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Ôðîáåíèóñîâî ìíîãîîáðàçèå Áìîäåëè ËàíäàóÃèíçáóðãà ïàðû(Ẽ8 , Z3 )ìîðôíî ôðîáåíèóñîâó ìíîãîîáðàçèþ òåîðèè ÃðîìîâàÂèòòåíà îðáèôîëäàèçî-P12,2,2,2 .Äîêàçàòåëüñòâî îñíîâàíî íà àêñèîìàõ Áìîäåëè ËàíäàóÃèíçáóðãà ïàðû(Ẽ8 , Z3 ),ïðåäëîæåííûõ â ïðåäûäóùåé ãëàâå, ÿâíûõ âû÷èñëåíèÿõ ÷àñòè ôðîáåíèóñîâà ïîòåíöèàëà òåîðèè ÃðîìîâàÂèòòåíà îðáèôîëäàP16,3,2è àíàëèçå íåêîòîðûõ äèôôåðåíöèàëüíûõóðàâíåíèé.Íàïîìíèì, ÷òî ãèïîòåçû çåðêàëüíîé ñèììåòðèè ïðåäïîëàãàþò íàëè÷èå ñåìåéñòâàôðîáåíèóñîâûõ ñòðóêòóð äëÿ ïàðû(Ẽ8 , Z3 ).Ïðèâåäåííàÿ âûøå òåîðåìà èäåíòèôèöè-ðóåò âñåãî ëèøü îäíó ôðîáåíèóñîâó ñòðóêòóðó (êîòîðóþ ìû îáîçíà÷àåì àááðåâèàòóðîéLCSL) èç ýòîãî ñåìåéñòâà. Äëÿ òîãî, ÷òîáû ïîñòðîèòü âñå ñåìåéñòâî, ìû èñïîëüçóåìíåêîòîðîå òåõíè÷åñêîå óòâåðæäåíèå, êîòîðîå àíàëèçèðóåòñÿ â ñëåäóþùåé ãëàâå.Ñîäåðæàíèå ãëàâû 6.Á.À.
Äóáðîâèí îïðåäåëèë ñòðóêòóðó ôðîáåíèóñîâà ìíîãîîáðàçèÿ íà ïðîñòðàíñòâåðàçâåòâëåííûõ íàêðûòèé ñôåðû. Òàêèå ôðîáåíèóñîâû ìíîãîîáðàçèÿ íîñÿò â íàñòîÿùååâðåìÿ íàçâàíèå ãóðâèö-ôðîáåíèóñîâûõ ìíîãîîáðàçèé.  äàííîé ãëàâå äîêàçûâàåòñÿñëåäóþùàÿ òåîðåìà, îïóáëèêîâàííàÿ àâòîðîì â [1].Ïóñòüz êîîðäèíàòà íà ýëëèïòè÷åñêîé êðèâîéÐàññìîòðèì ïðîñòðàíñòâî ôóíêöèéE2ω1 ,2ω2 , èìåþùåé ïåðèîäû 2ω1 , 2ω2 .H1,(2,2,2,2) := {λ : E2ω1 ,2ω2 → P1 },èìåþùèõ ñëåäó-þùèé îáùèé âèä.4 X1 ℘0 (z − ai ; 2ω1 , 2ω2 )si + c,λ(z) :=℘(z − ai ; 2ω1 , 2ω2 )ui +2 ℘(z − ai ; 2ω1 , 2ω2 )i=110ω1 , ω2 , ai , ui , si , c ïàðàìåòðû îòîáðàæåíèÿ λ. Ðàññìîòðèì òàêæå ïîäïðîñòðàíñòâîRH1,(2,2,2,2) ⊂ H1,(2,2,2,2) ñîñòîÿùåå èç òàêèõ λ, ÷òî:ãäåa1 = 0,a2 = ω1 + ω2 ,a3 = ω1 ,a4 = ω2 ,s1 = s2 = s3 = s4 = 0.Òåîðåìà (Òåîðåìà 1 â [1]).
