Sections of Lagrangian fibrations
on holomorphically symplectic manifolds
and degenerate twistorial deformations
Fedor A. Bogomolov111Fedor Bogomolov is partially supported by the HSE
University Basic Research Program, Russian Academic Excellence Project ’5-100’
and by EPSRC programme grant EP/M024830., Rodion N. Déev, Misha
Verbitsky222Partially supported by the Russian Academic Excellence Project
’5-100’, FAPERJ E-26/202.912/2018 and CNPq - Process 313608/2017-2.

Abstract

Let $(M,I,\Omega)$ be a holomorphically symplectic manifold equipped with a
holomorphic Lagrangian fibration $\pi:\;M\mapsto X$, and $\eta$ a closed form
of Hodge type (1,1)+(2,0) on $X$. We prove that
$\Omega^{\prime}:=\Omega+\pi^{*}\eta$ is again a holomorphically symplectic
form, for another complex structure $I^{\prime}$, which is uniquely determined
by $\Omega^{\prime}$. The corresponding deformation of complex structures is
called “degenerate twistorial deformation”. The map $\pi$ is holomorphic with
respect to this new complex structure, and $X$ and the fibers of $\pi$ retain
the same complex structure as before. Let $s$ be a smooth section of of $\pi$.
We prove that there exists a degenerate twistorial deformation
$(M,I^{\prime},\Omega^{\prime})$ such that $s$ is a holomorphic section.

###### Contents

  1. 1 Introduction
    1. 1.1 Complex structure obtained from a complex symplectic form
    2. 1.2 Lagrangian fibrations on hyperkähler manifolds
  2. 2 Complex structures via symplectic forms
    1. 2.1 C-symplectic structures
    2. 2.2 Sections of Lagrangian projections of c-symplectic vector spaces
  3. 3 Degenerate twistorial deformation
  4. 4 Holomorphic Lagrangian fibrations on a K3 surface

##  1 Introduction

###  1.1 Complex structure obtained from a complex symplectic form

A complex structure $I$ on a real vector space $V$ is uniquely determined by a
complex-linear symplectic form $\Omega$. Indeed, $\Omega$ has Hodge type
$(2,0)$ because it is complex-linear. The corresponding map
$\Omega:\;V\otimes_{\mathbb{R}}\mathbb{C}{\>\longrightarrow\>}V^{*}\otimes_{\mathbb{R}}\mathbb{C}$
is non-degenerate on the Hodge component $V^{1,0}$, and vanishes on $V^{0,1}$.
Since the complex structure on $V$ is uniquely determined by $V^{0,1}\subset
V\otimes_{\mathbb{R}}\mathbb{C}$, $\Omega$ determines $I$ uniquely.

This description is very beneficial on a manifold, when $\Omega$ is a complex-
valued differential form. In that case, $d\Omega=0$ implies that the almost
complex structure determined by $I$ is integrable (2.2). In [V2], this
observation was used to construct deformations of hyperkähler manifolds
admitting a Lagrangian fibration. It turns out that a holomorphic symplectic
form can be characterized intrinsically in terms of its rank and exterior
powers (2.1).

More precisely, a complex-valued exterior 2-form $\Omega$ on a
$4n$-dimensional real vector space is called c-symplectic if $\Omega^{n+1}=0$
and $\Omega^{n}\wedge\overline{\Omega}^{n}$ is non-degenerate. By 2.1, this is
equivalent to being a non-degenerate complex linear 2-form for some complex
structure on $V$.

Recall that holomorphic Lagrangian fibration on a holomorphically symplectic
manifold $(M,\Omega)$ is a holomorphic map with fibers which are Lagrangian
with respect to $\Omega$.

Given a Lagrangian fibration $\pi:\;M{\>\longrightarrow\>}B$ on a holomorphic
symplectic manifold $(M,\Omega)$ and a closed 2-form $\eta$ on $B$ of Hodge
type $(2,0)+(1,1)$, the sum $\Omega+\pi^{*}\eta$ is c-symplectic, hence
determines an almost complex structure $I_{\eta}$ on $M$ (2.1). This almost
complex structure is integrable when $\eta$ is closed (2.2). The corresponding
deformation of complex structures is uniquely determined by the cohomology
class of $\eta$; it is called degenerate twistor deformation ([V2]).

As shown in [V2], the map $\pi:\;M{\>\longrightarrow\>}B$ is a holomorphic
Lagrangian fibration with respect to $\eta$, and the complex structure on the
fibers of $\pi$ does not change; in dimension 1, this deformation gives “the
Tate-Shafarevich twist” of an elliptic fibration (that is, a deformation of a
fibration which retains the complex structures on the base and on the fibers).

###  1.2 Lagrangian fibrations on hyperkähler manifolds

When $M$ is a compact hyperkähler manifold of maximal holonomy,111This is the
same as “IHS”, irreducibly holomorphic symplectic manifold. essentially all
non-trivial holomorphic maps $M{\>\longrightarrow\>}B$ are Lagrangian
fibrations, by Matsushita’s theorem.

Theorem 1.1: Let $M$ be a compact hyperkähler manifold of maximal holonomy,
and $\pi:\;M{\>\longrightarrow\>}B$ a surjective holomorphic map, with $\dim
M>\dim B>0$. Then $\pi$ is a holomorphic Lagrangian fibration.

Proof: [Mat1].

Applying the Stein factorization theorem, we can factorize a given holomorphic
Lagrangian fibration through a fibration with connected fibers. Further on, we
shall tacitly assume that all Lagrangian fibrations we consider have connected
fibers. In this case the base $B$ is known to have the same rational
cohomology as a complex projective space, by a theorem of D. Matsushita
([Mat2]). J.-M. Hwang has shown that $B$ is a complex projective space
whenether it is smooth ([Hw]). It was conjectured that $B$ is biholomorphic to
$\mathbb{C}P^{n}$ when it is normal ([CMSB]).

Since the degenerate twistor deformation is uniquely determined by the
cohomology class of $\eta$, and $b_{2}(B)=1$, all degenerate twistor
deformations belong to a 1-dimensional holomorphic family determined by
$\Omega+t\pi^{*}\eta$, with $t\in\mathbb{C}$. This family is obtained as a
limit of twistor families, which explains the term “degenerate twistor
deformation”. Like in the usual twistor case, a generic fiber of a degenerate
twistor family is non-algebraic. However, the Moishezon fibers are dense in
the degenerate twistor family.

Much is still unknown about the degenerate twistor families. For example, it
is unknown whether all fibers of the degenerate twistor deformation are
Kähler. It is unclear when a degenerate twistor deformation of a Lagrangian
fibration admits a section.

In this paper we prove that existence of a section is essentially a
topological condition.

Theorem 1.2:  Let $\pi:\;M{\>\longrightarrow\>}B$ be a Lagrangian fibration on
a holomorphic symplectic manifold $(M,\Omega)$, and
$S:\;B{\>\longrightarrow\>}M$ its smooth section. Then there exists a closed
form $\eta\in\Lambda^{2,0}(B)+\Lambda^{1,1}(B)$ such that $S$ is holomorphic
Lagrangian with respect to the complex structure $I_{\eta}$ induced by
$\Omega+\pi^{*}\eta$.

Proof: 3.

Existence of holomorphic Lagrangian fibrations is a standard conjecture,
sometimes referred to as “hyperkähler SYZ conjecture”; see [V1] for the
history and a precise formulation. It is known that holomorphic Lagrangian
fibrations exist for all known classes of hyperkähler manifolds (see e. g.
[KV, Claim 1.20]). Any Lagrangian torus on a hyperkähler manifold $M$ is a
fiber of a rational Lagrangian fibration; moreover, this fibration becomes
holomorphic after replacing $M$ with another holomorphically symplectic
birational model ([HW, GLR]).

Existence of holomorphic sections would follow if we construct a smooth
section, which seems to be doable in many (or all) cases. This would bring us
closer to understanding the holomorphic Lagrangian fibrations, and (hopefully)
bring new classification results.

There is an obvious topological obstruction to existence of a smooth section:
a fibration with multiple fibers cannot have sections. It is easy to construct
holomorphic Lagrangian fibrations with multiple fibers if the general fiber is
not connected. However, this problem can be rectified by using the Stein
factorization theorem. Further on, we shall assume that all holomorphic
Lagrangian fibrations we consider have connected fibers.

In that case (as far as we know) there are no examples of Lagrangian
fibrations with multiple fibers on compact, maximal holonomy hyperkähler
manifolds. We conjecture that multiple fibers in this situation don’t occur.

For K3 surface non-existence of multiple fibers is implied by Kodaira’s
classification of singular elliptic fibers (it follows directly from the
canonical bundle formula, [BHPV, Thm 12.1]). We give a direct proof of this
fact in Section 4.

Multiple fibers of Lagrangian fibrations were classified in [HO] and [Mat3].
In particular, it is shown that the multiplicity of a general fiber is at most
6 ([HO, Theorem 1.1]).

It is clear from our construction that for any smooth section $S$ of a
Lagrangian fibration $\pi$, there exists a unique degenerate twistorial
deformation of $\pi$ such that $S$ is holomorphic. We don’t know the number of
degenerate twistorial deformations of $\pi$ admitting a holomorphic section;
it is unknown even whether this number is finite or infinite.

##  2 Complex structures via symplectic forms

###  2.1 C-symplectic structures

Definition 2.1: Let $V$ be a real vector space equipped with an operator
$I\in\operatorname{End}(V)$, $I^{2}=-\operatorname{Id}_{V}$. We say that $I$
is a complex structure operator on $V$, and $V$ is a complex vector space. The
operator $I$ can be understood as imaginary unit $\sqrt{-1}\in\mathbb{C}$
acting on $V$. Complex-linear maps of complex vector spaces are the same as
maps which commute with the complex structure.

Suppose $W$ is a complex vector space and
$\Omega\in\Lambda^{2}_{\mathbb{C}}W^{*}$ a complex symplectic form. Let us
forget about the complex structure on $W$, and consider its underlying real
vector space $W_{\mathbb{R}}$ with a 2-tensor
$\Omega\in\Lambda^{2}_{\mathbb{R}}W^{*}\otimes\mathbb{C}$ with complex
coefficients. Knowing just this tensor, one can uniquely reconstruct the
complex structure operator $I_{\Omega}\colon W_{\mathbb{R}}\to
W_{\mathbb{R}}$. In the present section we determine which forms can occur as
complex symplectic forms for some complex structure on $W_{\mathbb{R}}$.

Definition 2.2: Let $V$ be an $4n$-dimensional real vector space. A 2-tensor
$\Omega\in\Lambda^{2}V^{*}\otimes\mathbb{C}$ is called a c-symplectic form, if
for any nonzero vector $v\in V$ one has $\iota_{v}\Omega\neq 0\in
V^{*}\otimes\mathbb{C}$, and $\ker\Omega\subset V\otimes\mathbb{C}$ has rank
$2n$. A pair $(V,\Omega)$ in such situation is called a c-symplectic vector
space.

Remark 2.3: The kernel rank $\ker\Omega=2n$ is maximal possible for a form
which does not vanish on real vectors. Indeed, if
$\dim_{\mathbb{C}}\ker\Omega>2n$, equivalently,
$\dim_{\mathbb{R}}\ker\Omega>4n$ this space would intersect $V\subset
V\otimes\mathbb{C}$ of real dimension $4n$, and this is impossible.

