Papers by Samson Shatashvili
WORLD SCIENTIFIC eBooks, May 21, 2018
About 30 years ago, in a joint work with L. Faddeev we introduced a geometric action on coadjoint... more About 30 years ago, in a joint work with L. Faddeev we introduced a geometric action on coadjoint orbits. This action, in particular, gives rise to a path integral formula for characters of the corresponding group G. In this paper, we revisit this topic and observe that the geometric action is a 1-cocycle for the loop group LG. In the case of G being a central extension, we construct Wess-Zumino (WZ) type terms and show that the cocycle property of the geometric action gives rise to a Polyakov-Wiegmann (PW) formula with boundary term given by the 2-cocycle which defines the central extension. In particular, we obtain a PW type formula for the Polyakov's gravitational WZ action. After quantization, this formula leads to an interesting bulk-boundary decoupling phenomenon previously observed in the WZW model. We explain that this decoupling is a general feature of the Wess-Zumino terms obtained from geometric actions, and that in this case the path integral is expressed in terms of the 2-cocycle which defines the central extension.
Springer eBooks, 2021
Following Simpson we consider the integrable system structure on the moduli spaces of Higgs bundl... more Following Simpson we consider the integrable system structure on the moduli spaces of Higgs bundles on a compact Kähler manifold X. We propose a description of the corresponding spectral cover of X as the fiberwise projective dual to a hypersurface in the projectivization P(T X ⊕ O X) of the tangent bundle T X to X. The defining equation of the hypersurface dual to the Simpson spectral cover is explicitly constructed in terms of the Higgs fields.
American Mathematical Society translations, Aug 1, 2000
We propose a few tests of Seiberg-Witten solutions of N = 2 supersymmetric gauge theories by the ... more We propose a few tests of Seiberg-Witten solutions of N = 2 supersymmetric gauge theories by the instanton calculus in twisted gauge theories. We reexamine the low-energy effective abelian theory in the presence of sources and present the formalism which makes duality transformations transparent and easily fixes all the contact terms in a broad class of theories. We also discuss ADHM integration and its relevance to the stated problems.
arXiv (Cornell University), Mar 17, 2016
Following Simpson we consider the integrable system structure on the moduli spaces of Higgs bundl... more Following Simpson we consider the integrable system structure on the moduli spaces of Higgs bundles on a compact Kähler manifold X. We propose a description of the corresponding spectral cover of X as the fiberwise projective dual to a hypersurface in the projectivization P(T X ⊕ O X) of the tangent bundle T X to X. The defining equation of the hypersurface dual to the Simpson spectral cover is explicitly constructed in terms of the Higgs fields.
Russian Mathematical Surveys, Dec 31, 2017
This survey was written by students of L. D. Faddeev under the editorship of L.
Journal of High Energy Physics, Jun 10, 1999
Various exact two-dimensional conformal field theories with AdS 2d+1 target space are constructed... more Various exact two-dimensional conformal field theories with AdS 2d+1 target space are constructed. These models can be solved using bosonization techniques. Some examples are presented that can be used in building perturbative superstring theories with AdS backgrounds, including AdS 5 .
Nuclear Physics B, Jun 1, 1998
We consider M theory 5-branes with compact transverse dimensions. In certain limits the theory on... more We consider M theory 5-branes with compact transverse dimensions. In certain limits the theory on the 5-brane decouples and defines "little string theories" in 5 + 1 dimensions. We show that the familiar structure of IIA/IIB, M, F − theory in 10, 11, 12 dimensions respectively has a perfect parallel in a theory of strings and membranes in 6, 7, 8 dimensions. We call these theories a/b, m, f theories. They have a coupling constant but no gravity. This construction clarifies some mysteries in F-theory and leads to several speculations about the phase structure of M theory.
Springer eBooks, 1999
We propose a few tests of Seiberg-Witten solutions of N = 2 supersymmetric gauge theories by the ... more We propose a few tests of Seiberg-Witten solutions of N = 2 supersymmetric gauge theories by the instanton calculus in twisted gauge theories. We reexamine the low-energy effective abelian theory in the presence of sources and present the formalism which makes duality transformations transparent and easily fixes all the contact terms in a broad class of theories. We also discuss ADHM integration and its relevance to the stated problems.
