = {ω˜
bˇ′
aˇcˇ′}, (42). Let us consider
AD˜ =
canD˜+ ψ(τ) pe
a˜′Ia˜′,
where the canonical d–connection on 6X enabled with almost-Kähler structure is
canD˜ = { canω˜bˇ
′
aˇcˇ′ := ω˜
bˇ′
aˇcˇ′ +
1
2
( +
p Θ˜)
bˇ′
cˇ′aˇ′e
aˇ′
aˇ }.
8The warping factor h(y4) can be considered in a more generalized form h(xi, y3, y4) because the AFDM also allows us to
generate these classes of solutions. For simplicity, we shall consider factorizations and frame transforms when the warping factor
depends only on the coordinate y4. This allows us to reproduce, in explicit form, the results for Kähler internal spaces if the 6-d
metric structures do not depend on y4.
21
In these formulas, the matrices Ia˜′ = (I˜i′ , Iaˇ′) split into a basis/generators I˜i′ ⊂ s0(3) and generators Iaˇ′ for the
orthogonal components of su(3) in g ⊂ s0(7) satisfy the Lie algebra commutator
[Iaˇ′ , Ibˇ′ ] = f
i˜′
aˇ′ bˇ′
I˜
i′
+ f cˇ
′
aˇ′bˇ′
Icˇ′ ,
with respective structure constants f i˜
′
aˇ′bˇ′
and f cˇ
′
aˇ′bˇ′
(see formulas (3.9) in [1] for explicit parametrizations).
We can define and compute the curvature d–form
AF˜ =
1
2
[ AD˜, AD˜] := F(ψ)
=
canR˜+
1
2
ψ2f i˜
′
aˇ′ bˇ′
I˜
i′
pe
aˇ′ ∧
pe
bˇ′ +
∂ψ
∂τ
dτ ∧ Icˇ′
pe
cˇ′ +
1
2
(ψ − ψ2)Ibˇ′(
+
p Θ˜)
bˇ′
cˇ′aˇ′
pe
cˇ′ ∧
pe
aˇ′ ,
with parametric dependence on τ (57) via ψ(τ). Such an AF˜ is a solution of the nonholonomic instanton
equations (58) for any solution of the ’kink equation’
∂ψ
∂τ
= 2ψ(ψ − 1).
For the aforementioned types of d–connections, and for the LC-connection, we can consider two fixed points,
ψ = 0 and ψ = 1 and a non-constant solution
ψ(τ) =
1
2
(1− tanh |τ − τ0|) ,
where the integration constant τ0 fixes the position of the instanton in the τ direction but such an instanton
also encodes an almost-Kähler structure.
In heterotic supergravity, we can consider two classes of nonholonomic instanton configurations. The first
one is for the gauge-like curvature, AF˜ = F(
1ψ) and R˜ = R( 2ψ), [in the second case, we also solve the
condition R˜ ·ǫ = 0] where the values 1ψ and 2ψ will be defined below.
5.2 Static and/or dynamic SU(3) nonholonomic structures on almost Kähler configura-
tions
The transforms 6g˜aˇbˇ = [
ph]
2
6g˜aˇbˇ in d–metric (56) impose certain relations on the two pairs (
pJ˜,
pΘ˜)
and (J˜,Θ˜), where the last couple is subjected to the respective flow and structure equations, (53) and (54),
and define the 3–form Ĥ (52). We note that such relations are a mixing between real and imaginary parts
and nonholonomically constrained in order to adapt the Lie algebra symmetries to the almost-Kähler structure.
Like in [1], it is considered a y4 depending mixing angle
pβ(y
4) ∈ [0, 2π) when
J˜ = [ ph]
2
pJ˜,
+Θ˜ = [ ph]
3( +
p Θ˜ cos
pβ +
−
p Θ˜ sin
pβ),
−Θ˜ = [ ph]3(− +
p Θ˜ sin
pβ +
−
p Θ˜ cos
pβ).
