# An efficient application of Bayesian optimization to an industrial MDO
framework for aircraft design

Rémy Priem111Ph.D. Candidate, Information Processing and Systems Department &
Complexes Systems Engineering Department<EMAIL_ADDRESS>Student AIAA
Member ONERA, DTIS, Université de Toulouse, Toulouse, France  Hugo Gagnon
Bombardier Aviation, Dorval, Québec, H9P 1A2, Canada  222Engineering
Professional, Advanced Design<EMAIL_ADDRESS>Ian Chittick
Bombardier Aviation, Dorval, Québec, H9P 1A2, Canada  333Engineering
Specialist, Advanced Aerodynamics<EMAIL_ADDRESS>and
Stéphane Dufresne444Senior Engineering Specialist, Specialized Aircraft,
<EMAIL_ADDRESS>Bombardier Aviation, Dorval, Québec,
H9P 1A2, Canada  Youssef Diouane555Associate Professor, Complexes Systems
Engineering Department<EMAIL_ADDRESS>ISAE-SUPAERO,
Université de Toulouse, Toulouse, 31055 Cedex 4, France  Nathalie
Bartoli666Senior researcher, Information Processing and Systems Department,
<EMAIL_ADDRESS>AIAA Member. ONERA, DTIS, Université de Toulouse,
Toulouse, France

###### Abstract

The multi-level, multi-disciplinary and multi-fidelity optimization framework
developed at Bombardier Aviation has shown great results to explore efficient
and competitive aircraft configurations. This optimization framework has been
developed within the Isight software, the latter offers a set of ready-to-use
optimizers. Unfortunately, the computational effort required by the Isight
optimizers can be prohibitive with respect to the requirements of an
industrial context. In this paper, a constrained Bayesian optimization
optimizer, namely the super efficient global optimization with mixture of
experts, is used to reduce the optimization computational effort. The obtained
results showed significant improvements compared to two of the popular Isight
optimizers. The capabilities of the tested constrained Bayesian optimization
solver are demonstrated on Bombardier research aircraft configuration study
cases.

##  1 Nomenclature

$d$ = | the number of design variables
---|---
$m$ = | the number of inequality constraints
$\Omega$ = | the domain of the design variables, i.e., $\Omega\subset\mathbb{R}^{d}$
$f$ = | the objective function, i.e., $f:\mathbb{R}^{d}\mapsto\mathbb{R}$
$\bm{g}$ = | the constraints function, i.e., $\bm{g}:\mathbb{R}^{d}\mapsto\mathbb{R}^{m}$
$\mu_{s}^{(l)}$ = | the prediction of the Gaussian process of a given function $s:\mathbb{R}^{d}\mapsto\mathbb{R}$ build with $l$ samples
$\sigma_{s}^{(l)}$ = | the uncertainty of the Gaussian process of a given function $s:\mathbb{R}^{d}\mapsto\mathbb{R}$ build with $l$ samples
$\alpha_{f}^{(l)}$ = | the acquisition function of the objective function $f:\mathbb{R}^{d}\mapsto\mathbb{R}$
$\alpha_{\bm{g}}^{(l)}$ = | the feasibility criterion of the constraint functions $\bm{g}:\mathbb{R}^{d}\mapsto\mathbb{R}^{m}$
$\Omega^{(l)}_{\bm{g}}$ = | the approximated feasible domain defined by the feasibility criterion $\alpha_{\bm{g}}^{(l)}:\mathbb{R}^{d}\mapsto\mathbb{R}^{m}$
$C_{L_{max}}$ = | the low speed maximum lift coefficient

##  2 Introduction

Bombardier aviation multi-disciplinary optimization (MDO) framework has been
continuously evolving over the past decade to ensure that future development
programs result in highly efficient and competitive aircraft [1]. In order to
achieve this vision, Bombardier has developed a multi-level, multi-fidelity
MDO framework to mature an aircraft configuration from early design stage to
detailed design [2].

The first element of this framework is called Conceptual MDO (CMDO). The main
objective of the CMDO framework is to enable the design team to explore the
design space assuming a set of realistic constraints established from previous
aircraft development programs and marketing requirements. The main deliverable
of this process is an aircraft configuration, which includes the initial
sizing of the major structural components and systems. This conceptual
aircraft configuration then generates a set of design requirements that will
be validated using high fidelity frameworks, namely the Preliminary MDO (PMDO)
and the Detailed MDO (DMDO). The second element corresponds to the PMDO
framework analyzing the aerodynamic and structural characteristics of the
wing. Its main objectives are to refine the wing sizing and to validate the
wing related conceptual requirements, such as drag polar and wing weight. The
two deliverables of this process are (1) a detailed wing geometry, external
and internal, and (2) an initial definition of the wing structure.

The context of this research is about the design process of a new aircraft
configuration in an industrial setting. The main inputs to initiate the design
of a new aircraft platform, as well as inputs to the MDO framework, are a set
of requirements from the marketing team (e.g. cruise mach number, design
range, field performance and critical airport operations). For each of these
sets of requirements, or constraints to the MDO framework, there needs to be a
convergence between the CMDO and PMDO framework to ensure a robust initial
design point. Since there could be multiple sets of requirements to be
analyzed, the design team becomes rapidly limited by the time and number of
CPU available.

The CMDO framework is built on a large set of low fidelity modules
representing the aircraft disciplines. All these modules include their own set
of design variables, which increases the number of dimensions of the
optimization problem. Consequently, it makes the thorough exploration of the
design space expensive in terms of function calls and CPU time, in the order
of days. Similarly, the PMDO framework also includes a large number of design
variables required to define the wing geometry and the aero-structural models
are time consuming to converge, in the order of weeks. The motivation of this
research is then to apply Bayesian Optimization (BO) methods [3] to reduce the
number of functions calls and accelerate the convergence time for both MDO
frameworks. To do so, the SEGOMOE python tool-box is investigate against
historical optimizers used at Bombardier aviation. The main objective of this
paper is thus to demonstrate that all the optimizers of the tool-box has good
convergence properties and performs well on an industrial test case.

This paper is organized into four sections: (1) detailed description of the
models used in the CMDO and PMDO frameworks, (2) introduction to the
Bombardier Research Aircraft Configuration (BRAC) case study, (3) definition
of the optimizer selected for the analysis of the case study and finally (4)
the analysis of the results with the emphasis on the convergence in terms of
CPU time and number of evaluations.

##  3 An industrial multi-level, multi-fidelity and multi-disciplinary
optimization process

The MDO framework developed at Bombardier Aviation consists of three levels,
corresponding to the three traditional stages of aircraft design: conceptual,
preliminary, and detailed. Only a brief description of the first two levels is
given here. Refer to [2] for more information.

###  3.1 Conceptual multi-disciplinary optimization framework

Conceptual MDO (CMDO) is the first level of the multi-level MDO framework. It
accounts for all major aircraft-level disciplines, such as aerodynamics,
structures, systems, weight and balance, performance, stability and control,
and economics. Due to the high number of disciplines, the level of fidelity is
low and varies from empirical to simple physics-based methods. For example, in
the aerodynamics module, an in-house method is used to predict the low-speed
maximum lift coefficient $C_{L_{max}}$ while a vortex-lattice method coupled
with a 2D computational fluid dynamics code is used to compute high-speed trim
drag. Similarly, other disciplines can combine mixed-fidelity analyses
depending on the desired accuracy and turnaround times. Example usages of CMDO
are design space exploration, optimization of marketing requirements and
objectives, validation of business cases, assessment of technology insertion,
down selection of promising configurations, and definition of performance
targets [2].

Typical design variables of CMDO include the wing area and planform as well as
the engine scaling factor in the case of a fixed engine architecture. It is
also possible to vary the thickness-to-chord ratios of the wing and chordwise
positions of the spars at various spanwise locations. Typical constraints
include performance parameters such as the balanced field length (BFL),
approach speed ($V_{\textsubscript{ref}}$), initial cruise altitude (ICA),
range for different missions, and the ability to take-off and land at critical
airports. Systems and geometric considerations such as landing gear
integration and wingtip chord can also be included. Typical objectives include
maximum take-off weight (MTOW), cost, and climate impact, or any combination
thereof. For this study only MTOW is used since it combines the effects of
cost and climate impact through fuel burn.

The CMDO framework is implemented as a workflow in Isight [4]. Isight is a
process-integration, design-optimization software developed and distributed by
Dassault Systemes. It comes with its own set of optimizers.

The output of CMDO is an input to Preliminary MDO (PMDO), discussed next.

###  3.2 Preliminary multi-disciplinary optimization framework

The preliminary MDO (PMDO) is the second level of the multi-level MDO
framework. Compared to CMDO, the scope of the design space is narrowed, while
the fidelity of the underlying analysis tools is increased. PMDO focuses
primarily on the coupling and trade-off between aerodynamics and structures.
The objective is to obtain wing outer mold lines (OML) that are
aerodynamically efficient in both high-speed and low-speed flight, as well as
to provide adequate performance at critical off-design conditions. At the same
time, the OML must be structurally viable, yield low weight, and satisfy space
allocation requirements.

Piperni et al. [2] previously described the PMDO process in detail. Although
many of the components remain the same, the framework has continued to evolve,
and a complete description falls outside the scope of this paper. In brief,
the aircraft surfaces are updated in a parametric CATIA V5 model. Bombardier’s
in-house structured mesh generator, MBGRID [5], and Navier-Stokes flow solver,
FANSC [6], are used to compute the flow solutions at several high-speed design
and static load case conditions. The external loads are transferred to the
wing structural mesh generated by AWSOM [7], Bombardier’s in-house wing sizing
tool, which minimizes the primary wing-box weight while respecting various
margin of safety constraints. The low-speed characteristics are assessed using
the semi-empirical Valarezo criteria [8] in conjunction with VSAERO, a 3D
panel code [9]. Finally, the high-speed drag and structural weight are related
through a fractional change equation, as described in [10].

The architecture of the PMDO framework has changed significantly with the
maturation of BOOST, an adjoint-based optimization framework used at
Bombardier [11, 12]. The adjoint method is an efficient tool for computing the
gradients needed for aerodynamic shape optimization, which involves long
running simulation times and large numbers of design variables. In order to
integrate the aerodynamic adjoint in a multidisciplinary environment, a
hybrid-adjoint MDO framework was developed [13], as depicted in Figure 1. In
the first stage, an aero-structural optimization is performed in Isight by
varying the wing plan-form, wing twist, and maximum thickness distribution (by
scaling the wing profiles). This is followed by an aero-shape optimization in
BOOST, where the drag is minimized by varying the airfoil shapes at fixed
thickness to maximum chord ratio and wing span-load. This process is repeated
in a sequential manner until convergence is achieved. This paper will examine
the top-level optimizer used to drive the aero-structural optimization,
highlighted in bold in Figure 1, which is the current bottleneck in the PMDO
process.

Figure 1: Simplified schematic of the hybrid-adjoint PMDO workflow.

###  3.3 Bombardier research aircraft configuration

The test case considered for this work is representative of a small business
aircraft, and is referred to as the Bombardier research aircraft configuration
(BRAC). It is based on an early design of the Challenger 300 platform, and was
developed to be shared with academic partners and for publication purposes. An
image of the baseline BRAC geometry is shown in Figure 2.

Figure 2: The baseline BRAC configuration.

##  4 Bombardier research aircraft configuration optimization case

In this section, the two level optimization process is introduced. First, we
present the optimization of the BRAC model into the CMDO framework. Then, the
previously obtained result is used in the PMDO of the BRAC model.

###  4.1 BRAC CMDO problem

The BRAC CMDO problem is based on a modified version of the initial Challenger
300 marketing requirements and objectives. It is thus representative of a
typical industrial problem, yet here the design space is purposefully relaxed
to test the global optimization capabilities of each optimizer. For
confidentiality reasons the exact bounds on the design variables and
constraints are not given here; instead, in the following sections their value
will be normalized between $0$ and $1$.

Formally, the CMDO problem is:

$\min\limits_{\bm{x}\in[0,1]^{d}}\left\\{\text{MTOW}(\bm{x})~{}\text{s.t.}~{}\bm{g}(\bm{x})\leq\bm{B}\right\\},$
(1)

where $\bm{x}$ is the design variable vector of $[0,1]^{d}$ described in Table
2, $d$ the number of design variables, $\bm{g}$ is the inequality constraints
described in Table 3 and $\bm{B}$ the associated bounds. Due to
confidentiality reason $\bm{B}$ is not given. Finally for the CMDO
optimization problem (1), we consider here 12 design variables and 8
constraints.

Also note that the long-range cruise (i.e. nominal) mission range is satisfied
implicitly by the workflow. The baseline BRAC (see Figure 2) as defined in the
CMDO workflow is unfeasible. This is inconsequential, since that initial
design is not used as a starting point by any of the optimizers. Section 5
explains which starting point each optimizer uses instead. For industrial
reasons, the CMDO of BRAC cannot take more than $8$ hours.

Table 2: Design optimization space definition.