ÏðîñòðàíñòâîRH1,(2,2,2,2)èìååò ñòðóêòóðó ôðîáåíèóñîâàìíîãîîáðàçèÿ, èçîìîðôíóþ ôðîáåíèóñîâîé ñòðóêòóðå òåîðèè ÃðîìîâàÂèòòåíà îð-P12,2,2,2 .áèôîëäàÄîêàçàòåëüñòâî òåîðåìû òåõíè÷åñêîå.Ñîäåðæàíèå ãëàâû 7. ýòîé ãëàâå ìû âîçâðàùàåìñÿ ê âîïðîñó ïîñòðîåíèÿ ñåìåéñòâà ôðîáåíèóñîâûõñòðóêòóð äëÿ ïàðû(Ẽ8 , Z3 ).Çàìåòèì, îäíàêî, ÷òî èñïîëüçîâàííûå ìåòîäû ìîãóò áûòüòàêæå ïðèìåíåíû äëÿ áîëåå îáùèõ ñëó÷àåâ(Wσ , G),ãäåWσçàäàåò ïðîñòóþ ýëëèïòè-÷åñêóþ îñîáåííîñòü.G = {id}Íàïîìíèì, ÷òî ïðèñåìåéñòâî ôðîáåíèóñîâûõ ñòðóêòóð áûëî ïîñòðîåíîÊ.Ñàèòî â ïîìîùüþ ò.í. ïðèìèòèâíûõ ôîðì. Ïî íàñòîÿùèé ìîìåíò òåîðèÿ ïðèìèòèâíûõ ôîðì äëÿ îðáèôîëäîâûõ ìîäåëåé ËàíäàóÃèíçáóðãà åùå íå ïîñòðîåíà. Ââèäóýòîãî â ãëàâå 7 ìû ïðåäëàãàåì ðàññìîòðåòü ýôôåêò îò èçìåíåíèÿ ïðèìèòèâíîé ôîðìûòîëüêî â êëàññå ôðîáåíèóñîâûõ ìíîãîîáðàçèé.
Ñ ïîìîùüþ çåðêàëüíîé ñèììåòðèè òèïàCYLG, äîêàçàííîé â òåîðåìå , ìû èìååì â ÿâíîì âèäå ôðîáåíèóñîâ ïîòåíöèàë îäíîãî ïðåäñòàâèòåëÿ âñåãî ñåìåéñòâà ïàðûäåéñòâèå(τ0 ,ω0 )A(ãäåτ0 ∈ H, ω0 ∈ C∗(Ẽ8 , Z3 ). ãëàâå 7 ìû ïðåäëàãàåì íåêîòîðîå ïàðàìåòðû) íà ïðîñòðàíñòâå ôðîáåíèóñîâûõñòðóêòóðû, ýêâèâàëåíòíîå çàìåíå ïðèìèòèâíîé ôîðìû ïðèÍàïîìíèì òàêæå, ÷òî ôðîáåíèóñîâà ñòðóêòóðà ïàðûG = {id}.(Ẽ8 , Z3 ), çàäàþùàÿ çåðêàëüíóþñèììåòðèþ òèïà LGLG, â ñîîòâåòñòâèè ñ ãèïîòåçîé çåðêàëüíîé ñèììåòðèè, äîëæíàáûòü ñîãëàñîâàíà ñ ïðèìèòèâíîé ôîðìîé â ñïåöèàëüíîé òî÷êå. Òàêîå ïîíÿòèå òàêæåíå îïðåäåëåíî äëÿ îðáèôîëäîâûõ ìîäåëåé ËàíäàóÃèíçáóðãà. Àíàëîãè÷íî çåðêàëüíîéñèììåòðèè ñ òðèâèàëüíîé ãðóïïîé ñèììåòðèè ìû ïðåäëàãàåì íàçûâàòü ñïåöèàëüíûìèòå òî÷êè, äëÿ êîòîðûõ√τ0 ∈ Q −Däëÿ íåêîòîðîãî(W, {id})↔çàìåíà ïðèìèòèâíîé ôîðìû↔D ∈ N+ .