Definition 2.4: Let $(V,\Omega)$ be a c-symplectic vector space. A complex
structure operator $I\colon V\to V$ is called an induced by the c-symplectic
form $\Omega$, if it makes $\Omega$ a complex linear symplectic form in
$\Lambda^{2}_{\mathbb{C}}V^{*}$, where the complex exterior power is taken w.
r. t. the complex structure $I$.

Proposition 2.5:  For any c-symplectic vector space $(V,\Omega)$ there exists
a unique induced complex structure $I$ on $V$, that is, a complex structure
operator $I\in\operatorname{End}_{\mathbb{R}}(V)$ such that $\Omega$ is a non-
degenerate form of Hodge type (2,0)

Proof: Since $\ker\Omega\subset V\otimes\mathbb{C}$ has maximal possible
dimension, any real vector $v\in V$ can be represented as $v=v^{1,0}+v^{0,1}$,
where $v^{0,1}\in\ker\Omega$ and $v^{1,0}\in\overline{\ker\Omega}$. Moreover,
this representation is unique, since $\ker\Omega$ contains no real vectors and
hence does not intersect its complex conjugate subspace. One can define an
operator $I$ as multiplication by $-\sqrt{-1}$ on $\ker\Omega$ and by
$\sqrt{-1}$ on $\overline{\ker\Omega}$. Since it is self-conjugate, it is
defined over reals, it is well-defined because of the existence and uniqueness
of the above decomposition, and it obviously squares to
$-\operatorname{Id}_{V}$. Moreover, one has
$\Omega(Iu,v)=\Omega(\sqrt{-1}u^{1,0}-\sqrt{-1}u^{0,1},v)=\sqrt{-1}\Omega(u^{1,0},v)=\sqrt{-1}\Omega(u,v)$.
Therefore $I$ is an induced complex structure; in particular, at least one
induced complex structure exists.

On the other hand, any induced complex structure must have $\ker\Omega$ as its
$-\sqrt{-1}$-eigenspace, and this determines a complex structure operator in a
unique way.

We obtain that any c-symplectic vector space is of nature prescribed above: it
is the underlying real space of some complex symplectic space. In what
follows, we denote the induced complex structure of a c-symplectic vector
space $(V,\Omega)$ by $I_{\Omega}$.

Definition 2.6: Let $(V,\Omega)$ be a c-symplectic vector space. A real
subspace $U\subset V$ is called c-isotropic if for any $u,u^{\prime}\in U$ one
has $\Omega(u,u^{\prime})=0$, and c-Lagrangian if it is c-isotropic and is not
contained in any proper c-isotropic supersubspace.

Proposition 2.7: ([Hi, Proposition 1])
Any c-Lagrangian subspace of a c-symplectic vector space is preserved by the
induced complex structure.

Proof: Let $L\subset V$ be a c-Lagrangian subspace, and $u\in L$ be any
vector. Consider the linear span of the space $L$ and the vector
$I_{\Omega}u$, denote it by $L_{u}$. What is the restriction of the form
$\Omega$ on this span? By definition of a c-isotropic subspace, one has
$\Omega(u,v)=0$ for any $v\in L$, so it is completely determined by the 1-form
$\left(\iota_{I_{\Omega}u}\Omega\right)|_{L_{u}}$. Moreover, since the form
$\Omega$ has Hodge type $(2,0)$ w. r. t. $I_{\Omega}$, one has
$0=\sqrt{-1}\cdot 0=\sqrt{-1}\Omega(u,v)=\Omega(I_{\Omega}u,v)$, and since
$\Omega$ is skew-symmetric, one has $\Omega(I_{\Omega}u,I_{\Omega}u)=0$.
Therefore the form $\Omega$ vanishes identically on the subspace $L_{u}$, i.
e. $L_{u}$ is c-isotropic, and since $L$ is contained in no proper c-isotropic
supersubspace, it must be equal to $L$. This implies that $I_{\Omega}u\in L$,
and since $u$ is arbitrary, it means that the operator $I_{\Omega}$ maps the
subspace $L$ to itself.

In particular, quotient by a c-Lagrangian vector subspace inherits the complex
structure. Note that this complex structure on the quotient is not in general
determined by any c-symplectic form: indeed, the quotient may have odd complex
dimension. In what follows, we shall refer to the complex structures on
quotients of c-symplectic vector spaces by their c-Lagrangian subspaces which
makes the projection map complex linear as to inherited ones.

Proposition 2.8: (existence of a c-symplectic basis). Let $V$ be a
c-symplectic space of real dimension $4n$. Then $V$ possesses a basis in which
its c-symplectic form is given by a block diagonal matrix with equal blocks
$Q$ on the diagonal, where $Q$ stands for a $4\times 4$-block given by

$Q=\begin{pmatrix}0&0&1&\sqrt{-1}\\\ 0&0&\sqrt{-1}&-1\\\ -1&-\sqrt{-1}&0&0\\\
-\sqrt{-1}&1&0&0\end{pmatrix}.$

Proof: The Proposition is proved by running an analogue of the symplectic
Gram–Schmidt process. Indeed, $I_{\Omega}$ be the induced complex structure,
and $z_{1},...,z_{2n}\in V$ a basis in $(V^{*},I)$, considered as a complex
vector space, such that $\Omega=\sum_{i=1}^{n}z_{2i-1}\wedge z_{2i}$. Then
$z_{1},I_{\Omega}(z_{1}),...,z_{2n},I_{\omega}(z_{2i})$ is a real basis in $V$
such that $\Omega$ is written as the above block matrix.

For the convenience of the reader, we give a direct argument constructing such
a basis explicitly.

For the first two vectors in the basis we may choose an arbitrary nonzero
vector $u_{1}$ and its image under the induced complex structure,
$I_{\Omega}u_{1}$. Orthogonals of these vectors (i. e. the kernels of the
forms $\iota_{u_{1}}\Omega$ and $\iota_{I_{\Omega}u_{1}}\Omega$) coincide by
the definition of the induced complex structure. If one picks a vector $u_{2}$
outside this orthogonal, then, by outstretching it and adding to it a multiple
of the vector $I_{\Omega}u_{2}$ in case of necessity, we can make $u_{2}$ such
that

$\Omega(u_{1},u_{2})=1.$

One can deduce from complex linearity of $\Omega$ w. r. t. $I_{\Omega}$ and
the above relation that the restriction of the form $\Omega$ onto the four-
dimensional subspace $U\subset V$ spanned by
$\\{u_{1},I_{\Omega}u_{1},u_{2},I_{\Omega}u_{2}\\}$ is given by the matrix $Q$
in these coordinates.

Now we can proceed, exercising the same procedure in the orthogonal to $U$ (i.
e. the set $U^{\perp}$ of vectors $w\in V$ s. t.
$\left(\iota_{w}\Omega\right)|_{U}=0$), because $U\cap U^{\perp}=\\{0\\}$.

Claim 2.9:  Let $V$ be a real vector space of real dimension $4n$, and
$\Omega\in\Lambda^{2}V^{*}\otimes\mathbb{C}$ be a complex-valued skew-
symmetric 2-form. The following are equivalent:

  1. 1.

$\dim_{\mathbb{C}}\ker\Omega{\left|{}_{{\phantom{|}\\!\\!}_{V\otimes\mathbb{C}}}\right.}=2n$,

  2. 2.

$(\Omega\wedge\overline{\Omega})^{n}$ is nonzero and $\Omega^{n+1}=0$.

Proof: Suppose that $\dim\ker\Omega=2n$. Then for any decomposable polyvector
$a\in\Lambda^{2n+2}V\otimes\mathbb{C}$ the corresponding subspace in $V$ needs
to intersect $\ker\Omega$, so $\iota_{a}\Omega^{n+1}$ vanishes. Hence
$\Omega^{n+1}=0$. An easy direct calculation shows that, in notation of the
2.1, $(Q\wedge\overline{Q})(u_{1},v_{1},u_{2},v_{2})=4$, so the top power of
$\Omega\wedge\overline{\Omega}$ cannot be zero.

Suppose that $\Omega^{n+1}=0$. If the top power of a skew-symmetric form on
some vector space is zero, then this form has a nontrivial kernel. That’s why
the form $\Omega$ has nontrivial kernel when restricted to any $(2n+2)$-plane
inside $V\otimes\mathbb{C}$ (and, moreover, any $(2n+2k)$-plane for any
$k>0$). Restriction of a symplectic form onto a maximal subspace tranversal to
its kernel is non-degenerate, so $\dim\ker\Omega\geqslant 2n$. If
$(\Omega\wedge\overline{\Omega})^{n}\neq 0$, it cannot be greater, because in
this case $\ker\Omega$ needs to intersect $\ker\overline{\Omega}$, thus giving
a real vector in $\ker\Omega$, substitution of which would vanish
$(\Omega\wedge\overline{\Omega})^{n}$.

###  2.2 Sections of Lagrangian projections of c-symplectic vector spaces

In this subsection, we will deal with the following situation. Let
$(V,\Omega)$ be a $4n$-dimensional real vector space space equipped with a
c-symplectic structure, and $L\subset V$ a c-Lagrangian subspace. The complex
linear surjective map $V{\>\longrightarrow\>}V/L$ is called a c-Lagrangian
projection. This is a linearization of the Lagrangian fibrations which are
common in holomorphic symplectic geometry (Subsection 1.2).

Proposition 2.10:  Let $V$ be a real vector space, $\Omega$ a c-symplectic
form on it, $L$ a c-Lagrangian subspace and $\sigma\colon V/L\to V$ a real
section (not necessarily complex linear w. r. t. the induced complex
structures). Define the form
$\Omega_{\sigma}\in\Lambda^{2}(V/L)^{*}\otimes\mathbb{C}$ by the rule
$\Omega_{\sigma}(u_{1},u_{2})=\Omega(\sigma(u_{1}),\sigma(u_{2}))$. In other
words, $\Omega_{\sigma}$ is a restriction of $\Omega$ onto the subspace
$\sigma(V/L)$ after the identification $\pi|_{\sigma(V/L)}\colon\sigma(V/L)\to
V/L$). Then $\Omega_{\sigma}$ has Hodge type $(2,0)+(1,1)$ w. r. t. the
inherited complex structure on the quotient $V/L$.

Proof: Suppose that $\sigma=\sigma_{0}$ is complex linear. Then the form
$\Omega_{\sigma}$ is of type $(2,0)$, since the type is preserved by complex
linear maps.

Now let $\sigma=\sigma_{0}+\tau$, where $\sigma_{0}$ is a complex linear
section and $\tau\colon V/L\to L$ a real perturbation. For any vectors
$u,v\in(V/L)$,

$\Omega_{\sigma}(u,v)=\Omega(\sigma(u),\sigma(v))=\\\
=\Omega(\sigma_{0}(u),\sigma_{0}(v))+\Omega(\sigma_{0}(u),\tau(v))+\\\
+\Omega(\tau(u),\sigma_{0}(v))+\Omega(\tau(u),\tau(v)).$ (2.1)

Since one has $\tau(u),\tau(v)\in L$ and $L$ is a c-Lagrangian subspace w. r.
t. $\Omega$, the last term vanishes. The first term is a $(2,0)$ form, since
$\sigma_{0}$ is a complex linear section. The term
$\Omega(\sigma_{0}(u),\tau(v))+\Omega(\tau(u),\sigma_{0}(v))$ vanishes if
$u,v$ are both of type $(0,1)$, since $\sigma_{0}(u)$ and $\sigma_{0}(v)$ are
vectors of type $(0,1)$ in this case, and therefore annihilate the form
$\Omega$. Therefore, all terms of (2.1) are of type $(2,0)+(1,1)$.

