#### by Albert Schwarz

This is not a narrative of my life. I will not talk about events of my life that were not directly connected to my work. I will talk only briefly about the relation of my papers to the work of other people.

More information about my life and my work can be found in my and E. Witten’s interviews for the AIP oral history project (www.aip.org/history-programs/niels-bohr-library/oral-histories).1

I started as a topologist, later switched to mathematical physics or,
better to say, to physical mathematics. (The notion of mathematical
physics is not very well defined, but physical mathematics has the
following definition given in corresponding section of Nuclear Physics
B: Mathematical ideas which are of current interest for their
potential application to theoretical physics. Of course in this
definition physical mathematics is a time dependent notion. ~~Prof. F.
Hirzebruch~~ told me that his work on topological questions of algebraic
geometry was considered as very abstract, very far from any
applications. Calabi–Yau manifolds also were regarded as a purely
mathematical toy.)

I worked in many directions, applying tools from various branches of mathematics, changing my field of interest every 4–5 years. Let me list some directions of my research.

- Geometry of uniform continuity. Volume invariant (growth) of a group (1952–1955).
- Topological questions of the calculus of variations. Space of closed curves. Genus of fiber space (1956–1962).
- Duality of functors in autonomous categories (1962–1968, 1971).
- Scattering matrix in quantum field theory (adiabatic definition, axiomatic theory) (1971–1974).
- Particle-like solutions to classical equations of motion and quantum particles. Topological charges. Magnetic monopoles. Symmetric gauge fields (1975–1978, 1980).
- Instantons. Quantum fluctuations of instantons (1975–1979).
- Topological quantum field theories (1978–1979, 1987, 1996, 2000).
- Topologically nontrivial string-like fields. Alice strings (1981–1982).
- Geometry of supergravity. Field-space democracy (1981–1986).
- Superconformal manifolds. Supermoduli spaces. Multiloop contribution to superstring theory (1985–1991).
- Mathematical problems of 2D gravity. String equation (1990–1991).
- Geometry of Batalin–Vilkovisky formalism (1992–1994).
- Noncommutative algebraic equations (1995, 1997, 2000).
- Noncommutative geometry and its applications to string/M-theory (1997–).
- Maximally supersymmetric gauge theories (2003–).
- Applications of arithmetic geometry to physics (2005–).
- Homology of moduli spaces of algebraic curves and of infinite-dimensional Grassmannian (2011–).
- Homology of Lie algebras of supersymmetries and super-Poincaré Lie algebra, their applications (2010–).
- Quantum curves (1991, 2013 ).
- Inclusive scattering matrix (2019–).
- Geometric approach to quantum theory. Jordan algebras in quantum theory. (2019–).

This list does not cover all directions of my work (for example, my papers on combinatorial invariance of Pontryagin classes and on topology of infinite-dimensional spaces are not covered); it includes only the fields that drew my attention for significant periods of time.

I never worked in fashionable directions, but some directions of my work became fashionable (geometric group theory, topological methods in physics, topological quantum field theories, noncommutative geometry in physics, etc).

My first paper (joint with N. Ramm) was written in 1952 under the guidance of Professor V. A. Efremovich and published in 1953 [1]. I was fortunate to have Professor V. A. Efremovich as my advisor — he was an outstanding mathematician and remarkable human being. Most of his work was devoted to the geometry of uniform continuity. In 1935 he proved that a map of metric spaces is uniformly continuous if and only if it transforms two sets with zero distance between them into two sets having the same property. This observation led him to introduce proximity spaces (spaces with a relation of proximity between two sets; in metric spaces this relation corresponds to zero distance.) Another axiomatization of the notion of uniform continuity was given at the same by A. Weil who introduced uniform spaces. My first paper was devoted to the analysis of the relation between proximity spaces and uniform spaces. Unfortunately, Efremovich’s results were not published on time: he was arrested during the purges of 1937 and was freed in 1944 because he was considered terminally ill. Against the odds he survived and returned to mathematics. However, in 1949 he was fired from his Moscow job in a new wave of purges; this was the reason why he landed as Professor in Ivanovo (300 km from Moscow where his family lived).

My story was a little bit similar. My parents were imprisoned in 1937. My father was sentenced to 10 years without the right to write letters. It was discovered later that this sentence was a euphemism for death penalty. My mother was arrested as a spouse of “an enemy of the people” and after several years in prison camp was exiled to Kazakhstan. Even my grandmother was arrested; after a couple of months in jail she found me in an orphanage under a changed name. In 1941 my grandmother and I joined my mother and in 1948 we were able to leave Kazakhstan.2

We settled in Ivanovo. There I met Professor V. A. Efremovich while still a high school student — he organized “mathematical circles” and olympiads. I entered Ivanovo Pedagogical Institute in 1951. (My application to Moscow University was predictably unsuccessful — I was a son of “an enemy of the people”.)

By the time I graduated from high school I knew already a little bit
of mathematics: calculus, elements of Lobachevsky geometry. Therefore
in my freshman year I was able to study general topology and work in
this direction. This was an ideal starting point from which Professor
V. A. Efremovich led me to more geometrical questions. My first
serious work
[2]
was inspired by Efremovich’s remark that the “volume
invariant” of universal covering of a compact manifold is a
topological invariant of the manifold. (If two compact manifolds are
homeomorphic, then the natural homeomorphism between universal
coverings is uniformly continuous. Efremovich proved that under
certain conditions the growth of the volume of a ball with radius
tending to infinity is an invariant of uniformly continuous
homeomorphisms.) I proved that the volume invariant of universal
covering can be expressed in terms of the fundamental group of the
original manifold; in modern language
it is determined by the *growth of the fundamental group*. I
also gave estimates for volume invariants of manifolds with
nonpositive and with negative curvature. Thirteen years later
J. Milnor
published a paper containing the same results with the only
difference that Milnor was able to use in his proofs some theorems
derived after the appearance of my paper. At the moment of writing his
first paper in this direction Milnor did not know about my work, but
his second paper contained corresponding references. The notion of
growth of a group (volume invariant of a group in my terminology) was
studied later in numerous papers (one should mention, in particular,
the results by
Gromov
and
Grigorchuk).
A new interesting field — geometric group theory — was born from these papers.

