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Geometry, Topology , an d Mathematical Physic s S. P . Novikov' s Seminar : 2006-2007
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American Mathematica l Societ y
TRANSLATIONS Series 2 • Volum e 22 4
Advances in the Mathematical Sciences—61 {Formerly Advances in Soviet Mathematics)
Geometry, Topology , an d Mathematical Physic s S. P . Novikov' s Seminar : 2006-2007
V. M. Buchstabe r I. M. Kricheve r Editors
American Mathematica l Societ y Providence, Rhode Islan d
http://dx.doi.org/10.1090/trans2/224
A D V A N C E S I N T H E M A T H E M A T I C A L S C I E N C E S E D I T O R I A L C O M M I T T E 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 .
Library o f Congres s Car d Numbe r 91-64074 1 ISBN-13: 978-0-8218-4674- 2
ISSN 0065-929 0
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Contents
Preface vi i
Hurwitz Number s fo r Regula r Covering s o f Surface s b y Seame d Surface s an d Cardy-Probenius Algebra s o f Finit e Group s
A. V . ALEXEEVSK I an d S . M . NATANZO N 1
Equivariant Comple x Structure s o n Homogeneous Space s and Thei r Cobordis m Classes
VICTOR M . BUCHSTABE R an d SVJETLAN A TERZI C 2 7
On Universalit y o f Critica l Behaviou r i n Hamiltonia n PDE s BORIS DUBROVI N 5 9
On th e geometr y o f V-system s M. FEIGI N an d A . P . VESELO V 11 1
Spectral Conservatio n Law s fo r Periodi c Nonlinea r Equation s o f the Melniko v Type
P. G . GRINEVIC H an d I . A . TAIMANO V 12 5
An Equivarian t Versio n o f the Monodrom y Zet a Functio n S. M . GUSEIN-ZADE , I . LUENGO , an d A . MELLE-HERNANDE Z 13 9
Symplectic Aoo -Algebras an d Strin g Topolog y Operation s ALASTAIR HAMILTO N an d ANDRE Y LAZARE V 14 7
Differential Form s an d Od d Symplecti c Geometr y HOVHANNES M. KHUDAVERDIAN and THEODORE TH. VORONOV 159
Abelian Solution s o f the K P Equatio n I. KRICHEVE R an d T . SHIOT A 17 3
Deformations o f the Whitha m System s i n th e Almos t Linea r Cas e A. YA . MALTSE V 19 3
Frobenius Manifold s a s a Specia l Clas s o f Submanifold s i n Pseudo-Euclidea n Spaces
O. I . MOKHO V 21 3
Integrability o f the Gibbons-Tsare v Syste m MAXIM V . PAVLO V 24 7
2D Tod a Chai n an d Associate d Commutato r Identit y A. K . POGREBKO V 26 1
CONTENTS
On Certai n Curren t Algebra s Relate d t o Finite-Zon e Integratio n OLEG K . SHEINMA N 27 1
Preface
Sergey Petrovic h Noviko v i s on e o f th e mos t outstandin g mathematician s o f our time . Th e article s i n this book , writte n b y member s o f his famous semina r an d covering man y aspect s o f seemingl y unrelate d area s o f moder n mathematic s an d mathematical physics , reflec t th e breadt h o f Novikov' s scientifi c interest s an d ar e dedicated t o hi m o n th e occasio n o f his 70t h birthday. 1
We briefly describ e th e paper s include d i n thi s volume .
In the pape r b y Alexeevski i and Natanzon , a n analog o f classical Hurwitz num -bers fo r regula r covering s o f surface s wit h marke d point s b y seame d surface s i s in -troduced. Seame d surface s ar e no t actua l surfaces . A simpl e exampl e o f a seame d surface i s a book-lik e seame d surface : severa l rectangle s ar e glue d b y edge s lik e sheets i n a book . Th e author s sho w tha t suc h number s defin e a ne w exampl e o f Klein Topologica l Fiel d Theory . I t i s know n tha t suc h theorie s ar e i n one-to-on e correspondence wit h a certai n clas s o f algebra s calle d Cardy-Frobeniu s algebras . The algebra s correspondin g t o covering s wit h finite grou p actio n ar e describe d i n terms o f th e grou p an d it s subgroups . A s a result , a n algebrai c formul a fo r th e proposed generalizatio n o f Hurwit z number s i s obtained .