Fiberwise application of the above construction allows one to obtain an almost
complex structure on a $4n$-dimensional manifold $X$ from a non-degenerate
complex-valued 2-form $\Omega$ on it such that
$(\Omega\wedge\overline{\Omega})^{n}$ is nowhere zero and $\Omega^{n+1}=0$; by
2.1, this condition gives a c-symplectic form on the tangent bundle.

Definition 2.11: An almost c-symplectic form on a manifold $X$ of real
dimension $4n$ is a form $\Omega\in\Gamma(\Lambda^{2}T^{*}X\otimes\mathbb{C})$
such that the top degree form $(\Omega\wedge\overline{\Omega})^{n}$ is nowhere
zero and $\Omega^{n+1}=0$. A c-symplectic form is a closed almost c-symplectic
form.

We shall denote the induced almost complex structure obtained from the form
$\Omega$ as the induced complex structure on each fiber by $I_{\Omega}$.

Proposition 2.12: [[V2] , Theorem 3.5]
Let $\Omega$ be a c-symplectic form on a manifold, and
$I_{\Omega}\in\operatorname{End}(TM)$ the corresponding almost complex
structure. Then $I_{\Omega}$ is integrable.

Proof: From Cartan’s formula it follows immediately that for any closed
$k$-form $\Phi$ and two vectors $X,Y$ such that the contractions of $\Phi$
with $X,Y$ vanish, $\Phi\hskip 2.0pt\raisebox{1.0pt}{\text{$\lrcorner$}}\hskip
2.0ptX=\Phi\hskip 2.0pt\raisebox{1.0pt}{\text{$\lrcorner$}}\hskip 2.0ptY=0$,
one also has $\Phi\hskip 2.0pt\raisebox{1.0pt}{\text{$\lrcorner$}}\hskip
2.0pt([X,Y])=0$. However, the space $T^{0,1}_{I_{\Omega}}M\subset
TM\otimes\mathbb{C}$ is defined as

$T^{0,1}_{I_{\Omega}}M=\\{X\in TM\otimes\mathbb{C}\ \ |\ \ \Omega\hskip
2.0pt\raisebox{1.0pt}{\text{$\lrcorner$}}\hskip 2.0ptX=0\\},$

which gives $[T^{0,1}_{I_{\Omega}}M,T^{0,1}_{I_{\Omega}}M]\subset
T^{0,1}_{I_{\Omega}}M$. By Newlander-Nirenberg, this condition is equivalent
to integrability of $I_{\Omega}$.

Note that converse is not generally true: if $\Omega$ is a c-symplectic form
and $f$ is a non-vanishing function, then $f\Omega$ is not closed unless $f$
is constant; however, $I_{f\Omega}=I_{\Omega}$.

##  3 Degenerate twistorial deformation

Though this section is self-contained, its results generalise and simplify the
results of [V2].

Proposition 3.1:  Let $\Omega\in\Lambda^{2}V^{*}\otimes\mathbb{C}$ be a
c-symplectic form on a real vector space $V$, $L$ a c-Lagrangian subspace in
$V$ and $\pi\colon V\to V/L$ the projection. Then for any complex valued form
$\gamma\in\Lambda^{2}(V/L)^{*}\otimes\mathbb{C}$ of Hodge type $(2,0)+(1,1)$
w. r. t. the inherited complex structure on $V/L$, the form
$\Omega_{\gamma}=\Omega+\pi^{*}\gamma$ is c-symplectic.

Proof: We shall proceed in two steps: first, we prove that
$\dim\ker\Omega_{\gamma}$ is at least half of $\dim V$, and then we prove that
$\dim\ker\Omega_{\gamma}$ does not exceed half of $\dim V$.

Assertion 1: $\dim\ker\Omega_{\gamma}\geqslant\frac{1}{2}\dim V$. Clearly,
this would follow if we prove that $\Omega_{\gamma}^{n+1}=0$, where
$4n=\dim_{\mathbb{R}}V$. Let $V_{\mathbb{C}}:=V\otimes_{\mathbb{R}}\mathbb{C}$
be the complexification of $V$, $L\oplus K=V$ a direct sum decomposition, with
$K,L\subset V$ complex vector spaces, and $L_{\mathbb{C}},K_{\mathbb{C}}$
complexifications of these spaces. Denote by
$L_{\mathbb{C}}=L_{\mathbb{C}}^{1,0}\oplus L_{\mathbb{C}}^{0,1}$ and
$K_{\mathbb{C}}=K_{\mathbb{C}}^{1,0}\oplus K_{\mathbb{C}}^{0,1}$ their Hodge
decompositions. Consider a linear automorphism
$T_{\lambda}:\;V_{\mathbb{C}}{\>\longrightarrow\>}V_{\mathbb{C}}$ acting as
multiplication by a scalar $\lambda\in\mathbb{C}$ on $K_{\mathbb{C}}^{1,0}$
and as identity on $L_{\mathbb{C}}$ and $K_{\mathbb{C}}^{0,1}$. Since $\Omega$
is a pairing between $K_{\mathbb{C}}^{1,0}$ and $L_{\mathbb{C}}^{1,0}$, one
has $T_{\lambda}(\Omega)=\lambda\Omega$. Since $\pi^{*}\gamma$ vanishes on
$L$, one has
$T_{\lambda}(\pi^{*}\gamma)=\lambda^{2}(\pi^{*}\gamma)^{2,0}+\lambda(\pi^{*}\gamma)^{1,1}$.
Using this, we write the weight decomposition for the action of $T_{\lambda}$
on $\Omega_{\gamma}^{n+1}$ as follows:

$T_{\lambda}(\Omega_{\gamma}^{n+1})=\sum_{i=0}^{n}\sum_{j=0}^{n-i}\lambda^{n+1+j}\Omega^{i}\wedge(\gamma^{2,0})^{j}\wedge(\gamma^{1,1})^{n-i-j+1}$

However, $T_{\lambda}$ cannot act on $2n+2$-forms with weight $\geqslant n+1$
because $\dim_{\mathbb{C}}L_{\mathbb{C}}^{1,0}=n$. Therefore,
$\Omega_{\gamma}^{n+1}=0$.

Assertion 2: $\dim\ker\Omega_{\gamma}\leqslant\frac{1}{2}\dim V$, or,
equivalently, ${\rm rk}\Omega_{\gamma}\geqslant{\rm rk}\Omega$. Clearly, the
point $\Omega\in\Lambda^{2}(V_{\mathbb{C}})$ has a neighbourhood $U$ such that
all 2-forms $y\in U$ have rank $\geqslant{\rm rk}\Omega$. Assertion 2 is
trivial when $\Omega_{\gamma}\in U$. Consider, as in the previous step, the
decomposition

$V_{\mathbb{C}}=L_{\mathbb{C}}^{1,0}\oplus L_{\mathbb{C}}^{0,1}\oplus
K_{\mathbb{C}}^{1,0}\oplus K_{\mathbb{C}}^{0,1},$

and let $R_{\lambda}\in\operatorname{Aut}(V_{\mathbb{C}})$ act on
$K_{\mathbb{C}}^{0,1}$ and $L_{\mathbb{C}}^{0,1}$ as identity, on
$L_{\mathbb{C}}^{1,0}$ as multiplication by $\lambda^{-1}$ and on
$K_{\mathbb{C}}^{1,0}$ as multiplication by $\lambda$. Since $\Omega$ is a
pairing between $L_{\mathbb{C}}^{1,0}$ and $K_{\mathbb{C}}^{1,0}$, one has
$R_{\lambda}(\Omega)=\Omega$. Since $\gamma$ vanishes on $L$, one has
$R_{\lambda}(\gamma)=\lambda^{2}\gamma^{2,0}+\lambda\gamma^{1,1}$. Then
$\lim_{\lambda\mapsto 0}R_{\lambda}(\Omega_{\gamma})=\Omega$, hence for
$\lambda$ sufficiently small, the form $R_{\lambda}(\Omega_{\gamma})$ belongs
to $U$, and satisfies ${\rm rk}\Omega_{\gamma}\geqslant{\rm rk}\Omega$. We
proved 3.

Proposition 3.2:  Let $\Omega\in\Lambda^{2}V^{*}\otimes\mathbb{C}$ be a
c-symplectic form on a real vector space $V$, $L$ a c-Lagrangian subspace in
$V$ and $\pi\colon V\to V/L$ the projection. Consider a (2,0)+(1,1)-form
$\gamma$ on $V/L$, and let $\Omega_{t}:=\Omega+t\pi^{*}\gamma$, for some
$t\in\mathbb{C}$. Then the subspace $L$ is c-Lagrangian w. r. t. all the forms
$\Omega_{t}$, and restrictions
$I_{\Omega_{t}}{\left|{}_{{\phantom{|}\\!\\!}_{L}}\right.}$ of induced complex
structures $I_{\Omega_{t}}$ coincide. Moreover, the complex structures on the
quotient $V/L$ inherited from $I_{\Omega_{t}}$ also coincide, for all
$t\in\mathbb{C}$.

Proof: The first claim is obvious from the construction of $\Omega_{t}$. The
prove the second claim, take $v\in(V/L)^{0,1}_{I_{\Omega}}$, and let
$u\in\pi^{-1}(v)$ be a vector in its preimage. Then $\Omega_{t}\hskip
2.0pt\raisebox{1.0pt}{\text{$\lrcorner$}}\hskip 2.0ptu=t\pi^{*}(\gamma\hskip
2.0pt\raisebox{1.0pt}{\text{$\lrcorner$}}\hskip 2.0ptv)$. Therefore,

$u\in V^{0,1}_{I_{\Omega_{t}}}\ \ \Leftrightarrow\ \ \forall z\in V,\ \
\Omega(u,z)=-t\pi^{*}(\gamma\hskip
2.0pt\raisebox{1.0pt}{\text{$\lrcorner$}}\hskip 2.0ptv)(z))$ (3.1)

The 1-form $z{\>\longrightarrow\>}t\pi^{*}(\gamma\hskip
2.0pt\raisebox{1.0pt}{\text{$\lrcorner$}}\hskip 2.0ptv)(z)$ vanishes on $L$
and has type (1,0) on $(V,I_{\Omega})$, because $\gamma$ is of type
(2,0)+(1,1).

Since $\Omega$ is a non-degenerate pairing between $L^{0,1}_{I_{\Omega}}$ and
$(V/L)^{0,1}_{I_{\Omega}}$, for any (1,0)-form $\xi$ on $V/L$ there exists a
vector $x\in L$ such that $\xi=\Omega\hskip
2.0pt\raisebox{1.0pt}{\text{$\lrcorner$}}\hskip 2.0ptx$.

This gives a vector $\zeta_{t}\in L^{0,1}_{I_{\Omega}}$ such that such that

$\Omega(u+\zeta_{t},z)=-t\pi^{*}(\gamma\hskip
2.0pt\raisebox{1.0pt}{\text{$\lrcorner$}}\hskip 2.0ptv)(z))$

for all $z\in V$. By (3.1), $u+\zeta_{t}\in V^{0,1}_{I_{\Omega_{t}}}$ is a
vector which projects to $v$.

This implies that the space of (0,1)-vectors in $(V/L)_{I_{\Omega_{t}}}$ is
independent from $t$.

Theorem 3.3:  Suppose that $X\xrightarrow{\pi}B$ is a Lagrangian fibration on
a holomorphically symplectic manifold $(X,\Omega)$, and
$\eta\in\Omega_{\mathrm{cl}}(B)$ a closed $(2,0)+(1,1)$-form on the base. Then
the forms $\Omega_{t}=\Omega+t\pi^{*}\eta$ on $X$ are c-symplectic, and this
deformation (called degenerate twistorial deformation) preserves the
Lagrangian fibration and the base.