(W, G)äåéñòâèå√↔ τ0 ∈ Q −D,ñïåöèàëüíàÿ òî÷êàA(τ0 ,ω0 )D ∈ Z+ . ñëåäóþùèõ ðàçäåëàõ ìû ìîòèâèðóåì èñïîëüçîâàíèå äåéñòâèÿA(τ0 ,ω0 )äëÿ çàìåíûïðèìèòèâíîé ôîðìû.Äëÿ ïðîñòîé ýëëèïòè÷åñêîé îñîáåííîñòèêðèâûõEσ := {x ∈ C3 | Wσ (x) = 0}.Wσðàññìîòðèì ñåìåéñòâî ýëëèïòè÷åñêèõÏîñêîëüêó ïðèìèòèâíàÿ ôîðìàζïðîñòîé ýëëèï-òè÷åñêîé îñîáåííîñòè ôèêñèðóåòñÿ âûáîðîì ðåøåíèÿ óðàâíåíèÿ ÏèêàðàÔóêñà êðèâîé11Eσäëÿ âñÿêîãîσ,äëÿ êëàññèôèêàöèè âñåõ ïðèìèòèâíûõ ôîðì íåîáõîäèìî è äîñòà-òî÷íî êëàññèôèöèðîâàòü âñå ðåøåíèÿÇàìåòèì, ÷òî âñå ðåøåíèÿπAπAñîîòâåòñòâóþùåãî óðàâíåíèÿ ÏèêàðàÔóêñà.ìîãóò áûòü ïðåäñòàâëåíû â âèäåZπA (σ) :=resEσAσäëÿ íåêîòîðîãî ôèêñèðîâàííîãî êëàññàπ∞ ,dx1 ∧ dx2 ∧ dx3dWσAσ ∈ H1 (Eσ ).Ôèêñèðóåì îäíî òàêîå ðåøåíèåòîãäà âñå îñòàëüíûå ðåøåíèÿ ìîãóò áûòü ïîëó÷åíû äåéñòâèåì ãðóïïûïðîñòðàíñòâå ãîìîëîãèé êðèâîéEσ .GL(2, C)íà ãëàâå 7 ìû ïðîäîëæàåì ýòî äåéñòâèå äî äåé-H1;(2,2,2,2) è òåîðèè ÃðîìîâàÂèòòåíà îðáèôîëäà1P2,2,2,2 .
 ïåðâîì ñëó÷àå äåéñòâèå âîçíèêàåò êàíîíè÷åñêè èç îïðåäåëåíèÿ ïðîñòðàí-ñòâèÿ íà ôðîáåíèóñîâûõ ñòðóêòóðàõñòâà ÐèìàíàÃóðâèöà. Òåîðåìà îá èçîìîðôèçìå ôðîáåíèóñîâûõ ñòðóêòóð èç ãëàâû 6ïðîäîëæàåò ýòî äåéñòâèå íà òåîðèþ ÃðîìîâàÂèòòåíà îðáèôîëäàP12,2,2,2 .Èäåÿ òàêîãîäåéñòâèÿ áûëà ðàçðàáîòàíà àâòîðîì â ñîâìåñòíîé ðàáîòå ñ À.Òàêàõàøè [2]. Ïîëó÷àåìîå äåéñòâèå ýêâèâàëåíòíî ñëåäóþùåìó äåéñòâèþ íà ïðîñòðàíñòâå ðåøåíèé ñèñòåìûÀëüôàíà:XkA (t)det(A):=Xk(ct + d)2at + bct + dc−,ct + dA=!abcd∈ GL(2, C).Îäíàêî äëÿ äåéñòâèÿ íà ïðîñòðàíñòâå ðåøåíèé óðàâíåíèÿ ÏèêàðàÔóêñà äîñòàòî÷íîîãðàíè÷èòüñÿ ìàòðèöàìè âèäàτ̄0 4πω0 Im(τ0 ):= 14πω0 Im(τ0 )A(τ0 ,ω0 )ω0 τ0ω0τ0 ∈ H, ω0 ∈ C∗ . ðåçóëüòàòå ìû ïîëó÷àåì ñåìåéñòâî øåñòèìåðíûõ ôðîáåíèóñîâûõ ñòðóêòóðäåéñòâèåìA(τ0 ,ω0 )(τ0 ,ω0 )M6íà ôðîáåíèóñîâîé ñòðóêòóðå òåîðèè ÃðîìîâàÂèòòåíà îðáèôîëäàP16,3,2 .
