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\DOI{10.5802/crphys.223}
\datereceived{2024-09-24}
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\begin{document}

\begin{noXML}

\CDRsetmeta{articletype}{history-of-sciences}

\title{Adventures in physics}

\alttitle{Aventures en physique}

\author{\firstname{Valentin} \lastname{Po\'{e}naru}}
\address{Professeur \'em\'erite \`a l'Institut des hautes \'etudes scientifiques, Bures sur Yvette, France}
\email{valpoe@hotmail.com}

\dedicatory{To the memory of my friend G\'{e}rard Toulouse}

\keywords{\kwd{Symmetry breaking}
\kwd{Non-commutativity}
\kwd{Bouligand rings}}

\altkeywords{\kwd{Brisure de sym\'{e}trie}
\kwd{Non-commutativit\'{e}}
\kwd{Anneaux de Bouligand}}

\begin{abstract}
In the 1970s G\'{e}rard Toulouse and myself we wrote three papers on
how the homotopy groups of the manifold of internal states influenced
the defects of ordered media. Specifically, we showed how the algebra
of homotopy groups acted on the dynamics of defects.
\end{abstract}

\begin{altabstract}
Dans les ann\'{e}es~1970, G\'{e}rard Toulouse et moi-m\^{e}me avons
\'{e}crit trois articles sur la fa\c{c}on dont les groupes d'homotopie
de la vari\'{e}t\'{e} des \'{e}tats internes influen\c{c}aient les
d\'{e}fauts des milieux ordonn\'{e}s. Plus pr\'{e}cis\'{e}ment, nous
avons montr\'{e} comment l'alg\`{e}bre des groupes d'homotopie agissait
sur la dynamique des d\'{e}fauts.
\end{altabstract}

\maketitle

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\vspace*{1pc}

\xsection{}
I am a mathematician, and my personal field of research is geometric
topology, more precisely low-dimensional manifolds and geometric group
theory. In my life I have worked a lot on the Poincar\'{e} conjecture
and related topics.

Over the years, however, my mathematical interests were quite varied. I
also worked on the singularities of mappings, under the effect of a
symmetry group, and I happened to stumble on the Higgs potential,
although I did not understand its meaning at the time. But there, I
quite naturally met the topic of symmetry breaking, something which
fits perfecly with what I am about to tell you.

In the early 1970s, I was a professor in the mathematics department of
the Paris-Orsay University; our neighbors and friends, the Williamses,
an Anglo-Belgian couple, worked in the condensed matter physics
department of the same university. Claudine Williams was organizing
monthly lectures on general scientific topics for the members of her
department. And in about 1973, I gave there an introductory lecture on
topology.

As soon as I finished my lecture, two members of the audience
approached me, saying that the kind of things I had talked about was
exactly what they needed. The two were G\'{e}rard Toulouse and Maurice
Kl\'{e}man. I understood from them that topology was discovered to help
understand the defects of ordered media. At that time, only two groups
in the world had started to work on this topic. One in Paris, more
precisely in Orsay and Bures-sur-Yvette, with Toulouse, Kl\'{e}man and
Louis Michel from IHES, and the other in Moscow, with Volovik and
Mineev who worked in the team of Dzialoshinsky. Incidentally, a younger
physicist, Misha Monastyrsky joined them a bit later. And after some
years, Misha started coming to Bures-Orsay, we met and we became
friends.

{\advance\baselineskip by .1pt

This was how things were in those old days. But, about thirty years
later, the situation changed drastically. With the work of Ed Witten,
Barry Simon (who, incidentally, had been my student in a topology
course at Harvard, in the early 1960s) and others, that led to
topological quantum field theory, a topic that fascinates me, and with
other things like that, topology became a big hot topic in physics.