Proof: From 3 it follows that $\Omega_{t}$ is c-symplectic. The fibers stay
Lagrangian, and the complex structure thereof, as well as that of the base,
remains unchanged by 3.

Remark 3.4: This result was proven for compact manifolds in [V2, Theorem
1.10].

For a hyperkähler manifold $X$, the degenerate twistorial deformation produces
an entire curve in the period space of $X$. In terms of the oriented 2-plane
Grassmannian (see e. g. Section 3 in [De]), the plane corresponding to
$(X,\Omega_{x+\sqrt{-1}y})$ is spanned by $(\Omega+\overline{\Omega})+2x\eta$
and $\sqrt{-1}(\Omega-\overline{\Omega})-2y\eta$. Thus one can define
degenerate twistorial curves in an abstract situation: namely, for a 2-plane
$\tau\in\operatorname{Gr}_{++}(V,q)$ and a vector $e\in\tau^{\perp}\subset V$
with $q(e,e)=0$, the subvariety
$\operatorname{Deg}_{\tau}(e)=\operatorname{Gr}_{++}(\mathrm{span}(\tau,e),q|_{\mathrm{span}(\tau,e)})\subseteq\operatorname{Gr}_{++}(V,q)$
is an entire curve, and in the case when $V=H^{2}(X,\mathbb{R})$, $q$ is the
Bogomolov–Beauville–Fujiki form and $e=\pi^{*}[\omega]$ is the inverse image
of the Kähler class on the base, this curve is exactly the base of the
degenerate twistorial deformation.

Theorem 3.5:  Let $X\xrightarrow{\pi}B$ be a holomorphic Lagrangian fibration
on a holomorphically symplectic manifold $(X,\Omega)$, and $\sigma\colon B\to
X$ a smooth section of $\pi$. Then there exists a degenerate twistorial
deformation $(X,\Omega^{\prime})$ of $(X,\Omega)$ s. t. the fibers stay
Lagrangian, the complex structure on the fibers and the base stays the same,
and $\sigma$ is a holomorphic map.

Proof: Consider the form $\eta=\sigma^{*}\Omega\in\Omega^{2}(B)$. By
Proposition 2.2, it is of type $(2,0)+(1,1)$. Then by Proposition 3 the forms
$\eta$ gives rise to a deformation with desired properties. For $t=-1$ one has
$\Omega_{t}|_{\sigma(B)}=\left(\Omega-\pi^{*}\sigma^{*}\Omega\right)|_{\sigma(B)}=\Omega|_{\sigma(B)}-\Omega|_{\sigma(B)}=0$.
By Hitchin’s lemma (2.1), this means that the submanifold $\sigma(B)$ is a
complex submanifold. Since the projection is a holomorphic map, the section
$\sigma$ is also holomorphic.

##  4 Holomorphic Lagrangian fibrations on a K3 surface

In this section we apply 3 to obtain sections of holomorphic Lagrangian
fibrations on K3 surfaces.

It is not hard to see that the elliptic fibrations on a K3 surface $X$ are in
bijective correspondence with primitive numerically effective classes
$e\in\operatorname{NS}(X)$ with $(e,e)=0$. Indeed, its linear system has no
basepoints and establishes a map to $\mathbb{C}P^{1}$ with generic fiber
elliptic curve. Cross-sections of the fibration determined by $e$ correspond
to classes $s\in\operatorname{NS}(X)$ with $(s,e)=1$. Our first goal is to
find an effective class $s\in H^{2}(X,\mathbb{Z})$ with $(s,e)=1$ and a
deformation in which $s$ would have type $(1,1)$.

Claim 4.1:  For any isotropic primitive vector $e$ in an even unimodular
lattice $\Lambda$ there exists a vector $a\in\Lambda$ such that one has
$(a,e)=1$ and $(a,a)=-2$.

Proof: Since the lattice $\Lambda$ is unimodular, one can pick some vector
$b\in\Lambda$ with $(b,e)=1$. The number $(b,b)$ is even since $\Lambda$ is
even, so the vector $a=b-\left(1-(b,b)/2\right)e$ is integral. One has
$(a,e)=(b,e)-(1+{(b,b)}/2)(e,e)=1$ and
$(a,a)=(b-(1+{(b,b)}/2)e,b-(1+{(b,b)}/2)e)=(b,b)-2(1+{(b,b)}/2)(b,e)=(b,b)-2-(b,b)=-2$.

The following claim is trivial.

Claim 4.2:  Suppose that $X$ is a K3 surface with an elliptic fibration
determined by an isotropic class $e$. Then the pullback of the Fubini–Study
form from the base represents $e$.

Proof: The Fubini–Study form on the base represents a Poincaré dual to the
class of a point. Then its inverse image represents the Poincaré dual to the
class of the inverse image of the point, i. e. of the fiber.

Lemma 4.3:  Suppose that $X$ is a K3 surface with elliptic fibration
$X\xrightarrow{\pi}\mathbb{C}P^{1}$ determined by an isotropic class $e$,
$\omega_{\operatorname{FS}}$ the Fubini–Study form on its base, and $s\in
H^{2}(X,\mathbb{Z})$ is such that one has $(s,e)\neq 0$. Then there exists a
unique degenerate twistorial deformation $X^{\prime}$ of $(X,\pi)$ in such way
that $s$ belongs to $H^{1,1}(X^{\prime})$.

Proof: The forms $\Omega-t\pi^{*}\omega_{\operatorname{FS}}$,
$t\in\mathbb{C}$, are c-symplectic by Proposition 3. Denote by $I_{t}$ the
complex structure induced by $\Omega-t\pi^{*}\omega_{\operatorname{FS}}$
(2.1). The homology class of $s$ is of type $(1,1)$ if and only if
$(s,[\Omega-t\pi^{*}\omega_{\operatorname{FS}}])=0$. Using 4, one can rewrite
this equation as $(s,[\Omega])=t(s,e)$. It has a unique solution $t$ whenever
$(s,e)\neq 0$.

Proposition 4.4:  Let $X$ be a K3 surface with elliptic fibration
$X\xrightarrow{\pi}\mathbb{C}P^{1}$. Then there exists a degenerate twistorial
deformation $X^{\prime}$ of $(X,\pi)$ admitting a holomorphic section.

Proof: Since the lattice $H^{2}(X,\mathbb{Z})$ for a K3 surface $X$ is even
and unimodular, by 4, one can find a vector $s\in H^{2}(X,\mathbb{Z})$ such
that $(s,e)=1$ and $(s,s)=-2$. By 4, we can deform the complex structure on
$X$ in such way that $s$ would have type $(1,1)$. By Lefschetz theorem on
$(1,1)$-classes, there exists a line bundle $L\to X$ with $c_{1}(L)=s$.
Riemann–Roch theorem for this bundle $L$ reads
$\chi(L)=\chi(\mathscr{O}_{X})+\frac{L\cdot(L\otimes K_{X}^{*})}{2}$. By
Serre’s duality one has $h^{2}(L)=h^{0}(L^{*}\otimes K_{X})$, and, as soon as
$K_{X}$ is trivial and $\chi(\mathscr{O}_{X})=2$, we can rewrite it as
$h^{0}(L)-h^{1}(L)+h^{0}(L^{*})=2+\frac{-2}{2}=1$. Since $L$ is nontrivial,
either $h^{0}(L)$ or $h^{0}(L^{*})$ does not vanish, and either $L$ or $L^{*}$
is effective. If $L^{*}$ is, then $-s$ is represented by a curve, and
$0<(-s,e)=-(s,e)=-1$, hence $L$ is effective and $s$ is represented by a
curve. Suppose that $s=\sum_{i}s_{i}$, where $s_{i}$ are the classes of
irreducible curves. One has $1=(s,e)=(\sum_{i}s_{i},e)=\sum_{i}(s_{i},e)$. All
the numbers $(s_{i},e)$ are positive integers with sum $1$, so exactly one of
them, say $(s_{0},e)$, equals $1$. The curve represented by the class $s_{0}$
intersects each fiber at one point, i. e. is a section of the fibration.

In particular, this implies that elliptic fibrations on K3 surfaces cannot
have multiple fibers.

#### Acknowledgements.

We are grateful to Dmitri Kaledin for insightful discussions.

## References

  * [BHPV] Barth, Wolf P. ; Hulek, Klaus ; Peters, Chris A. M. ; Ven, Antonius Van d., Compact complex surfaces, second edition, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge.. Bd. 4: Berlin : Springer-Verlag, 2004
  * [CMSB] Koji Cho, Yoichi Miyaoka, and N. I. Shepherd-Barron, Characterizations of Projective Space and Applications to Complex Symplectic Manifolds, Adv. Stud. Pure Math. Higher Dimensional Birational Geometry, S. Mori and Y. Miyaoka, eds. (Tokyo: Mathematical Society of Japan, 2002), 1 - 88
  * [De] R. Déev. Compact fibrations with hyperkähler fibers Journal of Geometry and Physics Volume 123, pp. 372–378, 2018.
  * [GLR] D. Greb, C. Lehn, S. Rollenske, Lagrangian fibrations on hyperkähler manifolds - On a question of Beauville, Annales scientifiques de l’École Normale Supérieure, Serie 4, Volume 46 (2013) no. 3, p. 375-403
  * [Hi] N. Hitchin, The moduli space of complex Lagrangian submanifolds, Surveys in Differential Geometry, 2000, Vol. VII. International Press, pp. 327—345/
  * [Hw] Jun-Muk Hwang, Base manifolds for fibrations of projective irreducible symplectic manifolds, arXiv:0711.3224, Inventiones mathematicae, vol. 174, issue 3, pp. 625-644.
  * [HW] Hwang J.-M., Weiss, R. M., Webs of Lagrangian tori in projective symplectic manifolds, Invent. Math. 192 (2013) 83-109.
  * [HO] J.-M. Hwang and K. Oguiso, Multiple fibers of holomorphic Lagrangian fibrations, Commun. Contemp. Math. 13 (2011), no. 2, 309-329.
  * [KV] Kamenova, L., Verbitsky, M., Families of Lagrangian fibrations on hyperkaehler manifolds, Adv. Math. 260 (2014) 401 - 413; arXiv:1208.4626 [math.AG].
  * [Mat1] Matsushita, D., On fibre space structures of a projective irreducible symplectic manifold, 38 (1999), No. 1, 79-83, see also Addendum, Topology 40 (2001) No. 2, 431 - 432.
  * [Mat2] Matsushita, D., Higher direct images of Lagrangian fibrations, Amer. J. Math. 127 (2005), arXiv:math/0010283.
  * [Mat3] Daisuke Matsushita, A canonical bundle formula for projective Lagrangian fibrations, preprint arXiv:0701.0122.
  * [V1] Verbitsky, M., Hyperkahler SYZ conjecture and semipositive line bundles, arXiv:0811.0639, 21 pages, GAFA 19, No. 5 (2010) 1481-1493.
  * [V2] M. Verbitsky. Degenerate twistor spaces for hyperkähler manifolds. Journal of Geometry and Physics Volume 91, pp. 2–11, 2015

Fedor A. Bogomolov
Department of Mathematics
Courant Institute, NYU
251 Mercer Street
New York, NY 10012, USA,
<EMAIL_ADDRESS>also:
National Research University, Higher School of Economics, Moscow, Russia,

Rodion N. Déev
Independent University of Moscow
119002, Bolshoy Vlasyevskiy Pereulok 11
Moscow, Russian Federation
<EMAIL_ADDRESS>also:
Department of Mathematics
Courant Institute, NYU
251 Mercer Street
New York, NY 10012, USA,
<EMAIL_ADDRESS>

Misha Verbitsky
Instituto Nacional de Matemática Pura e Aplicada (IMPA)
Estrada Dona Castorina, 110
Jardim Botânico, CEP 22460-320
Rio de Janeiro, RJ - Brasil
also:
Laboratory of Algebraic Geometry,
National Research University HSE,
Department of Mathematics, 7 Vavilova Str. Moscow, Russia,
<EMAIL_ADDRESS>


# An interdisciplinary overview of developmental indices and behavioral
measures of the minimal self ††thanks: This work was funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) - 402790442
(”Prerequisites for the Development of an Artificial Self”) within the SPP
”The Active Self” (SPP 2134). The work of GS has partially received funding
from the European Union’s Horizon 2020 research and innovation programme under
the Marie Sklodowska-Curie grant agreement No. 838861 (Predictive Robots).