Variable Variable Name Variable Variable Variable Name Variable index symbol
index symbol $\bm{x}[0]$ Engine scaling factor DV_0 $\bm{x}[4-5]$ Wing rear
spar chord-wise locations DV_4-5 $\bm{x}[1]$ Wing aspect ratio DV_1
$\bm{x}[6]$ Wing sweep DV_6 $\bm{x}[2]$ Wing area DV_2 $\bm{x}[7]$ Wing taper
DV_7 $\bm{x}[3]$ Wing inboard trailing edge sweep DV_3 $\bm{x}[8-11]$ Wing
maximum thickness-to-chord ratios DV_8-11

Table 3: Constraints definition. Constraint | Constraint Name | Bounds | Constraint | Constraint Name | Bounds
---|---|---|---|---|---
index | index | index | index
$\bm{g}[0]$ | BFL | $\bm{B}[0]$ | $\bm{g}[4]$ | Climb performance | $\bm{B}[4]$
$\bm{g}[1]$ | ICA | $\bm{B}[1]$ | $\bm{g}[5]$ | High-speed mission range | $\bm{B}[5]$
$\bm{g}[2]$ | Vref | $\bm{B}[2]$ | $\bm{g}[6]$ | Landing gear spacing | $\bm{B}[6]$
$\bm{g}[3]$ | Excess fuel | $\bm{B}[3]$ | $\bm{g}[7]$ | Wingtip chord | $\bm{B}[7]$

###  4.2 BRAC PMDO problem

As mentioned earlier, the PMDO portion of this study will consider the aero-
structural workflow, shown on the left half of Figure 1. Similar to CMDO, the
objective is to minimize MTOW, as in problem (1). The wing and winglet design
variables considered are listed in Table 4. The wing area and engine size are
kept constant at the PMDO level, and the wing area of the CMDO optimum is used
as an input to the plan-form generator. Similarly, the spar locations are
fixed using the CMDO result, since the high-lift systems integration is not
considered. The constraints imposed during PMDO are listed in Table 5. The
volume availability factor is a measure of the fuel volume available in the
wing compared to the volume required to meet the range requirements. The
maximum wingtip twist deformation is a proxy for a dynamics constraint, and is
enforced to prevent an overly flexible wing. Again, for confidentiality
reasons, the variable and constraint bounds are omitted. Finally for the PMDO
optimization problem (1), we consider here 19 design variables and 5
constraints which is representative of a typical PMDO industrial application.

Table 4: PMDO design optimization space definition.

Variable Variable Name Variable Variable Variable Name Variable index symbol
index symbol $\bm{x}[0]$ Wing span DV_0 $\bm{x}[8-13]$ Wing max thickness-to-
chord ratios DV_8-13 $\bm{x}[1]$ Wing leading edge sweep DV_1 $\bm{x}[14]$
Wing inboard trailing edge sweep DV_14 $\bm{x}[2]$ Wing break loation DV_2
$\bm{x}[15]$ Winglet cant DV_15 $\bm{x}[3]$ Wingtip chord DV_3 $\bm{x}[16]$
Winglet span DV_16 $\bm{x}[4-7]$ Wing twist distribution DV_4-7
$\bm{x}[17-18]$ Winglet twist DV_17-18

Table 5: PMDO constraints definition. Constraint | Constraint Name | Bounds | Constraints | Constraint Name | Bounds
---|---|---|---|---|---
index | index | index | index
$\bm{g}[0]$ | Volume availability factor | $\bm{B}[0]$ | $\bm{g}[3]$ | Landing gear spacing | $\bm{B}[3]$
$\bm{g}[1]$ |  $C_{L_{max}}$ retracted | $\bm{B}[1]$ | $\bm{g}[4]$ | Wingtip twist deformation | $\bm{B}[4]$
$\bm{g}[2]$ | Cruise deck angle | $\bm{B}[2]$ |  |  |

##  5 Optimizers

In this section, we introduce six optimizers to solve the following
constrained problem:

$\min\limits_{\bm{x}\in\Omega}\left\\{f(\bm{x})~{}~{}\mbox{s.t.}~{}~{}\bm{g}(\bm{x})\geq
0\right\\},$ (2)

where $f:\mathbb{R}^{d}\mapsto\mathbb{R}$ is the objective function,
$\bm{g}:\mathbb{R}^{d}\mapsto\mathbb{R}^{m}$ gives the inequality constraints
which are expensive to evaluate and $\bm{x}\in\Omega\subset\mathbb{R}^{d}$ is
the vector of design variables. Two of the optimizers are included in the
Isight software, which is mandatory to use because of CMDO and PMDO
frameworks. We developed an Isight interface for the SEGOMOE python tool-box
in which the other optimizers are implemented.

###  5.1 Bayesian optimizers

This section introduces the constrained Bayesian optimization (CBO) framework
[3, 14] that aims to solve the optimization problem (2) with a minimal number
of calls. To do so, one uses Gaussian Process (GP) (also known as Kriging)
[15, 16] trained with a pre-computed design of experiments (DoE) (i.e. set of
designs evaluated on the objective and constraints functions) of $l$ points.
GPs are then used to provide, with a cheap computational cost, a prediction
$\mu_{s}^{(l)}:\mathbb{R}^{d}\mapsto\mathbb{R}$ and an associated uncertainty
$\sigma_{s}^{(l)}:\mathbb{R}^{d}\mapsto\mathbb{R}$ for each point of
$\bm{x}\in\Omega$ where $s:\mathbb{R}^{d}\mapsto\mathbb{R}$ can be either $f$
and $g_{i}$ for a given constraint component $i$. Concerning the objective
function, these information are combined in an acquisition function
$\alpha_{f}^{(l)}:\mathbb{R}^{d}\mapsto\mathbb{R}$ [17, 18, 19] coding the
trade-off between exploration of the highly uncertain domain that can hide a
minimum and exploitation of the minimum of the GP prediction. For the
constraints, these information are joined to produce a feasibility criterion
$\alpha_{\bm{g}}^{(l)}:\mathbb{R}^{d}\mapsto\mathbb{R}^{m}$ [17, 20, 21] which
is generally explicit. The point $\bm{x}^{(l+1)}$, solving the constrained
maximization trade-off sub-problem:

$\bm{x}^{(l+1)}=\arg\min\limits_{\bm{x}\in\Omega}\left\\{\alpha_{f}^{(l)}(\bm{x})~{}~{}\mbox{s.t.}~{}~{}\bm{x}\in\Omega_{\bm{g}}^{(l)}\right\\},$
(3)

where $\Omega_{\bm{g}}^{(l)}$ is the approximated feasible domain defined by
the feasibility criterion $\bm{\alpha}_{\bm{g}}^{(l)}$, is thus iteratively
added to the DoE until a maximum number of iterations max_nb_it is reached.
The solution provided by CBO to the problem (2) is eventually the best point
in the DoE (i.e with the minimal feasible value of $f$). The main steps of CBO
are finally summarized in Algorithm 1.

0: : Objective and constraints functions, initial DoE for objective and
constraints, a maximum number of iterations max_nb_it

1: for $l=1$ to max_nb_it do

2: Build the surrogate models using GPs

3: Find $\bm{x}^{(l+1)}$ a solution of the enrichment maximization sub-problem

4: Evaluate the objective and constraints functions at $\bm{x}^{(l+1)}$

5: Update the DoE

6: end for

6: : The best point found in the DoE

Algorithm 1 The constrained Bayesian optimization framework.

In this context, Bartoli et al. [18] implemented a CBO python tool-box based
on the super efficient global optimization (SEGO) algorithm of Sasena et al.
[22] which has been enhanced by the use of mixture of experts (MOE) [23, 24],
the kriging with Partial Least Squares (KPLS) for high dimensional problems
[25], additional acquisition functions (e.g. WB2S [18]) for highly multimodal
objective functions, supplementary feasibility criteria (e.g. the upper trust
bound (UTB) [20, 21]) for non-linear constraint functions and a
multiprocessing ability to speed-up the optimization process. The tool-box
[18] is thus named the super efficient global optimization with mixture of
experts (SEGOMOE) tool-box. We also developed an interface of the python tool-
box to allow its use in the Isight software in which BRAC is implemented.

In this paper, the four following variants of SEGO (available with the SEGOMOE
tool-box) are tested to show their common ability on industrial MDO problems:

  * •

SEGO [22]: it can be seen as a basic implementation of the super efficient
global optimization framework, it was developed for standard constrained
optimization problems.

  * •

SEGO-UTB [21]: it includes a decreasing upper trust bound in the SEGO
framework. In the context of this paper and to encourage the exploration of
the feasible domain, we used an exponential decrease for the constraints
learning rate, see [21, Fig. 3].

  * •

SEGOMOE [18]: it combines the use of multiple models (local GPs) with the SEGO
framework in order to mitigate the high non-linearity of the targeted
optimization problem.

  * •

SEGOMOE-UTB: similarly to SEGO-UTB, it combines the use of the UTB strategy
and the SEGOMOE framework. Again, we choose to work with an exponential
decrease for the constraints learning rate.

We note that for all the four variants we used the WB2S acquisition function
combined with the KPLS models as the dimension of the design space is larger
than 12.

###  5.2 Isight optimizers

The Isight software [4] is delivered with many ready-to-use constrained
optimizers. In this paper, we focus on two of the popular derivative-free
Isight constrained optimizers: the evolutionary optimization algorithm (Evol)
[26] and a variant of the Pointerdog algorithm (Pointer-2) [4].

Evol is an evolutionary strategy that mutates iteratively the best known point
by adding a normally distributed perturbation to the design variables. The
standard deviation of the normal distribution is adapted during the
optimization process to solve the regarded problem with a minimal number of
evaluations. This simple algorithm is enhanced by different features such as:
(1) a repeat calculation check to ensure that all the points computed are
different, (2) a standard deviation expansion process when the same point is
always evaluated, (3) a consecutive variable search allowing the exploration
in a single canonical direction of the design space, and (4) a parallel
execution accelerating the optimization process.

The Pointer-2 strategy is an optimization framework based on an Isight
proprietary algorithm that managed a set of well-known optimizers: Evol [26],
the Hooke-Jeeves direct search method [27], the non-linear programming by
quadratic Lagrangian (NLPQL) algorithm [28], the downhill simplex algorithm
[29], the multi-function optimization system tool (MOST) technique [4] and the
multi-objective particle swarm optimization (MOPSO) algorithm [30]. The choice
of the set of optimizers used in the optimization process is made with respect
to a classification system using the information available on the problem
(i.e. often given by the user). The best optimizer, and its settings, are then
updated thanks to the information collected all along the optimization
process. Furthermore, the optimization method can be used in two ways. First,
the optimizers and their settings are selected to have the higher improvement
leading to the best solution in the shortest time. On the contrary, one
searches a robust solution to uncertainties which leads to a slower
optimization process due to a higher number of calls. The Pointer-2 strategy
periodically performs a surrogate based optimization to speed up the
convergence. In fact, the Pointer-2 is able to solve a wide range of
constrained optimization problems and thus allows non-specialist to solve an
optimization problem only providing the objective and constraints functions,
the design variables, and a targeted optimization time.

##  6 Results

In this section, we introduce and comment the different tests realized on the
BRAC CMDO and PMDO problems. We also recall the aim here is to find the best
solution to the optimization problem, the fastest, and the different
optimizers are tested in this scope. In the following, the six previously
introduced optimizers (see Section 5) are tested on the BRAC CMDO and PMDO.
For the sake of comparison, we will produce convergence and parallel plots to
assess the efficiency of the SEGOMOE tool-box solvers compared to the Isight
optimizers.

###  6.1 BRAC CMDO results

####  6.1.1 Tests details

The optimizers, introduced in Section 5, are compared with the following test
plan. We perform $10$ independent runs for each solver using $10$ different
initial DoEs build with the Latin hyper-cube sampling method. For each run,
all the solvers are initiated with the same DoE. The size of the initial DoEs
is set to $n_{start}=13$ (i.e $d+1$ where $d$ is the dimension of BRAC CMDO
problem). Note that Evol and Pointer-2 need a single point to launch the
optimization process, we thus provide the point with the best valid value of
the DoE. If there is no valid value in the DoE, the point with the minimal
constraints violation is provided.

The maximum number of evaluations of the SEGO-like solvers is set to
$\text{max\\_it\\_nb}=227$ (i.e. $20d-n_{start}$ meaning a total of $20d=240$
evaluations is performed). Concerning Evol and Pointer-2, we used the
historical options settings of the CMDO framework. The maximum number of
evaluations of Evol is set to $\text{max\\_it\\_nb}=960$. For each run, Evol
thus performed $973$ function evaluations, which takes approximately 8 hours
to perform. We respectively fix the maximum allowable job time and the
topography type options of Pointer-2 to $8$ hours and nonlinear. All the
remaining options of Evol and Pointer-2 are kept as default. Finally, the best
solution among the ten optimizations performed for each optimizer is kept to
initiate the BRAC PMDO problem as explained in Section 3.2.

####  6.1.2 Convergence plots

To assess the performance of the introduced optimizers, we build two kinds of
convergence plots. The first one, named evaluation convergence plot, displays
the average and the standard deviation of the best valid value over the $10$
runs for increasing number of evaluations. The best valid value is defined as
the best valid value if there is, at least, one valid point in the DoE,
otherwise, a penalization replaces the obtained invalid value. The
penalization is the highest feasible value of the objective function ever
found. Because of confidentiality reasons, we scale the value of the
convergence plots between $0$ and $1$. The second convergence plot, named time
convergence plot, also shows the average and the standard deviation of the
best valid value over the $10$ runs along the optimization time. The
penalization is kept the same as in the evaluation convergence plot.

(a) Evaluation convergence plot.  (b) Time convergence plot.

Figure 3: Convergence plots for the CMDO of BRAC. In the time convergence
plot, the vertical red-dashed line indicates the maximum CPU time allowed for
an industrial optimization process (8 hours for this test case).

Figure 3 shows that SEGO-like solvers clearly outperform Evol and Pointer-2 in
term of the optimum value. In term of number of evaluations, Figure 3a
displays that SEGO is converging the fastest to the optimum value even if
SEGOMOE has a fastest convergence rate at the beginning of the optimization.
Furthermore, note that SEGO-UTB and SEGOMOE-UTB are not converging as fast as
SEGO and SEGOMOE because of their extensive exploratory behaviour. When
looking to the optimization time, Figure 3b reveals that the SEGO-like solvers
converge in approximately $2$ hours meanwhile Evol and Pointer-2 does not
converge after $8$ hours.

To conclude, these convergence plots have shown that the SEGO-like solvers
provide a better solution in less running CPU-time than Evol and Pointer-2. We
also note that the use of the UTB option within SEGO and SEGOMOE seems to not
lead to any improvement on the obtained performance which suggests that the
exploration of the feasible domain was not a difficult task to handle on this
test case.

