# Interpretable Distribution Shift Detection using Optimal Transport

Neha Hulkund  Nicolo Fusi  Jennifer Wortman Vaughan  David Alvarez-Melis

###### Abstract

We propose a method to identify and characterize distribution shifts in
classification datasets based on optimal transport. It allows the user to
identify the extent to which each class is affected by the shift, and
retrieves corresponding pairs of samples to provide insights on its nature. We
illustrate its use on synthetic and natural shift examples. While the results
we present are preliminary, we hope that this inspires future work on
interpretable methods for analyzing distribution shifts.

Machine Learning, ICML

##  1 Introduction and Related Work

As machine learning models become more widely used in high-stakes domains such
as healthcare and education, it is important to analyze situations when models
fail. This often occurs with distribution shift, e.g., when train and test
distributions differ. For example, a model could be trained on MRI data from
one machine and tested on data from another, leading to performance drops that
negatively affect patient care and derail trust. Detecting and identifying
these distribution shifts are important steps in deciding whether to apply
mitigation techniques, such as domain adaption or stronger regularization.
While out-of-distribution detection techniques identify which samples are
different than the training distribution, they give little reason as to why
the shifts might be occurring (Liang et al., 2018; Liu et al., 2020; Chen et
al., 2020). Much previous research has been focused on interpreting model
decisions from analyses on model gradients (Selvaraju et al., 2016; Zintgraf
et al., 2017; Shrikumar et al., 2016) to post-hoc model explanations like
feature summary statistics (Fisher et al., 2018; Rodríguez-Pérez & Bajorath,
2020). However, there has been comparatively less work in interpretability
from the dataset point of view, especially under the conditions of dataset
distribution shift. In this work, we propose a method to identify and
characterize distribution shift through examples using optimal transport (OT)
(Peyré & Cuturi, 2019).

##  2 Methodology

Consider two datasets $\\{\mathbf{x}_{a}^{i}\\}_{i=1}^{n}$ and
$\\{\mathbf{x}_{b}^{j}\\}_{j=1}^{m}$ with associated probability weights
$\\{\mathbf{a}\in\Delta_{n},\mathbf{b}\in\Delta_{m}\\}$. When interpreted as
empirical distributions $\alpha_{n},\beta_{m}$, OT provides a way to compare
them by solving the problem

$\mathrm{OT}(\alpha_{n},\beta_{m})=\min_{\pi\in\Pi(\alpha,\beta)}\sum_{i,j=1}^{n,m}d(\mathbf{x}_{a}^{i},\mathbf{x}_{b}^{j})\pi_{ij},$

where $\Pi(\alpha,\beta)$ is the set of transport couplings (joint
distributions) with $\mathbf{a}$ and $\mathbf{b}$ as marginals. The minimizing
$\pi^{*}$ can be interpreted as the ‘best’ (least-cost) soft matching between
the two samples, and its total cost, i.e.,
$\mathrm{OT}(\alpha_{n},\beta_{m})$, yields a notion of distance between them.
Although typically formulated between unlabeled samples, a recent extension of
OT allows for comparison of classification datasets, called the optimal
transport dataset distance (OTDD) (Alvarez-Melis & Fusi, 2020).

Here, we propose to use OTDD to provide examples of distribution shift. We
first compute OTDD between the original and shifted datasets, and then use the
optimal coupling $\pi^{*}$ to identify corresponding images across the two
datasets, ranking them by their dissimilarity. Specifically, we select pairs
$(i,j)$ for which $\pi^{*}_{ij}$ is high, and visualize them ranked according
to the value of $d(\mathbf{x}_{a}^{i},\mathbf{x}_{b}^{j})\cdot\pi^{*}_{ij}$.
Farthest and closest pairs indicate where there is most and least shift,
respectively. Furthermore, the fact that these pairs are the ‘best
corresponding’ (in an OT sense) across the datasets controls for natural
variation and highlights shift-induced change, potentially making them
interpretable and providing insights on the nature of the shift.

##  3 Synthetic MNIST example

As a proof of concept, we use a synthetic shift of Gaussian noise applied to
one class of the handwritten digits dataset MNIST.

Figure 1: MNIST shift dataset, where the top row is the classic training
dataset and the second row is the shifted testing dataset with Gaussian noise
added to class 1.

From the heatmap of matches in our coupling matrix $\pi^{*}$ in Figure 2, we
find that most classes match well, with the exception of class 1 from the
noisy dataset to class 8 from the original dataset. This indicates that a
noisy 1 looks more similar to an noiseless 8 than an noiseless 1, representing
a type of dataset shift. We can now examine this class mismatch more closely,
taking the OT coupling between the two classes to obtain the closest and
farthest ranked pairs Figure 3. From the pairings, the closest pairs of
noiseless images from class 8 have less definitive shape to them, possibly
creating mismatch between the classes. Furthermore, we found our mismatch
finding to be consistent with classifier prediction, where noiseless 8’s are
often mislabeled as 1’s under this setting. We believe that further analyses
on shape similarities on image datasets between classes can lead to insight
about class-level shifts.

Figure 2: Coupling matches for synthetic and natural examples. Figure 3: MNIST
pairs between class 1 and 8.

##  4 Day/Night Natural example

As a natural shift, we use the Common Objects Day and Night (CODaN) dataset,
an image classification dataset of 10 common object classes recorded in both
day and nighttime (Lengyel et al., 2021). The training dataset is the 10
classes taken from the day distribution and the testing dataset is 9 classes
taken from the day distribution and 1 class (bicycle) from the night
distribution, as seen in Figure 4. In the heatmap of matched classes in Figure
2, we can see a mismatched class pair $\\{0,2\\}$, or the night bicycle/day
motorcycle pair. This indicates that a bicycle at night is matched more
closely to a motorcycle in the day than it is to itself in the day. The
coupling pairs in Figure 5 illustrate which bicycle-motorcycle pairs are
closest and farthest in the dataset, implying that at night with lower image
brightness, bicycles look darker, similar to motorcycles during the daytime.

Figure 4: CODaN dataset shift, with class 0 (bicycle) images taken during the
day in the training set and at night in the testing set. Figure 5: CODaN pairs
between night-bicycle and day-motorcyle.