Yasmin Kim Georgie  Adaptive Systems Group
Humboldt-Universität zu Berlin
Berlin, Germany
<EMAIL_ADDRESS>Guido Schillaci  The BioRobotics
Institute
Scuola Superiore Sant’Anna, Pisa, Italy
and Adaptive Systems Group
Humboldt-Universität zu Berlin
<EMAIL_ADDRESS>Verena Vanessa Hafner  Adaptive Systems Group
Humboldt-Universität zu Berlin
Berlin, Germany
<EMAIL_ADDRESS>
###### Abstract

In this review paper we discuss the development of the minimal self in humans,
the behavioural measures indicating the presence of different aspects of the
minimal self, namely, body ownership and sense of agency, and also discuss
robotics research investigating and developing these concepts in artificial
agents. We investigate possible avenues for expanding the research in robotics
to further explore the development of an artificial minimal self.

###### Index Terms:

Models of self and agency; Sensorimotor development; Machine Learning methods
for robot development

##  I Introduction

For centuries, philosophers and scholars from different disciplines have been
debating the nature of subjective experience and self-consciousness. A recent
account brings back a phenomenological view on this debate, researching self-
consciousness in its minimal form, that is studying subjective experiences in
their immediate and first-personal ”mineness”. According to Gallagher, this
so-called minimal self refers to the ”consciousness of oneself as an immediate
subject of experience, unextended in time” [1]. Aspects of this minimal self
involve the sense of agency—the sense of the self as the one causing or
generating an action, and the sense of ownership—the sense of the self as the
one subjected to an experience [1].

This view is distinct from more elaborated aspects of the self, such as the
reflexive self and the narrative self [2]. The minimal self is closer to a
minimalist level of subjective experience, where the focus is more on the
contribution of the here and now, bodily experience in its construction [3].
Other low-level notions of the self have been proposed in the literature, such
as the proto-self and the immunological self [4]. Hereby, we do not enter the
debate about which of them constitutes the lowest level of consciousness, as
researchers have not yet converged to an agreement. We commit to the notion of
the minimal self, as this aspect of consciousness is perhaps the most easily
accessible in terms of experimental exploration and quantification, and is in
fact receiving greater attention from disciplines such as neuroscience,
behavioural and cognitive sciences, and developmental psychology [5, 6].

Developmental psychologists consider the emergence of a sense of the self as a
key step in cognitive development. By the second year of life, thus few months
after having acquired basic linguistic skills, toddlers are capable of using
self-referential language such as I, me, my—suggesting that the acquisition of
a self-concept has started earlier. The minimal self is pre-linguistic and
non-conceptual, and is suggested to unfold already during early developmental
stages [7].

This paper presents an interdisciplinary overview of developmental indices and
behavioural measures of the minimal self. The minimal self is argued to
include two main aspects—a sense of agency and a sense of body ownership—which
are thought to be dependent on an internal body representation maintained by
our brain. This manuscript thus starts with discussing the development of body
representations as a necessary condition for the emergence of the senses of
body ownership and of agency (section II). Behavioural paradigms and measures
indicating the presence of different aspects of the minimal self (section III)
are analysed. In particular, we survey studies on self-touch, intentional
binding and sensory attenuation, and the rubber hand illusion.

Alongside the survey on the developmental aspects of the minimal self and the
aforementioned behavioural paradigms and measures, the main goal of this work
is to discuss the most prominent related studies in robotics. In fact, there
is a growing interest in the developmental robotics community in implementing
processes capable of enabling the experience of the self in artificial agents.
Self-awareness may clearly improve adaptivity and autonomy in robots and, as a
result, reduce human intervention in their programming. Using robots as test-
beds for studying the minimal self may also shed light on the cognitive
mechanisms underling subjective experience. Nonetheless, the investigation on
the artificial self is still young and fragmented. This work contributes with
identifying current knowledge gaps and limitations in robotics studies and
with suggesting research directions for the implementation of behavioural
paradigms and measures for the artificial self.

##  II The development of body representations

In order to effectively interact with the environment, an embodied agent must
form and maintain an internal representation of its own body situated within
the environment. The term representation may sound controversial. In fact,
scholars usually take either a representationalist or a sensorimotor position
in the philosophical study of bodily awareness (see [8], section 2, for a
review), which may lead to the definition of body image—the mental
representation of the body, constituted by a combination of sensorimotor
experience and social and psychological concepts about it, and body schema—the
integrated, neural organisation of multimodal stimuli coming from the
different parts of the body, which is essential for movements. Here, we commit
to the latter interpretation of the term body representation, that is a
mapping of the body in its various modalities (tactile, proprioceptive,
visual, motor, etc.), which is operated in a nonconscious way [9]. The neural
foundation for these representations are the so-called cortical ”homunculi” in
the primary sensory (S1) and motor (M1) cortices. These are neurological
representations of the different anatomical divisions of the body, mapped onto
brain areas charged with sensory and motor processing along S1 and M1,
respectively. These specialized areas are organised in a somatotopic map where
adjacent body parts are represented closely together. The extent of cortex
dedicated to a body region is not proportional to its size in the body, but
rather to the density of innervation in that specific part (e.g. the mouth and
palms). The establishment of the somatotopic organisation in S1 and M1 is
driven by genetic factors that are later elaborated through changes in
connectivity driven by embodied interactions both before and after birth [10].

Body representations dynamically integrate information from different sensory
modalities: tactile, proprioceptive, vestibular, motor, and visual. Studies
suggest that the first sense to emerge in the foetus is the somatosensory
sense[11], where foetuses are in a state of constantly being touched by their
environment. In addition, they engage in self-touch in the womb: often
touching their mouth and feet—body parts that are highly innervated and
therefore most sensitive to touch—and later on, other parts of the body. The
early inclination for movements and self-touch in parts of the body that are
more sensitive, suggests that the foetus shows a preference towards movements
that induce more informative sensations [12]

From as early as 19 weeks, foetuses can anticipate hand-to-mouth touch, in
opening their mouth prior to contact [13, 14], indicating the early presence
of a sort of sensorimotor mapping and inference. From 22 weeks, movements seem
to show a form of intentional movement, as they become more direct dependening
on the action goal [15]. Evidence from neural development studies suggests
that even before birth, the prenatal brain should be able to perceive
information arising from the body, while higher level (multimodal)
representations are possibly formed during the first year after birth, in
accordance with the development of association areas [16].

At around 2 months of age, the dominant control of behaviours transitions from
subcortical to higher order cortical systems [17]: PET studies show dominant
metabolic activity in subcortical regions and the sensorimotor cortex in
infants under 5 weeks after birth, and by 3 months, an increase in metabolic
activity in the parietal, temporal, and dorsolateral occipital cortices [18].
Hand-mouth coordination continues to develop after birth, and from birth to 6
months, infants display self-touch in a progressive manner throughout their
body, from frequently touching rostral parts such as the head and trunk, to
more caudal parts of the body such as the hips, legs, and feet later on [19].
The development of goal-directed reaching considerably speeds up at about 5
months of age. Reaching to the own body is thought to be the product of
interactions between multiple subsystems. The body representation is
constructed through reaching to the body because, in this, the body that is
used to act upon the environment becomes the target itself, and therefore
needs to be modeled. In certain cases, the self-touch process seems to bypass
vision, as when the target is the face, relying only on somatosensory
information.

In summary, genetically predetermined cortical maps—the ”homunculi” in the
primary sensory (S1) and motor (M1) cortices—facilitate the formation of body
representations through cortical learning of sensorimotor contingencies—i.e.
the statistical connections between sensory and motor information, and
sensorimotor integration—integrating this information into common percepts.
This learning of sensorimotor contingencies both drives and is rendered
through interactions between brain-body-environment. Specifically, interlinked
with the neural ontogenetic process (brain maturation, brain-body
interaction), self-exploration (body babbling), self-touch, and goal-directed
reaching are considered the necessary behavioural conditions that facilitate
and reflect this process. Importantly, this process is thought to be driven
and progressively refined by the reduction of prediction errors between
predicted sensory outcomes and motor actions, such that the agent learns not
only to predict the outcomes of its actions on the environment, but also to
predict the (sensory) outcomes of its (motor) actions on its own body [20,
21].

Rochat [22] describes the idea that infants’ self-exploration, and
interactions with the environment, give rise to the sense of body ownership
through the ”ability to detect intermodal invariants and regularities in their
sensorimotor experience, which specify themselves as separate entities agent
in the environment.” Therefore, self-exploration (body babbling), self-touch,
and goal-directed reaching are necessary conditions for the development of
motor control and the emergence of body ownership and sense of agency.

##  III Behavioural paradigms and measures of the self

A challenging task in the study of the minimal self is to experimentally
quantify the attribution of subjective experience. A number of behavioural
paradigms and attempts for objective measures have been proposed. This section
reviews the most prominent ones. In particular, we analyse studies on self-
touch, intentional binding and sensory attenuation, and the rubber hand
illusion. There are different reasons why we consider them important in this
study:

  * A.

Self-touch is likely to contribute to the formation of initial sensorimotor
representations, and may therefore constitute one of the very first cues for
subjective experience during early developmental stages.

  * B.

The way the brain interprets action effects has been shown to differ depending
on whether the sensory perception is self- or externally produced, with
respect both to the perceived timing of their occurrence (intentional binding)
and to their intensity (sensory attenuation) [23].

  * C.

The rubber hand illusion has been extensively used as a paradigm for
investigating the mechanisms underlying the sensorimotor minimal self.

An obvious advantage of conducting robotics research as test-beds for
cognitive models in humans is that using robots allows unmediated access to
the actual process (the algorithm) and information that is registered and
processed in the system. In human research, even with advanced neuroimaging
and analysis methods, researchers could only ever have an approximation of the
actual cognitive processes underlying behaviour. This is especially true when
investigating subjective experiences such as the illusion of a dummy hand or
object being ”mine”, which is measured by observing behaviour or with
subjective self-reports from human participants, whereas the difficulty lies
with developing and most importantly validating objective measures of
cognitive processes (e.g. neuroimaging, proprioceptive drift). The question of
how we can measure a ”subjective” experience in a robot then arises.

###  III-A Self-Touch

How does self-touch relate to body ownership?

Throughout development, the brain must establish links between sensory and
motor maps. Refinement of these links eventually leads to goal directed
actions. Establishing the basis of this lies in forming sensorimotor
contingencies—the statistical links between sensory and motor information.

