Antti J. Niemi y. P.O. Box 803, S-75108, Uppsala, Sweden z. and. Research Institute for Theoretical Physics. Pirjo Pasanen

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1 UU-ITP HU-TFT hep-th/ ON QUANTUM COHOMOLOGY AND DYNAMICAL SYSTEMS Antti J. Niemi y Deprtment of Theoreticl Physics, Uppsl University P.O. Box 803, S-7508, Uppsl, Sweden z nd Reserch Institute for Theoreticl Physics P.O.Box 9, FIN-0004 University of Helsinki, Finlnd Pirjo Psnen Reserch Institute for Theoreticl Physics P.O. Box 9, FIN-0004 University of Helsinki, Finlnd We investigte spects of quntum cohomology nd Floer cohomology in the context of generic clssicl Hmiltonin system. In prticulr, we show tht Floer's instnton eqution is relted to quntum Euler chrcter in the quntum cohomology dened by topologicl nonliner -model. This reltion is n innite dimensionl nlogy with the reltion between Poincre-Hopf nd Guss-Bonnet-Chern formule in clssicl Morse theory. By pplying locliztion techniques to functionl integrls we then show tht for Khler mnifold this quntum Euler chrcter lso coincides with the Euler chrcter determined by the derhm cohomology of the trget spce. Our results re consistent with the Arnold conjecture which estimtes periodic solutions to clssicl Hmilton's equtions in terms of derhm cohomology of the phse spce. z permnent ddress y Supported by Gorn Gustfsson Foundtion for Science nd Medicine nd by NFR Grnt F-AA/FU E-mil: NIEMI@TETHIS.TEORFYS.UU.SE E-mil: PIRJO.PASANEN@HELSINKI.FI

2 The methods of quntum eld theory tht were originlly developed to understnd prticle physics, hve since proven useful lso in sttisticl physics. Recently it hs been noticed tht these methods could even be succesfully pplied to clssicl Hmiltonin dynmics. There, one of the intriguing open problems is the Arnold conjecture [], [2] which sttes, tht on compct phse spce the number of periodic solutions to Hmilton's equtions is bounded from below by the sum of Betti numbers. In this Letter we shll be interested in developing functionl integrl techniques to ddress issues such s the Arnold conjecture. In prticulr, we rgue tht the methods of topologicl quntum eld theories when combined with functionl locliztion techniques pper quite eective lso in the cse of clssicl dynmicl systems. We shll consider Hmilton's equtions on phse spce which is compct symplectic mnifold X with locl coordintes.we re interested in T -periodic trjectories tht solve Hmilton's equtions, i.e. re criticl points of the clssicl ction S cl = T 0 d(# _ H(; )) () Here # re components of the symplectic potentil corresponding to the symplectic two-form! = d#. We ssume tht the Hmiltonin depends explicitly on time in T -periodic mnner H(; 0) = H(; T ), so tht energy is not necessrily conserved. Hmilton's equtions re _! b H(; ) = _ X H =0 (2) with T -periodic boundry condition (0) = (T ). Without ny loss of generlity we shll ssume tht such periodic solutions re nondegenerte. When energy is conserved so tht H does not hve explicit dependence on ech criticl point ofhgenertes trivilly T -periodic trjectory. According to the clssicl Morse theory their number is bounded from below by the sum of Betti numbers on X nd consequently Arnold's conjecture is vlid: f #periodic trjectories g X B k = X dim H k (X; R) (3) However, if H depends explicitly on time so tht energy is not conserved, the criticl points of H do not solve (2) nd the methods of nite dimensionl Morse theory re no longer pplicble. In order to show tht (3) nevertheless remins vlid we need n innite dimensionl generliztion of Morse theory. Unfortuntely this is not very esy: There is no minimum for (), nd periodic solutions of (2) re sddle points of

