# Lars Brink: November 12, 1943 - October 29, 2022

Bengt E.W. Nilsson and Björn Jonson
Department of Physics,
Chalmers University of Technology,
SE-412 96 Göteborg, Sweden
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We give some personal reflections on the person and scientist Lars Brink and
on some of his scientific achievements. Our relations to Lars are briefly
described in Nilsson  and Jonson , while the sources relevant for this text
are summarised in sources .

##  1 Lars Brink, the person

Our friend and colleague Professor LARS BRINK passed away on October 29, 2022,
a short time before his 79th birthday. Between the early 1970s and late 1980s
Lars made a number of fundamental contributions to the early developments of
dual models and string theory. One subarea of string theory that he was
particularly proud of was maximally supersymmetric Yang-Mills theory and the
many novel results he was responsible for in works with several different
people. Some of these will be briefly described below.

Later in his career he became deeply interested in the role of theoretical
physics in general and its growing difficulties to stay relevant in a world
focusing more and more on entrepreneurship and applied sciences. Lars
therefore got involved in supporting and helping many scientific institutions
outside Sweden, like the Max Planck Institute, the Solvay Institutes in
Brussels etc. Lars was also influential as a player behind the scene in the
high energy physics community. The fact that he knew personally the key people
in physics including many Nobel laureates in physics gave him ample
opportunities to have a say in many global as well as local activities in the
community. In Sweden he was for many years very active in the main funding
agency, NFR later renamed VR. He was also a member of the Royal Academies KVA
in Stockholm since 1997 and KVVS in Göteborg since 2004. In the former case
Lars was also for several years a member of the Nobel Committee for Physics
and, during the last year 2013, also acting as its chairman. Lars was during
these years directly involved in the process of selecting the Nobel Prize
laureates in physics.

Figure 1: Lars Brink’s doctoral thesis, December 1973.

##  2 Lars Brink, the career

Lars Brink was born on November 12, 1943, in Uddevalla, a small town north of
Göteborg, but grew up in Göteborg. After graduating from high school in 1962
he continued his physics studies at Chalmers Technical University (Chalmers)
receiving his master’s degree in 1967. His doctoral studies resulted half-way
in a licentiate degree in 1970 supervised by professor Jan S. Nilsson.

In June 1969 Lars attended, as a second year doctoral student, the Lund
International Conference on Elementary Particles. This probably helped him in
selecting research topics in the field of high energy particle physics and led
in 1971 - 1972 to his first four papers on aspects of scattering amplitudes
involving pions and protons as well as other hadrons. Judging from the many
comments in this direction we have heard over the years, he probably never
entirely stopped thinking about hadronic physics.

Lars then went to CERN, the European accelerator facility outside Geneva,
where he held a position as fellow 1971 - 1973, impressively enough before
having received his doctor’s degree. At CERN his career really took off and in
a short period of time he wrote most of the 14 papers that in December 1973
made up his doctoral thesis. The extremely high quality of the thesis gave
Lars the best possible grade and he was immediately awarded the title
docent111Lars was one of the last students in Sweden to graduate with a
doctor’s degree. Soon after his thesis defence the transition to the PhD
system was completed with a more strict study plan and new rules for the
dissertation etc. The grade on the thesis defence was dropped and the
respondent either passed or failed. . During his time at CERN Lars met many of
his life-long friends, some of whom would also become his collaborators. Two
individuals from this period who had a huge impact on his future career were
David Olive and Holger Bech Nielsen, the latter a legendary researcher from
the Niels Bohr Institute in Copenhagen. Another very gifted young researcher
who also participated in these early collaborations and whom Lars greatly
admired was Joël Scherk. Joël tragically died a few years later, in 1980, a
fact that Lars repeatedly returned to in conversations with his friends.

During the academic year of 1976 - 1977 Lars was a research associate at the
California Institute of Technology (Caltech) in Pasadena, Ca, USA. This turned
out to be a very productive time where he wrote his first paper on
${\mathcal{N}}=4$ super-Yang-Mills theory with John Schwarz and Joël Scherk,
his most cited paper222L. Brink, J.H. Schwarz and J. Scherk, Supersymmetric
Yang-Mills theories, Nucl. Phys. B121 (1977), 77-92.. He was then given the
opportunity to visit Caltech two months per year for a number of years (1977 -
1985) which gave rise to a long-time collaboration with Mike Green and John
Schwarz. During these years Lars also met Pierre Ramond which was the start of
a life-long and very close friendship. These visits resulted in several
papers, three of them together with Murray Gell-Mann, Pierre Ramond and John
Schwarz on supergravity and the relation between the component and superspace
descriptions of the ${\mathcal{N}}=1$ theory. Thus Lars worked for many years
closely together with the people who later gave one of the most important
strings its name, the Ramond-Neveu-Schwarz-superstring (RNS). Later, Lars
received funds from a Swedish foundation to visit Chile and Claudio
Teitelboim’s (now Bunster) institute in Santiago. At one of these visits,
which took place yearly in the period 1986 - 2000, Lars met another life-long
friend and collaborator, Marc Henneaux. Lars and Marc were, in 2004,
instrumental in the revival of the Solvay Institutes and their famous
conference programme.

On February 1, 1986, Lars was appointed professor by the Swedish government, a
position that was funded by the Research Council (NFR) and placed at Chalmers
in Göteborg. This was one of a very small number of pure research positions at
this level that existed in Sweden. However, Lars used the total freedom
inherent in this position to get involved in teaching at both lower and higher
levels. One reason for this was his huge interest in interactions with young
students and to discuss all kinds of physics with them. The students were so
appreciative of his relation to the students that they gave him a life-long
membership and an honorary title, Inspector, in their student organisation at
Chalmers. During his active time as a supervisor Lars guided 15 students to
their PhD degrees, and as an administrator he spent six years as the vice dean
of the Physics Department.

In 1984 Lars accepted the task to work for the Research Council as a member of
the physics committee which he continued doing for ten years, and when the
Research Council was reorganised in 2001 he became the chairman of the physics
subcommittee that dealt with the funding of particle physics and some other
more or less related areas in physics. At the international level, Lars was
the Swedish delegate to the board of the Nordic institute Nordita during the
years 1987 to 1995 and acted as its chairman 1990 to 1992. At the European
level Lars was also very active, in particular during the years 1991 - 1995
and 2000 - 2008 when he led the EU-network ”Superstrings” which involved a
large number of research groups all over Europe.