These sensorimotor contingencies are thought to be established through body
babbling where the infant moves its body in an exploratory manner, whereas the
brain initiates actions and organises the resulting sensory outcomes
continuously, refining its ability to predict sensory outcomes from motor
actions. The process of establishing sensorimotor contingencies is gradual due
to the large amount of information that is being processed—sensory inputs,
motor outputs, and the statistical correlations between them.

Hoffmann et al. [24] suggest that the most systematic correlations are those
that will emerge most easily, and therefore, the links between motor actions
and proprioceptive changes are presumably the simplest to be extracted,
followed by the links between motor actions and tactile input. Hoffman [16]
suggests that the redundant information arising from the configurations of
self-touch in the proprioceptive-tactile space facilitates learning the body
model in space. He asserts that pre-natal self-touch likely contributes to the
formation of initial somatosensory representations. Evidence for an early
integration between modalities comes from the instances of hand-to-mouth
anticipation already in the womb [13, 14]. However, the formation of more
comprehensive multimodal body representations probably occurs after birth,
from 2-3 months, to include the visual modality and its connections to
tactile-proprioceptive modalities. These are learned through self-exploration
including self-touch within the environment, which involves learning temporal
contingencies, spatial congruence, and redundancies of information coming from
different sensory modalities.

Rochat et al. [25] provide evidence that early on infants are capable of
discriminating perceptual events—tactile stimuli—that are either self- or not
self-produced. The authors tested the rooting behaviour—a reflex behaviour
that is triggered by touching the cheek of the infant—in 24h and 4 weeks old
newborns, and reported that infants tended to manifest rooting responses
almost three times more often in response to external compared to self-
stimulation. This suggests that infants pick up already at birth sensorimotor
contingencies (single touch or double touch) that specify self- versus
external stimulation.

In a longitudinal study, Hofmann et al. [24] observed how infants between 3
and 21 months react to vibrotactile stimulation applied to different body
parts. They report responses that varied between particular movement in the
stimulated body part and successfully reaching and removing the buzzer. They
found an overall developmental progression from general to specific movement
patterns, especially from 3 to 12 months. Specifically, their results suggest
that at 3-4 months, the infant responds to the buzzer in a non-specific way by
moving its whole body, rather than moving the particular limb that was
stimulated. However, between 4-12 months, the limb-specific buzzer-oriented
reaching develops.

Robotics research

Yamada et al. [26] worked on an embodied brain model of a human foetus in
order to examine the causal link between sensorimotor experiences through
embodied interactions and cortical learning of body representations. The
embodied brain model was based on anatomical and physiological data and
included a cortex, a spinal circuit, and musculoskeletal body with sensory
receptors for proprioception, tactile perception, and vision, within a model
of a uterine environment. The results of this study showed that embodied
interaction between brain-body-environment within the womb help to guide the
cortical learning of body representations through regularities in sensorimotor
experiences. Also, the embodied interactions inside the womb provided a better
arena for forming cortical body representations when compared to extrauterine
embodied interactions. These findings support the notion suggested by previous
studies on animal newborns and preterm infants that the formation of body
representations begins even before birth. In addition, their findings suggest
that embodied interactions inside the womb set the stage for visual-
somatosensory integration after birth [26].

Expanding on this line of research but in a real robot body, Noda et al. [27]
equipped a humanoid robot with soft skin sensors and left it engaging in
human-robot touch interactions. The tactile data generated through such
interactions were used by the authors to form a self-organising somatosensory
map, where the feature space was composed by spatially-adjacent sensor pairs.
Computational body representations based on self-organising maps and
multimodal integration through Hebbian connections have been proposed by
Schillaci et al. [28], although not in the context of self-touch. Nonetheless,
the model enabled a humanoid robot with predictive capabilities, fundamental
for the study of sensory attenuation processes and sense of agency (see next
section).

In [29] Hoffman et al. targeted the development of self-organising body
representations in the iCub humanoid robot, learning somatosensory
representations of the tactile space. Hoffmann et al. examined how the iCub
robot, equipped with artificial skin, could form a topographic body
representation by learning from tactile stimulations over the surface of its
skin. They used modified SOMs that restrict the size of the maximum receptive
field (MRF) of neuron groups at the output layer in order to reproduce the
genetically predetermined organisation of somatotopic cortical maps. The
formation of a tactile map organisation (predetermined with the MRF-SOM) has
been reproduced by using training data obtained from a ”double-touch”
procedure, in which two experimenters provided tactile stimulation in two
different places on the artificial skin.

Current limitations in robotics research

The authors in [29] focused on mimicking the formation process of the cortical
”homunculus” in a real robot, determined a-priori. The training data for the
model was not generated by the robot itself, but rather with two methods:
either with simulated data or with a human (or two, for studying multi-touch)
experimenter touching the robot. Indeed, investigating self-touch in robotics
would require the robot itself generating the behaviour that drives the
formation of the body representation, which would incorporate the work on
proprioceptive representations in [30]. Another possible expansion could be to
implement this in a predictive coding framework, such that the self-touch
behaviour would be driven by prediction error minimization.

###  III-B Intentional binding and sensory attenuation

How do intentional binding and sensory attenuation relate to agency?

The human brain interprets action effects differently depending on whether the
sensory perception is self-produced or externally generated, with respect both
to the perceived timing of their occurrence (intentional binding) and to their
intensity (sensory attenuation) [23].

In 2002, Haggard et al. studied the perceived time of intentional actions and
their sensory consequences, while investigating action awareness [31].
Interestingly, they found that voluntary actions and their effects are
perceived as closer in time, compared to the perceived time shift between
involuntary movements—induced by transcranial magnetic stimulation
(TMS)—followed by the same effects. Specifically, subjects perceived voluntary
movements as occurring later than when they actually occurred, and their
sensory consequences as occurring earlier. This effect, known as intentional
binding, has attracted the attention of many scholars interested in shedding
light on the nature of the sense of agency (see [32] for a review). Engel and
Singer reported several pieces of evidence from animal and human studies
suggesting that temporal dynamics in neuronal activity may be critically
involved in conscious states, in particular that synchronisation may be
involved in the generation and maintenance of sensory awareness [5].

Sensory attenuation refers to the partial cancellation, or reduction, of the
perceived intensity of the effects of a self-initiated action. Several studies
show similar effects. Blakemore et al., for instance, found that self-produced
tactile stimulation was perceived as less intense compared to when the same
stimulus was produced externally [20]. In the experiment, participants moved
the arm of a robot with their left hand in order to produce tactile stimuli on
their right hand via a second robot. The authors found that varying the delay
between the movements of the left hand and the resulting movements producing
the tactile stimuli on the right hand, and varying the degree of trajectory
perturbation all had an effect on the rating of tickliness sensations.
Participants perceived the stimuli produced by more delayed and more
perturbated movements as more tickling, suggesting that self-produced
movements attenuate resulting sensations and that a prerequisite for this
attenuation is that stimuli and their causal motor commands correspond in time
and space.

Robotics research

In [33], Michel et al. experimented the incremental learning of characteristic
time delay inherent in the action-perception loop from a sequence of random
arm motions within the visual field in a humanoid robot. Interestingly, the
study showed that the learned time delay can be successfully used to identify
own body parts in the visual field.

Lang, Schillaci and Hafner [34] studied how a humanoid robot can learn,
through a self-exploration behaviour, the sensory outcomes (in the visual
domain) of self-generated movements. The sensorimotor experience gathered
during this process was used as training data for a deep convolutional neural
network that mapped proprioceptive and motor data (e.g. initial arm joint
positions and applied motor commands) onto the visual outcomes of these
actions. The authors then used such a forward model in two experiments. First,
for generating visual predictions of self-generated movements, which were
compared to actual visual perceptions and then used to compute a prediction
error. The system generated higher prediction errors when an external subject
was performing actions in front of the robot, compared to situations in which
the robot was observing only itself doing the same arm movements. The authors
also showed how predictions can be used to attenuate self-generated movements,
and thus create enhanced visual perceptions, where the sight of
objects—originally occluded by the robot body—was still maintained. This
suggests that similar processes may shed light also on the understanding of
the sense of object permanence and of short term memory systems in humans.

In [28], Schillaci et al. presented a biologically inspired model for learning
multimodal body representations in artificial agents in the context of
learning and predicting robot ego-noise, i.e. the auditory noise produced by
the robot’s motors while it moves. The authors performed an ego-noise
attenuation experiment, which showed the effects in the ego-noise suppression
performance of coherent and incoherent proprioceptive and motor information
passed as inputs to the predictive process implemented by a forward model. In
line with the aforementioned studies, the experiments showed that ego-noise
attenuation was more pronounced when the robot was the owner of the action.
Sensory attenuation was less pronounced when the robot (not moving itself) was
listening to a simulated moving robot, as the incongruence of the
proprioceptive and motor information with the perceived ego-noise generated
larger prediction errors.

Other robotics studies can be found in the literature implementing top-down
processes for interpreting bottom-up sensory streams. An example is the
interesting work of Jun Tani [35], who implemented an incremental learning
system based on recurrent neural networks and self-organising networks that
evolves by showing steady and unsteady phases. The author explains these
fluctuations as a result of the interaction between top-down and bottom-up
processes, and makes a parallel between them and phenomenological
observations.

Current limitations in robotics research

For many years, sense of agency has been measured using explicit self-reported
judgements (e.g. [36]). However, the self-report approach has limitations, as
it is sensitive to different biases [37]. Moreover, this is not a feasible
approach in current robotics technologies if we look at it from a
phenomenological perspective, as the only level of judgement available in
robots comes at the point when the experimenter observes and interprets the
internal states of the machine.

Tani [35], for instance, compares the dynamical structure of the system
created in his robotic study to the structure of the ”self”, and observes that
”the ”self” is made aware when the unsteady phase appears in the course of the
time-development of the system”. Although we recognise the quality of the
proposed model, we believe that similar statements on subjective mind and self
in robots—as reported in the paper—are prone to criticisms about their
objective validity.

In the work of Michel et al. on learning time delays in action-perception
loops in humanoid robots [33], the learning mechanism was simply looking for
regularities in the delays between the motor activations and the detections of
moving objects on the screen. The authors performed a basic processing of the
visual input, which was at any stage mapped to other sensory modalities (e.g.
to proprioceptive modalities). The authors pointed out the need for a more
robust method of visual recognition of the robot body. However, we find as
more critical the lack of multimodal integration, which as to the proposed
study consisted only in the analysis of the timestamps of intermodal events.
In fact, as discussed in section II, the development of multimodal body
representations is a necessary condition for the emergence of a minimal self.
Further works on intentional binding effects in the investigation of sense of
body ownership and of agency in robots should thus consider also this aspect.

In general, there is a need for further investigation on the intentional
binding action effects, especially in robotics. In fact, there are studies
[38] showing that motor prediction seems to not modulate that, casting doubts
on the assumption that intentional binding of action effects is linked to an
internal forward predictive process. These studies suggest that just the
temporal control of a stimulus, by means of a voluntary action, might be
sufficient to trigger the binding effect.

Regarding the sensory attenuation measure, most of the robotics studies
addressing this topic (e.g. [28, 34]) adopt comparator computation models for
sensory prediction and attenuation. Nonetheless, more recent probabilistic
computational proposals—such as the predictive coding framework described in
the next section—would represent a more biologically plausible approach.

###  III-C Rubber Hand Illusion (RHI)

How does the RHI relate to body ownership?