Áóäåì îáîçíà÷àòü ÷åðåç(τ ,ω )∞ñòâèåì A 0 0 íà òðîéêå Xk .(τ0 ,ω0 )Xkñîîòâåòñòâóþùèå ôóíêöèè, ïîëó÷åííûå äåé-Ñîäåðæàíèå ãëàâû 8.Ñëåäóþùàÿ òåîðåìà äîêàçûâàåò ãèïîòåçó çåðêàëüíîé ñèììåòðèè òèïà LGLG è ãèïîòåçó î ñîîòâåòñòâèè CY/LG äëÿ ïàðû(Ẽ8T , ZT3 ).Òåîðåìà 0.2. Ôðîáåíèóñîâî ìíîãîîáðàçèå îðáèôîëäîâîé Àìîäåëè ËàíäàóÃèíçáóðãàïàðû(Ẽ8T , ZT3 )èçîìîðôíî ôðîáåíèóñîâó ìíîãîîáðàçèþω0 :=Γ√( −1,ω0 )M6, ãäå1 2434π 2.Ãëàâà 8 ïîñâÿùåíà åå äîêàçàòåëüñòâó. Çàìåòèì, ÷òî ââèäó ðåçóëüòàòîâ ãëàâû 7 äëÿäîêàçàòåëüñòâà äàííîé òåîðåìû íåîáõîäèìî ðàññìîòðåòü îðáèòó ôðîáåíèóñîâà ìíîãîîáðàçèÿ òåîðèè ÃðîìîâàÂèòòåíà îðáèôîëäà12P16,3,2ïîä äåéñòâèåìA(τ0 ,ω0 )äëÿ âñåõτ0 ∈ H, ω0 ∈ C∗.Ïîñêîëüêó îäíîé èç àêñèîì îðáèôîëäîâîé Àìîäåëè ËàíäàóÃèíçáóðãà ÿâëÿåòñÿ óñëîâèè îïðåäåëåííîñòè ôðîáåíèóñîâà ïîòåíöèàëà íàäãëàâå ðàññìàòðèâàþòñÿ óñëîâèÿ íàäåëåí íàäQ, â äàííîé(τ0 ,ω0 )τ0 ,ω0 , ãàðàíòèðóþùèå, ÷òî ïîòåíöèàë M6îïðå-Q.Èç ÿâíîãî âèäà ñèñòåìû Àëüôàíà ñëåäóåò, ÷òî ïîòåíöèàë(τ0 ,ω0 )òîãäà è òîëüêî òîãäà, êîãäà Xk(τ )∈ Q[[τ ]]îïðåäåëåí íàä2 ≤ k ≤ 4,÷òî, â ñâîþ(τ0 ,ω0 )î÷åðåäü, èìååò ìåñòî òîãäà è òîëüêî òîãäà, êîãäà çíà÷åíèÿ â íóëå Xk(0) ∈ Q äëÿQâñåõäëÿ âñåõ(τ0 ,ω0 )M62 ≤ k ≤ 4.Çàìåòèì, ÷òî äëÿ âñÿêîé òðîéêè(τ0 ,ω0 )ðåøåíèåì ñèñòåìû Àëüôàíà, ôóíêöèÿ(τ0 ,ω0 )(τ ,ω )(τ ), X4 0 0 (τ )}, ÿâëÿþùåéñÿP(τ ,ω )4γ (τ0 ,ω0 ) (τ ) := 23 k=2 Xk 0 0 (τ ) ÿâëÿåòñÿ ðåøå-{X2(τ ), X3íèåì óðàâíåíèÿ Øàçè:γ 000 = 6γγ 00 − 9(γ 0 )2 .Êëàññèôèêàöèÿ ðåøåíèé óðàâíåíèÿ Øàçè âèäàγ (τ0 ,ω0 ) ,èìåþùèõ ðàçëîæåíèå íàä√Q, áûëà ïîëó÷åíà àâòîðîì âìåñòå ñ À.