Lars became, after some time as an employee at Chalmers, very interested in
the conditions under which fundamental research was conducted at Swedish
universities. During the years 2002 - 2008 he was chairman of the Faculty
Senate which is a body whose main role is to guard the academic perspective in
the running of the university. This gave Lars ample opportunities to act on
the threat, as he saw it, from industry and other external forces that tended
to inhabit the university boards to a larger and larger extent for each
reorganisation imposed by the government. Lars was not afraid of making his
opinion on these matters very clear; among other things, this was obvious from
a sticker he had on his office door saying Not an entrepreneur and proud of
it!. As the chairman of the Faculty Senate at Chalmers he also held a seat on
the governing board of the entire university. In this capacity he became in
2008 involved in the selection of a new head of the university (rector) as
well as in the reorganisation of the whole university that took place shortly
before. In the year 2018 Lars was awarded the Chalmers medal for his hard work
and efforts to improve Chalmers as a technical university.

Figure 2: Lars Brink’s speech at the Nobel ceremony 2013.

The next important step in Lars’ academic career is connected to him being
elected member of KVA, the Swedish Royal Academy of Sciences, in 1997. During
the year 2001 he was an adjoint member of the Nobel Committee for Physics,
which happened again in 2004, a year which ended with him delivering the
speech at the Prize ceremony in the Stockholm Concert Hall. Lars was then a
full member of the Physics Committee during 2008 - 2013. The 2008 prize in
Physics was close to his own research so he once again delivered the speech at
the Prize ceremony. Since the three laureates at this occasion were all from
Japan333The three physics laureates 2008 were M. Kobayashi, T. Maskawa and Y.
Nambu. Lars decided to end his speech in their native tongue which he did
fluently to the huge surprise of everyone present. The last year as a member
of the Committee he was also its chairman. The third time Lars delivered the
speech at the ceremony was in 2013 (see Figure 2)444The two physics laureates
2013 were F. Englert and P. Higgs..

After his retirement in 2010 Lars continued to work part-time at Chalmers for
another three years. Besides his engagement in the Nobel committee he was now
as emeritus free to accept invitations as guest professor to a number of
institutes all around the globe. He visited for instance the Institute of
Advanced Studies in Princeton, CERN as a special guest of the General Director
of CERN, LMU in Munich and the Institute of Advanced Studies in Singapore. In
the latter case it gave Lars the opportunity to organise a number of
conferences and edit volumes of collected works lifting the importance of the
achievements of key people in high energy physics, some of them Nobel Prize
laureates. Among the people who were honoured this way were S. Mandelstam, A.
Salam, C.N. Yang and Y. Nambu. His special relation to Y. Nambu will be
commented upon below.

Lars’ deep interest in assisting, as an advisor, different physics institutes
eventually gave him two honorary doctorates, first at the University of
Craiova, Romania in 2012, and later, in 2014, at the University of Florida. An
especially important effort was done in 2004 and subsequent years together
with Marc Henneaux at UBL, Brussels, in connection with the revival of the
Solvay Institutes and their famous conferences. That these conferences have
had an enormous impact historically is without doubt. One merely needs to
recall the one in physics held in 1927. The conference photo (on Wikipedia) is
classic: It contains 17 Nobel Prize laureates in physics and chemistry. Lars’
international engagements included being part of several advisory committees,
in some cases as chairman, at, e.g., Albert Einstein Institute in Potsdam,
Werner-Heisenberg Institute in Munich, Centro de Estudios de Scientificos in
Valdivia, Chile and the International Center for Fundamental Physics in Moscow
(besides the Solvay Institutes).

When the Covid-19 pandemic started in the spring of 2020 Lars stopped
travelling and spent most of his time in his home in Torslanda, a suburb
outside Göteborg. Lars was shortly before the outbreak informed about his
incurable disease.

##  3 Lars Brink, the scientist

Lars gave important contributions to the following six research topics as will
be briefly described below:
1\. Dual string models: zero point sum, projection operators (1971-1973)
2\. Supergravity and supersymmetry on the worldsheet (1976, 1977, 1987)
3\. Supergravity and supersymmetry in spacetime (1978 - 1980, 2005, 2008)
4\. String theory (1976, 1982, 1983)
5\. Light-cone: N=4 super-Yang-Mills and N=8 supergravity (1977, 1983, 2008)
6\. Light-cone: Higher spins (1983, 2000)

Lars Brink was one of the real pioneers of the field of string theory and gave
several fundamental contributions to its early developments. As a young
doctoral student at Chalmers he attended, as mentioned above, the Lund
International Conference on Elementary Particles held in Lund, Sweden, at the
end of June 1969 as a scientific secretary. The proliferation of unstable
strongly interacting hadronic particles appearing on so called Regge
trajectories (see below) and emerging ideas about partons, including quarks,
were discussed at the meeting. It seems likely that being present at this
meeting influenced him when selecting research topics for his graduate
studies. Four years later his doctoral thesis, defended in December 1973,
contained 14 papers some of which provided key results in dual string models
which at the time were theories of hadrons. Lars thanks in his thesis his
supervisor professor Jan S. Nilsson for making his doctoral studies so
immensely successful. In the years following Lars’ dissertation dual models
developed into string theory, starting with the realisation in 1974 of T.
Yoneya555T. Yoneya, Connection of Dual Models to Electrodynamics and
Gravidynamics, Prog. Theor. Phys. 51 (1974), 1907 - 1920. and of two of Lars’
close friends Joël Scherk and John Schwarz666J. Scherk and J.H. Schwarz, Dual
Models and Non-hadrons, Nucl. Phys. B81 (1974), 118 - 144. that the spectrum
contained a massless spin 2 particle and hence was actually a model of
gravity. Subsequently it was also understood that string theory even was a
consistent theory of quantum gravity. These steps from dual models to string
theory, that is from hadron physics to quantum gravity, turned out to
constitute a paradigm shift for both high energy particle physics and
gravitational research. Lars certainly played a central role in making this
transition possible.