The rubber hand illusion (RHI) [39], along with other body ownership
illusions, is a widely used paradigm for investigating the mechanisms
underlying the (sensorimotor) minimal self. In the rubber hand illusion, an
observer sees a dummy hand receiving touch (e.g. brush strokes) while at the
same time receiving tactile stimulation on their real hidden hand at the same
location. This usually elicits an illusory experience of sensing the
stimulation on the dummy hand [39, 40], and as a result, incorporating the
fake hand as a part of the observer’s own body [41, 42]. This effect was found
to be reflected behaviourally as a fear response when the fake hand is being
threatened [43, 44] or in perceiving the location of the real hand as closer
towards where the dummy hand is located (a ”proprioceptive drift”) [39, 41],
suggesting that the dummy hand is treated by the brain as part of the own body
as a result of the multisensory stimulation [42]. These illusory effects are
thought to be an indicator for the presence of multimodal, integrated body
representations. RHI studies show that both top-down and bottom-up processes
influence the embodiment of the dummy hand. For example, top-down expectations
about the appearance of the human hand—stronger illusory effects are
experienced when the dummy hand closely resembles a human hand—are thought to
result from internal body representations, and bottom-up sensory inputs where
illusory effects are dependent on spatiotemporal congruence of the stimulation
and on the proximity of the dummy hand to the real hand [41].

A predictive coding account of the RHI

Different explanations to the RHI effects have been proposed in the cognitive
neuroscience literature. The predictive coding [21] account, that is recently
receiving great support, proposes that this results from the fact that, to
reduce uncertainty, the brain makes inferences about causes of sensory events
in a probabilistic-Bayesian manner: prior beliefs (bias) represented in
internal models generate predictions about sensory input (top-down). When
predictions contradict actual sensory input, this generates ”prediction
errors” that propagate up the hierarchy—to unimodal, multimodal,
representational areas (bottom-up). The contradiction results in ”surprise”
that needs to be ”explained away” by updating the model, thus reducing the
prediction error (see [45] for a review on the minimal self within the
predictive coding framework).

During the RHI, the co-occurrence of the visual input that comes from
observing touch on the dummy hand together with the tactile input that comes
from the stimulation of the real hand, evokes—according to the predictive
coding account—a prediction error (or surprise), because this spatiotemporal
congruence is not predicted by the initial generative forward model. According
to [46], the illusion is induced when the probability of the dummy hand being
”me” exceeds the probability of the real hand being ”one’s own”, given the
sensory input. If the prediction error can be explained away by adjusting the
body model to incorporate the dummy hand, then the RHI will be induced. The
explanation that the dummy hand is ”mine” is equivalent to mean that the
visual and tactile perceptual inputs occur at the same location and arise from
a common cause.

RHI and sense of agency: ”passive” versus ”active” RHI

In the classic RHI paradigm, the participants are not allowed to move their
hand. It has been observed that if they do move their hand, and they do not
observe congruent movement in the dummy hand, then the illusion will be
immediately abolished. The proposed explanation is that, as the participant
moved their hand in order to ”test” the body ownership of the not moving dummy
hand, the prediction error cannot resolve in favour of perceiving the dummy
hand as one’s own [45]. This classic RHI paradigm is therefore named as
”passive”, and while it induces the illusion of body ownership, the effect it
has on the sense of agency can not be directly examined.

In ”active” RHI, in addition to the concurrent multisensory stimulation, the
participant also moves their real hand while observing a dummy hand that moves
along it [47, 48]. In this version of the RHI paradigm (which could also be
induced in a virtual environment), the illusion of embodying a dummy hand (or
object) is induced as a result of the congruence between the participant’s
motor actions and the sensory outcomes of said actions, namely, the perception
of the movement of the dummy hand, rather than as a result of the multimodal
sensory integration alone. Also of note, in this case, the visual properties
of the dummy hand or object, do not necessarily have to resemble those of a
real human hand [47, 49]. In ”active” RHI, one can directly manipulate the
sense of agency, or even possibly disassociate agency from body ownership
[48]. In line with this, there is evidence that body ownership or embodiment
of an object, even one which is anatomically implausible, can still be
successfully induced given systematic synchrony between visual input when
observing the object and one’s own movement. In [50], Ma and Hommel induced in
a virtual setting body ownership of virtual 2-D shapes (a virtual balloon
changing in size, and a virtual square changing in size and color) when the
changes in the 2-D shapes were systematically congruent with participants’
actions. In addition to the concurrent multisensory stimulation, the induced
illusion of body ownership is thought to be cultivated by the congruence
between predicted sensory outcomes of motor actions and the actual sensory
input, pointing to the role of agency in body ownership. This is reminiscent
of the manner in which the sense of body ownership emerges in the ontogenetic
developmental process.

In another study in a virtual setting [49], Ma and Hommel manipulated the
similarity of the object-to-be-embodied (end-effector) to the real hand, the
synchrony between stimulation or movement of the end-effector and the
stimulation or movement of participant’s real hand, and the degree of agency,
operationalized by the level of control over the end-effector. They found that
agency strongly effected synchrony-induced body ownership, but not similarity.
However, both similarity and agency induced a bias towards body ownership of
the end-effector. This shows that agency contributes to body ownership.

Robotics research

When humans are subjected to the RHI, they show a perceptual drift in the
location of the real hand toward the dummy hand, which suggests an update in
the body representation. Using a multisensory robotic arm, Hinz et al. [51]
replicated these drifting patterns in both human and robot experiments with
the classic (”passive”) RHI paradigm. The learning and estimation algorithm
[52] used in the study was based on the framework of predictive coding [21].
Specifically, Lanillos and Cheng [52] developed a method for integrating
different sources of information (tactile, visual, and proprioceptive) that
drives the robot priors to infer its body configuration. This computational
perceptual model enables a multisensory robot to learn, make inferences, and
update its body configuration from its sensors. They modeled the robot body
estimation as a process of minimizing the prediction error between the body
configuration ”belief” (prediction) and the observed posterior, and minimizing
the variational free energy [21] by using the sensory prediction error. Using
the algorithm in [52], Hinz et al. [51] showed that body configuration
estimation can be done through minimization of prediction error as one process
that involves both predictive coding and causal inference. The results from
the human and robot experiments suggest that the perceived locations of both
the real and the dummy hand drift to a common location between them. In human
data, in fact, illusion scores (self-report) were not correlated with the
proprioceptive drift, suggesting that the drift and body-ownership illusions
are related, but different processes [53].

Current limitations in robotics research

Many studies can be found in the literature which use robotic or virtual hands
in active RHI experiments, that is in scenarios where participants move their
real hand while observing a dummy robotic or virtual hand that moves along it.
However, the investigation on RHI experienced by artificial systems is very
scarce. To the best of our knowledge, the work of Hinz et al. [51] is the only
study on replicating the rubber hand illusion on a robot.

Another concern is related to the ”proprioceptive drift” as the classic
”objective measure” for the RHI, as used in the experiment mentioned above
[51]. Both the human data from this study, as well as previous work in humans
[53, 54], failed to find a correlation between the proprioceptive drift and
the self-report of the participants, casting doubt on the validity of the
proprioceptive drift as an objective measure for body ownership. Further
investigation is therefore suggested. Also, after reproducing the classic,
”passive” RHI in a robot using free energy minimization [51], it could be
suggested to examine the algorithm in an ”active” RHI experimental setup,
which would allow to distinctly examine sense of agency in a robot apart from
body ownership.

##  IV Conclusions

This manuscript presented an interdisciplinary overview of developmental
indices and behavioural measures of the minimal self. The fundamental role of
the development of body representation in the emergence of body ownership and
sense of agency has been discussed. This work also addressed the task of
experimentally quantifying the attribution of subjective experience, and
surveyed a number of behavioural paradigms and measures indicating the
presence of different aspects of the minimal self, namely self-touch,
intentional binding and sensory attenuation, and the rubber hand illusion.

Self-touch is likely to contribute to the formation of initial sensorimotor
representations, and may therefore constitute one of the very first cues for
subjective experience during early developmental stages. Moreover, the way in
which our brain interprets action effects has been shown to differ depending
on whether the sensory perception is self-produced or externally triggered,
with respect both to the perceived timing of their occurrence (intentional
binding) and to their intensity (sensory attenuation). Finally, we addressed
the rubber hand illusion as it has been extensively used as a paradigm for
investigating the mechanisms underlying the sensorimotor minimal self.

We reviewed the most prominent studies addressing these paradigms and measures
from the literature in neuroscience, cognitive and developmental sciences. For
each of these topics, we presented related robotics studies. Equipping robots
with self-awareness and studying the possibility of subjective experience in
artificial systems is, in fact, of high interest for the cognitive and
developmental robotics communities. This manuscript contributed to this quest
by identifying current knowledge gaps and limitations in robotics. In the next
section, we conclude this work by highlighting the most critical gaps and by
suggesting further research directions.

Further research directions in robotics

The development of multimodal body representations has been discussed as
fundamental in the emergence of self-awareness. Further robotics research
should address the implementation of multimodal integration through online
developmental processes. Current research on self-touch and on self-
organisation of somatosensory maps in robots do not explicitly consider the
active role that the robot should have in the generation of sensorimotor
experience. We therefore encourage further experimentation considering self-
exploration behaviours in such a developmental process.

Recent proposals on predictive processes represent promising research lines
that go beyond their higher level of biological plausibility. Prediction error
minimization processes could result in intelligent robot exploration
behaviours, where the intrinsic motivation of reducing uncertainty would
generate artificial curiosity and goal-directed behaviours—both prerequisites
for motor and cognitive development.

The intentional binding effects and sensory attenuation processes are
recognised by the neuroscience and cognitive science communities as important
measures for the definition of self-boundaries. Current studies, however,
mostly focus on explicit judgement from human participants. This self-report
approach is clearly not feasible in robotics. Nonetheless, robots allow
experimenters to inspect their internal states, the flowing sensorimotor data
and the predictive processes implemented by their computational models. These
data is for obvious reasons not accessible in humans. Robots represent,
therefore, promising tools for the investigation of intentional binding
effects and sensory attenuation processes. Beside encouraging further
investigations in robotics, we also suggest more experimentations considering
the effects of the developmental path in the performance of such measures. In
particular, how do developmental stages—for instance, the levels of multimodal
integration reached after a certain stage of sensorimotor exploration—affect
predictive performances, and consequently sensory attenuation and intentional
binding effects? Can this be linked to stages in early development of the
minimal self in humans?

The RHI represents a well-established paradigm to measure subjective
experience in humans, and it makes therefore very much sense to extend this to
the investigation of subjective experiences in robots. Further usage of this
measure in the study of the artificial self is thus encouraged. In particular,
we suggest testing the ”active” RHI paradigm in robots in order to investigate
also the sense of agency. Moreover, similar effects as the ones mentioned
above could be studied for the RHI paradigm. Further studies could address
whether and how the perceptual drift measure is affected by the developmental
stage in which the agent finds themselves.

## References

  * [1] S. Gallagher, “Philosophical conceptions of the self: implications for cognitive science,” _Trends in cognitive sciences_ , vol. 4, no. 1, pp. 14–21, 2000.
  * [2] B. Martin, M. Wittmann, N. Franck, M. Cermolacce, F. Berna, and A. Giersch, “Temporal structure of consciousness and minimal self in schizophrenia,” _Frontiers in psychology_ , vol. 5, p. 1175, 2014.
  * [3] F. Ferri, F. Frassinetti, M. Costantini, and V. Gallese, “Motor simulation and the bodily self,” _PloS one_ , vol. 6, no. 3, p. e17927, 2011.
  * [4] A. Damasio, “Mental self: The person within,” _Nature_ , vol. 423, no. 6937, p. 227, 2003.
  * [5] A. K. Engel and W. Singer, “Temporal binding and the neural correlates of sensory awareness,” _Trends in cognitive sciences_ , vol. 5, no. 1, pp. 16–25, 2001.
  * [6] F. Crick and C. Koch, “Towards a neurobiological theory of consciousness,” in _Seminars in the Neurosciences_ , vol. 2. Saunders Scientific Publications, 1990, pp. 263–275.
  * [7] P. Rochat, “Five levels of self-awareness as they unfold early in life,” _Consciousness and cognition_ , vol. 12, no. 4, pp. 717–731, 2003.
  * [8] F. de Vignemont, “Bodily awareness,” in _The Stanford Encyclopedia of Philosophy_ , spring 2018 ed., E. N. Zalta, Ed. Metaphysics Research Lab, Stanford University, 2018.
  * [9] S. Gallagher, “Body image and body schema: A conceptual clarification,” _The Journal of Mind and Behavior_ , pp. 541–554, 1986.
  * [10] S. Dall’Orso, J. Steinweg, A. Allievi, A. Edwards, E. Burdet, and T. Arichi, “Somatotopic mapping of the developing sensorimotor cortex in the preterm human brain,” _Cerebral Cortex_ , vol. 28, no. 7, pp. 2507–2515, 2018.
  * [11] R. M. Bradley and C. M. Mistretta, “Fetal sensory receptors,” _Physiological Reviews_ , vol. 55, no. 3, pp. 352–382, 1975.
  * [12] J. Fagard, R. Esseily, L. Jacquey, K. O’Regan, and E. Somogyi, “Fetal origin of sensorimotor behavior,” _Frontiers in neurorobotics_ , vol. 12, 2018.
  * [13] M. Myowa-Yamakoshi and H. Takeshita, “Do human fetuses anticipate self-oriented actions? a study by four-dimensional (4d) ultrasonography,” _Infancy_ , vol. 10, no. 3, pp. 289–301, 2006.
  * [14] N. Reissland, B. Francis, E. Aydin, J. Mason, and B. Schaal, “The development of anticipation in the fetus: a longitudinal account of human fetal mouth movements in reaction to and anticipation of touch,” _Developmental psychobiology_ , vol. 56, no. 5, pp. 955–963, 2014.
  * [15] S. Zoia, L. Blason, G. D’Ottavio, M. Bulgheroni, E. Pezzetta, A. Scabar, and U. Castiello, “Evidence of early development of action planning in the human foetus: a kinematic study,” _Experimental Brain Research_ , vol. 176, no. 2, pp. 217–226, 2007.
  * [16] M. Hoffmann, “The role of self-touch experience in the formation of the self,” _arXiv preprint arXiv:1712.07843_ , 2017.
  * [17] M. B. McGraw, “The neuromuscular maturation of the human infant.” _Yale Journal of Biology and Medicine_ , vol. 15, no. 6, 1943.
  * [18] H. T. Chugani, “Development of regional brain glucose metabolism in relation to behavior and plasticity.” _Human behavior and the developing brain_ , pp. 153–175, 1994.
  * [19] B. L. Thomas, J. M. Karl, and I. Q. Whishaw, “Independent development of the reach and the grasp in spontaneous self-touching by human infants in the first 6 months,” _Frontiers in psychology_ , vol. 5, p. 1526, 2015.
  * [20] S.-J. Blakemore, C. D. Frith, and D. M. Wolpert, “Spatio-temporal prediction modulates the perception of self-produced stimuli,” _Journal of cognitive neuroscience_ , vol. 11, no. 5, pp. 551–559, 1999.
  * [21] K. Friston, “A theory of cortical responses,” _Phil. trans. of the Royal Society B: Biological sciences_ , vol. 360, no. 1456, pp. 815–836, 2005.
  * [22] P. Rochat, “Self-perception and action in infancy,” _Experimental brain research_ , vol. 123, no. 1-2, pp. 102–109, 1998.
  * [23] G. Hughes, A. Desantis, and F. Waszak, “Mechanisms of intentional binding and sensory attenuation: The role of temporal prediction, temporal control, identity prediction, and motor prediction.” _Psychological bulletin_ , vol. 139, no. 1, p. 133, 2013.
  * [24] M. Hoffmann, L. K. Chinn, E. Somogyi, T. Heed, J. Fagard, J. J. Lockman, and J. K. O’Regan, “Development of reaching to the body in early infancy: From experiments to robotic models,” in _IEEE Int. Conf. on Development and Learning and Epigenetic Robotics_. IEEE, 2017.
  * [25] P. Rochat and T. Striano, “Perceived self in infancy,” _Infant behavior and development_ , vol. 23, no. 3-4, pp. 513–530, 2000.
  * [26] Y. Yamada, H. Kanazawa, S. Iwasaki, Y. Tsukahara, O. Iwata, S. Yamada, and Y. Kuniyoshi, “An embodied brain model of the human foetus,” _Scientific reports_ , vol. 6, p. 27893, 2016.
  * [27] T. Noda, T. Miyashita, H. Ishiguro, and N. Hagita, “Super-flexible skin sensors embedded on the whole body, self-organizing based on haptic interactions,” _Human-Robot Interaction in Social Robotics_ , p. 183, 2012\.
  * [28] B. Lara, V. V. Hafner, C.-N. Ritter, and G. Schillaci, “Body representations for robot ego-noise modelling and prediction. towards the development of a sense of agency in artificial agents,” in _Proc. of the Artificial Life Conference 2016 13_. MIT Press, 2016, pp. 390–397.
  * [29] M. Hoffmann, Z. Straka, I. Farkaš, M. Vavrečka, and G. Metta, “Robotic homunculus: Learning of artificial skin representation in a humanoid robot motivated by primary somatosensory cortex,” _IEEE Trans. on Cognitive and Developmental Systems_ , vol. 10, no. 2, pp. 163–176, 2018.
  * [30] M. Hoffmann and N. Bednarova, “The encoding of proprioceptive inputs in the brain: knowns and unknowns from a robotic perspective,” _arXiv preprint arXiv:1607.05944_ , 2016.
  * [31] P. Haggard, S. Clark, and J. Kalogeras, “Voluntary action and conscious awareness,” _Nature neuroscience_ , vol. 5, no. 4, p. 382, 2002.
  * [32] J. W. Moore and S. S. Obhi, “Intentional binding and the sense of agency: a review,” _Consciousness and cognition_ , vol. 21, no. 1, pp. 546–561, 2012\.
  * [33] P. Michel, K. Gold, and B. Scassellati, “Motion-based robotic self-recognition,” in _IEEE/RSJ International Conference on Intelligent Robots and Systems_ , vol. 3. IEEE, 2004, pp. 2763–2768.
  * [34] C. Lang, G. Schillaci, and V. V. Hafner, “A deep convolutional neural network model for sense of agency and object permanence in robots,” in _Joint IEEE International Conference on Development and Learning and Epigenetic Robotics_ , 2018.
  * [35] J. Tani, “An interpretation of the ‘self’from the dynamical systems perspective: A constructivist approach,” _Journal of Consciousness Studies_ , vol. 5, no. 5-6, pp. 516–542, 1998.
  * [36] E. Daprati, S. Wriessnegger, and F. Lacquaniti, “Kinematic cues and recognition of self-generated actions,” _Experimental Brain Research_ , vol. 177, no. 1, pp. 31–44, 2007.
  * [37] A. Cavazzana, “Sense of agency and intentional binding: How does the brain link voluntary actions with their consequences?” _Ph.D. thesis, Dipartimento di Psicologia Generale, University of Padua, Italy_ , 2016.
  * [38] A. Desantis, G. Hughes, and F. Waszak, “Intentional binding is driven by the mere presence of an action and not by motor prediction,” _PLoS one_ , vol. 7, no. 1, p. e29557, 2012.
  * [39] M. Botvinick and J. Cohen, “Rubber hands ‘feel’touch that eyes see,” _Nature_ , vol. 391, no. 6669, p. 756, 1998.
  * [40] H. H. Ehrsson, C. Spence, and R. E. Passingham, “That’s my hand! activity in premotor cortex reflects feeling of ownership of a limb,” _Science_ , vol. 305, no. 5685, pp. 875–877, 2004.
  * [41] M. Tsakiris and P. Haggard, “The rubber hand illusion revisited: visuotactile integration and self-attribution.” _Journal of Exp. Psychology: Human Perception and Performance_ , vol. 31, no. 1, p. 80, 2005.
  * [42] M. Tsakiris, “My body in the brain: a neurocognitive model of body-ownership,” _Neuropsychologia_ , vol. 48, no. 3, pp. 703–712, 2010\.
  * [43] K. C. Armel and V. S. Ramachandran, “Projecting sensations to external objects: evidence from skin conductance response,” _Proc. of the Royal Society of London. Series B: Biological Sciences_ , vol. 270, no. 1523, pp. 1499–1506, 2003.
  * [44] H. H. Ehrsson, K. Wiech, N. Weiskopf, R. J. Dolan, and R. E. Passingham, “Threatening a rubber hand that you feel is yours elicits a cortical anxiety response,” _Proc. of the National Academy of Sciences_ , vol. 104, no. 23, pp. 9828–9833, 2007.
  * [45] J. Limanowski and F. Blankenburg, “Minimal self-models and the free energy principle,” _Frontiers in human neuroscience_ , vol. 7, p. 547, 2013.
  * [46] M. A. Apps and M. Tsakiris, “The free-energy self: a predictive coding account of self-recognition,” _Neuroscience & Biobehavioral Reviews_, vol. 41, pp. 85–97, 2014.
  * [47] M. Slater, D. Pérez Marcos, H. Ehrsson, and M. V. Sanchez-Vives, “Inducing illusory ownership of a virtual body,” _Frontiers in neuroscience_ , vol. 3, p. 29, 2009.
  * [48] A. Kalckert and H. H. Ehrsson, “Moving a rubber hand that feels like your own: a dissociation of ownership and agency,” _Frontiers in human neuroscience_ , vol. 6, p. 40, 2012.
  * [49] K. Ma and B. Hommel, “The role of agency for perceived ownership in the virtual hand illusion,” _Consciousness and cognition_ , vol. 36, pp. 277–288, 2015.
  * [50] ——, “Body-ownership for actively operated non-corporeal objects,” _Consciousness and cognition_ , vol. 36, pp. 75–86, 2015.
  * [51] N.-A. Hinz, P. Lanillos, H. Mueller, and G. Cheng, “Drifting perceptual patterns suggest prediction errors fusion rather than hypothesis selection: replicating the rubber-hand illusion on a robot,” _arXiv preprint arXiv:1806.06809_ , 2018.
  * [52] P. Lanillos and G. Cheng, “Adaptive robot body learning and estimation through predictive coding,” _arXiv preprint arXiv:1805.03104_ , 2018.
  * [53] Z. Abdulkarim and H. H. Ehrsson, “No causal link between changes in hand position sense and feeling of limb ownership in the rubber hand illusion,” _Attention, Perception, & Psychophysics_, vol. 78, no. 2, pp. 707–720, 2016\.
  * [54] H. Holle, N. McLatchie, S. Maurer, and J. Ward, “Proprioceptive drift without illusions of ownership for rotated hands in the “rubber hand illusion” paradigm,” _Cognitive neuroscience_ , vol. 2, no. 3-4, pp. 171–178, 2011\.