I am very grateful to P. S. Alexandrov and V. A. Efremovich for the possibility to spend several months at Moscow University, in the best Mathematics Department of Russia, remaining a student at Ivanovo. (The status of visiting student, not uncommon in the US, was unheard of in Russia. ) In 1955 I graduated from Ivanovo Pedagogical Institute and was nominated for graduate school. (Without such a nomination I would be obliged to teach mathematics and physics in high school.) In the same year I was admitted to graduate school at Moscow University. I was very lucky — in 1954, after Stalin’s death, my father was exonerated. Moreover, 1955 was the best year for a Jew to enter the University: the discrimination of Stalin’s time was abolished, the new period of discrimination started later. My advisor at Moscow was P. S. Alexandrov, but at this moment I was already able to work independently (I wrote 7 papers in my undergraduate years) and he gave me complete freedom.

My years in graduate school were quite productive. I would like to
mention my paper with
V. Rokhlin
containing a proof of combinatorial invariance of rational Pontryagin
characteristic classes
[3].
(At the same
time this result was obtained by
R. Thom.)
Most of my papers of this
time were devoted to topological problems that appear in the calculus
of variations. In one of my papers I studied
the topology of the space of
closed curves on a manifold
[6]
This space is a infinite-dimensional
orbifold (in modern terminology). The fact that it is not a manifold
was not understood in preceding papers by
M. Morse
and
R. Bott;
this
led to some errors. I corrected them and used new methods to study the
homology of this space. (By the way, this space appeared later in
papers by
Sullivan
and his collaborators in relation to string theory;
he uses the name string homology for its homology. It was considered
also in my paper about A-model and generalized Chern–Simons theory.)
In another paper
[4]
I introduced and studied a notion of genus of a fiber
space; it generalized the notions of the Lusternik–Schnirelman
category and of Krasnoselsky genus of a covering.
It was analyzed further in my papers
[5],
[7]
[9]
The same notion was
rediscovered (under another name) 25 years later by
S. Smale
and used
to estimate topological complexity of algorithms. S. Smale found the
simplest estimates of genus and applied them to estimate topological
complexity of algorithm of solution of algebraic equation having
degree __\( n \)__. His estimate from below was __\( (\log_2n)^{2/3} \)__.
V. Vasiliev,
applying my estimates of genus, strengthened this result and proved
that the complexity is close to __\( n \)__. (If __\( n = 2^k + a \)__ where __\( 0 \leq a
< 2^k \)__ then the complexity is between __\( 2^k - 1 \)__ and __\( n - 1 \)__). Later
the notion of genus was applied to robot motion planning by
M. Farber
and other scholars. My results were based on methods of topology that
were new at that time, especially on the use of spectral sequences.
When I entered
graduate school Moscow mathematicians did not have
a working knowledge of these methods; I was one of the first Russian
mathematicians to successfully apply them. I gave numerous talks
explaining these methods, later
V. G. Boltyansky,
M. M. Postnikov
and
I organized a seminar on Geometric Topology devoted mostly to
applications of spectral sequences. The seminar was very successful;
it attracted many brilliant young mathematicians. (First of all I
should mention
S. Novikov,
who after receiving
the Fields Medal for his
work in topology, made very important contributions to integrable
systems and mathematical physics.) The seminar announcement contained
the following description of geometric topology: General topology
studies simple properties of complicated
spaces; geometric topology
studies complicated properties of simple spaces (manifolds,
polyhedra). Experts in general topology, including P. S. Alexandrov,
one of creators of this field, were insulted by this description. It
seems that P. Alexandrov thought that this was my fault (although
among the sponsors of the seminar were two
*full professors* and I was a
graduate student). I don’t remember who invented the “criminal”
sentence (the announcement was written by Boltyansky and me), but in
any case we did not have any intention to insult anybody.

In 1958 I defended my Candidate of Sciences (__\( = \)__ PhD) dissertation.
At this time I was already an author of 15 papers; many of them were
well known. (In a review book *Mathematics in USSR: 40 years*
published in 1957 my results were mentioned many times.) However, it
was not simple for me to find a job after graduation. The main
obstacles came from the KGB. It did not give me clearance in two
places that wanted to hire me (the Laboratory for Theoretical Physics
in
Joint Institute for Nuclear Research
(JINR), Dubna, and the Division of Applied Mathematics of
the Steklov
Institute). I accepted an offer from Voronezh University. At this time
the Department of Mathematics at Voronezh University was a lively
place. It was informally headed by
Professor M. Krasnoselsky,
who
created a very active group working in nonlinear functional analysis
and using topological methods in their work. In Voronezh I continued
my work on the genus of fiber space that was close to the interests of
Voronezh group. In 1960 I defended my Doctor dissertation in Moscow
University; it was based on
my work on genus. (A Doctor degree in
Russia corresponds to the German Habilitation; a
full professor as a rule
should have this degree.) I am very grateful to Krasnoselsky who
convinced me that my results
were
sufficient for the Doctor
dissertation and,
what was more important, explained to me that I
should become a Doctor if I would like to get an apartment from the
university. The defense of the dissertation is not a final step in the
Russian system; the dissertation must be certified by a special
organization, the so-called
VAK. In my case everything went smoothly,
but several years later the VAK became a major tool in the fight
against Soviet mathematicians of Jewish origin; I was very lucky that
I did not postpone submission of my dissertation.