Dual models was initially an attempt to understand the ramifications of the
huge spectrum of hadronic resonances discovered in the 1950s and 1960s. These
states seem to appear on so called Regge trajectories given by linear
relations between spin and mass-squared, $J=\alpha^{\prime}M^{2}+\alpha(0)$
where $\alpha^{\prime}$ is the Regge slope parameter (about 1 $(GeV)^{-2}$ in
hadron physics) and $\alpha(0)$ is the intercept. The key feature of the
$2\rightarrow 2$ scattering amplitude $A(s,t)$ for pions was argued to be that
it should satisfy the duality condition $A(s,t)=A(t,s)$, that is, be symmetric
(dual) between the $s$\- and $t$-channels. This way one could perhaps explain
how an infinite set of more and more massive hadrons exchanged in the
$t$-channel might avoid breaking unitarity by summing up to poles
corresponding to themselves appearing in the $s$-channel. Thus only one of the
channels was needed to get the full result a feature reminiscent of what
happens in string theory.

The first expression discovered that satisfies this requirement was proposed,
but not derived, by Veneziano in 1968777Actually, the full amplitude for the
four-pion scattering is the sum $A(s,t,u)=A(s,t)+A(s,u)+A(t,u)$.:

$A_{o}(s,t)=\frac{\Gamma(-\alpha(s))\Gamma(-\alpha(t))}{\Gamma(-\alpha(s)-\alpha(t))},$
(1)

where $\alpha(s)=\alpha^{\prime}s+\alpha(0)$. In the hadronic context
discussed here the Regge trajectories were assumed to fit the experimental
observations. For example, the leading trajectory with at least five data
points ($J=1,2,..,5$), the so called $\rho$-trajectory, is quite well
described by the linear $J(M^{2})$ relation888For the numerical values see the
textbook discussion in B. Zwiebach, A first course in string theory (2nd Ed.,
Cambridge , 2009), page 527.

$J=0.87702(GeV)^{-2}M^{2}+0.47188.$ (2)

Later a slightly different expression was introduced by Virasoro (1969)

$A_{c}(s,t,u)=\frac{1}{stu}\frac{\Gamma(1-\frac{\alpha^{\prime}s}{4})\Gamma(1-\frac{\alpha^{\prime}t}{4})\Gamma(1-\frac{\alpha^{\prime}u}{4})}{\Gamma(1+\frac{\alpha^{\prime}s}{4})\Gamma(1+\frac{\alpha^{\prime}t}{4})\Gamma(1+\frac{\alpha^{\prime}u}{4})}.$
(3)

The index notation on the left-hand side of these two expressions refers to
the fact that they later came to be associated with open and closed strings.
That dual models mathematically could be connected to string-like objects was
realised first by Y. Nambu, and within a year later also by L. Susskind and
independently H.B. Nielsen. This idea emerged from the fact that the spin-
energy relation of Regge trajectories could be found also in rotating rigid
rods999See also Nambu’s own account of the origin of his understanding of
these facts in Nambu, A Foreteller of Modern Physics, Eds. T. Eguchi and M.Y.
Han, World Scientific Series in 20th Century Physics, Vol 43 (2014).. Note
that, as presented above, $A_{c}(s,t)$ is the form obtained from string
theory, i.e., the Regge slope parameter is now related to the Planck length
and the intercept is such that the first excitation is a massless spin 2
state101010This result from string theory seems to have been conjectured first
in C. Lovelace, Phys. Lett. 34B (1971) 457. This paper may also be the first
one to mention the critical dimension $D=26$. . Clearly, dual string theory
could no longer be associated to hadron physics once the intercept was
computed mathematically. It is very interesting that already in his thesis, as
a very last comment, Lars compared the Planck’s constant $\hbar$ to
$\alpha^{\prime}$ when he asked if the introduction of the latter might in the
future turn out to be as significant as that of $\hbar$.

The important step of connecting dual models to strings was a result of trying
to answer the question of how to generate the two dual amplitudes above from
some field theoretic models. The answer was indicated by the integral version
of the four-point amplitude:

$A_{o}(s,t)=\frac{\Gamma(-\alpha(s))\Gamma(-\alpha(t))}{\Gamma(-\alpha(s)-\alpha(t))}=\int_{0}^{1}dxx^{-\alpha(s)-1}(1-x)^{-\alpha(t)-1},$
(4)

and entailed a factorisation of the amplitude which Nambu found could be done
using an infinite set of harmonic oscillators put together into expressions
like

$F(p)=e^{i\sqrt{2\alpha^{\prime}}p^{\mu}\Sigma_{n=1}^{\infty}\frac{a^{n}_{\mu}x^{n}}{\sqrt{n}}},\,\,\,G(p)=e^{i\sqrt{2\alpha^{\prime}}p^{\mu}\Sigma_{n=1}^{\infty}\frac{a^{n\dagger}_{\mu}}{\sqrt{n}}},$
(5)

where the operators commute as
$[a_{\mu}^{m},a_{\nu}^{n\dagger}]=\eta_{\mu\nu}\delta^{mn}$. Using the
standard Baker-Housdorff-Campbell (BHC) formula applied to operators whose
commutator is a c-number one readily finds

$(1-x)^{-2\alpha^{\prime}p_{2}\cdot p_{3}}=\langle
0|F(p_{2})G(p_{3})|0\rangle.$ (6)

A very interesting account of these early developments (taking place September
to December 1968) is the one presented by P. Frampton in the 2016 memorial
volume for Y. Nambu that Lars was partly responsible for111111P. Frampton, in
Memorial Volume for Y. Nambu, Eds. L. Brink, L.N. Chang, M.-Y. Han and K.K.
Phua, World Scientific (2016)..

Nambu also introduced the important concept of a vertex operator121212See the
contribution of Peter Freund in Memorial Volume for Y. Nambu, Eds. L. Brink,
L.N. Chang, M.-Y. Han and K.K. Phua, World Scientific (2016). which is an
obvious generalisation of the expressions in (5). In modern notation the
vertex operator related to the emission of a tachyon state in the closed
bosonic string is simply (now with
$[\alpha_{\mu}^{m},\alpha_{\nu}^{n\dagger}]=m\eta_{\mu\nu}\delta^{mn}$ and
similarly for $\bar{\alpha}_{m}^{\mu}$)

$V(k^{\mu};(z,\bar{z}))=:e^{ik_{\mu}X^{\mu}(z,\bar{z})}:\equiv
e^{ik_{\mu}X^{(+)\mu}(z,\bar{z})}e^{ik_{\mu}X^{(-)\mu}(z,\bar{z})},$ (7)

where we have made the normal ordering relative a zero-momentum and oscillator
vacuum state $|0\rangle_{p}$ explicit: $X^{(+)\mu}(z)|0\rangle_{p}=0$. This
follows from the definitions

