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Moscow Semina r o n Mathematical Physics , I I
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American Mathematica l Societ y
TRANSLATIONS Series 2 • Volum e 22 1
Advances in the Mathematical Sciences—6 0 (Formerly Advances in Soviet Mathematics)
Moscow Semina r o n Mathematical Physics , I I
Yu. Nereti n M. Olshanetsk y A. Rosl y Editors
American Mathematical Societ y k Providence , Rhode Island
&NDED
http://dx.doi.org/10.1090/trans2/221
ADVANCES I N TH E MATHEMATICA L SCIENCE S EDITORIAL COMMITTE E
V. I . ARNOL D S. G . GINDIKI N V. P . MASLO V
2000 Mathematics Subject Classification. P r i m a r y 00B25 ; Secondar y 37-06 .
Library o f Congres s Car d Numbe r 91-64074 1 ISBN 0-8218-4371- 0
ISSN 0065-929 0
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Contents
Mathematical Physic s Semina r i n ITE P Yu. NERETIN , M . OLSHANETSKY , an d A . ROSL Y vii
Hecke-Tyurin Parametrizatio n o f the Hitchi n an d KZ B System s B. ENRIQUE Z an d V . RUBTSO V 1
Combinatorics an d Geometr y o f Higher Leve l Weyl Module s B. FEIGIN , A . N . KIRILLOV , an d S . LOKTE V 3 3
Cosh-Gordon Equatio n an d Quasi-Fuchsia n Group s V. V . FOC K 4 9
On a Clas s o f Representation s o f Quantu m Group s an d It s Application s A. GERASIMOV , S . KHARCHEV , D . LEBEDEV , an d S . OBLEZI N 5 9
On Syzygie s o f Highes t Weigh t Orbit s A. L . GORODENTSEV , A . S . KHOROSHKIN , an d A . N . RUDAKO V 7 9
On Fines t an d Modula r t-Stabilitie s A. L . GORODENTSE V an d S . A . KULESHO V 12 1
Hochschild Homolog y an d Gabber' s Theore m D. KALEDI N 14 7
Method o f Projections o f Drinfeld Current s SERGEY KHOROSHKI N an d STANISLA V PAKULIA K 15 7
On Rationa l an d Ellipti c Form s o f Painleve V I Equatio n A. LEVI N an d A . ZOTO V 17 3
Determinantal Poin t Processe s an d Fermioni c Foc k Spac e YURI A . NERETI N 18 5
On Adeli c Mode l o f Boson Foc k Spac e YURI A . NERETI N 19 3
Hypercomplex Manifold s wit h Trivia l Canonica l Bundl e an d Thei r Holonom y MISHA VERBITSK Y 20 3
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Mathematical Physic s Semina r i n ITE P
This volum e contain s proceeding s o f seminar s o f th e Mathematica l Physic s Group o f the Institut e fo r Theoretica l an d Experimenta l Physic s (ITEP) . Pro m it s initiation b y Alexe i Morozo v i n 1990 , th e Mathematica l Physic s Grou p continue s to b e a n origina l researc h cente r an d a n importan t plac e o f collaboratio n betwee n physicists and mathematicians . Thi s volume is the second such collection publishe d by th e America n Mathematica l Society , th e firs t bein g Moscow Seminar in Math-ematical Physics (A . Yu . Morozo v an d M . A . Olshanetsky , editors) , Amer . Math . Soc, Providence , RI , 1999 .
Papers o f thi s collection s ar e devote d t o variou s mathematica l topic s tha t strongly influence d moder n physics . Thes e topic s ar e quit e diverse . W e hope , however, tha t th e choic e o f topic s show s wh y th e ITE P semina r o n mathemati -cal physic s ha s bee n know n t o an d wa s attende d b y a larg e grou p o f participants . We ar e gla d t o us e thi s opportunit y t o cordiall y than k al l who participate d i n ou r seminar.