3 () with n innite Morse index. Due to such diculties, for explicitly -dependent Hmiltonins the conjecture hs only been proven in certin specil cses [2]. In the pproch to Arnold conjecture developed by Floer [3], [2] one strts by dening grdient ow in the spce of closed loops (0) = = gb S cl b (4) where g b is Riemnnin metric on X. Using this metric nd the symplectic two-form! b we set I b = g c! cb Since I ci c b = (4) becomes b this denes n lmost complex structure on the mnifold X + I b@ b H(; ): (5) This eqution is dened on cylinder S R with locl coordintes nd. It describes the ow ofloops() onx, nd the bounded orbits tend symptoticlly to the periodic solutions of Hmilton's eqution (2). Using (5), Floer constructs complex with the solutions to (2) being the vertices nd the trjectories (5), so-clled pseudo-holomorphic instntons, connecting them s the edges. He proves tht for generic Hmiltonin the cohomology of this complex is in fct independent of H(; ). Subsequently Witten [4] found tht Floer's cohomology hs connections to quntum cohomology which is generted by the quntum ground sttes of topologicl -model. Using the more generl Novikov ring structure Sdov [5] then estblished tht these two cohomologies in fct coincide. According to the Arnold conjecture (3) these cohomologies should lso be intimtely connected with the stndrd derhm cohomology of the underlying phse spce. Witten's quntum cohomology is bsed on solutions of Cuchy-Riemnn equtions for holomorphic + I b@ b =0 (6) This corresponds to the (denegerte) specil cse of (5) with H = 0, which is not generic from the point of view of Hmiltonin dynmics. Consequently it is not cler how the topologicl -model, even if it describes Floer's cohomology, could be pplied to understnd Arnold's conjecture. For this, one needs to extend the topologicl - model so tht it ccounts for n rbitrry nontrivil Hmiltonin H(; ). Such issues hve been ddressed by Sdov [5]. In the present Letter we shll continue his work 2

4 by explining how functionl integrls nd locliztion techniques, when pplied to the topologicl -model, cn be used to derive Morse-theoretic reltions for generic Hmiltonin system. In prticulr, we shll explin how the stndrd, nite dimensionl De Rhm cohomology reltes to quntum cohomology by studying n innite dimensionl version of Poincre-Hopf nd Guss-Bonnet-Chern formul for (5). Topologicl -model [4] is theory of mps from Riemnn surfce with metric nd lmost complex structure to mnifold X with Riemnnin metric g b nd lmost complex structure I b.we ssume tht the lmost complex structures re both comptible with the metrics, so tht for exmple on X we hveg b = I c I d bg cd. Moreover, if we hve D c I b=@ c I b+ cd I d b d cb I d =0 (7) I b is n integrble complex structure nd g b is Khler. However, in the following we do not necessrily ssume (7). The bsic elds re mps :!X; =::: dim X, which correspond to locl coordintes on X. Anticommuting elds re sections of TX, the pullbck of the tngent bundle of X. Anticommuting elds ; ( =;2), re one-forms on with vlues on TX, so they re sections of the bundle E = TX T. Commuting uxiliry elds F re sections of the sme bundle s. Becuse the rnk of E is innitely bigger thn dimension of the spce of mps from to X we restrict to subbundle, the self-dul prt E +. This mens tht nd F constrint both stisfy the self-dulity = I b b (8) The elds hve grding which t the clssicl level corresponds to bosonic symmetry with chrges 0; ; ; 0 for ; ; nd F, respectively. The ction of topologicl -model cn be constructed in the following wy: Consider nilpotent fermionic opertor ~ Q of degree ~Q = d 2 x " i (x) (x) + F (x) (x) constructed from the elds of the theory where summtion over lwys implies n integrtion over = (x) ; = # (x) : + F ; (9) 3

5 This we identify s dierentil opertor d + in the superspce dened on the complex (E) (E). Here E mens tht the coordintes nticommute. Now introduce cnonicl conjugtion of ~ Q Q = e ~ Qe = ~ Q + f ~ Q;g+ 2 ff ~ Q;g;g+ (0) The cohomologies dened by the opertors ~ Q nd Q re the sme, nd we still hve Q 2 = 0. A suitble conjugtion is dened by where = i c b ( = bc F () : 2 D c I b); In coordinte free lnguge this cn be written s = ^ b = ^ b E = b 2 DI b i(; ^) with being connection -form nd d denoting the bsis of -forms on the spce of mps! X. Notice tht the covrint derivtive of the self-dul sub-bundle E + cn be written using the modied connection A strightforwrd clcultion gives Q = or in short Here c d b F i c b ( bc D + = 2 [d + ^ +(d+^) I]: 2 D c I b) if b c ( bc R bcd + 2 D d I e bd c I e + 2 I e br ecd Q = i( ;@)+(F +i^;) i(f^;) 2 ^R = d^ +^^^ 2 (^R; ): 2 D c I b) 2 I er e bcd is the Riemnn curvture 2-form corresponding to the connection ^ b. In components, ^R b 2 = ( 2 R b 4 DIe bdi e) + 4 ( I er e b I e br e ) : () 4