In Voronezh I worked also on topological questions arising in
functional analysis. I analyzed the topology of several classes of
operators. In particular, I calculated homotopy groups of the space of
Fredholm operators; my results agreed with results of
Atiyah
and
Janich,
who studied this space from the viewpoint of K-theory at the
same time. I proved that the embedding of finite-dimensional unitary
group __\( U (n) \)__ into the infinite-dimensional unitary group is homotopic
to zero and conjectured that the infinite-dimensional unitary group is
aspheric; this conjecture was proven by
Kuiper.
I constructed homotopy
invariants of nonlinear Fredholm operators in Banach spaces and gave
a talk about these results in 1961 at the Congress of Mathematicians
of the USSR (Leningrad) but never published these results. I was
convinced that they were incomplete: for operators of index 0 I
obtained a residue __\( \text{mod }2 \)__ instead of an integer as an
invariant. Moreover, Professor
Yu. Borisovich informed me that
Caccioppoli
studied the degree of Fredholm operators of index zero in
the thirties. Later S. Smale independently found constructions that
were very similar to my work. He also did not have a complete picture
(the construction of invariants in terms of oriented cobordism classes
was invented later by
Elworthy
and
Tromba),
but he did not know about
the paper of Caccioppoli and he published his results.

One more direction of my work in Voronezh was related to category theory. Being a graduate student in Moscow I advised several very gifted undergraduate students. A couple of them were willing to continue to work under my guidance after my departure from Moscow. I was reluctant to agree — at that time email did not exist, long distance phone calls were expensive. However, I agreed to supervise informally D. Fuchs’ work on his PhD dissertation. As I understand now I was following the advice of one tutor that prepared high school students to university entrance examination: “To be a very successful tutor you should work with students that are able to pass the examination without your help”. D. Fuchs found the topic of his dissertation by himself. Moreover, for some time I worked in the direction he started. Fuchs’ goal was to understand Spanier duality in topology in terms of duality of functors. Together with D. Fuchs, B. Mityagin and my Voronezh students (V. Kuznetsov and R. Pokazeeva), I studied duality of functors in autonomous categories (in categories where the set of morphisms is equipped with the structure of an object of the category — we used the word D-category for this notion) [8], [10], [11], [13], [12], [14], [16].

In 1960, during the week I spent in Moscow after the defense of my Doctor of Sciences dissertation, I met my future wife, L. Kissina. We married several months later. My wife was born in Moscow and did not want to leave her city. I convinced her to give it a try and to live in Voronezh for a year. She found that Voronezh was great, but it was a great village and Moscow was a great city. As I promised I started to look for a job in Moscow. One of our friends learned from physicists working in MEPhI (Moscow Engineering Physics Institute) that they were not satisfied with the level of mathematicians there. This friend mentioned me as a young Doctor of Sciences looking for a job in Moscow. (Probably, I was at this moment the youngest Doctor of Sciences in the USSR.) In 1964 I was offered a position as Professor of Theoretical Physics at MEPhI. The original idea of the physicists was that at some moment I would be transferred to the Department of Mathematics to strengthen it. This never happened. Mathematicians were not eager to have me in their department. I enjoyed teaching various courses in Theoretical Physics and was eager to stay in the Physics Department and physicists were not disappointed with my work.

I was interested in physics for a long time. The possibility to work with physicists was very exciting for me. My first goal was to understand thoroughly the main physical theories. I thought a lot about the foundations of quantum mechanics, especially about measurement theory. However, conversations with physicists convinced me that these questions were not interesting for them and I turned my attention to other things. After years of work on the more concrete problems of theoretical physics I have also lost interest in foundations. There was some evolution in physicists’ attitude to measurement theory; the modern view on these problems does not contradict my old ideas. (I published some of my ideas together with Yu. Tyupkin in 1987 [66]. Very recently, in 2019, I used the results of this paper in a geometric approach to quantum theory.)

I believe that physicists were right in avoiding thorny questions of measurement theory. The quantum probabilities can be derived from decoherence; the subtle problems of measurement theory are not relevant in this derivation.

The last fifty years were marked by very intensive and fruitful interaction between mathematics and theoretical physics. However, in the sixties this process had only just started — in Russia slightly earlier than in the West. A handful of Russian mathematicians started to study seriously theoretical physics and to work on related mathematical problems; I belonged to this small group that included F. Berezin, R. Dobrushin, L. Faddeev, Yu. Manin, R. Minlos, S. Novikov, and Ya. Sinai.

We followed the footsteps of A. Kolmogorov and I. Gelfand. Later many very talented young Russian mathematicians moved in the same direction; I would like to mention A. Beilinson, I. Bernstein, V. Drinfeld, B. Feigin, D. Fuchs, and D. Kazhdan.

At the same time many extraordinary young physicists moved toward mathematics; first of all Sasha Polyakov and three other Sashas (Belavin, Migdal, Zamolodchikov), V. Knizhnik, A. Morozov, M. Shifman, and A. Vainstein.

Together with F. Berezin I organized a seminar devoted to mathematical problems of physics in Moscow University. Later R. Dobrushin, R. Minlos and Ya. Sinai joined us as organizers.

My attention was concentrated mostly on attempts to understand quantum field theory. Together with my students V. Fateev, V. Likhachev, and Yu. Tyupkin I wrote several papers on the scattering matrix in quantum field theory. We studied the scattering matrix in the axiomatic approach [18] and the relation of an adiabatic S-matrix to the scattering of physical particles (see, for example, [17]). (Much later the ideas of an axiomatic approach based on asymptotic commutativity were used in my paper “Space and time from translation symmetry” [122]. They are relevant for string field theory.)

I was not satisfied with textbooks in quantum field theory. Of course,
I understood that there is no reason to require mathematical rigor
from such a book. My main concern was that the main concepts, such as
the notion of scattering matrix, were ill-defined, that the rules of
the game were changed in the middle of the game. Physics students were
convinced that renormalization is necessary only to remove
divergences,
and did not understand that it appears already in statistical physics
where all Feynman diagrams are convergent. I have written two books
devoted to
the foundations of quantum field theory. The exposition in
these books was not rigorous, but it was based on clear definitions.
The second edition of
my book *Mathematical Foundations of Quantum Field Theory* was
published in 2018. Recently, in 2020,
the English translation of the book
(with
the addition of some new results) was published by World
Scientific.