The wor k o f Buchstabe r an d Terzi c i s on th e borde r o f algebrai c topolog y an d complex geometry . A n effectiv e computatio n o f th e universa l tori c genu s o f th e complex an d stabl e comple x structure s o n homogeneou s space s i s provided. A s a n application, explici t formula s fo r th e cobordis m classe s an d characteristi c number s of th e flag manifolds , Grassman n manifold s an d som e othe r interestin g example s are obtained . I n particular , the y completel y solve d th e proble m tha t characteristi c Chern number s ma y diffe r fo r comple x structure s o n the sam e homogeneou s space . It i s wel l know n tha t Pontryagi n number s d o no t d o this . Th e first resul t i n tha t problem wa s obtaine d b y Bore l an d Hirzebruch . I n 1958 , the y gav e a n exampl e of tw o comple x structure s o n 10-dimensiona l homogeneou s spac e havin g differen t numbers c\ .
S. P. Novikov i s one of the founder s o f cobordism theory . I n 1967 , he publishe d a pape r whic h helpe d t o determin e th e futur e direction s o f cobordis m theory . Th e paper o f Buchstabe r an d Terzi c i s on e mor e demonstratio n o f th e effectivenes s o f the method s develope d b y S . P . Novikov .
1 The thre e previou s volume s o f Novikov' s semina r wer e publishe d b y th e America n Mathe -matical Societ y i n th e serie s Advance s i n th e Mathematica l Sciences , Amer . Math . Soc . Transla -tions, Ser . 2 .
1. "Topic s i n Topolog y an d Mathematica l Physics" , vol . 170 , 1995 . 2. "Solitons , Geometr y an d Topology : O n th e Crossroad" , vol . 179 , 1997 . 3. "Geometry , Topology , an d Mathematica l Physics , S . P . Novikov' s Seminar : 2002-2003" ,
vol. 212 , 2004 .
vii
viii PREFACE
The pape r o f Feigi n an d Veselo v i s devoted t o th e stud y o f a geometr y o f cer -tain collection s o f vectors in a linear spac e calle d V-systems . The y wer e introduce d earlier b y th e secon d name d autho r i n relatio n wit h a certai n clas s o f solution s of th e generalize d Witten-Dijkgraaf-Verlinder-Verlinde r (WDVV ) o r associativit y equations, playin g a n importan t rol e i n 2D topologica l field theorie s an d super -symmetric gaug e theories . I t i s show n tha t thes e system s ar e close d unde r th e natural operation s o f restrictio n an d takin g subsystems . A detaile d analysi s o f a special class of V-systems related t o generalized roo t system s and basic classical Lie superalgebras i s presented .
Gusein-Zade, Lueng o an d Melle-Hernande z i n thei r pape r introduc e a n equi -variant versio n o f th e classica l monodrom y zet a function . A numbe r o f invariant s of singularitie s hav e a n equivarian t versio n fo r singularitie s invarian t wit h respec t to a n actio n o f a finite grou p G. Fo r example , a n equivarian t versio n o f the Milno r number i s a n elemen t o f the rin g o f virtua l representation s o f the group . I t i s no t so clea r wha t shoul d b e considere d a s th e equivarian t versio n o f th e monodrom y zeta function . Th e tw o mai n ingredient s o f th e definitio n propose d i n th e pape r are equivarian t Lefschet z number s an d th e A structur e o n the Grothendiec k rin g of finite G-sets .
The work of Taimanov and Grinevich is devoted to the study of a certain class of nonlinear equation s considere d i n the framework o f soliton theory in which Noviko v has mad e man y profoun d contributions . Th e author s sho w tha t th e characteristi c feature o f thes e equations , ofte n calle d equation s wit h self-consisten t sources , i s a special typ e o f deformatio n o f th e spectra l curve s o f auxiliar y linea r operators . I t turns ou t tha t despit e th e fac t tha t th e spectra l curve s ar e not preserved , the y stil l provide man y conservatio n law s fo r th e system .
Dubrovin's pape r analyze s th e singularitie s o f solution s t o th e system s o f first order quasilinea r PDE s an d thei r perturbation s containin g highe r derivatives . Th e work i s a n essentia l ste p i n th e furthe r developmen t o f th e author' s approac h t o the classificatio n o f integrabl e systems . Th e stud y i s focuse d o n th e subclas s o f Hamiltonian PDE s wit h on e spatia l dimension . Fo r th e system s o f orde r on e o r two th e loca l structur e o f singularitie s o f a generi c solutio n t o th e unperturbe d system nea r th e poin t o f "gradien t catastrophe " i s describe d i n term s o f standar d objects o f classica l singularit y theory . Th e autho r argue s tha t thei r perturbe d companions mus t b e give n b y certai n specia l solution s o f Painlev e equation s an d their generalizations .