6 This opertor Q is exctly the sme s in [4] when we tke into ccount the self-dulity condition (8) for nd F. We shll be interested in cohomologicl ctions of the form S = fq; g (2) Such ctions re utomticlly invrint under the BRST-trnsformtion generted by Q nd consequently the prtition function = [d ][ df ][ d ][ ]eis (3) should remin invrint under rbitrry locl vritions of. If we select =(; s) 4 (; F ) = g b s b where s [] is section of E nd is prmeter, we get S = " i D c(g b s b ) c + F g bs b + 6 D ci ed d I e b c d b 8 R bcd c d b 4 g b F b ; (4) i 2 b D ci b c g d s d 4 F F # : (5) specilizing to s [] =@ nd = then gives the usul ction [4] of topologicl -model. Since the prtition function (3) is (formlly) invrint under locl vritions of we conclude tht it must be independent of. Indeed, if we eliminte the uxiliry eld F, the prtition function yields n innite dimensionl version of the Mthi- Quillen formlism [6], [7]: In this formlism, one hs section of bundle E over the mnifold X, nd n (ordinry) integrl over X of the so clled Thom clss MQ = X dexp [ 4 (; ) ir R]: (6) 4 This integrl is independent of, nd s! 0 it loclizes to nite dimensionl integrl over the moduli spce M of solutions to the eqution = 0. On the other hnd, s!(6) is nothing but the integrl of the Pfn of the curvture which is the sme s the Euler chrcter of the bundle. The integrl (6) thus yields n interpoltion between the Poincre-Hopf nd Guss-Bonnet-Chern formule. 5

7 In the present cse, elimintion of the uxiliry eld F gives n innite dimensionl functionl integrl version of the Mthi-Quillen formlism: Using () the ction becomes S = 4 (s + I bs b )(s + I b s ) b # 4 ^R bcd c d b i 2 D c(s + I bs b ) c which is clerly of the sme functionl form s the integrnd tht ppers in (6), the relevnt bundle being E + nd the section (7) = s + I bs b (8) Thus we my view (3) s n innite dimensionl version of the integrl of the Thom clss (6). Since (3) is (formlly) independent of, we cn consider its!limit. According to the nite dimensionl Mthi-Quillen formlism, this limit should be relted to the Euler chrcter of the functionl spce. For this, we specilize the world-sheet to be torus with coordintes nd such tht the metric is unit mtrix with comptible complex structure = the prtition function evlutes to! = =. In the!limit we then nd tht [d ][ d ] Pf( ^R b ): (9) At lest formlly, this is the Euler chrcter of the innite dimensionl bundle E +.In prticulr, ll dependence on T hs vnished from the lst integrl, it only depends on the Euler chrcter of TX. Formlly, this innite dimensionl quntity is topologicl invrint nd s such does not depend on how we choose the connection. It is the Euler chrcter in the quntum cohomology dened by the quntum ground sttes of the topologicl -model, nd counts the dierence in the number of bosonic vcu (even forms) nd fermionic vcu (odd forms) in the quntum theory. In nlogy with nite dimensionl Morse theory, we next relte the (forml) innite dimensionl Euler chrcter (9) to n lternting sum over criticl points of functionl describing the Floer cohomology. For this we consider the limit! 0, gin on torus with locl coordintes ;. As! 0, the integrl obviously concentrtes round the zeroes of s + I bs b 6

8 For simplicity we shll ssume tht these zeroes re non-degenerte. (A generliztion to the degenerte cse is strightforwrd, see for exmple [8].) Let 0 be such tht [ 0 ] = 0 nd write = 0 + ^. In the bsence of degenercies, the rst term in the c ( )^ c + O(^ 2 ) does not vnish. The corresponding expnsion of the ction is S = + 4 i 2 g b@ c ( ) c i 2 g b@ c ( ) c ( )g d ( b )+@ c ( )g d ( b ) ^c ^d + O(^ 3 ) Using the self-dulity of nd the fct tht ner 0 we hve = I b b S = i g b@ c ( ) c + c( )g b@ d ( b )^ c ^d + O(^ 3 ) As! 0, we cn then evlute the prtition function which yields!0 = = = X [d 0][ d^ ][ d ][ ][ ] det [d 0 ] det 2 (g) det 2 i 2 ( g) exp[is] 2 c ( )g b@ d ( b )i det g c b : (20) this gives : (2) =0 sign det jj@ b jj (22) In prticulr, if we select s nd tke to be self-dul Hmiltonin vector eld, i.e. = H (;) (23) where H () re two priori rbitrry Hmiltonin functions on X relted by the self-dulity condition for,we nd tht + I b@ H (;) = 0 which is Floer's instnton eqution (5). Note tht demnding 's to be self-dul together with (23) implies tht I b must be complex structure so tht X is now Khler mnifold. 7