Introducing such values into the relations for (J˜, +Θ˜, −Θ˜), we obtain (compare to (55))
Ĥ = ℓvol[ 3g] +
phdy
4 ∧ (ℓ ph− 4 sin
pβ) pJ˜
+[ ph]
2
[
−
1
2
ph(∂4φ) cos
pβ + 3 sin
2
pβ −
7
8
ℓ ph sin
pβ
]
+
p Θ˜
+[ ph]
2
[
−
1
2
ph(∂4φ) sin
pβ − 3 sin
pβ cos
pβ +
7
8
ℓ ph cos
pβ
]
−
p Θ˜.
22
The above formula involves the conditions
+T1 = 2( ph)
−1 cos
pβ and
−T1 = −2( ph)−1 sin
pβ
which allows us to fix
α˜(y4) = 3 −T1 +
15
8
ℓ.
There are additional conditons on the scalar functions
ph,
pβ, ℓ and φ which must be imposed on coupled
nonholonomic instanton solutions 1ψ and 2ψ satisfying the Bianchi conditions, the nonholonomic BPS equations
and the time–like components of the equations of motion. We omit such an analysis because it is similar to
that of the pure Kähler configurations, see sections 4 and 5 in [1]. The priority of nonholonomic almost-Kähler
variables is so that we can work with respect to N-adapted frames in a form which is very similar to that for
complex and symplectic structures.
5.3 Equations of motion of heterotic supergravity in nonholonomic variables
In order to apply the AFDM, we have to rewrite the motion equations (generalized Einstein equations) in
nonholonomic variables. The formal procedure is to take such equations written for the LC–connection with
respect to coordinate frames and re–write them for the same metric structure, but for corresponding geometric
objects with "hats " and "waves" and with respect to N–adapted frames on corresponding shells.
In this
way, including terms of order α′, the N–adapted equations of motion of heterotic nonholonomic supergravity
considered in [1] can be written in such a form:
R̂µsνs + 2(
sD̂d̂φ̂)µsνs −
1
4
ĤαsβsµsĤ
αsβs
νs
+
α′
4
[
R˜µsαsβsγsR˜
αsβsγs
νs
− tr
(
F̂µsαsF̂
αs
νs
)]
= 0,
(59)
sR̂+ 4̂φ̂− 4|d̂φ̂|2 −
1
2
|Ĥ|2 +
α′
4
tr
[
|R˜|2 − |F̂|
]
= 0,
(60)
e2φ̂d̂∗̂(e−2φ̂F̂) + Â ∧ ∗̂F̂− ∗̂F̂ ∧ Â+ ∗̂Ĥ ∧ F̂ = 0,
(61)
d̂∗̂(e−2φ̂Ĥ) = 0,
(62)
where the Hodge operator ∗̂, sD̂ = {D̂µs} (12), the canonical nonholonomic d’Alambert wave operator ̂ :=
ĝµsνsD̂µsD̂νs , R̂µsνs (17),
sR̂ (18) are all determined by a d–metric ĝ (10). The curvature d–tensor R˜µsαsβsγs
is taken for an almost-Kähler structure θ˜ (40) defined by corresponding nonholonomic distributions which are
stated up to frame transforms by the N–connection structure and components of d–metric on shells s = 1, 2, 3
as we described above. The gauge field  corresponds to the N–adapted operator
s
AD̂ =
sD̂+ 1ψ(y4)[ea1Ia1 + e
a2Ia2 + e
a3Ia3 ] =
sD̂+ 1ψ(y4)Icˇ′
pe
cˇ′ = d̂+ Â
and curvature F̂ = F( 1ψ) via a map constructed above (for sD̂|T̂ =0 →
s∇, see details in [1]). For instance,
the LC–configurations of (59) are determined by equations
Rµν + 2(∇dφ)µν −
1
4
HαβµH
αβ
ν
+
α′
4
[
R˜µαβγR˜
αβγ
ν
− tr
(
F̂µαF̂
α
ν
)]
= 0
(63)
with standard 10-d indices α, β, ... = 0, 1, 2, ...9 and geometric values determined by ∇. Unfortunately, the
system of nonlinear PDEs (63) can not be decoupled and integrated in any general form if we do not consider
shell N–adapted frames/coordinates and generalized connections which can be nonholonomically constrained to
LC-configurations.