####  6.1.3 Parallel plots

The parallel plots introduced in the following depict the behaviour of the
tested solvers along the optimization process in which targeted values are
displayed (e.g., the explored design variables). Here, the plotted values are
the number of iterations, the design variables values, the objective function
value and the constraints violation. Furthermore, several colors are used to
distinguish the reference design in red (i.e. the best feasible design found
so far by all the tested optimizers, here SEGO), the optimum design found by
the regarded solver in black, the feasible explored designs in green, and the
unfeasible designs in blue. Due to the stochastic nature of our tests, we
build our parallel plots using a median run in the following way. For each run
of the optimizer, we store the best valid objective value; if none of the runs
converges to a feasible point, we collect the minimal violation explored by
the optimizer. The median run is then selected based on the stored values for
all runs.

Figures 4 and 5 display the parallel plots of the median run of the tested
optimizers. First, note that all the SEGO-like optimizers are converging to
the optimum. On the contrary, Evol and Pointer-2 do not converge to the
optimum, as depicted by Figures 4a and 4b. Furthermore, Evol is not able to
explore the entire domain as implied by the grouped green lines of Figure 4a.
Secondly, note the different solutions of the SEGO-like solvers even if they
all converge approximately to the same objective function value. Indeed, the
values for DV_8 and DV_11 (i.e. wing maximum thickness-to-chord ratios) are
different from an optimizer to another. This can be due to either a local
optimum or an inactive design variable (i.e. the objective function does not
change along the design variable). From the expert point of view, the behavior
observed on the BRAC CMDO problem is due to inactive design variables.

(a) Evol.

(b) Pointer-2.

(c) SEGO.

Figure 4: Parallel plots using the median run for the CMDO of BRAC for Evol,
Pointer-2 and SEGO. In blue: the unfeasible designs; in green: the feasible
designs; in black: the optimum; in red: the reference design.

(a) SEGO-UTB.

(b) SEGOMOE-UTB.

(c) SEGOMOE.

Figure 5: Parallel plots using the median run for the CMDO of BRAC for
SEGOMOE, SEGOMOE-UTB and SEGO-UTB. In blue: the unfeasible designs; in green:
the feasible designs; in black: the optimum; in red: the reference design.

###  6.2 PMDO results

####  6.2.1 Tests details

The optimizers are again compared on the BRAC PMDO problem. Because of the
computation time of the BRAC PMDO problem (ca. $25min$), we only performed one
optimization for each solver. Each of the optimizers is initiated with the
same DoE of $n_{start}=20$ sample points using the Latin hyper-cube sampling
method. To mimic the introduced MDO process, we also add to the initial DoE
the point corresponding to the best solution $\bm{x}^{*}$ of all the BRAC CMDO
process. As in Section 6.1.1, Evol and Pointer-2 only need a single point to
launch the optimization, we thus provide the best solution $x^{*}$ of all the
BRAC CMDO cases.

The maximum number of evaluations of the $4$ SEGO-like solvers is set to
$\text{max\\_it\\_nb}=170$ (i.e. $10d-n_{start}-1$ meaning a total of
$10d=190$ evaluations). The maximum number of evaluations of Evol and
Pointer-2 is set to $510$ to follow historical values used in the PMDO
framework, and the parallel batch size is set to $4$ for both. In addition, a
smooth topography type is selected for Pointer-2. All the other settings are
kept to default values.

####  6.2.2 Convergence plots

The convergence plots defined in this Section are slightly different from the
ones introduced in Section 6.1.2. Indeed, only one optimization is here
performed for each of the $6$ optimizers. Thus Figure 6 displays the best
valid values of the regarded solvers for increasing time or number of
evaluations.

(a) Evaluation convergence plot.  (b) Time convergence plot.

Figure 6: Convergence plots for the PMDO of BRAC. The red dashed line is the
value of the BRAC CMDO design in the BRAC PMDO process.

First, Figure 6a shows that none of the solvers converge to the same optimum
value, with the best value obtained with SEGO. Similar values are also
achieved with Evol and SEGOMOE. Furthermore, Pointer-2 provides the worst
value of all the solvers with an optimum at $0.1$. The other algorithms find
better solution values than Pointer-2, but worse than SEGO, SEGOMOE and Evol.
Note that SEGO and SEGOMOE are able to find a similar solution as Evol with 30
fewer evaluations.

Secondly, in terms of convergence time, Evol and Pointer-2 benefit from the
batch evaluation of the objective function which is not the case for the SEGO-
like solvers. Indeed, the 220 necessary evaluations of Evol to converge are
done in less than 40 hours, whereas the SEGO-like solvers need 80 hours for
190 evaluations.

To conclude, Figure 6 shows that the UTB feasibility criterion is not useful
for the BRAC CMDO problem, whereas SEGO and SEGOMOE show great performances
with a small budget. Moreover, the batch evaluation capability of Evol speeds
up the convergence time.

####  6.2.3 Parallel plots

Here, no run selections are needed for the parallel plots. Indeed, only a
single run is performed for the BRAC PMDO problem for each solver. They are
produced in the same way as in Section 6.1.3. Figures 7 and 8 show the
parallel plots for Evol, Pointer-2, SEGO, SEGOMOE, SEGO-UTB and SEGOMOE-UTB.

(a) Evol.

(b) Pointer-2.

(c) SEGO.

Figure 7: Parallel plots for the PMDO of BRAC for Evol, Pointer-2 and SEGO. In
blue: the unfeasible designs; in green: the feasible designs; in black: the
optimum; in red: the reference design.

(a) SEGO-UTB.

(b) SEGOMOE-UTB.

(c) SEGOMOE.

Figure 8: Parallel plots for the PMDO of BRAC for SEGO-UTB, SEGO-UTB and
SEGOMOE. In blue: the unfeasible designs; in green: the feasible designs; in
black: the optimum; in red: the reference design.

As mentioned, SEGO finds the best design, followed closely by Evol and
SEGOMOE. These three solutions share similar main plan-form features, namely
the wing span (DV_0) and sweep (DV_1), while there is some variation in the
wing twist distribution (DV_4-7) and maximum thickness-to-chord ratios
(DV_8-13), as shown in Figures 7a and 8c. Only SEGO converges to the lower
thickness bound for the two most outboard profiles (DV_12 and DV_13). The
winglet has a secondary impact on the MTOW, and as a result even greater
variation is seen in the optimized winglet parameters (DV_15-18). The best
solutions of the other three optimizers deviate even further from the SEGO
optimum. It is possible that all the optimizers converge to different local
optima, but the variability in the solutions is more likely due to incomplete
convergence of the algorithms, since the number of iterations was limited to
meet the time restrictions of an industrial application.

##  7 Conclusion

The multi-level optimization framework developed at Bombardier Aviation aims
to help the design of efficient and competitive aircraft. The two optimization
levels, implemented within the Isight software, were discussed in this paper:
conceptual and preliminary multi-disciplinary optimization (CMDO and PMDO).
The use of the ready-to-use Isight optimizers have shown great results in the
past but within a minimum of 8 hours for the CMDO study case and a week for
the PMDO one. The SEGOMOE python tool-box was investigated in this paper and
led to significant improvements on Bombardier research aircraft configuration
test cases. In particular, on the CMDO test case, we showed that all the
tested solvers (implemented within the SEGOMOE tool-box) have outperformed two
of the best Isight optimizers both in terms of the required computational
effort as well as on the CPU running time. On the PMDO test case, the results
are more contrasted. SEGO is able to provide a better solution than Evol in
less evaluations but with more computational time. This is due to the batch
evaluation capability of Evol. The batch evaluation capability of the SEGOMOE
tool-box is a future improvement work to perform.

## Acknowledgments

This work is part of the activities of ONERA - ISAE - ENAC joint research
group in a context of a partnership between ONERA, ISAE-SUPAERO and Bombardier
Aviation. In addition, we thank Jasveer Singh for his work on the BRAC CMDO
specification.

## References

  * Piperni et al. [2004] Piperni, P., Abdo, M., and Kafyeke, F., “The application of multi-disciplinary optimization technologies to the design of a business jet,” _10th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference_ , 2004, p. 4370.
  * Piperni et al. [2013] Piperni, P., DeBlois, A., and Henderson, R., “Development of a Multilevel Multidisciplinary-Optimization Capability for an Industrial Environment,” _AIAA Journal_ , Vol. 51, 2013, pp. 2335–2352.
  * Močkus [1975] Močkus, J., “On Bayesian Methods for Seeking the Extremum,” _Optimization Techniques IFIP Technical Conference_ , 1975, pp. 400–404.
  * Van der Velden and Koch [2010] Van der Velden, A., and Koch, P., “Isight design optimization methodologies,” _ASM handbook_ , Vol. 22, 2010, p. 79.
  * Piperni and Boudreau [2003] Piperni, P., and Boudreau, J., “The Evolution of Structured Grid Generation at Bombardier Aerospace,” _50th Canadian Air and Space Institute Annual General Meeting and Conference_ , 2003.
  * Mohamed et al. [2009] Mohamed, K., Laurendeau, E., and Nadarajah, S., “Implementation and Validation of Detached Eddy Simulation in the Bombardier Aerospace Navier-Stokes Flow Solver,” _56th Canadian Air and Space Institute Aeronautics Conference_ , 2009.
  * Deblois and Abdo [2010] Deblois, A., and Abdo, M., “Multi-Fidelity Multidisciplinary Design Optimization of Metallic and Composite Regional and Business Jets,” _13th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference 2010_ , 2010.
  * Valarezo and Chin [1994] Valarezo, W., and Chin, V., “Method for the Prediction of Wing Maximum Lift,” _Journal of Aircraft_ , Vol. 31, 1994, pp. 103–109.
  * Maskew [1982] Maskew, B., “PROGRAM VSAERO: A computer program for calculating the non-linear aerodynamic characteristics of arbitrary configurations: User’s manual,” 1982.
  * Piperni et al. [2007] Piperni, P., Abdo, M., Kafyeke, F., and Isikveren, A. T., “Preliminary Aerostructural Optimization of a Large Business Jet,” _Journal of Aircraft_ , Vol. 44, 2007, pp. 1422–1438.
  * Sermeus et al. [2010] Sermeus, K., Mohamed, K., Laurendeau, E., and Nadarajah, S., “Development of an Industrial Aerodynamic Shape Optimization Strategy Using a Discrete Adjoint Method,” _18th Annual Conference of the CFD Society of Canada_ , 2010, p. 6.
  * Reist et al. [Submitted 2019] Reist, T. A., Koo, D., Zingg, D. W., Bochud, P., Castonguay, P., and Leblond, D., “Cross-Validation of High-Fidelity Aerodynamic Shape Optimization Methodologies for Aircraft Wing-Body Optimization,” _AIAA Journal_ , Submitted 2019.
  * Deblois [2013] Deblois, A., “A New Hybrid-Adjoint MDO Framework for High-Fidelity Aerostructural Optimization,” _SAE 2013 Aerotech Congress and Exhibition_ , 2013.
  * Jones et al. [1998] Jones, D. R., Schonlau, M., and Welch, W. J., “Efficient Global Optimization of Expensive Black-Box Functions,” _Journal of Global optimization_ , Vol. 13, 1998, pp. 455–492.
  * Rasmussen and Williams [2006] Rasmussen, C. E., and Williams, C. K. I., _Gaussian Processes for Machine Learning_ , The MIT Press, 2006.
  * Krige [1951] Krige, D. G., “A Statistical Approach to Some Basic Mine Valuation Problems on the Witwatersrand,” _Journal of the Southern African Institute of Mining and Metallurgy_ , Vol. 52, 1951, pp. 119–139.
  * Frazier [2018] Frazier, P. I., “A tutorial on Bayesian optimization,” _aarXiv:1807.02811_ , 2018.
  * Bartoli et al. [2019] Bartoli, N., Lefebvre, T., Dubreuil, S., Olivanti, R., Priem, R., Bons, N., Martins, J., and Morlier, J., “Adaptive modeling strategy for constrained global optimization with application to aerodynamic wing design,” _Aerospace Science and technology_ , Vol. 90, 2019, pp. 85–102.
  * Wang and Jegelka [2017] Wang, Z., and Jegelka, S., “Max-Value Entropy Search for Efficient Bayesian Optimization,” _34th International Conference on Machine Learning_ , Vol. 70, 2017, pp. 3627–3635.
  * Lam et al. [2015] Lam, R., Allaire, D., and Willcox, K., “Multifidelity optimization using statistical surrogate modeling for non-hierarchical information sources,” _56th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference_ , 2015, p. 143.
  * Priem et al. [2019] Priem, R., Bartoli, N., and Diouane, Y., “On the Use of Upper Trust Bounds in Constrained Bayesian Optimization Infill Criteria,” _AIAA Aviation 2019 Forum_ , 2019, p. 2986.
  * Sasena et al. [2002] Sasena, M. J., Papalambros, P., and Goovaerts, P., “Exploration of Metamodeling Sampling Criteria for Constrained Global Optimization,” _Engineering optimization_ , Vol. 34, 2002, pp. 263–278.
  * Hastie et al. [2005] Hastie, T., Tibshirani, R., Friedman, J., and Franklin, J., “The elements of statistical learning: data mining, inference and prediction,” _The Mathematical Intelligencer_ , Vol. 27, 2005, pp. 83–85.
  * Bettebghor et al. [2011] Bettebghor, D., Bartoli, N., Grihon, S., Morlier, J., and Samuelides, M., “Surrogate modeling approximation using a mixture of experts based on EM joint estimation,” _Structural and multidisciplinary optimization_ , Vol. 43, No. 2, 2011, pp. 243–259.
  * Bouhlel et al. [2016] Bouhlel, M. A., Bartoli, N., Otsmane, A., and Morlier, J., “Improving Kriging Surrogates of High-Dimensional Design Models by Partial Least Squares Dimension Reduction,” _Structural and Multidisciplinary Optimization_ , Vol. 53, 2016, pp. 935–952.
  * Schwefel [1975] Schwefel, H., “Evolutionsstrategie und numerische Optimierung [Evolution strategy and numerical optimization],” Ph.D. thesis, Fachbereich Verfahrenstechnik, Technische Universität Berlin, 1975.
  * Hooke and Jeeves [1961] Hooke, R., and Jeeves, T. A., ““Direct Search” Solution of Numerical and Statistical Problems,” _Journal of the ACM_ , Vol. 8, 1961, pp. 212–229.
  * Schittkowski [1986] Schittkowski, K., “NLPQL: A FORTRAN subroutine solving constrained nonlinear programming problems,” _Annals of operations research_ , Vol. 5, 1986, pp. 485–500.
  * Nelder and Mead [1965] Nelder, J. A., and Mead, R., “A simplex method for function minimization,” _The computer journal_ , Vol. 7, 1965, pp. 308–313.
  * Coello et al. [2004] Coello, C. A., Pulido, G. T., and Lechuga, M. S., “Handling multiple objectives with particle swarm optimization,” _IEEE Transactions on evolutionary computation_ , Vol. 8, 2004, pp. 256–279.