$X^{(+)\mu}(z,\bar{z})\equiv-i\frac{\alpha^{\prime}}{2}(\log
z\bar{z})\,p^{\mu}+i\sqrt{\frac{\alpha^{\prime}}{2}}\Sigma_{n=1}^{\infty}\frac{1}{n}(\alpha_{n}^{\mu}z^{-n}+\bar{\alpha}_{n}^{\mu}{\bar{z}}^{-n}),$
(8) $X^{(-)\mu}(z,\bar{z})\equiv
x^{\mu}-i\sqrt{\frac{\alpha^{\prime}}{2}}\Sigma_{n=1}^{\infty}\frac{1}{n}(\alpha^{\mu\dagger}_{n}z^{n}+\bar{\alpha}^{\mu\dagger}_{n}{\bar{z}}^{n}),$
(9)

which thus in addition to the oscillators introduced above also contain the
(operator) zero modes $x^{\mu}$ and $p^{\mu}$, i.e., the center of mass
parameters of the string131313These were first introduced in S. Fubini and G.
Veneziano, Nuovo Cimento A 67, (1970) 29. For an early use of the oscillator
expressions see also Phys. Lett. B29 (1969), 679 by the same authors together
with D. Gordon.. Here we have expressed it in terms of the complex variables
$z=e^{-i(\sigma+i\tau)}$ where $\sigma$ and $\tau$ are the world-sheet
coordinates. The simplicity of these last two expressions was noticed by Nambu
and can be seen by computing the expectation value of the product of two such
operators (today known as an operator product expansion (OPE)):

${}_{x}\langle
0|V(k_{1}^{\mu},z_{1})V(k_{2}^{\mu},z_{2})|0\rangle_{p}=e^{[ik_{1\mu}X^{(+)\mu}(z_{1},\bar{z}_{1}),ik_{2\nu}X^{(-)\nu}(z_{2},\bar{z}_{2})]}=e^{\alpha^{\prime}k_{1}\cdot
k_{2}\log|z_{2}-z_{1}|}=|z_{2}-z_{1}|^{\alpha^{\prime}k_{1}\cdot k_{2}},$ (10)

which, again, is an immediate consequence of the BHC formula. In fact, the
contraction, or correlator, of two string fields $X^{\mu}(\tau,\sigma)$ is
given by the normal ordered product
$:X^{\mu}(z_{2})X^{\mu}(z_{1}):=-\frac{\alpha^{\prime}}{2}(\log(z_{2}-z_{1})+\log(\bar{z}_{2}-\bar{z}_{1}))$
which is the $1+1$ dimensional propagator. The N-point version is then easily
obtained and leads to what is known as the Koba-Nielsen formula.

It is quite remarkable that Nambu shortly after these discoveries also was the
first one to introduce the world-sheet area (in July 1969, see the last two
footnotes above), spanned by coordinates $(\tau,\sigma)$, as the two-
dimensional action principle that generates amplitudes of the kind above, the
so-called Nambu-Goto action141414See Nambu’s lecture notes for Copenhagen
1970, and T. Goto, Prog. Theor. Phys 46 (1971), 1560.

$S[X^{\mu}]=\frac{1}{4\pi\alpha^{\prime}}\int d\tau
d\sigma\left(\sqrt{(\dot{X}\cdot
X^{\prime})^{2}-(\dot{X})^{2}(X^{\prime})^{2}}\right),\,\text{where}\,{\dot{X}}^{\mu}\equiv\frac{\partial
X^{\mu}}{\partial\tau},\,X^{\prime\mu}\equiv\frac{\partial
X^{\mu}}{\partial\sigma}.$ (11)

Lars was fascinated by the enormous impact that Nambu had on the early
foundations of string theory and was later instrumental in putting together a
memorial volume to honour Nambu151515 Memorial Volume for Y. Nambu, Eds. L.
Brink, L.N. Chang, M.-Y. Han and K.K. Phua, World Scientific (2016).. In 2008
Nambu was awarded the Nobel prize for something unrelated to string theory,
namely broken symmetries and the Nambu-Goldstone boson, although he
practically single-handedly invented string theory as described above. The
factorisation of the Virasoro amplitude was within a year after Nambu also
obtained by L. Susskind and independently by H.B. Nielsen. The action
principle for the string was soon after Nambu constructed also by Goto and is
today known as the Nambu-Goto action. Neither of these two key results by
Nambu were published and only appeared in talks and lecture notes (APS meeting
in Chicago, January 1970, summer school in Copenhagen 1970) and an unpublished
manuscript (for the 1970 Rochester conference in Kiev)161616See Nambu’s own
contribution to his collected works, Nambu, A Foreteller of Modern Physics,
Eds. T. Eguchi and M.Y. Han, World Scientific Series in 20th Century Physics,
Vol 43 (2014).. More details about the life of Y. Nambu and his scientific
achievements can be found in the book that Lars coauthored with Pierre
Ramond171717Lars Brink and Pierre Ramond, The Essence of a Genius; A Tribute
to Yoichiro Nambu, World Scientific Series in 20th Century Physics: Volume 46,
June 2023..

One of Lars’ early papers that had a huge impact on the understanding of the
string concerned the physical origin of the critical dimension and the
intercept, $D=26$ and $a=-1$. These facts, which actually contradict the
hadron interpretation, were derived prior to Lars’ work together with Holger
Bech Nielsen181818L. Brink and H.B. Nielsen, A simple physical interpretation
of the critical dimension of space-time in dual models, Phys. Lett. 45B (1973)
332. but they showed that $D=26$ and $a=-1$ could be related to the zero point
energy of the bosonic string with similar results for other string models. In
fact, by performing a physical regularisation of the naively infinite zero
point energy (involving the infinite sum over all positive integers) somewhat
similar to the first calculations of the Casimir energy, Lars and Holger
arrived at the following result in $D$ spacetime dimensions:

$a=-\frac{D-2}{24}.$ (12)

An interesting episode related to this is that at a summer meeting in Aspen in
1974 Lars was asked by Nambu to visit him in his office. Lars thought it was
to discuss the latest developments in dual string theory but, in fact, it was
to congratulate him on the beautiful paper he had written with Holger Bech
Nielsen on getting the critical dimension from summing up the zero point
fluctuations of the string. The standard attitude today is to attribute the
fact that

$1+2+3+4+...=-\frac{1}{12},$ (13)

to a classic result in the theory of analytic functions: It follows directly
from the Riemann zeta function
$\zeta(s)=\Sigma_{n=1}^{\infty}\frac{1}{n^{s}},\,Re(s)>1,$ analytically
continued and evaluated at $s=-1$. As Lars repeatedly pointed out to his
friends, the physics community was largely unaware of this mathematical result
in 1973 which made his paper with Holger even more extraordinary. Today this
fact is an absolutely fundamental insight for anyone with an interest in
string theory and can be found in all textbooks on the subject.