Let u s giv e a brie f descriptio n o f the paper s i n thi s volume . Benjamin Enrique s an d Vladimi r Rubtso v construc t a n explici t parametriza -
tion fo r th e modul i spac e o f 2-dimensiona l vecto r bundle s ove r a n algebrai c curv e and appl y i t t o th e Hitchi n an d Knizhnik-Zamolodchikov-Bernar d system s an d Beilinson-Drinfeld operators .
The subjec t o f the wor k o f Bori s Feigin , Anato l Kirillov , an d Serge i Lokte v i s "Weyl modules " ove r th e algebra s o f polynomial current s i n tw o variables .
The pape r b y Vladimir Foc k shows how many aspect s o f 2-dimensional geome -try com e t o one' s min d b y lookin g a t th e cosh-Gordo n equatio n (whe n on e looks) .
The wor k o f Anto n Gerasimov , Serge i Kharchev , Dmitr y Lebedev , an d Serge i Oblezin i s devote d t o a realizatio n o f on e clas s o f representation s o f Li e algebra s and quantu m group s base d o n Gelfand-Tsetli n schemes .
Alexei Gorodentsev , Anto n Khoroshkin , an d Alexe i Rudako v investigat e syzy -gies of flag manifolds fro m th e cohomology of certain infinite-dimensiona l Li e super -algebras. Thei r approac h allow s the m t o comput e th e syzygie s explicitl y i n som e special cases .
The subject o f the paper by Alexei Gorodentsev an d Serge i Kuleshov i s Harder-Narasimhan filt rat ions on categories .
The pape r b y Dmitr y Kaledi n i s aime d t o clarif y th e Gabbe r theore m o n th e involutivity o f the singula r suppor t o f D-modules .
The wor k o f Sergey Khoroshki n an d Stanisla v Pakulia k i s devoted t o the Bore l subalgebras i n th e quantu m affin e algebra s an d th e metho d o f Drinfeld projectors . It i s also a surve y o f the subject .
vii
viii MATHEMATICA L PHYSIC S SEMINA R I N I T E P
The pape r b y Andre y Levi n an d Andre i Zoto v i s devote d t o variou s form s o f the Painlev e V I equation .
In th e first not e o f Yuri Neretin , th e autho r construct s a canonica l embeddin g of th e spac e L 2 wit h respec t t o a determinanta l poin t stochasti c proces s (a s th e Dyson sine-process ) t o th e fermioni c Foc k space .
In th e secon d note , Nereti n discusse s a realizatio n o f th e spac e o f function s o f a rea l variabl e a s a spac e o f distribution s o n th e adeli c produc t o f al l th e p-adi c fields.
Misha Verbitsk y consider s a certain clas s o f hyper complex manifold s an d give s a criterio n tha t th e holonom y o f a canonica l torsion-fre e connectio n i s contained i n the quaternioni c SL .
We are grateful t o the American Mathematica l Societ y and especiall y to Simo n Gindikin fo r a n opportunity t o publish thi s collection. W e also thank Andre y Levi n and Andre i Marshakov , wh o helpe d u s i n ou r wor k o n thi s volume . Th e wor k i s supported b y th e Federa l Agenc y fo r Atomi c Energ y o f Russia , an d b y th e grant s SciSch-8004.2006.2, SciSch-8065.2006.2 .
Yu. Neretin, 1 M . Olshanetsky , an d A . Rosl y
Also affiliate d wit h Vienn a University , gran t FWF , projec t 19064 .