The main goal of the work of Hamilton an d Lazarev is to establish th e existenc e of th e so-calle d strin g structure s o n th e ordinar y an d equivarian t homolog y o f th e free loo p spac e o n a manifol d or , mor e generally , a forma l Poincar e dualit y space . The authors ' metho d i s base d o n th e obstructio n theor y o f certai n algebra s an d rational homotop y theory . I t i s shown tha t th e resulting strin g topology operation s are manifestl y homotop y invariant .
Krichever an d Shiot a i n thei r pape r introduc e a genera l notio n o f th e abelia n solutions o f solito n equations . Th e theor y o f suc h solution s unifie s th e idea s o f the theor y o f th e Calogero-Mose r typ e system s connecte d wit h th e theor y o f pol e systems o f meromorphi c solution s o f a variet y o f solito n equations , an d th e idea s underlying Novikov' s remarkabl e conjecture : th e Jacobain s o f curve s ar e exactl y the indecomposabl e principall y polarize d abelia n varietie s whos e theta-function s provide explici t solution s o f th e K P equation . Thi s conjectur e wa s prove d b y th e
P R E F A C E i x
second name d autho r an d unti l recentl y ha s remaine d th e mos t effectiv e solutio n of the famou s Riemann-Schottk y problem . Th e mai n resul t o f the presen t pape r i s that al l solution s o f th e K P equatio n expressibl e i n term s o f holomorphi c section s of a lin e bundl e o n a n abelia n (no t necessaril y principall y polarized ) variet y ar e algebraic-geometrical. Evidenc e for the existence of a new type of integrable syste m on th e space s o f higher orde r theta-function s i s provided .
Mokhov's paper i s devoted to a further stud y of interconnections between classi-cal differential geometr y and the Hamiltonian theor y of hydrodynamic-type system s originated b y Dubrovi n an d Novikov . I t i s shown tha t th e associativit y equation s of two-dimensiona l topologica l quantu m fiel d theor y ca n b e identifie d wit h certai n reductions o f th e fundamenta l nonlinea r equation s o f th e theor y o f submanifold s in pseudo-Euclidea n spaces . A notio n o f potentia l submanifold s i s introduced . I t is proved tha t eac h semisimpl e Probeniu s manifol d ca n b e locall y represente d a s a potential submanifold .
Maltsev i n hi s pape r present s a detaile d stud y o f deformation s o f Whitham -type system s i n th e almos t linea r case . Th e Whi t ham equation s ar e a cornerston e of th e perturbatio n theor y o f algebraic-geometrica l solution s o f solito n equations . The genera l structur e o f thes e system s i s wel l understoo d b y now . Nevertheless , it turn s ou t tha t i n th e cas e whe n th e initia l dat a ar e clos e t o th e linea r case , th e deformation approac h shoul d b e modifie d i n orde r t o mak e al l th e construction s stable i n th e linea r limit .
Pavlov's pape r i s devote d t o a constructio n o f particula r solution s fo r th e fa -mous Benne y hydrodynamica l chain . Connection s betwee n th e famou s Lowne r equation an d th e Gibbons-Tsare v system s tha t ar e specia l reduction s o f th e Ben -ney chai n ar e discussed .
The main goal of the paper by Pogrebkov is to explore further a n earlier author' s observation tha t som e well-known integrabl e system s ar e in mutual correspondenc e with som e simpl e commutato r identitie s i n associativ e algebras . I t turn s ou t tha t under som e genera l assumptions , th e nonlinea r equation s an d thei r La x pair s ca n be reconstructe d fro m representation s o f thes e algebras . I n th e presen t pape r th e construction i s generalized fo r th e lattic e case .
The purpos e o f Sheinman' s wor k i s t o presen t recen t result s o n La x operato r algebras an d thei r applications . Thes e ne w type s o f almos t grade d algebra s repre -sent furthe r development s o f ideas goin g back t o a theory o f high ran k solution s o f integrable equation s develope d b y Noviko v an d Krichever .
Victor Buchstaber , Igo r Kricheve r
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Titles i n Thi s Subserie s
61 V . M . Buchstabe r an d I . M . Krichever , Editors , Geometry , topology , an d
mathematical physic s (S . P . Novikov' s seminar : 2006-2007 ) (TRANS2/224 )
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 . Polotovskii , 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 , Max i m Kontsevich , an d Anto n Zorich , Editors , Pseudoperiodi c
topology (TRANS2/197 )
45 Ya . Eliashberg , D . Fuchs , T . Rat iu , an d A . Weinstein , 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 . Khovanskii , 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 )
TITLES I N THI S SUBSERIE S
35 G . I . Olshanski , Editor , Kirillov' s Semina r o n Representatio n Theor y (TRANS2/181 )
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 . Ne t 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 . Pes in , 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 Advance s 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 D m i t r 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 . Samborskii , 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