But let us go back to the early 1970s. G\'{e}rard and I started talking
a lot together, he taught me physics and I taught him topology. And
since I was having some problems with my own mathematical work at the
time, I welcomed a temporary change of horizon. G\'{e}rard was a
charming personality, full of enthusiasm, inspiration, and clever
ideas. He was also very good at communicating these things. Of the many
physicists I have met in my life, he was the one with whom I could
interact best. 

Concerning our work together, it was already known at the time that
defects of ordered media were connected with the homotopy groups of the
manifold of internal states, a space that somehow encodes symmetry
breaking. In particular, the existence of defect lines was connected
with the fundamental group of that manifold of internal states, and
similarly, the defect points had to do with the second homotopy group.

But G\'{e}rard and I looked at the dynamics of defects, and we
discovered that the non-commutativity of that fundamental group, when
it exists, prevented the defect lines from crossing~\cite{1}. Many
physicists outside the topic of condensed matter became interested in
our result, partly I think, because non-commutativity is so fundamental
in quantum theory.

Actually, the famous Russian field theorist Sasha Polyakov suggested
that we look at what happens to our little theory when one tries to
extend it to higher dimensions; that was clearly outside of condensed
matter physics.

And there we found a beautiful surprise. The totality of homotopy
groups, when endowed with the Whitehead product, becomes a
supersymmetric algebra. And we found out that the non-commutativity of
the Whitehead product is the obstruction to the crossing of
higher-dimensional defects~\cite{2}. So, somehow, supersymmetry is now
involved. G\'{e}rard told me that he had never experienced such joy at
a discovery. 

I will introduce here a little anecdote, which I find amusing. In the
early 1970s, Jean Dieudonn\'{e}, the famous French mathematician, wrote
a book on the mathematics of Bourbaki. And when it came to topology, he
stated that this topic had no connection with physics. When he
published a second edition a few years later, he corrected that
statement, and he also quoted G\'{e}rard, myself, and Louis Michel.

Our third and last work together, a piece done by us two, together with
Yves Pomeau, Bernard Derrida and Yves Bouligand, concerned the
following puzzling issue. Bouligand, who was a biologist but also a
distinguished physicist, had discovered what are now called Bouligand
rings in his study of biological tissues. These were defect rings,
existing in a situation when the potential obstruction to crossing was
living at the center of the fundamental group of the manifold of
internal states, and which nevertheless stayed hooked together,
refusing to cross. But non-commutativity was not the explanation.

What we discovered was that the explanation for this paradoxical
situation was the emergence of a new player in the game, the third
homotopy group of the manifold of internal states. The deep reason for
the appearance of this totally new ingredient, was that we were now in
the context of textures, where long-range effects also become 
important~\cite{3}. And I think that the mathematics involved here is
not unrelated to the still conjectural magnetic monopoles of t'Hooft
and Polyakov.

}

We have now reached the late 1970s. I had managed to overcome the
difficulties which were bogging me down 
earlier, so I went back to my
mathematical research. Actually, I had another, shorter incursion into
physics, when I helped Geoff Chew from Berkeley to develop a
topological bootstrap theory.

However, those years of collaboration with G\'{e}rard Toulouse and the
others were a very exciting and interesting period in my scientific
career. G\'{e}rard was a very important figure in my scientific life.

\section*{A short appendix on Bouligand rings}
Yves Bouligand developed an experimental technique for studying the
cholesteric order. He found defect lines, called now the Bouligand
rings, which stayed linked, although the homotopical obstruction to
their crossing lived in the center of the fundamental group of the
manifold of internal states. Non-commutativity was therefore not what
kept them linked.

We are actually here in the presence of textures, which are ordered
media where the order parameter is defined everywhere, and the region
where it is not constant plays the role of defects. It is now a
completely new actor, the third homotopy group of the manifold of
internal states, which is the obstruction to crossing. For more
information, see~\cite{4}.

\section*{Declaration of interests}
The author does not work for, advise, own shares in, or receive funds
from any organization that could benefit from this article, and has
declared no affiliations other than his research institution.

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