Another kind of basic contributions by Lars in the year of 1973 appeared in
several papers together with David Olive, and in a couple of cases also with
J. Scherk and C. Rebbi, on the construction of unitary string loop diagrams.
From previous works by Brower and by Thorn it was known that the Veneziano
amplitude was free of negative norm states (ghosts) provided $D<26$ and the
intercept $a=-1$. The problem was to get only physical modes to propagate also
in the loop diagrams which requires the insertion of projection
operators191919L. Brink and D. Olive, The physical state projection operators
in dual resonance models for the critical dimension of space-time, Nucl. Phys.
B56 (1973), 253.. This step (for $D=10$) was particularly complicated in the
case of the fermionic dual string models. Here one also needed to know the
proper set of gauge conditions which Lars investigated in a paper together
with D. Olive, C. Rebbi and J. Scherk202020L. Brink, D. Olive, C. Rebbi and J.
Scherk, The missing gauge conditions for the dual fermion emission vertex and
their consequences., Phys. Lett. 45B (1973), 379.. In order to solve these
problems Lars and his collaborators used Feynman’s tree theorem to define loop
amplitudes by sewing tree graphs without propagating ghosts. Lars often
mentioned the tree theorem as an example of the greatness of Feynman212121Lars
kept for his whole life a note on his office wall where Feynman thanks him for
using the tree theorem..

Up to this point the papers by Lars mentioned above were all part of his
doctoral thesis defended, as mentioned already, in December of 1973.

The next set of key contributions by Lars and his collaborators took place in
1976 which turned out to be a very interesting year with several parallel
developments involving local supersymmetry. As already mentioned, in 1974
Yoneya, followed by Scherk and Schwarz, developed the non-hadronic, that is
the Yang-Mills and gravitational picture, of the open and closed string
models. In order to understand the physics of string theory (in particular
spacetime fermions) it became crucial to develop a fermionic version of the
bosonic dual string. The necessary ingredients had been discovered already in
1971, first by P. Ramond and for a different sector of the theory by A. Neveu
and J. Schwarz. Together these constitute what we today know as the RNS
superstring. The fact that modern string theory could be turned into a success
story relied heavily on this development and the fact that the tachyonic
particles that appear can be seen to decouple and thus eliminated from the
theory as first shown in 1977 by F. Gliozzi, J. Scherk and D.I. Olive222222F.
Gliozzi, J. Scherk and D.I. Olive, Supergravity, Supergravity Theories and the
Dual Spinor Model, Nucl. Phys. B122 (1977) 253.. The problematic nature of the
tachyon is discussed at length at the end of Lars’ doctoral thesis. In fact,
on page 45 in his thesis Lars is making the remarkable prediction that the
elimination of the unphysical tachyon should happen through a decoupling
instead of the construction of a new tachyon-free dual string model.

The existence of the fermionic dual string models called for a supersymmetric
action principle in $1+1$ dimensions, that is on the worldsheet of the string.
To find such an action Lars first developed (submitted to Physics Letters in
August of 1976 and thus prior to the discovery of spacetime supergravity (see
below)) the simpler case of a locally supersymmetric relativistic particle in
collaboration with also Stanley Deser, Bruno Zumino, Paolo Di Vecchia and Paul
Howe232323L. Brink, S. Deser, B. Zumino, P. Di Vecchia and P. Howe, The local
symmetry of spinning particles, Phys. Lett. B64 (1976) 435 - 438.. The
fermionic string, or spinning string, action was obtained the month after this
and demonstrated to be locally supersymmetric in $1+1$ dimensions, in a
classic paper together with Paolo Di Vecchia and Paul Howe. The action they
found 242424L. Brink, P. Di Vecchia and P. Howe, A locally Supersymmetric and
Reparametrization Invariant Action for the Spinning String, Phys. Lett. 65B
(1976) 471 - 474. is written in terms of the string coordinates
$X^{\mu}(\tau,\sigma)$ and their anticommuting (world-sheet) spinorial
partners $\psi^{\mu}(\tau,\sigma)$. It reads

$S[X^{\mu},\psi^{\mu};e_{a}{}^{\alpha},\chi_{a}]=\int d\tau
d\sigma\,e(-h^{\alpha\beta}\frac{1}{2}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}\eta_{\mu\nu}-\frac{i}{2}\bar{\psi}^{\mu}\gamma^{\alpha}\partial_{\alpha}\psi^{\nu}\eta_{\mu\nu}$
$\frac{i}{2}\bar{\chi}_{\alpha}\gamma^{\beta}\gamma^{\alpha}\psi^{\mu}\partial_{\beta}X^{\nu}\eta_{\mu\nu}+\frac{1}{8}(\bar{\chi}_{\alpha}\gamma^{\beta}\gamma^{\alpha}\psi^{\mu})(\bar{\chi}_{\beta}\psi^{\nu}\eta_{\mu\nu})).$
(14)