At the end of eighties I
published
a book
*Quantum Field Theory and Topology* containing a short introduction to topology
and to its applications in quantum field theory; the second edition
was published in 2018. An English translation of this book was
published by Springer Verlag as
two volumes
(*Quantum Field Theory and Topology* and
*Topology for Physicists*.)

In the sixties topology did not play any significant role in quantum field theory. I continued to work on some topological and algebraic problems, but this work was not related to physics in any way. The relations between physics and modern mathematics changed drastically in the seventies. The change was prompted by discovery that soliton solutions of classical equations of motion are related to quantum particles and that topological considerations can be used to guarantee stability of solitons and corresponding quantum particles. I will not discuss in detail the history of this discovery that started with almost unnoticed papers by Skyrme. My interest in this subject was inspired by influential papers by G. ’t Hooft and A. Polyakov who independently suggested in 1974 that classical solutions in one of the models describing interactions of gauge fields and scalar fields (the Georgi–Glashow model) can be interpreted as magnetic monopoles. In a paper joint with V. Fateev and Yu. Tyupkin [21] we proved that the space of finite energy fields in gauge theories is disconnected in general and the components of this space are labeled by numbers that may be called topological charges or topological invariants of motion. Topological charges can be identified with elements of some homotopy groups; one can calculate these groups by means of homotopy theory. If the gauge group is simply connected and the group of unbroken gauge symmetries is one-dimensional then there exists only one topological charge and it can be identified with magnetic charge [28]. This statement leads to a prediction that magnetic monopoles exist in any grand unification theory, i.e., in a gauge theory describing strong, weak and electromagnetic interaction at the same time.

Magnetic monopoles and other topologically nontrivial fields were
considered in many of my papers,
for example in
[19],
[20],
[23],
[27],
[26],
[28],
[25],
[29],
[40]
with the help of homotopy theory,
homology theory, characteristic classes and index theory. The joint
paper of 1975 with
A. Belavin,
A. Polyakov,
and
Yu. Tyupkin
[24]
was especially
influential. In this paper we introduced and analyzed topologically
nontrivial extrema of euclidean action in gauge theory
(*pseudoparticles* in the terminology
of our paper is the same as *gauge instantons*
in modern terminology). Instantons were studied later in numerous
papers, including papers written by me (often in collaboration with my
students
V. Fateev,
I. Frolov,
V. Romanov,
and
Yu. Tyupkin). In
particular, I calculated the dimension of the moduli space
[30]
of
instantons by means of index theory and analyzed quantum fluctuations
of instantons
[31],
[32],
[37],
[38],
[36],
[43].
A slightly different calculation of the dimension of
the instanton moduli space was given by Atiyah,
Hitchin
and
Singer.
It
seems that it was inspired by my preprint, that did not contain
complete results, but their paper appeared after publication of mine
(and refers to it). Both of these papers are devoted to consideration
of instantons on the four-dimensional sphere, however, it was clear
from the very beginning that moduli space of instantons on more
general four-dimensional manifolds can be studied
in a similar way.
Atiyah, Hitchin, Singer worked in this direction and their results
formed a basis of famous papers by
Donaldson
on classification of
four-dimensional manifolds.

I also studied one-dimensional topologically stable configurations in quantum field theory (topologically stable string-like objects) [41], [55], [46]. I have shown that such configurations appear in gauge theories if the gauge group is connected but the group of unbroken symmetries is disconnected and that a particle that goes around topologically stable string-like object can change its type. The last statement found an experimental confirmation in condensed matter physics. It led also to the statement that in grand unification theories describing weakly interacting “real” and “mirror” worlds there exist quite unusual string-like objects (Alice strings) having the property that a particle going around such a string becomes a mirror particle.3 Electric charge is not localized in the presence of Alice strings — people say that it becomes a Cheshire charge, remembering the Cheshire cat from L. Carroll’s book. In principle Alice strings can be detected by means of astronomical observations; there exist some observations that can be explained by means of Alice strings very naturally; however, all of them have more complicated, but less exotic explanations.

When I invented Alice strings my first thought was that they give wonderful material for
science fiction. It seems that other people shared this opinion. J. Gribbin wrote a book
*The Alice Encounter* based on this idea. This is an excerpt from the synopsis of this book:

There is about 10 times more dark matter (DM, also known here as

Alice matter) than bright stuff in our Galaxy. The DM is spread out in a roughly uniform sphere (a spherical distribution of Alice stars), with our flattened disk Galaxy embedded in it. The Alice matter is a kind of mirror image shadow stuff; the termlooking glass matterhas been used by some scientists. Alice matter can be turned into ordinary matter (and vice versa) by sending it through a loop of Alice string, a naturally occurring cosmic phenomenon.

I enjoyed my work on topological integrals of motions and topologically nontrivial solutions. My research in topology gave me an excellent background for this work and I was happy to work in a new and fast-moving field. I gave several talks with titles like “Applications of topology to physics”, and these talks were interesting both to physicists, who realized there exists a powerful new tool, and to mathematicians, who discovered a new rich source of exciting mathematical problems.

I was even happier when in 1978 I found a way to use ideas from physics in topology [34], [35]. I realized that action functionals that do not depend explicitly on the metric should give rise to topological invariants. In lectures about these new ideas I talked about “applications of physics to topology”. Of course, this was not a precise statement; however, the very fact that one could use ideas from physics to obtain topological results was fascinating. I have found that Reidemeister torsion (or, more precisely, its differential counterpart defined by Ray and Singer) can be obtained as a partition function of some metric-independent action functional. The gauge transformations for this functional were not independent (in today’s terminology the theory was reducible), hence it was impossible to perform quantization using the Faddeev–Popov approach; I developed techniques allowing quantization of theories of this kind.