$Spin(7)$ and generalized $SO(8)$ instantons in eight dimensions.

A.V. Smilga

SUBATECH, Université de Nantes, 4 rue Alfred Kastler, BP 20722, Nantes 44307,
France.

We present a simple compact formula for a topologically nontrivial map
$S^{7}\to Spin(7)$ associated with the fiber bundle
$Spin(7)\stackrel{{\scriptstyle G_{2}}}{{\to}}S^{7}$. The homotopy group
$\pi_{7}[Spin(7)]=\mathbb{Z}$ brings about the topologically nontrivial
8-dimensional gauge field configurations that belong to the algebra $spin(7)$.
The instantons are special such configurations that minimize the functional
$\int{\rm Tr}\\{F\wedge F\wedge\star(F\wedge F)\\}$ and satisfy non-linear
self-duality conditions, $F\wedge F\ =\ \pm\star(F\wedge F)$.

$Spin(7)\subset SO(8)$, and $Spin(7)$ instantons represent simultaneously
$SO(8)$ instantons of a new type. The relevant homotopy is
$\pi_{7}[SO(8)]=\mathbb{Z}\times\mathbb{Z}$, which implies the existence of
two different topological charges. This also holds for all groups $SO(4n)$
with integer $n$. We present explicit expressions for two topological charges
and calculate their values for the conventional 4-dimensional and
8-dimensional instantons and also for the 8-dimensional instantons of the new
type.

Similar constructions for other algebras in different dimensions are briefly
discussed.

##  1 Introduction and Summary

Eight-dimensional instantons of two varieries have been discussed in the
literature. There are the so-called octonionic or secular instantons [1] that
satisfy certain modified linear self-duality conditions (these instantons are
outside the scope of this paper) and there are $SO(8)$ instantons where the
gauge fields satisfy the nonlinear self-duality conditions

$\displaystyle F\wedge F\ =\ \pm\star(F\wedge F)\,.$ (1.1)

$SO(8)$ instantons [2, 3] are associated with topologically nontrivial
mappings $S^{7}\to SO(8)$. In Refs. [2, 3], the simplest such mapping was
discussed:

$\displaystyle g_{8}\ =\
\exp\\{i\alpha_{m}\Gamma_{m}\\}=\mathbb{1}\cos\alpha+i\frac{\sin\alpha}{\alpha}\alpha_{m}\Gamma_{m}\,,$
(1.2)

where $\alpha=\|\alpha_{m}\|\leq\pi$ and $\Gamma_{m=1,\ldots,7}$ are purely
imaginary antisymmetric 8-dimensional matrices satisfying the Clifford
algebra,

$\displaystyle\Gamma_{m}\Gamma_{n}+\Gamma_{n}\Gamma_{m}\ =\
2\delta_{mn}\mathbb{1}\,,$ (1.3)

and the relation

$\displaystyle\Gamma_{1}\Gamma_{2}\Gamma_{3}\Gamma_{4}\Gamma_{5}\Gamma_{6}\Gamma_{7}\
=\ i\mathbb{1}\,.$ (1.4)

The mapping $g_{8}$ is associated with the instanton satisfying $F\wedge F\ =\
\star(F\wedge F)$ and the mapping $g_{8}^{-1}$ with the antiinstanton
satisfying $F\wedge F\ =\ -\star(F\wedge F)$. The explicit expressions for
these instantons are quite analogous to the familiar 4-dimensional BPST
instantons [4]. Multiinstanton solutions, the analogs of ADHM solutions [5],
are also known [6].

The $SO(8)$ instantons satisfying the nonlinear self-duality conditions (3.18)
have several interesting physical applications. They were discussed in the
string theory context [7] and in the context of the 8-dimensional quantum Hall
effect [8].

All these configurations are characterized by a nonzero integer values of the
fourth Chern class 111A “physical” way to fix the coefficient in (1.6) is to
require that the Dirac equation on the gauge field background $A_{M}(x)$ is
well defined and has a single zero mode. For the Chern class of order $D/2$ in
$R^{D}$, one derives [9] $\displaystyle q\ =\
\pm\frac{1}{(2\pi)^{D/2}(D/2)!}\int_{R^{D}}{\rm
Tr}\,\\{\overbrace{F\wedge\cdots\wedge F}^{D/2}\\}$ (1.5) with the sign
depending on the convention.

$\displaystyle q\ =\ \frac{1}{384\pi^{4}}\int_{R^{8}}{\rm Tr}\,\\{F\wedge
F\wedge F\wedge F\\}\,.$ (1.6)

With this convention, $q=1$ for the instanton and $q=-1$ for the
antiinstanton. The construction of Ref. [2] is quite parallel to the
construction of the ordinary 4-dimensional instantons, which are associated
with the mappings $S^{3}\to SU(2)$ and have an integer topological charge
(second Chern class or Pontryagin number),

$\displaystyle q\ =\ \frac{1}{8\pi^{2}}\int_{R^{4}}{\rm Tr}\,\\{F\wedge
F\\}\,.$ (1.7)

However, for $SO(8)$ gauge configurations in 8 dimensions, the topology is
more complicated than in the BPST case. The relevant homotopy group is [10]

$\displaystyle\pi_{7}[SO(8)]=\mathbb{Z}\times\mathbb{Z}\,,$ (1.8)

so that a generic gauge field configuration is characterized by two different
topological charges. We will show later that the second topological charge is
expressed as [11]

$\displaystyle\tilde{q}\ =\ \frac{1}{3\cdot
2^{11}\cdot\pi^{4}}\varepsilon^{a_{1}\cdots a_{8}}\int F^{a_{1}a_{2}}\wedge
F^{a_{3}a_{4}}\wedge F^{a_{5}a_{6}}\wedge F^{a_{7}\,a_{8}}\,.$ (1.9)

This construction can be generalized to $D=4n$ dimensions with any $n$. The
presence of the second topological charge for the $4n$-dimensional gauge
fields belonging to the algebra $so(4n)$ can be understood by tracing the
analogy with $4n$-dimensional gravity. The charge (1.9) has the same structure
(and the same coefficient!) as the Gauss-Bonnet invariant, giving the Euler
characteristic of an even-dimensional manifold. For $D=4n$,

$\displaystyle\chi\ =\
\frac{1}{(2n)!\pi^{2n}2^{4n}}\varepsilon^{A_{1}A_{2}\cdots
A_{4n-1}A_{4n}}\int\,R^{A_{1}A_{2}}\wedge\cdots\wedge R^{A_{4n-1}\,A_{4n}}\,,$
(1.10)

where $R^{AB}=\frac{1}{2}\,R^{AB}_{\ \ MN}\,dx^{M}\wedge dx^{N}$ is the
curvature 2-form — the field density for the spin connection form.

The integral representation for the Euler characteristic, the analog of
(1.10), can be written for any even-dimensional manifold, also for $D=4n+2$.
Similarly, the topological charge of the type (1.9) can also be defined for
$SO(4n+2)$ gauge fields in $4n+2$ dimensions. However, the Chern class of
order $4n+2$, the analog of the integral (1.5) with an odd number of $F$
factors, vanishes in this case due to the skew symmetry of $F$.222One can
notice at this point that, while the topological charge (1.9) has the same
structure as the Gauss-Bonnet invariant, the Chern classes (1.7) and (1.6)
have the same structure as the Hirzebruch signature invariant,
$\displaystyle\tau\ \propto\ \int{\rm Tr}\\{R\wedge\ldots\wedge R\\}\,,$
(1.11) which is nonzero only for $D=4n$. This matches well with the
topological fact that $\pi_{D-1}[SO(D)]$ has two $\mathbb{Z}$ factors only for
$D=4n$.

The expression (1.2) gives a mapping of $S^{7}$ in $SO(8)$. Hovewer, it will
be more convenient for us to treat $g_{8}$ as a matrix that rotates
8-dimensional 8-component real spinors. The set of all such matrices form the
so-called $Semispin(8)$ group [12]. By triality, this group is isomorphic to
$SO(8)$ and has the center $Z_{2}$ (the matrices $g=\mathbb{1}$ and
$g=-\mathbb{1}$). The double cover of $Semispin(8)$ is $Spin(8)$ with the
center $Z_{2}\times Z_{2}$.

$Semispin(8)\simeq SO(8)$ has a $Spin(7)$ subgroup, and one can be interested
in the tolopogically nontrivial mappings $S^{7}\to Spin(7)$, which exist due
to nontrivial $\pi_{7}[Spin(7)]\ =\ \mathbb{Z}$, and the associated
8-dimensional instantons with $Spin(7)$ gauge group. We may observe at this
point that the mapping (1.2) is not such mapping — the commutators of the
generators $\Gamma_{m}$ in (1.2) give 21 generators of $SO(7)$ and, together
with $\Gamma_{m}$, this completes the full $so(8)$ algebra.

As we will see later, the simplest map $S^{7}\to Spin(7)$ has a somewhat more
complicated than (1.2) form:

$\displaystyle g\ =\
\exp\left\\{-\frac{1}{2}\alpha_{m}f_{mnk}\Gamma_{n}\Gamma_{k}\right\\}\,,$
(1.12)

where $f_{mnk}$ are the structure constants of the octonion algebra and
$\alpha_{m}$ is a 7-dimensional vector with a norm not exceeding $\pi$.

We will also see that the map (1.12) represents a local section of the fiber
bundle $Spin(7)\stackrel{{\scriptstyle G_{2}}}{{\to}}S^{7}$. This map gives
rise to the associated $Spin(7)$ instantons, which, evidently, are also
$SO(8)$ instantons.

The expressions (1.6) and (1.9) for the topological charges are written for
gauge field configurations. There are associated expressions for the
topological charges describing the maps $S^{7}\to SO(8)$. They read

$\displaystyle q\ =\ -\frac{1}{2^{7}\cdot 3\cdot 35\,\pi^{4}}\int_{S^{7}}{\rm
Tr}\,\\{(g^{-1}dg)^{7}\\}\,.$ (1.13)

and

$\displaystyle\tilde{q}\ =\ -\frac{i}{3\cdot 2^{11}\cdot
35\pi^{4}}\,\varepsilon^{abcdefgh}\int_{S^{7}}A^{ab}\wedge(A\wedge
A)^{cd}\wedge(A\wedge A)^{ef}\wedge(A\wedge A)^{gh}\,.$ (1.14)

We will calculate these charges for the maps (1.2) and (1.12) and also for the
maps $g=g_{7}g_{8}$ and $g=g_{7}g_{8}^{-1}$. The results are quoted in the
table below. The charge $q$ for both $g_{8}$ and $g_{7}$ is unity [this is so
due to our choice of the sign in (1.12)]. The topological charges of $g^{-1}$
are opposite to the charges of $g$. The charges for the map $g_{8}g_{7}$ is
the same as for $g_{7}g_{8}$, etc. It is interesting that the charge
$\tilde{q}$ takes the value $\tilde{q}=1$ for the map $g_{8}$, but it is
negative for the map $g_{7}$, involving the extra factor 3.

map |  $q$ |  $\tilde{q}$
---|---|---
$g_{8}$ |  1 |  1
$g_{7}$ |  1 |  -3
$g_{7}g_{8}$ |  2 |  -2
$g_{7}g_{8}^{-1}$ |  0 |  -4

Table 1. Topological charges of the maps $S^{7}\to SO(8)$ and of the
associated $SO(8)$ instantons.

The topological charges for a generic composite mapping $g=g_{8}^{n}g_{7}^{m}$
with integer $n,m$ are

$\displaystyle q(g_{8}^{n}g_{7}^{m})\ =\
n+m,\qquad\tilde{q}(g_{8}^{n}g_{7}^{m})\ =\ n-3m\,.$ (1.15)

The plan of the subsequent discussion is the following. In the next section,
we first remind the well-known BPST construction for the 4-dimensional $SU(2)$
instantons and then discuss the second topological charge for the $SO(4)$
gauge configurations and present the simple expressions for two types of
instantons.

In Sect. 3, we describe the conventional $SO(8)$ instantons associated with
the mapping (1.2).

The heart of the paper is Sect. 4, where we discuss in detail the instanton
constructions for $Spin(7)$ and generalized instantons in $SO(8)$.

Sect. 5 is devoted to a brief discussion of topologically nontrivial gauge
field configurations in other dimensions with other gauge groups. For $D=4n$
with any $n$, there exist two types of instantons for the groups $SO(4n)$ and
only one such type for the groups $SO(4n-1)$. The corresponding homotopies are
$\pi_{4n-1}[SO(4n)]=\mathbb{Z}\times\mathbb{Z}$ and
$\pi_{4n-1}[Spin(4n-1)]=\mathbb{Z}$.

Topological nontrivial configurations exist also in other cases. We will
discuss in particular 6-dimensional fields with $SU(3)$ gauge group. Their
nontrivial topology is associated with $\pi_{5}[SU(3)]=\mathbb{Z}$ and with
the fiber bundle $SU(3)\stackrel{{\scriptstyle
SU(2)}}{{\longrightarrow}}S^{5}$. We present the known explicit expression for
the topologically nontrivial map $S^{5}\to SU(3)$.