WORLD SCIENTIFIC eBooks, Feb 14, 2016
The functional integral for the quantization of the coadjoint orbits of the unitary and orthogona... more The functional integral for the quantization of the coadjoint orbits of the unitary and orthogonal groups is given by means of an explicit construction of the corresponding~Darbouiø> variables.
Springer eBooks, 1989
Investigation of 2d conformal field theory in terms of geometric quantization is given. We quanti... more Investigation of 2d conformal field theory in terms of geometric quantization is given. We quantize the so-called model space of the compact Lie group, Virasoro group and Kac-Moody group. In particular, we give a geometrical interpretation of the Virasoro discrete series and explain that this type of geometric quantization reproduces the chiral part of CFT (minimal models, 2d-gravity, WZNW theory). In the appendix we discuss the relation between classical (constant) r-matrices and this geometrical approach.
arXiv (Cornell University), Jul 18, 2023
This is a summary of my lecture at Igor Frenkel's 70th birthday conference. I give a brief review... more This is a summary of my lecture at Igor Frenkel's 70th birthday conference. I give a brief review of my almost forty years of scientific interactions with Igor. I focus here on three topics of joint interests: 2-cocycles, coadjoint orbits and W ZW 4. At the end of each topic I comment on new developments, if any.
Reviews in Mathematical Physics, Jul 1, 2018
About 30 years ago, in a joint work with L. Faddeev we introduced a geometric action on coadjoint... more About 30 years ago, in a joint work with L. Faddeev we introduced a geometric action on coadjoint orbits. This action, in particular, gives rise to a path integral formula for characters of the corresponding group G. In this paper, we revisit this topic and observe that the geometric action is a 1-cocycle for the loop group LG. In the case of G being a central extension, we construct Wess-Zumino (WZ) type terms and show that the cocycle property of the geometric action gives rise to a Polyakov-Wiegmann (PW) formula with boundary term given by the 2-cocycle which defines the central extension. In particular, we obtain a PW type formula for the Polyakov's gravitational WZ action. After quantization, this formula leads to an interesting bulk-boundary decoupling phenomenon previously observed in the WZW model. We explain that this decoupling is a general feature of the Wess-Zumino terms obtained from geometric actions, and that in this case the path integral is expressed in terms of the 2-cocycle which defines the central extension.
Theoretical and Mathematical Physics, Nov 1, 1987
A simple method is proposed for calculating the anomalous commutator in the theory of interacting... more A simple method is proposed for calculating the anomalous commutator in the theory of interacting chiral fermions and Yang-Mills field.
Annales Henri Poincaré, May 20, 2023
Following Nag-Sullivan, we study the representation of the group Diff + (S 1) of diffeomorphisms ... more Following Nag-Sullivan, we study the representation of the group Diff + (S 1) of diffeomorphisms of the circle on the Hilbert space of holomorphic functions. Conformal welding provides triangular decompositions for the corresponding symplectic transformations. We apply Berezin formalism and lift this decomposition to operators acting on the Fock space. This lift provides quantization of conformal welding, gives a new representative of the Bott-Virasoso cocycle class, and leads to a surprising identity for the Takhtajan-Teo energy functional on Diff + (S 1).
Uspekhi Matematicheskikh Nauk, 2017
Russian Mathematical Surveys, 2017
In February 2017 Ludwig Dmitrievich Faddeev, one of the greatest modern mathematicians and theore... more In February 2017 Ludwig Dmitrievich Faddeev, one of the greatest modern mathematicians and theoretical physicists, passed away. Faddeev's death is a heavy loss to Russian and international science. His work has largely reshaped modern mathematical physics, to the development of which he devoted his life. Many of his results have become classical, belonging to the gold reserve of pure mathematics, and at the same time playing a decisive role in the development of the most important areas of theoretical physics. He also made enormous contributions to filling the decades-long gap between mathematics and physics and forming several generations of Russian and international theorists. For more than 60 years, Faddeev's scientific life was associated with the Leningrad (later St. Petersburg) Branch of the Steklov Mathematical Institute (LOMI for short, and later POMI), where he went from a postgraduate student to the head of a laboratory he himself organized, and then to the director of the institute. Faddeev was as proud of the creation of the Laboratory of Mathematical Problems in Physics, where he brought together students who had grown under his guidance, as of his purely scientific results. In 1976,
Les Houches - Ecole d’Ete de Physique Theorique
I review the properties of superstrings on the manifolds of exceptional holonomy.