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Titles i n Thi s Subserie s
60 Yu . Neret in , M . Olshanetsky , an d A . Rosly , Editors , Semina r o n Mathematica l physics, I I (TRANS2/221 )
59 M . S . Birma n an d N . N . Uraltseva , Editors , Nonlinea r equation s an d spectra l theor y
(TRANS2/220)
58 V . Kaimanovic h an d A . Lodkin , Editors , Representatio n theory , dynamica l systems , and asymptoti c combinatoric s (TRANS2/217 )
57 M . V . Karasev , Editor , Quantu m algebra s an d Poisso n geometr y i n mathematica l
physics (TRANS2/216 )
56 Ernes t Vinberg , Editor , Li e group s an d invarian t theor y (TRANS2/213 ) 55 V . M . Buchstabe r an d I . M . Krichever , Editors , Geometry , topology , an d
mathematical physic s (S . P . Novikov' s seminar : 2002-2003 ) (TRANS2/212 )
54 S . G . Gindikin , Editor , Li e group s an d symmetri c spaces . I n memor y o f F . I . Karpelevich (TRANS2/210 )
53 M . V . Karasev , Editor , Asymptoti c method s fo r wav e an d quantu m problem s
(TRANS2/208)
52 Yu . M . Suhov , Editor , Analyti c method s i n applie d probabilit y (TRANS2/207 ) 51 M . A . Shubi n an d M . S . Agranovich , Editors , Partia l differentia l equation s (Mar k
Vishik's seminar ) (TRANS2/206 )
50 V . Turae v an d A . Vershik , Editors , Topology , ergodi c theory , rea l algebrai c geometry .
Rokhlin's memoria l (TRANS2/202 )
49 Michae l Semenov-Tian-Shansky , Editor , L . D . Faddeev' s semina r o n mathematica l physics (TRANS2/201 )
48 L . Lerman , G . Polotovskit , an d L . Shilnikov , Editors , Method s o f qualitativ e theor y
of differentia l equation s an d relate d topic s (TRANS2/200 )
47 R . A . Minlos , Seny a Shlosman , an d Yu . M . Suhov , Editors , O n Dobrushin' s way . From probabilit y theor y t o statistica l physic s (TRANS2/198 )
46 Vladimi r Arnold , M a x i m Kontsevich , an d A n t o n Zorich , Editors , Pseudoperiodi c
topology (TRANS2/197 )
45 Ya . Eliashberg , D . Fuchs , T . Rat iu , an d A . Weinste in , Editors , Norther n
California symplecti c geometr y semina r (TRANS2/196 )
44 Alexande r Astashkevic h an d Serg e Tabachnikov , Editors , Differentia l Topology , Infinite-Dimensional Li e Algebras , an d Application s (D . B . Fuchs ' 60t h Anniversar y Collection) (TRANS2/194 )
43 A . Yu . Morozo v an d M . A . Olshanetsky , Editors , Mosco w Semina r i n Mathematica l Physics (TRANS2/191 )
42 S . Tabachnikov , Editor , Differentia l an d Symplecti c Topolog y o f Knot s an d Curve s
(TRANS2/190)
41 V . Buslaev , M . Solomyak , an d D . Yafaev , Editors , Differentia l Operator s an d Spectral Theor y (M . Sh . Birman' s 70t h anniversar y collection ) (TRANS2/189 )
40 M . V . Karasev , Editor , Coheren t Transform , Quantization , an d Poisso n Geometr y
(TRANS2/187)
39 A . Khovanskn , A . Varchenko , an d V . Vassiliev , Editors , Geometr y o f Differentia l Equations (TRANS2/186 )
38 B . Feigi n an d V . Vassiliev , Editors , Topic s i n Quantu m Group s an d Finite-Typ e Invariants (Mathematic s a t th e Independen t Universit y o f Moscow ) (TRANS2/185 )
37 Pete r Kuchmen t an d Vladimi r Lin , Editors , Voronez h Winte r Mathematica l School s (Dedicated t o Seli m Krein ) (TRANS2/184 )
36 V . E . Zakharov , Editor , Nonlinea r Wave s an d Wea k Turbulenc e (TRANS2/182 )