##  2 Four dimensions

###  2.1 $SU(2)$

Consider a 4-dimensional Euclidean gauge field $A(x)=A_{\mu}(x)dx_{\mu}\in
su(2)$. Assume that the field density form

$\displaystyle F(x)=dA(x)-iA(x)\wedge A(x)\ \equiv\
\frac{1}{2}F_{\mu\nu}\,dx_{\mu}\wedge dx_{\nu}\,,$ (2.1)

where
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}-i[A_{\mu},A_{\nu}]$,
decays fast enough at $x\to\infty$, so that the integral (1.7) for the
Pontryagin index converges. As is very well known, (1.7) is integer in this
case. This integer is associated with the nontrivial
$\pi_{3}[SU(2)]=\mathbb{Z}$.

Indeed, the form Tr$\\{F\wedge F\\}$ is exact:

$\displaystyle{\rm Tr}\\{F\wedge F\\}\ =\ d\omega_{3},\qquad{\rm
where}\qquad\omega_{3}\ =\ {\rm Tr}\left\\{A\wedge F+\frac{i}{3}A\wedge
A\wedge A\right\\}\,.$ (2.2)

Then the integral (1.7) boils down to the surface term,

$\displaystyle q\ =\ \frac{1}{8\pi^{2}}\int_{S^{3}}\omega_{3}\,,$ (2.3)

where $S^{3}$ is a 3-sphere at large distance from the origin. Bearing in mind
that $F$ rapidly decays at large $x$, the potential $A$ acquires at large
distances a pure gauge form333See Ref. [13] for a mathematical proof.

$\displaystyle A\ =\ ig^{-1}dg$ (2.4)

with $g\in SU(2)$. We derive

$\displaystyle q\ =\ \frac{1}{24\pi^{2}}\int_{S^{3}}{\rm
Tr}\,\\{g^{-1}dg\,g^{-1}dg\,g^{-1}dg\\}\,.$ (2.5)

The simplest nontrivial mapping $S^{3}\to SU(2)$ may be presented as

$\displaystyle g\ =\ \frac{\mathbb{1}x_{4}-ix_{m}\sigma_{m}}{r}\,,$ (2.6)

where $\mathbb{1}$ is the unity matrix, $\sigma_{m=1,2,3}$ are the Pauli
matrices and $x_{\mu}=(x_{m},x_{4})$ marks a point on the distant sphere of
radius $r=\sqrt{x_{\mu}x_{\mu}}$. The gauge field (2.4) acquires the form

$\displaystyle A_{\mu}\ =\
\frac{\eta^{a}_{\mu\nu}x_{\nu}\,\sigma^{a}}{r^{2}}\,,$ (2.7)

where $\eta^{a}_{\mu\nu}$ is the ‘t Hooft symbol [14],

$\displaystyle\eta^{a}_{m4}\ =\ -\eta^{a}_{4m}\ =\
\delta_{am},\qquad\eta^{a}_{mn}=\varepsilon_{amn}\,.$ (2.8)

To calculate the integral (2.5), we represent it in a little bit more explicit
form

$\displaystyle q\ =\
\frac{1}{24\pi^{2}}\int_{S^{3}}\varepsilon_{\mu\nu\rho\sigma}\,\frac{x_{\mu}}{r}\,{\rm
Tr}\,\\{g^{-1}\partial_{\nu}g\,g^{-1}\partial_{\rho}g\,g^{-1}\partial_{\sigma}g\\}\,\,dV_{S^{3}}$
(2.9)

with the convention $\varepsilon_{1234}=1$. Pick up the north pole of $S^{3}$,
$x_{\mu}=(\vec{0},r)$. Then the integrand in (2.9) reads

$\displaystyle-\frac{i}{24\pi^{2}r^{3}}\,\varepsilon_{jkl}\,{\rm
Tr}\,\\{\sigma_{j}\sigma_{k}\sigma_{l}\\}\ =\ \frac{1}{2\pi^{2}r^{3}}\,.$
(2.10)

Due to rotational symmetry, the value of the integrand is the same at all
other points of the sphere. Multiplying (2.10) by $V_{S^{3}}=2\pi^{2}r^{3}$,
we reproduce the result $q=1$.

Any 4-dimensional gauge field with the asymptotics (2.7) has the topological
charge $q=1$. But the configurations realizing the minimum of the action
functional,

$\displaystyle S\ =\ \int_{R^{4}}{\rm Tr}\,\\{F\wedge*F\\}\ =\ \frac{1}{2}\int
d^{4}x\,{\rm Tr}\,\\{F_{\mu\nu}F_{\mu\nu}\\}\,,$ (2.11)

represent a particular physical interest. Such configurations with $q=1$
satisfy the self-duality condition, $F=*F$ or

$\displaystyle F_{\mu\nu}\ =\
\frac{1}{2}\varepsilon_{\mu\nu\rho\sigma}F_{\rho\sigma}\,.$ (2.12)

The standard BPST self-dual instanton with the center at the origin has the
form

$\displaystyle A_{\mu}\ =\
\frac{\eta^{a}_{\mu\nu}x_{\nu}\,\sigma^{a}}{r^{2}+\rho^{2}}\,,\qquad
F_{\mu\nu}\ =\
-\frac{2\rho^{2}\eta^{a}_{\mu\nu}\sigma^{a}}{(r^{2}+\rho^{2})^{2}}\,,$ (2.13)

where $\rho$ is a parameter having the meaning of the instanton size.

There are also self-dual or anti-self-dual configurations with an arbitrary
integer topological charge $q$ [5]. The values of $q$ give the degrees of the
corresponding maps.

####  2.1.1 Exponential parameterization

Note (it will be helpful for us later) that one can also describe the mapping
$S^{3}\to SU(2)$ in an alternative way:

$\displaystyle g\ =\
\exp\\{-i\alpha_{m}\sigma_{m}\\}=\mathbb{1}\cos\alpha-\frac{i\sin\alpha}{\alpha}\alpha_{m}\sigma_{m}\,,$
(2.14)

where $0\leq\alpha=\|\vec{\alpha}\|\leq\pi$. The relation to the standard
parameterization (2.6) is

$\displaystyle\alpha_{m}\ =\
\frac{x_{m}}{\|\vec{x}\|}\arccos\left(\frac{x_{4}}{r}\right)\,.$ (2.15)

The 2-sphere $\|\vec{\alpha}\|=\pi$ maps onto one point of $S^{3}$ — its south
pole, $g=-\mathbb{1}$.

In these variables,

$\displaystyle q\ =\ -\frac{1}{4\pi^{2}}\int_{\alpha\leq\pi}d^{3}\alpha\,{\rm
Tr}\,\\{g^{-1}\partial_{1}g\,g^{-1}\partial_{2}g\,g^{-1}\partial_{3}g\\}\,,$
(2.16)

where

$\displaystyle g^{-1}\partial_{m}g=-iA_{m}\ =\
-i\left[\frac{\cos\alpha\sin\alpha}{\alpha}\sigma_{m}-\frac{\sin^{2}\alpha}{\alpha^{2}}\alpha_{k}\varepsilon_{kml}\sigma_{l}+\frac{\alpha_{m}\alpha_{k}}{\alpha^{2}}\left(1-\frac{\cos\alpha\sin\alpha}{\alpha}\right)\sigma_{k}\right]\,.$
(2.17)

The result of the calculation is, of course, the same:

$\displaystyle q\ =\ \frac{2}{\pi}\int_{0}^{\pi}\sin^{2}\alpha\,d\alpha\ =\
1\,.$ (2.18)

###  2.2 $SO(4)$

Suppose now that we are interested in the gauge configurations belonging to
$so(4)$ rather than $su(2)$ algebra. $so(4)$ is a direct sum of two $su(2)$
summands with the generators $(\Sigma_{1}^{A})_{ab}=-i\eta^{A}_{ab}/2$ and
$(\Sigma_{2}^{A})_{ab}=-i\bar{\eta}^{A}_{ab}/2$.444Then
$\displaystyle[\Sigma^{A}_{1},\Sigma^{B}_{1}]=i\varepsilon^{ABC}\Sigma_{1}^{C},\qquad[\Sigma^{A}_{2},\Sigma^{B}_{2}]=i\varepsilon^{ABC}\Sigma_{2}^{C},\qquad[\Sigma_{1}^{A},\Sigma_{2}^{B}]=0\,,$
$\displaystyle{\rm Tr}\\{\Sigma_{1}^{A}\Sigma_{1}^{B}\\}={\rm
Tr}\\{\Sigma_{2}^{A}\Sigma_{2}^{B}\\}=\delta^{AB}\,,\qquad{\rm
Tr}\\{\Sigma_{1}^{A}\Sigma_{2}^{B}\\}=0\,.$ (2.19) An arbitrary $so(4)$ gauge
field can be presented as
$(A_{\mu})_{ab}=B_{\mu}^{A}(\Sigma_{1}^{A})_{ab}+C_{\mu}^{A}(\Sigma_{2}^{A})_{ab}$.

There are thus two types of instantons with the field densities

$\displaystyle F_{\mu\nu}^{ab}\ =\
\frac{2i\rho^{2}\eta^{A}_{\mu\nu}\eta^{A}_{ab}}{(r^{2}+\rho^{2})^{2}}\quad{\rm
and}\quad F_{\mu\nu}^{ab}\ =\
\frac{2i\rho^{2}\eta^{A}_{\mu\nu}\bar{\eta}^{A}_{ab}}{(r^{2}+\rho^{2})^{2}}\,.$
(2.20)

In addition, there are two types of the antiinstantons:

$\displaystyle F_{\mu\nu}^{ab}\ =\
\frac{2i\rho^{2}\bar{\eta}^{A}_{\mu\nu}\eta^{A}_{ab}}{(r^{2}+\rho^{2})^{2}}\quad{\rm
and}\quad F_{\mu\nu}^{ab}\ =\
\frac{2i\rho^{2}\bar{\eta}^{A}_{\mu\nu}\bar{\eta}^{A}_{ab}}{(r^{2}+\rho^{2})^{2}}\,.$
(2.21)

Now note that, for the both types of instantons, the topological charge (1.7),
rewritten as

$\displaystyle q\ =\
-\frac{1}{64\pi^{2}}\,\varepsilon_{\mu\nu\alpha\beta}\int_{R^{4}}F^{ab}_{\mu\nu}F^{ab}_{\alpha\beta}\,d^{4}x$
(2.22)

is equal to $q=1$, while for the both types of antiinstantons it is $q=-1$.
Consider now the second topological charge,

$\displaystyle\tilde{q}\ =\
-\frac{1}{16\pi^{2}}\varepsilon^{abcd}\int_{{\mathbb{R}}^{4}}F^{ab}\wedge
F^{cd}\ =\
-\frac{1}{128\pi^{2}}\varepsilon^{abcd}\varepsilon_{\mu\nu\alpha\beta}\int
F_{\mu\nu}^{ab}F_{\alpha\beta}^{cd}\,d^{4}x\,.$ (2.23)

Like the charge (1.7), $\tilde{q}$ is a gauge invariant, as follows from the
identity

$\displaystyle\varepsilon^{\tilde{a}\tilde{b}\tilde{c}\tilde{d}}O^{a\tilde{a}}O^{b\tilde{b}}O^{c\tilde{c}}O^{d\tilde{d}}\
=\ \varepsilon^{abcd}$ (2.24)

for any matrix $O\in SO(4)$. The sign in (2.23) is chosen so that
$\tilde{q}=1$ for the first instanton in (2.20) (with $F\sim\eta\eta$). It is
not difficult then to see that, for the second instanton, $\tilde{q}=-1$.
Similarly, for the antiinstantons: $\tilde{q}$ have different signs for the
different types.

configuration |  $q$ |  $\tilde{q}$
---|---|---
First instanton |  1 |  1
First antiinstanton |  -1 |  -1
Second instanton |  1 |  -1
Second antiinstanton |  -1 |  1

Table 2. Topological charges of the $SO(4)$ instantons.

Note that the coefficient in the integral (2.23) for $\tilde{q}$ is the same
as in the Gauss-Bonnet invariant,

$\displaystyle\chi^{D=4}\ =\
\frac{1}{128\pi^{2}}\,\varepsilon^{abcd}\varepsilon^{\mu\nu\alpha\beta}\int\sqrt{g}\,d^{4}x\,R^{ab}_{\mu\nu}\wedge
R^{cd}_{\alpha\beta}\,.$ (2.25)

As for the Pontryagin index (1.7), it has the same structure as the Hirzebruch
signature invariant,

$\displaystyle\tau\ =\ -\frac{1}{24\pi^{2}}\int_{M}{\rm Tr}\\{R\wedge R\\}\,,$
(2.26)

but the coefficients are different.

##  3 Conventional $SO(8)$ instantons

We now go to eight dimensions and consider the gauge fields $A_{M=1,\ldots,8}$
belonging to the algebra $so(8)$. The 8-dimensional counterpart of the
topological charge (1.7) is the fourth Chern class (1.6).

The construction is quite parallel to the standard instanton construction of
the previous section. We note that the form Tr$\\{F\wedge F\wedge F\wedge
F\\}$ is exact:

$\displaystyle{\rm Tr}\\{F^{4}\\}\ =\ d\omega_{7},$ (3.1)

where555The expression (3.3) follows from the general formula [15]
$\displaystyle\omega_{2n-1}\ =\ n\int_{0}^{1}dt\,t^{n-1}{\rm Tr}\\{A(dA-
itA^{2})^{n-1}\\}\,.$ (3.2)

$\displaystyle\omega_{7}\ =\ {\rm
Tr}\left\\{AF^{3}+\frac{2i}{5}F^{2}A^{3}+\frac{i}{5}AFA^{2}F-\frac{1}{5}FA^{5}-\frac{i}{35}A^{7}\right\\}$
(3.3)

(the products are understood here as wedge products).

Let $F$ decay rapidly at large distances so that the integral (1.6) converges.
Then $A$ represents at large distances a pure gauge (2.4). The integral is
reduced to (1.13).