I understood very well that the idea to construct topological invariants from metric-independent action functionals (such as the Chern–Simons functional) was quite general and it should lead to new interesting invariants; moreover, it was clear that this is also a way to find invariants of other structures (say, invariants of complex manifolds). However, at that time I was able to analyze nonquadratic action functionals only in semiclassical approximation or in the framework of perturbation theory. It was easy to write down perturbation series for invariants, but it was very difficult to analyze these series. Much later, in 1987 V. Turaev told me that V. Jones constructed an invariant of knots that can be considered as a noncommutative generalization of the Alexander polynomial. One can express the Alexander polynomial in terms of Reidemeister torsion. V. Turaev asked me whether it is possible to modify the action functional I used to obtain Reidemeister torsion in a way that leads to the Jones polynomial. Pretty soon I came to a conjecture that the role of such a modification should be played by the Chern–Simons action functional; I talked about this conjecture at the topological conference in Baku [70]. I was not able to prove this conjecture; the perturbative expressions I tried to analyze together with Yu. Tyupkin were too complicated. The difficulties arising in this way were overcome only in 1992 by Axelrod–Singer [e4] and Kontsevich (in an unpublished talk given at Harvard in the 1991–92 academic year). However, in 1988 E. Witten independently conjectured that the Jones polynomial was related to the Chern–Simons action functional and gave a (heuristic) proof of this conjecture [e2]. My 1978 paper [34] was a precursor to a series of remarkable papers where the methods of theoretical physics were applied to topology, algebraic geometry, symplectic geometry and other branches of mathematics. Many of those papers were related to topological quantum field theories (my paper gave the first examples of such theories). The axioms of topological quantum field theories were suggested by M. Atiyah, who conjectured that Donaldson invariants and Floer homology can be obtained from topological quantum field theories. This conjecture was proven by E. Witten, who found a new way to construct topological quantum field theories. (I considered metric-independent action functionals; E. Witten has shown that many very interesting theories can be constructed if we allow BRST-trivial dependence on the metric.)

Today topological quantum field theories play an important role both in mathematics and physics. I returned to them several times (for example, in 2000, when I was invited to give a plenary lecture on TQFT at International Congress for Mathematical Physics). In my paper with S. Gukov and C. Vafa [114] I gave an interpretation of Khovanov and Khovanov–Rozansky invariants of knots in terms of topological strings.

In the eighties I studied the geometry of supergravity (together with my students M. Baranov, A. Gayduk, O. Khudaverdian, V. Romanov, and A. Rosly) [39], [42], [44], [44], [44], [47], [49], [50], [57], [56], [52], [54], [60], [58]. This work led me to an idea that fields should by considered on an equal footing with space-time coordinates [39], [42]. Following the suggestion of E. Brezin I called this idea field-space democracy. Although explicit references to my papers on field-space democracy are rare the idea is alive. (In particular, field-space democracy was used by Townsend to construct action functionals of D-branes.) An unexpected consequence of my paper on field-space democracy was an invitation to attend a conference on parapsychology in Italy. (The organizers of the conference had a goal to present parapsychology as a science, therefore they included a section devoted to new ideas in the theory of space and time. A. Salam was listed as one of the speakers; perhaps, the organizers knew my name from him.) Of course, I did not attend the conference.

My work in supergravity was based on the theory of G-structures, in particular on the theory of Cauchy–Riemann structures (CR-structures). CR-structures were used later by A. Rosly to formulate theories in terms of isotopic analogs of twistors (isotwistors). I also worked in this direction together with A. Rosly. Rosly’s idea of isotwistors (under the name of harmonic space) was used by V. Ogievetsky and his group (Sokatchev, Ivanov, Kalitzin, and Galperin) to obtain very interesting explicitly supersymmetric formulations of some supersymmetric theories. Later, I came back to this circle of ideas studying maximally supersymmetric gauge theories together with M. Movshev.

In the second half of eighties I worked on geometric questions arising in superstring theory (together with M. Baranov, I. Frolov, A. Rosly, and A. Voronov) [53], [59], [63], [61], [64], [62], [65], [67], [71], [69], [68], [74]. I introduced a notion of a superconformal manifold (this notion was introduced independently by D. Friedan and S. Shenker under the name “super Riemann surface”). We studied the moduli space of superconformal manifolds (supermoduli space), that appears in the calculation of multiloop contributions for superstrings. Recently the results of these papers were popularized by E. Witten who has written several review papers on related topics. I would like to mention my paper about universal moduli space [68]; it seems that this space deserves further study. I talked about my results in this direction in an invited lecture at the International Congress of Mathematicians (Kyoto, 1990).

In [51] I have a definition of superspace in terms of functors on the category of Grassmann algebras. This definition can be considered as a mathematical formalization of the standard definition used by physicists; it is especially convenient for infinite-dimensional supermanifolds. Later together with A. Konechny I generalized this definition [91], [102]. The more general definition is convenient when we are dealing with two-dimensional supermanifolds with independent left-moving and right-moving modes and with corresponding moduli spaces.

At this moment I have to interrupt the discussion of my work to talk about political changes in the Soviet Union and about changes in my life. This was the time of Gorbachev’s perestroika. All of my colleagues were fascinated by new possibilities and afraid of imminent dangers. For me, the first sign of new times was an almost successful attempt to go to Swansea. I was invited to give a lecture at the International Congress for Mathematical Physics and the KGB had no objections. I was excluded from the Soviet delegation by bureaucrats in the Academy.