35 G . I . Olshanski , Editor , Kirillov' s Semina r o n Representatio n Theor y (TRANS2/181 )
TITLES I N THI S SUBSERIE S
34 A . Khovanskii , A . Varchenko , an d V . Vassiliev , Editors , Topic s i n Singularit y
Theory (TRANS2/180 )
33 V . M . Buchstabe r an d S . P . Novikov , Editors , Solitons , Geometry , an d Topology : O n
the Crossroa d (TRANS2/179 )
32 R . L . Dobrushin , R . A . Minlos , M . A . Shubin , an d A . M . Vershik , Editors , Topics i n Statistica l an d Theoretica l Physic s (F . A . Berezi n Memoria l Volume )
(TRANS2/177)
31 R . L . Dobrushin , R . A . Minlos , M . A . Shubin , an d A . M . Vershik , Editors ,
Contemporary Mathematica l Physic s (F . A . Berezi n Memoria l Volume ) (TRANS2/175 )
30 A . A . Bolibruch , A . S . Merkur'ev , an d N . Yu . Net sve taev , Editors , Mathematic s
in St . Petersbur g (TRANS2/174 )
29 V . Kharlamov , A . Korchagin , G . Polotovskii , an d O . Viro , Editors , Topolog y o f
Real Algebrai c Varietie s an d Relate d Topic s (TRANS2/173 )
28 L . A . Bunimovich , B . M . Gurevich , an d Ya . B . Pesin , Editors , Sinai' s Mosco w
Seminar o n Dynamica l System s (TRANS2/171 )
27 S . P . Novikov , Editor , Topic s i n Topolog y an d Mathematica l Physic s (TRANS2/170 )
26 S . G . Gindiki n an d E . B . Vinberg , Editors , Li e Group s an d Li e Algebras : E . B .
Dynkin's Semina r (TRANS2/169 )
25 V . V . Kozlov , Editor , Dynamica l System s i n Classica l Mechanic s (TRANS2/168 )
24 V . V . Lychagin , Editor , Th e Interpla y betwee n Differentia l Geometr y an d Differentia l
Equations (TRANS2/167 )
23 Yu . Ilyashenk o an d S . Yakovenko , Editors , Concernin g th e Hilber t 16t h Proble m
(TRANS2/165)
22 N . N . Uraltseva , Editor , Nonlinea r Evolutio n Equation s (TRANS2/164 )
Published Earlie r a s Advances i n Sovie t Mathematic s 21 V . I . Arnold , Editor , Singularitie s an d bifurcations , 199 4
20 R . L . Dobrushin , Editor , Probabilit y contribution s t o statistica l mechanics , 199 4
19 V . A . Marchenko , Editor , Spectra l operato r theor y an d relate d topics , 199 4
18 Ole g Viro , Editor , Topolog y o f manifold s an d varieties , 199 4
17 Dmi tr y Fuchs , Editor , Unconventiona l Li e algebras , 199 3
16 Serge i Gelfan d an d Simo n Gindikin , Editors , I . M . Gelfan d seminar , Part s 1 and 2 ,
1993
15 A . T . Fomenko , Editor , Minima l surfaces , 199 3
14 Yu . S . Il'yashenko , Editor , Nonlinea r Stoke s phenomena , 199 2
13 V . P . Maslo v an d S . N . SamborskiT , Editors , Idempoten t analysis , 199 2
12 R . Z . Khasminskii , Editor , Topic s i n nonparametri c estimation , 199 2
11 B . Ya . Levin , Editor , Entir e an d subharmoni c functions , 199 2
10 A . V . Babi n an d M . I . Vishik , Editors , Propertie s o f globa l attractor s o f partia l
differential equations , 199 2
9 A . M . Vershik , Editor , Representatio n theor y an d dynamica l systems , 199 2
8 E . B . Vinberg , Editor , Li e groups , thei r discret e subgroups , an d invarian t theory , 199 2
7 M . Sh . Birman , Editor , Estimate s an d asymptotic s fo r discret e spectr a o f integra l an d
differential equations , 199 1
6 A . T . Fomenko , Editor , Topologica l classificatio n o f integrabl e systems , 199 1
For a complet e lis t o f t i t le s i n thi s series , visi t t h e AMS Bookstor e a t w w w . a m s . o r g / b o o k s t o r e / .