##  5 Discussion

The method we propose characterizes distribution shifts through OT distances
and image correspondences across datasets. Since these examples are best
corresponding, there is potential that they might offer qualitative
interpretability about dataset shift. In a domain such as healthcare, examples
of such pairings could then be provided to domain experts to gain insights of
possible causes of the data shift when model explanations are not sufficient.
While other methods illustrate examples of data that occur on an example
level, this method could be potentially useful for detecting class level
drifts within a dataset. Further work remains to evaluate how this method
works on other datasets and whether it does indeed lead to greater insight.

## References

  * Alvarez-Melis & Fusi (2020) Alvarez-Melis, D. and Fusi, N.  Geometric dataset distances via optimal transport.  _NeurIPS_ , 2020.
  * Chen et al. (2020) Chen, J., Li, Y., Wu, X., Liang, Y., and Jha, S.  Robust out-of-distribution detection in neural networks.  _CoRR_ , abs/2003.09711, 2020.  URL https://arxiv.org/abs/2003.09711.
  * Fisher et al. (2018) Fisher, A. J., Rudin, C., and Dominici, F.  Model class reliance: Variable importance measures for any machine learning model class, from the ”rashomon” perspective.  2018\.
  * Lengyel et al. (2021) Lengyel, A., Garg, S., Milford, M., and van Gemert, J. C.  Zero-shot day-night domain adaptation with a physics prior.  _ICCV_ , 2021.
  * Liang et al. (2018) Liang, S., Li, Y., and Srikant, R.  Enhancing the reliability of out-of-distribution image detection in neural networks.  _arXiv: Learning_ , 2018.
  * Liu et al. (2020) Liu, W., Wang, X., Owens, J. D., and Li, Y.  Energy-based out-of-distribution detection.  _ArXiv_ , abs/2010.03759, 2020.
  * Peyré & Cuturi (2019) Peyré, G. and Cuturi, M.  Computational optimal transport.  _Foundations and Trends in ML_ , 2019.  ISSN 1935-8237.  doi: 10.1561/2200000073.
  * Rodríguez-Pérez & Bajorath (2020) Rodríguez-Pérez, R. and Bajorath, J.  Interpretation of machine learning models using shapley values: application to compound potency and multi-target activity predictions.  _Journal of Computer-Aided Molecular Design_ , 34:1013 – 1026, 2020.
  * Selvaraju et al. (2016) Selvaraju, R. R., Das, A., Vedantam, R., Cogswell, M., Parikh, D., and Batra, D.  Grad-cam: Why did you say that? visual explanations from deep networks via gradient-based localization.  _CoRR_ , abs/1610.02391, 2016.  URL http://arxiv.org/abs/1610.02391.
  * Shrikumar et al. (2016) Shrikumar, A., Greenside, P., Shcherbina, A., and Kundaje, A.  Not just a black box: Learning important features through propagating activation differences.  _CoRR_ , abs/1605.01713, 2016.  URL http://arxiv.org/abs/1605.01713.
  * Zintgraf et al. (2017) Zintgraf, L. M., Cohen, T. S., Adel, T., and Welling, M.  Visualizing deep neural network decisions: Prediction difference analysis.  _CoRR_ , abs/1702.04595, 2017.  URL http://arxiv.org/abs/1702.04595.


Richard J. Szabo 11institutetext: Department of Mathematics, Heriot-Watt
University
Maxwell Institute for Mathematical Sciences, Edinburgh
11email<EMAIL_ADDRESS>
# Gravity versus Noncommutative Gauge Theory: A Double Copy Perspective

Richard J. Szabo

###### Abstract

We discuss how Moyal deformations of gauge theories, which arise naturally
from open string theory, fit into the paradigm of colour-kinematics duality
and the double copy of gauge theory to gravity. Along the way we encounter
novel noncommutative scalar field theories with rigid colour symmetry that
have no interacting commutative counterparts. These scalar theories offer new
perspectives on old ideas that rank one noncommutative gauge theories are
gravitational theories. This is rendered explicit in four dimensions where
they and their double copy images yield deformations of integrable theories
describing the self-dual sectors of Yang-Mills theory and gravity.

##  1 Introduction

It is an endearing pleasure for me to write this contribution on the occasion
of Gordon Semenoff’s 70th birthday. Gordon was my MSc and then PhD supervisor
at the University of British Columbia from 1990–1995. From him I learned how
to become a careful and independent researcher. The present article describes
interactions between quantum field theory, string theory, gravity and
mathematical physics, which are themes that have permeated Gordon’s research
interests.

The basic premise of this contribution is the double copy of noncommutative
Yang-Mills theory, as it appears in low-energy effective field theories on
D-branes in constant $B$-fields, to gravity, which we describe in §2 below. It
is based on the lengthy recent paper [1], which is engulfed in the modern
language of the homotopy algebraic approach to quantum field theory. In the
remaining sections we give a concise summary of some of the main points of
that work, in a manner which we hope is more palatable to a broader audience.
We skip most technical details, while referring to [1] for all mathematical
constructions as well as more thorough and complete citations to the relevant
literature.

##  2 Double Copying Noncommutative Yang-Mills Theory

### KLT Double Copy Relations.

The double copy of gauge theory to gravity has its origins in string theory,
going back to the mid-1980s with the famous relations between tree-level
scattering amplitudes of closed and open strings discovered by Kawai, Lewellen
and Tye (KLT) [2]. These relations reflect the topological property that a
sphere, representing a closed string worldsheet, can be realised by gluing
together two discs, representing open string worldsheets:

In the field theory limit $\alpha^{\prime}\longrightarrow 0$, the left-hand
side represents a tree-level amplitude for graviton scattering while the
right-hand side consists of amplitudes of gluons. This results in the famous
double copy slogan:

$\textrm{Gravity}\ =\ (\textrm{Gauge Theory})^{2}\ .$ (1)

At a more quantitative level, the KLT double copy relations between $n$-point
amplitudes read

$M_{n}^{L\otimes R}\ =\ \sum_{w,w^{\prime}}\,A_{n}^{L}(w)\
\bm{S}_{n}(w|w^{\prime})\ A_{n}^{R}(w^{\prime})\ ,$ (2)

where the sums run over orderings of $n$ external gluons, and $A_{n}$ are
colour-stripped amplitudes appearing in the full gluon amplitudes

${\mathcal{A}}_{n}\ =\ \sum_{S_{n}/\mathbbm{Z}_{n}}\,\>{\rm
Tr}\,(T^{a_{1}}\cdots T^{a_{n}})\ A_{n}(1,\dots,n)$ (3)

expressed as a sum over cyclic orderings of external particles.

The quantity $\bm{S}_{n}(w|w^{\prime})$ which glues left and right gauge
theory amplitudes together is called the _KLT momentum kernel_. It was
realised almost 30 years after the advent of the KLT relations that they can
be expressed as the inverses

$\bm{S}_{n}(w|w^{\prime})\ =\ \bm{m}_{n}(w|w^{\prime})^{-1}$ (4)

of bi-coloured scalar amplitudes $\bm{m}_{n}(1,\dots,n|1,\dots,n)$ in a field
theory known as the biadjoint scalar theory [3], which we shall discuss in
more detail later on.

### Noncommutative Field Theory.

In the late 1990s it was realised that, on ${\mathbbm{R}}^{d}$ with a constant
NS–NS $B$-field and flat background metric $g$, the low-energy interactions of
open strings ending on stacks of $N$ D-branes are described by a non-local
deformation of $U(N)$ Yang-Mills theory to a _noncommutative gauge theory_ [4,
5, 6, 7, 8, 9, 10]. Open strings see an effective metric $G$ and a bivector
$\theta$ related to the closed string background $(g,B)$, such that $G=g$ and
$\theta=0$ when $B=0$.

Colour-stripped open string amplitudes then have a simple $\theta$-dependence
given by phase factors

$A_{n}(1,\dots,n)_{G,\theta}\ =\ {\,\rm e}\,^{-{\,{\rm
i}\,}\sum_{i<j}\,p_{i}\times p_{j}}\ A_{n}(1,\dots,n)_{G,\theta=0}\ ,$ (5)

where $k\times q\ =\ \mbox{$\frac{1}{2}$}\,k_{\mu}\,\theta^{\mu\nu}\,q_{\nu}$
and $p_{i}$ are the external momenta. The right-hand side involves the open
string amplitude in the absence of the $B$-field but computed with the open
string metric $G$. In the Seiberg-Witten scaling limit where
$\alpha^{\prime}\longrightarrow 0$ with
$g\sim(\alpha^{\prime})^{2}\longrightarrow 0$, these phase factors are
captured by replacing the commutative pointwise product
$\mu(\phi_{1}\otimes\phi_{2}):=\phi_{1}\,\phi_{2}$ of fields
$\phi_{1},\phi_{2}\in\Omega^{0}({\mathbbm{R}}^{d},{\mathfrak{u}}(N))$ with
their associative Moyal product defined by

$\phi_{1}\star\phi_{2}\ =\ \mu\circ\exp\big{(}\mbox{$\frac{{\,{\rm
i}\,}}{2}$}\,\theta^{\mu\nu}\,\partial_{\mu}\otimes\partial_{\nu}\big{)}(\phi_{1}\otimes\phi_{2})\
.$ (6)

### KLT Relations with $\bm{B}$-Fields.

The $B$-field modifications of the KLT relations are trivial: we simply
rewrite the original KLT formula (2) by substituting (5) to get

$\displaystyle M_{n}^{L\otimes R}$ $\displaystyle\ =\ $
$\displaystyle\sum_{w,w^{\prime}}\,A_{n}^{L}(w)_{G,\theta}\
\bm{S}_{n}(w|w^{\prime})_{G;\theta,\bar{\theta}}\
A_{n}^{R}(w^{\prime})_{G,\bar{\theta}}\ ,$ (7)

where the gravitational amplitudes on the left-hand side are now computed with
the background metric $G$. The new momentum kernel is

$\displaystyle\bm{S}_{n}(w|w^{\prime})_{G;\theta,\bar{\theta}}$
$\displaystyle\ =\ $ $\displaystyle\Theta_{n}(w)^{-1}\
\bm{S}_{n}(w|w^{\prime})_{G;\theta=0,\bar{\theta}=0}\
\bar{\Theta}_{n}(w^{\prime})^{-1}\ ,$ (8)

where

$\quad\Theta_{n}(1,\dots,n)\ =\ {\,\rm e}\,^{-{\,{\rm
i}\,}\sum_{i<j}\,p_{i}\times p_{j}}\ .$ (9)

We have allowed for different bivectors $\theta$ and $\bar{\theta}$ in the
left and right gauge theory amplitudes, which appear when distinct gauge
fluxes on the worldvolumes of different stacks of D-branes are turned on.

So far this is a boring result: it simply modifies the slogan (1) to

$\textrm{Gravity}\ =\ (\textrm{Noncommutative Gauge Theory})^{2}\ .$ (10)

That is, the double copy of noncommutative Yang-Mills theory is _ordinary_
general relativity, rather than some deformation of gravity as one may have
anticipated. However, this result raises several theoretical questions; for
example:

  * Question 1: What is the scalar theory sourcing the relation $\bm{S}_{n}=\bm{m}_{n}^{-1}$ for the $B$-field modified KLT momentum kernel?

The answers to this and other questions lead to new perspectives on the double
copy relations, which require careful considerations of non-locality, with
interesting answers and applications. These are the topic of the remaining
sections.

##  3 Colour and Kinematics in Moyal-Deformed Theories

### Colour-Kinematics Duality.

Fastforwarding ahead 20 years from the KLT paper, Bern, Carrasco and Johansson
(BCJ) [11] posit a stripping of colour from kinematic factors in numerators of
tree-level gauge theory amplitudes in the form

${\mathcal{A}}_{n}^{L/R}\ =\
\displaystyle\sum\limits_{I}\,\frac{c_{I}\,n_{I}^{L/R}}{D_{I}}\ ,$ (11)

where the sum runs over all planar trivalent Feynman graphs with $n$ external
edges, $c_{I}$ are colour factors constructed from structure constants
$f^{abc}$ of the gauge algebra ${\mathfrak{g}}$, $n_{I}$ depend only on purely
kinematic data (momentum, spin, flavour, etc.), and $D_{I}$ are propagator
factors associated to internal edges.

The colour-stripping in (11) depends on a choice of basis, which is dictated
by demanding that it be compatible with _colour-kinematics duality_ : any
relation of the schematic form $c_{I}+c_{J}+c_{K}+\cdots=0$ involving only
colour factors, such as the Jacobi identity among $f^{abc}$ for the Lie
algebra ${\mathfrak{g}}$, is mirrored by a relation
$n_{I}+n_{J}+n_{K}+\cdots=0$ among kinematic factors. These are called
generalized Jacobi identities. The duality predicts the existence of an
infinite-dimensional kinematic algebra underlying scattering amplitudes, whose
general structure is currently one of the main open problems in the double
copy programme.

When the colour-stripping is compatible with colour-kinematics duality, then
the double copy relations replace colour by kinematics $c_{I}\longmapsto
n_{I}^{R/L}$ in (11) to give amplitudes of a theory with linearized
diffeomorphism symmetry via two copies of gauge theory kinematic factors:

$M_{n}^{L\otimes R}\ =\
\displaystyle\sum\limits_{I}\,\frac{n_{I}^{L}\,n_{I}^{R}}{D_{I}}\ .$ (12)

The BCJ relations ensure that the amplitudes are independent of the chosen
basis for the decomposition (11).

This operation also allows one to replace kinematic factors
$n_{I}^{L/R}\longmapsto\bar{c}_{I}$ in (11) by colour factors constructed from
structure constants $\bar{f}^{\bar{a}\bar{b}\bar{c}}$ of another gauge algebra
$\bar{\mathfrak{g}}$, regarded as a kinematic Lie algebra since the resulting
amplitudes trivially satisfy colour-kinematics duality. This gives the
amplitudes of a _biadjoint scalar theory_ , called the “zeroth copy” of the
gauge theory because it acts as the identity theory in the space of copyable
theories: the double copy of any field theory with the biadjoint scalar theory
simply returns the original theory.

These considerations now bring us to our second question:

  * Question 2: What is the meaning of colour-kinematics duality in noncommutative gauge theories?

To understand why this is a question, we review the well-known mixing between
gauge and kinematic degrees of freedom in Moyal-deformed field theories.

### Colour-Kinematics Mixing.

Consider the noncommutative associative algebra
$\Omega^{0}({\mathbbm{R}}^{d},{\mathfrak{u}}(N))$ with the Moyal product. The
star-commutator

$[\phi_{1}\stackrel{{\scriptstyle\star}}{{,}}\phi_{2}]\ =\
\phi_{1}\star\phi_{2}-\phi_{2}\star\phi_{1}$ (13)

makes it an infinite-dimensional Lie algebra, which we denote as
${\mathfrak{u}}_{\star}(N)$. As a vector space,
$\Omega^{0}({\mathbbm{R}}^{d},{\mathfrak{u}}(N))\simeq{\mathfrak{u}}(N)\otimes\Omega^{0}({\mathbbm{R}}^{d})$,
but not as the Lie algebra ${\mathfrak{u}}_{\star}(N)$ unless $\theta=0$,
because expanding the fields in a basis $\\{T^{a}\\}$ of ${\mathfrak{u}}(N)$
shows

$[\phi_{1}\stackrel{{\scriptstyle\star}}{{,}}\phi_{2}]\ =\
\mbox{$\frac{1}{2}$}\,\big{(}f^{abc}\,\\{\phi_{1}^{a}\stackrel{{\scriptstyle\star}}{{,}}\phi_{2}^{b}\\}+{\,{\rm
i}\,}d^{abc}\,[\phi_{1}^{a}\stackrel{{\scriptstyle\star}}{{,}}\phi_{2}^{b}]\big{)}\,T^{c}\
,$ (14)

where $[T^{a},T^{b}]=f^{abc}\,T^{c}$ and $\\{T^{a},T^{b}\\}={\,{\rm
i}\,}d^{abc}\,T^{c}$. This loss of factorization spoils standard colour-
stripping and colour-kinematics duality.

### Kinematic Lie Algebras.

On the other hand, this mixing provides a novel realisation of the kinematic
algebras for some theories. The momentum space basis $e_{k}^{a}={\,\rm
e}\,^{{\,{\rm i}\,}k\cdot x}\,T^{a}$ of
$\Omega^{0}({\mathbbm{R}}^{d},{\mathfrak{u}}(N))$ obeys the star-commutation
relations

$\displaystyle\big{[}e_{k}^{a}\stackrel{{\scriptstyle\star}}{{,}}e_{p}^{b}\big{]}$
$\displaystyle\ =\ $ $\displaystyle F^{abc}(k,p,k+p)\ e^{c}_{k+p}\ ,$ (15)

with the momentum dependent structure constants

$\displaystyle F^{abc}(k,p,k+p)$ $\displaystyle\ =\ $ $\displaystyle
f^{abc}\cos(k\times p)+{\,{\rm i}\,}d^{abc}\sin(k\times p)\ .$ (16)

For $N=1$ this makes $\Omega^{0}({\mathbbm{R}}^{d})$ into the _kinematic Lie
algebra ${\mathfrak{u}}_{\star}(1)$_ with

$[e_{k}\stackrel{{\scriptstyle\star}}{{,}}e_{p}]=F(k,p,k+p)\
e_{k+p}\quad,\quad F(k,p,k+p)=2{\,{\rm i}\,}\sin(k\times p)\ .$ (17)

In the semi-classical limit, i.e. to first order in the deformation bivector
$\theta$, the star-commutator truncates to the Poisson bracket
$\\{\phi_{1},\phi_{2}\\}_{\theta}=\theta^{\mu\nu}\,\partial_{\mu}\phi_{1}\,\partial_{\nu}\phi_{2}$,
and ${\mathfrak{u}}_{\star}(1)$ reduces to the Lie algebra
${\mathfrak{sdiff}}({\mathbbm{R}}^{d})$ of _symplectic diffeomorphisms_. The
full deformation ${\mathfrak{u}}_{\star}(1)$ of
${\mathfrak{sdiff}}({\mathbbm{R}}^{d})$ is related to the infinite unitary
algebra ${\mathfrak{u}}(\infty)$ as well as a certain $W_{1+\infty}$-algebra
known as the symplecton algebra, see e.g. [12, 13, 14, 15]. For a description
of the $N>1$ kinematic Lie algebras ${\mathfrak{u}}_{\star}(N)$, see [16].

### Twisted Colour-Stripping.

The way to handle the mixing issue comes from a new perspective on colour-
stripping [1], and ultimately a new notion of colour-kinematics duality,
tailored to the momentum dependent structure constants of these inherently
non-local field theories. It is based on the beautiful formalism of twisted
tensor products of homotopy algebras developed by Borsten _et al._ [17], which
has the additional power to provide off-shell formulations of colour-
kinematics duality and the double copy relations.

Skipping the formal definitions and details, the upshot is that the Lie
algebra

${\mathfrak{u}}_{\star}(N)\ =\
{\mathfrak{u}}(N)\otimes_{\tau}\Omega^{0}({\mathbbm{R}}^{d})$ (18)

can be factorized as a _twisted tensor product_ of the Lie algebra
${\mathfrak{u}}(N)$ with the commutative algebra
$(\Omega^{0}({\mathbbm{R}}^{d}),\mu)$ of functions, where

$\tau:{\mathfrak{u}}(N)\otimes{\mathfrak{u}}(N)\longrightarrow{\mathfrak{u}}(N)\otimes{\rm
End}\big{(}\Omega^{0}({\mathbbm{R}}^{d})\big{)}\otimes{\rm
End}\big{(}\Omega^{0}({\mathbbm{R}}^{d})\big{)}$ (19)

is the _twist datum_ defined by

$\displaystyle\tau(T^{a},T^{b})$ $\displaystyle\ =\ $
$\displaystyle[T^{a},T^{b}]\otimes\cos\big{(}\mbox{$\frac{1}{2}$}\,\theta^{\mu\nu}\,\partial_{\mu}\otimes\partial_{\nu}\big{)}$
(20) $\displaystyle\qquad+\,{\,{\rm
i}\,}\\{T^{a},T^{b}\\}\otimes\sin\big{(}\mbox{$\frac{1}{2}$}\,\theta^{\mu\nu}\,\partial_{\mu}\otimes\partial_{\nu}\big{)}\
.$

Then in Sweedler notation

$\displaystyle[T^{a}\otimes\phi_{1}^{a}\stackrel{{\scriptstyle\star}}{{,}}T^{b}\otimes\phi_{2}^{b}]$
$\displaystyle\ =\ $ $\displaystyle\tau^{(1)}(T^{a},T^{b})$ (21)
$\displaystyle\qquad\otimes\
\mu\big{(}\tau^{(2)}(T^{a},T^{b})(\phi^{a}_{1})\,,\,\tau^{(3)}(T^{a},T^{b})(\phi^{b}_{2})\big{)}\
.$

This strips colour from kinematics, with each factor of the twisted tensor
product a conventional algebraic object: noncommutativity is now completely
absorbed into the twist datum $\tau$. For a given theory, this twisted
factorization must be compatible with a form of colour-kinematics duality in
order to be able to apply the double copy relation. This answers Question 2
above.

### Noncommutative Biadjoint Scalar Theory.

Let us illustrate this operation by constructing a new non-local scalar field
theory of relevance to our ensuing discussions. The _biadjoint scalar theory_
is a massless cubic scalar field theory with rigid symmetry under a pair of
quadratic Lie algebras ${\mathfrak{g}}$ and $\bar{\mathfrak{g}}$. Its fields

$\phi\ =\ T^{a}\otimes\bar{T}^{\bar{a}}\otimes\phi^{a\bar{a}}\ \in\
\Omega^{0}({\mathbbm{R}}^{d},{\mathfrak{g}}\otimes\bar{\mathfrak{g}})\ \simeq\
{\mathfrak{g}}\otimes\big{(}\bar{\mathfrak{g}}\otimes\Omega^{0}({\mathbbm{R}}^{d})\big{)}$
(22)

propagate according to the action functional

$\displaystyle S_{{\rm bi}}[\phi]\ =\ \int\,{\rm d}^{d}x\
\frac{1}{2}\,\phi^{a\bar{a}}\,\square\,\phi_{a\bar{a}}-\frac{g}{3!}\,f^{abc}\,\bar{f}^{\bar{a}\bar{b}\bar{c}}\,\phi^{a\bar{a}}\,\phi^{b\bar{b}}\,\phi^{c\bar{c}}\
.$ (23)

Deform the biadjoint scalar theory by replacing the Lie algebra
$\bar{\mathfrak{g}}\otimes\Omega^{0}({\mathbbm{R}}^{d})$ with the Lie algebra
${\mathfrak{u}}_{\star}(\bar{N})\simeq{\mathfrak{u}}(\bar{N})\otimes_{\bar{\tau}}\Omega^{0}({\mathbbm{R}}^{d})$.
The new action functional is

$S_{{\rm bi}}^{\star}[\phi]\ =\ \int\,{\rm d}^{d}x\
\frac{1}{2}\,\phi^{a\bar{a}}\star\square\,\phi_{a\bar{a}}-\frac{g}{3!}\,f^{abc}\,\bar{\lambda}^{\bar{a}\bar{b}\bar{c}}\,\phi^{a\bar{a}}\star\phi^{b\bar{b}}\star\phi^{c\bar{c}}\
,$ (24)

where
$\bar{\lambda}^{\bar{a}\bar{b}\bar{c}}:=\bar{f}^{\bar{a}\bar{b}\bar{c}}+{\,{\rm
i}\,}\bar{d}^{\bar{a}\bar{b}\bar{c}}$. This theory has the standard scalar
propagator, but now the 3-point Feynman vertex

comes from replacing the structure constants $\bar{f}^{\bar{a}\bar{b}\bar{c}}$
of $\bar{\mathfrak{g}}$ with the momentum dependent structure constants of
$\bar{F}^{\bar{a}\bar{b}\bar{c}}(k,p,q)$ of ${\mathfrak{u}}_{\star}(\bar{N})$
in the usual biadjoint scalar vertex.

##  4 Noncommutative Chern-Simons Theory

The simplest example in which to illustrate the new double copy prescription
is the Moyal deformation of Chern-Simons gauge theory in three dimensions. The
gluon fields are ${\mathfrak{u}}(N)$-valued 1-forms
$A\in\Omega^{1}({\mathbbm{R}}^{3},{\mathfrak{u}}(N))$ with action functional

$S_{{\rm CS}}^{\star}[A]\ =\
\int\,\mathrm{Tr}\,\Big{(}\frac{1}{2}\,A\wedge_{\star}{\rm
d}A+\frac{g}{3!}\,A\wedge_{\star}[A\stackrel{{\scriptstyle\star}}{{,}}A]\Big{)}\
,$ (25)

where the Moyal-deformed exterior product $\wedge_{\star}$ of forms is defined
analogously to (6) by replacing partial derivatives with Lie derivatives along
the coordinate directions. The propagator is the same as in ordinary Chern-
Simons theory, but now the 3-gluon Feynman vertex is

$\hskip 85.35826pt\ =\ -{\,{\rm i}\,}g\,\epsilon_{\mu\nu\rho}\,F^{abc}(k,p,q)$

### Twisted Colour-Kinematics Duality.

Like its commutative version, noncommutative Chern-Simons theory has no
propagating degrees of freedom and so all gluon scattering amplitudes vanish.
However, one can go slightly off-shell and consider amplitudes of ‘harmonic’
gluon states [1]. Then numerators of these amplitudes in Lorenz gauge are
built from the Lie brackets of the twisted tensor product
${\mathfrak{u}}(N)\otimes_{\tau}\mathfrak{vdiff}({\mathbbm{R}}^{3})$ with the
infinite-dimensional Lie algebra $\mathfrak{vdiff}({\mathbbm{R}}^{3})$ of
volume-preserving diffeomorphisms of ${\mathbbm{R}}^{3}$. The generators are

$\varepsilon^{a}_{\mu}(p)\ =\ e_{p}^{a}\ \Pi_{\mu\nu}(p)\,\partial^{\nu}\ ,$
(26)

where $\Pi_{\mu\nu}$ are transverse projection operators with
$\Pi_{\mu\nu}(p)\,p^{\mu}=0$. The commutation relations are

$[\varepsilon^{a}_{\mu}(k)\stackrel{{\scriptstyle\star}}{{,}}\varepsilon^{b}_{\nu}(p)]\
=\ {\tt F}_{\mu\nu\rho}^{abc}(k,p,k+p)\ \varepsilon^{c}_{\rho}(k+p)\ ,$ (27)

where the momentum dependent structure constants are given by

${\tt F}_{\mu\nu\rho}^{abc}(k,p,k+p)\ =\ F^{abc}(k,p,k+p)\
\Pi_{\mu\mu^{\prime}}(k)\,\epsilon_{\mu^{\prime}\nu^{\prime}\rho}\,\Pi_{\nu^{\prime}\nu}(p)\
.$ (28)

### Double Copy.

As before, using Lie derivatives we factorize the (differential graded) Lie
algebra
$\big{(}\Omega^{\bullet}({\mathbbm{R}}^{3},{\mathfrak{u}}(N)),[-\\!\stackrel{{\scriptstyle\star}}{{,}}\\!-]\big{)}$
as the twisted tensor product
${\mathfrak{u}}(N)\otimes_{\tau}\Omega^{\bullet}({\mathbbm{R}}^{3})$ with the
(differential graded) commutative algebra
$\Omega^{\bullet}({\mathbbm{R}}^{3})$. This factorization is compatible with
twisted colour-kinematics duality.

Replacing now the full _twisted_ colour-stripping by kinematic factors leads
to the same double copy theory as that of ordinary Chern-Simons theory with
itself [18]. This is a non-local theory of fields
$H_{\bar{\mu}\mu}=A_{\bar{\mu}}\otimes A_{\mu}$, with action functional

$\widehat{S}_{{\rm CS}}[H]\ =\ \int\,\mathrm{d}^{3}x\
\epsilon^{\bar{\mu}\bar{\nu}\bar{\rho}}\,\epsilon^{\mu\nu\rho}\,\Big{(}H_{\bar{\mu}\mu}\,\frac{1}{\square}\,\partial_{\bar{\nu}}\,\partial_{\nu}H_{\bar{\rho}\rho}+\frac{g}{3}\,H_{\bar{\mu}\mu}\,H_{\bar{\nu}\nu}\,H_{\bar{\rho}\rho}\Big{)}\
.$ (29)

This example illustrates a general result, which in particular explains what
we found in §2: Double copies of Moyal-deformed gauge theories coincide with
the ordinary double copy theories of their commutative counterparts.

##  5 Coloured Moyal Deformations of Cubic Scalar Theory

Among other things, in this section we will answer Question 1 from §2.

### Adjoint Scalar Theory.

Setting $\bar{N}=1$ in the noncommutative biadjoint scalar theory from §3
leads to a non-local theory of a massless scalar field
$\phi\in\Omega^{0}({\mathbbm{R}}^{d},{\mathfrak{g}})$ with rigid colour
symmetry and action functional

$S_{{\rm ad}}[\phi]\ =\ \int\,{\rm d}^{d}x\
\frac{1}{2}\,\phi^{a}\star\square\,\phi_{a}-\frac{{\,{\rm
i}\,}g}{3}\,f^{abc}\,\phi^{a}\star\phi^{b}\star\phi^{c}\ .$ (30)

This theory has a standard scalar propagator and the 3-point Feynman vertex

An unusual feature of the adjoint scalar theory is that it becomes free when
$\theta=0$ (hence the omission of the adjective ‘noncommutative’). This is not
what normally happens in Moyal-deformed scalar theories, where the effect of
noncommutativity is to replace local interactions of an ordinary scalar field
theory with non-local interactions induced by the Moyal product.

The adjoint scalar theory possesses manifest colour-kinematics duality:
numerators of its amplitudes factorize according to the untwisted tensor
product ${\mathfrak{g}}\otimes{\mathfrak{u}}_{\star}(1)$. The kinematic
algebra is the Lie algebra ${\mathfrak{u}}_{\star}(1)$.

Following the relations of the twist formalism, the zeroth copy of the adjoint
scalar theory replaces the Lie algebra
${\mathfrak{u}}_{\star}(1)={\mathfrak{u}}(1)\otimes_{\tau}\Omega^{0}({\mathbbm{R}}^{d})$
with $\bar{\mathfrak{g}}\otimes\Omega^{0}({\mathbbm{R}}^{d})$. This results in
the commutative biadjoint scalar theory from §3.

On the other hand, the double copy relation applied to the adjoint scalar
theory replaces ${\mathfrak{g}}\otimes{\mathfrak{u}}_{\star}(1)$ with
${\mathfrak{u}}(1)\otimes_{\bar{\tau}}\big{(}{\mathfrak{u}}(1)\otimes_{\tau}\Omega^{0}({\mathbbm{R}}^{d})\big{)}$.
This results in a complicated cubic theory of a scalar field
$\phi\in\Omega^{0}({\mathbbm{R}}^{d})$ deformed by two bivectors $\theta$ and
$\bar{\theta}$, with action functional

$\displaystyle\widehat{S}_{{\rm ad}}[\phi]\ =\ \int\,{\rm d}^{d}x$
$\displaystyle\\!\frac{1}{2}\,\phi\star\square\,\phi-\frac{{\,{\rm
i}\,}g}{3}\,\sum_{n=0}^{\infty}\,\frac{(-1)^{n}}{(2n+1)!}\,\bar{\theta}^{\mu_{1}\nu_{1}}\cdots\bar{\theta}^{\mu_{2n+1}\nu_{2n+1}}$
$\displaystyle\hskip 85.35826pt\times\ \phi\
[\partial_{\mu_{1}}\cdots\partial_{\mu_{2n+1}}\phi\stackrel{{\scriptstyle\star}}{{,}}\partial_{\nu_{1}}\cdots\partial_{\nu_{2n+1}}\phi]\
.$

### Binoncommutative Biadjoint Scalar Theory.

Taking things one step further, one can modify the double copy of the adjoint
scalar theory to include colour: the scalar fields now live in the double
twisted tensor product

$\mathfrak{u}(N)\otimes_{\tau}\big{(}\mathfrak{u}(\bar{N})\otimes_{\bar{\tau}}\Omega^{0}({\mathbbm{R}}^{d})\big{)}\
.$ (32)

The interactions in this theory involve a binoncommutative product which is a
binonabelian extension of the product used to construct the double copy theory
for the adjoint scalar. This theory is the _zeroth copy_ of the noncommutative
Chern-Simons theory from §4.

This scalar field theory is quite complicated when written down in position
space in terms of an action functional, but it has a simple representation in
momentum space as the standard scalar propagator with the Feynman vertex

The $n$-point colour-stripped tree-level binoncommutative biadjoint scalar
amplitudes $\bm{m}_{n}^{\star\bar{\star}}$ can be obtained by dressing the
corresponding amplitudes $\bm{m}_{n}$ of the usual biadjoint scalar theory
from §3 with the phase factors (9) as [1]

$\bm{m}_{n}^{\star\bar{\star}}(w|w^{\prime})\ =\ \Theta_{n}(w)\
\bm{m}_{n}(w|w^{\prime})\ \bar{\Theta}_{n}(w^{\prime})\ .$ (33)

### Noncommutative Yang-Mills Theory.

We finally return to our story from §2 and reconsider the Moyal deformation of
Yang-Mills theory on ${\mathbbm{R}}^{d}$. It is defined by the action
functional

$S^{\star}_{{\rm YM}}[A]\ =\
\frac{1}{2}\,\int\,\mathrm{Tr}\,\big{(}F_{A}^{\star}\,\wedge_{\star}\ast_{\textrm{\tiny$G$}}\,F_{A}^{\star}\big{)}\
,$ (34)

where $F_{A}^{\star}={\rm
d}A+\mbox{$\frac{g}{2}$}\,[A\stackrel{{\scriptstyle\star}}{{,}}A]$ is the
noncommutative field strength of a gauge field
$A\in\Omega^{1}(\mathbbm{R}^{d},\mathfrak{u}(N))$, and
$\ast_{\textrm{\tiny$G$}}$ denotes the Hodge duality operator with respect to
the open string metric $G$.

The detailed analysis of this theory is rather involved and was discussed
extensively in [1]. The double copy relations require a _strictification_ of
the theory, which amounts to replacing the quartic interactions of gluons by
cubic interactions through the introduction of suitable auxiliary fields. This
is possible to do in a way that twisted colour-stripping is compatible with
colour-kinematics duality at each multiplicity order. The corresponding tree-
level scattering amplitudes obey the appropriate versions of the BCJ
relations.

From (4), (8) and (33) it follows that the $B$-field modified KLT momentum
kernel $\bm{S}_{n}(w|w^{\prime})_{G;\theta,\bar{\theta}}$ of the double copy
relations for noncommutative Yang-Mills theory is sourced by the
binoncommutative biadjoint scalar theory as

$\bm{S}_{n}(w|w^{\prime})_{G;\theta,\bar{\theta}}\ =\
\bm{m}^{\star\bar{\star}}_{n}(w|w^{\prime})_{G}^{-1}\ .$ (35)

This answers Question 1 from §2. The novel scalar theories introduced in this
section have further interesting applications which we now describe.

##  6 $\bm{{\mathfrak{u}}_{\star}(1)}$ Gauge Theories are Double Copy Theories

The rank $N=1$ limits of noncommutative gauge theories are interesting
interacting theories of photons. Like the adjoint scalar theory, they become
free theories in the commutative limit $\theta=0$. They have no colour
symmetry, but they are invariant under the kinematic Lie algebra
${\mathfrak{u}}_{\star}(1)$ of noncommutative gauge transformations. These
theories have long been believed to contain gravitation. Evidence is provided
for instance through the interplay between gauge transformations and spacetime
diffeomorphisms discussed in §3, as well as emergent gravity phenomena; see
e.g. [19, 20] for reviews.

The double copy relations offer a new perspective on the emergence of
gravitation: a rank one noncommutative gauge theory can be realised as the
double copy of the adjoint scalar theory from §5 with the ordinary non-abelian
version of the gauge theory in question, for any gauge algebra
${\mathfrak{g}}$. Symbolically

$\textrm{${\mathfrak{u}}_{\star}(1)$ Gauge Theory}\ =\ \textrm{Adjoint Scalar
Theory}\ \otimes\ \textrm{Gauge Theory}\ .$ (36)

For example, in the case of Chern-Simons theory, the left-hand side of this
relation realises a theory with local symmetry under a subalgebra of spacetime
diffeomorphisms, along the lines discussed in §4.

### Self-Dual Yang-Mills Theory and Gravity.

The general relation (36) encompasses recent discussions of the best
understood example of the double copy. Set $d=4$ and work in coordinates
$(x^{+},x^{-},z,\bar{z})$ on
${\mathbbm{R}}^{4}\simeq{\mathbbm{R}}^{2}\times{\mathbbm{C}}$. Turn on
bivectors $\theta$ and $\bar{\theta}$ whose only non-zero components are equal
to $\pm\,1$ along the null $(z,x^{-})$-plane.

Then the semi-classical limit in $\bar{\theta}_{-z}$ of the $\bar{N}=1$
binoncommutative biadjoint scalar theory from §5 is the usual integrable
deformation of self-dual Yang-Mills theory given by the noncommutative
instanton equations [21]

$\square\,\phi-{{\,{\rm
i}\,}g}\,[\partial_{z}\phi\stackrel{{\scriptstyle\star}}{{,}}\partial_{-}\phi]\
=\ 0\ ,$ (37)

where $\phi\in\Omega^{0}({\mathbbm{R}}^{4},{\mathfrak{u}}(N))$ is the Leznov
prepotential representing the single polarization state of a gluon that
remains after projection to the self-dual sector. The self-dual sector of
noncommutative Yang-Mills theory has a non-trivial commutative limit to
ordinary self-dual Yang-Mills theory and arises (in Leznov gauge) from
quantizing open ${\cal N}=2$ strings in a constant $B$-field background [22].

For $N=1$, this theory is the double copy of the adjoint scalar theory
constructed in §5 and (37) coincides with the deformed Plebański equation for
noncommutative gravity [23, 24]. In the semi-classical limit in $\theta_{-z}$
it reduces to Plebański’s equation
$\square\,\phi=g\,\\{\partial_{z}\phi,\partial_{-}\phi\\}_{\theta}$ for self-
dual gravity, and the theory with action functional (5) is related to the
closed string field theory of ${\cal N}=2$ strings [25]. Thus self-dual
${\mathfrak{u}}_{\star}(1)$ Yang-Mills theory is noncommutative self-dual
gravity.

The semi-classical limit of the adjoint scalar theory itself is self-dual
Yang-Mills theory in Leznov gauge [26], with equation of motion
$\square\,\phi=g\,[\partial_{z}\phi,\partial_{-}\phi]$. It reproduces the
well-known double copy relation in the self-dual sector

$\textrm{Self-Dual Gravity}\ =\ (\textrm{Self-Dual Yang-Mills Theory})^{2}\ ,$
(38)

where the kinematic algebra is the infinite-dimensional Lie algebra
$w_{1+\infty}$ of area-preserving diffeomorphisms of the null
$(z,x^{-})$-plane [27]. Thus the adjoint scalar theory is a deformation of
self-dual Yang-Mills theory consistent with colour-kinematics duality;
however, it is not integrable [15]. On the other hand, the double copy
relation (36) now reads

$\stackrel{{\scriptstyle\textrm{Noncommutative}}}{{\textrm{ Self-Dual
Gravity}^{\phantom{{\dagger}}}}}{}_{\phantom{dag}}^{\bm{\ =\
}}\stackrel{{\scriptstyle\textrm{Adjoint Scalar}}}{{\textrm{
Theory}^{\phantom{\ast}}}}\ \ {}_{\phantom{dag}}^{\bigotimes}\
\stackrel{{\scriptstyle\textrm{Self-Dual}}}{{\textrm{ Yang-Mills
Theory}^{\phantom{{\dagger}}}}}\ {}_{\phantom{dag}}^{\bm{.}}$ (39)

In both instances (38) and (39) the integrability of gravity is “inherited”
from the gauge theory [15, 28].

##  7 Outlook: A Braided Double Copy?

In this contribution we have described recently developed twisted versions of
colour-stripping and colour-kinematics duality that can handle the non-
locality of Moyal-deformed gauge theories which induce momentum dependent
structure constants in amplitudes [1]. The result of the double copy relations
is then classical, in that ordinary gravitational theories are always
produced. It is natural to wonder whether there is a way in which a
noncommutative theory of gravity is related to noncommutative gauge theory. We
assert that the answer is affirmative, through a suitable variant of the
double copy.

Moyal-deformed theories of gravity have been developed with “twisted”
diffeomormphism symmetry, though it is unclear whether these models can arise
in string theory; see e.g. [19] for a review. These theories lie in the broad
class of ‘braided quantum field theories’, which are theories with braided
symmetries that can be defined and treated in the new language of _braided
homotopy algebras_ [29, 30]. The key point is that if one works with braided
homotopy algebras, then _untwisted_ colour-stripping and factorization are
possible, with the kinematic factors now retaining information about
noncommutativity in the form of (differential graded) braided commutative
algebras.

Hence it should be possible to realise twisted noncommutative gravity theories
through

  * Conjecture: There is a braided version of the double copy construction in terms of factorizations into braided kinematic algebras and braided noncommutative field theories such that

$\textrm{Twisted Noncommutative Gravity}\ =\ (\textrm{Braided Gauge
Theory})^{2}\ .$ (40)

### Acknowledgements.

The author thanks Guillaume Trojani for the collaboration on [1]. This article
is based upon work from COST Actions CaLISTA CA21109 and THEORY-CHALLENGES
CA22113 supported by COST (European Cooperation in Science and Technology).

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