I received many invitations to spend some time in the West; I could not accept any of them until 1989. In early eighties L. Michel told me that I had a standing invitation from IHES, but a new formal letter would be sent only if I have a chance to come. Before “perestroika” I was allowed to go abroad only twice — I visited Bulgaria in 1980 and Czechoslovakia in 1981. In 1988 I was able to attend a conference in Poland. There I met K. Gawedzki and asked him to inform the administration of IHES that the situation had changed. I never received the promised invitation letter. Later, in Italy, K. Gawedzki explained to me that usually IHES invited about 20 mathematicians from Russia every year and only a couple of invitations were accepted. That year all twenty invited scholars came; IHES did not have any possibility to send an additional invitation. I came to IHES for the first time in 1995. Later I spent many summers in this wonderful institute. IHES has extraordinarily strong permanent members; I was fortunate to collaborate with several of them (with A. Connes, M. Douglas, M. Kontsevich, N. Nekrasov).

In 1989 I was able to accept an invitation from the
International Center for Theoretical Physics (ICTP)
to spend 1 1/2 months
in
Trieste. I was immediately invited to come back in July to
participate in the Supermembrane Conference. I explained to
Prof. Sezgin
that there was no chance for me to be allowed to make another
formal trip to Italy on such short notice, but that I could try to
come as a private citizen. By my request Prof. Sezgin formally invited
me and my wife to visit him in Italy. My family considered the attempt
to come back to Italy in two months as completely hopeless, but things
were moving very fast in the Soviet Union. On ~~July 17,~~ 1989 I left the
Soviet Union — forever.

Very positive changes in the Soviet Union went hand in hand with dangerous developments. Gorbachev’s economic policy was disastrous. Instead of radical changes on a small scale he accepted large scale half-baked reforms. As a result the economy, that was already in bad shape before Gorbachev, rapidly deteriorated. Nationalistic sentiments both in Soviet republics and Russia, suppressed under communist rulers, flourished in more democratic times; the government antisemitism declined, but the appearance of openly antisemitic organizations and newspapers was very disturbing. My children did not see any future for them in the Soviet Union. I decided to leave Russia fearing for my family.

I would like to express my sincere gratitude to all people that helped me to come to the US and settle here, first of all to D. Gross, A. Jaffe, J. Schwarz, I. Singer, E. Witten and B. Zumino. I spent the academic year 1989–1990 in Princeton (in the Institute for Advanced Study) and in Cambridge (at Harvard and MIT). Since 1990 I have worked in the Department of Mathematics at UC Davis. This is now a very strong department; it was completely transformed due to the efforts of its chairmen, first of all A. Krener, then C. Tracy, M. Mulase, J. Hunter, B. Nachtegaele, J. Hass, D. Romik, and A. Thompson.

When I came to the US I realized that it is necessary to make a choice: should I look for a position in a department of mathematics or in physics department? I started as a mathematician, but I worked in physics department in Russia and I always characterized myself citing a Russian proverb “Gentle calf sucks milk of two cows”. I decided that nobody can say that I am not a mathematician, but some people can say that I am not a physicist. My decision to join the Department of Mathematics in Davis was a good one; now we have several excellent scholars working on the interface between mathematics and physics.

Russian science was concentrated mostly in Moscow and Leningrad; American mathematicians and physicists are dispersed over the whole country. Happily now, using internet and telephone, scholars are able to collaborate even across the ocean; American scholars used this possibility earlier than scholars of other countries. I started such a “long distance collaboration” immediately after coming to the West. My first joint paper after leaving Russia was written together with A. Sen from India (our collaboration started when we met in Trieste for a couple of days); it was devoted to boundary conformal field theory [73]. I collaborated with V. Kac, H. S. La and P. Nelson, K. Anagnostopoulos and M. Bovick, A. Tseytlin, L. Friedlander, M. Kontsevich, A. Connes, S. Novikov, M. Douglas, M. Rieffel, N. Nekrasov, B. Pioline, A. Polishchuk, S. Gukov, C. Vafa, V. Vologodsky, A. Mikhailov, D. Krefl, J. Walcher, Xiaojun Liu and with many people from Davis: D. Fuchs, M. Penkava, O. Zaboronsky, A. Konechny, M. Alexandrov, A. Astashkevich, M. Dieng, M. Luu, M. Movshev, Yujun Chen, Xiang Tang, I. Shapiro, Jia-Ming Liou, and Renjun Xu.

I would like to thank all of my coauthors.

As usual I worked in various directions. One of my first papers in
America was written in collaboration with V. Kac
[72];
we studied the
partition function of 2D Gravity from the viewpoint of geometry of
infinite-dimensional Grassmannian. More complete results in this
direction were obtained later in my papers and in a joint paper with
K. Anagnostopoulos and M. Bovick
[75].
In particular, I have studied in
detail the so called string equation:
__\( [A, B] = \text{const} \)__, when
__\( A \)__ and __\( B \)__ are differential operators on a line. I came back to this
thematics later in
[131].
Several of my papers were devoted to the geometry of
Batalin–Vilkovisky quantization
[76],
[77],
[79],
[84]. I am convinced
that the BV-formalism should be used not only as a tool, that allows
us to quantize degenerate Lagrangians, but also a starting point in
classical and quantum field theory. This idea together with the
geometric approach to BV-formalism was applied in a paper
written
jointly with M. Kontsevich and my students
M. Alexandrov and O. Zaboronsky
[84]
to give a geometric construction of a BV sigma-model
(now called AKSZ model) containing many important theories as
particular cases. Later M. Kontsevich used this construction in his
famous paper on quantization of Poisson manifolds.

The geometric approach to the BV-formalism is based on the notion
of a Q-manifold (__\( = \)__ supermanifold equipped with an odd vector field
with square equal to zero). This notion is closely related to the
notion of an __\( L_{\infty} \)__-algebra that can be regarded as formal
Q-manifold; __\( A_{\infty} \)__-algebras can be characterized as formal
noncommutative Q-manifolds. Hochschild cohomology corresponds to
__\( A_{\infty} \)__-deformations and cyclic cohomology corresponds to
deformations of __\( A_{\infty} \)__-algebras with invariant inner product
[82].
My work with
M. Movshev on supersymmetric gauge theories
[125]
is based
on these relations.

The notion of Q-manifold was one of the precursors of derived algebraic geometry, a new field that is important both in mathematics and physics.

