素粒子論研究 ･ 電子版 Vol. 32 (2020) No. 2

Soryushiron Kenkyu

2019 ೥౓म࢜࿦จ
Double Field Theory ʹ͓͚ΔήʔδରশੑͱLie ѥ୅਺ରʹΑΔ Vaisman ѥ୅਺ͷߏ੒๺ཬେֶେֶӃ ཧֶ‫ڀݚ‬Պ෼ࢠՊֶઐ߈ ྔࢠ෺ཧֶߨ࠲

৿ང֓ཁ

Double Field Theory (DFT) ͸ɼT ૒ରੑΛ໌നͳରশੑͱͯ࣋ͭ͠ॏྗཧ࿦Ͱ͋Γɼͦͷഎ‫ʹܠ‬͸ഒԽ‫ز‬Կֶ (doubled geometry) ͷଘࡏ͕ߟ͑ΒΕ͍ͯΔɽ·ͨɼDFT ͷήʔδରশੑ͸ Vaisman ͓Αͼ Courant ѥ୅਺ (algebroid) ʹΑͬͯ‫ه‬ड़͞ΕΔɽຊम࢜࿦จͰ
͸ɼLie ୅਺ͷ Drinfel’d double Λग़ൃ఺ͱͯ͠ɼVaisman ѥ୅਺͕;ͨͭͷ Lie ѥ୅਺ͷ૊͔ΒಘΒΕΔ͜ͱΛࣔ͢ɽ͞ΒʹɼഒԽ‫ز‬Կֶͷ۩‫ݱ‬Խͱͯ͠ύϥΤϧϛʔτଟ༷ମΛಋೖ͠ɼ͜ͷଟ༷ମ্ʹ Vaisman ѥ୅਺Λߏங͢Δɽ

Soryushiron Kenkyu

Soryushiron Kenkyu

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໨࣍
ୈ1ষ

͸͡Ίʹ

5

ୈ2ষ

Double Field Theory

9

2.1

DFT ʹ‫ݱ‬ΕΔ‫ز‬Կֶ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.2

DFT ͷ࡞༻ͱ C ‫ ހׅ‬. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

2.3

෺ཧత߆ଋ৚݅ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12ѥ୅਺ (Algebroid) ͷಋೖ

15

3.1

Lieʢ૒ʣ୅਺ͱ Drinfel’d double . . . . . . . . . . . . . . . . . . . . . . . . .

15

3.2

Lie ѥ୅਺ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

3.3

Courant ѥ୅਺ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

3.4

Double ʹΑΔ Vaisman ѥ୅਺ͷߏ੒ . . . . . . . . . . . . . . . . . . . . . .

23

Vaisman algebroid ͕‫ݱ‬ΕΔ‫ز‬Կֶ

31

4.1

ύϥෳૉߏ଄ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

4.2

ύϥΤϧϛʔτଟ༷ମ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

4.3

ύϥυϧϘʔίϗϞϩδʔ . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

4.4

DFT ʹ͓͚Δ Vaisman ѥ୅਺ . . . . . . . . . . . . . . . . . . . . . . . . . .

37

4.5

Derivaiton ৚݅ͷഁΕ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

4.6

Ұൠ‫ز‬Կֶͱͷؔ࿈ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

ୈ5ষ

·ͱΊ

43

෇࿥ A

‫ࢉܭ‬ϊʔτ

46

A.1

C ‫ހׅ‬ͷ Jacobiator ͷಋग़ . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

A.2

C ‫͔ހׅ‬Β Courant ‫΁ހׅ‬ͷ reduction . . . . . . . . . . . . . . . . . . . . .

48

A.3

T (e1 , e2 , e3 ) ͷؔ܎ࣜ (3.36) ͷಋग़ . . . . . . . . . . . . . . . . . . . . . . . .

49

A.4

T (e1 , e2 , e3 ) ͷؔ܎ࣜ (3.37) ͷಋग़ . . . . . . . . . . . . . . . . . . . . . . . .

51

A.5

Axiom C1 ͷ֬ೝ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

A.6

(A.45) ࣜͷಋग़ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

A.7

(A.46) ࣜͷಋग़ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

A.8

Axiom C2 ͷ֬ೝ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

A.9

Axiom C3 ͷ֬ೝ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

ୈ3ষ

ୈ4ষ

Soryushiron Kenkyu

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A.10

Axiom C4 ͷ֬ೝ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

A.11

Axiom C5 ͷ֬ೝ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

A.12

NK = NP + NP̃ Ͱ͋Δ͜ͱͷ֬ೝ . . . . . . . . . . . . . . . . . . . . . . . .

63ࢀߟจ‫ݙ‬

65

Soryushiron Kenkyu

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ୈ1ষ

͸͡Ίʹ
‫ݭ‬ཧ࿦͸ɼϋυϩϯΛ෼ྨ͢Δ͜ͱΛ໨తʹɼ‫͍ڧ‬૬‫༻࡞ޓ‬Λѻ͏ͨΊͷཧ࿦ͱͯ͠ 1960 ೥୅ʹੜ·Εͨɽ‫ݭ‬ཧ࿦Ͱ͸ɼ෺࣭ͷࠜ‫ݯ‬తͳ୯Ґͱ࣭ͯ͠఺ͷ୅ΘΓʹ 1 ࣍‫ݩ‬ͷ‫( ݭ‬string) Λ༻͍ͯཧ࿦Λߏங͢Δɽ‫ݭ‬ͷৼಈϞʔυͷҧ͍ʹΑͬͯɼҟͳΔछྨͷϋυϩϯΛ౷Ұͯ͠‫ه‬ड़Ͱ͖Δɼͱ͍͏΋ͷͰ͋ͬͨɽ͕ͩɼ1970 ೥୅ʹͳΔͱྔࢠ৭ྗֶ (quantum chromodynamics,

QCD) ͕։ൃ͞Εɼ‫͍ڧ‬૬‫༻࡞ޓ‬Λ‫ه‬ड़͢Δཧ࿦ͱͯ͠ਖ਼౰ͳͷ͸ QCD Ͱ͋Δͱೝࣝ͞Εͨɽ͔͠͠ɼดͨ͡‫ݭ‬ͷྭ‫ى‬ঢ়ଶ͕ॏྗʹؔ܎͢Δ͜ͱ͕ࢦఠ͞Εɼ‫ݭ‬ཧ࿦͸ɼࠓ౓͸ྔࢠॏྗཧ
࿦ͷީิͱͯ͠࠶ͼ೔ͷ໨ΛΈΔ͜ͱͱͳͬͨɽॏྗΛ‫ه‬ड़͢Δཧ࿦ͱͯ͠‫ݭ‬ཧ࿦Λղऍ͢Δʹ͋ͨΓɼҎԼͷΑ͏ͳΑ͍ಛ௃͕͋Δ͜ͱ͕Θ͔ͬͨɽ• ॏྗࢠ (graviton) ͕ཧ࿦ͷεϖΫτϧʹ‫ݱ‬ΕΔɽ• ௒‫ݭ‬ཧ࿦Ͱ͸ɼۭ࣌ͷ࣍‫ ͕ݩ‬10 ࣍‫ʹݩ‬ఆ·Δɽ• ‫ݭ‬ཧ࿦ʹ͓͚Δਓҝతͳύϥϝʔλʔ͸‫ݭ‬ͷுྗ T ͷΈͰ͋Δɽۭ࣌ͷ࣍‫ݩ‬ͷ਺͕ఆ·ΔҰํɼզʑ͕ೝࣝ͢Δۭ࣌͸ 4 ࣍‫͋Ͱݩ‬Δɽ͕ͨͬͯ͠ɼ༨৒࣍‫ݩ‬ͷॲཧΛߟ͑ͳͯ͘͸ͳΒͳ͍ɽ༨৒࣍‫΁ݩ‬ͷରॲ๏ͷͻͱͭ͸ɼ௿ΤωϧΪʔͰ͸‫ݕ‬ग़͞Εͳ͍Α͏ʹɼۭؒΛίϯύΫτԽ͢Δ͜ͱͰ͋ΔɽίϯύΫτԽͷ֓೦͸ɼKaluza-Klein (KK) ཧ࿦ [1, 2]ͰߟҊ͞Εͨɽ‫ݭ‬ཧ࿦ͷಛ௃͸ɼ૬‫Ͱ఺͕༻࡞ޓ‬͸ͳۭ࣌͘ͷ֦ுྖҬͰ‫͜ى‬Δ͜ͱͰ͋Δɽ৔
ͷྔࢠ࿦ (Quantum Field Theory, QFT) Ͱ͸ɼॏྗͷઁಈ‫ࢉܭ‬Λߦ͏ͱൃࢄ͕‫͏·ͯͬ͜͠ى‬ɽ͜Ε͸ɼQFT Ͱ͸෺࣭ͷ࠷খ୯ҐʹମੵΛ࣋ͨͳ͍఺ཻࢠΛ༻͍͍ͯΔͨΊʹൃੜ͢Δ໰୊Ͱɼ‫ݭ‬ཧ࿦Ͱ͸ຊ࣭తʹճආͰ͖Δɽ௒‫ݭ‬ཧ࿦ͷ௿ΤωϧΪʔ‫ݶۃ‬͸௒ॏྗཧ࿦Ͱ͋Δɽ‫͑׵͍ݴ‬Ε͹ɼ௒‫ݭ‬ཧ࿦͸‫ط‬ଘͷ෺ཧͱͯ͠௒ॏྗཧ࿦Λ಺แ͢Δ [3]ɽ‫ݭ‬ཧ࿦͕զʑʹ΋ͨΒͨ͠ൃ‫ݟ‬͸ɼۭ࣌ͦͷ΋ͷͷੑ࣭ʹؔͯ͠ଟ‫ʹذ‬ΘͨΔɽ͜ΕΒͷൃ‫ݟ‬ͷ‫ͨͬͳͱݩ‬ͷ͸ ʮ૒ରੑʯͰ͋Δɽ࠷΋༗໊ͳ૒ରੑ͸ɼAdS/CFT ରԠͰ͋Ζ͏ [4]ɽ͜Ε͸ɼʮD ࣍‫ݩ‬ͷ൓υɾδολʔ (Anti-de Sitter, AdS) ۭ࣌എ‫Ͱܠ‬ఆٛ͞Εͨॏྗཧ࿦͕ɼD − 1࣍‫ݩ‬ͷ‫ܗڞ‬৔ཧ࿦ (Conformal Field Theory, CFT) ͱ૒ରͰ͋Δʯͱ͍͏༧૝Ͱ͋Δɽ͜͜Ͱʮ૒ରͰ͋Δʯͱ͸ɼ
ʮ2 ͭͷҟͳΔཧ࿦͕ಉ͡෺ཧΛද͢͜ͱʯΛҙຯ͢Δɽ͜ͷ༧૝ͷ༗༻ͳ఺͸ɼҰํͷཧ࿦ͷ݁߹͕‫͖ͱ͍ڧ‬ɼ૒ରͳ΋͏ยํͷཧ࿦Ͱ͸݁߹͕ऑ͍͜ͱʹ͋Γɼ‫ݭ‬ཧ࿦ͷඇઁಈతͳཧղΛਂΊΔ͜ͱͱͳͬͨɽ‫ݭ‬ཧ࿦ʹ‫ݱ‬ΕΔ૒ରੑ͸ɼଞʹ΋ S ૒ରੑ [5]ɼU ૒ର
ੑ [6]ɼͦͯ͠ຊम࢜࿦จͷத৺ͱͳΔ T ૒ରੑ [7] ͕͋ΔɽU ૒ରੑ͸ɼT ૒ରੑͱ S ૒ରੑΛ

Soryushiron Kenkyu

6ֻ͚߹ΘͤͨΑ͏ͳରশੑͰ͋Δɽ௒‫ݭ‬ཧ࿦͸ɼ౳Ձͳཧ࿦͕ 5 ͭ͋Γɼ͜ΕΒͷ૒ରੑΛ௨ͯ͠ҠΓ͋͏͜ͱ͕஌ΒΕ͍ͯΔɽಛʹ T ૒ରੑ͸ɼ‫ݭ‬ͷ௕͞ʹ༝དྷ͢Δɼ‫ݭ‬ཧ࿦ʹಛ༗ͳ૒ରੑ
Ͱ͋ΔɽT ૒ରੑ͕‫ݱ‬ΕΔ࠷΋؆୯ͳྫ͸ɼ1 ࣍‫ ͚ͩ޲ํݩ‬S 1 ʹίϯύΫτԽۭͨ࣌͠Ͱ͋Δɽ‫ݭ‬͸ 1 ࣍‫ݩ‬ͷ෺ମͰ͋Δ͜ͱ͔ΒɼίϯύΫτԽ͞Εۭͨؒ S 1 ʹ‫͘෇͖ר‬ɽ͜ͷ‫͖෇͖ר‬ͷΤωϧΪʔ͸ S 1 ͷ൒‫ܘ‬Λ R ͱ͠ɼ‫ݭ‬ͷுྗΛઃఆ͢Ε͹‫͖Ͱࢉܭ‬Δɽ͜ͷ݁Ռ͸‫ݭ‬ͷ࣭ྔεϖΫτϧʹ‫د‬༩͠ɼ‫ݭ‬ͷ࣭ྔεϖΫτϧ M 2 ͸֓ͶҎԼͷΑ͏ʹॻ͘͜ͱ͕Ͱ͖Δɽ2

M ∝! n "2
R

+

#

mR
α′

$2

,

n, m ∈ Z.

(1.1)

Ұ߲໨͸‫ݭ‬ͷӡಈྔʹɼೋ߲໨͸‫ݭ‬ͷுྗʹ༝དྷ͢ΔΤωϧΪʔΛද͢ɽͨͩ͠ɼα′ ͸‫ݭ‬ͷுྗΛܾΊΔύϥϝʔλͰ͋Δɽ͜ͷ‫ݭ‬ͷεϖΫτϧ͸൒‫ ܘ‬R ⇔ α′ /R ͷೖΕସ͑ͷԼͰɼ‫ݭ‬ͷӡಈྔ n ͱ‫ ਺͖෇͖ר‬m ͷ໾ׂΛަ‫͢׵‬Δɽ͜ͷೖΕସ͑ͷԼͰཧ࿦͕ෆมͱͳΔ͜ͱΛ T ૒ରੑ
ͱ͍͏ɽT ૒ରੑʹΑΓɼ‫ݭ‬ཧ࿦Ͱ͸‫ݱ͕ྔ͏͍ͱ਺͖͖ͭר‬Εɼ‫ͱࢠཻ఺͕ݭ‬͸ҟͳΔํ๏Ͱۭ࣌Λ‫ه‬ड़͢Δ͜ͱ͕ಡΈऔΕΔɽ͔͠͠ɼ‫ݭ‬ཧ࿦ʹ͓͍ͯ T ૒ରੑ͸ӅΕͨରশੑͰ͋ΔɽT
૒ରੑΛ໌നͳରশੑͱ͢Δཧ࿦͸ߟ͑ΒΕͳ͍ͩΖ͏͔ʁ

Double Field Theory (DFT) ͸ɼT ૒ରੑΛཧ࿦ͷରশੑͱ͢Δ৽͍͠ॏྗཧ࿦Ͱ͋Δ [8]ɽDFT ͷಛ௃͸ɼ‫ݭ‬ͷӡಈྔʹ Fourier ‫ڞ‬໾ͳ࠲ඪͱ‫ݭ‬ͷ‫ ʹ਺͖͖ͭר‬Fourier ‫ڞ‬໾ͳ࠲ඪΛ౳Ձʹѻ͏͜ͱͰɼۭ࣌࠲ඪΛ double ʹ֦ு͠ɼ৔Λ O(D, D) ‫ڞ‬มͳ‫ʹܗ‬૊Μͩ͜ͱʹ͋Δɽ͜ΕʹΑΓɼT ૒ରੑΛ໌നͳରশੑͱ࣮ͯ͠‫͢ݱ‬ΔɽDFT ͕֦ு͞Εۭͨ࣌Ͱఆٛ͞ΕΔҰํͰɼ௒‫ݭ‬ཧ࿦Ͱ͸ۭ࣌ͷ࣍‫ ͕ݩ‬10 ࣍‫͍ͯͬ·ܾͱݩ‬ΔɽDFT ͕෺ཧతʹਖ਼͍ࣗ͠༝౓Λ࣋ͪɼཧ࿦
ͱͯ͠ഁ୼͠ͳ͍Α͏ʹ͢ΔͨΊʹ͸ɼԿΒ͔ͷ߆ଋ৚݅Λ༩ۭ͑ͯ࣌ͷ࣍‫ݩ‬ͷ਺Λ߹ΘͤΔඞ
ཁ͕͋Δɽ͜ͷ৚݅Λ෺ཧ section ৚݅ͱ͍͏ɽ͜ͷ৚݅ʹΑΓɼ֦ுۭ࣌ͷத͔Β෺ཧతʹҙຯ͕͋Δ෦෼ۭؒΛࢦఆ͢ΔɽDFT Ͱ͸ɼ2D ࣍‫ݩ‬ͷഒԽ (doubled) ۭؒͷத͔ΒɼD ࣍‫ݩ‬ͷ෦෼ۭؒΛબͿ͜ͱʹͳΔɽDFT Ͱ section ৚݅Λຬͨ͢ͻͱͭͷํ๏͸ɼഒԽۭؒʹఆٛ͞Εͨ͋ΒΏΔ৔΍ύϥϝʔλ͔Βɼ‫࠲͖͖ͭר‬ඪͷґଘੑΛऔΓআ͘͜ͱͰ͋Δɽ͜ͷΑ͏ͳɼ૒ରੑʹ‫͍ͨͮج‬ཧ࿦͸Ұ‫ݟ‬ඒ͘͠‫͑ݟ‬ɼ͜Ε·Ͱѻ͑ͳ͔ͬͨߴΤωϧΪʔεέʔϧͰͷۭ࣌ͷ༷ࢠΛ‫ه‬ड़͍ͯ͠ΔΑ͏ʹࢥ͑Δɽ͔͠͠ɼ‫ݭ‬ཧ࿦ͷΤωϧΪʔεέʔϧ͸‫ݱ‬ࡏͷٕज़Ͱ͸࣮‫ূݕͰݧ‬ՄೳͳϨϕϧΛ௒͑ͨͱ͜Ζʹ͋ΔͨΊɼཧ࿦͕ਖ਼౰͔Ͳ͏͔Λ‫͢ূݕ‬Δ͜ͱ͸೉͍͠ɽͦ͜Ͱɼগͳ͘ͱ΋ߏஙͨ͠ཧ࿦ʹഁ୼͕ͳ͍͜ͱΛɼ਺ֶͷྗΛआΓ͔ͯ֬Ί͍ͨɽۭؒΛѻ͍ɼͦͷੑ࣭Λௐ΂Δ਺ֶͱ͍͑͹ɼ‫ز‬ԿֶͰ͋Δɽ
Ұൠ૬ରੑཧ࿦͕ Einstein ʹΑͬͯ‫׬‬੒͞Εͨͷ͸ɼ1915 ೥ͷ͜ͱͰ͋Δ [10]ɽ͜ͷཧ࿦ʹΑΔ࠷΋ॏཁͳओு͸ɼʮॏྗͱ͸ۭ࣌ؒͷ࿪ΈͰ͋Δʯͱ͍͏͜ͱͰ͋Δɽ͕ͨͬͯ͠ɼҰൠ૬ରੑཧ࿦ΛఆࣜԽ͢Δʹ͸ɼ‫ۭؒ )࣌( ͕ͨͬۂ‬Λ‫ه‬ड़͢Δ‫ز‬Կֶ͕ඞཁෆՄܽͰ͋ͬͨɽ͜ͷͨΊɼ(ٖ) ϦʔϚϯ‫ز‬Կֶ͸ॏྗΛ‫ه‬ड़͢Δπʔϧͱͯ͠ɼҰൠ૬ରੑཧ࿦ͷ‫׬‬੒ͱͱ΋ʹ࠶஫໨͞ΕΔΑ͏ʹͳͬͨɽҰൠ૬ରੑཧ࿦ͱϦʔϚϯ‫ز‬Կֶͷ૊Έ߹Θͤ͸ɼॏྗཧ࿦Λߏங͠ɼ࣌
ۭߏ଄Λཧղ͢Δʹ͋ͨͬͯ‫ز‬Կֶ͕‫ͳྗڧ‬෢‫͋Ͱث‬Δ͜ͱΛࣔͨ͠ࣄྫͱ‫͑ݴ‬ΔͩΖ͏ɽ७ਮ਺ֶͱͯ͠ͷ‫ز‬Կֶ͸ɼ‫ݱ‬୅΋ൃలΛଓ͚͍ͯΔɽ୅දతͳྫ͸ɼ2004 ೥ʹ Hitchin ΒʹΑͬͯఏҊ͞ΕͨɼҰൠԽ‫ز‬Կֶ (generalized geometry) Ͱ͋Δ [11]ɽҰൠԽ‫ز‬ԿֶͰ͸ɼଟ༷ମͷ઀ଋ (tangent bundle) Λɼ઀ଋͱ༨઀ଋ (cotangent bundle) ͷ࿨ʹ֦ு͢ΔɽҰൠԽ‫ز‬Կֶ΋·ͨɼ͋Δछͷॏྗཧ࿦Λ‫ه‬ड़͢Δπʔϧͱͯ͠΋‫׆‬༂͢ΔɽҰൠ૬ରੑཧ࿦ͷ௒ରশੑΛ

Soryushiron Kenkyu

7‫ؚ‬Ί֦ͨுͱͯ͠ɼ௒ॏྗཧ࿦͕ଘࡏ͢ΔɽҰൠԽ‫ز‬Կֶ͸ɼ௒‫ݭ‬ཧ࿦͔ΒಘΒΕΔ௒ॏྗཧ࿦
ͷ͏ͪɼಛʹ NS-NS sector ʹରͯ͠ͷ‫ز‬Կֶతͳඳ૾Λ༩͑Δɽ
͜ΕΒͷલྫʹΑͬͯߟ͑ΔͱɼDFT ΋௒ॏྗཧ࿦ͷ T ૒ରෆมͳఆࣜԽΛ໨తͱͯ͠ߏங
͞Εͨॏྗཧ࿦Ͱ͋Δ͔Βɼͦͷഎ‫͋ʹܠ‬Δ‫ز‬Կֶతͳඳ૾ʹΑͬͯఆࣜԽ͞ΕΔͩΖ͏ɽDFTͷഎ‫͋ʹܠ‬Δ‫ز‬Կֶͱͯ͠஫໨͞Ε͍ͯΔͷ͕ɼഒԽ‫ز‬Կֶ (doubled geometry) [12] Ͱ͋ΔɽഒԽ‫ز‬Կ্ֶ͕ड़ͷϦʔϚϯ‫ز‬Կֶ΍ҰൠԽ‫ز‬ԿֶͱҟͳΔͷ͸ɼ਺ֶతʹਖ਼౰ʹߏங͖͠Ε͍ͯͳ͍఺Ͱ͋Δɽ͜ͷҙຯͰɼഒԽ‫ز‬Կֶ͸‫ݱ‬ஈ֊Ͱ͸ DFT ʹ‫ݱ‬ΕΔ਺ֶతͳߏ଄ͷ૯শͱ‫ͬݴ‬ͯ΋Α͍ɽ਺ֶͱͯ͠ ഒԽ‫ز‬ԿֶΛ੔උ͢Δ͜ͱͰɼDFT ͷཧ࿦ߏ଄ΛཧղͰ͖Δͱ‫ظ‬଴͞Ε
Δɽ͜Ε͸ɼT ૒ରੑΛ௨ͯ͠ɼ‫͍ͯݟ͕ݭ‬ΔߴΤωϧΪʔۭ࣌ͷ࢓૊ΈΛඥղ͘େ͖ͳख͕͔
ΓʹͳΔͩΖ͏ɽຊम࢜࿦จ͸ɼ2020 ೥ 1 ݄ʹग़൛ͨ͠࿦จ [13] ͷ಺༰ΛϕʔεʹɼDFT ͷ‫ز‬Կֶతͳඳ૾ʹண໨ͯ͠ҎԼͷΑ͏ʹߏ੒ͨ͠΋ͷͰ͋Δɽ
ୈ 2 ষ ͜ͷষͰ͸ɼDFT ͱɼͦͷഎ‫͋ʹܠ‬Δ ഒԽ‫ز‬ԿֶΛಋೖ͢ΔɽॳΊʹɼDFT Ͱ༻͍Δ
ഒԽ࠲ඪΛఏࣔ͢Δɽ࣍ʹɼҰൠԽ Lie ඍ෼Λఆٛ͠ɼDFT ͷήʔδରশੑΛ‫ه‬ड़͢Δ

C ‫ހׅ‬Λ༩͑Δɽͦͯ͠ɼ‫ޙ‬൒ͷষͰٞ࿦ͷओ୊ͱͳΔ‫߆͍ڧ‬ଋ৚݅ͱ͸Կ͔ʹ͍ͭͯड़΂Δɽ
ୈ 3 ষ લͷষͰड़΂ͨɼC ‫ن͕ހׅ‬ఆ͢Δ୅਺ߏ଄ʹ͍ͭͯɼ਺ֶతͳ‫͔఺؍‬Βٞ࿦͢ΔɽC‫ن͕ހׅ‬ఆ͢Δ୅਺ߏ଄͸ Vaisman ѥ୅਺ (algebroid) ͱ‫ݺ‬͹ΕΔɽLie ୅਺ (algebra)Λग़ൃ఺ͱ͠ɼ୅਺ߏ଄ΛঃʑʹҰൠԽ͍ͯ͘͜͠ͱͰ Vaisman ѥ୅਺ͷఆٛΛ༩͑Δɽ
·ͨɼ୅਺ֶͷ෼໺ͰΑ͘஌ΒΕ͍ͯΔ Lie ୅਺ͷ Drinfel’d double ͱ͍͏ૢ࡞Λ‫ʹݩ‬ɼ;ͨͭͷ Lie ѥ୅਺͔Β Vaisman ѥ୅਺ΛߏஙͰ͖Δ͜ͱΛূ໌͢Δɽ
ୈ 4 ষ ୈ 3 ষͷ݁ՌΛɼDFT ʹؐ‫͢ݩ‬ΔɽഒԽ‫ز‬Կֶͷ਺ֶతͳ۩‫ݱ‬Խ͸ύϥΤϧϛʔτ(para-Hermitian) ଟ༷ମʹΑͬͯͳ͞ΕΔͱ͍͏ࢦఠ͕͋Δ [14, 15]ɽ͜ͷ݁Ռʹ‫͍ͮج‬ͯɼύϥΤϧϛʔτଟ༷ମ্ʹɼલͷষͰఆٛͨ͠ɼdouble ʹΑͬͯಘΒΕΔ Vaismanѥ୅਺Λߏங͢Δɽ·ͨɼ੒෼‫ࢉܭ‬ͷ݁Ռ͔Βɼ‫߆͍ڧ‬ଋ৚݅ͷ୅਺తͳ‫͍ͯͭʹݯى‬ٞ࿦͢ΔɽՃ͑ͯɼഒԽ‫ز‬ԿֶͱҰൠԽ‫ز‬Կֶͷؔ܎ʹ͍ͭͯ΋ٞ࿦͢Δɽ
ୈ 5 ষ Ҏ্ͷٞ࿦Λ·ͱΊΔɽ·ͨɼࠓ‫ޙ‬ͷ՝୊΍ల๬ʹ͍ͭͯड़΂Δɽຊจதʹ‫ݱ‬Εͨ਺ࣜ΍ূ໌ʹؔ͢Δ‫ࢉܭ‬ͷৄࡉ͸ɼAppendix ʹճͨ͠ɽಛʹɼୈ 3 ষͰߦͬͨ

Vaisman ѥ୅਺ͷߏ੒ʹඞཁͳ‫ࢉܭ‬͸શͯ‫ͨ͠ࡌه‬ɽஅΓ͕ͳ͍‫ݶ‬Γɼຊจதͷ‫߸ه‬͸ҎԼͷҙຯͰ༻͍Δɽ༻‫ޠ‬ͷఆٛ౳͸ඞཁʹԠͯ͡ຊจதͰ༩͑Δɽ

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M

ଟ༷ମ

M

ഒԽۭؒ (·ͨ͸ύϥΤϧϛʔτଟ༷ମ)E (L, L̃)

Lie ѥ୅਺ (·ͨ͸ Lagrangian ෦෼ۭؒ, Dirac ߏ଄)

g

E

∗

Lie algebraE ͷ૒ରۭؒɼ·ͨ͸૒ରଋρ• : • → T M

Anchor map

V

Vaisman ѥ୅਺[·, ·]E,L,L̃

E, L, L̃ ্ͷ Lie ‫ހׅ‬[·, ·]cC ্ͷ Courant ‫ހׅ‬[·, ·]CC ‫ހׅ‬C

Courant ѥ୅਺[·, ·]

T M ্ͷ Lie ‫ހׅ‬[·, ·]SSchouten(-Nijenhuis) ‫ހׅ‬[·, ·]VV ্ͷ Vaisman ‫ހׅ‬⟨·, ·⟩(·, ·)

಺ੵ૒ઢ‫ࣜܗܕ‬Soryushiron Kenkyu

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ୈ2ষ

Double Field Theory͜ͷষͰ͸ɼDouble Field Theory ͷ‫ૅج‬తͳ෦෼ʹ͍ͭͯ͘͝؆୯ʹड़΂Δɽಛʹɼ3 ষҎ߱ͰඞཁͱͳΔɼDFT ʹ͓͚Δήʔδ୅਺ʹ͍ͭͯৄࡉʹ‫͍ͯ͘ݟ‬ɽ·ͣ͸ɼDFT ʹ‫ݱ‬ΕΔഒԽ
‫ز‬ԿֶͷཁૉΛ༩͑Δɽ࣍ʹɼDFT ʹ͓͚Δ࡞༻ͱήʔδม‫׵‬Λ༩͑ɼ͜ΕΛఆٛ͢Δ‫ͱੵހׅ‬ͯ͠ɼC ‫ހׅ‬Λಋೖ͢Δɽήʔδม‫׵‬ͷ୅਺͕ด͡ΔΑ͏ʹɼ‫߆͍ڧ‬ଋ৚݅Λಋೖ͢Δɽ

2.1 DFT ʹ‫ݱ‬ΕΔ‫ز‬Կֶ
DFT ͸ɼtype II ௒ॏྗཧ࿦ͷ NS-NS sector Λ໌നʹ T ૒ରෆมఆࣜԽͨ͠ཧ࿦Ͱ͋Δ [8]ɽ͜ͷཧ࿦͸ɼmassless ৔ͱͯ͠‫ ྔܭ‬gµν ɼKalb-Ramond ৔ (B ৔) Bµν ɼdilaton φ Λ‫ؚ‬ΉɽT૒ରੑͷԼͰ͸ɼ‫ྔܭ‬ͷҰൠ࠲ඪม‫ͱ׵‬ɼB ৔ͷ U (1) ήʔδม‫͢߹͕ࠞ׵‬Δɽ͜ΕΛ‫ز‬Կֶతʹද‫͢ݱ‬Δ͜ͱ͸ɼ௨ৗͷ઀ۭؒ (‫ݭ‬ͷӡಈྔϞʔυ) ͱ༨઀ۭؒ (winding Ϟʔυ) Λ૊Έ߹ΘͤͨɼҰൠԽ઀ۭؒͷಋೖΛଅ͢ɽҰൠԽ઀ଋ (generalized tangent bundle) E ͸ɼଟ༷ମ M ͷ઀ଋ T M ͱ ༨઀ଋ T ∗ M ͷ௚࿨ʹΑͬͯߏ੒͞ΕΔɽ

E = T M ⊕ T ∗ M.

(2.1)

ҰൠԽ઀ଋ͸ҰൠԽ‫ز‬ԿֶͰऔΓѻΘΕΔɽҰൠԽ‫ز‬Կֶ͸ɼ2004 ೥ʹ Hitchin [11] ͱ Gualtieri[16] ʹΑͬͯఏҊ͞ΕͨɽϕΫτϧΛ઀ଋ T M ͷ੾அ Γ(T M ) ͱ͢Δͱɼඍ෼ 1 ‫ࣜܗ‬͸༨઀ଋT ∗ M ͷ੾அ Γ(T ∗ M ) Ͱ͋Δɽ͕ͨͬͯ͠ɼΓ(E) ͸ϕΫτϧͱඍ෼ 1 ‫ࣜܗ‬Λ‫ࣜܗ‬తʹ଍ͨ͠ྔͱ
ͳΓɼ͜ΕΛҰൠԽϕΫτϧͱ͍͏ɽҰൠԽ‫ز‬ԿֶͰ͸ɼඍ෼ಉ૬ࣸ૾ (diﬀeomorphism) ͱήʔδม‫׵‬Λ૊Έ߹ΘͤͯɼҰൠԽ઀ۭؒʹର͢ΔҰൠԽඍ෼ಉ૬ࣸ૾Λಋೖ͢Δɽͦͷ݁ՌɼCourantѥ୅਺ [17] ͱ͍͏ɼରশੑΛ‫ه‬ड़͢Δ্Ͱͷ‫ͳྗڧ‬ಓ۩Λఏ‫͢ڙ‬Δɽจ‫[ ݙ‬17] ͷ Courant ѥ୅਺͸ɼҎԼͷ‫Ͱੵހׅ‬༩͑ΒΕΔ୅਺ߏ଄ͷҰछͰ͋ΔɽXi ∈ Γ(T M ), ξi ∈ Γ(T ∗ M ) ͱͨ͠ͱ͖ɼҎԼͷ Courant ‫ހׅ‬1
[X1 + ξ1 , X2 + ξ2 ]c = [X1 , X2 ] + (LX1 ξ2 − LX2 ξ1 ) + d0 (ξ1 (X2 ) − ξ2 (X1 )),
2

(2.2)

ʹΑͬͯ༩͑ΒΕΔɽCourant ѥ୅਺ʹؔͯ͠͸ 3 ষͰΑΓৄࡉʹड़΂Δɽ͞Βʹ, ‫ݭ‬ͷ‫ڞͱ਺͖͖ͭר‬໾ͳม਺ x̃ Λಋೖ͠ɼ௨ৗͷۭ࣌࠲ඪ x ͱಉ౳ʹѻ͏͜ͱͰɼඪ
తۭؒ (ۭ࣌ଟ༷ମ) ͷ࣍‫ݩ‬΋ 2 ഒʹͳΓɼഒԽ‫ز‬Կֶ (doubled geometry) ʹ౸ୡ͢ΔɽഒԽ‫ز‬

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10Կֶͷ਺ֶతͳ۩ମతߏ੒͸͍·ͩʹࢼߦஈ֊ʹ͋Δɽ‫Ͳͳ཰ۂ‬ͷ‫ز‬ԿֶతͳྔΛఆٛ͠ɼ਺ֶ
తʹద੾ͳഒԽ‫ز‬ԿֶΛߏங͢ΔࢼΈ͸ [18, 19] ͳͲͰߦΘΕ͍ͯΔɽ
ҎԼɼഒԽۭؒͷ࠲ඪΛ༩͑ɼDFT Ͱѻ͏‫ྔܭ‬΍৔ͷఆٛΛ༩͑ΔɽҎ߱ɼ௨ৗͷ௒ॏྗཧ࿦
͕ఆٛ͞ΕΔۭ࣌ͷ࣍‫ݩ‬Λ D ࣍‫͠ͱݩ‬ɼDFT ͷഒԽۭؒ͸ 2D ࣍‫͋Ͱݩ‬Δͱ͢ΔɽҰൠʹɼ௒‫ݭ‬ཧ࿦͕ఆٛ͞ΕΔۭ͕࣌ 10 ࣍‫͋Ͱݩ‬Δ͜ͱ͔ΒɼഒԽۭؒ͸ 20 ࣍‫͋Ͱݩ‬Δͱߟ͑ΒΕΔɽഒ
Խۭؒͷ࠲ඪ (ഒԽ࠲ඪ) xM ͸ҎԼͷΑ͏ʹॻ͚Δ΋ͷͱ͢Δɽxµ ͸ӡಈྔͱ Fourier ‫ڞ‬໾ͳ࠲ඪɼx̃µ ͸‫ ͱ਺͖͖ͭר‬Fourier ‫ڞ‬໾ͳ࠲ඪͰ͋Δɽ

x

M

=

#

$
x̃µ
,
xµ

(M = 1, . . . , 2D; µ = 1, . . . , D).

(2.3)

T ૒ରΛͱΔ͜ͱ͸ɼxµ ͱ x̃µ ͷೖΕସ͑ʹରԠ͢Δɽ·ͨɼഒԽۭؒͰͷඍ෼ԋࢉࢠ͸ҎԼͷΑ͏ʹද͢ɽ

# µ$
∂˜
,
∂µ

∂
∂M =
=
∂xM

∂µ =

∂
∂
, ∂˜µ =
.
µ
∂x
∂ x̃µ

(2.4)

ҰൠʹɼD ࣍‫ݩ‬ίϯύΫτۭؒʹ͓͚Δ T ૒ରม‫( ܈׵‬T-duality ‫ )܈‬͸ O(D, D) Ͱ༩͑ΒΕΔɽ
ഒԽۭؒ͸ O(D, D) ෆมͳߏ଄ η Λ࣋ͭɽ্ड़ͷ xM ͷఆ͔ٛΒɼη ͱͦͷ‫ٯ‬͸ҎԼͷΑ͏ʹॻ͚Δɽ

ηM N =

#

δµ ν
0

0

δµ ν

$

,

η

MN

=

#

δµ ν
0

0

δµ ν

$

.

(2.5)

ഒԽ࠲ඪ্ͷҰൠԽςϯιϧͷ଍ͷ্͛Լ͛͸ɼ͢΂ͯ η ʹΑͬͯߦΘΕΔɽ
࠷΋؆୯ͳ DFT ͸ɼtype II ௒ॏྗཧ࿦ͷ NS-NS sector Λ‫ʹݩ‬ఆࣜԽ͞ΕΔɽ͜ͷཧ࿦ʹ‫ݱ‬ΕΔ‫ج‬ຊతͳཁૉ͸ɼ‫ ྔܭ‬gµν ɼKalb-Ramond ৔ Bµν ɼdilaton φ ͷ 3 ͭͰ͋Δɽ͜ΕΒͷ৔ΛT ૒ରੑ͕໌നͳཧ࿦Ͱѻ͏ʹ͸ɼO(D, D) ‫ڞ‬มͱͳΔΑ͏ʹ૊Έ௚͢ඞཁ͕͋Δɽ͜ΕΛҰൠԽ‫( ྔܭ‬generalized metric) HM N ͱ͍͏ɽHM N =#g µν
Bµρ g ρν−g µρ Bρνgµν − Bµρ g ρσ Bσν

$

.

(2.6)

·ͨɼHM N ͸ҎԼͷؔ܎Λຬͨ͢ɽ

HM N = η M K η N L HKL .

(2.7)

Ճ͑ͯɼdilaton ΋ҰൠԽ͞ΕΔɽDFT dilaton (ҰൠԽ dilaton) d ͸ҎԼͷΑ͏ʹͳΔɽ

e−2d =√−ge−2φ .

(2.8)

ͨͩ͠ɼg ͸‫ྔܭ‬ͷߦྻࣜͰ͋Δɽ͜ͷ૊Έ௚͠ʹΑΓɼHM N ΍ d ͸ DFT ʹ͓͍ͯ O(D, D)ม‫׵‬Λड͚Δɽ·ͨɼഒԽۭؒͷؔ਺ HM N (xM ), d(xM ) ͱͳΔɽ

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2.2 DFT ͷ࡞༻ͱ C ‫ހׅ‬લઅͰఏࣔͨ͠৔Λ༻͍ͯɼO(D, D) ෆมͳ DFT ͷ࡞༻͸ɼҰൠԽ Ricci εΧϥʔͱ͍͏ྔ
ʹΑͬͯҎԼͷΑ͏ʹ༩͑ΒΕΔ [18].

SDFT =
R=%

d2D x e−2d R(H, d),

(2.9)

1 MN
1
H
∂M HKL ∂N HKL − HM N ∂M HKL ∂K HN L
8
2
+ 4HM N ∂M ∂N d − ∂M ∂N HM N − 4HM N ∂M d ∂N d + 4∂M HM N ∂N d.

(2.10)

͜ͷ࡞༻͕ O(D, D) εΧϥʔ͔ͭҰൠԽήʔδεΧϥʔͰ͋Δ͜ͱ΋ɼ[18] ʹ͓͍ͯࣔ͞Εͨɽ࡞༻ (2.9) ͸ɼେҬతͳ O(D, D) ରশੑ

HM N → HLK MLM MKN ,xM → xN MNM(2.11)

ʹରͯ͠໌നʹෆมͰ͋ΔɽͰ͸ɼ͜ΕΛ΋ͱʹ‫ॴہ‬ରশੑ͸ͲͷΑ͏ʹ‫ن‬ఆ͞ΕΔͩΖ͏͔ʁҰൠԽϕΫτϧ V ʹର͢ΔҰൠԽඍ෼ಉ૬ࣸ૾ͷແ‫ݶ‬খ൛ɼ͢ͳΘͪഒԽ࠲ඪͷແ‫ݶ‬খม‫׵‬͸ҎԼͷΑ͏ʹఆٛ͞ΕΔ [8]ɽV M → V M + δΞ V M ,x M → xM − Ξ M .

(2.12)

ͨͩ͠ɼδΞ ͸ҎԼͷΑ͏ͳҰൠԽ Lie ඍ෼Ͱ༩͑ΒΕΔɽॏΈ w(V ) Λ࣋ͭҰൠԽϕΫτϧ V MͷҰൠԽ Lie ඍ෼͸ҎԼͷΑ͏ʹఆٛ͞ΕΔɽ

L&Ξ V M = ΞK ∂K V M + (∂ M ΞK − ∂K ΞM )V K + w(V )V M ∂K ΞK .

(2.13)

Ұ߲໨͸௨ৗͷ Lie ඍ෼ͱಉ͡ҠಈΛද߲͢ɼೋ߲໨ͷ‫ހׅ‬ͷதͷ߲͸ O(D, D) ม‫׵‬Λද͢ɽH

ͷॏΈ͸ w(H) = 0 Ͱ͋ΓɼҰൠԽ dilaton ͷॏΈ͸ w(e−2d ) = 1 Ͱ͋Δɽ·ͨɼҰൠԽ Lie ඍ෼͸ O(D, D) ߏ଄Λอͭɽ͢ͳΘͪɼҎԼͷؔ܎͕੒ཱ͢Δɽ

L&Ξ ηM N = L&Ξ η M N = 0.

(2.14)[L&Ξ1 , L&Ξ2 ] = L&[Ξ1 ,Ξ2 ]C + F (Ξ1 , Ξ2 , ·)(2.15)

ҰൠԽ Lie ඍ෼͸ɼDFT ʹ͓͚Δήʔδରশੑͷ୅਺ߏ଄Λఆٛ͢Δɽ͜ͷ୅਺ߏ଄͸ C ୅਺Λͳ͠ɼC ‫ހׅ‬

K
M
K M L
[Ξ1 , Ξ2 ]M
C = 2Ξ[1 ∂K Ξ2] − ηKL Ξ[1 ∂ Ξ2]
1
M
K
M
K M L
K M L
= ΞK
(2.16)
1 ∂K Ξ2 − Ξ2 ∂K Ξ1 − ηKL (Ξ1 ∂ Ξ2 − Ξ2 ∂ Ξ1 ).
2ʹΑͬͯࢧ഑͞ΕΔɽͨͩ͠ɼΞ[1 Ξ2] = (Ξ1 Ξ2 − Ξ2 Ξ1 )/2 Ͱ͋Δɽ·ͨɼF ͸ҎԼʹࣔ͢Α͏ͳ༨෼ͳ߲Ͱ͋Δɽ

F (Ξ1 , Ξ2 , V )M =

1
P L
K P L
M
K
P M
K
ηKL (ΞK− (∂ P ΞM
1 ∂ Ξ2 − Ξ2 ∂ Ξ1 )∂P V
1 ∂P Ξ2 − ∂ Ξ2 ∂P Ξ1 )VK .
2(2.17)F ͕ଘࡏ͢Δ͜ͱ͔ΒɼҰൠԽ Lie ඍ෼ͷަ‫ੵࢠ׵‬͸ C ‫Ͱހׅ‬ด͡ͳ͍ɽ·ͨɼC ‫ހׅ‬͸ Jacobi཯Λຬͨ͞ͳ͍ɽF Λফ‫͠ڈ‬ɼC ‫ن͕ހׅ‬ఆ͢Δ୅਺Λด͡Δʹ͸ɼ࣍અͰઆ໌͢Δ‫߆͍ڧ‬ଋ৚݅ (strong constraint) Λ՝ͤ͹े෼Ͱ͋Δɽ

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12

2.3 ෺ཧత߆ଋ৚݅
‫ݭ‬ཧ࿦ͷ level-matching ৚݅ʹ༝དྷͯ͠ɼDFT ʹ‫ݱ‬ΕΔ৔͸ɼҎԼͷऑ͍߆ଋ৚݅ (weakconstraint) Λຬ଍͢Δඞཁ͕͋Δɽ͢ͳΘͪɼ೚ҙͷ৔ͱήʔδύϥϝʔλΛ Φ ʹର͠ɼ∂µ ∂˜µ Φ = 0(2.18)

ͱ͍͏৚͕݅՝͞ΕΔɽҰൠʹɼ৔ͷੵ͸͜ͷ৚݅Λຬͨ͞ͳ͍͕ɼࣹӨԋࢉࢠΛಋೖ͢Δ͜ͱͰ͜ΕΛิਖ਼͢Δɽ࣮ࡍʹɼऑ͍߆ଋ৚݅ʹ‫ ͍ͨͮج‬DFT ͸ɼओʹ [22] Ͱٞ࿦͞Ε͍ͯΔɽ໰୊͸ɼࣹӨԋࢉࢠΛ༻͍ͨ‫·͕͋ࢉܭ‬Γʹ΋൥ࡶ͗͢Δ͜ͱͰ͋ͬͨɽ
࠷ॳ‫ظ‬ͷ DFT ͸ɼ‫ݭ‬ͷ৔ͷཧ࿦Λ༻͍ͯߏங͞Εͨ [8]ɽ͜Ε͸ɼmassless ৔ͷΏΒ͗ͱ 2 ֊·Ͱͷඍ෼Λ‫ؚ‬Ήɼ৔ͷ 3 ࣍·ͰͷήʔδෆมੑΛ࣋ͭཧ࿦Ͱ͋Δɽ࠷ॳ‫ظ‬ͷ DFT ΛɼΑΓߴ࣍ͷήʔδෆมੑΛ࣋ͭཧ࿦ʹൃలͤ͞Δ͜ͱ͸ɼ্ड़ͷࣹӨԋࢉࢠͷଘࡏʹΑΓࠔ೉ͱͳͬͨɽ
͜ͷঢ়‫گ‬Λଧ։͢Δʹ͸ɼDFT ৔ͱήʔδύϥϝʔλͱʹՃ͑ͯɼͦΕΒͷ೚ҙͷੵʹର͢ΔੵΛ∂µ ∂˜µ (ΦΨ) = 0(2.19)

Ͱফ໓ͤ͞Δඞཁ͕͋Δɽ(ΦΨ) ͕ɼDFT ৔ͱήʔδύϥϝʔλͷ೚ҙͷੵΛද͢ɽ(2.19) Λ‫ڧ‬͍߆ଋ৚݅ (strong constraint) ͱ͍͏ɽ‫߆͍ڧ‬ଋ৚݅͸ɼ໊শͷ௨Γ৔ʹରͯ͠‫੍͍ڧ‬໿Λ༩͑
Δ͜ͱ͔ΒɼҎԼͷ͜ͱ͕ূ໌Ͱ͖Δ [23]ɽ‫߆͍ڧ‬ଋΛຬͨ͢৔ͷू߹͕༩͑ΒΕͨͳΒ͹ɼ৔͕ x ࠲ඪͷΈʹґଘ͢ΔΑ͏ͳഒԽ࠲ඪ (x, x̃) ͷऔΓํ͕ඞͣଘࡏ͢Δɽ‫͑׵͍ݴ‬Δͱɼ‫߆͍ڧ‬ଋ৚݅Λ՝ͨ͠ͱ͖ཧ࿦͸ਅʹഒԽ͞Εͳ͍ɽ‫߆͍ڧ‬ଋ৚݅ͱऑ͍߆ଋ৚݅Λ͋Θͤͯɼ෺ཧత section ৚݅ͱ‫Ϳݺ‬ɽ͜ΕΒͷ৚݅Λղ͘࠷΋؆୯ͳํ๏͸ɼ৔͕࠲ඪͷ൒෼͚ͩʹґଘ͢ΔΑ͏ʹ͢Δ͜ͱͰ͋Δɽಛʹɼۭ࣌࠲ඪ xµ ʹґଘ͠ɼ‫࠲͖͖ͭר‬ඪ x̃µ ʹґଘ͠ͳ͍ (∂˜µ Φ = 0) ৔߹ɼDFT ͸ҰൠԽ‫ز‬ԿֶʹԊͬͨ௒ॏྗཧ࿦ͷఆࣜԽʹࢸΔɽxµ ͱ x̃µ ͷࠞࡏΛߟ͑Δ͜ͱ͸ՄೳͰ͸͋Δ͕ɼη M N ͰੵΛ‫ܗ‬੒͢Δ࠲ඪͱͯ͠
࣮‫͢ݱ‬Δͷ͸ɼxµ ·ͨ͸ x̃µ ͷ͍ͣΕ͔Ұํʹґଘ͢Δ৔߹͚ͩͰ͋Δɽ
͜͜Ͱɼ෺ཧత section ৚݅ʹ͍ͭͯ਺ֶతͳଆ໘͔Βগ͠ߟ࡯͓ͯ͜͠͏ɽͨͩ͠ɼഒԽ‫ز‬Կֶͷ਺ֶతʹ‫׬‬શͳ۩‫ݱ‬Խ͸࣮‫ʹͱ͍͜ͳ͍ͯ͠ݱ‬஫ҙ͢Δɽ‫߆͍ڧ‬ଋ৚݅Λ՝͢͜ͱͰɼཧ
࿦ͷ਺ֶతഎ‫ܠ‬͸ഒԽ‫ز‬Կֶ͔Β ҰൠԽ‫ز‬Կֶ΁Ҡߦ͢Δɽ͜ͷҙຯͰɼഒԽ‫ز‬Կֶ͕࣮‫͢ݱ‬Δ
ͳΒ͹ɼͦΕ͸‫ط‬ଘͷҰൠԽ‫ز‬ԿֶΛ಺แ͍ͯ͠Δ΂͖Ͱ͋Δɽ·ͨɼ2D ࣍‫ݩ‬ͷඪతۭؒ (ۭ࣌ଟ༷ମ) ͸ η ಺ੵʹରͯ͠࠷େ౳ํ (maximally isotropic) ͳೋͭͷ෦෼ۭؒʹ෼ׂ͞ΕΔɽ
໰୊͸ɼͲͪΒΛ෺ཧతۭ࣌ͱΈͳ͔͢ (xµ ଆͱΈͳ͔͢) Ͱ͋ΔɽO(D, D) ߏ଄ͷ‫ ྔܭ‬η ʹΑͬͯ࡞ΒΕΔ෦෼ۭؒͱͯ͠ͷۭ࣌ͷಛఆ͸ɼ[24] Ͱ‫͕͋ٴݴ‬ΔɽഒԽ‫ز‬Կֶͷ਺ֶతͳඳ૾
ͱͯۙ͠೥஫໨͞Ε͍ͯΔͷ͕ɼύϥΤϧϛʔτ‫ز‬Կֶ (para-Hermitian geometry) [14, 15, 25]΍ɼͦͷ֦ுͰ͋ΔϘϧϯ‫ز‬Կֶ (Born geometry) [26, 27] Ͱ͋Δɽ͜ΕΒͷ‫ز‬Կֶ͸ɼύϥෳૉ (para-complex) ߏ଄ʹΑͬͯࣗવʹ 2D ࣍‫ݩ‬ͷఈۭؒ (ۭ࣌ଟ༷ମ) ͷ઀ۭؒͷ෼ׂΛ༩͑Δͱ͍͏ɼഒԽ‫ز‬Կֶͷ‫ه‬ड़ʹ͋ͨΓඇৗʹΑ͍ੑ࣭Λ΋ͭɽৄࡉ͸ 4 ষͰٞ࿦͢Δɽ
͍ͭͮͯɼήʔδ୅਺ʹؔͯ͠΋‫͏͓ͯ͜͠ٴݴ‬ɽऑ͍߆ଋ৚݅͸ɼO(D, D) ‫Ͱࣜܗ‬͸ҎԼͷ

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13Α͏ʹද͞ΕΔɽΦ Λ೚ҙͷ৔΍ήʔδύϥϝʔλͱͯ͠η M N ∂M ∂N Φ = 0.

(2.20)

ಉ༷ʹɼ‫߆͍ڧ‬ଋ৚݅͸ҎԼͷΑ͏ʹද͞ΕΔɽΦi Λ೚ҙͷ৔΍ήʔδύϥϝʔλͱͯ͠η M N ∂M Φ1 ∂N Φ2 = 0.

(2.21)

લઅͰड़΂ͨ௨Γɼ‫߆͍ڧ‬ଋ৚݅Λ՝͢͜ͱ͸ɼ(2.17) ͷ F Λফ‫͠ڈ‬ɼDFT ͷήʔδ୅਺͕ด
͡ΔͨΊͷे෼৚݅Ͱ͋Δɽ೚ҙͷ‫·[ ੵހׅ‬, ·] ͷ Jacobiator ͸ɼҎԼͷΑ͏ʹఆٛ͞ΕΔɽ

Jac(Ξ1 , Ξ2 , Ξ3 ) = [[Ξ1 , Ξ2 ], Ξ3 ] + [[Ξ2 , Ξ3 ], Ξ1 ] + [[Ξ3 , Ξ1 ], Ξ2 ].

(2.22)

͜ͷఆٛʹͷͬͱΓ‫͢ࢉܭ‬ΔͱɼҰൠʹ C ‫ހׅ‬ͷ Jacobiator JC ͸ҎԼͷΑ͏ʹͳΔɽಋग़͸

Appendix A.1 Λࢀরͷ͜ͱɽJC (Ξ1 , Ξ2 , Ξ3 ) = ∂ • NC (Ξ1 , Ξ2 , Ξ3 ) + SCC (Ξ1 , Ξ2 , Ξ3 ).

(2.23)

ͨͩ͠ɼ∂ • ͸ɼη M N Ͱ଍্͕͕ͬͨඍ෼ԋࢉࢠͰ͋Δɽ·ͨɼNC ͸ Ξi ʹؔͯ͠‫׬‬શ൓ରশͳҎԼͷྔNC (Ξ1 , Ξ2 , Ξ3 ) =(1'NηM N [Ξ1 , Ξ2 ]M
C Ξ3 + c.p.
3(2.24)

Ͱ͋ΓɼSCC ͸‫߆͍ڧ‬ଋ৚݅Λ՝ͤ͹ফ͑Δ߲Ͱ͋Δɽ‫߆͍ڧ‬ଋ৚݅Λ՝ͨ͠‫ޙ‬ɼ(2.23) ͷӈล͸Ұൠʹ 0 Ͱ͸ͳ͍ɽC ‫ހׅ‬͸ Jacobi ཯Λຬͨͣ͞ɼগͳ͘ͱ΋ DFT ͷήʔδ୅਺͸ Lie ୅਺Ͱ͸ͳ͍͜ͱ͕௚ͪʹΘ͔Δɽ·ͨɼ௨ৗͷ Lie ඍ෼ͷަ‫ੵࢠ׵‬ͷ֦ுͱͯ͠ɼҎԼͷΑ͏ʹ D‫[ ހׅ‬18] Λಋೖͯ͠΋ɼC ‫ͱހׅ‬ಉ͡୅਺ߏ଄͕ಘΒΕΔɽ
M&[Ξ1 , Ξ2 ]M
D = LΞ 1 Ξ 2 .

(2.25)D ‫ ͱހׅ‬C ‫ހׅ‬͸ҎԼͷΑ͏ʹؔ܎͢Δɽ

1 MN
M
L
[Ξ1 , Ξ2 ]M∂N (ηKL ΞK
D = [Ξ1 , Ξ2 ]C + η
1 Ξ2 ).
2(2.26)

D ‫ހׅ‬͸൓ରশͰ͸ͳ͍͕ɼJacobi ཯Λຬͨ͢ҰํɼC ‫ހׅ‬͸൓ରশͰ Jacobi ཯Λຬͨ͞ͳ͍ɽ
D ‫ ͱހׅ‬C ‫ހׅ‬͸ੑ࣭తʹදཪͷؔ܎ʹ͋Δɽຊम࢜࿦จͰ͸ɼC ‫ʹހׅ‬ΑΔఆٛΛ࠾༻͢ΔɽC ‫ހׅ‬ͷ‫ن‬ఆ͢Δήʔδ୅਺͸ɼྫ͑͹ॳ‫ظ‬ͷจ‫Ͱݙ‬͸ [29,30] ͰѻΘΕ͍ͯΔ΄͔ɼ[19–21] ͳͲͰٞ࿦͞Ε͍ͯΔɽC ‫ʹހׅ‬ରͯ͠ ∂˜µ Φ = 0 Λ՝͠ɼ‫߆͍ڧ‬ଋ৚݅Λࣗ໌ʹղ͘ͱɼC ‫ހׅ‬͸(2.2) ࣜͷ Courant ‫[ ހׅ‬17] ʹมԽ͢Δɽಉ༷ʹɼD ‫ހׅ‬͸ Dorfman bracket operator [33,34,53]
ʹมԽ͢ΔɽC ‫͔ހׅ‬Β Courant ‫ހׅ‬ΛಘΔաఔ͸ɼAppendix A.2 ʹࣔͨ͠ɽ͜ͷΑ͏ʹಘΒ
ΕΔ Courant ‫ހׅ‬΍ Dorfman ‫ހׅ‬͸ɼ͍ͣΕ΋ Courant ѥ୅਺ͱ‫ݺ‬͹ΕΔߏ଄Λఆٛ͢Δ [34]ɽ
ΑͬͯɼC ‫ހׅ‬͸ҰൠԽ‫ز‬Կֶ [11, 31] ʹ‫ݱ‬ΕΔ Courant ‫[ ހׅ‬17] Λ O(D, D) ‫ڞ‬มԽͨ͠΋ͷ
ͱ͍͑Δ [32]ɽ·ͨɼ(2.16) Ͱࣔͨ͠ C ‫ހׅ‬ͷ‫ܗ‬͸ɼ[35] Ͱఆٛ͞Εͨ Courant ‫ࣜܗͱހׅ‬తʹ͸ಉ͡‫͋Ͱܗ‬Δɽ͔͠͠ɼ[35] ͷ Courant ‫ހׅ‬ͷ Jacobiator ͷ‫݁ࢉܭ‬Ռ͸ (2.23) ͱ͸ҟͳΔ
ͨΊɼC ‫ࣗހׅ‬਎͸ Courant ѥ୅਺Λఆٛ͠ͳ͍ɽۙ೥ɼഒԽۭؒΛύϥΤϧϛʔτ‫ز‬Կֶʹ

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14Αͬͯ਺ֶతʹ࣮‫ͤ͞ݱ‬ΔࢼΈ͕͋ΓɼͦͷաఔͰ C ‫ن͕ހׅ‬ఆ͢Δߏ଄͸ metric ѥ୅਺Ͱ͋Δͱ͍͏ࢦఠ͕ͳ͞Εͨ [14, 15]ɽ·ͨɼmetric ѥ୅਺ͱຊ࣭తʹ͸ಉ͡ߏ଄͕ɼ[36] ʹ͓͍ͯ

pre-DFT ѥ୅਺ͱͯ͠༩͑ΒΕ͍ͯΔɽࠞཚΛආ͚ΔͨΊɼຊम࢜࿦จͰ͸ metric ѥ୅਺ͷ͜ͱΛɼൃ‫ऀݟ‬ͷ໊Λͱͬͯ Vaisman ѥ୅਺ͱ͍͏‫ݺ‬শͰ౷Ұ͢ΔɽVaisman ѥ୅਺͸ Courantѥ୅਺Λ͞ΒʹҰൠԽͨ֓͠೦Ͱ͋Δɽ࣍ষҎ߱ͰɼVaisman ѥ୅਺͕಺แ͢Δߏ଄΍ੑ࣭ʹ͍ͭͯड़΂ɼ‫߆͍ڧ‬ଋ৚݅ͷ਺ֶతͳग़ࣗʹ͍ͭͯٞ࿦͍ͯ͘͠ɽ

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15

ୈ3ষ

ѥ୅਺ (Algebroid) ͷಋೖ͜ͷষͰ͸ɼछʑͷ୅਺ߏ଄ͱɼ(classical) Drinfel’d double ͱ͍͏ૢ࡞ʹ͍ͭͯ঺հ͢Δɽ

Drinfel’d double ͸‫ݩ‬ʑ Hopf ୅਺ʹରͯ͠ߟҊ͞Εɼ૒ରͳ Hopf ୅਺ͷ௚࿨ʹΑͬͯɼ৽ͨͳ
Hopf ୅਺͕ಘΒΕΔ͜ͱΛࢦ͢ɽHopf ୅਺͸Ұ‫͑ݴͰݴ‬͹ Lie ୅਺Λύϥϝʔλม‫ͨ͠ܗ‬΋ͷͰɼύϥϝʔλʔʹରͯ͋͠Δ‫ݶۃ‬ΛͱΔͱ Lie ୅਺͕ಘΒΕΔɽHopf ୅਺͔Β Lie ୅਺ΛಘΔૢ࡞Λ ‫ݹ‬య (classical) ‫ݶۃ‬ΛͱΔͱ͍͏ɽHopf ୅਺ͷ Drinfel’d double ͷ‫ݹ‬య‫ͯ͠ͱݶۃ‬ɼLie୅਺ͷ (classical) Drinfel’d double Λߟ͑ΒΕΔɽ͜Εʹ͍ͭͯ͸ɼจ‫[ ݙ‬37, 38] ʹৄ͍͠ɽ͜
ͷૢ࡞͸ɼDFT ͷ‫߆͍ڧ‬ଋ৚݅ͷ୅਺తͳ‫ݯى‬Λ஌ΔͨΊͷ‫ͳͱૅج‬ΔɽLie ୅਺ͷ Drinfel’ddouble ͷ֦ுͱͯ͠ɼจ‫[ ݙ‬35] ʹ͓͍ͯ Lie ѥ୅਺ (algebroid) ͷ Drinfel’d double ͕ߟҊ͞Εͨɽզʑ͸ɼͦͷ͞ΒʹҰൠԽͱͯ͠ C ୅਺͕‫ن‬ఆ͢Δ Vaisman ѥ୅਺͕ɼLie ѥ୅਺ʹରͯ͠

Drinfel’d double ʹྨࣅͨ͠ૢ࡞Λߦ͏͜ͱͰಘΒΕΔ͜ͱΛࣔͨ͠ [13]ɽ͜ͷૢ࡞ΛɼจதͰ͸ͨͩ୯ʹ “double” ͱ‫Ϳݺ‬ɽ

3.1 Lieʢ૒ʣ୅਺ͱ Drinfel’d doubleॳΊʹɼLie ୅਺ΛҎԼͷΑ͏ʹఆٛ͢Δɽ
✓
Lie ୅਺ gV Λ ମ K ্ͷϕΫτϧۭؒͱ͢ΔɽV ͷ‫ݩ‬ΛҾ਺ͱ͢Δ൓ରশ͔ͭ૒ઢ‫ͳܕ‬ೋ߲ԋࢉͱ͠✏

ͯɼLie ‫( ހׅ‬bracket) [·, ·] : V × V → V ͕༩͑ΒΕΔͱ͖ɼV ͱ Lie ‫ހׅ‬ͷ૊ (V, [·, ·]) Λ

Lie ୅਺ͱ͍͍ɼg ͱද‫͢ه‬ΔɽLie ‫ ހׅ‬͸ Jacobi ཯Λຬͨ͢ɽ͢ͳΘͪɼx, y, z ∈ g Ͱ͋Δͱ͖✒[[x, y], z] + [[y, z], x] + [[z, x], y] = 0.
∗

(3.1)
∗

∗

V ͷ૒ର (dual) ϕΫτϧۭؒ V ʹରͯ͠΋ɼಉ༷ʹ Lie ୅਺ΛఆٛͰ͖ΔɽV ͱ V ͷ‫ݩ‬ΛҾ਺ͱ͢Δ Lie ‫·[ ހׅ‬, ·]∗ : V ∗ × V ∗ → V ∗ ͷ૊ (V ∗ , [·, ·]∗ ) Λɼg ʹର͢Δ૒ରͳ Lie ୅਺ g∗

ͱ͍͏ɽg ͱ g∗ ͷؒʹ͸ࣗવʹ಺ੵ ⟨·, ·⟩ ͕ఆٛ͞ΕΔɽ⟨·, ·⟩ ͸ K ্ͷ஋ΛͱΔɽ࣍ʹɼϕΫτϧۭؒ R ʹΑΔ Lie ୅਺ g ͷද‫ ݱ‬ϱ Λߟ͑Δɽx ∈ g, m ∈ R ͱ͢Δͱɼx ͸Ұൠʹɼ m ʹରͯ͠ ϱ(x) · m ͱ࡞༻͢Δɽಛʹɼg ͕ g ࣗ਎΁࡞༻͢Δ৔߹Λߟ͑Δ͜ͱͰɼg ͷ
ਵ൐ (adjoint) ද‫ ݱ‬ad : x ∈ g +→ adx ∈ End g Λ‫ن‬ఆͰ͖Δɽਵ൐ද‫ݱ‬͸ɼLie ‫ހׅ‬Λ༻͍Δͱ

✑

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16ҎԼͷΑ͏ʹॻ͖௚ͤΔɽadx (y) = [x, y],x, y ∈ g.

(3.2)

x ∈ g ͸೚ҙͷςϯιϧੵ ⊗p g = g ⊗ g ⊗ · · · ⊗ g ΁ɼҎԼͷΑ͏ʹ࡞༻͢Δɽϱ(x) · (y1 ⊗ · · · ⊗ yp ) = ad(p)x (y1 ⊗ · · · ⊗ yp )= adx (y1 ) ⊗ y2 ⊗ · · · ⊗ yp + y1 ⊗ adx (y2 ) ⊗ · · · ⊗ yp + · · ·· · · + y1 ⊗ · · · ⊗ adx (yp ).

(3.3)͕ͨͬͯ͠ɼad ͸ Leibniz ଇΛຬͨ͢ɽLie ‫ੵހׅ‬ͷ Jacobi ཯ (3.2) ͸ɼad ͷ Leibniz ଇͱͯ͠΋ղऍͰ͖Δɽadz ([x, y]) = [adz (x), y] + [x, adz (y)].

(3.4)ಉ༷ʹɼp ࣍ͷ֎ੵ୅਺ ∧p g ΁ͷ x ∈ g ͷ࡞༻Λߟ͑Δɽ‫׬‬શ൓ରশͳςϯιϧੵ ⊗p g ʹର͢Δ

࡞༻Λߟ͑Ε͹Α͘ɼҎԼͷΑ͏ʹఆٛͰ͖Δɽϱ(x) · y1 ∧ y2 = [x, y1 ] ∧ y2 + y1 ∧ [x, y2 ].

(3.5)͜ͷͱ͖ɼଟ༷ମͷ༨઀ଋ্ͷ֎ඍ෼ԋࢉࢠ d ͱಉ͡Α͏ʹߟ͑Ε͹ɼԋࢉࢠ d : ∧p g∗ → ∧p+1 g∗ΛఆٛͰ͖Δɽd2 = 0 Ͱ͋Δɽಉ༷ͷखॱͰɼg∗ ͷද‫ͦͱݱ‬ͷ࡞༻Λߟ͍͑ͯ͘͜ͱͰɼd ͱ૒ରͳ֎ඍ෼࡞༻ૉ d∗ : ∧p g → ∧p+1 g ΋ఆٛͰ͖Δɽd2∗ = 0 Ͱ͋Δɽ͜ΕΒΛ༻͍Δ͜ͱͰɼgʹରͯ͠ɼLie ୅਺ίϗϞϩδʔΛఆٛͰ͖Δ [39]ɽ

Lie ‫ހׅ‬͸ɼҾ਺͕ߴ࣍ ∧p g ͷ৔߹΁ͱ֦ுͰ͖Δɽ͜ΕΛ Schouten-Nijenhuis ‫·[ ހׅ‬, ·]S ͱ͍͏ [40]ɽ[·, ·]S ͸ҎԼͷΑ͏ʹఆٛ͞ΕΔɽͨͩ͠ɼa ∈ ∧p g, b ∈ ∧q g, c ∈ ∧r g Ͱ͋Δɽ(i) [a, b]S = −(−)(p−1)(q−1) [b, a]S .

(ii) [a, b ∧ c]S = [a, b]S ∧ c + (−)(p−1)q b ∧ [a, c]S .

(iii) (−)(p−1)(r−1) [a, [b, c]S ]S + (−)(q−1)(r−1) [b, [c, a]S ]S + (−)(r−1)(q−1) [c, [a, b]S ]S = 0.
(iv) [·, ·]S ͷҾ਺ͷɼগͳ͘ͱ΋ͲͪΒ͔Ұํ͕ ∧0 g = K Ͱ͋Δͱ͖ɼͦͷ݁Ռ͸ 0 ʹͳΔ.
Schouten-Nijenhuis ‫ހׅ‬͸ɼGerstenhaber ୅਺Λ੒͢ [41]ɽ࣍ʹɼg ͱ g∗ ͷؔ܎Λௐ΂ΔɽҰൠʹ Lie ‫·[ ހׅ‬, ·] ͸ɼg ͷ;ͨͭͷ‫͔ݩ‬Βผͷ g ͷ‫ݩ‬ΛಘΔͷͰɼ૒ઢ‫( ૾ࣸܗ‬bilinear map) ͱΈͳͤΔɽLie ‫͕ހׅ‬൓ରশͳ͜ͱΛߟྀͯ͠ɼ[·, ·] Λ

µ : ∧2 g → g ͱ͢Δͱɼ͜Εʹ૒ରͳ Lie ‫·[ ހׅ‬, ·]∗ ͸ µ∗ : ∧2 g∗ → g∗ ͱॻ͚Δɽµ ͷ co-bracketΛ δ : g → ∧2 g ͢Δɽx ∈ g, ξ ∈ ∧2 g∗ ͱ͢Δͱɼµ∗ ͷਵ൐ µ∗∗ : g → ∧2 g ͸಺ੵΛ௨ͯ͠⟨x, µ∗ (ξ)⟩ = ⟨µ∗∗ (x), ξ⟩(3.6)

ͱ༩͑ΒΕΔ͜ͱ͔Βɼδ ͸ µ∗∗ ͰఆٛͰ͖Δɽ૒ରͳ Lie ‫ ހׅ‬µ∗ ͕ Jacobi ཯Λຬͨ͢͜ͱ͔
Βɼδ ͸ Jacobi ཯Λຬͨ͢ɽ‫͑׵͍ݴ‬Δͱɼ૒ରͳ Lie ‫ ހׅ‬µ∗ ͸ɼδ ∗ ʹΑͬͯఆٛͰ͖Δɽ͠

͕ͨͬͯɼLie ୅਺ g = (V, [·, ·]) ͱɼ[·, ·] ͷ co-bracket δ Λఆ͓͚ٛͯ͠͹ɼ૒ରͳ Lie ୅਺g∗ = (V ∗ , [·, ·]∗ ) ͸಺ੵʹΑͬͯࣗવʹ༠ಋͰ͖Δɽ

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Lie ૒୅਺

Lie ୅਺ (V, [·, ·]) Λఆٛ͠ɼ[·, ·] ͷ co-bracket Λ δ ͱ͢Δɽಛʹɼco-bracket δ ͕ ࣍ͷ✏

1-cocycle ৚݅Λຬͨ͢৔߹ɼ

(2)
δ([x, y]) = ad(2)x δ(y) − ady δ(x),x, y ∈ g,(3.7)(g, µ, δ) ͷ૊ͷ͜ͱΛ Lie ૒୅਺ͱ͍͏ɽ
✒(g, µ, δ) ͕ Lie ૒୅਺ͳΒ͹ɼ(g∗ , µ∗ , δ ∗ ) ΋ಉ͡ Lie ૒୅਺Λ༩͑ΔɽΑͬͯɼࠓ‫ޙ‬͸ Lie ૒୅✑਺ͷ͜ͱΛ (g, g∗ ) ͱද‫͢ه‬Δɽ
࣍ʹɼLie ૒୅਺ʹରͯ͠ Drinfel’d double Λߦ͏ɽLie ૒୅਺ (g, g∗ ) Λߏ੒͍ͯ͠Δ g, g∗ ʹରͯ͠ɼͦͷ௚࿨ d = g ⊕ g∗ Λߟ͑Δͱɼd ্ʹඇୀԽ (non-degenerate) Ͱରশͳ૒ઢ‫ࣜܗܕ‬(bilinear form) (·, ·) ΛҎԼͷΑ͏ʹఆٛͰ͖Δɽ
(x, y) = (ξ, η) = 0,x, y ∈ g, ξ, η ∈ g∗ .

(x, ξ) = ⟨ξ, x⟩,(3.8)ଓ͍ͯɼ্ड़ͷ૒ઢ‫ࣜܗܕ‬ΛෆมʹอͭΑ͏ʹ d ্ʹ‫·[ ੵހׅ‬, ·]d Λఆٛ͢Δɽ͜͜Ͱɼ૒ઢ‫ܕ‬‫ࣜܗ‬Λෆมʹอͭͱ͸ҎԼͷؔ܎͕੒ཱ͢Δ͜ͱΛࢦ͢ɽ(y, [x, ξ]d ) = ([y, x]d , ξ).

(3.9)

·ͣɼg, g∗ ͕ͦΕͧΕ d ͷ෦෼୅਺ (subalgebra) Ͱ͋Δ͜ͱ͔Βɼ[x, y]d = [x, y],[ξ, η]d = [ξ, η]∗ ,x, y ∈ g, ξ, η ∈ g∗ .

(3.10)

ͱ͢Δͷ͕ࣗવͰ͋Δɽ࣍ʹɼg ͱ g∗ ͕Ϋϩε͢Δ৔߹ [x, ξ]d Λߟ͑Δɽ

(y, [x, ξ]d ) = ([y, x]d , ξ)= ([y, x], ξ) = ⟨ξ, [y, x]⟩ = ⟨ξ, −adx (y)⟩ = ⟨ad∗x ξ, y⟩ = (y, ad∗x ξ).

(3.11)

͜͜Ͱɼ1 ஈ໨ͷ౳߸͸‫·[ ੵހׅ‬, ·]d ͕૒ઢ‫ࣜܗܕ‬Λෆมʹอͭͱ͍͏ཁ੥͔Β੒ཱ͢Δɽ·ͨɼ

2 ஈ໨ͷ࠷΋ࠨͷ౳߸͸ g ͕ Lie ෦෼୅਺ͳͷͰɼ(3.10) ʹΑΓ੒ཱ͢Δɽ·ͨɼ2 ஈ໨ͷӈ͔Β
2 ൪໨ͷ౳߸͸ɼad ͷ co-adjoint Λ ad∗x = −(adx )∗ ͱఆٛͨ͜͠ͱʹΑΓ੒ཱ͢Δɽಉ༷ͷٞ࿦͔Βɼ(η, [x, ξ]d ) = −(η, ad∗ξ x) Ͱ͋Δ͜ͱ΋ಋ͚ΔɽΑͬͯɼ[x, ξ]d ͸ҎԼͷΑ͏ʹॻ͚Δɽ

[x, ξ]d = −ad∗ξ x + ad∗x ξ.

(3.12)

ࣜ (3.7), (3.10), (3.12) ͔Βɼ[·, ·]d ΋ Jacobi ཯Λຬͨ͢͜ͱ͕Θ͔ΔɽΑͬͯɼg ͱ g∗ ͷؒʹ

1-cocycle ৚݅ (3.7) ͕੒ཱ͍ͯͯ͠ɼ(g, g∗ ) ͕ Lie ૒୅਺Λͳ͢ͱ͖ɼద੾ʹ [·, ·]d ΛఆΊΕ͹(d = g ⊕ g∗ , [·, ·]d ) ͸ Lie ୅਺ͱͳΔɽ͜ͷΑ͏ʹɼ৽ͨͳ Lie ୅਺ d Λߏங͢Δૢ࡞ͷ͜ͱΛLie ૒୅਺ͷ (classical) Drinfel’d double ͱ͍͏ɽඇୀԽͳ૒ઢ‫ͱࣜܗܕ‬ɼͦΕΛෆมʹอͭΑ͏ͳ Lie ‫ʹހׅ‬ΑΔ Lie ୅਺ͱͷ૊Λɼಛʹ quadratic Lie ୅਺ͱ͍͏ɽd ͸ࣗવͱ quadratic

Lie ୅਺ͱͳΔɽ
Drinfel’d double ͷ‫ٯ‬Λߟ͑Δ͜ͱ΋Ͱ͖Δɽp Λ quadratic Lie ୅਺ͱ͢Δɽa, b ͕ p ͷ෦෼୅਺Ͱ͋Γɼ͔ͭ p ͷ૒ઢ‫ʹࣜܗܕ‬ରͯ͠౳ํత (isotropic) Ͱ͋Δ*1 ͱ͖ɼ(p, a, b) ͷ૊Λ

Manin triple ͱ͍͏ [42]ɽ্ड़ͷ (d, g, g∗ ) ͷ૊͸ɼ౰વ Manin triple Ͱ͋Δɽ
*1 g ͕ (·, ·) ʹରͯ͠౳ํతͱ͸ɼ೚ҙͷ x, y ∈ g Ͱ (x, y) = 0 Ͱ͋Δ͜ͱΛ͍͏ɽ

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18

3.2 Lie ѥ୅਺͜͜Ͱ͸ɼLie ୅਺ͷҰൠԽͱͯ͠ଟ༷ମ্ͷϕΫτϧଋΛ༻͍ͨ Lie ѥ୅਺ͷఆٛ [43] Λ༩✓Lie ѥ୅਺ E✏

͑Δɽͦͷ‫ޙ‬ɼLie ୅਺ͱͷେ·͔ͳҧ͍ʹ͍ͭͯड़΂ΔɽҎԼͷߏ଄ͷ૊ (E, [·, ·]E , ρ) ΛɼLie ѥ୅਺ͱ͍͏ɽ
π• M ্ͷϕΫτϧଋ E −→M• E ͷ੾அ (section) Γ(E) ΛҾ਺ͱ͢Δ Lie ‫·[ ހׅ‬, ·]E : Γ(E) × Γ(E) → Γ(E)• E ͔Β઀ଋ T M ΁ͷଋࣸ૾ (bundle map) ρ : E → T M .

ρ ͱ [·, ·]E ͷؒʹɼҎԼͷؔ܎͕੒ཱ͢Δ͜ͱΛཁ‫͢ٻ‬Δɽ
ρ([X, Y ]E ) = [ρ(X), ρ(Y )],X, Y ∈ Γ(E).

(3.13)

ρ Λ anchor map ͱ‫Ϳݺ‬ɽ
Lie ‫·[ ހׅ‬, ·]E ͸ Jacobi ཯Λຬͨ͢ɽ·ͨɼLeibniz ଇΛຬͨ͢ɽ͢ͳΘͪɼؔ਺ f ∈ C ∞ (M )ʹରͯ͠ɼ[X, f Y ]E = (ρ(X) · f )Y + f [X, Y ]E .

(3.14)(ρ(X) · f ) ͸ɼρ(X) ͕ f ΁ͷඍ෼ͱͯ͠࡞༻͢Δ͜ͱΛҙຯ͢Δɽ
✒E ͷ੾அͰಘΒΕΔϕΫτϧ Γ(E) ͸ɼM ͷ֤఺ͱϕΫτϧۭ͕ؒҰରҰͰରԠͨ͠΋ͷͱΈ✑ͳͤΔɽҰํͰɼલઅͰड़΂ͨ௨ΓɼLie ୅਺͸ ϕΫτϧۭؒʹରͯ͠ఆٛ͞ΕΔɽ͕ͨͬͯ͠

Lie ѥ୅਺͸ɼ M ͷ֤఺ͱ Γ(E) ্ͷ Lie ୅਺ΛҰରҰͰରԠͤͯ͞࡞ͬͨɼ Lie ୅਺ͷू߹Λѻ͍ͬͯΔͱΈͳͤΔɽ͜ͷҙຯͰɼLie ѥ୅਺͸ Lie ୅਺ͷҰൠԽͰ͋ΔɽM ͕Ұ఺Ͱ͋Γɼ

anchor ͕ ρ = 0 Ͱ͋Δͱ͖ Lie ѥ୅਺͸ Lie ୅਺ͱͳΔɽ
Lie ୅਺ g ʹରͯ͠ɼ૒ରͳ Lie ୅਺ g∗ ͕ఆٛͰ͖ͨΑ͏ʹɼE ʹ૒ରͳϕΫτϧଋ E ∗ Λ༻͍ͯɼಉ͡ଟ༷ମ্ʹ૒ରͳ Lie ѥ୅਺ E ∗ = (E ∗ , ρ∗ , [·, ·]E ∗ ) ΛఆٛͰ͖Δɽ[·, ·]E ∗ : Γ(E ∗ ) ×

Γ(E ∗ ) → Γ(E ∗ ) ͸ [·, ·]E ʹରͯ͠૒ରͳ Lie ‫͋Ͱހׅ‬Δɽ·ͨɼanchor map ρ∗ : E ∗ → T M

Ͱ͋ΔɽE ͱ E ∗ ͷؒʹ͸ɼࣗવʹ಺ੵ ⟨·, ·⟩ ͕ఆٛͰ͖Δɽ

Lie ୅਺ͷ৔߹͸ɼLie ‫ހׅ‬Λ Schouten-Nijenhuis ‫·[ ހׅ‬, ·]S ʹҰൠԽͰ͖ͨɽLie ‫ހׅ‬ͷ

co-bracket ͕ [·, ·]S Λ‫ؚ‬Μͩ৚݅ࣜͰ͋Δ 1-cocycle ৚݅ (3.7) Λຬͨ͢ͷͰ͋Ε͹ Lie ૒୅਺͕ߏ੒Ͱ͖ɼ૒ରͳ Lie ୅਺Λ༠ಋͰ͖ͨɽͦͷྨਪͱͯ͠ɼLie ѥ୅਺͔Β Lie ૒ѥ୅਺(bialgebroid) ͕ಘΒΕΔ͜ͱΛ֬ೝ͢ΔɽͦͷͨΊʹɼLie ‫·[ ހׅ‬, ·]E ͷҰൠԽΛߦ͏ɽ·ͣ͸ɼඞཁͱͳΔԋࢉΛઌʹఆ͓ٛͯ͘͠ɽ

Γ(T M ) ΍ Γ(T ∗ M ) ͔Β֎ੵ୅਺Λߏங͢Δͷͱಉ͡Α͏ʹɼΓ(E) ΍ Γ(E ∗ ) ͔Β p-vectorΓ(∧p E) ΍ p-form Γ(∧p E ∗ ) Λఆٛ͢Δɽ·ͣɼΓ(∧p E) ΍ Γ(∧p E ∗ ) ʹରͯ͠ɼ௨ৗͷඍ෼ p ‫ܗ‬ࣜͱಉ༷ʹ֎ඍ෼ԋࢉࢠ dɼd∗ ΛఆٛͰ͖Δɽͨͱ͑͹ɼd ͸ɼξ ∈ Γ(∧p E ∗ ) ʹରͯ࣍͠ͷΑ͏

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p+1
)
i=1

+

'
(
(−)i+1 ρ(Xi ) · ξ(X1 , . . . , X̌i , . . . , Xp+1 ))(−)i+j ξ([Xi , Xj ]E , X1 , . . . , X̌i , . . . , X̌j , . . . , Xp+1 ).

(3.15)

i<jͨͩ͠ɼX̌i ͸ i ൪໨ͷ X Λফ‫͢ڈ‬Δͱ͍͏ҙຯͰ͋Δɽ͜ͷఆ͔ٛΒɼd ͸ҎԼͷΑ͏ͳੑ࣭Λຬͨ͢ɽX, Y ∈ Γ(E), ξ, η ∈ Γ(E ∗ ) ͱͨ͠ͱ͖ɼd(ξ ∧ η) = dξ ∧ η − ξ ∧ dη,df (X) = ρ(X) · f,

(3.16)dξ([X, Y ]E ) = ρ(X) · (ξ(Y )) − ρ(Y ) · (ξ(X)) − ξ([X, Y ]E ).
ಉ༷ʹɼΓ(E) ʹର͢Δ಺෦ੵ ιX : Γ(∧p E ∗ ) → Γ(∧p−1 E ∗ ) ͸࣍ͷΑ͏ʹఆٛͰ͖ΔɽιX (ξ(Y1 , . . . , Yp−1 )) = ξ(X, Y1 , . . . , Yp−1 ).

(3.17)

͜ΕΒͷ૊Έ߹Θ͔ͤΒɼΓ(E) ʹର͢Δ Lie ඍ෼ LX : Γ(∧p E ∗ ) → Γ(∧p E ∗ ) ͸ҎԼͷΑ͏ʹఆٛͰ͖Δɽ

LX (ξ)(Y1 , . . . , Yp ) = ρ(X) · (ξ(Y1 , . . . , Yp )) −p)

ξ (Y1 , . . . , [X, Yi ]E , . . . , Yp ) .

(3.18)i=1͜ΕΒͷԋࢉࢠ͸ɼҎԼͷΑ͏ͳؔ܎ࣜΛຬͨ͢ɽX, Y ∈ Γ(E), f ∈ C ∞ (M ), ξ ∈ Γ(∧p E ∗ ) ͱͨ͠ͱ͖ɼL[X,Y ]E = LX · LY − LY · LX ,
ι[X,Y ]E = LX · ιY − ιY · LX ,LX = d · ιX + ιX · d,

(3.19)

Lf X (ξ) = f LX (ξ) + df ∧ ιX (ξ),

LX ⟨ξ, Y ⟩ = ⟨LX ξ, Y ⟩ + ⟨ξ, LX Y ⟩,ಉ༷ʹɼΓ(E ∗ ) ʹର͢Δ಺෦ੵͱ Lie ඍ෼΋ఆٛͰ͖Δɽ͜ΕͰɼΓ(∧p E) ΍ Γ(∧p E ∗ ) ΛऔΓѻ͑ΔΑ͏ʹͳͬͨɽ࣍ʹɼLie ‫ހׅ‬ΛҾ਺͕ Γ(∧p E) ͷ৔߹΁ͱ֦ு͢Δɽ[·, ·]E ͷҰൠԽͱͯ͠ɼSchouten-Nijenhuis ‫·[ ހׅ‬, ·]S ͸ҎԼͷؔ܎ࣜΛຬͨ͢΋ͷͱͯ͠ఆٛ͞ΕΔɽA ∈ Γ(∧p E), B ∈ Γ(∧q E) ͱͨ͠ͱ͖ɼ

(i) [A, B]S = −(−)(p−1)(q−1) [B, A]S .

(ii) ಛʹɼp = 1 ͷͱ͖ [A, f ]S = ρ(A) · f .

(iii) [X, · ]S ͸ɼΓ(∧q E) ʹରͯ͠ɼp ࣍ͷඍ෼ͱͯ͠࡞༻͢Δɽ

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Lie ૒ѥ୅਺ (E, E ∗ )Lie ૒ѥ୅਺͸ҎԼͷΑ͏ʹఆٛ͞ΕΔɽE Λ Lie ѥ୅਺ɼE ∗ Λ E ͱ૒ରͳ Lie ѥ୅਺ͱ✏

͢ΔɽE ͷ‫͔ੵހׅ‬Βߏ੒ͨ͠ Schouten-Nijenhuis ‫·[ ހׅ‬, ·]S ͱɼΓ(∧p E) ʹର͢Δ֎ඍ෼ԋࢉࢠ d∗ ͷؒʹɼҎԼͷ derivation ৚݅d∗ [X, Y ]S = [d∗ X, Y ]S + [X, d∗ Y ]S

(3.20)

͕੒ཱ͍ͯ͠Δ৔߹ɼE, E ∗ ͸ Lie ૒ѥ୅਺ (E, E ∗ ) Λͳ͢ [44]ɽ

✒

∗

✑

Lie ૒ѥ୅਺͸ɼLie ૒୅਺ͷҰൠԽͰ͋Δɽ্ड़ͷఆٛͷ୅ΘΓʹɼE Λ‫ ʹݩ‬SchoutenNijenhuis ‫·[ ހׅ‬, ·]∗S Λߏ੒͠ɼΓ(∧p E ∗ ) ʹର͢Δ֎ඍ෼ԋࢉࢠ d ͱͷؒʹ derivation ৚݅Λ՝ͯ͠΋ɼಉ͡ Lie ૒ѥ୅਺ (E, E ∗ ) ͕ಘΒΕΔɽderivation ৚݅͸ɼલઅͷ Lie ୅਺ͷؒͷ

1-cocycle ৚݅ʹରԠ͢ΔɽM ͕Ұ఺Ͱɼ֤ѥ୅਺ͷ anchor ͕ 0 ͷͱ͖ɼLie ૒ѥ୅਺͸ Lie ૒୅਺ͱͳΔɽLie ૒୅਺ͷ৔߹ͱಉ༷ʹɼLie ૒ѥ୅਺ʹରͯ͠΋ Drinfel’d double Λߦ͑Δ͕ɼͦͷ݁ՌಘΒΕΔߏ଄͸ɼLie ѥ୅਺Ͱ͸ͳ͘ɼ࣍અͰड़΂Δ Courant ѥ୅਺Ͱ͋Δ [35]ɽ

3.3 Courant ѥ୅਺਺ֶʹ͓͍ͯɼ৽͍͠‫ڀݚ‬ର৅͕Α͘஌ΒΕͨ֓೦ͷ֦ு΍ҰൠԽ͔ΒಘΒΕΔ͜ͱ͸ଟ͍ɽઌड़ͷ Lie ୅਺΍ Lie ѥ୅਺΋ྫ֎Ͱ͸ͳ͍ɽඍ෼‫ز‬Կֶͷ෼໺Ͱٞ࿦͞Ε͍ͯͨ໰୊ͷͻͱͭ͸ɼʮϕΫτϧଋͷ֦ுʯͰ͋ΔɽϕΫτϧۭؒ V ͱ ૒ରϕΫτϧۭؒ V ∗ ͷ௚࿨ۭؒ V ⊕ V ∗ʹ͸ɼLie ‫ހׅ‬Λ༩͑ͯ Lie ୅਺ΛఆٛͰ͖ͨɽͰ͸ɼϕΫτϧଋ E ͱ૒ରϕΫτϧଋ E ∗ ͷ௚
࿨ E ⊕ E ∗ ʹରͯ͠͸ɼͲͷΑ͏ͳ‫͕ੵހׅ‬༩͑ΒΕɼͲͷΑ͏ͳߏ଄͕ಘΒΕΔͩΖ͏͔ɽಛʹଟ༷ମ M ͷ઀ଋ T M ͱ༨઀ଋ T ∗ M ͷ௚࿨ T M ⊕ T ∗ M Λྫʹߟ͑ΔͱɼCourant ѥ୅਺ͱ͍͏৽͍͠ߏ଄͕‫ݱ‬ΕΔ͜ͱ͕ 1990 ೥ʹࣔ͞Εͨ [17]ɽจ‫[ ݙ‬17] Ͱఏࣔ͞Εͨ Courant ‫ހׅ‬(2.2) ʹ‫͍ͯͮج‬ɼͷͪʹҰൠԽ઀ଋ T M ⊕ T ∗ M Λ‫ڀݚ‬ର৅ͱ͢ΔɼҰൠԽ‫ز‬Կֶ (generalized

geometry) ͕։ൃ͞ΕΔ͜ͱͱͳΔ [11]ɽҰൠԽ‫ز‬ԿֶͰ͸ɼఈۭؒ (ۭ࣌ଟ༷ମ) ͸ഒԽͤͣɼ
઀ଋ͚͕ͩഒԽ͞ΕΔɽҰൠԽ‫ز‬Կֶʹ͍ͭͯ͸ɼ‫ݪ‬࿦จ [11] ͷஶऀͰ͋Δ Hitchin ʹΑΔߨٛϊʔτ͕͋Δ [45] ΄͔ɼจ‫[ ݙ‬16, 31, 46] ͳͲʹৄ͍͠ɽ͜ΕΒͷจ‫Ͱݙ‬͸ɼۭؒͷ‫ܗ‬ঢ়Λ‫ن‬ఆ͢ΔྔͰ͋Δ‫཰ۂ‬΍ᎇ཰ʹ͍ͭͯ΋ٞ࿦͞Ε͍ͯΔɽ·ͨɼҰൠԽ‫ز‬Կֶͷ‫ه‬ड़ͱͯ͠ϙΞιϯ(Poisson) ଟ༷ମ্Ͱͷ‫ه‬ड़΋ߟ͑ΒΕ͍ͯΔ [47, 48]ɽ෺ཧతʹ͸௒‫ݭ‬ཧ࿦͔ΒಘΒΕΔ௒ॏྗཧ࿦ͷ͏ͪɼNS-NS sector ʹରͯ͠ͷ‫ز‬Կֶతͳඳ૾Λ༩͑ΔɽCourant ‫ʹހׅ‬Αͬͯ༩͑ΒΕΔͷ͸ɼඍ෼ಉ૬ࣸ૾ͱ B ৔ʹΑΔήʔδม‫׵‬Λ૊Έ߹ΘͤͨɼҰൠԽ઀ଋʹର͢ΔҰൠԽඍ෼ಉ૬ࣸ૾Ͱ͋Δɽจ‫[ ݙ‬49] Ͱ͸ɼ্ड़ͷϙΞιϯଟ༷ମ্ͷҰൠԽ‫ز‬ԿֶΛ༻͍ͯɼβ ௒ॏྗཧ࿦ʹ͍ͭͯ΋ٞ࿦͞Ε͍ͯΔɽΑΓҰൠʹɼE ⊕ E ∗ ͷ৔߹΋ɼLie ૒ѥ୅਺ͷ Drinfel’d double ʹΑͬͯ Courant ѥ୅਺ΛߏஙͰ͖Δ͜ͱ͕ 1997 ೥ʹࣔ͞Εͨ [35]ɽจ‫[ ݙ‬17] Ͱఏࣔ͞Εͨ Courant ‫ͱހׅ‬ɼจ‫[ ݙ‬35] Ͱఏࣔ͞Εͨ Courant ‫ހׅ‬ͷ‫ܗ‬ঢ়͸ҟͳΔɽจ‫[ ݙ‬35] ͷ Courant ‫ހׅ‬͸ɼ‫ࣜܗ‬తʹ͸ DFT ͷ C‫( ހׅ‬2.16) ͱಉ͡‫͕ͩܗ‬ɼJacobi ཯Λ‫݁ͨ͠ࢉܭ‬Ռ͕ҟͳΔͷͰɼ C ‫ن͕ހׅ‬ఆ͢Δ୅਺ߏ଄͸ Courant ѥ୅਺Ͱ͸ͳ͍ɽ͜ͷઅͰ͸ɼॳΊʹҰൠతͳ Courant ѥ୅਺ͷఆٛΛ༩͑ΔɽCourant ‫ހׅ‬ͷఆٛํ๏͸ 2 ௨

Soryushiron Kenkyu

21Γͷ౳Ձͳํ๏͕͋Δ͜ͱΛ঺հ͢Δɽͦͯ͠ɼจ‫[ ݙ‬35] ͷϨϰϡʔΛߦ͏ɽ͜ͷจ‫Ͱݙ‬ఏࣔ͞Εͨ Courant ‫ހׅ‬Λ঺հͨ͠ͷͪɼLie ૒ѥ୅਺ͷ Drinfel’d double Λߦ͏ɽ·ͣɼCourant ѥ୅਺ͷఆٛΛࣔ͢ɽ
✓
Courant ѥ୅਺✏ҎԼͷ 4 ͭͷߏ଄ͷ૊ (C, [·, ·]c , ρc , (·, ·)) Λߟ͑Δɽπ• M ্ͷϕΫτϧଋ C −→M• C ͷ੾அ Γ(C) ΛҾ਺ͱ͢Δ൓ରশͳ Courant ‫·[ ހׅ‬, ·]c : Γ(C) × Γ(C) → Γ(C)• C ͔Β઀ଋ T M ΁ͷ anchor map ρc : C → T M .

• ඇୀԽͰରশͳ૒ઢ‫·( ࣜܗܕ‬, ·)͜ͷ૊ (C, [·, ·]c , ρc , (·, ·)) ͸ɼҎԼʹࣔ͢ 5 ͭͷެཧ Axiom C1-C5 Λຬͨ͢৔߹ɼCourantѥ୅਺Λ੒͢ [35]ɽͨͩ͠ɼD ͸ Γ(C) ʹ࡞༻͢Δ֎ඍ෼ԋࢉࢠͰ͋Δɽ

Axiom C1. e1 , e2 , e3 ∈ Γ(C) ͱ͢ΔɽCourant ‫ހׅ‬ͷ Jacobiator ͸ҎԼͷΑ͏ʹͳΔɽ
[[e1 , e2 ]c , e3 ]c + c.p. = DT (e1 , e2 , e3 ).

(3.21)

ͨͩ͠ɼT (e1 , e2 , e3 ) = 13 (([e1 , e2 ]c , e3 ) + c.p.) Ͱ͋Δɽc.p. ͸ cyclic permutation Λҙຯ͢ΔɽT (e1 , e2 , e3 ) ͸ e ͷೖΕସ͑ʹؔͯ͠‫׬‬શ൓ରশͰ͋Δɽ

Axiom C2. e1 , e2 ∈ Γ(C) ʹରͯ͠ɼρc ([e1 , e2 ]c ) = [ρc (e1 ), ρc (e2 )].

(3.22)

ͨͩ͠ɼ[·, ·] ͸ T M ্ͷ Lie ‫ހׅ‬Λࢦ͢.

Axiom C3. Courant ‫ʹހׅ‬ରͯ͠ɼमਖ਼͞Εͨ Leibniz ଇ͕੒ཱ͢Δɽ͢ͳΘͪɼe1 , e2 ∈Γ(C), f ∈ C ∞ (M ) ʹରͯ͠ɼ[e1 , f e2 ]c = f [e1 , e2 ]c + (ρc (e1 ) · f )e2 − (e1 , e2 )Df.

(3.23)

Axiom C4. ρc · D = 0 Ͱ͋Δɽ‫͑׵͍ݴ‬Δͱɼf, g ∈ C ∞ (M ) ʹରͯ͠ɼ(Df, Dg) = 0.

(3.24)

Axiom C5. (·, ·) ͱ ρc ͸ཱ྆৚݅Λຬͨ͢ɽ͢ͳΘͪɼe1 , e2 , e3 ∈ Γ(C), ͱͨ͠ͱ͖
✒ρc (e1 ) · (e2 , e3 ) = ([e1 , e2 ]c + D(e1 , e2 ), e3 ) + (e2 , [e1 , e3 ]c + D(e1 , e3 )).

(3.25)จ‫[ ݙ‬50] ʹΑΔͱɼ֤ެཧͷ͏ͪɼAxiom C5 ͸ Axiom C3 ͔Βಋग़Ͱ͖ɼAxiom C2 ͔Β✑

Axiom C4 Λಋग़Ͱ͖Δɽ‫׬‬શʹಠཱ͍ͯ͠Δͷ͸ Axiom C1 ͷΈͰ͋ΔɽҎԼͷٞ࿦Ͱ͸ɼ
Axiom C1-C5 ͸શͯಠཱͳ΋ͷͱͯ͠ѻ͏ɽ͜͜ͰɼCourant ѥ୅਺ͷͷఆٛʹ͍ͭͯίϝϯτ͢Δɽ্ड़ͷ൓ରশͳ Courant ‫·[ ހׅ‬, ·]cͷ୅ΘΓʹɼDorfman bracket operator ͱ͍͏ Γ(C) ʹର͢Δೋ߲ԋࢉ ◦ Λ༻͍ͯఆٛ͢Δํ๏΋͋Δ [34, 53]ɽ͜ͷ৔߹ɼ্ʹࣔͨ͠ Axiom C1-C5 ͸ҎԼͷΑ͏ʹ Axiom C1′ -C5′ ʹஔ͖‫׵‬

Soryushiron Kenkyu

22ΘΔɽ͜ΕΒͷఆٛํ๏͕౳ՁͰ͋Δ͜ͱ͸ɼจ‫[ ݙ‬34] Ͱূ໌͞Ε͍ͯΔɽAxiom C1′ ͔ΒΘ͔ΔΑ͏ʹɼ◦ ͸ Jacobi ཯ͷ୅ΘΓʹɼࠨ Leibniz ཯Λຬͨ͢ɽ൓ରশͳ‫·[ ੵހׅ‬, ·]c ʹΑΔఆ͔ٛΒɼCourant ѥ୅਺͸ strong homotopy Lie ୅਺ (L∞ ୅਺) ͱͯ͠ղऍ͢Δ͜ͱ͕Ͱ͖Δ [51]ɽ
ҰํͰɼೋ߲ԋࢉ ◦ ʹΑΔఆ͔ٛΒɼCourant ѥ୅਺͸ Leibniz ѥ୅਺ͱͯ͠ղऍ͢Δ͜ͱ΋Ͱ͖Δɽ

e1 ◦ (e2 ◦ e3 ) = (e1 ◦ e2 ) ◦ e3 + e2 ◦ (e1 ◦ e3 )(Axiom C1′ )e1 ◦ (f e2 ) = f (e1 ◦ e2 ) + (ρc (e1 )f )e2(Axiom C3′ )ρc (e1 ◦ e2 ) = [ρc (e1 ), ρc (e2 )]e1 ◦ e1 = D(e1 , e1 )ρc (e3 )(e1 , e2 ) = (e3 ◦ e1 , e2 ) + (e1 , e3 ◦ e2 )(Axiom C2′ )(Axiom C4′ )(Axiom C5′ )ຊ࿦จͰ͸ɼ൓ରশͳ‫·[ ੵހׅ‬, ·]c ʹΑΔఆٛʹΑͬͯٞ࿦Λߦ͏*2 ɽ

ଓ͍ͯɼ[35] ʹͷͬͱΓ Lie ૒ѥ୅਺ͷ Drinfel’d double ʹΑͬͯಘΒΕΔ Courant ѥ୅਺ʹ͍ͭͯड़΂ΔɽલͷઅͰఆٛͨ͠௨Γɼ૒ରͳ;ͨͭͷ Lie ѥ୅਺ E, E ∗ Λಋೖ͢ΔɽE, E ∗ ͷ੾அͰಘΒΕΔϕΫτϧΛɼͦΕͧΕ Γ(E) = Xi , Γ(E ∗ ) = ξi ͱ͢Δɽ࣍ʹɼE ͱ E ∗ ͷ௚࿨ʹΑΓ৽ͨͳϕΫτϧଋ C = E ⊕ E ∗ Λఆٛ͢Δɽei ∈ Γ(C)ɼ͢ͳΘͪ ei = Xi + ξi ͱͨ͠ͱ͖ɼ

Γ(C) ্ͷඇୀԽͳ૒ઢ‫·( ࣜܗܕ‬, ·)± Λ(e1 , e2 )± =1(⟨ξ1 , X2 ⟩ ± ⟨ξ2 , X1 ⟩)2(3.28)

ͷΑ͏ʹఆٛ͢Δɽ(e1 , e2 )+ ͸ରশ, (e1 , e2 )− ͸൓ରশͰ͋Δɽ·ͨɼρ ͸ҎԼͷΑ͏ʹఆٛ͢Δɽρc (ei ) = ρ(Xi ) + ρ∗ (ξi ).

(3.29)

Γ(C) ʹର͢Δඍ෼ԋࢉࢠ D : C ∞ (M ) → Γ(C) ͸(Df, ei ) =1ρc (ei )f2(3.30)

ͷΑ͏ʹఆٛ͢Δɽ্ड़ͷ ρ ͷఆٛͱ߹ΘͤΔͱɼD = d + d∗ ͱද͢͜ͱ͕Ͱ͖Δɽͦͯ͠ɼ

Courant ‫·[ ހׅ‬, ·]c Λ۩ମతʹҎԼͷΑ͏ʹఆΊΔɽ[e1 , e2 ]c = [X1 , X2 ]E + Lξ1 X2 − Lξ2 X1 − d∗ (e1 , e2 )−+ [ξ1 , ξ2 ]E ∗ + LX1 ξ2 − LX2 ξ1 + d(e1 , e2 )− .

*2

(3.31)

Courant ‫ހׅ‬͸ɼDorfman bracket operator Λ൓ରশʹ૊Ή͜ͱͰఆٛͰ͖Δɽ
[e1 , e2 ]c =1((e1 ◦ e2 ) − (e2 ◦ e1 )).
2(3.26)

ҰൠԽ‫ز‬Կֶͷจ຺Ͱ͸ɼCourant ‫ͱހׅ‬ͷରൺͱͯ͠ Dorfman bracket Λରশʹ૊Ή͜ͱͰ “Dorfman

bracket”[e1 , e2 ]d =1((e1 ◦ e2 ) + (e2 ◦ e1 ))2(3.27)Λఆٛ͢Δ৔߹͕͋ΔɽDFT ͷจ຺ͰɼD ‫߆͍ڧʹހׅ‬ଋ৚݅Λ՝͢ͱಘΒΕΔͷ͸ɼ“Dorfman bracket” Ͱ͸ͳ͘ Dorfman bracket operator Ͱ͋Δɽ

Soryushiron Kenkyu

23͜ͷΑ͏ʹఆ͓ٛͯ͘͠ͱɼ(E ⊕ E ∗ , [·, ·]C , (·, ·)+ , ρc ) ͸ Courant ѥ୅਺ͱͳΔ͜ͱ͕จ‫[ ݙ‬35]
Ͱূ໌͞Εͨɽ[e1 , e2 ]c ͸ DFT ͷ C ‫ࣜܗͱހׅ‬తʹಉ͡‫͋Ͱܗ‬Δ*3 ɽจ‫[ ݙ‬35] Ͱ͸ɼDrinfel’d double ͷ‫ٯ‬΋ٞ࿦͞Ε͍ͯΔɽ͋Δ Courant ѥ୅਺Λఆٛ͢ΔϕΫτϧଋ C ʹண໨͢ΔɽC ͸ɼ૒ઢ‫·( ࣜܗܕ‬, ·)+ ʹؔͯ͠౳ํతͳ෦෼ଋ (subbundle) E, E ∗ ʹ෼ׂͰ͖ɼͳ͓͔ͭ Courant ‫·[ ހׅ‬, ·]c ͕ Γ(C) Ͱด͍ͯ͡Δ৔߹ɼ(E, E ∗ ) ͸ࣗવʹ Lie ૒ѥ୅਺ͷߏ଄Λ࣋ͭ͜ͱ͕ূ໌͞Ε͍ͯΔɽ͜͜ͰɼҎ߱ͷٞ࿦ͰඞཁͱͳΔ Dirac ߏ଄Λಋೖ͢Δɽ

Definition 3.3.1. E, E ∗ ͕ҎԼͷ;ͨͭͷ৚݅Λຬͨ͢ͱ͖ɼC ͷ Dirac ߏ଄Ͱ͋Δͱ͍͏ɽ• ࠷େ౳ํతͰ͋Δɼ͢ͳΘͪ dim E = dim E ∗ = 12 dim C Ͱ͋Δɽ

• E ্ʹดͨ͡‫·[ ੵހׅ‬, ·] (Lie ‫ )ހׅ‬ΛఆٛͰ͖Δɽ͢ͳΘͪɼ೚ҙͷ X, Y ∈ Γ(E) ʹରͯ͠ɼ[X, Y ] ∈ Γ(E) Ͱ͋ΔΑ͏ͳ [·, ·] ͕ଘࡏ͢Δɽ

E, E ∗ ͕ C ͷ Dirac ߏ଄Ͱ͋Δͱ͖ɼ(E, E ∗ , E ⊕ E ∗ ) ͷ૊͸ѥ୅਺ߏ଄ʹΑΔ Manin tripleͱ‫͑ݴ‬Δɽ΋ͪΖΜɼશͯͷ Courant ѥ୅਺͕ Lie ѥ୅਺ͷ Drinfel’d double ʹΑͬͯಘΒΕΔ
Θ͚Ͱ͸ͳ͍ɽ‫ʹٯ‬ɼCourant ѥ୅਺͕ඞͣ Dirac ߏ଄ʹΑͬͯ෼ׂͰ͖Δͱ͸‫ݶ‬Βͳ͍ɽ

3.4 Double ʹΑΔ Vaisman ѥ୅਺ͷߏ੒
Vaisman ѥ୅਺͸࣍ͷΑ͏ʹఆٛ͞ΕΔɽ
✓Vaisman ѥ୅਺ҎԼͷ 4 ͭͷߏ଄ͷ૊ (V, [·, ·]V , ρV , (·, ·)) Λߟ͑Δɽ✏π• M ্ͷϕΫτϧଋ V −→M• V ͷ੾அ Γ(V) ΛҾ਺ͱ͢Δ Vaisman ‫·[ ހׅ‬, ·]V : Γ(V) × Γ(V) → Γ(V)• V ͔Β઀ଋ T M ΁ͷ anchor map ρV : V → T M .

• ඇୀԽͰରশͳ૒ઢ‫·( ࣜܗܕ‬, ·)͜ͷ૊ (V, [·, ·]V , ρV , (·, ·)) ͸ɼҎԼʹࣔ͢ 2 ͭͷఆٛࣜ Axiom V1, V2 Λຬͨ͢৔߹ɼ

Vaisman ѥ୅਺Λ੒͢ɽͨͩ͠ɼD ͸ Γ(V) ʹ࡞༻͢Δ֎ඍ෼ԋࢉࢠͰ͋Δɽ
Axiom V1. [e1 , f e2 ]V = f [e1 , e2 ]V + (ρV (e1 ) · f )e2 − (e1 , e2 )Df .

Axiom V2. ρV (e1 ) · (e2 , e3 ) = ([e1 , e2 ]V + D(e1 , e2 ), e3 ) + (e2 , [e1 , e3 ]V + D(e1 , e3 )).
✒Courant ѥ୅਺ͷఆٛͱൺֱ͢ΔͱɼAxiom V1 ͸ Axiom C3 ͱɼAxiom V2 ͸ Axiom C5 ͱ✑

ରԠ͢Δɽ͕ͨͬͯ͠ɼVaisman ѥ୅਺͸ Courant ѥ୅਺ΛߋʹҰൠԽͨ͠ߏ଄Ͱ͋Δͱ͍͑
Δɽ·ͨɼAxiom C5 ͔Β Axiom C3 ͕ಋ͔ΕΔͱ͢Δจ‫[ ݙ‬50] ͷٞ࿦ͱ΋ໃ६͠ͳ͍ɽจ‫[ ݙ‬35] ͷ Courant ѥ୅਺ʹؔ͢Δ݁Ռ͔ΒɼVaisman ѥ୅਺΋ԿΒ͔ͷߏ଄ͷ double ͰಘΒΕΔ΋ͷͱਪଌ͞ΕΔɽզʑ͸ɼAxiom V1, V2 Λຬͨ͢ Vaisman ѥ୅਺͕ɼCourant ѥ୅਺ͱྨࣅͨ͠ double Խͷखଓ͖ʹΑͬͯಘΒΕΔ͜ͱΛࣔͨ͠ [13]ɽ‫ͳͱݤ‬Δͷ͸ɼLie ૒ѥ୅*3 ࣌‫ͯ͠ͱྻܥ‬͸ [35] Ͱ਺ֶతʹൃ‫͞ݟ‬Εͨͷ͕ઌͰ͋Δɽ

Soryushiron Kenkyu

24਺ʹ͓͚Δ derivation ৚݅ (3.20) Ͱ͋ΔɽҎԼͰ͸ɼderivation ৚݅Λ՝ͣ͞ʹ;ͨͭͷ Lie ѥ୅਺ͷ double ΛͱΔ͜ͱͰɼVaisman ѥ୅਺͕ಘΒΕΔ͜ͱΛࣔ͢ɽ·ͨɼdouble ʹΑͬͯߏ੒ͨ͠ Vaisman ѥ୅਺ʹରͯ͠ɼ‫͔ޙ‬Β derivation ৚݅Λ՝͢ͱจ‫[ ݙ‬35] ͷ Courant algeroid
͕ಘΒΕΔɽ
·ͣɼ૒ରͳ;ͨͭͷ Lie ѥ୅਺ E = (E, [·, ·]E , ρ) ͱ E ∗ = (E ∗ , [·, ·]E ∗ , ρ∗ ) Λఆٛ͠ɼ௚࿨ʹΑͬͯϕΫτϧଋ V = E ⊕ E ∗ Λ༩͑Δɽͨͩ͠ɼ(E, E ∗ ) ͸ Lie ૒ѥ୅਺Λͳ͞ͳ͍΋ͷͱԾ

ఆ͢Δɽ͢ͳΘͪɼ‫ͱੵހׅ‬ඍ෼ͷؒʹ derivation ৚݅ (3.20) ͸੒ཱ͍ͯ͠ͳ͍ͱ͢ΔɽΓ(V)্ͷඇୀԽͳ૒ઢ‫·( ࣜܗܕ‬, ·)± Λ(e1 , e2 )± =1(⟨ξ1 , X2 ⟩ ± ⟨ξ2 , X1 ⟩)2(3.32)

ͷΑ͏ʹఆٛ͢Δɽ·ͨɼanchor ρV ͸ҎԼͷΑ͏ʹఆٛ͢ΔɽρV (ei ) = ρ(Xi ) + ρ∗ (ξi ).

(3.33)

Γ(V) ʹର͢Δඍ෼ԋࢉࢠ D : C ∞ (M ) → Γ(V) ͸(Df, ei ) =1ρV (ei )f2(3.34)

ͷΑ͏ʹఆٛ͢Δɽ·ͨɼD = d + d∗ ͱද͢͜ͱ͕Ͱ͖ΔɽVaisman ‫ހׅ‬͸จ‫[ ݙ‬52] ΛࢀߟʹҎԼͷΑ͏ͳ൓ରশͳ‫ͯ͠ͱੵހׅ‬ఆٛ͢Δɽ

[e1 , e2 ]V = [X1 , X2 ]E + Lξ1 X2 − Lξ2 X1 − d∗ (e1 , e2 )−+ [ξ1 , ξ2 ]E ∗ + LX1 ξ2 − LX2 ξ1 + d(e1 , e2 )− .

(3.35)

͜Ε͸ɼDFT ͷ C ‫ͱހׅ‬ಉ͡‫͋Ͱܗ‬Δɽ·ͨɼ[35] ͷ Courant ‫ͱހׅ‬ಉ͡‫͋Ͱܗ‬Δɽ͜ͷΑ͏ʹఆٛͨ͠ 4 ͭͷߏ଄ͷ૊ (V, [·, ·]V , ρV , (·, ·)+ ) ͕ Courant ѥ୅਺Ͱ͸ͳ͘ Vaismanѥ୅਺Λఆٛ͢Δ͜ͱΛҎԼʹࣔ͢ɽԾఆ͔Βɼderivation ৚݅ (3.20) ͕੒ཱ͍ͯ͠ͳ͍͜ͱʹ஫ҙͯ͠ɼAxiom C1-C5 ͕੒ཱ͍ͯ͠Δ͔Ͳ͏͔ΛͦΕͧΕ͔֬ΊΔɽຊจதʹ͸ɼ֤‫ࢉܭ‬ͷཁ఺͚ͩΛࡌͤΔɽৄࡉͳ‫ࢉܭ‬͸ Appendix A.5-A.11 Λࢀরͷ͜ͱɽAxiom C3, C5 ͷΈ͕੒ཱ͠ɼଞͷ Axiom ͕ഁΕ͍ͯΕ͹ (V, [·, ·]V , ρV , (·, ·)) ͕ Vaisman ѥ୅਺Ͱ͋Δͱ͍͑Δɽূ໌ͷաఔͰඞཁͱͳΔิॿతͳؔ܎ࣜ͸ɼҎԼͷ 2 ࣜͰ͋Δɽ1(([e1 , e2 ]c , e3 )+ + c.p.)31= {⟨ξ3 , [X1 , X2 ]E ⟩ + ⟨X3 , [ξ1 , ξ2 ]E ∗ ⟩2+ ρ(X3 )(e1 , e2 )− − ρ∗ (ξ3 )(e1 , e2 )− } + c.p.,

T (e1 , e2 , e3 ) ≡([e1 , e2 ]V , e3 )− + c.p. = T (e1 , e2 , e3 )*+ {ρ(X3 )(e1 , e2 )− + 2ρ∗ (ξ3 )(e1 , e2 )−+− ⟨[ξ1 , ξ2 ]V , X3 ⟩} + c.p. .

͜ΕΒͷؔ܎ࣜͷಋग़΍ɼূ໌ʹؔΘΔશ‫ࢉܭ‬͸ Appendix A.5 ʹऩ࿥ͨ͠ɽ

(3.36)(3.37)

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Axiom C1 ͷ֬ೝ·ͣɼAxiom C1 Λ֬ೝ͢ΔͨΊʹɼVaisman ‫·[ ހׅ‬, ·]V ͷ Jacobiator Λ‫͢ࢉܭ‬Δɽ(3.21) ࣜͷࠨลΛల։͢ΔͱɼҎԼͷ݁Ռ͕ಘΒΕΔɽ

[[e1 , e2 ]V , e3 ]V + c.p. = I1 + I2 ,(3.38)

I1 = [[ξ1 , ξ2 ]E ∗ , ξ3 ]E ∗ + [LX1 ξ2 − LX2 ξ1 , ξ3 ]E ∗ + [d(e1 , e2 )− , ξ3 ]E ∗+ L[X1 ,X2 ]E +Lξ1 X2 −Lξ2 X1 −d∗ (e1 ,e2 )− ξ3− LX3 [ξ1 , ξ2 ]E ∗ − LX3 LX1 ξ2 + LX3 LX2 ξ1− LX3 d(e1 , e2 )− + d([e1 , e2 ]V , e3 )− + c.p.,(3.39)

I2 = [[X1 , X2 ]E , X3 ]E + [Lξ1 X2 − Lξ2 X1 , X3 ]E − [d∗ (e1 , e2 )− , X3 ]E+ L[ξ1 ,ξ2 ]E∗ +Lξ1 X2 −Lξ2 X1 +d(e1 ,e2 )− X3− Lξ3 [X1 , X2 ]E − Lξ3 Lξ1 X2 + Lξ3 Lξ2 X1+ Lξ3 d∗ (e1 , e2 )− − d∗ ([e1 , e2 ]V , e3 )− + c.p.

(3.40)

ͨͩ͠ɼΓ(E ∗ ) ʹଐ͢Δ߲͸ I1 ʹɼΓ(E) ʹଐ͢Δ߲͸ I2 ʹ·ͱΊͨɽҎ߱ͷ‫ࢉܭ‬͸ɼI1 ʹؔͯ͠ͷΈߦ͏ɽX ͱ ξ ΛೖΕସ͑Δ͜ͱͰɼI2 ʹؔ͢Δ‫ࢉܭ‬΋࠶‫͖Ͱݱ‬Δɽ্‫ه‬ͷ I1 ʹ͸ɼLie‫ހׅ‬ͷ Jacobi ཯ʹΑͬͯফ͑Δ߲͕‫·ؚ‬Ε͓ͯΓɼͦΕΒΛ੔ཧ͢Ε͹ҎԼͷ‫ͳͱܗ‬Δɽ

I1 = [LX1 ξ2 − LX2 ξ1 , ξ3 ]E ∗ + [d(e1 , e2 )− , ξ3 ]E ∗ + LLξ1 X2 −Lξ2 X1 ξ3 − Ld∗ (e1 ,e2 )− ξ3− LX3 [ξ1 , ξ2 ]E ∗ − LX3 d(e1 , e2 )− + d([e1 , e2 ]V , e3 )− + c.p.

(3.41)

͜͜ͰɼI1 ΁ҎԼͷެࣜΛద༻͢Δɽ͜ͷެࣜͷಋग़͸ Appendix A.7 ʹࡌͤͨɽ

LX3 [ξ1 , ξ2 ]E ∗ + c.p. = [LX1 ξ2 − LX2 ξ1 , ξ3 ]E ∗ + LLξ1 X2 −Lξ2 X1 ξ3+ 2[d(e1 , e2 )− , ξ3 ]E ∗ + 2d(ρ∗ (ξ3 ) · (e1 , e2 )− ) − d⟨[ξ1 , ξ2 ]E ∗ , X3 ⟩+ ιX3 (d[ξ1 , ξ2 ]E ∗ − Lξ1 dξ2 + Lξ2 dξ1 ) + c.p.

(3.42)

͢Δͱɼ࠷ऴతʹ I1 ͸ҎԼͷΑ͏ʹ·ͱ·Δɽ

I1 = dT (e1 , e2 , e3 ) − {K1 + K2 } + c.p.

(3.43)

ͨͩ͠ɼ

K1 = ιX3 (d[ξ1 , ξ2 ]E ∗ − Lξ1 dξ2 + Lξ2 dξ1 ),K2 = Ld∗ (e1 ,e2 )− ξ3 + [d(e1 , e2 )− , ξ3 ]E ∗ .

(3.44)

ಉ༷ʹɼI2 ʹؔͯ͠‫͢ࢉܭ‬ΔͱɼҎԼͷ݁Ռ͕ಘΒΕΔɽ

I2 = d∗ T (e1 , e2 , e3 ) − {K3 + K4 } + c.p,

K3 = ιξ3 (d∗ [X1 , X2 ]E − LX1 d∗ X2 + LX2 d∗ X1 ),"!
K4 = − Ld(e1 ,e2 )− X3 + [d∗ (e1 , e2 )− , X3 ]E .

(3.45)[[e1 , e2 ]V , e3 ]V + c.p. = I1 + I2 = DT (e1 , e2 , e3 ) − (J1 + J2 + c.p.).

(3.46)

͕ͨͬͯ͠ɼVaisman ‫·[ ހׅ‬, ·]V ͷ Jacobiator ͷ‫݁ࢉܭ‬Ռ͸࣍ͷΑ͏ʹͳΔɽ

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ͨͩ͠ɼJ1 , J2 ͸ҎԼͷΑ͏ͳ߲Ͱ͋Δɽ

J1 = K1 + K3
= ιX3 (d[ξ1 , ξ2 ]E ∗ − Lξ1 dξ2 + Lξ2 dξ1 ) + ιξ3 (d∗ [X1 , X2 ]E − LX1 d∗ X2 + LX2 d∗ X1 ),J2 = K2 + K4

= Ld(e1 ,e2 )− ξ3 + [d(e1 , e2 )− , ξ3 ]E ∗ + Ld∗ (e1 ,e2 )− X3 + [d∗ (e1 , e2 )− , X3 ]E .

(3.47)

Ұൠʹɼ೚ҙͷ Xi , ξi , f ʹର͠ (J1 + J2 + c.p.) ͸ 0 Ͱ͸ͳ͍ɽΑͬͯɼ(V, [·, ·]V , ρV , (·, ·)+ ) ʹ͓͍ͯɼAxiom C1 ͸ഁΕ͍ͯΔɽ

Axiom C2 ͷ֬ೝಉ༷ʹɼAxiom C2 ʹؔͯ͠‫͢ࢉܭ‬Δɽ·ͣɼ(3.22) ͷ֤ลΛ f ʹରͯ͠࡞༻ͤ͞ΔͱɼҎԼͷ͕ࣜಘΒΕΔɽρV ([e1 , e2 ]V ) · f = [ρV (e1 ), ρV (e2 )]f.

(3.48)

ࣜ (3.48) ͷࠨล͸࣍ͷΑ͏ʹల։͞ΕΔɽρV ([e1 , e2 ]V ) · f

1= [ρ(X1 ), ρ(X2 )] · f + ρ(Lξ1 X2 ) · f − ρ(Lξ2 X1 ) · f − ρρ∗∗ d0 (⟨ξ1 , X2 ⟩ − ⟨ξ2 , X1 ⟩) · f
2
1+ [ρ∗ (ξ1 ), ρ∗ (ξ2 )] · f + ρ∗ (LX1 ξ2 ) · f − ρ∗ (LX2 ξ1 ) · f + ρ∗ ρ∗ d0 (⟨ξ1 , X2 ⟩ − ⟨ξ2 , X1 ⟩) · f,
2
(3.49)

ͨͩ͠ɼd∗ = ρ∗∗ d0 , d = ρ∗ d0 Ͱ͋Γɼd0 ͸ Γ(T ∗ M ) ͷ֎ඍ෼ԋࢉࢠͰ͋Δɽࣜ (3.48) ͷӈล͸࣍ͷΑ͏ʹల։͞ΕΔɽ

[ρV (e1 ), ρV (e2 )]f= [ρ(X1 ), ρ(X2 )]f + [ρ∗ (ξ1 ), ρ(X2 )]f + [ρ(X1 ), ρ∗ (ξ2 )]f + [ρ∗ (ξ1 ), ρ∗ (ξ2 )]f.

(3.50)

ӈลͷ 2 ߲໨ɼ3 ߲໨ͷ X, ξ ͷ cross term ͸ɼҎԼͷެࣜρ∗ (ξ)ρ(X) · f = −ρρ∗∗ d0 ⟨ξ, X⟩ · f + ⟨ξ, Ldf X⟩ + ⟨df, Lξ X⟩.

(3.51)

Λ༻͍Δ͜ͱͰɼ

[ρ(X), ρ∗ (ξ)]f = (ρ(X)ρ∗ (ξ) − ρ∗ (ξ)ρ(X)) · f= ρ(X)ρ∗ (ξ) · f + (ρρ∗∗ d0 ⟨ξ, X⟩) · f − ⟨ξ, Ldf X⟩ − ρ(Lξ X) · f(3.52)

ͱॻ͚Δɽ͕ͨͬͯ͠ɼ(3.48) ͷࠨลͱӈลͷࠩ͸࣍ͷΑ͏ʹͳΔɽρV ([e1 , e2 ]V ) · f − [ρV (e1 ), ρV (e2 )]f((''= −⟨ξ1 , Ldf X2 − [X2 , d∗ f ]E ⟩ + ⟨ξ2 , Ldf X1 − [X1 , d∗ f ]E ⟩"1! ∗ρρ∗ + ρ∗ ρ∗ d0 (⟨ξ1 , X2 ⟩ − ⟨ξ2 , X1 ⟩) · f.
+
2

(3.53)

Ұൠʹɼ೚ҙͷ Xi , ξi ʹର͠ӈล͸ 0 ʹͳΒͳ͍ɽΑͬͯɼ(V, [·, ·]V , ρV , (·, ·)+ ) ʹ͓͍ͯɼ

Axiom C2 ͸ഁΕ͍ͯΔɽ

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Axiom C3 ͷ֬ೝ(3.23) ࣜͷࠨล͸ɼei = Xi + ξi ͔ΒɼҎԼͷΑ͏ʹ෼ׂͯ͠ॻ͚Δɽ
[e1 , f e2 ]V = [X1 , f X2 ]V + [X1 , f ξ2 ]V + [ξ1 , f X2 ]V + [ξ1 , f ξ2 ]V .

(3.54)

͜͜ͰɼVaisman ‫ހׅ‬ͷఆ͔ٛΒɼ(3.54) ͷӈล͸ҎԼͷΑ͏ʹల։͞ΕΔɽ

[X1 , f X2 ]V = [X1 , f X2 ]E ,1
[X1 , f ξ2 ]V = f [X1 , ξ2 ]V + (ρ(X1 ) · f )ξ2 − Df ⟨ξ2 , X1 ⟩,2
1[ξ1 , f X2 ]V = f [ξ1 , X2 ]V + (ρ∗ (ξ1 ) · f )X2 − Df ⟨ξ1 , X2 ⟩,2[ξ1 , f ξ2 ]V = [ξ1 , f ξ2 ]E ∗ .

(3.55)(3.55) ͷ 4 ͭͷࣜͷӈลΛશͯ଍͢ͱɼҎԼͷ݁Ռ͕ಘΒΕΔɽ
[e1 , f e2 ]V = f [e1 , e2 ]V + (ρV (e1 )f )e2 − Df (e1 , e2 )+ .

(3.56)

Αͬͯɼ(V, [·, ·]V , ρV , (·, ·)+ ) ʹ͓͍ͯɼAxiom C3 ͸੒ཱ͢Δɽ

Axiom C4 ͷ֬ೝ(3.24) ࣜͷࠨล͸ҎԼͷΑ͏ʹల։͞ΕΔɽ
(Df, Dg)+ =1(ρ∗ ρ∗ d0 f + ρρ∗∗ d0 f )g.
2(3.57)

Ұൠʹ anchor ρ ͸൓ରশͰ͸ͳ͍ɽ͢ͳΘͪ ρρ∗∗ = −ρ∗ ρ∗ ͸੒ཱ͠ͳ͍ɽΑͬͯɼ(3.57) ͷӈล͸ 0 ʹͳΒͣɼAxiom C4 ͸ (V, ρV , [·, ·]V , (·, ·)) ʹ͓͍ͯഁΕΔɽͨͩ͠ɼanchor ʹ෇͍͍ͯΔ্෇͖ͷ ∗ ͸ɼadjoint operator ͷҙຯͰ͋Γɼ಺ੵΛ௨ͯ͠‫ݩ‬ͷԋࢉࢠͷసஔͰఆٛ͞ΕΔɽρ : E → TM

ρ∗ : E ∗ → T M

ρ∗ : T ∗ M → E ∗ρ∗∗ : T ∗ M → E(3.58)

͕ͨͬͯ͠ɼρρ∗∗ : T ∗ M → T M ɼρ∗ ρ∗ : T ∗ M → T M Ͱ͋ΔɽDerivation ৚͕݅੒ཱ͍ͯ͠Δ৔߹͸ඞͣ anchor ͸൓ରশʹͳΔ͜ͱ͕ [44] ͷ Proposition

3.4 ͷ಺༰Λ‫ͤࣔʹݩ‬ΔɽҰൠʹɼLie ѥ୅਺ͷੑ࣭͔ΒҎԼͷؔ܎͕ࣜಘΒΕΔɽ
(Ldf X + [d∗ f, X]E ) ∧ Y!
" !
"= −f d∗ [X, Y ]E + LY d∗ X − LX d∗ Y + d∗ [X, f Y ]E − LX d∗ (f Y ) + Lf Y d∗ X .

(3.59)

Proposition 3.4 Ͱ͸ɼ(3.59) ͷӈล͕ derivation ৚݅Λ՝͢͜ͱͰ 0 ͱͳΔ͜ͱΛ͍ࣔͯ͠Δɽ͢ͳΘͪɼderivation ৚͕݅ຬͨ͞Εͨ৔߹ɼ෇ਵͯ͠ҎԼͷؔ܎͕ࣜ੒ཱ͢Δ͜ͱ͕֬ೝͰ͖Δɽ

Ldf X + [d∗ f, X]E = 0.

(3.60)(3.60) ͕ຬͨ͞Εͳ͍‫ݶ‬Γɼρ ͸൓ରশͱͳΒͳ͍͜ͱ͕ࣔ͞ΕΔɽৄࡉ͸ Appendix A.10 ࢀরͤΑɽ͕ͨͬͯ͠ɼderivation ৚͕݅੒ཱ͢ΔͳΒ͹ඞͣ ρ ͸൓ରশͱͳΓɼAxiom C4 ͕੒ཱ
͢Δɽ‫ʹٯ‬ɼderivation ৚݅Λ՝͞ͳ͍ͳΒ͹ɼanchor ͕൓ରশͰͳ͍৔߹ΛഉআͰ͖ͳ͍ɽ

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Axiom C5 ͷ֬ೝষͷ๯಄Ͱ༩͑ͨ T (e1 , e2 , e3 ) ͷؔ܎ࣜ (3.36) ͔ΒɼҎԼͷ;ͨͭͷ͕ࣜ੒ཱ͢Δɽ11([e, e1 ]V , e2 )+ = T (e, e1 , e2 ) + ρV (e) · (e1 , e2 )+ − ρV (e1 ) · (e, e2 )+ ,22
11
(e1 , [e, e2 ]V )+ = T (e, e2 , e1 ) + ρV (e) · (e2 , e1 )+ − ρV (e2 ) · (e, e1 )+ .
2
2

(3.61)(3.62)

ҎԼͷΑ͏ʹɼࣜ (3.61)ɼ(3.62) ͷࠨลಉ࢜ɼӈลಉ࢜Λ଍ͯ͠΋౳߸͸อͨΕΔɽ

([e, e1 ]V , e2 )+ + (e1 , [e, e2 ]V )+1
1= T (e, e1 , e2 ) + ρV (e) · (e1 , e2 )+ − ρV (e1 ) · (e, e2 )+221
1+ T (e, e2 , e1 ) + ρV (e) · (e2 , e1 )+ − ρV (e2 ) · (e, e1 )+ .
2
2

(3.63)

ࣜ (3.63) ͸ɼT ͷ൓ରশੑͳͲ͔ΒҎԼͷΑ͏ʹ੔ཧͰ͖Δɽ11ρV (e) · (e1 , e2 )+ = ([e, e1 ]V , e2 )+ + (e1 , [e, e2 ]V )+ + ρV (e1 ) · (e, e2 )+ + ρV (e2 ) · (e, e1 )+22([e, e1 ]V + D(e, e1 )+ , e2 )+ + (e1 , [e, e2 ]V + D(e, e2 )+ )+ .
(3.64)
Αͬͯɼ(V, [·, ·]V , ρV , (·, ·)+ ) ʹ͓͍ͯɼAxiom C5 ͸੒ཱ͢Δɽ͜ΕΒͷ‫ࢉܭ‬ͷৄࡉ͸ Appendix

A.9, A.11 ʹࣔͨ͠ɽ͜ΕͰɼ૒ରͳ 2 ͭͷ Lie ѥ୅਺ͷ double ʹΑͬͯɼVaisman ѥ୅਺͕ಘΒΕΔ͜ͱ͕ࣔͤͨɽAxiom C5 ͱ Axiom C3 ͕੒ཱ͢Δঢ়‫گ‬͸ɼจ‫[ ݙ‬50] ͷٞ࿦ͱ΋ໃ६͠ͳ͍ɽͰ͸ɼ‫ ʹٯ‬Vaisman ѥ୅਺ͷ෼ׂํ๏Λߟ͑ͯΈΑ͏ɽ[35] ͰߦΘΕ͍ͯΔ Courant ѥ୅਺ͷ Dirac ෦෼ଋʹΑΔ෼ׂʹ‫͍ͯͮج‬ٞ࿦͢ΔͨΊʹɼ·ͣ͸ [35] ͷ֘౰෦෼ͷϨϰϡʔΛߦ͏ɽCourant ѥ୅਺ (C, [·, ·]c , ρc , (·, ·)) ͸ (·, ·) ʹؔͯ͠࠷େ౳ํతͰ͋Γɼͳ͓͔ͭ Courant ‫ׅ‬‫·[ ހ‬, ·]c ͕ Γ(C) Ͱด͍ͯ͡Δ৔߹ɼDirac ෦෼ଋ L, L̃ ʹΑͬͯ෼ׂͰ͖Δɽ·ͨɼ(L, L̃) ͸ࣗવʹ Lie ૒ѥ୅਺ͱͳΔɽ͜ΕΛࣔͨ͢Ίʹඞཁͱͳͷ͸ɼҎԼʹఏࣔ͢Δ Proposition 2.3 ͱ

Lemma 5.2 Ͱ͋Δɽ
Proposition 2.3 in [35]. L ͕ɼCourant ѥ୅਺ (C, [·, ·]c , ρc , (·, ·)+ ) ͷ૒ઢ‫·( ࣜܗܕ‬, ·)+ ʹ͍ͭͯ౳ํతͳ෦෼ଋͰɼ͔ͭੵ෼Մೳ*4 ͳͱ͖, (L, [·, ·]c , ρc |L ) ͸ Lie ѥ୅਺ͱͳΔɽͨͩ͠ɼ
ʮ(·, ·) ʹ͍ͭͯ౳ํͳ෦෼ଋʯͱ͸ɼ೚ҙͷ X, Y ∈ Γ(L) ʹରͯ͠ɼ(X, Y ) = 0 ͕੒

ཱ͢ΔΑ͏ͳ෦෼ଋΛࢦ͢ɽρc |L ͸ɼanchor ρc ͔Β L ʹ࡞༻͢Δ෦෼͚ͩΛൈ͖ग़ͨ͠΋ͷͰ
͋ΔɽProposition 2.3 ͸ɼAxiom C1 ͷ (3.21) ࣜ[[e1 , e2 ]c , e3 ]c + c.p. = DT (e1 , e2 , e3 )(3.21)

͔ΒɼL ʹ༝དྷ͢Δ෦෼ (X ͚ͩΛ‫ؚ‬Ή߲) Λൈ͖ग़͢͜ͱͰ͙͢ʹ֬ೝͰ͖Δɽ(3.21) ࣜͷࠨล͸ɼL ্ͷ‫·[ ੵހׅ‬, ·]L ʹΑΔ Jacobiator ͱͳΔɽҰํɼӈล͸ T ͷఆٛT (e1 , e2 , e3 ) =1([e1 , e2 ]c , e3 ) + c.p.
3(3.65)*4 Մੵ෼ੑʹ͍ͭͯͷৄࡉ͸࣍ͷষͰ༩͑Δɽͻͱ͜ͱͰ‫ͱ͏ݴ‬ɼ෦෼ଋ L ্ͷ Lie ‫͕ހׅ‬ด͍ͯ͡Δ͜ͱΛࢦ͢ɽ

Soryushiron Kenkyu

29͔ΒɼL ͕ (·, ·) ʹؔͯ͠౳ํతͰ͋Δ͜ͱΛԾఆ͢Δͱඞͣ 0 ʹͳΔɽΑͬͯɼ[·, ·]L ͕ Jacobi཯Λຬͨ͢͜ͱ͕͔֬ΊΒΕɼL ͸ Lie ѥ୅਺ͱͳΔɽProposition 2.3 ͔ΒɼCourant ѥ୅਺ͷ

Dirac ߏ଄ L, L̃ ͸ͦΕͧΕ Lie ѥ୅਺ͱͳΔɽL ͷ anchor ͸ ρc |L ͰɼL̃ ͷ anchor ͸ ρc |L̃ Ͱఆٛ͞ΕΔɽ

Proposition 2.3 Λཁ੥͓ͯ͘͠ͱɼL ͕ (·, ·) ʹؔͯ͠౳ํతͳ͜ͱΛ༻͍ͯ Axiom C5 ͷ

(3.25) ͔ࣜΒҎԼͷؔ܎͕ಋ͚ΔɽX ∈ Γ(L), ξ ∈ Γ(L̃) ͱͨ͠ͱ͖ɼ

1
1[X, ξ]c = −Lξ X + d∗ ⟨ξ, X⟩ + LX ξ − d⟨ξ, X⟩.
2
2(3.66)

Lemma 5.2 in [35]. L, L̃ ΛɼCourant ѥ୅਺ C ʹରͯ͠ɼC = L ⊕ L̃ ͱͳΔΑ͏ͳ Dirac ߏ଄Ͱ͋Δͱ͢Δɽ͜ͷͱ͖ɼҎԼͷؔ܎͕ࣜ੒ཱ͢Δɽͨͩ͠ɼ

Ld∗ f ξ = −[df, ξ]L̃ ,Ldf X = −[d∗ f, X]L .

(3.67)

͜͜Ͱɼd, d∗ ͸ͦΕͧΕ Γ(∧p L̃), Γ(∧p L) ʹ࡞༻͢Δ֎ඍ෼ԋࢉࢠͰ͋Δɽ

Lemma 5.2 ͸ҎԼͷΑ͏ʹࣔͤΔɽ·ͣɼAxiom C4 ͔ΒɼҎԼͷؔ܎͕੒ཱ͢Δɽρ∗ · d = −ρ · d∗ .

(3.68)

͜ΕΛɼAxiom C2 ΁Ԡ༻͢Δ͜ͱͰɼҎԼͷؔ܎͕Θ͔Δɽ[ρ∗ (ξ), ρ(X)] = ρc [ξ, X]c

11= ρc (Lξ X − d∗ ⟨ξ, X⟩ − LX ξ + d⟨ξ, X⟩)
22= ρ(Lξ X) − ρ∗ (LX ξ) + ρ∗ (d⟨ξ, X⟩).

(3.69)

ͨͩ͠ɼࣜ (3.69) ͷ 1 ஈ໨͔Β 2 ஈ໨΁ͷม‫Ͱܗ‬ɼࣜ (3.66) Λ༻͍ͨɽҰํͰɼLie ѥ୅਺ͷੑ࣭͔Βɼρ∗ (d⟨ξ, X⟩) · f = [ρ∗ (ξ), ρ(X)]f − ρ(Lξ X) · f + ρ∗ (LX ξ) · f + ⟨Ld∗ f ξ + [df, ξ]L̃ , X⟩. (3.70)
ࣜ (3.69) ͱ (3.70) ͷൺֱ͔ΒɼLd∗ f ξ = −[df, ξ]L̃ Ͱ͋Δɽࣜ (3.69) ͱ (3.70) ͷ ξ ͱ X ΛೖΕସ͑ͯ࠶‫͢ࢉܭ‬Ε͹ɼLdf X = −[d∗ f, X]L ΋ࣔͤΔɽ

Lemma 5.2 ͷؔ܎ࣜΛ೦಄ʹஔ͖ͭͭɼ΋͏Ұ౓ Courant ‫ހׅ‬ͷ Jacobiator ͷ‫݁ࢉܭ‬Ռʹ஫໨͢Δɽ

[[e1 , e2 ]c , e2 ]c + c.p. = DT (e1 , e2 , e3 ) − (J1 + J2 + c.p.),(3.71)

ͨͩ͠ɼJ1 , J2 ͸ (3.47) Ͱࣔͨ͠ͷͱಉ͡΋ͷͰ͋Δ.

!
"!
"
J1 = ιX3 d[ξ1 , ξ2 ]V − Lξ1 dξ2 + Lξ2 dξ1 + ιξ3 d∗ [X1 , X2 ]V − LX1 d∗ X2 + LX2 d∗ X1 ,!
" !
"J2 = Ld∗ (e1 ,e2 )− ξ3 + [d(e1 , e2 )− , ξ3 ]V − Ld(e1 ,e2 )− X3 + [d∗ (e1 , e2 )− , X3 ]V .
(3.47)

͍·ɼC = L ⊕ L̃ ͸ Axiom C1 Λຬͨ͢ͷͰɼඞવతʹ J1 + J2 + c.p. = 0 Ͱ͋ΔɽՃ͑ͯɼ

Lemma 5.2 ͔ΒɼJ2 = 0 Ͱ͋ΔɽΑͬͯ J1 + c.p. = 0 ͕ཁ‫͞ٻ‬ΕΔɽ͕ͨͬͯ͠ɼe1 = X1 ,e2 = X2 , e3 = ξ3 ͱஔ͘ͱɼҎԼͷ৚݅ΛಘΔɽ
d∗ [X1 , X2 ]L − LX1 d∗ X2 + LX2 d∗ X1 = 0.

(3.72)

Soryushiron Kenkyu

30͜Ε͸ɼLie ૒ѥ୅਺ͷ derivation ৚݅ (3.20) ͦͷ΋ͷͰ͋ΔɽΑͬͯɼ(L ⊕ L̃) ͸ Lie ૒ѥ୅਺ͱͳΔɽલઅͰड़΂ͨ௨ΓɼCourant ѥ୅਺ C ͷ Dirac ߏ଄ L, L̃ ͸ɼશମͰ Manin triple(C, L, L̃) Λߏ੒͢Δɽ͜ͷࣄ࣮Λ΋ͱʹɼVaisman ѥ୅਺ͷ෼ׂΛߟ͑ΔɽVaisman ѥ୅਺ͷ Dirac ߏ଄͸ɼCourantѥ୅਺ͷ৔߹ͱಉ༷ʹɼΓ(V) ΛҾ਺ͱ͢Δ૒ઢ‫·( ࣜܗܕ‬, ·) ʹର͢Δɼ࠷େ౳ํͳ෦෼ଋͰఆٛ
Ͱ͖ΔɽL, L̃ ͸ V = L ⊕ L̃ ͱͳΔΑ͏ͳ V ͷ Dirac ߏ଄Ͱ͋Δͱ͢Δɽͨͩ͠ɼV ͸ Vaismanѥ୅਺ͳͷͰɼAxiom C3, C5 ͚͕ͩ੒ཱ͢Δɽ·ͣɼ[35] ͷ Proposition 2.3 ͸ɼಋग़͢Δࡍʹ

Axiom C1 Λ༻͍ΔͷͰɼVaisman ѥ୅਺Λલఏͱ͢Δͱ੒ཱ͠ͳ͍ɽΑͬͯɼL, L̃ ͷ‫ੵހׅ‬͸Ұൠʹ Jacobi ཯Λຬͨ͢ͱ͸‫ݶ‬ΒͣɼL, L̃ ͸ Lie ѥ୅਺ʹͳΒͳ͍৔߹΋ߟ͑ΒΕΔɽՃ͑ͯɼ
Ծʹ L, L̃ ͕ Lie ѥ୅਺Ͱ͋ͬͨͱͯ͠΋ Axiom C2, C4 ͕ഁΕ͍ͯΔͨΊɼLemma 5.2 ͸੒ཱ͠ͳ͍ɽ͕ͨͬͯ͠ɼ(L, L̃) ͷؒʹ derivation ৚݅͸੒ཱͤͣɼ(L, L̃) ͸શମͰ Lie ૒ѥ୅਺Λͳ͞ͳ͍ɽ͜ΕΛɼDFT ͷηοτΞοϓͰ۩ମతʹ੒෼‫݁ͨ͠ࢉܭ‬Ռ͸ ࣍ষͷ‫ޙ‬൒Ͱࣔ͢ɽ૒ରͳ

Lie ѥ୅਺ͷ૊ L ͱ L∗ ͕ Lie ૒ѥ୅਺ʹͳΔͷ͸ɼ(L, L∗ ) ͕શମͰ diﬀerential Gerstenhaber୅਺Λͳ͢ͱ͖ͷΈͰ͋Δ [54]ɽ͜Ε͸ɼ֎ඍ෼ԋࢉࢠ d (·ͨ͸ d∗ ) ͕ Schouten-Nijenhuis ‫ׅ‬‫ཱ྆͢ͱހ‬Δͱ͖Ͱ͋Δɽ

Soryushiron Kenkyu

31

ୈ4ষ

Vaisman algebroid ͕‫ݱ‬ΕΔ‫ز‬ԿֶύϥΤϧϛʔτ (para-Hermitian) ଟ༷ମʹ͸ɼDFT ͷഒԽۭ͕ؒࣗવʹ‫ݱ‬ΕΔ͜ͱ͕ [14, 15,

26–28] ͳͲͰٞ࿦͞Ε͍ͯΔɽ͜ͷষͰ͸ɼύϥΤϧϛʔτଟ༷ମ্ʹલઅͰࣔͨ͠ Vaisman ѥ୅਺Λ࠶‫͠ݱ‬ɼͦͷ Dirac ߏ଄ʹ‫͍ͯͮج‬ɼDFT ͷήʔδରশੑʹ͍ͭͯٞ࿦͢Δɽ͜ͷষͷલ൒Ͱ͸ɼѥ୅਺ͷ‫ه‬ड़ʹඞཁͳ਺ֶతߏ଄ͷఆٛΛ༩͑Δɽ·ͣɼύϥΤϧϛʔτଟ༷ମ M Λ༩͑ɼ઀ଋ T M ͷ෼ׂ͕ύϥෳૉ (para-complex) ߏ଄͔ΒࣗવʹಋೖͰ͖Δ͜ͱΛ֬ೝ͢ΔɽͦΕʹ෇ਵ͢Δ༿૚ߏ଄ (foliation) ߏ଄ [56] Λಋೖ͢ΔɽՃ͑ͯɼDFT ্ͷύϥυϧϘʔίϗϞϩδʔ (para-Dolbeault cohomology) ʹ͍ͭͯٞ࿦͠ɼѥ୅਺ߏ଄ͷ࠶‫ʹݱ‬ඞཁͳԋࢉࢠΛ੔ཧ͢Δɽষͷ‫ޙ‬൒Ͱ͸ɼ্ड़ͷఆٛΛ༻͍࣮ͯࡍʹ੒෼‫ࢉܭ‬Λߦ͍ɼύϥΤϧϛʔτଟ༷ମ্ʹ Vaisman ѥ୅਺Λߏங͢Δɽͦͯ͠ɼલͷষͰࣔͨ͠छʑͷѥ୅਺ߏ଄͕ɼDFT Ͱ͸ͲͷΑ͏ʹ‫ݱ‬ΕΔ͔Λௐ΂Δɽಛʹɼdeivation ৚݅ͱ DFT ͷ‫߆͍ڧ‬ଋ৚݅ͷؔ܎Λߟ࡯͢Δɽ

4.1 ύϥෳૉߏ଄
DFT ͷഒԽۭؒ M ͷ࠲ඪ‫ ܥ‬xM = (xµ , x̃µ ) ͸ɼKK Ϟʔυʹ Fourier ‫ڞ‬໾ͳ࠲ඪ xµ ͱ

winding Ϟʔυʹ Fourier ‫ڞ‬໾ͳ࠲ඪ x̃µ Λ૊Έ߹ΘͤΔ͜ͱͰಋೖ͞ΕΔɽจ‫[ ݙ‬14, 15] ʹ͓͍ͯɼ͜ͷΑ͏ͳߏ଄ͷ࠲ඪ͸ɼύϥΤϧϛʔτଟ༷ମʹࣗવʹ૊Έࠐ·Ε͍ͯΔͱ͍͏ࢦఠ͕ͳ
͞ΕͨɽύϥΤϧϛʔτଟ༷ମ M Λఆٛ͢Δʹ͸ɼύϥෳૉߏ଄͕ඞཁͱͳΔɽ·ͣɼ֓ύϥෳૉଟ༷ମ͸ҎԼͷΑ͏ʹఆٛ͞ΕΔɽ

Definition 4.1.1. ֓ (almost) ύϥෳૉଟ༷ମͱ͸ɼҎԼͷ֓ύϥෳૉߏ଄ K ͕ఆٛ͞Εͨଟ༷ମ M ͷ͜ͱͰ͋ΔɽK ͸ɼK 2 = 1 Λຬͨ͢Α͏ͳ T M ্ͷࣗ‫ݾ‬ಉ‫͋Ͱ૾ࣸܕ‬Δɽ

௨ৗͷෳૉߏ଄ J 2 = −1 Ͱ͸ Nijenhuis ςϯιϧ͕ 0 ʹͳΔͱ J ʹ͍ͭͯੵ෼Մೳͱͳ

Δ [57]ɽͦΕͱಉ༷ʹɼҎԼͷྔNK (X, Y ) =1([K(X), K(Y )] + [X, Y ] − K([K(X), Y ] + [X, K(Y )])
4

(4.1)

͕ 0 ʹͳΔͱ͖ɼ֓ύϥෳૉߏ଄ K ͕Մੵ෼Ͱ͋Δͱ͍͏ɽՄੵ෼ͳ֓ύϥෳૉߏ଄͸ύϥෳૉ
ߏ଄Ͱ͋ΔɽK ͕Մੵ෼ͳͱ͖ (K, M) Λύϥෳૉଟ༷ମͱ͍͏ɽ
֓ύϥෳૉߏ଄ K ͷ‫ݻ‬༗஋ ±1 ʹԠͯ͡ɼM ͷ઀ଋ T M Λ K ͷ‫ݻ‬༗෦෼ଋ L, L̃ ʹ෼ׂͰ

͖ΔɽT M ͷ෼ׂ͸ɼҎԼͷࣹӨԋࢉࢠ P, P̃ ʹΑͬͯߦΘΕΔɽP, P̃ ͸ɼT M ͔Β L, L̃ ΛબSoryushiron Kenkyu32ͿΑ͏ͳࣸ૾Ͱ͋ΓɼҎԼͷΑ͏ʹఆٛ͞ΕΔɽP =1(1 + K),2P̃ =1(1 − K).
2(4.2)

P, P̃ ʹΑͬͯಘΒΕΔ෦෼ଋ L, L̃ ͸ɼdistribution Ͱ͋ΔɽDefinition 4.1.2. Distribution ͸ɼҎԼͷΑ͏ʹఆٛ͞ΕΔɽҰൠʹɼM Λ m ࣍‫ݩ‬ͷ C ∞ ‫ڃ‬ଟ༷ମͱ͢ΔɽM ্ͷ೚ҙͷ఺ x ͷ઀ۭؒ Tx M ʹରͯ͠ɼx ʹؔͯ͠‫׈‬Β͔ͳ n ࣍‫( ݩ‬n ≤ m)ͷ෦෼ۭؒ ∆x Λߟ͑Δɽ∆ Λ M ্ͷશͯͷ఺ʹର͢Δ෦෼ۭؒ ∆x ͷू߹ͱ͢Δͱɼ∆ ͸ n࣍‫ݩ‬ͷ distribution ͱͳΔɽͨͩ͠ɼx ʹؔͯ͠‫׈‬Β͔ͱ͸ɼ∆x ʹଐ͢ΔϕΫτϧ͔Βɼx ۙ๣ͷશͯͷ఺ʹରͯ͠ n ࣍‫ݩ‬ͷϕΫτϧ৔Λ࡞ΕΔɼͱ͍͏ҙຯͰ͋Δɽ

Distribution ͷ࠷΋؆୯ͳྫ͸ɼn = 1 ͷ৔߹Ͱ͋ΔɽM ʹର͢Δ 1 ࣍ͷ distribution ͸ɼ
T M ͷ੾அʹΑΔϕΫτϧ৔ Γ(T M ) ͷ‫ͳͱ੻ي‬Δɽ೚ҙͷ X, Y ∈ Γ(L) ʹରͯ͠ɼLie ‫ʹހׅ‬ΑΔੵ [X, Y ]L ͕ඞͣ L ʹଐ͢Δͱ͖ɼdistribution L ͸ involutive Ͱ͋Δͱ͍͏ɽԾʹɼL, L͕̃ involutive Ͱ͋Ε͹ɼҎԼͷྔ͕ 0 ͱͳΔɽNP (X, Y ) = P̃ [P (X), P (Y )], NP̃ (X, Y ) = P [P̃ (X), P̃ (Y )].

(4.3)

͢ΔͱɼҎԼͷ Frobenius ͷఆཧ͔Βɼ(4.3) ࣜʹΑͬͯ L, L̃ ͷՄੵ෼ੑ͕ධՁͰ͖Δɽ

Theorem 4.1.3. ͋Δ distribution L ͕Մੵ෼ͱͳΔͷ͸ L ͕ involutive ͷͱ͖͚ͩͰ͋Δɽ·ͨɼ‫ٯ‬΋੒ཱ͢Δɽࣜ (4.3) Ͱఏࣔ͞Εͨ NP ͱ NP̃ Λ଍͢ͱɼࣜ (4.1) ͕ಘΒΕΔɽಋग़͸ Appendix A.12 ʹࣔͨ͠ɽͭ·ΓɼL ʹରͯ͠ͷՄੵ෼৚݅ͱ L̃ ʹରͯ͠ͷՄੵ෼৚݅͸ಠཱ͓ͯ͠ΓɼL, L̃ ͷ͏ͪL ͚͕ͩՄੵ෼ʹͳΔ৔߹ (NP (X, Y ) = 0) ΍ɼL̃ ͚͕ͩՄੵ෼ʹͳΔ৔߹ (NP̃ = 0) Λߟ͑Δ
͜ͱ͕Ͱ͖Δɽ͜ͷ఺͕ɼύϥෳૉߏ଄ͱ௨ৗͷෳૉߏ଄ͱͷେ͖ͳҧ͍Ͱ͋Δɽ͜ͷࣄ࣮͔ΒɼҎԼͷΑ͏ͳଟ༷ମΛఆٛͰ͖Δ [26, 27]ɽDefinition 4.1.4. (M, K) Λ֓ύϥෳૉଟ༷ମͱ͢ΔɽಛʹɼL ͚͕ͩ K ʹ͍ͭͯՄੵ෼ͳͱ
͖ɼ͜ΕΛ L-ύϥෳૉ (L-para-complex) ଟ༷ମͱ͍͏ɽL̃ ʹؔͯ͠΋ಉ༷Ͱ͋ΔɽಛʹɼL ͱ

L̃ ͕྆ํՄੵ෼Ͱ͋Ε͹ɼT M શମ͕ K ʹؔͯ͠Մੵ෼ (NK = 0) ͱͳΓɼ(M, K) ͸ύϥෳૉଟ༷ମʹͳΔɽ

4.2 ύϥΤϧϛʔτଟ༷ମ࣍ʹɼ֓ύϥΤϧϛʔτଟ༷ମΛఆٛ͠Α͏ɽ

Definition 4.2.1. (M, K) Λύϥෳૉଟ༷ମͱ͢Δɽ(M, K) ͕ neutral ͳ‫ ྔܭ‬η : T M ×

T M → R Λ΋ͭͱ͖ɼ(M, η, K) Λ֓ύϥΤϧϛʔτଟ༷ମͱ͍͏ɽη ΛύϥΤϧϛʔτ‫ͱྔܭ‬͍͍ɼη ͱ K ͷؒʹ͸ཱ྆৚݅ η(K·, K·) = −η(·, ·) ͕੒ཱ͢Δɽఆ͔ٛΒɼX, Y ∈ Γ(L) ͱͨ͠ͱ͖ɼη(X, Y ) = 0 Ͱ͋ΔɽՃ͑ͯɼη ͕ neutral ͳͷͰɼL, L̃ͷ rank ͸ 12 dim M ͱͳΔɽΑͬͯɼL, L̃ ͸ύϥΤϧϛʔτ‫ ྔܭ‬η ʹؔͯ͠࠷େ౳ํతͰ͋Δɽ

Soryushiron Kenkyu

33dω ̸= 0NK ̸= 0NK = 0dω = 0֓ύϥΤϧϛʔτ֓ύϥέʔϥʔ (almost para-Kähler)(֓γϯϓϨΫςΟοΫ)(γϯϓϨΫςΟοΫ)ύϥΤϧϛʔτύϥέʔϥʔ

(֓γϯϓϨΫςΟοΫ)(γϯϓϨΫςΟοΫ)ද 4.1

Մੵ෼৚݅ͱ ω ͷؔ܎.

֓ύϥΤϧϛʔτଟ༷ମ্ʹ͸ɼK ͱ η ͔ΒɼඇୀԽͳඍ෼ 2 ‫ ࣜܗ‬ω = ηK ΛఆٛͰ͖Δɽω͸Ұൠʹด‫ͱࣜܗ‬͸‫ݶ‬Βͳ͍ɽω ΛɼҎԼʹఆٛ͢ΔΑ͏ͳɼ֓γϯϓϨΫςΟοΫଟ༷ମͷγϯϓϨΫςΟοΫ 2 ‫ͱࣜܗ‬Έͳ͢͜ͱ͕Ͱ͖Δɽ

Definition 4.2.2. ֓γϯϓϨΫςΟοΫ (almost symplectic) ଟ༷ମͱ͸ɼ࣮ 2n ࣍‫ݩ‬ଟ༷ମ
M ͱɼM ্ͷඍ෼ 2 ‫ ࣜܗ‬ω ͷ૊ͷ͜ͱΛ͍͏ɽω Λ֓γϯϓϨΫςΟοΫ‫͏͍ͱࣜܗ‬ɽಛʹɼω ͕ด‫( ࣜܗ‬dω = 0) Ͱ͋Δ৔߹͸ɼγϯϓϨΫςΟοΫଟ༷ମͱͳΔɽ͜ͷͱ͖ɼdω = 0 Λʮ֓γϯϓϨΫςΟοΫ‫ʹࣜܗ‬ର͢ΔՄੵ෼৚݅ʯͱ͍͏ɽ
Αͬͯɼ֓ύϥΤϧϛʔτଟ༷ମ (M, η, K) ͸֓γϯϓϨΫςΟοΫଟ༷ମͰ΋͋ΓɼK ʹؔ
͢ΔՄੵ෼৚݅ͱ ω ʹؔ͢ΔՄੵ෼৚݅Λผ‫ʹݸ‬ѻ͑Δɽ͜ΕΛ༻͍ͯɼද 4.1 ͷΑ͏ʹෳ਺ͷଟ༷ମΛఆٛͰ͖Δɽಛʹɼࠓ‫ޙ‬ͷٞ࿦ͷத৺ͱͳΔύϥΤϧϛʔτଟ༷ମͷఆٛΛվΊͯఏ͓ࣔͯ͘͠ɽDefinition 4.2.3. M ͱɼύϥΤϧϛʔτ‫ ྔܭ‬η : T M × T M → Rɼύϥෳૉߏ଄ K ͷ૊Λߟ͑ΔɽK ͚͕ͩՄੵ෼Ͱɼω ͕Մੵ෼Ͱͳ͍ͱ͖ɼ(M, K, η) ͸ύϥΤϧϛʔτଟ༷ମͰ͋Δɽ
·ͨɼK ʹର͢ΔՄੵ෼৚͕݅ L, L̃ ʹରͯ͠ಠཱʹ՝ͤΔ͜ͱ͔ΒɼҎԼͷΑ͏ͳଟ༷ମ΋ఆٛՄೳͰ͋Δɽ

Definition 4.2.4. (M, K) Λ L-ύϥෳૉଟ༷ମͱ͢Δɽ͜ΕʹɼύϥΤϧϛʔτ‫ྔܭ‬ΛՃ͑ͨ(M, η, K) Λ L-ύϥΤϧϛʔτଟ༷ମͱ͍͏ɽL̃ ʹؔͯ͠΋ಉ༷ͰɼL̃-ύϥΤϧϛʔτଟ༷ମΛఆٛͰ͖Δɽ
্‫ه‬ͷΑ͏ʹ L, L̃ ͷ͍ͣΕ͔͚͕ͩՄੵ෼ͱͳΔΑ͏ʹଟ༷ମΛఆٛ͢Δ͜ͱ͸Ͱ͖Δ͕ɼͦ
Ε͕ഒԽۭؒΛ‫ه‬ड़͢Δʹ͋ͨͬͯͲͷΑ͏ʹӨ‫͢ڹ‬Δͷ͔͸͋·Γٞ࿦͞Ε͍ͯͳ͍ɽͨͩ͠ɼ

L-ύϥΤϧϛʔτଟ༷ମͷηοτΞοϓΛ༻͍ͯɼϦʔϚϯ‫ز‬ԿͰ‫ڞ‬มඍ෼Λಋೖ͢Δͷͱྨࣅͨ͠ྲྀΕͰ M ্ͷඍ෼Λิਖ਼͠ɼDFT ͷ D ‫ހׅ‬Λ‫ ྔܭ‬η ͕࿷‫͍ͯ͠ۂ‬Δ৔߹΁֦ுͰ͖Δͱ

͍͏ࢦఠ͕͋Γ [26]ɼยଆ͚ͩՄੵ෼ͱ͍͏ঢ়‫ޙࠓ͕گ‬༗༻ͱͳΔՄೳੑ͸े෼͋Γ͏Δɽຊ࿦จͰ͸Ҏ‫ޙ‬ɼL, L̃ ͷ྆ํ͕Մੵ෼ɼ͢ͳΘͪύϥΤϧϛʔτଟ༷ମΛ༻͍ͨഒԽۭؒͷ‫ه‬ड़Λٞ࿦͢ΔɽύϥΤϧϛʔτଟ༷ମ M ্ͷ distribution L, L̃ ͸ η ʹؔͯ͠࠷େ౳ํతɼ͢ͳΘͪ dim L =dim L̃ = 12 dim M Ͱ͋Δɽ͔ͭɼK ͕Մੵ෼Ͱ͋Δ͜ͱ͔Β (4.1) ͕ࣜ 0 ͱͳΓɼ(4.3) Ͱ༩͑ͨ NP , NP̃ ͕ 0 ͱͳΔɽ͕ͨͬͯ͠ɼL, L̃ ʹ involutiveɼͭ·ΓͦΕͧΕʹดͨ͡ަ‫ੵࢠ׵‬Λఆ

Soryushiron Kenkyu

34ٛͰ͖Δɽ3 ষͷ 3.3.1 અͷఆٛΛࢀর͢ΔͱɼL, L̃ ͸ T M ͷ Dirac ߏ଄Ͱ͋Δɽ·ͨɼL, L̃ ͸

Frobenius ͷఆཧ͔ΒɼՄੵ෼Ͱ͋ΔɽM ʹΑͬͯഒԽۭؒΛ਺ֶతʹ‫ه‬ड़͢Δʹ͋ͨΓɼL, L̃ ͕Մੵ෼Ͱ͋Δ͜ͱͷ༗༻ੑ͸ɼҎԼʹࣔ͢ Frobenius ͷఆཧͷผͷද‫ݱ‬Λద༻͢ΔͱΘ͔Δɽ

Theorem 4.2.5. ͋Δ෦෼ଋ E ⊂ T M ͕Մੵ෼ͱͳΔͷ͸ɼM ্ʹ༿૚ߏ଄ F ͕ఆٛͰ͖Δ৔߹ʹ‫ݶ‬ΒΕΔɽ‫ʹٯ‬ɼM ্ʹ༿૚ߏ଄ F ͕ఆٛͰ͖ΔͳΒ͹ɼՄੵ෼ͳ෦෼ଋ E ΛɼM ͷ༿૚ߏ଄ F ʹର͢Δ઀ଋͱͯ͠ඞͣ༩͑ΒΕΔɽ

͍·ɼL, L̃ ͸Մੵ෼Ͱ͋ΔɽFrobenius ͷఆཧ͔ΒɼL, L̃ ͕઀ଋͰ͋ΔΑ͏ͳ༿૚ߏ଄ F, F̃ΛఆٛͰ͖ΔɽL = TFandL̃ = T F̃.

(4.4)

Mp ͸ F ͷ༿Ͱ͋ΓɼM ্ͷ఺ p Λ௨ΔΑ͏ͳɼF ͷ෦෼ۭؒͰ͋ΔɽF ͸༿ Mp ͷ࿨ू߹,µ[p] M[p] Ͱ͋ΔɽL̃ ʹؔͯ͠΋ಉ༷Ͱɼ‫࠲ॴہ‬ඪ x ͸ɼMp ʹԊͬͯ༩͑ΒΕΔɽ͜ͷͱ͖ɼx̃µ ͸ Mp ্Ͱɼ͋Δఆ਺ΛͱΔɽ͜ͷྲྀΕ͸ɼਤ 4.1 ͷΑ͏ͳֆΛඳ͘ͱ૝૾͠΍͍͢ɽ͍·ɼM ͸ F (੺͍ઢͷํ޲) ͱɼF̃ ʹ‫( ޲ํ͍ͨͮج‬੨͍ઢͷํ޲) ͷ 2 ௨Γʹઍ੾ΓͰ͖Δɽਤ 4.1

Ͱࣔͨ͠ ੺͍ઢ͕ F ͷ༿ɼ੨͍ઢ͕ F̃ ͷ༿Ͱ͋Δɽ੺͍ઢͱ੨͍ઢͷҐஔؔ܎ΛܾΊΔͷ͕ɼ࠲ඪ xµ , x̃µ Ͱ͋Δɽxµ ͸੺͍ઢ (F ͷ༿) ʹԊͬͯɼx̃µ ͸੨͍ઢ (F̃ ͷ༿) ʹԊͬͯ༩͑ΒΕ

Δɽ͕ͨͬͯ͠ɼF ͔Β༿ΛҰຕબͿͱ x̃µ ͸ఆ਺ΛͱΔɽ

M

p

F
ਤ 4.1

F̃

M ʹ‫ݱ‬ΕΔ༿૚ߏ଄ F ΍ F̃ ɽ༿ͷҰ෦෼ΛઢͰࣔͨ͠ɽ

͜ͷઅͰ͔֬ΊΒΕͨ͜ͱΛ؆୯ʹ·ͱΊΔɽύϥΤϧϛʔτଟ༷ମ M ͷ࠷େͷಛ௃͸ɼ઀

ଋ T M Λ L, L̃ ʹ෼ׂͨ͠ͱ͖ʹɼL ͷՄੵ෼৚݅ͱ L̃ ͷՄੵ෼৚͕݅‫׬‬શʹಠཱ͢Δ͜ͱͩͬͨɽ͜ͷಛ௃͔ΒɼFrobenius ͷఆཧΛ༻͍ͯ M ্ʹ ೋछྨͷ༿૚ߏ଄ F ͱ F̃ ΛఆٛͰ͖ͨɽ

M ͷഒԽ࠲ඪ (xµ , x̃µ ) ͔Βɼ൒෼ͷ xµ ࠲ඪΛൈ͖ग़͢͜ͱ͸ɼM ্ͷ༿૚ߏ଄ F ͔Β༿ΛҰຕબͿ͜ͱͱͯ͠આ໌Ͱ͖ͨɽ͜Ε͕ɼύϥΤϧϛʔτଟ༷ମ M ͕ഒԽۭؒͷ਺ֶతͳ‫ه‬ड़ͱͯ͠༗ྗࢹ͞ΕΔཧ༝Ͱ͋Δɽ࠷‫ʹޙ‬ɼL̃ ͱ L ͷ૒ରଋͰ͋Δ L∗ ͷؔ܎ʹ͍ͭͯίϝϯτ͓ͯ͘͠ɽύϥΤϧϛʔτ‫ ྔܭ‬η
͸ɼT M = L ⊕ L̃ ͔Β T ∗ M = L∗ ⊕ L̃∗ ΁ͷࣸ૾ͱΈͳͤΔɽΑͬͯɼη ͸ҎԼͷ;ͨͭͷಉ‫ܕ‬

Soryushiron Kenkyu

35ࣸ૾Λఆٛ͢Δɽ

φ+ : L̃ → L∗

φ− : L → L̃∗ .

and

(4.5)

͜ΕΒͷࣸ૾͸ɼL̃ ্ͷϕΫτϧ͕ɼL∗ ಺ͷ‫ͱࣜܗ‬ಉҰࢹͰ͖Δ͜ͱΛҙຯ͢ΔɽL ্ͷϕΫτϧͱɼL̃∗ ্ͷ‫ͯؔ͠ʹࣜܗ‬΋ಉ༷Ͱ͋ΔɽΑͬͯɼҎԼͷࣸ૾͕༩͑ΒΕΔɽ

Φ+ : T M → L ⊕ L∗

Φ− : T M → L̃ ⊕ L̃∗ .

and

(4.6)

ಛʹɼΦ+ ͸ɼnatural isomorphism ͱ͍͍ɼഒԽ‫ز‬ԿֶͱҰൠԽ‫ز‬ԿֶΛ݁ͼ͚ͭΔͷʹར༻͞
ΕΔɽNatural isomorphism ͕ࣗવʹ༩͑ΒΕΔ͜ͱ΋ɼύϥΤϧϛʔτଟ༷ମ͕ഒԽ‫ز‬Կֶͷ਺ֶతͳ‫ه‬ड़ͱͯ͠༗ྗͳཧ༝Ͱ͋Δɽ

4.3 ύϥυϧϘʔίϗϞϩδʔલઅͰɼύϥΤϧϛʔτଟ༷ମ M ͕ഒԽۭؒͷ਺ֶతͳ‫ه‬ड़ͱͯ͠༗༻ͳ͜ͱ͕֬ೝͰ͖ͨɽ

͔͠͠ɼM ্Ͱ Vaisman ѥ୅਺Λ࣮ࡍʹߏங͢Δʹ͸ɼ֎ඍ෼΍಺෦ੵͳͲͷԋࢉࢠΛఆٛ͠ɼ੒෼‫͖Ͱࢉܭ‬ΔΑ͏ʹॻ͖Լ͢ඞཁ͕͋Δɽ͜ͷઅͰ͸ M ্ͷԋࢉࢠʹ͍ͭͯड़΂ɼ۩ମ‫ࢉܭ‬ͷͨΊʹඞཁͳ४උΛߦ͏ɽ͜ͷઅͰ͸ɼલઅͰఆٛͨ͠ M ͷ distribution L, L̃ ʹରͯ͠ɼύϥυϧϘʔίϗϞϩδʔΛఆٛ͢Δɽ͸͡ΊʹɼM ͷ઀ۭؒ T M ʹରͯ͠ɼ௨ৗͷ֎ੵ୅਺Λఆٛ͢Δɽk ࣍ͷ‫׬‬શ൓ରশςϯιϧੵͷ੾அ Γ(∧k T M) ΛɼÂk (M) ͱ͢Δɽ͢ΔͱɼT M = L ⊕ L̃ Ͱ͋Δ͜ͱ͔Βɼ

Γ(L) ͱ Γ(L̃) ্ʹ֎ੵ୅਺ΛఆٛͰ͖ΔɽAr,s (M) Λ Γ((∧r L) ∧ (∧s L̃)) ͱ͢ΔͱɼÂk (M) ͸ҎԼͷΑ͏ʹ෼ׂͰ͖Δɽ

Âk (M) =

-

k=r+s

Ar,s (M).

(4.7)·ͨɼ͜ͷ෼ׂΛ‫͢͜ى‬Α͏ͳਖ਼४ࣹӨԋࢉࢠͱͯ͠

π r,s : Âr+s (M) → Ar,s (M)(4.8)Λఆٛ͢Δɽπ r,s ͸ɼP, P̃ ʹΑͬͯ༠ಋ͞ΕΔɽ۩ମతͳදࣜ͸ҎԼͷΑ͏ʹ༩͑ΒΕΔɽ·ͣɼ

T M ͷ෼ׂΛ΋ͨΒࣹ͢ӨԋࢉࢠΛɼҎԼͷΑ͏ʹఆΊΔɽ
#
$
#
$
0 0
1 0
P =
, P̃ =
.
0 1
0 0

(4.9)

͔͜͜Β͸ɼྫͱͯ͠ r = 2, s = 0 ͷ৔߹Λߟ͑ΔɽT ∈ Â2 (M) ͸ɼ੒෼දࣔͰ͸ҎԼͷΑ͏ʹॻ͚Δɽ

T MN =

#

tµν
tµν

tµν
tµν

$

.

(4.10)ྫ͑͹ɼπ 2,0 ͸ P ʹΑͬͯҎԼͷΑ͏ʹఆٛ͞ΕΔɽ

P

M

KP

N

LT

KL

=
=

#

#

0 0
0 1
0
0

0

$#

tµν

$

tµν
tµν
.

tµν
tµν

$#

0 0
0 1

$
(4.11)

Soryushiron Kenkyu

36tµν ͸ Â2,0 (M) ͷ੒෼Ͱ͋Δɽ͜ΕʹΑΓɼπ 2,0 (T M N ) = tµν ͕༠ಋ͞ΕΔɽಉ༷ʹɼπ 1,1 , π 0,2΋ఆٛՄೳͰ͋Δɽ

࣍ʹɼL ͱ L̃ ʹ࡞༻͢Δ֎ඍ෼ԋࢉࢠΛҎԼͷΑ͏ʹ༩͑Δɽ

d̃ : Ar,s (M) → Ar+1,s (M),d : Ar,s (M) → Ar,s+1 (M).

(4.12)

͜ΕΒͷ֎ඍ෼࡞༻ૉ͸ɼύϥυϧϘʔ࡞༻ૉͱ‫ݺ‬͹Εɼ௨ৗͷෳૉଟ༷ମ্ͷυϧϘʔ࡞༻ૉͱಉ͘͡ɼҎԼͷੑ࣭Λ࣋ͭɽd2 = 0,dd̃ + d̃d = 0,d̃2 = 0.

(4.13)

͜ͷΑ͏ͳύϥυϧϘʔ࡞༻ૉͷϕΩྵੑ͔ΒɼύϥυϧϘʔίϗϞϩδʔΛఆٛͰ͖ΔɽՃ͑ͯɼA ∈ Γ(L) ͱ α ∈ Γ(L̃) ʹରͯͦ͠ΕͧΕ಺෦ੵΛఆٛͰ͖ΔɽιA : Ar,s (M) → Ar−1,s (M),ι̃α : Ar,s (M) → Ar,s−1 (M).

(4.14)

ΑͬͯɼҎԼͷΑ͏ʹೋछྨͷ Lie ඍ෼ΛఆٛͰ͖Δɽξ ∈ Ar,s (M) ʹରͯ͠ɼLA ξ = (dιA + ιA d)ξ,L̃α ξ = (d̃ι̃α + ι̃α d̃)ξ.

(4.15)

͜ΕͰɼM ্Ͱ Vaisman ѥ୅਺Λ࠶‫͢ݱ‬Δʹ͋ͨͬͯඞཁͳԋࢉࢠΛ͢΂ͯఆٛͰ͖ͨɽ

࣍ʹɼ͜ΕΒͷԋࢉΛ࠲ඪදࣔͰॻ͖Լ͢ํ๏Λߟ͑ΔɽύϥΤϧϛʔτ‫ ྔܭ‬η ʹΑͬͯɼA1,0(M) × A0,1 (M) ͷ C ∞ (M, R)-૒ઢ‫͕૾ࣸܕ‬ଘࡏ͢Δɽ⟨⟨α, A⟩⟩ ΛରশରΛͱΔɼͱ͍͏ૢ

࡞ͱͯ͠ఆٛ͢ΔɽϕΫτϧͱ 1 ‫ࣜܗ‬ͷ಺ੵ ⟨·, ·⟩ ͱࣅ͍ͯΔ͕ɼL ͱ L̃ ͕૒ରͰͳ͘ͱ΋ɼରশରΛͭ͘Δ͜ͱ͕Ͱ͖Δɽಛʹɼα ∈ A0,s (M), A1 , . . . , As ∈ A1,0 (M) ͱ͢Δͱɼ

⟨⟨α, A1 ∧ · · · ∧ As ⟩⟩ = α(A1 , . . . , As ).

(4.16)

ಉ༷ʹɼA ∈ Ar,0 (M), α1 , . . . , αr ∈ A0,1 (M) ͱ͢Δͱɼ

⟨⟨α1 ∧ · · · ∧ αs , A⟩⟩ = A(α1 , . . . , αr ).

(4.17)

͕ͨͬͯ͠ɼα ∈ A0,s (M) ͷͱ͖ɼιA α ͸ A0,s−1 (M) ʹ‫·ؚ‬ΕΔɽ͜ΕͰɼpairing ⟨⟨α, A⟩⟩ ʹ‫͍ͯͮج‬ɼΓ(∧r L̃) ͔Β Γ(∧r−1 L̃) ΛಘΔૢ࡞΍ɼΓ(∧s L) ͔Β Γ(∧s−1 L) ΛಘΔૢ࡞Λ‫ن‬ఆͰ͖ͨɽ͜Ε͸ɼ্ड़ͷ಺෦ੵͷఆٛͦͷ΋ͷͰ͋Δɽ಺෦ੵ͸ɼpairing Λ࢖ͬͯ۩ମతʹҎԼͷΑ͏ʹॻ͖ԼͤΔɽιA α(A1 ∧ · · · ∧ As−1 ) = α(A, A1 , . . . , As−1 ).

(4.18)

ಉ༷ʹɼA ∈ Ar,0 (M) ͷͱ͖ɼια A ͸ Ar−1,0 (M) ʹ‫·ؚ‬ΕΔɽΑͬͯɼι̃α A(α1 ∧ · · · ∧ αr−1 ) = A(α, α1 , . . . , αs−1 ).

(4.19)

·ͨɼα ∈ A0,s (M), β ∈ A0,• (M), A ∈ Ar,0 (M), B ∈ A•,0 (M) ͱͨ͠ͱ͖ɼιA , ι̃α ͸ͦΕͧΕҎԼͷΑ͏ʹ࡞༻͢Δɽ

ιA (α ∧ β) = (ιA α) ∧ β + (−1)s α ∧ ιA β,rι̃α (A ∧ B) = (ι̃α A) ∧ B + (−1) A ∧ ι̃α B.

(4.20)(4.21)

Soryushiron Kenkyu

37

4.4 DFT ʹ͓͚Δ Vaisman ѥ୅਺͜ͷઅͰ͸ɼલઅ·Ͱͷ಺༰Λ༻͍ͯɼC ‫͕ހׅ‬ఆٛ͢Δѥ୅਺ߏ଄ʹ͍ͭͯٞ࿦͢ΔɽॳΊʹɼഒԽۭؒΛύϥΤϧϛʔτଟ༷ମ M ʹΑͬͯ‫࠲ॴہ‬ඪ‫ܥ‬Λ༻͍ͨ‫۩Ͱܗ‬ମతʹ‫ن‬ఆ͢Δɽͦ

ͯ͠ɼલઅͰ L, L̃ ΛಘΔͷʹ༻͍ࣹͨӨ P, P̃ ͳͲΛߦྻͰ੒෼දࣔ͢Δɽ·ͨɼL, L̃ ͕ Lie ѥ୅਺Ͱ͋Δ͜ͱΛ֬ೝ্ͨ͠Ͱɼ3 ষͰ༩͑ͨ ֎ඍ෼ԋࢉࢠͳͲ΋͢΂ͯ੒෼දࣔ͢Δɽͦͯ͠ɼ

L ͱ L̃ ͷؒʹ derivation ৚͕݅੒ཱ͍ͯ͠ͳ͍͜ͱΛ֬ೝ͠ɼ‫߆͍ڧ‬ଋ৚݅ͱ derivation ৚݅ͷؔ܎Λௐ΂Δɽ

DFT ʹ‫ݱ‬ΕΔഒԽۭؒͱͯ͠ɼ2D ࣍‫ݩ‬ͷฏୱͳύϥΤϧϛʔτଟ༷ମ M Λ࠾༻͢Δ [26]ɽ

M ্ͷ‫࠲ॴہ‬ඪ‫ܥ‬Λ xM (M = 1, . . . , 2D) ͱ͢Δͱɼ઀ۭؒ T M ͸ ∂M ʹΑͬͯுΒΕΔɽલ

ͷઅͰड़΂ͨ௨ΓɼT M ͸ࣹӨԋࢉࢠ P, P̃ ʹΑͬͯ L, L̃ ʹ෼ׂ͞ΕΔɽͦΕʹԠͯ͡ೋछྨͷ༿૚ߏ଄ F ͱ F̃ ΛఆٛͰ͖ͨ͜ͱ͔ΒɼM ͷ࠲ඪ xM ͸ɼF ͷ༿ʹԊͬͨ࠲ඪ xµ ͱ F̃ ͷ༿ʹԊͬͨ࠲ඪ x̃µ ʹ෼ׂ͢Δ͜ͱ͕Ͱ͖ͨɽ͜ΕʹΑΓɼഒԽϕΫτϧ Ξ = ΞM ∂M ∈ Γ(T M) ͸ҎԼͷΑ͏ʹɼL ଆͷ੒෼ͱ L̃ ଆͷ੒෼ʹ෼ׂͯ͠ॻ͚ΔɽA ∈ Γ(L), α ∈ Γ(L̃) ͱ͢ΔͱɼΞM ∂M = Aµ (x, x̃)∂µ + αµ (x, x̃)∂˜µ .

(4.22)

·ͨɼύϥΤϧϛʔτଟ༷ମʹ͸ɼఆ͔ٛΒඞͣ neutral ‫ ྔܭ‬η ͕ଘࡏ͢ΔɽηM N =

#

0 1
1 0

$

.

(4.23)

Neutral ‫ྔܭ‬͸ɼͦͷ‫ݻ‬༗஋ͷූ߸͕ਖ਼ʹͳΔ੒෼ͱෛʹͳΔ੒෼͕ಉ਺ͳ‫ྔܭ‬Λࢦ͢ɽη ͸ɼ
DFT ʹ͓͚Δ O(D, D) ෆมͳߏ଄ (2.5) ʹରԠ͢Δɽ·ͨɼη ʹΑΓࣸ૾ L → L̃ ͕༠ಋ͞ΕɼഒԽϕΫτϧͷ଍ͷ্Լ ηM N AN = AM Λ‫ن‬ఆ͢Δɽ଍ͷ্ԼΛߟ͑ΔͱɼL̃ ͱ L ͷ૒ରଋ L∗ ͱͷؒʹಉ‫( ૾ࣸܕ‬4.6) ͕ଘࡏ͢Δ͜ͱ͕Θ͔Δɽη ʹΑͬͯɼഒԽϕΫτϧͷؒͷ಺ੵη(·, ·) = (·, ·)+ ΛఆٛͰ͖Δɽͨͩ͠ɼX, Y ∈ Γ(L) ʹରͯ͠ɼ⟨X, Y ⟩ = 0 Ͱ͋ΔɽL̃ ʹ͍ͭͯ΋ಉ༷Ͱ͋Δɽ͕ͨͬͯ͠ɼL, L̃ ͸ͦΕͧΕ T M ͷ࠷େ౳ํͳ෦෼ଋͰ͋Δɽ

ύϥΤϧϛʔτଟ༷ମ M Ͱ͸ɼL, L̃ ͸ involutive ͳͷͰɼͦΕͧΕดͨ͡ަ‫ ͯ͠ͱੵࢠ׵‬Lie‫ހׅ‬ΛఆٛͰ͖Δɽ·ͣɼL ্ͷ Lie ‫ހׅ‬ΛҎԼͷΑ͏ʹఆٛ͢ΔɽA, B ∈ Γ(L) ͱͨ͠ͱ͖ɼ

[A, B]L = [A, B]µL ∂µ = (Aν ∂ν B µ − B ν ∂ν Aµ )∂µ .

(4.24)[·, ·]L ͸ Jacobi ཯Λຬͨ͠ɼ͔ͭ Leibniz ଇΛຬͨ͢ɽ͞Βʹɼanchor ͱͯࣗ͠໌ͳଋࣸ૾

ρL = idL Λ࠾༻͢ΔͱɼL ͸ Lie ѥ୅਺ͱͳΔɽL̃ ʹ΋ಉ༷ʹɼLie ѥ୅਺ΛఆٛͰ͖Δɽ͜ΕͰɼM ্ʹ‫ݱ‬ΕΔ Lie ѥ୅਺ L, L̃ Λ۩ମతʹ༩͑ΒΕͨɽ࣍ʹɼ2 ষͰ multi vector Λಋೖ͠ɼLie ‫ހׅ‬Λ Schouten-Nijenhuis ‫ʹހׅ‬ҰൠԽͨ͠Α͏ʹɼ[·, ·]L ʹ‫ ͍ͯͮج‬Schouten-Nijenhuis ‫ހׅ‬Λ੒෼දࣔͰ༩͑Δɽ·ͣ͸ɼk ࣍ͷ multi vector ͱͯ͠ɼA ∈ Γ(∧k L) ΛҎԼͷΑ͏ʹ༩͑Δɽ

A=

1 µ1 ···µk
A∂ µ1 ∧ · · · ∧ ∂ µk .
k!

(4.25)

Soryushiron Kenkyu

38“odd ࠲ඪ” ͱͯ͠ɼζµ = ∂µ Λಋೖ͢Δɽζ ͸άϥεϚϯ਺ͱͯ͠ѻ͏͜ͱ͕Ͱ͖ɼ∂/∂ζ ͸ӈඍ෼ͱͳΔɽ͢ͳΘͪɼ←−∂∂(ζµ1 · · · ζµn · · · ζµk ) = (ζµ1 · · · ζµn · · · ζµk )= (−1)k−n ζµ1 · · · ζ̌µn · · · ζµk .
∂ζµn
∂ζµn

(4.26)

ͨͩ͠ɼζ̌µn ͸ ζµn Λফ‫͢ڈ‬Δɼͱ͍͏ҙຯͰ༻͍ͨɽζ Λ༻͍Δͱɼk-vector A ͸ҎԼͷΑ͏
ʹॻ͚Δɽ

A=

1 µ1 ···µkAζ µ1 · · · ζ µk .
k!

(4.27)͢ΔͱɼSchouten-Nijenhuis ‫ހׅ‬͸ҎԼͷΑ͏ʹॻ͚ΔɽA ∈ Γ(∧p L), B ∈ Γ(∧q L) ͱͨ͠ͱ͖ɼ

[A, B]S =

#

$
$
#
∂
∂
(p−1)(q−1)
A ∂µ B − (−1)
B ∂µ A.
∂ζµ∂ζµ(4.28)

ಉ༷ͷखॱͰɼ[·, ·]L̃ Λ֦ுͯ͠ɼΓ(∧k L̃) ΛҾ਺ͱ͢Δ Schouten-Nijenhuis ‫·[ ހׅ‬, ·]∗S ΋ఆٛͰ͖ΔɽҰൠʹɼଟ༷ମ্ͷ multi vector ͸ɼSchouten-Nijenhuis ‫ʹހׅ‬Αͬͯ Gerstenhaber୅਺Λͳ͢ [58]. ·ͨɼϕΫτϧଋ V ্ͷ Lie ѥ୅਺ V → M ͱɼmulti vector Γ(∧p V ) ʹΑΔ

Gerstenhaber ୅਺͸౳ՁͰ͋Δ [59]ɽ

k-vector Λ (k + 1)-vector ʹࣸ͢ͷ͸ɼલઅͰఆٛͨ͠ ύϥυϧϘʔ࡞༻ૉ d̃ : ∧k L → ∧k+1 L

Ͱ͋Δɽd̃ ͸ҎԼͷΑ͏ʹఆٛ͞ΕΔɽX ∈ Γ(∧k L), α ∈ Γ(L̃) ͱ͢Δͱɼ

d̃X(α1 , . . . , αk+1 ) =k+1)(−1)i+1 ρL̃ (αi )(X(α1 , . . . , α̌i , . . . , αk+1 ))

i=1

+

)

(−1)i+j X([αi , αj ]∗S , α1 , . . . , α̌i , . . . , α̌j , . . . , αk+1 ).

(4.29)

i<jͨͩ͠ α̌i ͸ i ൪໨ͷ α Λফ‫͢ڈ‬Δͱ͍͏ҙຯͰ͋Δɽಛʹɼ‫࠲ॴہ‬ඪΛ༻͍Δͱɼd̃ ͸ k-vector

X ʹରͯ͠ҎԼͷΑ͏ʹ࡞༻͢Δɽd̃X =1 ˜µ ν1 ···νk∂ X(x, x̃)∂µ ∧ ∂ν1 ∧ · · · ∧ ∂νk .
k!

(4.30)

͜͜Ͱɼಛʹ k = 1 ͷ৔߹Λ‫͢ࢉܭ‬Δ͜ͱͰɼd̃ ͱ [·, ·]∗S ཱ͕྆͢Δ͜ͱΛ֬ೝ͢Δɽd̃ ͷఆٛ

(4.29) ͔Βɼk = 1 ͷͱ͖d̃A(α1 , α2 ) = (−1)2 ρL̃ (α1 )(A(α2 )) + (−1)3 ρL̃ (α2 )(A(α1 )) + (−1)3 A([α1 , α2 ]∗S )= ρL̃ (α1 )(A(α2 )) − ρL̃ (α2 )(A(α1 )) − A([α1 , α2 ]∗S ).
ͨͩ͠ɼA ∈ Γ(L), α1 , α2 ∈ Γ(L̃) Ͱ͋ΔɽΑͬͯɼຊઅͷલ൒Ͱ༩͑ͨύϥΤϧϛʔτଟ༷ମ্ͷ‫࠲ॴہ‬ඪදࣔΛ༻͍Δͱɼಛʹ ρ∗ (α1 ) = α1µ ∂˜µ ͱॻ͚Δ͜ͱʹ஫ҙ͢Ε͹A([α1 , α2 ]∗S ) = ρL̃ (α1 )(A(α2 )) − ρL̃ (α2 )(A(α1 )) − d̃A(α1 , α2 )= α1µ ∂˜µ (Aν α2ν ) − α2ν ∂˜ν (Aµ α1µ ) − (∂˜µ Aν − ∂˜ν Aµ )α1µ α2ν
= Aµ (α1ν ∂˜ν α2µ − α2ν ∂˜ν α1µ ).

Soryushiron Kenkyu

39͕ͨͬͯ͠ɼd̃ ͱ [·, ·]∗S ͸ཱ྆͢Δ͜ͱ͕Θ͔Δɽಉ༷ͷखॱͰɼd ͱ [·, ·]S ΋ཱ྆͢Δ͜ͱ͕͔֬ΊΒΕΔɽ

࣍ʹɼLie ඍ෼Λ‫࠲ॴہ‬ඪදࣔ͢Δɽ಺෦ੵ͸࣍ͷΑ͏ʹ࡞༻͢ΔɽA, B ∈ A1,0 (M), α, β ∈A0,1 (M) ͱ͢ΔͱɼιA β = Aµ βµ ,ι̃α B = αµ B µ ,ιA dβ = (Aν ∂ν βµ − Aν ∂µ βν )∂˜µ ,ι̃α d̃B = (αν ∂˜ν B µ − αν ∂˜µ B ν )∂µ .

(4.31)

͢Δͱɼ(4.15) Ͱఆٛͨ͠ Lie ඍ෼͸ҎԼͷΑ͏ʹॻ͚Δɽ

LA β = (dιA + ιA d)β= d(ιA β) + ιA (dβ)= d(Aν βν ) + ιA (∂µ βν ∂˜µ ∧ ∂˜ν )= [(∂µ Aν )βν + Aν ∂µ βν ]∂˜µ + Aµ ∂µ βν ∂˜ν − Aν ∂µ βν ∂˜µ= (Aν ∂ν βµ + βν ∂µ Aν )∂˜µ .

(4.32)L̃α B = (d̃ι̃α + ι̃α d̃)B

= d̃(αν B ν ) + ι̃α (∂˜µ B ν ∂µ ∧ ∂ν )= [(∂˜µ αν )B ν + αν ∂˜µ B ν ]∂µ + αµ ∂˜µ B ν ∂ν − αν ∂˜µ B ν ∂µ= (αν ∂˜ν B µ + B ν ∂˜µ αν )∂µ .

(4.33)

4.5 Derivaiton ৚݅ͷഁΕ͜Ε·Ͱͷٞ࿦͔ΒɼύϥΤϧϛʔτଟ༷ମ M ্ʹ૒ରͳ;ͨͭͷ Lie ѥ୅਺ (L, [·, ·]L , ρL , d)ͱ (L∗ , [·, ·]L∗ , ρL∗ , d∗ ) Λ‫࠲ॴہ‬ඪදࣔͰಘΔ͜ͱ͕Ͱ͖ͨɽ͜ΕΒͷѥ୅਺Λ༻͍ͯɼલͷষͰఏࣔͨ͠ double Խ͢Δૢ࡞ʹΑͬͯಘΒΕΔߏ଄ (L, L̃) Λٞ࿦͍ͨ͠ɽલͷষͰड़΂ͨ௨ΓɼLie ૒ѥ୅਺͸ Lie ѥ୅਺ (L, [·, ·]L , ρL , d) ͱɼL ͱ૒ରͳ Lie ѥ୅਺ (L∗ , [·, ·]∗L , ρL , d∗ ) ͷؒʹ
derivation ৚͕݅੒ཱ͢Δͱ͖ʹఆٛͰ͖Δɽderivation ৚݅͸ҎԼͷΑ͏ͳ৚݅Ͱ͋Δɽd∗ [X, Y ]S = [d∗ X, Y ]S + [X, d∗ Y ]S ,X, Y ∈ Γ(∧• L).

(4.34)

લઅͷ (4.28) ͰɼSchouten-Nijenhuis ‫ހׅ‬Λ੒෼දࣔͨ͠ͷͰɼύϥΤϧϛʔτଟ༷ମ্ʹ࠲ඪΛ͋ΒΘʹͯ͠༩͑ͨ Lie ѥ୅਺ L, L̃ ʹରͯ͠ɼderivation ৚͕݅੒ཱ͍ͯ͠Δ͔Ͳ͏͔͔֬ΊΒΕΔΑ͏ʹͳͬͨɽ࣮ࡍʹ (4.34) ͷࠨลΛ‫͢ࢉܭ‬Δͱɼ࣍ͷ݁Ռ͕ಘΒΕΔɽ

d̃[A, B]S = ∂˜µ [A, B]νS ∂µ ∧ ∂ν= ∂˜µ (Aρ ∂ρ B ν − B ρ ∂ρ Aν )∂µ ∧ ∂ν= (∂˜µ Aρ ∂ρ B ν + Aρ ∂ρ ∂˜µ B ν − ∂˜µ B ρ ∂ρ Aν − B ρ ∂ρ ∂˜µ Aν )∂µ ∧ ∂ν .

(4.35)

Soryushiron Kenkyu40
ଓ͍ͯɼ(4.34) ͷӈลΛ‫͢ࢉܭ‬ΔͱҎԼͷ݁Ռ͕ಘΒΕΔɽ

#

$
#
$
∂
∂
0
[d̃A, B]S =B ∂ρ d̃A
d̃A ∂ρ B − (−1)∂ζρ∂ζρ= (∂˜µ Aρ ζµ − ∂˜ρ Aµ ζµ )∂ρ B ν ζν − B ρ ∂ρ ∂˜µ Aν ζµ ζν= (∂˜µ Aρ ∂ρ B ν − ∂˜ρ Aµ ∂ρ B ν − B ρ ∂ρ ∂˜µ Aν )∂µ ∧ ∂ν ,[A, d̃B]S = −[d̃B, A]S= −(∂˜µ B ρ ∂ρ Aν − ∂˜ρ B µ ∂ρ Aν − Aρ ∂ρ ∂˜µ B ν )∂µ ∧ ∂ν .

(4.36)

͕ͨͬͯ͠ɼderivation ৚݅͸࣍ͷΑ͏ʹഁΕ͍ͯΔɽ

d̃[A, B]S = [d̃A, B]S + [A, d̃B]S + (∂˜ρ Aµ ∂ρ B ν + ∂˜ρ B ν ∂ρ Aµ )∂µ ∧ ∂ν .

(4.37)

ӈลࡾ߲໨ͷଘࡏ͔ΒɼύϥΤϧϛʔτଟ༷ମ্ʹ‫ݱ‬ΕΔ;ͨͭͷ Lie ѥ୅਺ L ͱ L̃(≃ L∗ ) ͷؒʹɼderivation ৚݅͸੒ཱ͠ͳ͍͜ͱ͕Θ͔Δɽ͕ͨͬͯ͠ɼ(L, L̃) ͸ Lie ૒ѥ୅਺Λͳ͞ͳ͍ɽՃ͑ͯɼ 3 ষͰࣔͨ͠௨ΓɼL ͱ L̃ ͷ double L ⊕ L̃ ʹΑͬͯಘΒΕΔߏ଄͸ Vaisman ѥ୅਺Ͱ͋Δɽ͜ͷ Vaisman ѥ୅਺ͷ anchor ͸ ρ = ρL + ρL̃ ɼඍ෼ԋࢉࢠ͸ D = d + d̃ Ͱఆٛ͞ΕΔɽ·ͨɼඇୀԽͳ૒ઢ‫ࣜܗܕ‬͸ɼରশରΛ༻͍ͯҎԼͷΑ͏ʹఆٛ͞ΕΔɽΞi ∈ Γ(T M) ͱͨ͠ͱ͖ɼ(Ξ1 , Ξ2 ) = (A + α, B + β) =/1.
⟨⟨α, B⟩⟩ + ⟨⟨β, A⟩⟩ .
2(4.38)

·ͨɼVaisman ‫ހׅ‬͸ҎԼͷ‫ʹܗ‬ఆٛ͢Δɽ͜Ε͸ɼDFT ͷ C ‫ͱހׅ‬શ͘ಉ͡‫͋Ͱܗ‬Δɽ

1[Ξ1 , Ξ2 ]V = [A + α, B + β]V = [A, B]L + LA β − LB α − d(ιA β − ιB α)
21+ [α, β]L̃ + L̃α B − L̃β A + d̃(ιA β − ιB α).
2(4.39)(L ⊕ L̃, [·, ·]C , ρV , (·, ·)) ͷ૊͸ɼVaisman ѥ୅਺ͱͳΔɽ

͜͜Ͱɼ(4.37) ͷӈล 3 ߲໨ʹண໨͢Δɽ͜ͷ߲͸ɼderivation ৚݅ͷഁΕΛ‫͍ͯ͠ݱ‬Δ߲Ͱ͋Γɼ∂˜ρ Aµ ∂ρ B ν + ∂˜ρ B ν ∂ρ Aµ = η KL ∂K Aµ ∂L B ν .

(4.40)

ͱॻ͖௚͢͜ͱ͕Ͱ͖Δɽ͜Ε͸ɼ·͞ʹ‫߆͍ڧ‬ଋ৚݅ (2.21) Λ՝͢͜ͱͰফ͑Δ‫ܗ‬ͷ߲Ͱ͋Δɽͭ·Γɼ‫߆͍ڧ‬ଋ৚݅Λ՝͢͜ͱͰɼL ͱ L̃ ͷؒʹ derivation ৚͕݅੒ཱ͢ΔΑ͏ʹͳΔɽΑͬͯɼ͜ΕΒͷϖΞ (L, L̃) ͸ Lie ૒ѥ୅਺Λఆٛ͢Δɽ·ͨɼ3 ষͰͷٞ࿦͔Βɼ্‫ه‬ͷ L ⊕ L̃ʹରͯ͠ derivation ৚݅Λ՝͢ͱจ‫[ ݙ‬35] ͷ Courant ѥ୅਺ͱͳΔɽΑͬͯɼDFT ʹ͓͚Δ

‫߆͍ڧ‬ଋ৚݅ͷ୅਺తͳ‫ݯى‬͸ɼL ͱ L̃ ͷؒͷ derivation ৚݅Ͱ͋Δͱ‫͑ݴ‬Δɽ͜Ε͸ɼจ‫[ ݙ‬36] ʹ͓͚Δɼpre-DFT ѥ୅਺ʹରͯ͠‫߆͍ڧ‬ଋ৚݅Λ՝͢ͱ Courant ѥ୅਺͕ಘΒΕΔͱ
͍͏ࢦఠͱ΋߹க͢Δɽ
ҰํͰɼC ‫ʹހׅ‬ରͯ͠‫߆͍ڧ‬ଋ৚݅Λ՝͢ͱɼ(2.2) ࣜͰࣔͨ͠ [17] ͷ Courant ‫͕ހׅ‬ಘΒΕΔͷ͸ɼ2 ষͷ࠷‫Ͱޙ‬ड़΂ͨ௨ΓͰ͋Δɽจ‫[ ݙ‬35] ͷ Courant ‫͔ހׅ‬Β (2.2) ࣜͷ Courant ‫ׅ‬‫ހ‬ΛಘΔʹ͸ɼ΋͏গ͠ߟ࡯͕ඞཁͰ͋Δɽ‫߆͍ڧ‬ଋ৚݅ͱ derivation ৚݅ͷؒͷؔ܎͸ਤ 4.2
ʹࣔͨ͠΄͔ɼ࣍ͷઅͰҾ͖ଓ͖ٞ࿦Λߦ͏ɽ

Soryushiron Kenkyu

41

4.6 Ұൠ‫ز‬Կֶͱͷؔ࿈͜ͷઅͰ͸ɼલઅ·Ͱٞ࿦ͨ͠ L, L̃ ʹΑΔήʔδ୅਺ͱɼҰൠԽ‫ز‬Կֶͱͷؔ܎Λٞ࿦͢Δɽ

DFT ͷήʔδରশੑΛ‫ن‬ఆ͢Δ C ‫ހׅ‬͸ɼύϥΤϧϛʔτ‫ز‬Կֶʹ͓͚Δ Vaisman ‫ͱހׅ‬Ұ
க͢Δ͜ͱ͸ɼ[15, 25] Ͱࢦఠ͞Ε͍ͯΔɽC ‫ހׅ‬͸ɼҰൠ࠲ඪม‫׵‬ͷඍ෼ಉ૬ࣸ૾ͱɼB ৔ͷ

U (1) ήʔδม‫׵‬Λಉ࣌ʹѻ͍ɼO(D, D) ‫ڞ‬มͱͳΔΑ͏ʹ૊Μͩ΋ͷͰ͋ΔɽVaisman ‫ހׅ‬ΛύϥΤϧϛʔτଟ༷ମ্Ͱ࠶‫͢ݱ‬Δʹ͋ͨͬͯɼ‫߆͍ڧ‬ଋ৚݅Λཁ‫͢ٻ‬Δඞཁ͸ͳ͍ɽΑͬͯɼ

Vaisman ‫ހׅ‬͸‫߆͍ڧ‬ଋ৚݅Λ՝͞ͳ͍ঢ়ଶͰͷ DFT ͷήʔδରশੑɼ͢ͳΘͪ “oﬀ-shell” ͰͷήʔδରশੑΛ‫ه‬ड़͢Δɽ‫ز‬Կֶతʹ͸ɼഒԽۭؒΛ‫ه‬ड़͢ΔͨΊʹύϥෳૉߏ଄Λಋೖ͍ͯ͠ΔͨΊɼ઀ଋ T M ͸ L ⊕ L̃ ͷ‫ʹܗ‬෼ׂͰ͖ɼL ͱ L̃ ͸ͦΕͧΕ Lie ѥ୅਺ͱͳΔɽL, L̃ ͸ଞʹ΋Α͍ੑ࣭Λ࣋ͭɽL, L̃ ͸ distribution Ͱ͋ΓɼDirac ߏ଄Ͱ΋͋ΔɽL, L̃ ্ͷ‫ੵހׅ‬͸

involutive Ͱ͋ΓɼFrobenius ͷఆཧ͔Βੵ෼ՄೳͰ͋Δɽ͕ͨͬͯ͠ɼL, L̃ ͸ͦΕͧΕ M ্ͷ༿૚ߏ଄ F, F̃ ͷ઀ଋͱͳΔɽxµ Ͱಛ௃෇͚ΒΕΔ෺ཧతۭ࣌͸ɼ༿૚ߏ଄͔Β༿ΛҰຕબͿ͜ͱͰఆΊΒΕΔ (༿ΛબͿ͜ͱͰɼx̃µ ͸ఆ਺ʹͳΔ)ɽ·ͨɼύϥΤϧϛʔτ‫ ྔܭ‬η ʹΑͬͯఆٛ͞ΕΔಉ‫͔૾ࣸܕ‬ΒɼL̃ = T F ≃ L∗ = T ∗ F ͳͷͰɼL̃ ͷϕΫτϧ͸ɼL∗ ͷ 1 ‫ͱࣜܗ‬ಉҰࢹͰµ͖Δɽ͕ͨͬͯ͠ɼL ଆͷ Lie ‫ހׅ‬͸ϕΫτϧଆͷήʔδύϥϝʔλ ξi ʹΑͬͯҰൠ࠲ඪม‫׵‬Λࢧ഑͠ɼL̃ ଆͷ Lie ‫ހׅ‬͸ 1 ‫ࣜܗ‬ଆͷήʔδύϥϝʔλ ξ˜i,µ ʹΑͬͯ B ৔ͷήʔδม‫׵‬Λࢧ഑͢ΔɽҰൠʹɼ1 ‫ࣜܗ‬ଆͷ Lie ‫ހׅ‬͸ 0 ʹͳΒͳ͍͜ͱ͔ΒɼO(D, D) ‫ڞ‬มͳ B ৔ͷήʔδม‫׵‬͸ɼඇՄ‫“ ͳ׵‬oﬀ-shell” ରশੑ΁ͱ֦ு͞ΕΔɽ

C-bracket

Vaisman bracket

strong constraint ∂˜ = 0

derivation condition (14)Courant bracket in (21)[·, ·]∗ = 0

c-bracket
ਤ 4.2

c-bracket
֤‫ͱހׅ‬৚݅ͷؔ܎Λࣔͨ͠ɽͰ͸ɼ‫߆͍ڧ‬ଋ৚݅Λ՝ͨ͠‫ޙ‬ͷ͜ͱΛߟ͑ͯΈΑ͏ɽ͜ͷͱ͖ɼDFT ͷήʔδ୅਺͸ C ‫ހׅ‬
Ͱด͡Δɽ͕ͨͬͯ͠ɼC ‫ʹހׅ‬Αͬͯ‫ن‬ఆ͞ΕΔ୅਺͕ରশੑΛ‫ه‬ड़͢Δʹ͸ɼ‫߆͍ڧ‬ଋ৚͕݅ඞͣຬͨ͞ΕΔඞཁ͕͋Δɽ‫߆͍ڧ‬ଋ৚݅Λղ͘࠷΋؆୯ͳํ๏͸ɼwinding ࠲ඪʹ͔͔Δඍ

˜ = 0) ͜ͱͰ͋Δɽ͜ͷͱ͖ɼ1 ‫ࣜܗ‬ଆͷ Lie ‫ހׅ‬͸ 0 ͱͳΓɼC ‫ހׅ‬͸ (2.2)෼Λ 0 ʹ͢Δ (∂∗ࣜͷ c ‫·[ ހׅ‬, ·]c ʹม‫͢ܗ‬Δ (ਤ 4.2)ɽ͜Ε͸ɼDFT ৔΍ήʔδύϥϝʔλʹରͯ͠ ‫߆͍ڧ‬ଋ৚݅Λ՝͠ɼ“on-shell”ɼ͢ͳΘͪ෺ཧతͳ෦෼ۭؒΛఆٛ͢Δ͜ͱʹΑΓɼ“oﬀ-shell” Ͱ͸ඇՄ‫׵‬

ͩͬͨ B ৔ͷήʔδରশੑ͕ “on-shell” Ͱ͸Մ‫ͳʹ׵‬Δ͜ͱΛҙຯ͢Δɽ͜ͷҙຯͰɼC ‫ހׅ‬ͷ “on-shell” ൛ͱͯ͠ରԠ͢Δͷ͕ c ‫͍͑ͱހׅ‬Δɽ਺ֶతʹ͸ɼc ‫ހׅ‬͸࣍ͷखॱͰಘΒΕΔɽ

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42·ͣɼVaisman ‫ʹހׅ‬ରͯ͠ɼderivation ৚݅ (3.20) Λ՝͢ɽͦͯ͠ɼL̃ ଆͷ Lie ‫ހׅ‬Λ zerobracket ([·, ·]L̃ = 0) ͱ͢Ε͹Α͍ɽ੒෼‫ʹࢉܭ‬Αͬͯ໌ࣔతʹࣔͨ͠Α͏ʹɼC ‫ن͕ހׅ‬ఆ͢Δ

Vaisman ѥ୅਺ʹରͯ͠ derivation ৚݅Λ՝͢ͱɼͦͷߏ଄͸จ‫[ ݙ‬35] ͷ Courant ѥ୅਺ͱͳΔɽ·ͨɼ (L, L̃) ͸શମͰ Lie ૒ѥ୅਺Λ੒͢ɽ͞Βʹɼ[·, ·]L̃ = 0 ͱ͢Ε͹ C ‫ހׅ‬͸ (2.2) ࣜͷ c ‫ʹހׅ‬มԽ͢Δɽ͜ͷ c ‫ހׅ‬͸ɼҰൠԽ‫ز‬Կֶʹ‫ݱ‬ΕΔ Courant ‫ͳ͔΄ʹހׅ‬Βͳ͍ [11]ɽ
ύϥυϧϘʔίϗϞϩδʔΛ‫ن‬ఆͨ͜͠ͱͰɼ‫߆͍ڧ‬ଋ৚݅Λຬͨ͢Α͏ͳɼ“on-shell” ͷ

DFT ৔΍ήʔδύϥϝʔλ͸ɼύϥυϧϘʔ࡞༻ૉΛ࡞༻ͤ͞Δͱ 0ɼ͢ͳΘͪύϥਖ਼ଇ(para-holomorphic) ͳྔͱͳΔಛ௃͕͋Δɽ
para-holomorphic :d̃Φ = 0.

(4.41)Φ ͸ɼഒԽۭؒͷ೚ҙͷ৔΍ήʔδύϥϝʔλͰ͋Δɽ͜Ε͸ɼΦ ͕ F ͷ༿ʹ੍‫͞ݶ‬Ε͍ͯΔͷʹ౳͍͕͠ɼύϥυϧϘʔ࡞༻ૉ͕ d, d̃ ͷೋछྨ͋Δ͜ͱ͔Βɼ‫߆͍ڧ‬ଋ৚݅ͷղͱͯ͠ҰҙͰ͸ͳ͍ɽ‫߆͍ڧ‬ଋ৚݅ͷ΋͏Ұͭͷղ͸ɼΦ ͕ύϥ൓ਖ਼ଇ (anti-para-holomorphic) ͳྔͱͳΔ
৔߹Ͱ͋Δɽ

anti-para-holomorphic :dΦ = 0.

(4.42)

ύϥ൓ਖ਼ଇͳྔ͸ɼF̃ ͷ༿ʹ੍‫͞ݶ‬ΕΔɽͭ·Γɼxµ Λఆ਺ʹ͢Δ͜ͱͰఆٛͰ͖Δ windingۭؒʹଘࡏ͢ΔɽDFT ʹ͓͚Δӡಈํఔࣜͷղͱͯ͠ɼwinding ۭؒʹґଘ͢Δۭ࣌΋‫ݱ‬Ε
Δ [60–64]ɽ

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43

ୈ5ষ

·ͱΊ
ຊम࢜࿦จͰ͸ɼ‫ݭ‬ཧ࿦ʹ͓͚Δ T ૒ରੑΛ໌നʹ‫ؚ‬Μͩॏྗཧ࿦Ͱ͋Δ Double Field Theoryʹ‫͍ͯͮج‬ɼಛʹ DFT ͷۭ࣌ͷ‫ز‬Կֶతͳඳ૾΍ɼήʔδରশੑʹ‫ݱ‬ΕΔ୅਺ߏ଄ͳͲͷ਺ֶతͳଆ໘ʹ͍ͭͯٞ࿦ͨ͠ɽ
ୈ 2 ষͰ͸ɼDFT ʹ͍ͭͯɼඞཁ࠷௿‫ͳݶ‬෦෼͚ͩΛ͘͝؆୯ʹ·ͱΊͨɽ·ͣഒԽ࠲ඪΛఏࣔ͠ɼ‫ྔܭ‬΍ɼ‫ج‬ຊͱͳΔ DFT ৔ͳͲΛ঺հͨ͠ɽͦͯ͠ɼҰൠԽ Lie ඍ෼ͷ‫͔ࢉܭ‬Βɼಛʹ DFT ʹ‫ݱ‬ΕΔήʔδ୅਺ߏ଄Λੜ੒͢Δ C ‫ʹࡉৄ͍ͯͭʹހׅ‬ड़΂ɼຊ࿦จͷओ୊Ͱ͋Δ

Vaisman ѥ୅਺ͱ͍͏৽͍͠ߏ଄͕ఏࣔ͞ΕΔ͜ͱΛ঺հͨ͠ɽ·ͨɼ෺ཧత section ৚݅ʹ͍ͭͯड़΂ɼ‫߆͍ڧ‬ଋ৚݅Λ՝͢લ‫Ͱޙ‬ͷ C ‫ހׅ‬ͷ;Δ·͍ͷมԽ΍ɼDFT ͷ‫ز‬Կֶతͳ࿮૊Έͷҧ͍ʹ͍ͭͯ΋‫ͨ͠ٴݴ‬ɽ
ୈ 3 ষͰ͸ɼࠓ౓͸਺ֶతͳ‫͔఺؍‬Βɼ࠷΋‫ૅج‬తͳ୅਺ߏ଄Ͱ͋Δ Lie ୅਺Λग़ൃ఺ͱ͠ɼ
ঃʑʹߏ଄ΛҰൠԽ͍ͯ͘͜͠ͱͰ Vaisman ѥ୅਺ͷ‫ͳີݫ‬ఆٛΛ༩͑ͨɽ·ͨɼDrinfel’ddouble ͱ͍͏ૢ࡞ʹண໨͠ɼVaisman ѥ୅਺͸ɼશମͰ Lie ૒ѥ୅਺Λͳ͞ͳ͍Α͏ͳ Lie ѥ୅਺ͷϖΞ (E, E ∗ ) ͷ double ʹΑͬͯߏ੒Ͱ͖Δ͜ͱΛࣔͨ͠ɽՃ͑ͯɼ[35] ͷ Courant ѥ୅਺ʹؔ͢Δٞ࿦ʹ‫ ͍ͯͮج‬Vaisman ѥ୅਺ͷ Dirac ߏ଄ʹΑΔ෼ׂΛࢼΈͨɽ෼ׂʹΑͬͯ‫ݱ‬ΕΔ;ͨͭͷ Dirac ߏ଄ L, L̃ ≃ L∗ ͷؒʹ͸ derivation ৚݅͸੒ཱͤͣɼL, L̃ ≃ L∗ ͷ૊͸શମͰ

Lie ૒ѥ୅਺Λͳ͞ͳ͍͜ͱ͕͔֬ΊΒΕͨɽୈ 4 ষͰ͸ɼୈ 3 ষͰ໌Β͔ʹͨ͠ Vaisman ѥ୅਺ͷ double ߏ଄Λ༻͍ͯɼୈ 2 ষͰ঺հ͠
ͨ DFT ͷήʔδ୅਺ߏ଄ʹ͍ͭͯߟ࡯ͨ͠ɽ·ͣɼഒԽۭؒͷ਺ֶతͳඳ૾ͱͯ͠ɼ2D ࣍‫ݩ‬ͷύϥΤϧϛʔτଟ༷ମ M Λಋೖͨ͠ɽM ͸ɼύϥෳૉߏ଄ K Λ࣋ͭ͜ͱ͔Βɼ઀ଋ T M Λ࠷େ౳ํతͳ෦෼ଋ L, L̃ ΁ͱࣗવʹ෼ׂ͢Δ (T M = L ⊕ L̃)ɽ͜Εʹ൐͍ɼT M ͷஅ໘Ͱ༩͑ΒΕΔഒԽϕΫτϧ Ξ = ΞM ∂M ΛɼΞ = Aµ ∂µ + αµ ∂˜µ ʹ෼ׂͨ͠ɽ͜͜ͰɼA ∈ Γ(L), α ∈ Γ(L̃)Ͱ͋Δɽͦͯ͠ɼL, L̃ ͸ͦΕͧΕ Lie ѥ୅਺Ͱ͋Δ͜ͱΛ͔֬ΊͨɽL, L̃ ʹର͢Δ֎ੵ୅਺Λ‫ن‬ఆ͠ɼύϥυϧϘʔίϗϞϩδʔΛߏஙͨ͠ɽৄࡉͳ੒෼‫͔ࢉܭ‬ΒɼL ଆʹ࡞༻͢Δ֎ඍ෼ԋࢉࢠ d̃ ͕ɼLie ‫·[ ހׅ‬, ·]L ʹରͯ͠ derivation ৚݅Λຬͨ͞ͳ͍͜ͱΛ֬ೝ͠ɼL, L̃ ͸ҰൠʹશମͰ Lie ૒ѥ୅਺Λͳ͞ͳ͍͜ͱΛࣔͨ͠ɽୈ 2 ষͰͷٞ࿦͔ΒɼC ‫ن͕ހׅ‬ఆ͢Δ୅਺ߏ଄͸ Vaisman ѥ୅਺Ͱ͋Δ͜ͱΛվΊͯࣔͨ͠ɽ·ͨɼDFT ͷ‫߆͍ڧ‬ଋ৚݅ͷ୅਺తͳ‫͕ݯى‬ɼ

derivation ৚݅Ͱ͋Δ͜ͱΛࣔͨ͠ɽຊम࢜࿦จʹ͓͍ͯಘΒΕͨ৽ͨͳ݁Ռ͸ɼओʹ਺ֶతͳ‫͔఺؍‬ΒɼVaisman ѥ୅਺͕;ͨͭ

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ͷ Lie ѥ୅਺ͷ double ͰಘΒΕΔͱࣔͨ͜͠ͱͱɼͦͷ෺ཧతͳԠ༻ͱͯ͠ɼ‫߆͍ڧ‬ଋ৚݅ͷ୅਺తͳ‫ ͕ݯى‬derivation ৚݅Ͱ͋Δͱ໌Β͔ʹͨ͜͠ͱͷ;ͨͭͰ͋Δɽ͜ͷ݁Ռ͸ɼ[13] ʹ͓͍ͯग़൛ࡁΈͰ͋Δɽɹࠓ‫ޙ‬ͷల๬
ഒԽ‫ز‬ԿֶΛ௨ͯ͠ DFT ͷཧ࿦ߏ଄Λཧղ͠ɼT ૒ରੑ͕ۭ࣌ʹ༩͑ΔӨ‫ڹ‬Λ஌Γ͍ͨͱ͍͏͜ͱ͕ɼ‫ڀݚ‬ͷ໨తͰ͋Δɽ࠷‫ʹޙ‬ɼDFT ΍ ഒԽ‫ز‬Կֶʹؔ܎͢ΔຊจʹऔΓ্͛ͨҎ֎ͷ࿩୊΍ɼ͜ͷ໨తʹԊͬͯࠓ‫ޙ‬औΓ૊Έ͍ͨ໰୊Λ͍͔ͭ͘঺հ͍ͨ͠ɽ͢Ͱʹड़΂ͨ௨ΓɼDFT ʹ͓͍ͯ C ‫͕ހׅ‬ಛ௃෇͚ͨͷ͸ɼͦͷήʔδରশੑͰ͋Γɼ‫͑׵͍ݴ‬Ε͹ ແ‫ݶ‬খͷ(infinitesimal) ήʔδม‫͋Ͱ׵‬Δɽ͜Εʹର͠ɼ༗‫ݶ‬ͷ (finite) ήʔδม‫ͯؔ͠ʹ׵‬΋ٞ࿦͕ߦΘΕ͍ͯΔɽ୅දతͳ࿦จ͸ [19, 65–71] Ͱ͋Δɽಛʹ਺ֶతͳ‫͔఺؍‬Β‫ͱ͏ݴ‬ɼLie ‫ ͕܈‬Lie ୅਺͔Βߏ੒Ͱ͖Δͱ͍͏ࣄ࣮͔Βྨਪ͢ΔʹɼDFT ͷ༗‫ݶ‬ήʔδม‫׵‬͸ɼVaisman ѥ୅਺ͷ “ੵ෼ (integration)” ૢ࡞ʹΑͬͯಘΒΕΔͱߟ͑ΒΕΔɽগͳ͘ͱ΋ɼLie ૒ѥ୅਺ͷੵ෼ʹΑͬͯPoisson-Lie ‫͕܈‬ಘΒΕΔ [72] ͜ͱ΍ɼͦͷҰൠԽͰ Lie (૒) ѥ୅਺ͷੵ෼ʹΑͬͯ (Poisson-)Lie ѥ‫( ܈‬groupoid) ͕ಘΒΕΔ͜ͱ [43, 44] ͸͢Ͱʹ஌ΒΕ͍ͯΔɽ·ͨɼ͋Δछͷ Courant ѥ୅਺ͷੵ෼ʹΑͬͯѥ‫*܈‬1 ͕‫ݱ‬ΕΔͩΖ͏ɼͱ͍͏༧ଌ͕͋Δ [74, 75]ɽ͜ͷΑ͏ͳɼʮੵ෼ʹΑͬ
ͯɼ͋Δ୅਺ (·ͨ͸ѥ୅਺) ͔Β‫ͨ·( ܈‬͸ѥ‫ )܈‬ΛಘΔʯͱ͍͏໰୊͸ɼ“coquecigrue problem”*2 ͱ‫ݺ‬͹ΕΔɽ΋͠ɼVaisman ѥ୅਺ʹର͢Δ coquecigrue problem ͕ղܾ͢Ε͹ɼͦΕ͸਺ֶతʹ‫ڵ‬ຯਂ͍݁ՌͰ͋Δ͚ͩͰͳ͘ɼDFT ͷήʔδରশੑͷ‫ز‬Կֶతͳ‫ݯى‬Λ༩͑ΔͩΖ͏ɽ·ͨɼ‫߆͍ڧ‬ଋ৚݅Λ՝͢͜ͱ͸ɼDFT ͷดͨ͡ήʔδ୅਺ΛಘΔͨΊͷे෼৚݅Ͱ͋Γɼඞཁ৚݅Ͱ͸ͳ͍ɽΑΓҰൠతʹ͸ɼ‫߆͍ڧ‬ଋ৚݅Λඞཁͱ͠ͳ͍ DFT ηοτΞοϓ΋ߟ͑ΒΕ͍ͯΔ [76]ɽ͜ͷΑ͏ͳঢ়‫Ͱگ‬͸ɼຊम࢜࿦จͰѻͬͨ Vaisman ѥ୅਺͸ΑΓॏཁͳҙຯ߹͍Λ
࣋ͭͩΖ͏ɽ
ഒԽۭؒΛ਺ֶతʹ‫ه‬ड़͢Δʹ͋ͨͬͯɼ‫ݱ‬ஈ֊Ͱ͸༷ʑͳํ๏͕ߟ͑ΒΕ͍ͯΔɽࠓ‫͜ޙ‬ΕΒͷํ๏͕ͨͩͻͱͭͷਖ਼౰ͳ‫ه‬ड़ʹߜΓࠐ·Ε͍ͯ͘ͷ͔ɼ͋Δ͍͸ෳ਺ͷ౳Ձͳ‫ه‬ड़͕‫ͭݟ‬͔Δͷ͔͸ఆ͔Ͱ͸ͳ͍͕ɼ͍ͣΕʹͤΑঃʑʹͦΕͧΕͷํ๏ͷؔ܎ੑ΋੔ཧ͞Ε͍ͯͩ͘
Ζ͏ɽຊम࢜࿦จͰ͸ɼഒԽۭؒͷ࠶‫ͯ͠ͱݱ‬ύϥΤϧϛʔτଟ༷ମΛ࠾༻ͨ͠ɽ͜Εͱಠཱͨ͠Ξϓϩʔνͱͯ͠ graded ‫ز‬ԿֶΛ༻͍Δํ๏΋͋Δ [21, 77, 78]ɽ͜ΕΒͷ;ͨͭͷํ๏͸ɼ

Courant ѥ୅਺ͱ QP ଟ༷ମ͕౳ՁͰ͋Δ͜ͱ [79] ʹ༝དྷͯ͠ಉ౳ͳ΋ͷͰ͋ΔɽC ‫ހׅ‬ͷߏ଄ͱ derived ‫[ ੵހׅ‬21, 80] ͷؔ܎Λௐ΂Δͱɼgraded ‫ز‬ԿֶͱύϥΤϧϛʔτ‫ز‬Կֶͷؔ܎͕੔ཧ͞ΕΔ͸ͣͰ͋Δɽ·ͨɼύϥΤϧϛʔτ‫ز‬Կֶͷ֦ுͱͯ͠ɼϘϧϯ‫ز‬Կֶ͕ߟҊ͞Ε͍ͯ
Δ [26, 27]ɽϘϧϯ‫ز‬ԿֶͰ͸ɼᎇ཰΍‫ۭ͍ؒͨͬͱ཰ۂ‬ͷੑ࣭Λ‫ه‬ड़͢Δ‫ز‬ԿֶྔΛఆٛ͢ΔࢼΈ͕ߦΘΕ͓ͯΓɼଞͷํ๏ͱൺ΂ͯ։ൃ͕ҰาਐΜͰ͍ΔɽϘϧϯ‫ز‬ԿֶͰఆٛͨ͠ྔ͕ DFTʹͲͷΑ͏ʹ‫ݱ‬ΕΔ͔νΣοΫͰ͖Ε͹ɼഒԽۭؒͷ਺ֶతͳ‫ه‬ड़ͱͯ͠Ϙϧϯ‫ز‬Կֶͷ৴ጪੑ*1 ѥ‫܈‬͸ɼ୺తʹ͍͏ͱ୯Ґ‫ͨ͋ʹݩ‬Δ‫ݩ‬Λෳ਺΋ͭΑ͏ͳɼ‫܈‬Λ֦ுͨ͠ߏ଄Ͱ͋Δɽৄࡉ͸ɼ[73] ͳͲʹৄ͍͠ɽѥ‫( ܈‬groupoid) ʹؔͯ͠͸ɼจதͰ͸͜ΕҎ্ৄࡉʹ͸‫͍ͳ͠ٴݴ‬ɽ*2 coquecigrue ͸ɼ16 ੈ‫ل‬ͷϑϥϯεจֶͰ͋ΔʮΨϧΨϯνϡΞ (Gargantua) ෺‫ޠ‬ʯ,ʮύϯλάϦϡΤϧ

(Pantagruel)ʯʹ‫ݱ‬ΕΔۭ૝্ͷੜ෺Λࢦ͢ɽ

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T ૒ରੑͷ֦ுͰ͋Δ Poisson-Lie T-duality [81, 82] ͱͷؔ܎΋ٞ࿦͞Ε͍ͯΔɽtype II ௒ॏྗཧ࿦Λϕʔεʹ‫܈‬ଟ༷ମ্Ͱ DFT Λߏங͢ΔͱɼPoisson-Lie T-duality ͕ཧ࿦ͷ໌നͳର
শੑͱͳΔɽຊम࢜࿦จͰड़΂ͨ double ʹΑΔ Vaisman ѥ୅਺ͷߏங͸ఈۭؒͷछྨΛ໰Θͣߦ͑ΔͷͰɼ‫܈‬ଟ༷ମ্Ͱ࠶‫ͨ͠ݱ‬৔߹ʹͲͷΑ͏ͳ݁Ռ͕ಘΒΕΔ͔͔֬Ί͍ͨɽগͳ͘ͱ
΋ɼ[35] ͷ Drinfel’d double ʹΑΔ Courant ѥ୅਺͕‫ݭ‬ཧ࿦ʹ͓͍ͯ Poisson-Lie T-duality Λཧղ͢ΔͨΊͷ‫ͳͱݤ‬Δ [83–86]ɽ
࠷‫ʹޙ‬ɼDFT ࣗମͷૉ๿ͳ֦ுͱͯ͠ɼରশੑΛ T ૒ରੑ͔Β U ૒ରੑʹ্֨͛͢Δ͜ͱ͕ߟ͑ΒΕΔɽDFT ΛϕʔεʹɼU ૒ରੑΛ໌നͳରশੑͱ͢Δཧ࿦ͱͯ͠ Exceptional Field

Theory (EFT) ͕ߟҊ͞Εͨ [87]ɽEFT ͷήʔδରশੑ͸ [88] ͳͲͰٞ࿦͞Ε͍ͯΔ͕ɼDFTҎ্ʹ‫ز‬Կֶతͳඳ૾͕ෆ໌ྎͰ͋Δɽૉ๿ʹߟ͑Ε͹ɼEFT ͸ DFT ͷ֦ுͰ͋Δ͔ΒɼEFTͷ‫ز‬Կֶతඳ૾͸ ഒԽ‫ز‬ԿֶΛ಺แ͍ͯ͠Δ͸ͣͰ͋ΔɽԾʹഒԽ‫ز‬Կֶࣗମ͕਺ֶతʹ‫ه‬ड़Ͱ͖ͨͱͯ͠΋ɼ͞ΒʹͦΕ͕֦ு͞ΕΔՄೳੑ͸े෼࢒͍ͬͯΔɽࠓ‫ޙ‬ٞ࿦͍ͨ͠‫ڵ‬ຯਂ͍໰୊
͸ࢁੵΈͰ͋Δɽँֶࣙ෦͔ΒҾ͖ଓ͖ஸೡʹࢦಋͯͩͬͨ͘͠͞ߨࢣͷࠤʑ໦৳ઌੜʹਂ͘‫ँײ‬ਃ্͛͠·͢ɽࢲ͕ࠓͷ‫ڀݚ‬ςʔϚͱग़ձͬͨͷ΋ɼ૎Δ͜ͱ 3 ೥લʹʮҰൠԽ‫ز‬Կֶͱ͍͏໘നͦ͏ͳ୊ࡐ͕͋
ΔͷͰɼ΋͠ྑ͔ͬͨΒଔ‫Ͱڀݚۀ‬΍ͬͯΈͳ͍͔ʯͱ঺հΛड͚ͨ͜ͱ͕͖͔͚ͬͰͨ͠ɽ͋
ͷ΍ΓऔΓ͕ͳ͚Ε͹ࠓͷࢲ͸ଘࡏ͠·ͤΜɽ‫ڞ‬ಉ‫͋Ͱऀڀݚ‬Γɼଚ‫͖΂͢ܟ‬ઌഐͰ΋͋Δത࢜
ೋ೥ͷԘ୔݈ଠ͞Μʹ΋‫ँײ‬ਃ্͛͠·͢ɽӃʹਐֶͨ͠౰࣌͸ɼ·͔͞म࢜ऴྃ·Ͱʹ͓ೋਓͱ‫ڞ‬ಉ‫͖ͯͰ͕ڀݚ‬࿦จ͕ग़Δͱ͸ເʹ΋ࢥ͍ͬͯͳ͔ͬͨͰ͢ɽ͜Ε·ͰͷਓੜͰҰ൪‫͍͠خ‬ग़དྷࣄͰͨ͠ɽվΊͯਂ͘‫ྱޚ‬ਃ্͛͠·͢ɽՃ͑ͯɼ೔ࠒ͓ੈ࿩ʹͳ͍ͬͯΔ๺ཬେֶ͓Αͼଞେֶͷॾઌഐํɼಉ‫ڃ‬ੜɼ‫ޙ‬ഐͨͪʹ΋‫͢·͠ँײ‬ɽ࠷‫ʹޙ‬ɼओࠪΛ຿Ί͍͖ͯͨͩ·ͨ͠େ࠺ঁࢠେֶͷං઒ོ෉‫ڭ‬तʹ‫ँײ‬ਃ্͛͠·͢ɽ͓͔͛͞·Ͱɼਐֶͨ͠౰ॳ͸૝૾΋͠ͳ͔ͬͨΑ͏ͳɼඇৗʹೱີͳ म࢜ 2 ೥ؒΛա͢͜͝ͱ͕Ͱ͖·ͨ͠ɽͱͯ΋ָ͘͠ॆ࣮ͨ࣌ؒ͠Ͱͨ͠ɽΘͨ͠ʹͱͬͯҰੜͷๅͰ͢ɽ͜ͷΑ͏ʹॻ͘ͱɼ·ΔͰେֶӃੜ‫͕׆‬ऴΘΔ͔ͷΑ͏ͳࡨ֮Λ֮͑·͕͢ɼ΋͏Կ೥͔ಉ͡৔ॴͰ͓ੈ࿩
ʹͳΓ·͢ɽ‫͔ޙ‬Βࣗ෼Ͱɼ͜Ε͕ྑ͍൑அͩͬͨͱࢥ͑ΔΑ͏ʹ‫ؤ‬ுΓ·͢ɽվΊͯΑΖ͓͘͠‫͢·͍͍ͨ͠ئ‬ɽ

Soryushiron Kenkyu

46

෇࿥ A

‫ࢉܭ‬ϊʔτ
A.1 C ‫ހׅ‬ͷ Jacobiator ͷಋग़C ‫ހׅ‬ͷ Jacobiator (2.23) ͷಋग़Λߦ͏ɽC ‫ ͱހׅ‬D ‫͕ހׅ‬ҎԼͷΑ͏ʹؔ܎͢Δ͜ͱΛ༻
͍Δɽ

1 MN
M
L
[Ξ1 , Ξ2 ]M∂N (ηKL ΞK
D = [Ξ1 , Ξ2 ]C + η
1 Ξ2 )
2(A.1)

·ͨɼC ‫ހׅ‬͸ D ‫ʹހׅ‬ΑͬͯҎԼͷΑ͏ʹఆٛͰ͖Δɽ[Ξ1 , Ξ2 ]M
C =1
M([Ξ1 , Ξ2 ]M
D − [Ξ2 , Ξ1 ]D )
2

(A.2)

Ҏ্ͷࣄ࣮͔Βɼ·ͣ͸ D ‫͠ࢉܭͯؔ͠ʹހׅ‬ɼͦͷ݁ՌΛ C ‫ʹހׅ‬Ԡ༻͢ΔɽD ‫ހׅ‬͸ҎԼͷΑ͏ʹॻ͚ΔɽMM[Ξ1 , Ξ2 ]M+ ΞK
D = [Ξ1 , Ξ2 ]2 ∂ Ξ1K(A.3)

ͨͩ͠ɼ[·, ·] ͸ͨͩͷަ‫͋Ͱੵࢠ׵‬ΔɽҰൠʹɼLeibniz ཯͕ҎԼͷΑ͏ʹ༩͑ΒΕΔ͜ͱʹ‫ͮج‬

͍ͯɼD ‫ހׅ‬ͷ৔߹Λ‫͢ࢉܭ‬Δɽ[Ξ1 , [Ξ2 , Ξ3 ]] = [[Ξ1 , Ξ2 ], Ξ3 ] + [Ξ2 , [Ξ1 , Ξ3 ]](A.4)(A.3) ͷࠨล͸ɼҎԼͷΑ͏ʹ‫͖Ͱࢉܭ‬Δɽ[Ξ1 , [Ξ2 , Ξ3 ]D ]M
D
M L= [Ξ1 , [Ξ2 , Ξ3 ]D ]M + ηKL [Ξ2 , Ξ3 ]KD ∂ Ξ1
K N L
M
K M L
= [Ξ1 , [Ξ2 , Ξ3 ]]M + (ΞN
1 ∂N (ηKL Ξ3 ∂ Ξ2 ) − (ηKL Ξ3 ∂ Ξ2 )∂N Ξ1 )
I K J
M L
+ ηKL [Ξ2 , Ξ3 ]K ∂ M ΞL
1 + ηKL (ηIJ Ξ3 ∂ Ξ2 )∂ Ξ1 .

(A.5)

ҰํͰɼ(A.3) ͷҰ߲໨͸ҎԼͷΑ͏ʹ‫͖Ͱࢉܭ‬Δɽ

[[Ξ1 , Ξ2 ]D , Ξ3 ]MDML= [[Ξ1 , Ξ2 ]D , Ξ3 ]M + ηKL ΞK
3 ∂ [Ξ1 , Ξ2 ]D
N L
M
N
K M L
= [[Ξ1 , Ξ2 ], Ξ3 ]M + ((ηKL ΞK
2 ∂ Ξ1 )∂N Ξ3 − Ξ3 ∂N (ηKL Ξ2 ∂ Ξ1 ))
M
L
K M
I L J
+ ηKL ΞK
3 ∂ [Ξ1 , Ξ2 ] + ηKL Ξ3 ∂ (ηIJ Ξ2 ∂ Ξ1 ).

(A.6)

Soryushiron Kenkyu

47͕ͨͬͯ͠ɼ(A.3) ͷӈลશମ͸࣍ͷΑ͏ʹॻ͚Δɽ
M[[Ξ1 , Ξ2 ]D , Ξ3 ]M
D + [Ξ2 , [Ξ1 , Ξ3 ]D ]D
I K J
M L
N
K M L
= [Ξ1 , [Ξ2 , Ξ3 ]]M + ηKL ([Ξ2 , Ξ3 ]K ∂ M ΞL
1 + (ηIJ Ξ3 ∂ Ξ2 )∂ Ξ1 + Ξ1 ∂N (Ξ3 ∂ Ξ2 ))
N L
M
K N L
M
+ ηKL (ΞK2 ∂ Ξ1 ∂N Ξ3 − Ξ3 ∂ Ξ1 ∂N Ξ2 ).

(A.7)(A.5) Λ΋͏Ұ౓༻͍ͯ੔ཧ͢ΔͱɼD ‫ͯؔ͠ʹހׅ‬ҎԼͷؔ܎͕ಘΒΕΔɽ
MMM[Ξ1 , [Ξ2 , Ξ3 ]D ]M
D = [[Ξ1 , Ξ2 ]D , Ξ3 ]D + [Ξ2 , [Ξ1 , Ξ3 ]D ]D + SCD (Ξ1 , Ξ2 , Ξ3 ) ,

(A.8)

N L
M
K N L
M
K N L
M
SCD (Ξ1 , Ξ2 , Ξ3 )M = ηKL (ΞK
3 ∂ Ξ1 ∂N Ξ2 − Ξ2 ∂ Ξ1 ∂N Ξ3 − Ξ3 ∂ Ξ2 ∂N Ξ1 ).

(A.9)

SCD ͸ɼ‫߆͍ڧ‬ଋ৚݅Λ՝͢ͱফ͑Δ߲Ͱ͋Δɽ͜ͷ݁ՌΛ༻͍ͯɼC ‫ހׅ‬ͷ Jacobiator ͷ‫ʹࢉܭ‬ҠΔɽ(A.2) ͔ΒɼҎԼͷؔ܎͕ಘΒΕΔɽ

1([[Ξ1 , Ξ2 ]C , Ξ3 ]D − [Ξ3 , [Ξ1 , Ξ2 ]C ]D )2
1
= ([[Ξ1 , Ξ2 ]D , Ξ3 ]D − [[Ξ2 , Ξ1 ]D , Ξ3 ]D − [Ξ3 , [Ξ1 , Ξ2 ]D ]D + [Ξ3 , [Ξ2 , Ξ1 ]D ]D ).
4(A.10)[[Ξ1 , Ξ2 ]C , Ξ3 ]C =(A.8) ͱ (A.10) ͔Βɼ[[Ξ1 , Ξ2 ]C , Ξ3 ]C =1([Ξ1 , [Ξ2 , Ξ3 ]D ]D − [Ξ2 , [Ξ1 , Ξ3 ]D ]D − SCD (Ξ1 , Ξ2 , Ξ3 )4
− [Ξ2 , [Ξ1 , Ξ3 ]D ]D + [Ξ1 , [Ξ2 , Ξ3 ]D ]D + SCD (Ξ2 , Ξ1 , Ξ3 )− [Ξ3 , [Ξ1 , Ξ2 ]D ]D + [Ξ3 , [Ξ2 , Ξ1 ]D ]D ).

(A.11)(A.11) ͷ cyclic ෦෼Λ·ͱΊͳ͓͢ͱɼҎԼͷΑ͏ʹͳΔɽ[[Ξ1 , Ξ2 ]C , Ξ3 ]C + c.p.
1
= ([Ξ1 , [Ξ2 , Ξ3 ]D ]D − [Ξ2 , [Ξ1 , Ξ3 ]D ]D − SCD (Ξ1 , Ξ2 , Ξ3 ) + SCD (Ξ2 , Ξ1 , Ξ3 ) + c.p.). (A.12)
4
ӈลΛɼD ‫ހׅ‬ͷ Leibnitz ཯ (A.8) ͷ݁ՌΛ༻͍ͯஔ͖‫͑׵‬Δͱɼ࣍ͷΑ͏ʹͳΔɽ[[Ξ1 , Ξ2 ]C , Ξ3 ]C + c.p. =1([[Ξ1 , Ξ2 ]D , Ξ3 ]D − SCD (Ξ2 , Ξ1 , Ξ3 ) + c.p.).
4(A.13)͜͜ͰɼC ‫ ͱހׅ‬D ‫ހׅ‬ͷؔ܎ (A.1) ͔Βɼ[[Ξ1 , Ξ2 ]C , Ξ3 ]C = [[Ξ1 , Ξ2 ]C , Ξ3 ]D − ∂ • ([Ξ1 , Ξ2 ]C , Ξ3 )+= [[Ξ1 , Ξ2 ]D , Ξ3 ]D − [∂ • (Ξ1 , Ξ2 )+ , Ξ3 ]D − ∂ • ([Ξ1 , Ξ2 ]C , Ξ3 )+ .

(A.14)

N͜͜Ͱɼ∂ • ͸ɼη M N Ͱ଍্͕͕ͬͨඍ෼ԋࢉͰ͋Δɽ·ͨɼ(Ξ1 , Ξ2 )+ = (ηM N ΞM
1 Ξ2 )/2 Λҙຯ͢Δɽ(A.14) ͷӈลೋ߲໨͸ҎԼͷΑ͏ʹ‫͖Ͱࢉܭ‬Δɽ

"1! N K
M
K
M
KN∂ (Ξ1 Ξ2,K )∂N ΞM
+
(∂
∂
(Ξ
Ξ
)
−
∂
∂
(Ξ
Ξ
))Ξ
N
N
3
1 2,K
1 2,K
3
2
1
M
(A.15)= ∂ N (ΞK
1 Ξ2,K )∂N Ξ3 .
2[∂ • (Ξ1 , Ξ2 )+ , Ξ3 ]M
D =Soryushiron Kenkyu48
͜͜Ͱɼ(A.14) Λ࢖͏ͱɼC ‫ހׅ‬͸࣍ͷΑ͏ʹ‫͖Ͱࢉܭ‬Δɽ

ͨͩ͠ɼ

(
1'
([Ξ1 , Ξ2 ]C , Ξ3 )+ + c.p. ,3(
1' •SCC (Ξ1 , Ξ2 , Ξ3 ) = [∂ (Ξ1 , Ξ2 )+ , Ξ3 ]D − SCD (Ξ2 , Ξ1 , Ξ3 ) + c.p. .
3͕ͨͬͯ͠ɼ

(1'([Ξ1 , Ξ2 ]C , Ξ3 )+ + c.p. ,3(
1' •SCC (Ξ1 , Ξ2 , Ξ3 ) = [∂ (Ξ1 , Ξ2 )+ , Ξ3 ]D − SCD (Ξ2 , Ξ1 , Ξ3 ) + c.p. .
3NC (Ξ1 , Ξ2 , Ξ3 ) =NC (Ξ1 , Ξ2 , Ξ3 ) =JC (Ξ1 , Ξ2 , Ξ3 ) = ∂ • NC (Ξ1 , Ξ2 , Ξ3 ) + SCC (Ξ1 , Ξ2 , Ξ3 ).

(A.16)(A.17)(A.18)(A.19)(A.20)

͜ΕͰɼC ‫ހׅ‬ͷ Jacobiator ͕‫͖ͨͰࢉܭ‬ɽͨͩ͠ɼNC ͸ Nijenhuis ςϯιϧͰ͋ΓɼSCC ͸‫߆͍ڧ‬ଋ৚݅Ͱফ͑Δ߲Ͱ͋Δɽ

A.2 C ‫͔ހׅ‬Β Courant ‫΁ހׅ‬ͷ reduction
C ‫ހׅ‬Λ੒෼Ͱද‫͠ه‬ɼ͜Ε͸‫߆͍ڧ‬ଋ৚݅ͷ΋ͱͰ [17] Ͱఏࣔ͞Εͨ Courant ‫ހׅ‬1
[X1 + ξ1 , X2 + ξ2 ]c = [X1 , X2 ] + (LX1 ξ2 − LX2 ξ1 ) + d0 (ξ1 (X2 ) − ξ2 (X1 ))2(A.21)΁มԽ͢Δ͜ͱΛࣔ͢ɽͨͩ͠ Xi ͸ xµ ଆɼξi ͸ x̃µ ଆͷ੒෼Ͱ͋Δɽ·ͣɼgauge ύϥϝʔλΞ1 , Ξ2 ΛҎԼͷΑ͏ʹ੒෼Ͱ෼ղ͢Δɽ
# $
αµ
M
,
Ξ1 =
Aµ

ΞM
2 =

$
βµ
.
Bµ

#

(A.22)

͜ͷ੒෼දࣔʹΑͬͯɼC ‫ހׅ‬͸ҎԼͷΑ͏ʹॻ͖‫͑׵‬ΒΕΔɽ

1
K
M
K
M
K M L
K M L
[Ξ1 , Ξ2 ]M
C = Ξ1 ∂K Ξ2 − Ξ2 ∂K Ξ1 − ηKL (Ξ1 ∂ Ξ2 − Ξ2 ∂ Ξ1 )
2
ν
M
ν
M
˜ν M
= αν ∂˜ν ΞM
2 + A ∂ ν Ξ2 − βν ∂ Ξ1 − B ∂ ν Ξ11
− (αν ∂ M B ν + Aν ∂ M βν − βν ∂ M Aν − B ν ∂ M αν ).
2(A.23)

͜͜Ͱɼdoubled ‫ج‬ఈͱͯ͠ ∂ M = (∂µ , ∂˜µ ) Λಋೖ͢ΔͱɼC ‫ހׅ‬͸͞ΒʹҎԼͷΑ͏ʹॻ͚Δɽ
N[Ξ1 , Ξ2 ]C = [Ξ1 , Ξ2 ]M= αν ∂˜ν βµ ∂˜µ + Aν ∂ν βµ ∂˜µ − βν ∂˜ν αµ ∂˜µ − B ν ∂ν αµ ∂˜µC ηM N ∂+ αν ∂˜ν B µ ∂µ + Aν ∂ν B µ ∂µ − βν ∂˜ν Aµ ∂µ − B ν ∂ν Aµ ∂µ1− (αν ∂˜µ B ν + Aν ∂˜µ βν − βν ∂˜µ Aν − B ν ∂˜µ αν )∂µ2
1− (αν ∂µ B ν + Aν ∂µ βν − βν ∂µ Aν − B ν ∂µ αν )∂˜µ .
2(A.24)

Soryushiron Kenkyu

49͜͜Ͱɼ֤෦෼͸ҎԼͷΑ͏ʹ Lie ‫ͱ·ͰͲͳހׅ‬Ί௚ͤΔɽ

[A, B]L = [A, B]µL ∂µ = (Aν ∂ν B µ − B ν ∂ν Aµ )∂µ ,( µ'∂˜ = (αν ∂˜ν βµ − βν ∂˜ν αµ )∂˜µ ,[α, β] = [α, β]
L̃

L̃ µ

dιA β = d(A βν ) = ∂µ (Aν βν )∂˜µ = (βν ∂µ Aν + Aν ∂µ βν )∂˜µ ,d̃ιA β = d̃(Aν βν ) = ∂˜µ (Aν βν )∂µ = (βν ∂˜µ Aν + Aν ∂˜µ βν )∂µ ,ν

L̃α B = (αν ∂˜ν B µ + B ν ∂˜µ αν )∂µ ,LA β = (Aν ∂ν βµ + βν ∂µ Aν )∂˜µ .

(A.25)

͜͜ͰɼL ͸௨ৗͷϕΫτϧ৔ʹؔ͢Δ Lie ඍ෼Ͱ͋ΔɽҰํɼL̃ ͸ winding ࠲ඪ্ͷϕΫτϧ

৔ʹؔ͢Δ Lie ඍ෼Ͱ͋Δɽಉ༷ʹɼ[·, ·]L ͸௨ৗͷ Lie ‫͕ͩހׅ‬ɼ[·, ·]L̃ ͸ winding ࠲ඪ্ͷϕΫτϧ৔ʹؔ͢Δ Lie ‫͋Ͱހׅ‬Δɽ͕ͨͬͯ͠ɼC ‫ހׅ‬͸ҎԼͷΑ͏ʹ ∂µ ଆͱ ∂˜µ ଆʹ෼ׂͯ͠·ͱΊΒΕΔɽ
µ
ν
ν
˜µ
˜µ
˜µ
˜ν µ
˜ν µ
[Ξ1 , Ξ2 ]M
C ∂M = ([α, β]L̃ )µ ∂ + A ∂ν βµ ∂ − B ∂ν αµ ∂ + [A, B]L ∂µ + αν ∂ B ∂µ − βν ∂ A ∂µ1
− (2Aν ∂˜µ βν − (d̃ιA β)µ + (d̃ιB α)µ − 2B ν ∂˜µ αν )∂µ21− ((dιA β)µ − 2βν ∂µ Aν − (dιB α)µ + 2αν ∂µ B ν )∂˜µ
#2
$
1
µ
µ
µ
µ
= [A, B]L + L̃α B − L̃β A + (d̃(ιA β − ιB α)) ∂µ2#$1+ ([α, β]L̃ )µ + LA βµ − LB αµ − (d(ιA β − ιB α))µ ∂˜µ .
(A.26)
2͜ͷ‫ܗ‬͸ɼCourant ‫ހׅ‬Λ;ͨͭ଍͠߹ΘͤͨΑ͏ʹ‫͑ݟ‬Δ [9]ɽ1
[Ξ1 , Ξ2 ]C = [A + α, B + β]C = [A, B]L + LA β − LB α − d(ιA β − ιB α)
21+ [α, β]L̃ + L̃α B − L̃β A + d̃(ιA β − ιB α).
2(A.27)

˜ = 0) ͱɼӈลͷೋஈ໨͚͕ͩফ໓͠ɼC ‫ހׅ‬͸ Courant(A.27) ʹରͯ͠‫߆͍ڧ‬ଋ৚݅Λ՝͢ (∂∗͕ఏࣔͨ͠‫૆ݩ‬ͷ Courant ‫ހׅ‬ͷ‫[ ܗ‬17] ʹมԽ͢Δɽ

A.3 T (e1 , e2 , e3 ) ͷؔ܎ࣜ (3.36) ͷಋग़ҎԼͷࣜΛಋग़͢Δɽ1(([e1 , e2 ]V , e3 )+ + c.p.)31= ⟨ξ3 , [X1 , X2 ]E ⟩ + ⟨[ξ1 , ξ2 ]E ∗ , X3 + ρ(X3 )(e1 , e2 )− − ρ∗ (ξ3 )(e1 , e2 )− ⟩ (3.36)
2

T (e1 , e2 , e3 ) ≡͜Ε͸ɼจ‫[ ݙ‬35] ͷ Lemma 3.2 ʹ͋ͨΔؔ܎ࣜͰ͋ΓɼT (e1 , e2 , e3 ) ΛఆٛʹԊͬͯ‫͢ࢉܭ‬Δ͜
ͱͰࣔ͢͜ͱ͕Ͱ͖Δɽ‫ੵހׅ‬ͷ‫͕ܗ‬ಉ͡ͳͷͰɼຊจதͰఆٛͨ͠ Vaisman ‫·[ ހׅ‬, ·]V Ͱ‫ࢉܭ‬ͯ͠΋ɼจ‫[ ݙ‬35] ͷ Courant ‫·[ ހׅ‬, ·]c Ͱ‫ͯ͠ࢉܭ‬΋ಉؔ͡܎͕ࣜಘΒΕΔɽ

Soryushiron Kenkyu50
·ͣɼ([e1 , e2 ]V , e3 )+ ʹ஫໨͢Δͱɼ[·, ·]V , (·, ·)+ ͷఆٛΑΓ([e1 , e2 ]V , e3 )+ =1{⟨ξ3 , [X1 , X2 ]E ⟩ + ⟨ξ3 , Lξ1 X2 ⟩ − ⟨ξ3 , Lξ2 X1 ⟩ − ρ∗ (ξ3 )(e1 , e2 )−2⟨[ξ1 , ξ2 ]E ∗ , X3 ⟩ + ⟨LX1 ξ2 , X3 ⟩ − ⟨LX2 ξ1 , X3 ⟩ + ρ(X3 )(e1 , e2 )−(A.28)

͜͜ͰɼLie ඍ෼ͷ߲ʹண໨͢ΔͱɼLie ඍ෼ͷ෼഑ଇ͔ΒҎԼͷؔ܎͕ࣜ࢖͑Δɽ⟨ξ3 , Lξ1 X2 ⟩ = Lξ1 ⟨ξ3 , X2 ⟩ − ⟨[ξ1 , ξ3 ]E ∗ , X2 ⟩⟨LX1 ξ2 , X3 ⟩ = LX1 ⟨ξ2 , X3 ⟩ − ⟨ξ2 , [X1 , X3 ]E ⟩·ͨɼจ‫[ ݙ‬44] Ͱͷ d ͷఆ͔ٛΒɼ

LX1 ⟨ξ2 , X3 ⟩ = ιX1 d⟨ξ2 , X3 ⟩ = ρ(X1 )⟨ξ2 , X3 ⟩.

(A.29)

Lξ1 ⟨ξ3 , X2 ⟩ = ρ∗ (ξ1 )⟨ξ3 , X2 ⟩.

(A.30)

ಉ༷ʹɼΑͬͯɼ([e1 , e2 ]V , e3 )+ ͸ɼ(A.28) ͔ΒߋʹҎԼͷΑ͏ʹॻ͖‫͑׵‬ΒΕΔɽ

([e1 , e2 ]V , e3 )+ =1{⟨ξ3 , [X1 , X2 ]E ⟩ + ⟨[ξ1 , ξ2 ]E ∗ , X3 ⟩ + c.p.}2
1+ {ρ∗ (ξ1 )⟨ξ3 , X2 ⟩ − ρ∗ (ξ2 )⟨ξ3 , X1 ⟩ − ρ∗ (ξ3 )(e1 , e2 )−2+ ρ(X1 )⟨ξ2 , X3 ⟩ − ρ(X2 )⟨ξ1 , X3 ⟩ + ρ(X3 )(e1 , e2 )− }(A.31)

͜Εʹɼρ(X1 )(e2 , e3 )− , ρ(X2 )(e3 , e1 )− ͱ ρ∗ (ξ1 )(e2 , e3 ), ρ∗ (ξ2 )(e3 , e1 )− ΛͦΕͧΕ଍ͯ͠Ҿ͖ɼ८ճ͢Δ߲Λ੔ཧ͢Δͱɼρ ͷఆ͔ٛΒ ([e1 , e2 ]V , e3 )+ ͸࠷ऴతʹҎԼͷ‫·ͱ·ʹܗ‬Δɽ

([e1 , e2 ]V , e3 )+ =1{⟨ξ3 , [X1 , X2 ]E ⟩ + ⟨X3 , [ξ1 , ξ2 ]E ∗ ⟩ + ρ(X3 )(e1 , e2 )− − ρ∗ (ξ3 )(e1 , e2 )− c.p.}2
1
1(A.32)+ ρ(e1 )(e2 , e3 )+ − ρ(e2 )(e3 , e1 )+22͋ͱ͸ɼ([e1 , e2 ]V , e3 )+ ͷ଍Λ८ճͤͯ͞࿨ΛऔΕ͹ɼ(3.36) ͷӈลͱͳΔɽ1(([e1 , e2 ]V , e3 )+ + c.p.)31= {⟨[X1 , X2 ]E , ξ3 ⟩ + ⟨[ξ1 , ξ2 ]E ∗ , X3 ⟩ + ρ(X3 )(e1 , e2 )− − ρ∗ (ξ3 )(e1 , e2 )− c.p.}21+ {ρ(e1 )(e2 , e3 )+ − ρ(e2 )(e3 , e1 )+ + ρ(e2 )(e3 , e1 )+2− ρ(e3 )(e1 , e2 )+ + ρ(e3 )(e1 , e2 )+ − ρ(e1 )(e2 , e3 )+ }1
= {⟨ξ3 , [X1 , X2 ]E ⟩ + ⟨[ξ1 , ξ2 ]E ∗ , X3 ⟩2
(A.33)+ ρ(X3 )(e1 , e2 )− − ρ∗ (ξ3 )(e1 , e2 )− + c.p.}

T (e1 , e2 , e3 ) ≡Αͬͯɼ(3.36) ͕ࣔͤͨɽ

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A.4 T (e1 , e2 , e3 ) ͷؔ܎ࣜ (3.37) ͷಋग़ҎԼͷࣜΛࣔ͢ɽ

([e1 , e2 ]V , e3 )− + c.p. = T (e1 , e2 , e3 )+ [{ρ(X3 )(e1 , e2 )− + 2ρ∗ (ξ3 )(e1 , e2 )− − ⟨[ξ1 , ξ2 ]E ∗ , X3 ⟩} + c.p.](3.37)

͜Ε͸ [35] ͷ Lemma 3.4 ʹ͋ͨΔؔ܎ࣜͰ͋Γɼ(·, ·)± ͷఆ͔ٛΒࣔͤΔɽ

(·, ·)± ͷఆ͔ٛΒɼҎԼͷؔ܎͕ಘΒΕΔɽ

([e1 , e2 ]V , e3 )− + ([e1 , e2 ]V , e3 )+ = ⟨[ξ1 , ξ2 ]E ∗ , X3 ⟩ + ⟨LX1 ξ2 , X3 ⟩− ⟨LX2 ξ1 , X3 ⟩ + ⟨d(e1 , e2 )− , X3 ⟩.

(A.34)

͞Βʹɼ(A.29), (A.30) Λ࢖ͬͯӈลΛ੔ཧ͢Δͱɼ

([e1 , e2 ]V , e3 )− + ([e1 , e2 ]V , e3 )+ = ⟨[ξ1 , ξ2 ]E ∗ , X3 ⟩ + ρ(X1 )⟨ξ2 , X3 ⟩ − ⟨ξ2 , [X1 , X3 ]E ⟩− ρ(X2 )⟨ξ1 , X3 ⟩ + ⟨ξ1 , [X2 , X3 ]E ⟩ + ρ(X3 )(e1 , e2 )− .
(A.35)(A.35) ͷ଍Λ८ճͤͯ͞࿨ΛͱΔͱɼҎԼͷΑ͏ʹͳΔɽ్தɼ(3.36) Λ༻͍ͨɽ{([e1 , e2 ]V , e3 )− + ([e1 , e2 ]V , e3 )+ } + c.p.

= {([e1 , e2 ]V , e3 )− + c.p.} + 3T (e1 , e2 , e3 )= {⟨[ξ1 , ξ2 ]E ∗ , X3 ⟩ + ρ(X1 )⟨ξ2 , X3 ⟩ − ⟨ξ2 , [X1 , X3 ]E ⟩− ρ(X2 )⟨ξ1 , X3 ⟩ + ⟨ξ1 , [X2 , X3 ]E ⟩ + ρ(X3 )(e1 , e2 )− } + c.p.

(A.36)

Ұ୴ɼc.p. ͷ෦෼΋શͯ։͚ͯ‫͢ࢉܭ‬ΔͱɼҎԼͷΑ͏ʹ·ͱ·Δɽ

{([e1 , e2 ]V , e3 )− + c.p.} + 3T (e1 , e2 , e3 )= {⟨[ξ1 , ξ2 ]E ∗ , X3 ⟩ + 2⟨ξ3 , [X1 , X2 ]E ⟩ + 3ρ(X3 )(e1 , e2 )− } + c.p.

= 4T (e1 , e2 , e3 ) + [{ρ(X3 )(e1 , e2 )− + 2ρ∗ (ξ3 )(e1 , e2 )− − ⟨[ξ1 , ξ2 ]E ∗ , X3 ⟩} + c.p.](A.37)

࠷‫ޙ‬ͷߦ΁ͷม‫ܗ‬͸ɼ(3.37) ʹண஍͢ΔͨΊʹແཧ΍Γ 4T (e1 , e2 , e3 ) Λ͚ͭͬͨͩ͘Ͱ͋Δɽ

(A.37) ΛҠߦͯ͠੔ཧ͢Ε͹ɼ(3.37) ͱͳΔɽ

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A.5 Axiom C1 ͷ֬ೝ
Axiom C1 ͷࠨล [[e1 , e2 ]V , e3 ]V + c.p Λ‫͢ࢉܭ‬Δɽ[·, ·]V ͷఆ͔ٛΒɼ[[e1 , e2 ]V , e3 ]V + c.p. = I1 + I2 ,(A.38)

I1 = [[ξ1 , ξ2 ]E ∗ , ξ3 ]E ∗ + [LX1 ξ2 − LX2 ξ1 , ξ3 ]E ∗ + [d(e1 , e2 )− , ξ3 ]E ∗+ L[X1 ,X2 ]E +Lξ1 X2 −Lξ2 X1 −d∗ (e1 ,e2 )− ξ3− LX3 [ξ1 , ξ2 ]E ∗ − LX3 LX1 ξ2 + LX3 LX2 ξ1− LX3 d(e1 , e2 )− + d([e1 , e2 ]V , e3 )− + c.p.,(A.39)

I2 = [[X1 , X2 ]E , X3 ]E + [Lξ1 X2 − Lξ2 X1 , X3 ]E − [d∗ (e1 , e2 )− , X3 ]E+ L[ξ1 ,ξ2 ]E∗ +Lξ1 X2 −Lξ2 X1 +d(e1 ,e2 )− X3− Lξ3 [X1 , X2 ]E − Lξ3 Lξ1 X2 + Lξ3 Lξ2 X1+ Lξ3 d∗ (e1 , e2 )− − d∗ ([e1 , e2 ]V , e3 )− + c.p.

(A.40)

Γ(E ∗ ) ੒෼Λ I1 ɼΓ(E) ੒෼Λ I2 ͱ͓͘ɽI1 ͱ I2 ͸΄΅ಉ͡‫ͳͱࢉܭ‬ΔͷͰɼI1 ͚ͩऔΓग़ͯ͠‫͢ࢉܭ‬Δɽ

L[X1 ,X2 ]E = [LX1 , LX2 ]E(A.41)

L[X1 ,X2 ]E ξ3 − LX3 LX1 ξ2 + LX3 LX2 ξ1 + c.p. = 0(A.42)

Λ༻͍Δͱɼ

ͳͷͰɼI1 ͸ҎԼͷΑ͏ʹॻ͚Δɽ

I1 = {[[ξ1 , ξ2 ]E ∗ , ξ3 ]E ∗ + [LX1 ξ2 − LX2 ξ1 , ξ3 ]E ∗ − [d∗ (e1 , e2 )− , ξ3 ]E ∗+ L[X1 ,X2 ]E +Lξ1 X2 −Lξ2 X1 −d∗ (e1 ,e2 )− ξ3− LX3 [ξ1 , ξ2 ]E ∗ − LX3 LX1 ξ2 + LX3 LX2 ξ1 − Ld(e1 ,e2 )− + d([e1 , e2 ]V , e3 )− } + c.p.

= {[LX1 ξ2 − LX2 ξ1 , ξ3 ]E ∗ − [d∗ (e1 , e2 )− , ξ3 ]E ∗+ LLξ1 X2 −Lξ2 X1 −d∗ (e1 ,e2 )− ξ3− LX3 [ξ1 , ξ2 ]E ∗ − Ld(e1 ,e2 )− + d([e1 , e2 ]V , e3 )− } + c.p.

(A.43)

͜͜ͰɼLX3 [ξ1 , ξ2 ]E ∗ + c.p. ͷ߲ʹ஫໨͢Δɽ

LX3 [ξ1 , ξ2 ]E ∗ = (dιX3 + ιX3 d)[ξ1 , ξ2 ]E ∗= d⟨X3 , [ξ1 , ξ2 ]E ∗ ⟩ + iX3 d[ξ1 , ξ2 ]E ∗+ (ιX3 Lξ1 dξ2 − ιX3 Lξ1 dξ2 ) + (ιX3 Lξ2 dξ1 − ιX3 Lξ2 dξ1 )= d⟨X3 , [ξ1 , ξ2 ]E ∗ ⟩ + ιX3 Lξ1 dξ2 − ιX3 Lξ2 dξ1+ ιX3 (d[ξ1 , ξ2 ]E ∗ − Lξ1 dξ2 + Lξ2 dξ1 ).

(A.44)

͞ΒʹɼX ∈ Γ(E), ξ, η ∈ Γ(E ∗ ) ͱͨ͠ͱ͖ɼҰൠʹ LX3 [ξ1 , ξ2 ]E ∗ ʹରͯ͠

ιX Lξ dη = [ξ, LX η]E ∗ − LLξ X η + [d⟨η, X⟩, ξ]E ∗ + d(ρ∗ (ξ)⟨η, X⟩) − d⟨[ξ, η]E ∗ , X⟩

(A.45)

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͕੒ཱ͢Δɽ(A.45) ͷಋग़͸࣍ͷઅͰߦ͏ɽ͜ΕΛ iX3 Lξ1 dξ2 ͱ iX3 Lξ2 dξ1 ΁ద༻͢Δͱɼ

LX3 [ξ1 , ξ2 ]E ∗ + c.p. ͸ҎԼͷ‫ͳʹܗ‬Δɽ

LX3 [ξ1 , ξ2 ]E ∗ + c.p = {[LX1 ξ2 − LX2 ξ1 , ξ3 ]E ∗ + LLξ1 X2 −Lξ2 X1 ξ3+ 2[d(e1 , e2 )− , ξ3 ]E ∗ + 2d(ρ∗ (ξ3 )(e1 , e2 )− ) − d⟨[ξ1 , ξ2 ]E ∗ , X3 ⟩+ ιX3 (d[ξ1 , ξ2 ]E ∗ − Lξ1 dξ2 + Lξ2 dξ1 )} + c.p.

(A.46)(A.46) ͷಋग़͸࣍ͷ࣍ͷઅͰߦ͏ɽ
(A.46) Λ (A.43) ΁୅ೖ͢ΔͱɼI1 ͸͞Βʹ੔ཧͰ͖ͯɼ
I1 = {d{([e1 , e2 ]V , e3 )− − ρ(X3 )(e1 , e2 )− − 2ρ∗ (ξ3 )(e1 , e2 )− + ⟨[ξ1 , ξ2 ]E ∗ , X3 ⟩ − K1 − K2 } + c.p.,
K1 = ιX3 (d[ξ1 , ξ2 ]E ∗ − Lξ1 dξ2 + Lξ2 dξ1 ),K2 = Ld(e1 ,e2 )− ξ3 + [d(e1 , e2 )− , ξ3 ]E ∗ .

(A.47)

·ͨɼ(3.37) ͔Βɼ([e1 , e2 ]V , e3 )− + c.p. = T (e1 , e2 , e3 )+ [{ρ(X3 )(e1 , e2 )− + 2ρ∗ (ξ3 )(e1 , e2 )− − ⟨[ξ1 , ξ2 ]E ∗ , X3 ⟩ + c.p.}](3.37)Λ୅ೖ͢ΔͱɼI1 ͸ҎԼͷΑ͏ͳ‫ͳʹܗ‬Δɽ

I1 = dT (e1 , e2 , e3 ) − {K1 + K2 } + c.p.,(A.48)

ͨͩ͠ɼ

K1 = ιX3 (d[ξ1 , ξ2 ]E ∗ − Lξ1 dξ2 + Lξ2 dξ1 ),K2 = Ld∗ (e1 ,e2 )− ξ3 + [d(e1 , e2 )− , ξ3 ]E ∗ .

(A.49)

I2 ʹؔͯ͠΋ಉ༷Ͱ͋Δɽ
I2 = d∗ T (e1 , e2 , e3 ) − {K3 + K4 } + c.p.,

K3 = ιξ3 (d∗ [X1 , X2 ]E − LX1 d∗ X2 + LX2 d∗ X1 ),K4 = Ld∗ (e1 ,e2 )− X3 + [d∗ (e1 , e2 )− , X3 ]E .

(A.50)

Αͬͯɼ࠷ऴతʹ [[e1 , e2 ]V , e3 ]V + c.p ͷ‫݁ࢉܭ‬Ռ͸ҎԼͷΑ͏ʹಘΒΕΔɽ

[[e1 , e2 ]V , e3 ]V + c.p = I1 + I2
= DT (e1 , e2 , e3 ) − (J1 + J2 + c.p.).

(A.51)

ͨͩ͠ɼJ1 = K1 + K3
= ιX3 (d[ξ1 , ξ2 ]E ∗ − Lξ1 dξ2 + Lξ2 dξ1 ) + ιξ3 (d∗ [X1 , X2 ]E − LX1 d∗ X2 + LX2 d∗ X1 ),J2 = K2 + K4

= Ld(e1 ,e2 )− ξ3 + [d(e1 , e2 )− , ξ3 ]E ∗ + Ld∗ (e1 ,e2 )− X3 + [d∗ (e1 , e2 )− , X3 ]E .

(A.52)

Ұൠʹɼ೚ҙͷ Xi , ξi , f ʹର͠ (J1 + J2 + c.p.) ͸ 0 Ͱ͸ͳ͍ɽΑͬͯɼ(V, [·, ·]V , ρV , (·, ·)+ ) ʹ͓͍ͯɼAxiom C1 ͸ഁΕ͍ͯΔɽ

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A.6 (A.45) ࣜͷಋग़
ιX Lξ dη = [ξ, LX η]E ∗ − LLξ X η + [d⟨η, X⟩, ξ]E ∗ + d(ρ∗ (ξ)⟨η, X⟩) − d⟨[ξ, η]E ∗ , X⟩ ((A.45))
Λಋग़͢ΔɽࠨลΛ௚઀‫͢ࢉܭ‬Δͷ͸೉͍͠ͷͰɼ⟨ιX Lξ dη, Y ⟩ Λ‫͠ࢉܭ‬ɼҎԼͷΑ͏ʹӈลͱY Ͱ಺ੵΛͱͬͨ‫ͳʹܗ‬Δ͜ͱΛ͔֬ΊΔɽ⟨ιX Lξ dη, Y ⟩ = ⟨[ξ, LX η]E ∗ , Y ⟩ − ⟨LLξ X η, Y ⟩ − ⟨[ξ, d⟨η, X⟩]E ∗ , Y ⟩+ ⟨d(ρ∗ (ξ)⟨η, X⟩), Y ⟩ + ⟨d⟨[ξ, η]E ∗ , X⟩, Y ⟩(A.53)

·ͣɼLie ඍ෼ͷ෼഑ଇ͔Βɼ⟨ιX Lξ dη, Y ⟩ = Lξ (dη(X, Y )) − dη(Lξ X, Y ) − dη(X, Lξ Y ).

(A.54)

ӈลͷ 1 ߲໨͸ɼ(A.30) Λ࢖͏͜ͱͰ ρ∗ (ξ)dη(X, Y ) ʹஔ͖‫͑׵‬Δ͜ͱ͕Ͱ͖Δɽ·ͨɼ֎ඍ෼

࡞༻ૉͷఆ͔ٛΒɼdξ(X, Y ) = ρ(X)⟨ξ, Y ⟩ − ρ(Y )⟨ξ, X⟩ − ⟨ξ, [X, Y ]E ⟩.

(A.55)

͕੒ཱ͍ͯ͠ΔͷͰɼ͜ΕΛ֤߲ʹ౰ͯ͸ΊΔͱɼ(A.54) ͸ҎԼͷΑ͏ʹม‫͖Ͱܗ‬Δɽ

⟨ιX Lξ dη, Y ⟩ = ρ∗ (ξ)ρ(X)⟨ξ, Y ⟩ − ρ∗ (ξ)ρ(Y )⟨η, X⟩ − ρ∗ (ξ)⟨η, [X, Y ]E ⟩− ρ(Lξ X)⟨η, Y ⟩ + ρ(Y )⟨η, Lξ X⟩ + ⟨η, [Lξ X, Y ]E ⟩+ ρ(Lξ Y )⟨η, X⟩ − ρ(X)⟨η, Lξ Y ⟩ − ⟨η, [Lξ Y, X]E ⟩.

(A.56)

͞Βʹɼ(A.29),(A.30) ͱ (3.19) Λ࢖͏ͱɼ(A.56) ͷ֤ߦͷઌ಄ͷ߲͸ҎԼͷΑ͏ʹ෼ղͰ͖Δɽρ∗ (ξ)ρ(X)⟨ξ, Y ⟩ = ρ∗ (ξ)⟨LX η, Y ⟩ + ρ∗ (ξ)⟨η, [X, Y ]E ⟩ρ(Lξ X)⟨η, Y ⟩ = ⟨LLξ X η, Y ⟩ + ⟨η, [Lξ X, Y ]E ⟩ρ(Lξ Y )⟨η, X⟩ = ⟨LLξ Y η, X⟩ + ⟨η, [Lξ Y, X]E ⟩(A.57)(A.58)(A.59)

͜ΕΛ୅ೖͯ͠੔ཧ͢Δͱɼ͍͔ͭ͘ͷ߲͕ଧͪফ͋ͬͯ͠ҎԼͷΑ͏ʹ·ͱ·Δɽ

⟨ιX Lξ dη, Y ⟩ = ρ∗ (ξ)⟨LX η, Y ⟩ − ρ∗ (ξ)ρ(Y )⟨η, X⟩ − ⟨LLξ X η, Y ⟩+ ρ(Y )⟨η, Lξ X⟩ − ρ(X)⟨η, Lξ Y ⟩ + ⟨LLξ Y η, X⟩.

(A.60)

͜ͷ݁ՌΛɼanchor Λ࢖Θͣʹ Y ͱͷ಺ੵͷ‫Ͱܗ‬ද͢Α͏ʹॻ͖‫͑׵‬Δɽ·ͣ͸ɼρ(Y )⟨η, Lξ X⟩
ʹ஫໨͢Δɽ͜ͷ߲͸ɼ(A.29),(A.30) ͔Βρ(Y )⟨η, Lξ X⟩ = ⟨d(ρ∗ (ξ)⟨η, X⟩), Y ⟩ − ⟨d⟨[ξ, η]V , X⟩, Y ⟩ɽ

(A.61)

ͱදͤΔɽ͜ΕΛ୅ೖͯ͠⟨ιX Lξ dη, Y ⟩ = ⟨d(ρ∗ (ξ)⟨η, X⟩), Y ⟩ + ⟨d⟨[ξ, η]V , X⟩, Y ⟩ − ⟨LLξ X η, Y ⟩+ ρ∗ (ξ)⟨LX η, Y ⟩ − ρ∗ (ξ)ρ(Y )⟨η, X⟩ − ρ(X)⟨η, Lξ Y ⟩ + ⟨LLξ Y η, X⟩. (A.62)

Soryushiron Kenkyu55ಉ༷ʹɼ(A.29),(A.30) Λ࢖ͬͯ (A.62) ͷ 2 ஈ໨Λ੔ཧ͢ΔͱɼҎԼͷΑ͏ʹͳΔɽ

⟨ιX Lξ dη, Y ⟩ = ⟨d(ρ∗ (ξ)⟨η, X⟩), Y ⟩ + ⟨d⟨[ξ, η]V , X⟩, Y ⟩ − ⟨LLξ X η, Y ⟩+ ⟨[ξ, LX η]V , Y ⟩ − Lξ ⟨LY η, X⟩ − Lξ ⟨η, LY X⟩ − ⟨η, LX Lξ Y ⟩ + ⟨LLξ Y η, X⟩.
(A.63)͜͜Ͱɼ−⟨η, LX Lξ Y ⟩ + ⟨LLξ Y η, X⟩ = ⟨LLξ Y η, X⟩ + ⟨LLξ Y η, X⟩= ⟨d⟨η, X⟩, Lξ Y ⟩(A.64)

ͳͷͰɼ⟨ιX Lξ dη, Y ⟩ = ⟨d(ρ∗ (ξ)⟨η, X⟩), Y ⟩ + ⟨d⟨[ξ, η]V , X⟩, Y ⟩ − ⟨LLξ X η, Y ⟩+ ⟨[ξ, LX η]V , Y ⟩ − Lξ ⟨LY η, X⟩ − Lξ ⟨η, LY X⟩ + ⟨d⟨η, X⟩, Lξ Y ⟩.

(A.65)

Lie ඍ෼ͷ෼഑ଇ͔Βɼ⟨d⟨η, X⟩, Lξ Y ⟩ = Lξ ⟨d⟨η, X⟩, Y ⟩ − ⟨[ξ, d⟨η, X⟩]V , Y ⟩(A.66)

ͳͷͰɼ⟨ιX Lξ dη, Y ⟩ = ⟨d(ρ∗ (ξ)⟨η, X⟩), Y ⟩ + ⟨d⟨[ξ, η]E ∗ , X⟩, Y ⟩ − ⟨LLξ X η, Y ⟩+ ⟨[ξ, LX η]E ∗ , Y ⟩ − ⟨[ξ, d⟨η, X⟩]E ∗ , Y ⟩− Lξ ⟨LY η, X⟩ − Lξ ⟨η, LY X⟩ + Lξ ⟨d⟨η, X⟩, Y ⟩.

(A.67)

͜͜ͰɼҎԼͷ‫͔ࢉܭ‬Βࡾஈ໨͸ଧͪফ͋͠͏ɽ

Lξ ⟨d⟨η, X⟩, Y ⟩ = Lξ LY ⟨η, X⟩= Lξ ⟨LY η, X⟩ + Lξ ⟨η, LY X⟩.

(A.68)

Αͬͯɼ⟨ιX Lξ dη, Y ⟩ = ⟨d(ρ∗ (ξ)⟨η, X⟩), Y ⟩ + ⟨d⟨[ξ, η]E ∗ , X⟩, Y ⟩ − ⟨LLξ X η, Y ⟩+ ⟨[ξ, LX η]E ∗ , Y ⟩ − ⟨[ξ, d⟨η, X⟩]E ∗ , Y ⟩(A.69)

A.7 (A.46) ࣜͷಋग़લઅͷ‫͍ͨ༻Ͱࢉܭ‬ɼ

LX3 [ξ1 , ξ2 ]E ∗ + c.p. = [LX1 ξ2 − LX2 ξ1 , ξ3 ]V + LLξ1 X2 −Lξ2 X1 ξ3+ 2[d(e1 , e2 )− , ξ3 ]E ∗ + 2d(ρ∗ (ξ3 ) · (e1 , e2 )− ) − d⟨[ξ1 , ξ2 ]E ∗ , X3 ⟩+ iX3 (d[ξ1 , ξ2 ]E ∗ − Lξ1 dξ2 + Lξ2 dξ1 ) + c.p.

(A.46)

ͷಋग़Λߦ͏ɽ·ͣɼࠨลͷ LX3 [ξ1 , ξ2 ]E ∗ Λ‫Ͱͱ͋ͨ͠ࢉܭ‬ɼͦͷ८ճ࿨ΛͱͬͯӈลʹͳΔ͜

ͱΛ֬ೝ͢ΔɽLX3 [ξ1 , ξ2 ]E ∗ ͸ɼLie ඍ෼ͷఆٛʹ͕ͨͬͯ͠ҎԼͷΑ͏ʹల։Ͱ͖Δɽͨͩ͠ɼ

Soryushiron Kenkyu

56
2 ͭ໨ͷ౳߸ҎԼ͸ɼιX3 Lξ2 dξ1 ͷ߲ͱɼLξ2 dξ1 ͷ߲ΛͦΕͧΕΘ͟ͱ଍͔ͯ͠ΒҾ͍͍ͯΔɽ͜ͷૢ࡞ʹΑͬͯɼderivation ৚͕݅Ө‫͢ڹ‬Δ߲Λ໌ࣔ͢Δ͜ͱ͕Ͱ͖Δɽ

LX3 [ξ1 , ξ2 ]E ∗ = (dιX3 + ιX3 d)[ξ1 , ξ2 ]E ∗= d⟨[ξ1 , ξ2 ]E ∗ , X3 ⟩ + ιX3 Lξ1 dξ2 − ιX3 Lξ2 dξ1+ ιX3 (d[ξ1 , ξ2 ]E ∗ − Lξ1 dξ2 + Lξ2 dξ1 ).

(A.70)

Ҏ߱ͷ‫ࢉܭ‬͸ɼ1 ஈ໨ͷ ιX3 Lξ1 dξ2 − ιX3 Lξ2 dξ1 ͷ߲ʹ஫໨͢Δͱ؆୯ʹͳΔɽલઅͷ‫͔ࢉܭ‬ΒɼҰൠʹ

ιX Lξ dη = [ξ, LX η]E ∗ − LLξ X η + [d⟨η, X⟩, ξ]E ∗ + d(ρ∗ (ξ)⟨η, X⟩) − d⟨[ξ, η]E ∗ , X⟩.

(A.45)

ͱ͍͏ؔ܎͕੒ཱ͍ͯ͠Δɽ(A.45) Λ (A.70) ʹ୅ೖͯ͠ɼଧͪফ͋͠͏߲Λ੔ཧ͢ΔͱɼҎԼͷ‫͕ܗ‬ಘΒΕΔɽ

LX3 [ξ1 , ξ2 ]E ∗ = −d⟨[ξ1 , ξ2 ]E ∗ , X3 ⟩+ [ξ1 , LX3 ξ2 ]E ∗ − LLξ1 X3 ξ2 + [d⟨ξ2 , X3 ⟩, ξ1 ]E ∗ + d(ρ∗ (ξ1 )⟨ξ2 , X3 ⟩)− [ξ2 , LX3 ξ1 ]E ∗ + LLξ2 X3 ξ1 − [d⟨ξ1 , X3 ⟩, ξ2 ]E ∗ − d(ρ∗ (ξ2 )⟨ξ1 , X3 ⟩)+ ιX3 (d[ξ1 , ξ2 ]E ∗ − Lξ1 dξ2 + Lξ2 dξ1 ).

(A.71)

ଓ͍ͯɼ(A.71) ͷ८ճ࿨ LX3 [ξ1 , ξ2 ]V + c.p. Λ‫͢ࢉܭ‬Δɽ(A.71) ͷ 2 ஈ໨ͱ 3 ஈ໨ʹ͋ͨΔ߲͸ͦΕͧΕҎԼͷΑ͏ʹ·ͱ·Δɽ[ξ1 , LX3 ξ2 ]E ∗ − [ξ2 , LX3 ξ1 ]E ∗ + c.p. = [LX1 ξ2 − LX2 ξ1 , ξ3 ]E ∗ + c.p.,(A.72)[d⟨ξ2 , X3 ⟩, ξ1 ]E ∗ − [d⟨ξ1 , X3 ⟩, ξ2 ]E ∗ + c.p. = +2[d(e1 , e2 )− , ξ3 ]E ∗ + c.p.,(A.74)− LLξ1 X3 ξ2 + LLξ2 X3 ξ1 + c.p. = LLξ1 X2 −Lξ2 X1 ξ3 + c.p.,(A.73)d(ρ∗ (ξ1 )⟨ξ2 , X3 ⟩) − d(ρ∗ (ξ2 )⟨ξ1 , X3 ⟩) + c.p. = 2d(ρ∗ (ξ3 ) · (e1 , e2 )− ) + c.p.

(A.75)

͕ͨͬͯ͠ɼ

LX3 [ξ1 , ξ2 ]E ∗ + c.p. = −d⟨[ξ1 , ξ2 ]E ∗ , X3 + [LX1 ξ2 − LX2 ξ1 , ξ3 ]E ∗ + LLξ1 X2 −Lξ2 X1 ξ3+ 2[d(e1 , e2 )− , ξ3 ]E ∗ + 2d(ρ∗ (ξ3 ) · (e1 , e2 )− )+ ιX3 (d[ξ1 , ξ2 ]E ∗ − Lξ1 dξ2 + Lξ2 dξ1 ) + c.p.

(A.76)

ͱ͍͏݁Ռ͕ಘΒΕΔɽ͜Ε͸ɼ(A.46) ͷӈลͦͷ΋ͷͰ͋Δɽ

A.8 Axiom C2 ͷ֬ೝ௚઀‫͍ͳ͖Ͱࢉܭ‬ͷͰɼ(3.22) ͷ྆ลΛͦΕͧΕ೚ҙͷ f ∈ C ∞ (M ) ΁࡞༻ͤͯ͞ɼ

ρV ([e1 , e2 ]V )f = [ρV (e1 ), ρV (e2 )]f͕੒ཱ͍ͯ͠Δ͔Ͳ͏͔Λ֬ೝ͢Δɽ

(3.22’)

Soryushiron Kenkyu

57ࠨลΛల։͍ͯ͘͠ͱɼρ ͱ [·, ·]V ͷఆ͔ٛΒҎԼͷΑ͏ͳ‫ͳͱܗ‬ΔɽρV ([e1 , e2 ]V )f = ρV ([X1 + ξ1 , X2 + ξ2 ]V )f= ρ{[X1 , X2 ]E + Lξ1 X2 − Lξ2 X1 − d∗ (e1 , e2 )− }f+ ρ∗ {[ξ1 , ξ2 ]E ∗ + LX1 ξ2 − LX2 ξ1 + d(e1 , e2 )− }f= ρ([X1 , X2 ]E )f + ρ(Lξ1 X2 )f − ρ(Lξ2 X1 )f
1− ρρ∗∗ d0 (⟨ξ1 , X2 ⟩ − ⟨ξ2 , X1 ⟩)f
2+ ρ∗ ([ξ1 , ξ2 ]E ∗ )f + ρ∗ (LX1 ξ2 )f − ρ∗ (LX2 ξ1 )f
1+ ρ∗ ρ∗ d0 (⟨ξ1 , X2 ⟩ − ⟨ξ2 , X1 ⟩)f
2(A.77)

్தɼd∗ = ρ∗∗ d0 , d = ρ∗ d0 Ͱ͋Δ͜ͱΛ༻͍ͨɽ͜͜Ͱɼ[ρ(X1 ), ρ(X2 )] ͱ [ρ∗ (ξ1 ), ρ∗ (ξ2 )] ͸ɼρ, ρ∗ ͷఆ͔ٛΒ[ρ(X1 ), ρ(X2 )] = ρ([X1 , X2 ]E )[ρ∗ (ξ1 ), ρ∗ (ξ2 )] = ρ∗ ([ξ1 , ξ2 ]E ∗ )͕੒ཱ͢ΔͷͰɼ(A.77) ͸ҎԼͷΑ͏ʹॻ͖ͳ͓ͤΔɽͨͩ͠ɼ[·, ·] ͸ T M ্ͷ Lie ‫͋Ͱހׅ‬ΔɽρV ([e1 , e2 ]V )f = [ρ(X1 ), ρ(X2 )]f + ρ(Lξ1 X2 )f − ρ(Lξ2 X1 )f
1− ρρ∗∗ d0 (⟨ξ1 , X2 ⟩ − ⟨ξ2 , X1 ⟩)f
2+ [ρ∗ (ξ1 ), ρ∗ (ξ2 )]f + ρ∗ (LX1 ξ2 )f − ρ∗ (LX2 ξ1 )f
1+ ρ∗ ρ∗ d0 (⟨ξ1 , X2 ⟩ − ⟨ξ2 , X1 ⟩)f
2(A.78)·ͨɼ[ρ(X), ρ∗ (ξ)]f = −ρ(Lξ X)f + ρ∗ (LX ξ)f + (ρρ∗∗ d0 ⟨ξ, X⟩)f
− ