The Coulomb gauge in non-associative gauge theory
Sergey Grigorian
TL;DR
This paper extends the Coulomb gauge existence results from standard gauge theory to non-associative gauge theory based on smooth loops, motivated by $G_2$-geometry and torsion analysis.
Contribution
It introduces a framework for Coulomb gauge in non-associative gauge theory and proves existence of divergence-free torsion configurations under small torsion conditions.
Findings
Existence of Coulomb gauge configurations in non-associative gauge theory.
Construction of configurations with divergence-free torsion.
Applicability to $G_2$-geometry and torsion structures.
Abstract
The aim of this paper is to extend existence results for the Coulomb gauge from standard gauge theory to a non-associative setting. Non-associative gauge theory is based on smooth loops, which are the non-associative analogs of Lie groups. The main components of the theory include a finite-dimensional smooth loop , its tangent algebra a finite-dimensional Lie group , that is the pseudoautomorphism group of , a smooth manifold with a principal -bundle , and associated bundles and with fibers and , respectively. A configuration in this theory is defined as a pair , where is a section of and is a connection on . The torsion is the key object in the theory, with a role…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsBlack Holes and Theoretical Physics · Advanced Differential Geometry Research · Homotopy and Cohomology in Algebraic Topology
The Coulomb gauge in non-associative gauge theory
Sergey Grigorian
Abstract
The aim of this paper is to extend existence results for the Coulomb gauge from standard gauge theory to a non-associative setting. Non-associative gauge theory is based on smooth loops, which are the non-associative analogs of Lie groups. The main components of the theory include a finite-dimensional smooth loop , its tangent algebra a finite-dimensional Lie group , that is the pseudoautomorphism group of , a smooth manifold with a principal -bundle , and associated bundles and with fibers and , respectively. A configuration in this theory is defined as a pair , where is a section of and is a connection on . The torsion is the key object in the theory, with a role similar to that of a connection in standard gauge theory. The original motivation for this study comes from -geometry, and the questions of existence of -structures with particular torsion types. In particular, given a fixed connection, we prove existence of configurations with divergence-free torsion, given a sufficiently small torsion in a Sobolev norm.
Contents
1 Introduction
The main goal of this work is to extend results on existence of Coulomb gauge transformations from standard gauge theory to a non-associative setting. One of highly successful areas at the intersection of differential geometry, analysis, and mathematical physics is gauge theory, which, as it is well-known, is the study of connections on bundles with particular Lie groups as the structure groups. In [26], the author initiated a theory of smooth loops, which are non-associative analogs of Lie groups, and began the development of gauge theory based on loops, i.e. a non-associative gauge theory. The key example of a non-associative smooth loop is the loop of unit octonions. A non-associative gauge theory has the following components:
A finite-dimensional smooth (right) loop , i.e. a smooth manifold with a right multiplication diffeomorphism defined for every and a distinguished identity element with tangent algebra at identity. 2. 2.
A finite-dimensional Lie group that is the pseudoautomorphism group of , a non-associative generalization of an automorphism group. 3. 3.
A smooth manifold with a principal -bundle and associated bundles and with fibers and respectively. 4. 4.
A configuration is defined by a pair where is a section of and is a connection on Together they define the *torsion * which is an -valued -form on The torsion is then the key object in the theory, in the same way that a connection is the key object in standard gauge theory. 5. 5.
In addition to standard gauge transformations of by we now also have transformations of induced by loop multiplication. Both of these kinds of transformations induce transformations of the torsion.
The original motivation for studying non-associative gauge theories comes from -geometry [24]. A -structure on a -dimensional Riemannian manifold is a reduction of the structure group of the orthornormal frame bundle from to which is the automorphism group of the octonions. A further relationship between -structures and the octonions is that unit norm sections of an octonion bundle parametrize -structures that are associated with the same metric, also known as *isometric *-structures. A defining characteristic of a -structure is its torsion, and one of the unanswered questions in -geometry is which torsion types are admissible within a fixed metric class. One of the main goals in the theory of -structures is to obtain existence results for torsion-free -structures, similar to the Yau’s Theorem [49], that settled the existence question for Calabi-Yau manifolds. While this goal is formulated in terms of -structures, the real question is the existence of a Riemannian metric with holonomy group equal to The fact that for each metric there exists an entire family of compatible -structures presents a degeneracy in this problem. Some of the existing approaches involve Laplacian flows of -structures [4, 5, 7, 8, 9, 23, 21, 33, 38, 39, 40], with the hope of a flow eventually converging to a torsion-free -structure. As shown in [21], the Laplacian flow of a generic -structure has a component that moves within a metric class, and that component is precisely given by Laplacian flows have been more successful for *closed *-structures, in which case automatically vanishes, and thus the degeneracy is resolved. More generally, however, this degeneracy is the source of non-parabolicity of Laplacian flows, such as in the case of co-closed -structures [21]. Therefore, the condition can be regarded as a gauge-fixing condition. Moreover, in [24], it was found that on a compact manifold, -structures with are precisely the critical points of the -norm of the torsion when restricted to a fixed metric class. As shown in [24, 26], within the loop bundle framework, this is the precise analog of the Coulomb gauge condition.
Existence of -structures with divergence-free torsion has been studied from different perspectives by several authors: Bagaglini in [3]; Dwivedi, Gianniotis, and Karigiannis in [12]; the author in [25]; Loubeau and Sá Earp in [41]. All these approaches relied on a flow of isometric -structures (or more generally, geometric structures in [41] and [13]), and have shown existence of a -structure with divergence-free torsion as a long-term limit of the flow, given sufficiently small pointwise initial torsion or another quantity, the entropy.
The interpretation of isometric -structures as an octonionic non-associative gauge theory allows to adapt some gauge theory techniques in this setting. Moreover, without much additional effort, more general loops can be considered, with potential wider-reaching applications.
In gauge theory there are a number of versions of local and global existence results for connections in the Coulomb gauge, depending on the desired regularity [11, 14, 15, 16, 17, 47, 48]. In this paper we use the Quantitative Implicit Function Theorem for Banach Spaces, as given in [17], to prove the following main result.
Theorem A
Suppose is a smooth compact loop with tangent algebra and pseudoautomorphism group Let be a closed, smooth Riemannian manifold of dimension and let be a -principal bundle over and let be the associated vector bundle to with fibers isomorphic to . Let be a smooth connection on Also, suppose is a non-negative integer and such that Then, there exist constants and such that if is a smooth defining section for which
[TABLE]
then there exists a section such that
[TABLE]
and
[TABLE]
If moreover, then is smooth.
For -structures, this gives the following result for existence of smooth -structures with divergence-free torsion.
Theorem B
Suppose is a closed -dimensional manifold with a smooth -structure with torsion with respect to the Levi-Civita connection Suppose is the corresponding unit octonion bundle. Also, suppose is a positive integer and is a positive real number such that Then, there exist constants and such that if satisfies
[TABLE]
then there exists a smooth section such that
[TABLE]
and
[TABLE]
The results presented in this paper are of interest and importance in their own right, but perhaps even more crucially, they show that some well-known results and techniques from classical gauge theory can be reinterpreted and adapted in a non-associative setting. In particular, this may open the door to some analogues of Uhlenbeck compactness and a better understanding of the torsion of non-associative gauge theories. Furthermore, a non-associative version of Yang-Mills equations can be considered. Moreover, any such advances will give immediate results in -geometry.
The structure of this paper is the following. In Section 2, we give an overview of smooth loops, extending [26]. We give the basic properties of a smooth loop , define the pseudoautomorphism group and the tangent algebra at identity. The algebra is a generalization of a Lie algebra, but due to the non-associativity of does not satisfy the Jacobi identity. Similarly as for Lie algebras, there is a notion of an exponential map. There is however a family of brackets on , defined for each point For later use, we also give estimates for the exponential and adjoint maps. In particular, we analyze solutions of the following initial value problem for -valued maps
[TABLE]
where and
In Section 3, we switch attention to loop-valued maps. In particular, given a smooth manifold consider a map Using this map, we may define products of -valued maps and brackets of -valued maps. Then, using the right quotient, translating the differential to the tangent space at we obtain an -valued -form on which is the analogue of the Darboux derivative of Lie group-valued maps [45]. The differentials of various operations defined by are then expressed in terms of Suppose for some -valued map We show that satisfies a non-homogeneous version of (1.3).
Further, we define Sobolev spaces of maps from , and show in Lemma 3.9 that, similarly as for Lie groups, if and only if Using the evolution equation satisfied by then allows us to obtain Sobolev space estimates of and other quantities that satisfy equations based on (1.3).
Theorem C
Let be a compact Riemannian manifold and is a smooth compact loop. Suppose Let and and suppose Then,
[TABLE]
where
Similarly, if where is -parameter family of -valued maps that satisfies
[TABLE]
for -valued maps and , then,
[TABLE]
In Section 4, we introduce a principal -bundle over a compact manifold , and then apply the above results to -equivariant maps from to a loop and other related spaces. This immediately then allows to consider sections of bundles over that are associated to In particular, suppose we have a connection on and suppose is a section of the associated loop bundle with fibers diffeomorphic to It uniquely corresponds to a -equivariant map and thus we obtain the equivariant -valued -form on On the other hand, the connection defines a decomposition of into vertical and horizontal subspaces. Therefore, we may compose with the horizontal projection to obtain a basic, i.e. horizontal and equivariant -valued -form on This then corresponds to a section of a bundle over and gives us the *torsion * of the configuration Defining fiberwise loop multiplication, we see that all the possible configurations with a fixed may be obtained by multiplying by some section Therefore, the loop gauge transformations are precisely the transformations Moreover, as it was already known previously, [24, 25, 27, 26, 41], given appropriate algebraic conditions on the loop, the critical points of the functional are precisely the sections for which which relates to the previous discussion on divergence-free torsion and the Coulomb gauge.
Considering the transformations of of the form , for -valued sections , and using the loop exponential map, as developed in Section 2, the quantity is then shown to satisfy an ODE with the homogeneous part of the form (1.3). This allows to rewrite in terms of the evolution operator for (1.3). The equation is then written as a second-order PDE for This operator then defines a smooth functions between appropriate Banach spaces of sections of vector bundles, which allows to apply the Implicit Function Theorem from [17], to show existence of solutions for sufficiently small initial in an appropriate Sobolev norm, and thus prove Theorem A.
In Section 5, we carefully apply the general theory of smooth loops to the particular case of -structures, and then Theorem B follows as an immediate corollary of Theorem A.
Acknowledgements
The author is supported by the National Science Foundation grant DMS-1811754.
2 Smooth Loops
For a detailed introduction to smooth loops, the reader is referred to [26]. The reader can also refer to [29, 34, 43, 44, 46] for a discussion of these concepts.
Definition 2.1
*A *loop is a set with a binary operation with identity and compatible left and right quotients and , respectively.
In particular, existence of quotients is equivalent to saying that for any the left and right product maps and are invertible maps. Restricting to the smooth category, we obtain the definition of a smooth loop.
Definition 2.2
*A *smooth loop is a smooth manifold with a loop structure such that the left and right product maps are diffeomorphisms of
Remark 2.3
*In this paper we will not use the left quotient, so in fact everything that follows also holds true for smooth *right loops, i.e. where only the right quotient is defined, but the left product is not necessarily invertible. However, for brevity, we will keep referring to loops, rather than right loops. As Example 2.4 below shows, smooth right loops are plentiful and easy to construct.
Example 2.4
Suppose is a Lie group with a Lie subgroup and consider the left quotient K=\scalebox{-1.0}[1.0]{\nicefrac{{\scalebox{-1.0}[1.0]{}}}{{\scalebox{-1.0}[1.0]{}}}}. Suppose is a section of , regarded as a bundle over In particular, maps each right coset to a particular representative of that coset. Suppose A product structure on is then given by
[TABLE]
*Consider the equation Since is a section, we can see right away that we have a unique solution Thus, has *right *division, and is thus a *right loop [34, 43]. To define left division, and hence to obtain a full loop structure, more structure is needed.
Definition 2.5
A pseudoautomorphism of a smooth loop is a diffeomorphism for which there exists another diffeomorphism , known as the partial pseudoautomorphism corresponding to such that for any ,
[TABLE]
In particular, and The element is the companion of From (2.2), we also see the following property of with respect to quotients:
[TABLE]
It is easy to see that the sets of pseudoautomorphisms and partial pseudoautomorphisms are both groups. Denote the former by and the latter by . We also see that the *automorphism *group of the loop is the subgroup which is the stabilizer of
Remark 2.6
To avoid introducing additional notation, but at the risk of some ambiguity, we will use the same notation to denote the underlying manifold, the loop, and the -set with the full action of . However, since also admits the action of if a distinction between the -sets is needed, we will use to denote the set with the action of
Let , then we may define a modified product on via , so that equipped with product will be denoted by the corresponding quotient will be denoted by We have the following properties [26].
Lemma 2.7
Let . Then, for any
[TABLE]
Also, for any
[TABLE]
Lemma 2.8
Suppose and are smooth curves in with and , then
[TABLE]
Consider the tangent space at For any , the pushforward of the right translation map defines a linear isomorphism from to In particular, let us denote the linear map by , and correspondingly, by Similarly, for left multiplication, define On a smooth right loop, will not necessarily be invertible. The corresponding maps with respect to the product will be denoted by
Definition 2.9
For any define the fundamental vector field for any
The above definition of the fundamental vector field is the analog of a right-invariant vector field in Lie theory. However, in the loop case, although this vector field is canonical in some sense, it is not invariant under right translations. We use fundamental vector fields to define the loop exponential map.
Definition 2.10
Suppose is a smooth loop and suppose Then, given for sufficiently small define
[TABLE]
to be the solution of the equation
[TABLE]
Equivalently, satisfies
[TABLE]
Remark 2.11
In general, the solution will only be defined in a neighborhood of however as shown in [36, 42], if the loop is power-associative, so that powers of an element associate, then can be defined unambiguously. We will show this from a different perspective further below. This can then be used to define the solution for all and thus this extends to all of
Let us consider From the definition of , for any have
[TABLE]
In particular, is smooth and since the identity map is a linear isomorphism, by the Inverse Function Theorem, we have the following.
Lemma 2.12
For any the map is a local diffeomorphism around .
Remark 2.13
To distinguish the exponential map on from the exponential map on we will use a subscript to denote with respect to which element of the exponential map is used. The exponential map on will be without the subscript.
On smooth loops, we can define an analog of the Lie group Maurer-Cartan form.
Definition 2.14** ([26])**
The Maurer-Cartan form is an -valued -form on , such that for any vector field , and any Equivalently, for any
The loop Maurer-Cartan form allows us to define brackets on For each define the bracket given for any by
[TABLE]
As shown in [26, Theorem 3.7], we can equivalently define
[TABLE]
where, for , is the differential at of the conjugation map
Remark 2.15
In [26], the conjugation map was denoted by and its differential as However here we adopt notation that is more in line with standard usage in Lie theory.
Definition 2.16
*The vector space equipped with the bracket is known as the *loop tangent algebra
Define the *bracket function * to be the map that takes , so that is an -valued -form on , i.e. .
Definition 2.17
For any and the associator on given by
[TABLE]
Moreover, define mixed associators between elements of and . An -associator is defined for any and as
[TABLE]
and an -associator is defined for an and as
[TABLE]
*where we see that Similarly, for other combinations. Also define the *left-alternating associator given by
[TABLE]
which we can call the left-alternating associator.
Remark 2.18
*From the definitions of the associators, it is easy to see that if is power-associative, given associators with any combinations of and for any values of , in the three entries, will vanish. For example, *
[TABLE]
as well as any permutations.
*Similarly, if is left-power-associative, then associators with any combination of and in the first two entries will vanish, for example *
[TABLE]
for any and similarly with the third entry replaced by an element of
From [26] we cite several useful properties of these brackets and associators.
Theorem 2.19** ([26, Theorem 3.20])**
Suppose , and Then the bracket is related to via the expression
[TABLE]
Theorem 2.20** ([26, Theorem 3.10])**
The form satisfies
[TABLE]
where wedge product of -forms is implied. Also, for any we have
[TABLE]
It follows that the generalized Jacobi identity is satisfied:
[TABLE]
where
[TABLE]
Remark 2.21
Equation (2.19) is the loop Maurer-Cartan equation. The key difference from the Maurer-Cartan equation on Lie groups is that on non-associative loops, is non-constant on unlike on Lie groups, where there is a unique bracket on the Lie algebra, and hence is constant. In particular, the non-constant leads to a non-trivial associator (2.20) and the failure of the standard Jacobi identity to hold.
With respect to the action of the bracket and the associator satisfy the following properties.
Lemma 2.22
If and , then, for any ,
[TABLE]
If is a path on with and then for any ,
[TABLE]
Also, for any
[TABLE]
Proof. The first part is given in [26, Lemma 3.17]. To show (2.23), consider
[TABLE]
where we’ve used (2.6). Now,
[TABLE]
Hence,
[TABLE]
To show (2.24), we could use (2.20), but more directly, we can obtain it from the definition, using (2):
[TABLE]
Then, from (2.23),
[TABLE]
and from the definition (2.17), we obtain (2.24).
Let and Also let for in some interval I\subset\mathbb{R}\that contains [math]. Then consider a family that satisfies the following initial value problem:
[TABLE]
In other words, this is linear first-order ODE , so for all there exists an evolution operator , with such that
[TABLE]
From standard ODE theory, recall that if then is the evolution operator from to and is given by:
[TABLE]
The following properties of follow immediately.
Lemma 2.23
The evolution operator satisfies the following properties:
* for any and as long as and are both defined.* 2. 2.
** 3. 3.
If is compact, and is equipped with an inner product, then in a compatible operator norm, there exists a constant such
[TABLE]
Proof. Item 1 follows from a change of variables in (2.26). For item 2, consider
[TABLE]
Then,
[TABLE]
since Hence, but so for all .
For the estimate, from (2), we obtain
[TABLE]
Now, is a smooth real-valued map on a compact manifold, and is hence bounded. Therefore, there exists a constant and hence Thus,
[TABLE]
Remark 2.24
Since for brevity let us denote the operator by
If is a Lie algebra, then is independent of and then In the non-associative case, this is no longer true in general, but needs additional assumptions, as Theorem 2.25 below shows.
Theorem 2.25
Let , and Suppose is the evolution operator for the equation (2.26) as in (2.27). Then,
[TABLE]
Moreover,
If is compact, and is equipped with an inner product, then in a compatible operator norm, there exists a constant that depends only on such that,
[TABLE] 2. 2.
If is left-power-alternative, then
[TABLE] 3. 3.
If is both left-power-alternative and right-power-alternative, then
[TABLE]
Proof. Let and note that Then, consider the derivative of Let From (2.9) we have
[TABLE]
Then, using (2.6) we have
[TABLE]
Using (2.5), the second term in (2) becomes
[TABLE]
and similarly, the third term in (2) becomes
[TABLE]
Using (2) we then conclude that
[TABLE]
This is an inhomogeneous linear first order ODE. The homogeneous part is precisely (2.26), and hence we obtain precisely (2.30).
For the estimate, suppose is compact. Then, using (2.30) and (2.29), we have
[TABLE]
However, is a real-valued function on a compact manifold, and hence there exists a constant which is the supremum of this function over Hence,
[TABLE]
Renaming the constant we get (2.31).
Now if is left-alternative, the second term on the right hand side of (2.35) vanishes, since
[TABLE]
Then, satisfies the homogeneous equation, so the solution is just
From (2.18), we have
[TABLE]
If is left-power-alternative, then first of all, the associator is skew-symmetric in the first two entries, so
[TABLE]
however due to right-power-alternativity, Hence, in this case,
[TABLE]
Then, satisfies the first order homogeneous ODE with constant coefficients:
[TABLE]
so and hence the solution is now
[TABLE]
Corollary 2.26
If is power-associative, then
, 2. 2.
**
Proof. From (2.30),
[TABLE]
However, by power-associativity (2.16),
[TABLE]
Hence, using Lemma 2.23, we obtain
[TABLE]
For the second part, define
[TABLE]
Then, informally, we write
[TABLE]
Now using power-associativity, we see that associates with and since moreover and commute. Hence, we can rewrite
[TABLE]
The solution is thus
[TABLE]
so by uniqueness of solutions we then have the needed equality.
Remark 2.27
Corollary 2.26 thus shows that indeed, power-associativity allows to extend for all . This is a slightly different proof of this fact compared to [36, 42]. The result from Corollary 2.26 also allows to conclude that if is power-associative, then
[TABLE]
Theorem 2.28
Suppose is a path in with derivative , then
[TABLE]
Moreover,
If is left-power-alternative, then
[TABLE] 2. 2.
If is both left-power-alternative and right-power-alternative, then
[TABLE]
Proof. Using a similar approach as in the Lie group case, let
[TABLE]
We can write this (somewhat informally) as
[TABLE]
Note that Then, consider
[TABLE]
For each the homogeneous part of ODE is precisely (2.26), and since the initial condition is we find that the solution of the inhomogeneous equation is
[TABLE]
Setting , we obtain
[TABLE]
The special cases now follow immediately from (2.32) and (2.33).
Corollary 2.29
Let then
[TABLE]
Moreover, if is compact, given a norm on and a corresponding operator norm,
[TABLE]
where is a constant that depends on
Proof. The expression (2.39) follows directly from Theorem 2.28. We thus have
[TABLE]
From (2.41) we then obtain (2.40).
Let us now explore the dependence of on In particular, suppose we have a smooth -parameter family with and with For each satisfies
[TABLE]
Now for each so define
[TABLE]
so that in particular,
Lemma 2.30
Let , then the quantity is given by
[TABLE]
In particular, if is left-power-alternative, then for all
Proof. From (2.42), we have
[TABLE]
and therefore,
[TABLE]
where we used the derivative of the quotient formula (2.6) and also the fact that for all Noting that for any , consider the first term of (2):
[TABLE]
Now,
[TABLE]
The second term of (2) gives
[TABLE]
Overall, (2) becomes
[TABLE]
This is precisely the equation (2.35) satisfied by however with initial condition Therefore, the solution is
[TABLE]
If is left-power-associative, then from Theorem 2.25, and thus for all .
We will assume that group of pseudoautomorphisms of is a finite-dimensional Lie group, and suppose the Lie algebras of and are and , respectively. In particular, is a Lie subalgebra of . Also, we will assume that acts transitively on . The action of on \mathbb{L}\induces an action of the Lie algebra on , which we will denote by
Definition 2.31
Define the map such that for each and ,
[TABLE]
Lemma 2.32** ([26, Theorem 3.25])**
The map as in (2.46) is equivariant with respect to corresponding actions of in particular for , we have
[TABLE]
Moreover, the image of is and the kernel is , and hence, .
Lemma 2.33** ([26, Lemma 3.33 and Lemma 3.35])**
*Suppose and then *
[TABLE]
Similarly as for Lie groups, we may define a Killing form on . For , we have
[TABLE]
where is just composition of linear maps on and . Clearly is a symmetric bilinear form on In [26] it is shown that for and it satisfies .
General criteria for a loop algebra to admit a non-degenerate Killing form are currently not known, but it is known [37] that for a semisimple Malcev algebra, the Killing form is non-degenerate. A *Malcev *algebra is the tangent algebra of a Moufang loop and is an alternative algebra that also satisfies the following identity [36, 42]:
[TABLE]
Moreover, in this case, is -invariant and -invariant [26]. Suppose then from (2.24), we see that generally,
[TABLE]
In the special case of being a Moufang loop, and thus every being a Malcev algebra, we have the following.
Lemma 2.34
Suppose is a Moufang loop. Then, is independent of and for each then map is skew-adjoint with respect to
Proof. If is Moufang, then any is also a Moufang loop, and hence for any , is a Malcev algebra. Since a Malcev algebra is alternative, and is moreover totally skew-symmetric. In particular, the Malcev identity (2.50) can be written as
[TABLE]
In particular, taking the trace, we get
[TABLE]
Then, (2) gives This shows that is constant on
For the second part, from the generalized Jacobi identity (2.21), we obtain
[TABLE]
However, for an alternative algebra, this simplifies to
[TABLE]
The second line is symmetric in and , so it is sufficient to consider the case Indeed, for using (2.53), this vanishes, so we get
[TABLE]
Remark 2.35
Note that in Lemma 2.34, we only used the trace of the Malcev identity. The non-degeneracy of the Killing form in a semi-simple Malcev algebra also hinges on the property (2.56), same as for semi-simple Lie algebras. This suggests that weaker conditions could be sufficient for these key properties.
3 Loop-valued maps
Let be a smooth, -dimensional manifold and let be a smooth map. The map can be used to define a product on -valued maps from and a corresponding bracket on -valued maps. Indeed, let and be smooth maps, then at each , define
[TABLE]
In particular, the bracket defines the map We also have the corresponding associator and the left-alternating associator map Similarly, define the map
Then, similarly as for maps to Lie groups, we may define the (right) Darboux derivative of which is an -valued -form on given by the pull-back of the Maurer-Cartan form on [45]. In particular, at every ,
[TABLE]
and for any vector
[TABLE]
It is then clear that , being a pullback of , satisfies the loop Maurer-Cartan structural equation (2.19). In particular, for any vectors ,
[TABLE]
We can then calculate the derivatives of these maps (3.1).
Theorem 3.1** ([26, Theorem 3.51])**
*Let be a smooth manifold and suppose s\in C^{\infty}\left(M,\mathbb{L}\right)\and then *
[TABLE]
Suppose now , then
[TABLE]
The -valued map satisfies
[TABLE]
where \mathop{\rm id}\nolimits_{\mathfrak{p}}\is the identity map of and denotes the action of the Lie algebra on .
The Killing form satisfies
[TABLE]
Given and as shown in [26], we have the following expression for
[TABLE]
Moreover, let us consider the evolution equation satisfied by for for some This gives us the following.
Lemma 3.2
Let for then
[TABLE]
and hence
[TABLE]
Moreover, if is compact and given a metric on and an inner product on
[TABLE]
Proof. We will write symbolically
[TABLE]
so
[TABLE]
Solving this ODE, with , we find (3.10). To obtain the estimate, we first have
[TABLE]
but from (2.29), \left|U_{t\xi}^{\left(s\right)}\right|\leq e^{Ct\left|\xi\right|}\and so
[TABLE]
since and hence,
[TABLE]
and thus indeed, we obtain (3.11).
In a very similar fashion we obtain the same results for
Lemma 3.3
Let and then
[TABLE]
and hence
[TABLE]
Moreover, if is compact and given a metric on and an inner product on
[TABLE]
Thus we see that three important quantities , satisfy similar ODEs:
[TABLE]
Suppose we have an affine connection on , then by differentiating the above ODEs, we can obtain expressions for derivatives of , and However, first we have a helpful technical lemma.
Lemma 3.4
Suppose and and is a -linear map on Then, for
[TABLE]
where and are -linear maps on
In particular, given a metric on and a norm on we have the following pointwise bound
[TABLE]
where and f^{\left(k\right)}:\mathbb{L}\longrightarrow\mathbb{R}_{+}\is a continuous function for each .
Proof. Since depends on and , by chain rule we have
[TABLE]
where we have used (3.8). So now we can set
[TABLE]
Thus, we obtain (3.4).
From (3.5), we know that for
[TABLE]
where the alternating associator is a trilinear form on Hence, we have the following estimate
[TABLE]
where is some universal constant. However is smooth in , so is in particular a continuous real-valued function on Hence we can write
[TABLE]
for some positive real-valued function Now, as we have just shown,
[TABLE]
for some -linear form Therefore, for a vector field on ,
[TABLE]
so, we have the following estimate
[TABLE]
where is continuous.
Similarly, we obtain the expression for the second derivative of for a multilinear maps on
[TABLE]
where and are vector fields on . Hence,
[TABLE]
for a continuous function
Note that in these cases for we can symbolically write
[TABLE]
where are multilinear maps that depend on and Proceeding by induction, consider
[TABLE]
where are new multilinear forms. The form is obtained from and adds another Note that since replacing with increases this sum by .
The remaining terms in (3) are obtained from differentiating the derivatives of Symbolically,
[TABLE]
so differentiation of each term decreases the power of by one, and adds another Again, in the sum replacing with increases the sum by
Overall we can then rewrite (3) in the form
[TABLE]
where hence proving the inductive step. The estimate then follows immediately.
Lemma 3.5
We have the following pointwise estimates
[TABLE]
where
[TABLE]
with
In particular, for and we have
[TABLE]
Proof. By differentiating (3.9), we see that the -th covariant derivative of satisfies the following initial value problem
[TABLE]
for In particular,
[TABLE]
and thus the solution of the ODE is
[TABLE]
To estimate , consider
[TABLE]
using (2.29) and (3.12), and moreover,
[TABLE]
From (3.17), and using the fact that is compact, we know that
[TABLE]
where and is a constant that depends only on
To proceed with an induction argument, we now need to complete the base step. First, from (3.11), we know that
[TABLE]
Thus, for ,
[TABLE]
where we used the estimate (3.18). Moreover,
[TABLE]
Hence, for some new overall constant , we have
[TABLE]
Let us also complete the case.
[TABLE]
More explicitly,
[TABLE]
Let
[TABLE]
so that
[TABLE]
Using also (3.18) and (3.19) we have
[TABLE]
Since and are non-decreasing functions of , we can evaluate them at . In particular,
[TABLE]
and similarly for other terms. Overall, we find
[TABLE]
Suppose for each
[TABLE]
where
[TABLE]
where
Therefore,
[TABLE]
where
Now, from (3), we find
[TABLE]
where we have bounded since these functions are non-decreasing. Also, in the first integral, we bounded and in the second integral, we used for some new constant Further, we can bound
[TABLE]
Corollary 3.6
For k>0,\ b_{A\left(t\right)s}\ \satisfies
[TABLE]
where is given by (3.22).
Proof. From (3.17),
[TABLE]
where However, from Lemma 3.5,
[TABLE]
where
[TABLE]
with J_{j}=\left\{\left(i_{1},...,i_{j+1},k_{1},...,k_{j}\right)\in\mathbb{N}_{0}^{2j+1}:\sum_{m=1}^{j+1}mi_{m}+\sum_{m^{\prime}=1}^{j}m^{\prime}k_{m^{\prime}}=j+1\right\}.\So
[TABLE]
More generally, suppose we have -parameter family of -valued maps that satisfies
[TABLE]
where is also an -valued map. We know that
[TABLE]
and in particular,
[TABLE]
Differentiating (3.29), we obtain estimates for higher derivatives of
Lemma 3.7
Suppose is a -parameter family of -valued maps that satisfies (3.29). Then,
[TABLE]
with and for , is given by (3.22).
Proof. Differentiating (3.29), for , we get
[TABLE]
and hence,
[TABLE]
Let , so that
[TABLE]
Suppose for all ,
[TABLE]
where is non-decreasing. Then,
[TABLE]
For
[TABLE]
For ,
[TABLE]
Thus,
[TABLE]
Therefore, for ,
[TABLE]
Setting it is then easy to see that
[TABLE]
where
[TABLE]
and thus
[TABLE]
Therefore,
[TABLE]
with
We will need to be able to define loop-valued maps with Sobolev regularity. First, let us recall Sobolev spaces of functions between manifolds.
Lemma 3.8
Suppose is a compact -dimensional manifold and suppose is an -dimensional manifold. Let be a non-negative integer and such that Let be a smooth embedding (by Whitney Embedding Theorem) and suppose is an atlas for Suppose is a continuous map. Then, the following are equivalent:
** 2. 2.
** 3. 3.
* for any chart * 4. 4.
* in case when if a compact Lie group, with Lie algebra *\mathfrak{g}\and is the Maurer-Cartan form on
In particular, conditions (2) and (3) are independent of the choice of the embedding and the atlas respectively.
Note that the condition is needed in Lemma 3.8 due to the Sobolev embedding for We will prove a characterization of loop-valued -maps in terms of the loop Maurer-Cartan form that is similar to item (4) in Lemma 3.8.
Lemma 3.9
Suppose is a compact -dimensional manifold and suppose is a smooth loop of dimension with tangent algebra and -valued Maurer-Cartan form . Let be a non-negative integer and such that Suppose is a continuous map. Then, if and only if
Proof. Suppose By Lemma 3.8, if is an atlas for then for each chart Now, is an open cover of but using compactness of let be an finite subcover, and suppose is a smooth partition of unity subordinate to this subcover. Then, we can write
[TABLE]
For each , On the other hand is a smooth function, and hence composition with it is a continuous map (using [48, Lemma B.8]). Overall, we see that each term of this finite sum is bounded in the norm, and thus
Conversely, suppose now We will use item (2) in Lemma 3.8 to show that This adapts the proof of [48, Lemma B.5]. Let be a smooth embedding, so that is continuous. In particular, Now, let and consider
[TABLE]
where, for each we have the linear map , and the map is a smooth map from to Thus, we can write
[TABLE]
with being bounded in the operator norm, since is continuous. Hence, there exists a constant such that
[TABLE]
This shows that To show further that similar estimates are obtained by considering higher derivatives.
Theorem 3.10
Let be a compact Riemannian manifold. Suppose Let and and suppose Then,
[TABLE]
where
Similarly, if where is -parameter family of -valued maps that satisfies (3.29). Then,
[TABLE]
Proof. From Lemma 3.5, for each we have the pointwise estimate
[TABLE]
where
[TABLE]
with Thus,
[TABLE]
Now, from Lemma A.1, if then
[TABLE]
We can apply this to (3.33), with the weight for each or factor. Then,
[TABLE]
Since for each and, we obtain
[TABLE]
The right hand-side of (3.35) is thus a polynomial in and and from the definition of the degree of this polynomial is and the lowest order terms are and Hence, we can write
[TABLE]
where In particular,
[TABLE]
Now, since
[TABLE]
which gives us (3.32).
Now from Lemma 3.7, for each ,
[TABLE]
Hence,
[TABLE]
where
[TABLE]
Hence, from Lemma A.1,
[TABLE]
similarly as before. Hence, we conclude that
[TABLE]
Corollary 3.11
Suppose and , where Then, if and only if .
Proof. The map given by is smooth, hence the composition with is a continuous map from to If then since and hence from Lemma 3.9, and thus
Conversely, if then Since right division is a smooth map, and we conclude that
4 Gauge theory
Let be a smooth, finite-dimensional manifold with a -principal bundle
Definition 4.1
Let be an equivariant map. In particular, given the equivalence class defines a section of the associated bundle , where is the equivalence class with respect to the action of
[TABLE]
We will refer to as the defining map or defining section.
We will define several associated bundles related to As it is well-known, sections of associated bundles are equivalent to equivariant maps. With this in mind, we also give properties of equivariant maps that correspond to sections of these bundles. Let and, as before, denote by the partial action of .
[TABLE]
Given equivariant maps , define an equivariant product using , given for any by
[TABLE]
Due to Lemma 2.7, the corresponding map is equivariant, and hence induces a fiberwise product on sections of . Analogously, we define fiberwise quotients of sections of Similarly, we define an equivariant bracket and the equivariant map . Similarly, the Killing form is then also equivariant.
Suppose the principal -bundle has a principal Ehresmann connection given by the decomposition and the corresponding vertical -valued connection -form Given an equivariant map , define
[TABLE]
This is then a horizontal map since it vanishes on any vertical vectors. The map is moreover still equivariant, and hence induces a covariant derivative on sections of the associated bundle . If is a vector space, then this reduces to the usual definition of the exterior covariant derivative of a vector bundle-valued function and is a vector-bundle-valued -form.
Following [26], let us define the torsion of the defining map with respect to the connection .
Definition 4.2
The torsion of the defining map with respect to is a horizontal -valued -form on given by , where is Maurer-Cartan form of . Equivalently, at , we have
[TABLE]
Thus, is the horizontal component of We also easily see that it is -equivariant. Thus, is a basic (i.e. horizontal and equivariant) -valued -form on , and thus defines a -form on with values in the associated vector bundle
Recall that the curvature of the connection on is given by
[TABLE]
where wedge product is implied. Given the defining map , define to be the projection of the curvature to with respect to , such that for any
[TABLE]
Theorem 4.3** ([26, Theorem 4.19])**
* and satisfy the following structure equation*
[TABLE]
where a wedge product between the -forms is implied.
In the case of an octonion bundle over a -dimensional manifold, this relationship between the torsion and a curvature component has been shown in [24].
As discussed earlier, equivariant horizontal forms on give rise to sections of corresponding associated bundles over the base manifold So let us now switch perspective, and work in terms of sections of bundles. In particular, now we will consider to be a smooth section of the bundle so that we will say and will refer to it as the defining section. Similarly, we can also consider sections which admit the partial action of The product on elements of and then carries over to sections of bundles, so that we have a product
The connection on then induces connections on the associated bundles and correspondingly, covariant derivatives on sections of these bundles. The torsion , as defined earlier, was a horizontal and equivariant -form on with values in so it uniquely corresponds to a -form on with values in the bundle i.e., now we will consider
In standard gauge theory, the key object is the connection, however, in the non-associative theory, in addition to the connection we also the defining section We then make the following definition.
Definition 4.4
*A *non-associative gauge theory is defined by the following objects:
A smooth loop with a finite-dimensional pseudoautomorphism Lie group and tangent algebra at identity. 2. 2.
A smooth manifold with a principal -bundle and associated bundles with fibers , and respectively 3. 3.
*A *configuration , where is a defining section and is a connection on Each configuration carries torsion
As we see, the key components are the loop with its pseudoautomorphism group, and the corresponding principal bundle Up to a choice of the configuration everything else follows uniquely. In particular, the associated bundles are unique because particular actions of are used to define them.
The group acts via standard gauge transformations on and also acts on the section These actions are related in the following way, as shown in [26],
[TABLE]
where is a section of so is fiberwise in However, we will define loop gauge transformations in the following way.
Definition 4.5
*A *loop gauge transformation is a transformation of the defining section by right multiplication by a section such that and hence
With respect to a loop gauge transformation, the torsion and curvature transform in the following way.
Lemma 4.6** ([26, Theorem 4.28])**
*Suppose and . Then, *
[TABLE]
where denotes the infinitesimal action of on
Let us fix the connection , and suppose we have a path Then from Lemma 3.2, just by taking the horizontal projection, we immediately obtain that the corresponding one-parameter family of torsions satisfy a similar ODE.
Lemma 4.7
Suppose is the torsion with respect to a defining section and a connection Suppose then
[TABLE]
Using (4.10a) and (3.10), given we get
[TABLE]
Now suppose the base manifold is compact and Riemannian with a metric and also that the loop admits a non-degenerate Killing form on Then, define the functional
[TABLE]
where is a combination of the metric on and the Killing form on sections of Critical points then become analogues of the Coulomb gauge condition in gauge theory [10, 24, 25, 27, 26, 41]
Theorem 4.8
Suppose is a semisimple Moufang loop, then the critical points of the functional (4.13) with respect to deformations of the defining section are those for which
[TABLE]
Proof. From Lemma 2.34, we know that for a Moufang loop, is actually independent of Moreover, it is invariant under Let us consider deformations of . The semisimple condition implies is non-degenerate. Consider a path where Then,
[TABLE]
where we have used (4.11). Note that
[TABLE]
Hence,
[TABLE]
Thus critical points of with respect to deformations of satisfy
[TABLE]
Remark 4.9
In Theorem 4.8, we use the fact that the tangent algebra of a Moufang loop is a Malcev algebra, i.e. is alternative and satisfies the additional identity 2.50. Moreover, the semisimple condition implies that the Killing form is non-degenerate. As noted in Remark 2.35, the full Malcev algebra condition is likely to be too strong, and a weaker assumption may be sufficient to obtain these key properties and in fact obtain as the equation for critical points. On the other hand, other techniques, such as introducing a different metric (such as the Killing-Ricci form on Lie triple systems [35]) or introducing modified connections may produce similar results in other settings.
To prove existence of transformations of that lead to we will adapt the procedures from [17], and in particular will apply the Banach Space Implicit Function Theorem (Theorem A.2) . The relevant Banach spaces for us will be spaces of sections with appropriate regularity. The previously used notations and will always denote smooth sections and smooth bundle-valued forms, respectively. Given a smooth defining section and a smooth connection for any and denote by the Sobolev space of sections of with the norm given by
[TABLE]
for and
[TABLE]
Similarly we will denote as
By Definition 4.2, the torsion of is just the horizontal component of so we can immediately adapt the estimates from Section 3, we obtain the following estimates for torsion.
Lemma 4.10
Suppose is a smooth compact loop with tangent algebra and pseudoautomorphism group Let be a closed, smooth Riemannian manifold of dimension and let be a -principal bundle over with and let be the associated vector bundle to with fibers isomorphic to . Let be a smooth connection on \mathcal{P}\and let be a smooth defining section. Also, suppose is a non-negative integer and such that Let and suppose Then,
[TABLE]
where
Lemma 4.11
Now suppose that . Given other hypotheses the same as in Lemma 4.10, if and given such that
[TABLE]
then in fact is smooth.
Proof. Using the Whitney Embedding Theorem, suppose is smoothly embedded in some We can define a loop product and quotient on the image of the embedding. Hence the bundles and can be regarded as subbundles of a vector bundle over . In particular, since is smooth and is also a smooth map, we find that since and then Using (4.10a), we have
[TABLE]
where is the inner product on Thus, we can rewrite (4.17) as
[TABLE]
Now since , and is smooth, for any ,
[TABLE]
Also, Since we see that for any By the Sobolev Embedding Theorem, this shows that
[TABLE]
Thus, (4.21) shows that is bounded. By elliptic regularity, this implies that In particular, if then and thus Bootstrapping the elliptic regularity argument we then obtain the smoothness of . In particular, note that this does not depend on the choice of embedding.
Remark 4.12
The proof of Lemma 4.11 is an adaptation of the proof of [11, Proposition 2.3.4], where in particular the regularity of a gauge transformation to the Coulomb gauge was proved. In that case, and so the conditions and were equivalent since is an integer. More generally, the condition that is needed for smoothness is somewhat stronger than the one needed for continuity.
Then, we have the main theorem.
Theorem 4.13
Suppose is a smooth compact loop with tangent algebra and pseudoautomorphism group Let be a closed, smooth Riemannian manifold of dimension and let be a -principal bundle over with and let be the associated vector bundle to with fibers isomorphic to . Let be a smooth connection on Also, suppose is a non-negative integer and such that Then, there exist constants and such that if is a smooth defining section for which
[TABLE]
then there exists a section such that
[TABLE]
and
[TABLE]
If moreover, then is smooth.
Proof. Consider W_{\left(s,\omega\right)}^{k,r}\left(\mathcal{A}\right)\ \and For now, let us drop the subscript in function spaces. Since and satisfy by the Sobolev Embedding Theorem, embeds in . Define the function
[TABLE]
by
[TABLE]
The assumption that together with the smoothness of and the derivative maps, leads to the conclusion that is a smooth map of Banach spaces. Note that using (4), we can write
[TABLE]
Using the connection let us define the bundle-valued Hodge Laplacian
[TABLE]
On [math]-forms it reduces to It extends as an operator of Sobolev spaces as
[TABLE]
and by standard elliptic theory is Fredholm with index [math] and a closed range
[TABLE]
where denotes the -orthogonal complement.
To be able to apply the Implicit Function Theorem (Theorem A.2), in (4.23), let us constrain and we also see that . This can be seen immediately. Suppose for some and , then
[TABLE]
since on a compact manifold, if and only if Hence the image of is contained in which we’ll denote for brevity by , and so in fact,
[TABLE]
Now let us consider the differential of at in the direction
[TABLE]
since In particular, the partial derivative in the second direction is given by
[TABLE]
In Theorem A.2, let
[TABLE]
Then, the map is an isomorphism, and we define
[TABLE]
Let
[TABLE]
where is small enough such that
[TABLE]
Also define the constant as
[TABLE]
Then, by the conclusion of Theorem A.2, there exist an open set given by
[TABLE]
where , and a unique smooth map
[TABLE]
such that and
[TABLE]
In particular, for any with there exists a section with for which
[TABLE]
and
[TABLE]
Since is smooth and is a smooth map, this shows that
Now suppose and are such that then setting gives for which
[TABLE]
where From (4.16), we have
[TABLE]
where Now, using the estimate for in terms of , we get
[TABLE]
and since , Overall, combining the constants into a single constant we obtain
[TABLE]
and hence (4.22).
If then by Lemma 4.11, we see that is smooth.
5 -manifolds
The general picture considered in the previous sections can now be specialized to the case of manifolds with -structure. The 14-dimensional group is the smallest of the five exceptional Lie groups and is defined as the automorphism group of the loop of unit octonions . Let be a compact -dimensional manifold with vanishing first and second Stiefel-Whitney classes, so that the manifold is both orientable and admit a spin structure. Then, as it is well-known [18, 19], admits a -structure, that is a reduction of the structure group of the frame bundle to Since is a subgroup of , the -structure can be extended uniquely to an -structure, and thus defines a Riemannian metric and orientation on Equivalently, given a Riemannian metric , an -structure on lifts to a spin structure, which is a principal -structure. Given the spin structure, we can then construct a spinor bundle which will necessarily admit a nowhere vanishing section. Any such spinor section will then reduce the spin structure to a -structure on . Indeed, any unit spinor will hence define a -structure that is compatible with the metric
Recall that has three low-dimensional real irreducible representations: -dimensional representation -dimensional “vector” representation and the -dimensional “spinor” representation [2]. The representations and descend to representations of Moreover, the Clifford product gives the map
[TABLE]
Setting we can then extend this map to This product is non-degenerate, and fixing allows to identify with Both spaces are then identified with the octonions and the product then gives rise to octonion multiplication. The element is identified with The stabilizer of under the action of is isomorphic to Note that here then corresponds to the irreducible “vector” representation of while the two copies of are identified with the irreducible -dimensional chiral spinor representations of and thus gives the normed triality of [2]. Since the map preserves norms, it restricts to unit spheres in and which we will denote by and respectively, because they correspond to and in the general theory in Section 2. Clearly, is a compact smooth loop.The tangent space at to is then isomorphic to We thus have the following identification of objects.
[TABLE]
Therefore, on the manifold as above, the spin structure corresponds to a principal -bundle in the general theory, the unit spinor bundle corresponds to the bundle and the unit subbundle U\mathbb{O}M\of corresponds to This is precisely the octonion bundle introduced in [24]. Hence, we have the following dictionary relating objects in the general loop bundle theory and -geometry.
[TABLE]
-structures can also be described using differential forms since is alternatively defined as the subgroup of that preserves a particular -form [30].
Definition 5.1
Let be the standard basis for , and denote by . Then define to be the -form on given by
[TABLE]
Then is defined as the subgroup of that preserves .
It turns out that there is a - correspondence between -structures on a -manifold and smooth -forms for which the -form-valued bilinear form as defined by (5.3) is positive definite (for more details, see [6] and the arXiv version of [28]).
[TABLE]
Here the symbol denotes contraction of a vector with the differential form:
A smooth -form is said to be *positive *if is the tensor product of a positive-definite bilinear form and a nowhere-vanishing -form. In this case, it defines a unique Riemannian metric and volume form such that for vectors and , the following holds
[TABLE]
An equivalent way of defining a positive -form , is to say that at every point, is in the -orbit of . It can be easily checked that the metric (5.4) for is in fact precisely the standard Euclidean metric on . Therefore, every that is in the -orbit of has an associated Riemannian metric that is in the -orbit of The only difference is that the stabilizer of (along with orientation) in this orbit is the group , whereas the stabilizer of is . This shows that positive -forms forms that correspond to the same metric, i.e., are isometric, are parametrized by . Therefore, on a Riemannian manifold, metric-compatible -structures are parametrized by sections of an -bundle, or alternatively, by sections of an -bundle, with antipodal points identified. The precise parametrization of isometric -structures is given in Theorem 5.2.
Theorem 5.2** ([7])**
Let be a -dimensional smooth manifold. Suppose is a positive -form on with associated Riemannian metric . Then, any positive -form for which is also the associated metric, is given by the following expression:
[TABLE]
where is a pair with a scalar function on and a vector field such that
[TABLE]
The pair can in fact be also interpreted as a unit octonion section, where is the real part, and is the imaginary part. The relationship between octonion bundles and -structures was developed in detail in [24]. In particular, sections of a unit octonion bundle over parametrize -structures that are associated to the same metric.
Definition 5.3
*The *octonion bundle on is the rank real vector bundle given by
[TABLE]
where \Lambda^{0}\cong M\times\mathbb{R}\is a trivial line bundle. At each point ,
The definition (5.7) gives a natural decomposition of octonions on into real and imaginary parts. We may write or A=\left(\begin{array}[]{c}\mathop{\rm Re}A\\ \mathop{\rm Im}A\end{array}\right). Since is defined as a tensor bundle, the Riemannian metric on induces a metric on Let Then,
[TABLE]
The metric allows to define the subbundle of octonions of unit norm and allows allows to define a vector cross product on
Definition 5.4
*Given the -structure on we define a *vector cross product with respect to on Let and be two vector fields, then define
[TABLE]
for any vector field [20, 31].
Using the inner product and the cross product, we can now define the *octonion product *on .
Definition 5.5
*Let Suppose and . Given the vector cross product (5.9) on we define the *octonion product with respect to as follows:
[TABLE]
If there is no ambiguity as to which -structure is being used to define the octonion product, we will simply write to denote it. In particular,
Given a -structure with an associated metric , we may use the metric to define the Levi-Civita connection . The *intrinsic torsion *of a -structure is then defined by . Following [22, 32], we can write
[TABLE]
where is the full torsion tensor. Similarly, we can also write
[TABLE]
We can also invert (5.11) to get an explicit expression for
[TABLE]
This -tensor fully defines [22].
Remark 5.6
The torsion tensor as defined here is actually corresponds to in [24], in [22] and in [32]. Even though this requires extra care when translating various results, it will turn out to be more convenient.
Given a unit norm spinor section a -structure -form is defined in the following way:
[TABLE]
where denotes Clifford multiplication, are arbitrary vector fields and is the inner product on the spinor bundle. The Levi-Civita connection lifts to the spinor bundle giving the spinorial covariant derivative Then, the torsion of is given by [1, Definition 4.2 and Lemma 4.3]
[TABLE]
Note that in [1], the torsion endomorphism is denoted by
Comparing with Definition 4.2 and noting that the unit spinor bundle corresponds to the loop bundle we see that the torsion of the -structure precisely corresponds to the torsion of the section with respect to the Levi-Civita connection Similarly, given a unit octonion section is again a unit spinor which defines a -structure Considering both and as octonions in and respectively, this is just octonion multiplication and Therefore, all isometric -structures are given by for some unit octonion section The curvature component corresponds to the a particular component of the Riemann curvature tensor. These relationships are explored in detail in [24]. Thus we can reformulate Theorem 4.13 for -structures.
Theorem 5.7
Suppose is a closed -dimensional manifold with a smooth -structure with torsion with respect to the Levi-Civita connection Also, suppose is a positive integer and is a positive real number such that Then, there exist constants and such that if satisfies
[TABLE]
then there exists a smooth section such that
[TABLE]
and
[TABLE]
Remark 5.8
If we choose to work with Hilbert spaces, then for a smooth section we need so the condition on is to be sufficiently small in the -norm.
Appendix A Appendix
Lemma A.1
Let be positive integers and , for a positive real number and let be real-valued functions on a compact -dimensional Riemannian manifold . Also, suppose are non-negative integers and are positive integers such that then
[TABLE]
Proof. Let Then suppose for all for which so that and hence Thus, from Hölder’s inequality, we have
[TABLE]
Now note that using the definition of , and hence
[TABLE]
Since by assumption, , we obtain
[TABLE]
Using a version of the Sobolev Embedding Theorem, this shows that indeed,
[TABLE]
and (A.1) follows.
Theorem A.2** (Banach space uantitative implicit function theorem[17, Theorem F.1])**
Let be an integer or and let be real Banach spaces. Suppose and are open neighborhoods of points and and is a map such that and the partial derivative of at with respect to the second variable, is an isomorphism of Banach spaces. Define
[TABLE]
Let be small enough such that the open ball and and assume
[TABLE]
Then there exist a constant and unique map such that and
[TABLE]
Acknowledgement A.3
This work was supported by the National Science Foundation grant DMS-1811754.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] I. Agricola, S. G. Chiossi, T. Friedrich, and J. Höll. Spinorial description of SU ( 3 ) SU 3 {\rm SU}(3) - and G 2 subscript G 2 {\rm G}_{2} -manifolds. J. Geom. Phys. , 98:535–555, 2015, 1411.5663 .
- 2[2] J. Baez. The Octonions. Bull. Amer. Math. Soc. (N.S.) , 39:145–205, 2002. doi:10.1090/S 0273-0979-01-00934-X . · doi ↗
- 3[3] L. Bagaglini. The Energy Functional of G 2 subscript 𝐺 2 G_{2} -Structures compatible with a Background Metric. J. Geom. Anal. , 2019. doi:10.1007/s 12220-019-00264-6 . · doi ↗
- 4[4] L. Bagaglini, M. Fernández, and A. Fino. Laplacian coflow on the 7-dimensional Heisenberg group. 2017, 1704.00295 .
- 5[5] L. Bagaglini and A. Fino. The laplacian coflow on almost-abelian Lie groups. 2017, 1711.03751 .
- 6[6] R. L. Bryant. Metrics with exceptional holonomy. Ann. of Math. (2) , 126(3):525–576, 1987. doi:10.2307/1971360 . · doi ↗
- 7[7] R. L. Bryant. Some remarks on G 2 subscript 𝐺 2 {G}_{2} -structures. In Proceedings of Gökova Geometry-Topology Conference 2005 , pages 75–109. Gökova Geometry/Topology Conference (GGT), Gökova, 2006, math/0305124 .
- 8[8] R. L. Bryant and F. Xu. Laplacian Flow for Closed G 2 subscript 𝐺 2 G_{2} -Structures: Short Time Behavior. 2011, 1101.2004 .