My work on noncommutative algebraic equations started with a proof of
“matrix Vieta theorem” in a joint paper with D. Fuchs
[81].
We proved that
for generic roots __\( x_1 ,\dots, x_n \)__ of a matrix algebraic equation
__\[
x^n + a_1 x^{n-1} + \dots + a_n = 0
\]__
one has
__\[
\operatorname{Tr} x_1 +
\dots + \operatorname{Tr} x_n =
{}-\operatorname{Tr}a_1 , \quad \det x_1 \cdot\, \cdots \,\cdot \det x_n = \det
a_n.
\]__
The
coefficients __\( a_1 ,\dots, a_n \)__ can be expressed in terms of generic
roots __\( x_1 , \dots, x_n \)__; corresponding expressions can be considered
as noncommutative analogs of elementary symmetric functions and our
results can be formulated as a statement about these functions. We did
not think that our work was related to physics in any way, but several
years later physicists P. Aschieri, D. Brace, B. Morariu and B. Zumino
came to the noncommutative algebraic equations when studying
Born–Infeld-type Lagrangians. My paper
[99]
was inspired by their
work; it was based on another proof of the matrix Vieta theorem given
by Connes and me
[85].
The papers on noncommutative algebraic equations
are related to papers by I. Gelfand, V. Retakh and their
collaborators; all these papers belong to an emerging field that still
does not have a good name.

One of main directions of my work in recent years can be described as the application of Connes’ noncommutative geometry to string/M-theory.

The “second string revolution” of the nineties made clear that
superstring theory should be embedded into a more general theory that
was christened *M-theory*. It was conjectured that M-theory lives in
eleven-dimensional space; it should give superstrings after
compactification on a circle and eleven-dimensional supergravity in
its
low-energy limit. All existing superstring theories can be
obtained from M-theory in some limits; all of them are related by
dualities that can transform objects that allow a perturbative
description, into more exotic objects — branes.

A mathematical definition of M-theory is not known;
however in
1996
Banks,
Fischler,
Shenker
and
Susskind
suggested that M-theory can
be formulated in terms of a matrix model. This model (the BFSS model
or M(atrix) theory) is defined as a reduction of ten-dimensional
supersymmetric Yang–Mills theory to __\( (1+0) \)__-dimensional space (all
spatial dimensions are reduced to a point). I was very impressed by
the BFSS conjecture; it gave me a mathematical framework to work with
M-theory.

It is clear today that in some sense the BFSS conjecture is correct; however, the role of the BFSS model is not quite clear. I believe that the complete M-theory can be extracted from the BFSS model; it seems that my opinion is much more optimistic than the viewpoint of the creators of this model. It is possible, however, that the path from M(atrix) model to M-theory is very complicated and that there exist other, better approaches. (E. Witten and J. Schwarz independently expressed in conversations with me the same opinion: that it is clear now that M(atrix) model is correct, but it is not clear whether it is useful.)

When I started thinking about the BFSS model I realized that there exists another attractive matrix model that can be used to understand string/M-theory: the reduction of ten-dimensional supersymmetric Yang–Mills theory to a point. I found very soon that I was not the first scholar that came to this idea: this model was suggested by the Japanese physicists Ishibashi, Kawai, Kitazawa and Tsuchiya; it is now known under the name IKKT model.

The BFSS model can be obtained from IKKT by means of compactification on a circle and Wick rotation. This remark led me to consideration of the general problem of compactification of the matrix model on a circle and, more generally, on a torus. This problem was considered already in the original BFSS paper. However, I found that the logic used in this paper and other papers leads to more general compactifications; they can be described in terms of Connes’ noncommutative geometry. More precisely, these compactifications lead to noncommutative tori, which were thoroughly analyzed by A. Connes, M. Rieffel and other mathematicians. In a joint paper with A. Connes and M. Douglas [87] we have shown how noncommutative tori arise very naturally in matrix models and what is the physical meaning of compactifications on noncommutative tori. This paper paved the way to the application of beautiful theorems of noncommutative geometry proven by A. Connes and his followers to string/M-theory. Among numerous papers with applications of noncommutative geometry to physics, I would like to single out the very influential Seiberg–Witten paper [e5] that gave, in particular, a clear explanation of the appearance of noncommutative geometry in the framework of string theory, as well as an analysis of the relation between gauge fields on noncommutative spaces to ordinary gauge fields and the proof of background independence of noncommutative Born–Infeld theory. (A little bit earlier background independence was discovered in my paper with B. Pioline [95] at the level of the energy spectrum.)

I also have written many papers developing the ideas of
[87].
In
[93],
[89] it was shown that Morita equivalence of noncommutative tori is
related to physical duality of gauge theories on these tori and to
T-duality in string theory. I am convinced that the mathematical
notion of Morita equivalence of associative algebras and its
generalization for differential associative algebras should be
regarded as the mathematical foundation of dualities in
string/M-theory. Among other papers I would like to mention the paper
with N. Nekrasov
[90],
where we introduced and studied noncommutative
instantons, and papers with A. Konechny, devoted mostly to the study of
BPS-states on noncommutative tori and their orbifolds
(see
[96],
[94],
[101],
[100]).
Together with
A. Konechny I have written a review paper “Introduction to matrix
theory and noncommutative geometry”, published in *Physics
Reports*
(see
[107]).

More recent projects (together with M. Movshev) were devoted to
maximally supersymmetric gauge theories and their deformations. These
theories include ten-dimensional
super Yang–Mills theory and its
dimensional reductions, in particular, BFSS and IKKT matrix models.
Our methods are closely related to noncommutative supergeometry, more
precisely, to the theory of __\( A_{\infty} \)__-algebras ( __\( = \)__ formal
noncommutative Q-manifolds).

Our calculations led us to the study of homology groups of various super Lie algebras. We have written several papers (in collaboration with Renjun Xu and A. Mikhailov) devoted to the calculation of homology of the algebra of supersymmetries, of super Poincaré algebra and of other homology groups that appear in quantum field theory and string theory, in particular, in pure spinor formalism.