Here $e_{a}{}^{\alpha}(\tau,\sigma)$, $h_{\alpha\beta}$ and
$\chi_{a}(\tau,\sigma)$ are the 2-dimensional frame field (zwei-bein), metric
and the Rarita-Schwinger field, respectively, making this action a
2-dimensional supergravity theory252525Local supersymmetry is absolutely
necessary since without it one can not eliminate the negative normed states
coming from the operators in the expansion of $\psi^{0}$, the time component
of $\psi^{\mu}$.. Note that the Nambu-Goto action is here replaced by an
ordinary Klein-Gordon like kinetic term which seems to have been first done in
the earlier paper in 1976 mentioned above involving also S. Deser and B.
Zumino. This construction generalised to the supersymmetric case the fact that
the Klein-Gordon type action given by the first term in the above
$S[X^{\mu},\psi^{\mu};e_{a}{}^{\alpha},\chi_{a}]$ is classically equivalent
on-shell to the Nambu-Goto action. The paper by Brink, Di Vecchia and Howe was
submitted to Physics Letters B in September 1976, just a few months after the
breakthrough that had taken place concerning the construction of D=4
supergravity (with one supersymmetry) by Deser and Zumino, and independently
by Ferrara, Freedman, and van Nieuwenhuizen262626See the historical note by S.
Deser on these developments arXiv: 1704.05886v3 [physics.hist-ph].. That
string theory given by this action gives rise to a low energy effective field
theory in 10 spacetime dimensions that is also supersymmetric (in spacetime)
was known from arguments presented in September 1976 by Gliozzi, Scherk and
Olive272727F. Gliozzi, J. Scherk and D. Olive, Supergravity and the spinor
dual model, Phys. Lett. 65B (1976) 282.. The actions of the kind given in (3)
are today usually referred to as Polyakov actions as a result of his papers
presenting the quantum version from a path integral perspective282828A.M.
Polyakov, Quantum geometry of bosonic strings, Phys. Lett. 103B (1981), 207 -
210 and Quantum geometry of fermionic strings, Phys. Lett. 103B(1981) , 211 -
213..

Also in 1976 Lars, together with what he referred to as an Italian national
football team, namely 11 or 12 Italians, generalised in three papers the
fermionic spinning string models of Ramond and Neveu-Schwarz by adding new
degrees of freedom in an attempt to reduce the critical dimension to $D=4$.
This failed but the models constructed, having additional world-sheet local
supersymmetries, increased the understanding of string theory in general and
are heavily cited292929M. Ademollo et al, Supersymmetric strings and colour
confinement, Phys. Lett. B62, 105 (1976), Dual string with U(1) colour
symmetry Nucl. Phys. B111, 77 (1976), Dual models with non-abelian colour and
flavour symmetries Nucl. Phys. B114, 297 (1976).. This was probably one of the
last attempts to find a hadronic string theory.

Today string theory has created entirely new avenues towards this goal, now
via holography and AdS/CFT related methods. Lars seemed to the end of his life
to stick to the old beliefs that the critical dimension could one day be
reduced to four and thus that a directly physical dual string theory could be
found.

Remarkably, Lars most cited paper is not one of the previously mentioned ones
but instead another paper written in 1976 and submitted to Nuclear Physics in
December303030L. Brink, J.H. Schwarz and J. Scherk, Supersymmetric Yang-Mills
Theories, Nucl. Phys. B121 (1977) 77.. It concerned the construction of
maximally supersymmetric Yang-Mills theories in various dimensions, more
specifically313131They did not seem to be interested in three dimensions and
hence missed that interesting case. 2,4,6 and 10. One important point made in
the paper is that the $D=10$ theory is very simple. The action is

$S_{D=10}=\int
d^{10}x\left(-\frac{1}{4}F^{a}_{\mu\nu}F^{a\mu\nu}+i\bar{\lambda}^{a}\gamma^{\mu}D_{\mu}\lambda^{a}\right),$
(15)

where $D_{\mu}$ is the Yang-Mills covariant derivative in the adjoint irrep of
the gauge group. This theory is supersymmetric, but only on-shell, in $D=10$
for Majorana-Weyl spinors $\lambda$ due to the Fierz identity

$f^{abc}(\bar{\lambda}^{a}\gamma^{\mu}\lambda^{b})\bar{\lambda}^{c}\gamma_{\mu}=0.$
(16)

The second key result is that by dimensional reductions from $D=10$ to the
lower dimensions listed above one straightforwardly obtains the maximally
supersymmetric Yang-Mills theories in these dimensions which are a bit more
complicated. This paper seems to have been the start for Lars’ life-long
interest in these theories. The theory in $D=4$ space-time dimensions has four
supersymmetries and its action reads (with $i=1,2,3,4$)

$\displaystyle S_{D=4}$ $\displaystyle=$ $\displaystyle\int
d^{4}x(-\frac{1}{4}F^{a}_{\mu\nu}F^{a\mu\nu}+i\bar{\chi}^{ai}\gamma^{\mu}D_{\mu}\chi^{ai}+\frac{1}{2}D^{\mu}\Phi^{a}_{ij}D_{\mu}\Phi^{a}_{ij}$
(17) $\displaystyle-$
$\displaystyle\frac{i}{2}g\,f^{abc}(\bar{\chi}^{ai}\chi^{jb}\Phi^{c}_{ij})-\frac{1}{4}g^{2}f^{abc}f^{ade}\Phi^{b}_{bij}\Phi^{c}_{kl}\Phi^{dij}\Phi^{ekl}).$
(18)

We will have reason to return to this theory shortly due to a speculation in a
paper in 1982 where Lars, together with M. Green and J.H. Schwarz, derived the
maximally supersymmetric Yang-Mills and gravity theories in $D=4$ as limits of
the superstring. The speculation concerned the fact that since the above
super-Yang-Mills theory is obtained this way it might actually be finite to
all orders in perturbation theory323232L. Brink, M.B. Green and J.H. Schwarz,
$\mathcal{N}=4$ Yang-Mills and $\mathcal{N}=8$ supergravity as limits of
string theories., Nucl. Phys. B198 (1982) 474. It seems, according to Lars,
that this speculation was first articulated by M. Gell-Mann some time before
this paper appeared..

While local supersymmetry was necessary and directly used as a tool in the 1+1
dimensional theory it took some time before F. Gliozzi, D. Olive and J. Scherk
realised that the theory in spacetime, or rather its low energy effective
field theory, must also exhibit supersymmetry as already mentioned above. This
led to a race to find the Lagrangian which describes the interacting theory of
the graviton and the spin 3/2 field, the so called Rarita-Schwinger field.
These rather complicated results were obtained about the same time by two
different groups, Deser and Zumino and independently by Ferrara, Freedman and
van Nieuwenhuizen (see above). The rather complicated approaches used by these
groups called for more powerful techniques. One such was superspace which was
developed for supergravity by Wess and Zumino in 1976. The paper was submitted
to Physics Letters the day after Brink, Schwarz and Scherk submitted their
paper on supersymmetric Yang-Mills theories to Nuclear Physics at the very end
of the year. This marked the end of a truly remarkable year for dual string
models and supersymmetry, and for Lars with his many fundamental contributions
to these developments.

We may remark here that Lars, together with J.O. Winnberg, developed a
worldsheet superspace formalism in a paper submitted already in November
1975333333L. Brink and J.O. Winnberg, The superoperator formalism of the
Neveu-Schwarz-Ramond model, Nucl. Phys. B103 (1976)445., while Lars, now
together with John Schwarz, a few years later studied other features of
superspace and theta variables in their rather well-cited paper on Quantum
superspace343434L. Brink and J.H. Schwarz, Quantum superspace, Phys. Lett.
100B (1981) 310..

An alternative description of the superstring, but fully equivalent to the
$NSR$ model, was later developed by Green and Schwarz, and here Lars also made
important contributions during his frequent visits to Caltech. This new
formulation used to study the superstring is very close to what is known as
superspace in field theory, namely the superspace coordinates
$x^{\mu},\theta^{m}$ in supergravity field theory are turned into world-sheet
fields $(X^{\mu}(\tau,\sigma),\Theta^{m}(\tau,\sigma))$ providing an
alternative to the RNS formulation which contains a fermionic field which is a
space-time vector ($\psi^{\mu}$) instead of the fermionic world-sheet scalar
which is a space-time spinor ($\Theta^{m}$). Lars had a keen interest also in
the superspace description of both Yang-Mills and supergravity. In three
papers with M. Gell-Mann, P. Ramond and J.H. Schwarz Lars studied the
geometric approach to supergravity353535See, e.g., M. Gell-Mann, P. Ramond and
J. Schwarz, Supergravity as geometry of superspace, Phys. Lett. B74 (1978) 336
- 340.. This paved the way for two nice papers together with P. Howe, then a
postdoc in Lars’ group at Chalmers, in 1979 and 1980363636L. Brink and P.
Howe, $\mathcal{N}=8$ supergravity in superspace, Phys. Lett. B74 (1979) 268,
and Eleven-dimensional supergravity on the mass-shell in superspace, Phys.
Lett. B91 (1980) 384.. In these two papers Lars and Paul rederived the
maximally supersymmetric gravity theories in $D=4$ and $D=11$ dimensions from
the superspace Bianchi identities by imposing a suitable set of constraints on
the super-torsion tensor.

In the 1980s Lars continued, together with his students and collaborators at
Chalmers, to work on issues which grew out of questions and techniques he had
studied earlier in the string theory context. One such technique was the
light-cone approach to strings and field theories. A theory that he
contributed to in several papers was maximally supersymmetric Yang-Mills
theory which he regarded as his ”baby”, to quote his own words from an
interview with him for American Institute of Physics in May 2021373737 This
interview can be found on the English version of the Wikipedia site for Lars.
. The study of this theory started during a visit to Caltech in 1976 with a
paper with John Schwarz and Joel Scherk where the theory was constructed in
dimensions $D=10$ as already mentioned above. They also demonstrated how one
can relate the theories in different dimensions to each other by dimensional
reduction. Another important result in this context was obtained in 1982 when
Lars visited Caltech and showed in a paper with J.H. Schwarz and M. Green how
to, in some limits of string theory, obtain $\mathcal{N}=8$ supergravity as
well as $\mathcal{N}=4$ super-Yang-Mills (see above).

Shortly after this Lars, then back at Chalmers, together with Olof Lindgren
and Bengt E.W. Nilsson, applied supergraph techniques in the light-cone gauge
to this $\mathcal{N}=4$ super-Yang-Mills theory and proved it to be a quantum
mechanically finite theory, the first one of this kind and hence a result of
fundamental importance for the much of the formal high energy developments in
the following years. The proof was crucially based on the light-cone
superfield383838See L. Brink, O. Lindgren and B.E.W. Nilsson, N=4 Yang-Mills
Theory on the Light-Cone, Nucl. Phys. B212, 401 (1983). which collects all the
independent physical degrees of freedom393939Two photon states, six scalars
and four complex spin 1/2 states. Note that $i=1,2,3,4$ and that
$\Phi_{ij}(x)$ is complex self-dual representing six real scalar fields. of
the theory in one object

$\Phi^{a}(x,\theta)=\frac{1}{\partial^{+}}A^{a}(x)+\frac{i}{\partial^{+}}\theta^{i}\lambda^{a}_{i}(x)+\frac{i}{\sqrt{2}}\theta^{i}\theta^{j}\Phi^{a}_{ij}(x)+\frac{\sqrt{2}}{6}\theta^{i}\theta^{j}\theta^{k}\epsilon_{ijkl}\bar{\lambda}^{ak}(x)+\frac{1}{12}\partial^{+}(\theta)^{4}\bar{A}^{a}(x),$
(19)

where all fields are in the adjoint irrep of the gauge group as indicated by
$a$. That such a (complex) superfield exists relies on two facts: 1) The
momentum light-cone component $p^{+}\equiv\frac{1}{2}(p^{0}+p^{3})$, above
written as $\partial^{+}$, is declared positive definite, and 2) the light-
cone gauge, $A^{+}\equiv\frac{1}{2}(A^{0}+A^{3})=0$, that makes it possible to
explicitly solve for $A^{-}$ in terms of the two photon polarisations (here in
$A(x)\equiv A^{1}+iA^{2}$). In Yang-Mills loop-calculations, using so called
supergraphs, this superfield gives rise to combinations of a huge numbers of
ordinary Feynman diagrams resulting in loop-expressions which can then be
shown to be finite at any loop-order404040See L. Brink, O. Lindgren and B.E.W.
Nilsson, The Ultraviolet Finiteness of the N=4 Yang-Mills Theory, Phys. Lett.
B123, 323 (1983). A similar proof of finiteness was presented by S. Mandelstam
about the same time.. More historical details on these developments can be
found in Lars’ own 2015 account of the events414141L. Brink, Maximally
Supersymmetric Yang-Mills Theory, The Story of ${\mathcal{N}}=4$ Yang-Mills
Theory, arXiv 1511.0297 [hep-th].. Lars often emphasised the enormous
importance of this result for later developments in string theory, not least
for the emergence of the AdS/CFT correspondence at the end of the 1990s.

Soon after this important development Lars applied these light-cone methods to
higher spin theories which turned out to be a tremendous success. The
construction of higher spin three-point vertices in a flat space-time together
with two of his PhD students424242A.K.H. Bengtsson, I. Bengtsson and L. Brink,
Cubic interaction terms for arbitrary spin, Nucl. Phys. B227 (1983) 31 - 40
and Cubic interaction terms for arbitrarily extended supermultiplets, Nucl.
Phys. B227 (1983) 41 - 49. started an active subarea of field theory that has
been thriving ever since. More recently Lars tried to use these light-cone
methods to argue that supergravity with the maximal number of supersymmetries,
that is eight, for which a superfield similar to the one used in the super-
Yang-Mills theory can be constructed, is also a finite quantum field theory.
Despite the collective effort of many people in the field to show this it is
still far from being proven. Lars probably nourished the idea that maximally
supergravity in D=4 is finite for the rest of his life. In fact, he later
introduced some new techniques based on the scalar field coset structure in
these theories to study possible restrictions on counter terms and possible
implications on the issue of finiteness434343See, e.g., L. Brink, S.S. Kim and
P. Ramond, E7(7) on the light cone, JHEP 2008 (06), 034..

When Yoneya, and independently Scherk and Schwarz, in 1974 studied the
consequences of the massless spin-2 particle in the spectrum interpreting it
as a graviton it was clear that the Regge slope parameter should be connected
to the Planck scale instead of being 1 $(GeV)^{-2}$ as dictated by hadronic
physics. This meant that modern string theory was born. String theory not only
generated from basic principles (that is quantum mechanics in 1+1 dimensions)
a theory of spacetime gravity but it also solved one of the most profound
problems in our description of the Universe, the non-renormalizability issue
of Einstein’s theory of gravity based on the Einstein-Hilbert action
principle. Lars, together with most researchers in his generation, saw this as
a hint of a TOE, a theory of everything. In particular this implied that
string theory should be capable of predicting a unique vacuum and hence
provide a full fundamental explanation of the standard model including
particle masses etc. This attitude towards the role of string theory has
largely been abandoned in recent years in favour of the more formalistic point
of view similar to the one we have towards ordinary quantum mechanics. When
string theory, and in particular the spin-off AdS/CFT, eventually got applied
to everything from nuclear physics to superconductivity most people in the
field shifted attitude towards the new more formal one. Lars eventually came
to the conclusion that this shift in attitude was necessary but he probably
never really came to fully embrace it444444A most personal reflection by one
of the authors (BEWN)..

To make clear that Lars had a broader interest in physics than just the field
of high energy physics and string theory, we mention here the important paper
by Lars on the Calogero model where he and his collaborators managed to find
an explicit solution to the old N-body Calogero problem454545L. Brink, T.H.
Hansson and M.A. Vasiliev, Explicit solution to the N-body Calogero problem,
Phys. Lett. B286, 109..

##  4 Lars Brink, some final reflections

People close to Lars were often struck by his fantastic memory. He sometimes
took advantage of this for instance when he recited long poems by authors he
had a particular relation to, like Nils Ferlin. Besides the keen interest in
humanities that he demonstrated in this and other ways he was also a huge fan
of athletics, soccer and, in fact, sports in general. It could be quite
intimidating to realise that he remembered an enormous number of results by
athletes active decades ago and whose names were long forgotten. A memory
related to himself that he was particularly fond of was his visit, in 1958, to
the newly built soccer stadium in Göteborg to watch a soccer game in the then
on-going world championship in Sweden. It was probably one of the semi-finals.
From other sources (his family) we later learned that he was actually selling
candy at the stadium, a fact that he apparently “forgot” to inform his friends
about. His memory bank also contained an abundance of anecdotes, often of a
humorous kind, and frequently involving various well-known people in physics
like Murray Gell-Mann and many many others. Although recalled in a humorous
tone these anecdotes sometimes expressed in addition the enormous admiration
he felt for many of the giants in high energy physics.

With a slight smile on his face Lars could occasionally comment on the fact
that he did not get very far in life. What he had in mind was the fact that he
grew up in Johanneberg in central Göteborg and went to high school down the
street and to university across the same street where he also spent his whole
academic career, neither of these schools located more than a kilometer away
from his home.

After having been ill for a long time Lars passed away peacefully in October
2022. Despite being confined to his bed at home he was active giving talks via
internet464646 Via internet Lars honoured the achievements of Madame Wu in a
talk at the C.S. Wu Global Online Symposium, September 24th, 2022.and writing
articles until the very last weeks of his life474747Lars’ last paper appeared
on September 2, 2022, and was a final tribute to Y. Nambu: L. Brink and P.
Ramond, Nambu and the Ising Model, arXiv 2209.01122[hep-th].. We have lost a
very good friend, a most stimulating colleague and someone who fought fiercely
for curiosity-driven fundamental research of all kinds.

## Acknowledgement

We are very grateful to Paolo Di Vecchia, Marc Henneaux and Pierre Ramond for
a thorough reading of the manuscript and for a number of suggestions which
have improved the presentation.

## References

  * (1) B.E.W. Nilsson: I first met Lars when I started the PhD program in Theoretical Physics at the then joint department of Chalmers University of Technology, Göteborg, and Göteborg University in the fall of 1976 with Lars as my supervisor. After having graduated in September of 1981 I left a year later for three postdocs abroad, joining first the Weinberg Theory Group, Austin, Texas, and then CERN, Geneva and Imperial College, London. After returning to Chalmers in 1987 Lars and I were colleagues with our own research groups at the same department, and became close friends. We were both members of the Royal Society of Arts and Sciences in Göteborg from 2008 (Lars was a member since 2004).
  * (2) B. Jonson: We learnt to know each other in the early 1970s where we both worked on our doctor’s theses at CERN in Geneva, Switzerland. Lars worked at the TH Division and I at the EP Division. Since then, we have been close friends both in our private and professional lives. From 1985 we were active together at the Department of Physics at Chalmers University of Technology, Göteborg, in different positions. We were members of the Royal Society of Arts and Sciences in Göteborg from 1997 and in the Royal Swedish Academy of Sciences from 2004 (I since 1998). We worked together in the Nobel Committee for Physics from 2008 to 2012.
  * (3) Most of the sources for this text appear in the numerous footnotes. Lars Brink’s doctoral thesis, his CV, his list of publications and many of his key papers appear there. In addition some comments are picked up from (the arXiv), e.g., Lars’ own accounts of some historical events plus the Memorial Volume for Y. Nambu and Nambu’s collected works in ”Nambu, a foreteller of modern physics”. In addition, both authors have contributed with a number of personal recollections.