Òàêàõàøè â [2]. Çàìåòèì, ÷òî π −1E2 (τ ) =P42 k=2 Xk∞ (τ ). Ñ ïîìîùüþ äàííîãî ðàâåíñòâà ïðîáëåìà êëàññèôèêàöèè òàêèõ γ (τ0 ,ω0 )èìååò ñëåäóþùåå ðåøåíèå:Òåîðåìà0.3.Ôèêñèðóåìτ0∈ Hè∈ C\{0}.ω0Ñëåäóþùèåóòâåðæäåíèÿýêâèâàëåíòíû:(i) Ôóíêöèÿγ (τ0 ,ω0 ) (t)∗(ii) Âûïîëíåíî: E2 (τ0 )∗(iii) Âûïîëíåíî: E2 (τ0 )çäåñüE2 ,E4 ,E6èìååò ðàçëîæåíèå â ðÿä íàä∈∈Q;Qω02 , E4 (τ0 ) ∈ Qω04 , E6 (τ0 ) ∈ Qω06 ;Qω02 è ýëëèïòè÷åñêàÿ êðèâàÿ Eτ0 îïðåäåëåíà íàä ðÿäû Ýéçåíøòåéíà, àQ;E2∗ (τ ) := E2 (τ ) − 3/(πIm(τ )).Äàííàÿ òåîðåìà ïîçâîëÿåò ïðèìåíèòü òåõíèêó è ìåòîäû òåîðèè ÷èñåë äëÿ äîêàçàòåëüñòâà ãèïîòåçû çåðêàëüíîé ñèììåòðèè.
 ÷àñòíîñòè, âîïðîñ îïèñàíèÿ âñåõòàêèõ, ÷òî âûïîëíåíî óñëîâèåτ0 ∈ H,(ii) ïðèâåäåííîé âûøå òåîðåìû, ÿâëÿåòñÿ îòêðûòîé ïðî-áëåìîé. Òàêæå çàìåòèì, ÷òî ñîãëàñíî ãëàâå 7 äëÿ äîêàçàòåëüñòâà çåðêàëüíîé ñèììåòðèèòèïà LGLG äîñòàòî÷íî îãðàíè÷èòüñÿτ0 ∈ëåíòíî òîìó, ÷òî íà ýëëèïòè÷åñêîé êðèâîé√−DQEτ0äëÿD ∈ N+ .Ýòî óñëîâèå ýêâèâà-èìååòñÿ òàê íàçûâàåìîå êîìïëåêñíîåóìíîæåíèå. Ñ òî÷íîñòüþ äî èçîìîðôèçìà ñóùåñòâóåò ðîâíî 13 ýëëèïòè÷åñêèõ êðèâûõ,èìåþùèõ êîìïëåêñíîå óìíîæåíèå è îïðåäåëåííûõ íàäQ(ñì. [14]).Òàêèì îáðàçîì äëÿ êëàññèôèêàöèè øåñòèìåðíûõ ôðîáåíèóñîâûõ ñòðóêòóðîïðåäåëåííûõ íàäQ,äîñòàòî÷íî ðàññìîòðåòü ëèøü 13 çíà÷åíèé√τ0 ∈ SL(2, Z) −1 ñóùåñòâóåò íåíóëåâîå(τ ,ω )(τ ,ω )(τ ,ω )X2 0 0 , X3 0 0 , X4 0 0 ðàöèîíàëüíû.âàåòñÿ, ÷òî òîëüêî äëÿ÷òî âñå òðè ÷èñëàÇàìåòèì, ÷òî ïîëó÷åííîå ÷èñëîäóëåì√−1.ω0τ0 .(τ0 ,ω0 )M6, ãëàâå 8 äîêàçû-÷èñëîω 0 ∈ C∗ ,òàêîå,ÿâëÿåòñÿ ïåðèîäîì ýëëèïòè÷åñêîé êðèâîé ñ ìî-Òàêîé ïåðèîä îïðåäåëåí ñ òî÷íîñòüþ äî ìíîæèòåëÿ, îäíàêî â òåîðåìå îçåðêàëüíîé ñèììåòðèè òèïà LGLG äàííîå ÷èñëî ñòðîãî ôèêñèðîâàíî.13Ñïèñîê ïóáëèêàöèé ïî òåìå äèññåðòàöèè èç ñïèñêà ÂÀÊ1.
A. Basalaev. Orbifold GW theory as the HurwitzFrobenius submanifold . J. Geom.Phys., 77, (2014), pp. 3042; 1,5 ï.ë..2. A. Basalaev, A. Takahashi. On rational Frobenius Manifolds of rank three withsymmetries . J. Geom. Phys., 84, (2014), pp. 7386; 1,5 ï.ë. (âêëàä àâòîðà 1,3ï.ë.).Ñïèñîê ëèòåðàòóðû[1] A. Basalaev. Orbifold GW theory as the HurwitzFrobenius submanifold. J. Geom. Phys., 77, 2014,pp.