Together with Jia-Ming (Frank) Liou I have studied the homology groups of moduli spaces of algebraic curves and their relation to the homology of infinite-dimensional Grassmannian [124], [126], [132], [139]. Our papers were inspired by the old idea that nonperturbative string theory should be formulated in terms of infinite-dimensional Grassmannian (see [88] and references therein). They did not find yet any applications to physics, but they contain some interesting mathematical results.

I have studied two-dimensional topological quantum field theories
coupled to gravity ( string theories), in particular,
theories of this kind defined on bordered surfaces. I have applied
methods of arithmetic geometry in this study. Together with M.
Kontsevich and V. Vologodsky I have analyzed integrality of instanton
numbers using
__\( = \)__topological__\( p \)__-adic methods
[115].
This paper and subsequent papers written
in collaboration with V. Vologodsky
[120]
[117]
were based on the application of
Frobenius map on __\( p \)__-adic cohomology; in a paper written in
collaboration with I. Shapiro a construction of this map based on the
ideas of supergeometry was given
[118].
I am convinced that very soon the methods of number theory will play
an important role in quantum field theory and string theory.
In the
paper entitled “Twisted de Rham cohomology, homological definition of
integral and physics over a ring”
[119]
(with I. Shapiro) we argue that
together with conventional physical theories based on real or complex
numbers one can consider theories where the role of numbers is played
by elements of an arbitrary ring. The study of such unconventional
theories can be used to obtain interesting results in conventional
framework, but it is possible that these theories have direct physical
meaning.

I continued the work on applications of arithmetic geometry together with V. Vologodsky and J. Walcher [133].

It was suggested many years ago that one can consider a pair of
ordinary differential operators __\( P, Q \)__ obeying the string equation
__\( [P, Q] = \hbar \)__ as a quantum curve. (This suggestion is based on the
fact that commuting differential operators specify a classical
algebraic curve.) I
studied the string equation
in the 90s
using
the techniques of infinite-dimensional Grassmannian. More recently I
have obtained new results about string equation relating it to new
approaches to quantum curves in papers by
Eynard–Orantin,
Gukov–Sułkowski,
Mulase, etc. I have introduced a notion of discrete
quantum curve as a solution to the equation __\( KL = \lambda LK \)__ where
__\( K, L \)__ are difference operators and proved similar results for this
notion. Quantum curves are related to many problems of physics,
together with M. Luu I have analyzed their relation to __\( (p,
q) \)__-minimal models
coupled to 2D-gravity (see
[136]).

Together with A. Mikhailov [138], [127] I analyzed families of gauge conditions in BV formalism and the construction of string amplitudes in this setting.

My recent papers were devoted to scattering amplitudes.

Deep
connections of scattering amplitudes to positive Grassmannian and
cluster algebras were found in
[e6].
The positive
Grassmannian is closely related to the quantum Grassmannian. In the
paper written in collaboration with M. Movshev
[140]
we have shown using the
theory of quantum cluster algebras that the construction of
differential forms representing scattering amplitudes and defined on
cells of the positive Grassmannian can be “__\( q \)__-deformed” (there are
some objects on quantum Grassmannian generalizing these forms). We
conjecture that the
__\( q \)__-deformed amplitudes have physical meaning in
the framework of gauge theories on noncommutative spaces.

The paper [e6] and follow-up papers are based on the idea that quantum theory can be formulated not in terms of fields, but in terms of particles. These papers are dealing with four-dimensional theories; I am working now on generalization of this idea to any dimension.

Recently I have introduced the notion of inclusive scattering matrix
closely related to the notion of inclusive cross-section. This notion
came from the formulation of quantum theory in terms of
__\( L \)__-functionals given in my first physical paper published in 1967
[15].
To a density matrix __\( K \)__ in the representation of canonical commutation
relations (CCR) or canonical anticommutation relations (CAR) one can
assign a
functional
__\[
L(\alpha^{\ast} , \alpha) = \operatorname{Tr} e^{-\alpha a^{+}}
e^{\alpha^{\ast}a} K,
\]__
where
__\( \alpha a^{+} = \sum \alpha_k a^{+}_k \)__,
__\( \alpha^{\ast} a = \sum \alpha^{\ast}_k a_k \)__,
__\( a^{+}_k \)__, __\( a_k \)__ obey
CCR or CAR. This functional (called __\( L \)__-functional) can be considered as a positive functional
on the Weyl algebra or Clifford algebra. I have shown that the quantum
theory can be formulated in terms of __\( L \)__-functionals; this allows us
to avoid problems due to the existence of inequivalent representations
of CCR and CAR in the case of infinite numbers of degrees of freedom.

My
students Yu. Tyupkin, V. Fateev and I
wrote several papers
applying the formulation of quantum theory in terms of __\( L \)__-functionals
to various problems
(see, for example,
[e1],
[22]),
but we abandoned this direction in favor of
topological methods in physical problems.
In the
80s Umezawa rediscovered this formulation under the name of TFD
(thermofield dynamics) and applied it to equilibrium and
nonequilibrium statistical physics.

The inclusive scattering matrix for elementary excitations of any translation-invariant stationary state can be defined also in the framework of algebraic quantum field theory [141] and expressed in terms of generalized Green functions on shell [141], [143]. This fact should be important in quantum statistical physics. (The standard definition of scattering matrix works only for elementary excitations of ground state.) It can be used also to relate quantum scattering to the scattering of solitons.

Recently I have suggested a formulation of quantum theory where the convex set of states is taken as a starting point [144], [145]. I have shown that in this geometric approach one can prove decoherence and derive the formulas for probabilities from the first principles. The notion of inclusive scattering matrix appears very naturally in this approach [147]. The geometric approach is closely related to the formulation of quantum theory in terms of Jordan algebras [146].