The simplest topologically nontrivial mapping $S^{7}\to SO(8)\simeq
Semispin(8)$ has the form (1.2). One of many explicit representations for the
matrices $\Gamma_{m}$ satisfying (1.3) and (1.4) is

$\displaystyle\Gamma_{1}$ $\displaystyle=$
$\displaystyle-\sigma_{2}\otimes\sigma_{2}\otimes\sigma_{2};\quad\Gamma_{2}=\mathbb{1}\otimes\sigma_{1}\otimes\sigma_{2};\quad\Gamma_{3}=\mathbb{1}\otimes\sigma_{3}\otimes\sigma_{2};\quad\Gamma_{4}=-\sigma_{1}\otimes\sigma_{2}\otimes\mathbb{1};$
$\displaystyle\Gamma_{5}$ $\displaystyle=$
$\displaystyle\sigma_{3}\otimes\sigma_{2}\otimes\mathbb{1};\quad\Gamma_{6}=\sigma_{2}\otimes\mathbb{1}\otimes\sigma_{1};\quad\Gamma_{7}=\sigma_{2}\otimes\mathbb{1}\otimes\sigma_{3}\,.$
(3.4)

Seven matrices $\Gamma_{m}$ are among the generators of the group
$Semispin(8)$. 21 other generators are

$\displaystyle\Sigma_{mn}\ =\ \frac{i}{2}[\Gamma_{m},\Gamma_{n}]\,.$ (3.5)

One can also define

$\displaystyle\Gamma_{M}\ =\ (\vec{\Gamma},i),\qquad\Gamma_{M}^{\dagger}\ =\
(\vec{\Gamma},-i)\,,$ (3.6)

Then the whole set of the generators of $Semispin(8)$ can be presented as

$\displaystyle\Sigma_{MN}\ =\
\frac{i}{2}(\Gamma_{M}\Gamma_{N}^{\dagger}-\Gamma_{N}\Gamma_{M}^{\dagger})\,.$
(3.7)

The embedding (1.2) is quite analogous to (2.14), but the difference is that
in this case not all the generators of $Semispin(8)$ are engaged and it is an
injective rather than surjective map. The sign in (1.2) is chosen such that
the topological charge (1.13) is positive.

The matrix (1.2) treated as an element of $SO(8)$ describes the rotation by
the angle $\alpha$ in the plane spanned by the unit vectors $\vec{e}_{8}$ and
$\alpha_{j}\vec{e}_{j}/\alpha$. This brings the axis $Ox_{8}$ in a new
position. An arbitrary $SO(8)$ rotation can be presented as the superposition
of such rotation and an $SO(7)$ rotation “around” the new axis. Similarly, an
arbitrary element of $Semispin(8)$ can be presented as

$\displaystyle g\in Semispin(8)\ =\
\exp\\{i\alpha_{j}\Gamma_{j}\\}\times\left[g\in Spin(7)\right]\,,$ (3.8)

$0\leq\|\alpha_{j}\|\leq\pi$.

Note that, by the variable change

$\displaystyle\alpha_{m}\ =\
\frac{x_{m}}{\|\vec{x}\|}\arccos\left(\frac{x_{8}}{r}\right)\,,$ (3.9)

(1.2) can be presented in the form analogous to (2.6):

$\displaystyle g_{8}\ =\ \frac{i\Gamma_{M}^{\dagger}x_{M}}{r}\ =\
\frac{\mathbb{1}x_{8}+ix_{m}\Gamma_{m}}{r}$ (3.10)

with $r^{2}=(x_{M})^{2}$. The unit vector $n_{M}=x_{M}/r$ gives a natural
“round” parameterization of $S^{7}$.

The embedding (1.2), (3.10) represents a section of the fiber bundle

$\displaystyle Semispin(8)\ \stackrel{{\scriptstyle
Spin(7)}}{{\longrightarrow}}\ S^{7}\,.$ (3.11)

Note that, for $Semispin(8)$, the representation (3.8) is not unique. For
example, a unit element of $Semispin(8)$ can be represented in two ways:

$\displaystyle\mathbb{1}_{Semispin(8)}\ =\
\mathbb{1}_{S^{7}}\times\mathbb{1}_{Spin(7)}\ =\
(-\mathbb{1})_{S^{7}}\times(-\mathbb{1})_{Spin(7)}\,.$ (3.12)

However, an element of $Spin(8)$, the double covering of $Semispin(8)$, is
represented by two distinct matrices, $\exp\\{i\alpha_{j}\Gamma_{j}\\}$ and
$g_{7}\in Spin(7)$, rather than by their product, and this representation is
unique.

As a result, the whole group $Spin(8)$ represents the direct product
$S^{7}\times Spin(7)$, the bundle (3.11) is trivial and the embedding (1.2) is
its global section.666The triviality of the bundle (3.11) is a well known
fact. It follows also from a construction based on a representation of $SO(8)$
elements as certain octonion multiplications [16]. It is this fact that allows
one to derive the result (1.8).

Substituting (3.10) into (2.4), we derive the expression for the asymptotic
gauge field:

$\displaystyle A_{M}\ =\ \frac{\Sigma_{NM}x_{N}}{r^{2}}$ (3.13)

Now, pick up the north pole of $S^{7}$, $x_{M}=(\vec{0},r)$. At this point,
$A_{8}=0$ and

$\displaystyle g^{-1}\partial_{m}g=-iA_{m}\ =\ \frac{i\Gamma_{m}}{r}\,.$
(3.14)

The integrand in (1.13) is

$\displaystyle-(i)^{7}\frac{7!}{r^{7}}{\rm
Tr}\,\\{\Gamma_{1}\Gamma_{2}\Gamma_{3}\Gamma_{4}\Gamma_{5}\Gamma_{6}\Gamma_{7}\\}\
=\ -\frac{8!}{r^{7}}$ (3.15)

[the first minus sign in (3.15) comes from $\epsilon_{81234567}=-1$, cf. Eq.
(2.9)].

Multiplying it by the volume of $S^{7}$,

$\displaystyle V_{S^{7}}\ =\ \frac{\pi^{4}r^{7}}{3}\,,$ (3.16)

and substituting in (1.13), we derive

$\displaystyle q\ =\ 1\,.$ (3.17)

There is one particular field $A_{M}(x\in\mathbb{R}^{8})$ in the sector $q=1$
where the field density satisfies the nonlinear self-duality condition

$\displaystyle F\wedge F\ =\ \star(F\wedge F)$ (3.18)

It reads

$\displaystyle A_{M}(x)\ =\ \frac{\Sigma_{NM}x_{N}}{r^{2}+\rho^{2}}\,,$ (3.19)

going to (3.13) in the large $r$ limit.

The field density is

$\displaystyle F_{MN}\ =\ \partial_{M}A_{N}-\partial_{N}A_{M}-i[A_{M},A_{N}]\
=\ \frac{2\rho^{2}\Sigma_{MN}}{(r^{2}+\rho^{2})^{2}}\,.$ (3.20)

The gauge field configuration (3.19), (3.20) is what is called $SO(8)$
instanton [2, 3]. It realizes the minimum of the functional

$\displaystyle\int_{R^{8}}{\rm Tr}\,\\{F\wedge F\wedge\star(F\wedge F)\\}$
(3.21)

in the sector $q=1$. The analogy with the BPST instanton (2.13) is
straightforward.

Starting from the map

$\displaystyle g\ =\ \frac{\mathbb{1}x_{8}-ix_{m}\Gamma_{m}}{r}$ (3.22)

with the topological charge $q=-1$, we arrive at the antiinstanton

$\displaystyle A_{M}(x)\ =\
\frac{\tilde{\Sigma}_{NM}x_{N}}{r^{2}+\rho^{2}}\,,$ (3.23)

where
$\tilde{\Sigma}_{NM}=\frac{i}{2}(\Gamma_{M}^{\dagger}\Gamma_{N}-\Gamma_{N}^{\dagger}\Gamma_{M})$.
The corresponding field density is anti-self-dual, $F\wedge F\ =\
-\star(F\wedge F)$.

The configurations with an arbitrary topological charge are also known [6].
First of all, one can write a natural 8-dimensional generalisation of the ’t
Hooft Ansatz:

$\displaystyle A_{M}(x)\ $ $\displaystyle=$ $\displaystyle\
-\tilde{\Sigma}_{NM}\partial_{N}\ln\left(1+\sum_{i=1}^{q}\frac{\rho_{i}^{2}}{(x_{P}-a_{P}^{(i)})^{2}}\right)\qquad{\rm
and}$ $\displaystyle\quad A_{M}(x)\ $ $\displaystyle=$ $\displaystyle\
-\Sigma_{NM}\partial_{N}\ln\left(1+\sum_{i=1}^{q}\frac{\rho_{i}^{2}}{(x_{P}-a_{P}^{(i)})^{2}}\right)\,.$
(3.24)

The topological charges of these configurations are $q$ and $-q$,
respectively. When $q=1$, we reproduce the instanton of [2, 3], but written in
the singular gauge. One can also write more general multiinstanton solutions
generalizing the AHDM construction.

Note, however, that all these configurations are characterized by only one
integer (1.6), whereas the homotopy group (1.8) implies the existence of the
topological classes characterized by two integers. Such generalized $SO(8)$
instantons have not been studied so far. They are related to $Spin(7)$
instantons to be discussed in the next section.

##  4 $Spin(7)$ instantons

The generators of $Spin(7)\subset Semispin(8)$ are given in (3.5). As was
noted above, the embedding (1.2), which involves seven generators of
$Semispin(8)$ that are not in the set (3.5), is a global section in the fiber
bundle (3.11).

It is known, however, that the sphere $S^{7}$ also represents a base for
another fiber bundle

$\displaystyle Spin(7)\ \stackrel{{\scriptstyle G_{2}}}{{\longrightarrow}}\
S^{7}\,.$ (4.1)

Each fiber of this bundle represents a subgroup $G_{2}\subset Spin(7)$ that
does not transform a particular 8-component real spinor (it is one of the
definitions of $G_{2}$). The set of all such spinors of unit norm gives the
base $S^{7}$ of the bundle.

The bundle (4.1) implies a nontrivial homotopy

$\displaystyle\pi_{7}[Spin(7)]\ =\ \mathbb{Z}\,.$ (4.2)

Indeed, one can write the following exact sequence:

$\displaystyle\pi_{7}[G_{2}]\ \stackrel{{\scriptstyle 1}}{{\longrightarrow}}\
\pi_{7}[Spin(7)]\ \stackrel{{\scriptstyle 2}}{{\longrightarrow}}\
\pi_{7}[S^{7}]\stackrel{{\scriptstyle 3}}{{\longrightarrow}}\pi_{6}[G_{2}]$
(4.3)

or

$\displaystyle 0\ \stackrel{{\scriptstyle 1}}{{\longrightarrow}}\
\pi_{7}[Spin(7)]\ \stackrel{{\scriptstyle 2}}{{\longrightarrow}}\
\mathbb{Z}\stackrel{{\scriptstyle 3}}{{\longrightarrow}}Z_{3}$ (4.4)

(the homotopies of $G_{2}$ can be found in [10]).

The image of the map2 coincides with the kernel of the map 3, which is
$\mathbb{Z}$. Hence, $\pi_{7}[Spin(7)]$ is $\mathbb{Z}$ or larger. But the
kernel of the map 2 coincides with the image of the map 1, which is trivial.
Hence $\pi_{7}[Spin(7)]$ cannot be larger than $\mathbb{Z}$, and we are led to
(4.2).

In the previous section, we discussed the embedding (1.2) of $S^{7}$ into
$Semispin(8)$, which was topologically nontrivial. But only two elements of
(1.2): $g=\mathbb{1}$ and $g=-\mathbb{1}$ were also the elements of $Spin(7)$.
Probably, the main new result of this paper is an explicit construction for a
topologically nontrivial embedding $S^{7}\to Spin(7)$.

Figure 1: Fano graph.

Consider the expression

$\displaystyle g\ =\
\exp\left\\{-\frac{1}{2}\alpha_{m}f_{mnk}\Gamma_{n}\Gamma_{k}\right\\}\,,$
(4.5)

where $f_{mnk}$ are the structure constants of the octonion algebra,

$\displaystyle f_{165}\ =\ f_{134}\ =\ f_{127}\ =\ f_{235}\ =\ f_{246}\ =\
f_{367}\ =\ f_{475}\ =\ 1$ (4.6)

and all other nonzero elements of $f_{mnk}$ are restored by antisymmetry (see
the Fano graph on Fig. 1). (4.5) is an element of $Spin(7)$. Its logarithm
represents a linear combination of seven $Spin(7)$ generators, $T_{m}\ =\
\frac{1}{2}f_{mnk}\Sigma_{nk}$. Explicitly:

$\displaystyle T_{1}\ =\ \Sigma_{34}+\Sigma_{27}-\Sigma_{56},\qquad T_{2}\ =\
\Sigma_{35}+\Sigma_{46}-\Sigma_{17},\qquad T_{3}\ =\
\Sigma_{67}-\Sigma_{14}-\Sigma_{25},$ $\displaystyle T_{4}\ =\
\Sigma_{13}-\Sigma_{26}-\Sigma_{57},\qquad T_{5}\ =\
\Sigma_{23}+\Sigma_{16}+\Sigma_{47},$ $\displaystyle T_{6}\ =\
\Sigma_{24}-\Sigma_{15}-\Sigma_{37},\qquad T_{7}\ =\
\Sigma_{12}+\Sigma_{36}-\Sigma_{45}.$ (4.7)

Each $T_{m}$ is a linear combination of three commuting generators of
$Spin(7)$.

######  Theorem 1.

The expression (4.5) considered in the range $0\leq\|\vec{\alpha}\|\leq\pi$
represents an embedding of $S^{7}$ into $Spin(7)$. Locally, it represents a
section of the fiber bundle (4.1).

###### Proof.

To prove that (4.5) is an embedding of $S^{7}$, we only have to prove that,
for any $\vec{\alpha}$ of norm $\pi$, $g(\vec{\alpha})=-\mathbb{1}$, as it was
for the mappings (2.14) and (1.2). To this end, we notice that $f_{mnk}$ is an
invariant tensor of $G_{2}$ and hence (4.5) is a $G_{2}$ invariant depending
on a 7-dimensional vector $\alpha_{m}$. $G_{2}$ may be defined as the subgroup
of $SO(7)$ involving only the rotations leaving invariant the “mixed product”
of three different 7-vectors, $I\ =\ f_{mnk}A_{m}B_{n}C_{k}$. But we have only
one vector at our disposal, and (4.5) may depend only on the norm of
$\alpha_{m}$, but not on its orientation. Choosing
$\alpha_{m}=(\pi,0,0,0,0,0,0)$, we derive

$\displaystyle g(\|\alpha_{m}\|=\pi)=e^{i\pi
T_{1}}=\exp\\{-\pi\Gamma_{3}\Gamma_{4}\\}\exp\\{-\pi\Gamma_{2}\Gamma_{7}\\}\exp\\{\pi\Gamma_{5}\Gamma_{6}\\}\
=\ (-\mathbb{1})^{3}=-\mathbb{1}.$ (4.8)

To prove that (4.5) represents a local section of the bundle (4.1), we need to
prove that the generators (4) are not in the subalgebra $g_{2}\subset
spin(7)$, but are orthogonal to it:

$\displaystyle{\rm Tr}\\{T_{j}h\\}\ =\ 0,\qquad{\rm if}\ h\in g_{2}\,.$ (4.9)

Choosing $f_{mnk}$ as in (4.6), the $G_{2}$ invariant acquires the following
explicit form

$\displaystyle I\ =\ f_{jkl}A_{j}B_{k}C_{l}\ =\
[165]+[134]+[127]+[235]+[246]+[367]+[475],$ (4.10)

where

$\displaystyle[165]\ =\
(A_{1}B_{6}-A_{6}B_{1})C_{5}+(B_{1}C_{6}-B_{6}C_{1})A_{5}+(C_{1}A_{6}-C_{6}A_{1})B_{5}\,,$
(4.11)

etc. The explicit form of the generators of $G_{2}\subset SO(7)$, which do not
transform $I$, is

$\displaystyle\tilde{\Sigma}_{34}+\tilde{\Sigma}_{56},\quad\tilde{\Sigma}_{35}-\tilde{\Sigma}_{46},\quad\tilde{\Sigma}_{25}-\tilde{\Sigma}_{14},\quad\tilde{\Sigma}_{13}+\tilde{\Sigma}_{26},\
\tilde{\Sigma}_{23}-\tilde{\Sigma}_{16},\quad\tilde{\Sigma}_{15}+\tilde{\Sigma}_{24},\quad\tilde{\Sigma}_{12}-\tilde{\Sigma}_{36},$
$\displaystyle\tilde{\Sigma}_{34}-\tilde{\Sigma}_{27},\quad\tilde{\Sigma}_{35}+\tilde{\Sigma}_{17},\quad\tilde{\Sigma}_{14}+\tilde{\Sigma}_{67},\quad\tilde{\Sigma}_{13}+\tilde{\Sigma}_{57},\quad\tilde{\Sigma}_{23}-\tilde{\Sigma}_{47},\quad\tilde{\Sigma}_{37}-\tilde{\Sigma}_{15},\quad\tilde{\Sigma}_{12}+\tilde{\Sigma}_{45},\
\ \ \ $ (4.12)

where $\tilde{\Sigma}_{mn}$ are the counterparts of (3.5) in the vector
representation.

Consider e.g. the action of the operator $\tilde{\Sigma}_{34}$ on $I$. The
structures [165], [134] and [127] are left invariant. But four other
structures are not:

$\tilde{\Sigma}_{34}\,I\ \propto\ [245]-[236]+[467]-[375].$

One can observe, however, that this exactly coincides with
$-\tilde{\Sigma}_{56}\,I$ and $\tilde{\Sigma}_{27}\,I$, so that $I$ does not
transform under the action of the generators
$\tilde{\Sigma}_{34}+\tilde{\Sigma}_{56}$ or
$\tilde{\Sigma}_{34}-\tilde{\Sigma}_{27}$. On the other hand,

$(\tilde{\Sigma}_{34}+\tilde{\Sigma}_{27}-\tilde{\Sigma}_{56})\,I\ \propto\
3([245]-[236]+[467]-[375])\ \neq\ 0.$

The same is true for six other generators in (4) and their arbitrary linear
combinations.

The property (4.9) can be verified explicitly.

Hence the generators (4) together with (4) form an orthogonal basis in
$spin(7)$ and an arbitrary element of $Spin(7)$ can be represented as

$\displaystyle g\in Spin(7)\ =\ g_{7}\times\left[g\in G_{2}\right]\,,$ (4.13)

where $g_{7}$ is the embedding (4.5).

We have thus proven that (4.5) represents a local section of the bundle (4.1).
We will see a bit later that the representation (4.13) is not unique and hence
(4.5) is not its global section.

∎

By applying an approriate $G_{2}$ rotation, the exponential in (1.12) can be
disentangled for an arbitrary $\vec{\alpha}$, the condition
$\alpha\equiv\|\vec{\alpha}\|=\pi$ is not necessary. For $\alpha_{m}=\
(\alpha_{1},0,0,0,0,0,0)$, we derive

$\displaystyle g_{7}\ =\
(\cos\alpha_{1}\,\mathbb{1}-\sin\alpha_{1}\,\Gamma_{3}\Gamma_{4})(\cos\alpha_{1}\,\mathbb{1}-\sin\alpha_{1}\,\Gamma_{2}\Gamma_{7})(\cos\alpha_{1}\,\mathbb{1}+\sin\alpha_{1}\,\Gamma_{5}\Gamma_{6})$
$\displaystyle=\
\cos^{3}\alpha_{1}\,\mathbb{1}+i\cos^{2}\alpha_{1}\sin\alpha_{1}\,T_{1}+\cos\alpha_{1}\sin^{2}\alpha_{1}\,\Gamma_{1}T_{1}+i\sin^{3}\alpha_{1}\,\Gamma_{1}.$
(4.14)

The third and the fourth term in the R.H.S. of this identity were derived
using (1.4).

For an arbitrary $\vec{\alpha}$, we obtain

$\displaystyle g_{7}=\
\cos^{3}\alpha\,\mathbb{1}+i\cos^{2}\alpha\sin\alpha\,\frac{\alpha_{m}T_{m}}{\alpha}+\cos\alpha\sin^{2}\alpha\,\frac{\alpha_{m}\alpha_{n}\Gamma_{m}T_{n}}{\alpha^{2}}+i\sin^{3}\alpha\,\frac{\alpha_{m}\Gamma_{m}}{\alpha}.$
(4.15)

This form is convenient for rewriting $g$ in the Cartesian coordinates
$x_{M}$. Using (3.9), we derive

$\displaystyle g_{7}\ =\
\frac{x_{8}^{3}\,\mathbb{1}+ix_{8}^{2}\,x_{m}T_{m}+x_{8}\,x_{m}x_{n}\Gamma_{m}\,T_{n}+ix_{m}x_{m}\,x_{n}\Gamma_{n}}{r^{3}}.$
(4.16)

We can explain now why (1.12) is not a global section.

The subgroup $G_{2}\subset Spin(7)$ spanned by the generators
$\Sigma_{34}+\Sigma_{56}$ etc. does not transform the spinor

$\displaystyle\psi_{0}\ =\ \left(\begin{array}[]{c}0\\\
1\end{array}\right)\otimes\left(\begin{array}[]{c}0\\\
1\end{array}\right)\otimes\left(\begin{array}[]{c}0\\\ 1\end{array}\right)\,.$
(4.23)

Let us direct $\alpha_{m}$ along the first axis and consider the action of a
group element (4) on this spinor. One can derive

$\displaystyle g(\alpha_{1})\psi_{0}\ =\ \cos 3\alpha_{1}\,\psi_{0}+\sin
3\alpha_{1}\,\psi_{1}\,,$ (4.24)

where

$\displaystyle\psi_{1}\ =\ \left(\begin{array}[]{c}1\\\
0\end{array}\right)\otimes\left(\begin{array}[]{c}1\\\
0\end{array}\right)\otimes\left(\begin{array}[]{c}1\\\ 0\end{array}\right)\,.$
(4.31)

We see that three different elements in the set (4) with $\alpha_{1}=0,\pm
2\pi/3$, and many more different elements in the set (4.15) keep $\psi_{0}$
intact777Any element belonging to $S^{6}$ with $\|\alpha_{m}\|=2\pi/3$ does
so. and hence belong to one and the same $G_{2}$ fiber. For a global section,
this would not be possible. In contrast to the bundle (3.11) that could be
trivialized, the bundle (4.1) cannot. $Spin(7)$ is not homeomorphic to the
direct product $S^{7}\times G_{2}$ (indeed, homotopy groups of these two
spaces are different [10]) and a global section of the bundle (4.1) does not
exist.

Our next task is to evaluate the topological charge (1.13) for the map (4.15).
A nonzero value of the charge will assure the topological nontriviality of the
mapping.888Not any mapping of a sphere into a Lie group is topologically
nontrivial. For example, the expression  $\displaystyle g\ =\
\exp\\{i(\alpha_{1}\sigma_{1}+\alpha_{2}\sigma_{2})\\}$ (4.32) with
$0\leq\sqrt{\alpha_{1}^{2}+\alpha_{2}^{2}}\leq\pi$ realizes a mapping
$S^{2}\to SU(2)$ related to the fiber bundle $SU(2)\ \stackrel{{\scriptstyle
U(1)}}{{\longrightarrow}}\ S^{2}$. However, there is no topological charge
associated with the mapping (4.32) and this mapping is topologically trivial
(contractible to a point by a continuous deformation): $\pi_{2}[SU(2)]=0$.

The calculation of $q$ for the map (1.12) is somewhat more complicated than
for the map (1.2), but we can capitalize again on the rotational invariance of
the integrand, calculate it for $\alpha_{m}=\alpha\delta_{m1}$, multiply the
integrand by the volume of $S^{6}$ and then integrate it over $\alpha$. With
the help of Mathematica, we derived the result

$\displaystyle q[g_{7}]\ =\
\frac{512}{75\pi}\,\int_{0}^{\pi}\sin^{6}\alpha\cos^{2}\alpha(2\cos^{4}\alpha+5\cos^{2}\alpha+2)\,d\alpha\
=\ 1\,.$ (4.33)

For the embedding $g_{8}$, it was not necessary, but we could also calculate
the topological charge $q[g_{8}]$ by the same method to derive

$\displaystyle q[g_{8}]\ =\
\frac{16}{5\pi}\,\int_{0}^{\pi}\sin^{6}\alpha\,d\alpha\ =\ 1\,.$ (4.34)

For the maps $g=g_{7}^{2}$ and $g=g_{8}^{2}$, the topological charge is $q=2$.
One can be convinced that $q[g_{7}^{2}]$ and $q[g_{8}^{2}]$ are given by the
integrals where the integrands are related to the integrands in (4.33), (4.34)
as

$\displaystyle f_{2}(\alpha)\ =\ 2f_{1}(2\alpha)\,.$ (4.35)

Having the expression (4.15) for $g_{7}(x)$ in hand, one can derive the
expressions for the gauge potentials

$\displaystyle A_{M}=ig_{7}^{-1}\partial_{M}g_{7}$ (4.36)

at the distant 7-sphere. However, they are rather complicated and we will not
quote them here.

All topologically nontrivial $spin(7)$ gauge field configurations belonging to
the sector $q=1$ have the asymptotics (4.36). There is a distinguished
configuration realizing the minimum of the functional (3.21) and satisfying
the nonlinear self-duality conditions (3.18). This is the $Spin(7)$ instanton.
Its explicit form is, however, difficult to find.

The embedding

$\displaystyle g\ =\
\exp\left\\{\frac{1}{2}\alpha_{m}f_{mnk}\Gamma_{n}\Gamma_{k}\right\\}\,,$
(4.37)

has the topological charge $q=-1$. It brings about the anti-self-dual
antiinstanton.

Note that the maps $S^{7}\to Spin(7)$ are also maps of $S^{7}$ into $SO(8)$.
As was mentioned above, the generalized topologically nontrivial maps
$S^{7}\to SO(8)$ and the corresponding instantons are characterized by two
integer charges $(q,\tilde{q})$. The Chern classes $q$ of the instanton (3.19)
and of the instanton, corresponding to the embedding (1.12) are both equal to
one.

But the values of the second charge (1.14) are different for these two
embeddings.

Let us first explain how (1.14) is derived. The integrand in (1.9) is an
8-form, which is closed in $\mathbb{R}^{8}$. Due to the trivial topology of
$\mathbb{R}^{8}$, it is also exact,

$\displaystyle\varepsilon^{a_{1}\cdots a_{8}}\,F^{a_{1}a_{2}}\wedge
F^{a_{3}a_{4}}\wedge F^{a_{5}a_{6}}\wedge F^{a_{7}\,a_{8}}\ =\
d\tilde{\omega}_{7}\,,$ (4.38)

and the integral (1.9) only depends on the gauge field $A_{S^{7}}$ at the
distant 7-sphere.

We can find $\tilde{\omega}_{7}$ by the same method as $\omega_{7}$ in (3.3)
was found. Consider instead of $\mathbb{R}^{8}$ an open 8-dimensional ball of
unit radius and consider there the gauge potential $A_{B^{8}}=rA_{S^{7}}$,
where $r$ is the distance from the center. The field density form is then

$\displaystyle F_{B^{8}}\ =\ dr\wedge
A_{S^{7}}+rdA_{S^{7}}-ir^{2}A_{S^{7}}\wedge A_{S^{7}}\,.$ (4.39)

We derive

$\displaystyle\varepsilon^{a_{1}\cdots a_{8}}\,F^{a_{1}a_{2}}\wedge
F^{a_{3}a_{4}}\wedge F^{a_{5}a_{6}}\wedge
F^{a_{7}\,a_{8}}=4\varepsilon^{a_{1}\cdots a_{8}}dr\wedge
A_{S^{7}}^{a_{1}a_{2}}\wedge(rdA_{S^{7}}-ir^{2}A_{S^{7}}\wedge
A_{S^{7}})^{a_{3}a_{4}}$
$\displaystyle\wedge(rdA_{S^{7}}-ir^{2}A_{S^{7}}\wedge
A_{S^{7}})^{a_{5}a_{6}}\wedge(rdA_{S^{7}}-ir^{2}A_{S^{7}}\wedge
A_{S^{7}})^{a_{7}a_{8}}\,.$ (4.40)

We are interested only with $\tilde{\omega}_{7}$ at the boundary of the ball
and assume that the field density vanishes there. Using this, we can
substitute $iA_{S^{7}}\wedge A_{S^{7}}$ for $dA_{S^{7}}$ and integrate over
$r$. We derive

$\displaystyle\tilde{\omega}_{7}[{\rm boundary}]\ =\
-4i\int_{0}^{1}(r^{2}-r)^{3}\,dr\ \varepsilon^{a_{1}\cdots
a_{8}}A^{a_{1}a_{2}}\wedge$ $\displaystyle(A\wedge
A)^{a_{3}a_{4}}\wedge(A\wedge A)^{a_{5}a_{6}}\wedge(A\wedge A)^{a_{7}a_{8}}\
=$ $\displaystyle\frac{i}{35}\,\varepsilon^{a_{1}\cdots
a_{8}}A^{a_{1}a_{2}}\wedge(A\wedge A)^{a_{3}a_{4}}\wedge(A\wedge
A)^{a_{5}a_{6}}\wedge(A\wedge A)^{a_{7}a_{8}}\,.$ (4.41)

This gives (1.14).

The calculation gives $\tilde{q}[g_{8}]=1$, while

$\displaystyle\tilde{q}[g_{7}]\ =\
-\frac{512}{25\pi}\,\int_{0}^{\pi}\sin^{6}\alpha\cos^{2}\alpha(2\cos^{4}\alpha+5\cos^{2}\alpha+2)\,d\alpha\
=\ -3\,.$ (4.42)

We see the same integral as (4.33) but with an additional factor -3 !
Obviosly, the presence of this factor is related to the fact that (1.12) is a
local rather than global section of the bundle (4.1) and the presence of
triple intersections of the embedding (1.12) with any $G_{2}$ fiber, as
displayed in Eq.(4.24).

Consider now the composite map $g=g_{7}g_{8}$ and calculate its topological
charges. We derive

$\displaystyle q[g_{7}g_{8}]\ =\ -\tilde{q}[g_{7}g_{8}]\ =\
\frac{32}{5\pi}\,\int_{0}^{\pi}\sin^{6}(2\alpha)\,d\alpha\ =\ 2\,,$ (4.43)

the same for $g=g_{8}g_{7}$ and the same up to the sign for
$g=g_{8}^{-1}g_{7}^{-1}$.

Finally, for the map $g=g_{7}^{-1}g_{8}$ or $g=g_{8}g_{7}^{-1}$ the
topologocal charges are

$\displaystyle q[g_{7}g_{8}^{-1}]\ =\ \int_{0}^{\pi}0\,d\alpha\ =\ 0$ (4.44)

and

$\displaystyle\tilde{q}[g_{7}^{-1}g_{8}]\ =\
\frac{64}{5\pi}\,\int_{0}^{\pi}\sin^{6}(2\alpha)\,d\alpha\ =\ 4\,.$ (4.45)

Changing $g\to g^{-1}$ reverses the sign of the both charges.

These results were put together in Table 1.

We see that the topological charges of the composite maps coincide with the
sums of the individual topological charges, as it, of course, should be. For a
generic composition, $g=g_{7}^{n}g_{8}^{m}$,

$\displaystyle q(g_{8}^{n}g_{7}^{m})\ =\
n+m,\qquad\tilde{q}(g_{8}^{n}g_{7}^{m})\ =\ n-3m\,.$ (4.46)

##  5 Other groups and dimensions

As was mentioned, $\pi_{4n-1}[SO(4n)]=\mathbb{Z}\times\mathbb{Z}$, which
inplies that the mappings $S^{4n-1}\to SO(4n)$ and associated instantons are
characterized by two integer invariants. It is not too difficult to generalize
for higher $n$ the conventional $SO(8)$ instantons [3, 17]. All the formulas
are basically the same as in Sect. 3.1. where one has to replace 7-dimensional
gamma matrices by $4n-1$-dimensional gamma matrices
$\Gamma_{m=1,\ldots,4n-1}$. Note that these matrices have large dimension
$2^{2n-1}\times 2^{2n-1}$ and the formulas like (3.19), (3.20) describe now
gauge potentials and field densities in the spinor representation, which is
not equivalent for $n\neq 2$ to the vector representation.

The (anti-)instantons realize here the minimum of the functional

$\displaystyle\int_{R^{4n}}{\rm Tr}\,\\{\overbrace{F\wedge\ldots\wedge F}^{n}\
\star(\overbrace{F\wedge\ldots\wedge F}^{n})\\}$ (5.1)

The field densities satisfy the following nonlinear self-duality conditions

$\displaystyle\overbrace{F\wedge\ldots\wedge F}^{n}\ =\
\pm\star(\overbrace{F\wedge\ldots\wedge F}^{n})\,.$ (5.2)

There should exist also another type of $Spin(4n)$ instantons which are
simultaneosly $Spin(4n-1)$ instantons and which are associated with the
multidimensional analogs of the mapping (1.12). However, explicit expressions
for such mappings, not speaking of the explicit expressions for the instantons
are not known.

We can stay in $4n$ dimensions but consider the gauge group that is smaller
than $SO(4n)$. For example, one can take the group $SU(2n)$. In this case,
$\pi_{4n-1}[SU(2n)]=\mathbb{Z}$ and there is only one type of instanton [18].

Alternatively, one can take the still smaller gauge group $Sp(n)\subset
SU(2n)$, which also enjoys a nontrivial homotopy
$\pi_{4n-1}[Sp(n)]=\mathbb{Z}$ related to the fiber bundle

$Sp(n)\ \stackrel{{\scriptstyle Sp(n-1)}}{{\longrightarrow}}\ S^{4n-1}\,,$

which implies the presence of instantons.999One can notice in this regard that
$Sp(n)$ is the holonomy group of a hyper-Kähler manifold of dimension $4n$.
This suggests that the instantons with the gauge group $Sp(n)$ exist not only
in $R^{4n}$, but in any $4n$-dimensional hyper-Kähler manifold.

It would be interesting to make the ADHM-inspired construction of Ref. [18]
for $SU(2n)$ instantons more explicit and generalize it to the group $Sp(n)$.

One can also construct topologically nontrivial gauge field configurations in
$R^{4n-2}$. In 6 dimensions, they are associated with the nontrivial
$\pi_{5}[SU(3)]=\mathbb{Z}$ and the fiber bundle $SU(3)\stackrel{{\scriptstyle
SU(2)}}{{\longrightarrow}}S^{5}$. In this case, each fiber represents the
subgroup $SU(2)\subset SU(3)$ that leaves intact a unit complex vector,

$\displaystyle V\ =\ \left(\begin{array}[]{c}\alpha_{1}\\\ \alpha_{2}\\\
\alpha_{3}\end{array}\right),\qquad|\alpha_{1}|^{2}+|\alpha_{2}|^{2}+|\alpha_{3}|^{2}=1,$
(5.6)

and the base $S^{5}$ is the set of all such vectors. An explicit expression
for the mapping $S^{5}\longrightarrow SU(3)$ is not so nice as in (1.12), but
it is known [19]. If $|\alpha_{1}|>0$, the vector (5.6) maps into the
matrix101010There are two similar expressions $g_{2,3}$ for the regions
$|\alpha_{2}|>0$ and $|\alpha_{3}|>0$. $S^{5}$ is described as a union of
these three regions.

$\displaystyle g_{1}\ =\
\left(\begin{array}[]{ccc}-\frac{\alpha_{1}\alpha_{2}^{*}}{\sqrt{1-|\alpha_{2}|^{2}}}&-\frac{\alpha_{3}^{*}}{\sqrt{1-|\alpha_{2}|^{2}}}&\alpha_{1}\\\
\sqrt{1-|\alpha_{2}|^{2}}&0&\alpha_{2}\\\
-\frac{\alpha_{3}\alpha_{2}^{*}}{\sqrt{1-|\alpha_{2}|^{2}}}&\frac{\alpha_{1}^{*}}{\sqrt{1-|\alpha_{2}|^{2}}}&\alpha_{3}\end{array}\right)\
\in\ SU(3)\,.$ (5.10)

With this expression in hand, one can in principle calculate $A=ig^{-1}dg$ and
determine the topological charge of the mapping, which should give

$\displaystyle q\ =\ \pm\frac{1}{480\pi^{3}}\int_{S^{5}}{\rm
Tr}\\{(g^{-1}dg)^{5}\\}\ =\ \pm 1\,.$ (5.11)

The coefficient in (5.11) is obtained from the expression (1.5) for the third
Chern class,

$\displaystyle q\ =\ \pm\frac{1}{48\pi^{3}}\int{\rm Tr}\\{F\wedge F\wedge
F\\}\,,$ (5.12)

and the identity

$\displaystyle{\rm Tr}\\{F^{3}\\}\ =\ d\omega_{5}$ (5.13)

with

$\displaystyle\omega_{5}\ =\ {\rm
Tr}\left\\{AF^{2}+\frac{i}{2}A^{3}F-\frac{1}{10}A^{5}\right\\}$ (5.14)

What one cannot do in this case is to impose a self-duality condition. This
can be done and the distinguished instanton configurations can be built up
only in the space of dimension $D=4n$.

## Akcnowledgements

I am indebted to Robert Bryant, Jose Figueroa-O’Farrill, Maxim Kontsevich,
Olaf Lechtenfeld, Nikita Nekrasov, George Savvidy and Ilia Smilga for
illuminating discussions.

## References

  * [1] E. Corrigan, C. Devchand, D. Fairlie and J. Nuyts, First-order equations for gauge fields in spaces of dimension greater than four, Nucl. Phys. B214 (1983) 452; S. Fubini and H. Nicolai, The octonionic instanton, Phys. Lett. B155 (1985) 369.
  * [2] B. Grossman, T.W. Kephart and J.D. Stasheff, Solutions to gauge field equations in eight dimensions: Conformal invariance and the last Hopf map, Commun. Math. Phys. 96 (1984) 431.
  * [3] D.H. Tchrakian, Spherically symmetric gauge field configurations with finite action in 4p dimensions (p = integer), Phys. Lett. B150 (1985) 360.
  * [4] A.A. Belavin, A.M. Polyakov, A.S. Schwarz and Y.S. Tyupkin, Pseudoparticle solutions of the Yang-Mills equations, Phys. Lett. B59 (1975) 85.
  * [5] M.F. Atiyah, N.J. Hitchin, V.G. Drinfeld and Y.I. Manin, Construction of instantons, Phys. Lett. A65 (1978) 185; E. Corrigan, D.B. Fairlie, S. Templeton and P. Goddard, A Green’s function for the general selfdual gauge field, Nucl. Phys. B140 (1978) 31.
  * [6] A. Nakamula, S. Sasaki and K. Takesue, ADHM construction of (anti-)self-dual instantons in eight dimensions, Nucl. Phys. B910 (2016) 199, arXiv: 1604.01893 [hep-th]; E.K. Loginov, Octonionic instantons in eight dimensions, Phys. Lett. B816 (2021) 136244., arXiv: 2003.09601 [hep-th].
  * [7] M.J. Duff and J.X. Lu, Strings from five-branes, Phys. Rev. Lett. 66 (1991) 1402; R. Minasian, S.L. Shatashvili and P. Vanhove, Closed strings from $SO(8)$ Yang-Mills instantons, Nucl. Phys. B613 (2001) 87, arXiv: hep-th/0106096.
  * [8] B.A. Bernevig, J. Hu, N. Toumbas and S.-C. Zhang, The eight-dimensional quantum Hall effect and the octonions, Phys. Rev. Lett. 91 (2003) 236803, arXiv: cond-mat/0306045.
  * [9] See e.g. the last equation in [A.V. Smilga, Differential Geometry through Supersymmetric Glasses, World Scientific, 2020].
  * [10] M. Mimura, The homotopy groups of Lie groups of low rank, J. Math. Kyoto Univ., 6-2 (1967) 131.
  * [11] N.A. Nekrasov, private communication.
  * [12] See e.g. [B. McInnes, The semispin groups in string theory, J.Math.Phys. 40 (1999) 4699, arXiv:hep-th/9906059.]
  * [13] L.L. Uhlenbeck, Removable singularities in Yang-Mills fields, Commun. Math. Phys. 82 (1982) 11.
  * [14] G. ‘t Hooft, Computation of the quantum effects due to a four-dimensional pseudoparticle, Phys. Rev. D14 (1976) 3432.
  * [15] B. Zumino, Yong-Shi Wu and A. Zee, Chiral anomalies, higher dimensions and differential geometry, Nucl. Phys. B239 (1984) 477.
  * [16] See e.g. Theorem 8.6 in [N. Steenrod, Topology of Fibre Bundles, Princeton Univ. Press, 1999].
  * [17] D. O’Se and D. H. Tchrakian, Conformal properties of the BPST instantons of the generalized Yang-Mills system, Lett. Math. Phys. 13 (1987) 211.
  * [18] K. Takesue, ADHM construction of (anti-)self-dual Instantons in 4n dimensions, JHEP 07 (2017) 110, arXiv:1706.03518.
  * [19] G. Khanna, S. Mukhopadhyay, R. Simon and N. Mukunda, Geometric phases for $SU(3)$ representations and three-level quantum systems, Ann. Phys. 253 (1997) 55.