3042.[2] A. Basalaev, A. Takahashi. On rational Frobenius Manifolds of rank three with symmetries. J. Geom.Phys., 84, 2014, pp. 7386.[3] P. Berglund, M. Henningson. LandauGinzburg orbifolds, mirror symmetry and the elliptic genus. Nucl.Phys. B, 433, 1995, pp. 311332.[4] P. Candelas, X. De La Ossa, P.
Green, L. Parkes. A pair of Calabi-Yau manifolds as an exactly solublesuperconformal theory. Nucl. Phys. B, 359, 1991, pp. 2174.[5] A. Chiodo, Y. Ruan. A global mirror symmetry framework for the Landau-Ginzburg/Calabi-Yaucorrespondence. preprint arXiv: 1307.0939, 2013.[6] W. Ebeling, A. Takahashi. Mirror Symmetry between Orbifold Curves and Cusp Singularities withGroup Action. Int. Math. Res. Not., 2013(10), 2013, pp. 22402270.[7] H. Fan, T. Jarvis, Y. Ruan. The Witten equation, mirror symmetry, and quantum singularity theory.Ann.
Math., 178(1), 2013, pp. 1106.[8] K. Intriligator, C. Vafa. LandauGinzburg orbifolds. Nucl. Phys. B, 339(1), 1990, pp. 95120.[9] Y. Ishibashi, Y. Shiraishi, A. Takahashi. A Uniqueness Theorem for Frobenius Manifolds and GromovWitten Theory for Orbifold Projective Lines. preprint arXiv:1209.4870, 2012.[10] L. Katzarkov, M. Kontsevich, T. Pantev. Hodge theoretic aspects of mirror symmetry.
Proc. Symp.Pure Math., 78, 2008, pp. 87174.[11] R. Kaufmann. Singularities with Symmetries, orbifold Frobenius algebras and Mirror Symmetry.Contemp. Math., 403, 2006, pp. 146.[12] M. Krawitz. FJRW rings and Landau-Ginzburg Mirror Symmetry. preprint arXiv:0906.0796, 2009.[13] M. Krawitz, Y. Shen. LandauGinzburg/CalabiYau Correspondence of all Genera for Elliptic OrbifoldP1 . preprint arXiv:1106.6270, 2011.[14] D.
Lawden. Elliptic Functions and Applications. Appl. Math. Sci., Springer, 1989.[15] W. Lerche, C. Vafa, N. P. Warner. Chiral rings in N = 2 superconformal theories. Nucl. Phys. B, 324(2),1989, pp. 427474.[16] T. Milanov, Y. Ruan. GromovWitten theory of elliptic orbifold P1 and quasi-modular forms. preprintarXiv:1106.2321, 2011.[17] T. Milanov, Y. Shen. Global mirror symmetry for invertible simple elliptic singularities. preprintarXiv:1210.6862, 2012.[18] T.
Milanov, Y. Shen. The modular group for the total ancestor potential of Fermat simple ellipticsingularities. preprint arXiv:1401.2725, 2014.[19] K. Saito. Period mapping associated to a primitive form. Publ. Res. Inst. Math. Sci., 19, 1983, pp.12311264.[20] Ikuo Satake, A. Takahashi.
GromovWitten invariants for mirror orbifolds of simple elliptic singularities.Ann. Inst. Fourier, 61, 2011, pp. 28852907.14[21] A. Strominger. Mirror symmetry is Tduality. Nucl. Phys. B, 479(1-2), 1996, pp. 243259.[22] A. Varchenko, B. Blok. Topological Conformal Field Theories and the Flat Coordinates. Mod. Phys.Lett.
A, 7(07), 1992, pp. 14671490.[23] E. Witten. Mirror Manifolds And Topological Field Theory. preprint arXiv:hep-th/9112056, 1991.[24] E. Witten. Phases of N = 2 theories in two dimensions. Nucl. Phys. B, 403(1-2), 1993, pp. 159222.15.
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