The equations (61) and (62) can be solved for arbitrary almost-Kähler internal spaces by introducing cor-
responding classes of N-adapted variables as we proved in previous sections. Such solutions can be classified as
23
for pure Kähler spaces, for simplicity, considering a special case with −T1 = 0 and ℓ = 0 and in terms of ’kink’
solutons with
e2f = e2(τ−τ0) +
α′
4
[( 1ψ)2 − ( 2ψ)2], τ0 = const.
The NS 3–form flux is given by a simple formula
Ĥ(τ, ycˇ) = −
1
2
= [ ph]
3(∂4φ)(
+
p Θ˜) =
α′
4
[( 1ψ)2(2 1ψ − 3)− ( 2ψ)2(2 2ψ − 3)](τ)[ +
p Θ˜(y
cˇ)].
(64)
Here we reproduce the classification of 8 cases with fixed and/or kink configurations for almost-Kähler config-
urations,
Case
1ψ, 2ψ;
e2f = e2(τ−τ0) + α
′
4 [(
1ψ)2 − ( 2ψ)2], φ(τ) = φ0 + 2(f − τ)
1.
1ψ = 2ψ;
f = τ − τ0, φ = φ0 − 2τ
2.
1ψ = 2ψ = 0;
f = f0 :=
1
2 log(
α′
4 ), φ = φ0 + 2(f0 − τ)
3.
1ψ = 1, 2ψ = 0;
e2f = e2(τ−τ0) − α
′
4 , e
φ−φ0 = e−2τ0 − α
′
4 e
−2τ
4.
1ψ = 0, 2ψ = kink ;
e2f = e2(τ−τ0) + α
′
16 [1− tanh(τ − τ1)]
2, φ = φ0 + 2(f − τ)
5.
1ψ = kink , 2ψ = 0;
e2f = e2(τ−τ0) − α
′
16 [1− tanh(τ − τ1)]
2, φ = φ0 + 2(f − τ)
6.
1ψ = 1, 2ψ = kink ;
e2f = e2(τ−τ0) + α
′
16 [tanh(τ − τ1) + 1][tanh(τ − τ1)− 3]
7.
1ψ = kink , 2ψ = 1;
e2f = e2(τ−τ0) − α
′
16 [tanh(τ − τ1) + 1][tanh(τ − τ1)− 3]
8.
1ψ = kink , 2ψ = kink
with τ1
6= τ2;
e2f = e2(τ−τ0) + α
′
16 [tanh
2(τ − τ1)− 2 tanh(τ − τ1)−
tanh2(τ − τ2) + 2 tanh(τ − τ2)]
Such a classification can be used for parametrizing certain effective sources of Einstein–Yang-Mills-Higgs type
and preserved for constructing generic off-diagonal solutions following the AFDM [30, 31, 34, 29, 36, 37, 23, 24].
We emphasize that classes 1-8 distinguish different almost-Kähler structures encoding corresponding assump-
tions, that for identification of almost complex structure I with the complex unity i in the typical fiber of tangent
space use the same classification as for symplectic configurations introduced in [1, 2]. A unified classification
for internal (almost) Kähler spaces is possible with respect to corresponding N–adapted frames generated by a
conventional Lagrange function as we considered in formulas (30) and (40).
Finally, we note that we shall construct and analyze various classes of generic off-diagonal exact solutions
of the equations (59) and (60) in our associated work [3]. The main goal of that work is to prove that it is
possible to reproduce certain types of off–diagonal deformations of the Kerr metric in heterotic supergravity
in the 4-d sector, if the internal 6-d space is endowed with richer (than in Kähler geometry) structures. In
section 4, we proved that it is always possible to introduce such nonholonomic variables when the internal space
geometric data is parameterized via geometric objects of an effective almost-Kähler geometry. This allows us
to self-consistently solve the equations (61) and (62) following the same procedure and classification as in [1, 2]
when the domain walls were endowed with trivial pseudo-Euclidean structure warped nearly Kähler internal
spaces. The off–diagonal deformation techniques defined by the AFDM allow us to generalize the constructions
for nontrivial exact and parametric solutions in 4-d, 6-d and 10-d MGTs and string gravity.
6 Conclusions
In this work, we have studied how to generalize certain classes of ’prime’ solutions in heterotic string gravity
constructed in [1, 2] for arbitrary ’target’ ten dimensional, 10-d, metrics. The structure and classification of
prime configurations [with pseudo-Euclidean (1+3)–dimensional domain walls and 6-d warped nearly Kähler
manifolds in the presence of gravitational and gauge instantons] can be preserved for nontrivial curved 4-d
spacetime configurations. For instance, we can generate black hole/ellipsoid configurations if certain types
of nonholonomic variables with conventional 2+2+....=10 splitting are defined and respective almost-Kähler
24
internal configurations are associated for corresponding 2+2+2=3+3 double fibrations. The diadic "shell by
shell" nonholonomic decomposition of 4-d, 6-d and 10-d pseudo-Riemannian manifolds allows us to integrate
the motion equations (59)–(62) in very general forms using the AFDM, see further developments of this paper
in [3]. The 3+3 decomposition is used for constructing nonholonomic deformed instanton configurations which
are necessary for solving the Yang-Mills sector and the generalized Bianchi identity at order α′, when certain
generalized classes of solutions may contain internal configurations depending, in principle, on all 9 space-like
coordinates for a 10-d effective gravity theory. The triadic formalism is also important for associating SU(3)
structures in certain holonomic limits to well known solutions (with more "simple" domain wall and an internal
structures) in heterotic string gravity.
The geometric techniques elaborated in this paper (which is a development for the heterotic string gravity
results of a series of studies [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 9, 10, 40]) allows us to work with arbitrary
stationary generic off-diagonal metrics on 10-d spacetimes. Effective MGTs spacetimes can be enabled with gen-
eralized connections, depending on all possible 4-d and extra dimension space coordinates. One of main results
of the presented work is that the system (61)–(62) admits subclasses of solutions with warping on coordinate
y4 nearly almost-Kähler 6-d internal manifolds in the presence of nonholonomically deformed gravitational and
gauge instantons. The almost-Kähler structure is necessary if we want to generate in the 4-d spacetime part, for
instance, the Kerr metric with possible (off-) diagonal and nonholonomic deformations to black ellipsoid type
configurations characterized by locally anisotropic polarized physical constants, small deformations of horizons,
embedding into nontrivial extra dimension vacuum gravitational fields and/or gauge configurations [which are
considered in details in [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 3]]. In this work, we concentrate on 10-d configu-
rations preserving two real supercharges corresponding to N = 1/2 supersymmetry from the viewpoint of four
non-compact dimensions and various nonholonomic deformations.
Following methods of the geometry of nonholonomic manifolds and almost-Kähler spaces, applied for defor-
mation and/or A-brane quantization and geometric flows of gravity theories [9, 10, 40], we defined two pairs of
related ( pJ˜, pΘ˜) and (J˜,Θ˜) structures depending both on warping and other space coordinates. Such construc-
tions encode possible almost symplectic configurations determined by the 6-d internal sector of heterotic gravity
and BPS equations re-written in nonholonomic variables. This involves and mixes certain gauge like, i.e. a
NS 3-form for flux, gravitational solitons, and effective scalar fields. For additional constraints, such stationary
configurations transform into static SU(3) structures as in [1, 2]. This assumption is crucial for classifying new
types of nonolonomically deformed solutions following the same principles, as it was done originally for Kähler
internal spaces and standard instanton constructions.
Finally, we note that it remains to be seen how the AFDM generates certain classes of physically important
solutions of the motion equations (59)–(60) (like black holes, cosmological metrics etc.) in the 4-d sector, using
the nonholonomic solutions of (61)–(62) generated in this work. In the associated paper [3], the motion equations
in heterotic string gravity resulting in stationary metrics, in particular, in generic off-diagonal deformations of
the Kerr solution to certain ellipsoid like configurations are solved by integrating in very general off-diagonal
forms. Further developments in string MGTs with cosmological solutions of type [11, 12, 15, 16, 19, 25, 26, 27, 28]
are left for future work.
Acknowledgments: The SV research is for the QGR–Topanga with a former partial support by IDEI,
PN-II-ID-PCE-2011-3-0256 and DAAD. This work contains also a summary of results presented in a talk for
GR21 at NY.
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