Russian Mathematical Surveys, 2014
One of the leading contemporary mathematicians and theoretical physicists, Academician Ludvig Dmi... more One of the leading contemporary mathematicians and theoretical physicists, Academician Ludvig Dmitrievich Faddeev, turned 80 in March 2014. It is a difficult task to describe all his results in a short paper. Many of them are now part of the working tools of mathematics and theoretical physics. One of the most important discoveries of the great mathematicians of the past, from Euler and Gauss to Hilbert and Weyl and our contemporaries, was the unity of the whole of mathematics: algebra, geometry, analysis, number theory, and so on, without exceptions. Faddeev's work has added to this list several areas of modern theoretical physics, on a par with the others. In the course of a single generation, mathematical physics has completely changed its appearance, and Faddeev has invariably been one of the leading figures in this amazing development. While as recently as the 1960s there were grounds for the familiar joke about the marriage betweem mathematics and physics that took place in the 18th and 19th centuries ending in divorce, in a few years the whole situation changed completely, and one could speak about the reverse influence of physics on mathematics (an influence whose importance Faddeev never tires of stressing). It suffices to point out, for instance, that quantum field theory has become one of the most important recent discoveries in topology. (We note that the main tools used in the topological applications of quantum field theory are based on Faddeev's famous paper on the quantization of gauge fields.) L. D. Faddeev was born on 23 March 1934, in the family of eminent mathematicians Dmitrii Konstantinovich Faddeev and Vera Nikolaevna Faddeeva. Dmitrii Faddeev was one of the leading Soviet experts in algebra, who worked in the Mathematical Institute of the Academy of Sciences from the time it was founded (in 1934) and then became one of the first researchers in the Leningrad Department of the
Успехи математических наук, 2014
В марте 2014 г. исполнилось 80 лет одному из крупнейших математиков и физиков-теоретиков современ... more В марте 2014 г. исполнилось 80 лет одному из крупнейших математиков и физиков-теоретиков современности академику Людвигу Дмитриевичу Фаддееву. В короткой статье непросто описать все принадлежащие ему результаты; многие из них вошли в рабочий аппарат математики и теоретической физики. Одно из важнейших открытий великих математиков прошлогоот Эйлера и Гаусса до Гильберта, Германа Вейля и наших современников-состоит в единстве всей математики-алгебры, геометрии, анализа, теории чисел и т. д. без всяких исключений. Работы Л. Д. Фаддеева добавили к этому списку, в качестве его равноправной составной части, многие разделы современной теоретической физики. На протяжении жизни одного поколения математическая физика полностью изменила свое лицо, и Л. Д. Фаддеев неизменно был одним из ведущих участников этого увлекательного процесса. Если еще в 1960-е годы имела некоторое основание известная шутка о том, что брак математики и физики, заключенный в XVIII и XIX вв., окончился разводом, то спустя всего несколько лет положение полностью изменилось, причем можно говорить об обратном влиянии теоретической физики на математику (влиянии, важность которого Людвиг Дмитриевич всегда любит подчеркивать). Достаточно упомянуть, например, что одним из важнейших открытий последнего времени в топологии стала квантовая теория поля. (При этом основной аппарат, связанный с приложениями квантовой теории поля в топологии, основан на знаменитой работе Л. Д. Фаддеева о квантовании калибровочных полей.) Л. Д. Фаддеев родился 23 марта 1934 г. в семье замечательных математиков Дмитрия Константиновича и Веры Николаевны Фаддеевых. Д. К. Фаддеев, один из ведущих советских алгебраистов, был сотрудником МИАН со времени его основания (1934), а затем и одним из первых сотрудников Ленинградского отделения Математического института им. В. А. Стеклова (ЛОМИ) с момента его воссоздания в 1940 г. (после пятилетнего перерыва, вызванного переводом Академии в Москву). В. Н. Фаддеева заведовала в ЛОМИ созданной ею лабораторией вычислительной математики. Таким образом, жизнь Людвига оказалась уже с раннего детства связанной с Математическим институтом. Однако, поступая в Ленинградский университет, он выбрал
Physics Letters B, 1986
The explicit regularization of the Weyl fermion current leading to the anomalous commutation rela... more The explicit regularization of the Weyl fermion current leading to the anomalous commutation relation of the Gauss law constraint is established. This requires the modification of the canonical quantization of the corresponding gauge model. The modified form of functional integral quantization for this model is proposed.
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Papers by Samson Shatashvili