9 This result estblishes tht the quntum cohomology of the topologicl -model indeed describes the cohomology of Floer's instnton eqution, t lest in the sense of Poincre-Hopf nd Guss-Bonnet-Chern formul. The underlying ide in Floer's pproch to the Arnold conjecture is tht this cohomology should lso be intimtely relted to the derhm cohomology of the originl symplectic mnifold, i.e. the trget mnifold of the -model. Such reltion would then explin why n estimte such s (3) mkes sense s Morse inequlity. We shll now proceed to evlute our functionl integrl using locliztion methods to estblish tht the Euler chrcter (9) of quntum cohomology indeed coincides with the Euler chrcter of the derhm cohomology over the symplectic mnifold X. For this, we specilize to symplectic mnifold which iskhler. We select locl coordintes so tht I b = i b nd I b = i b. Self-dulity then implies tht F z = F = 0 so tht the only surviving components re F nd F, nd similrly for z z z. Using the (forml) invrince of (3) under locl vritions of, weintroduce the functionl = g b F b + g b (24) nd consider the pertinent ction (2). Explicitly (we set F z F etc.), S = g bf F b + R bc d c d b g e e + R bc d c d b g e e + (F g b@ z b + F g z b ) + ( ig b@ z ig d@ z e d be ) b + ( ig z ig z e d be ) b (25) We evlute the corresponding functionl integrl in the!limit, by seprting the z; z independent constnt modes (for exmple in Fourier decomposition) nd scle the non-constnt modes by p, (z; z) = o + ^ (z; z)! o + p ^ (z; z) F (z; z) = F o + ^F (z; z)! F o + p ^F (z; z) (z; z) = o +^ (z; z)! o + p ^ (z; z) (z; z) = o + ^(z; z)! o + p ^(z; z) (26) nd similrly for ; F ; ;. The Jcobin for this chnge of vribles in (3) is trivil, nd evluting the integrls in the!limit we end up with the Euler 8

10 chrcter of the phse spce X in the form = d o d o d o d o Pf(R bc d c o d o )Pf(R bc d c o d o ) (27) which lso exhibits the underlying complex structure on X. As consequence, we hve found tht the Euler chrcteres in quntum cohomology nd derhm cohomology coincide, estblishing n intimte reltionship between these two cohomologies. prticulr, the Floer instnton eqution dened over our torus obeys X =0 sign det jj@ c b jj = X k ( ) k B k with B k the Betti numbers of the symplectic mnifold X. Obviously this is fully consistent with (3). In conclusion, we hve studied three priori dierent cohomologies: Floer's cohomology which describes periodic solutions to Hmilton's equtions, Witten's quntum cohomology which describes the quntum ground stte structure of topologicl nonliner -model, nd stndrd nite dimensionl derhm cohomology. By investigting n innite dimensionl generliztion of the fmilir Poincre-Hopf nd Guss-Bonnet- Chern formul, we hve found tht these three cohomologies re intimtely relted. This result is consistent with the Arnold conjecture. In prticulr, it indictes tht topologicl eld theories nd functionl locliztion methods re useful tools lso in the study of clssicl dynmicl systems. In References [] V.I. Arnold, C.R. Acd. Pris 26 (965) 379; nd Uspeki Mth. Nuk. 8 (963) 9 [2] H. Hofer nd E. ehnder, Symplectic Invrints nd Hmiltonin Dynmics (Birkhuser Verlg, 994) [3] A. Floer, Commun. Mth. Phys. 20 (989) 575 [4] E. Witten, Commun. Mth. Phys. 8 (990) 4 9

11 [5] V. Sdov, hep-th/93053 [6] V. Mthi nd D. Quillen, Topology 25 (986) 85 [7] S. Cordes, G. Moore nd S. Rmgoolm, hep-th/9420; S. Wu, hep-th/ [8] A.J. Niemi nd K. Plo, hep-th/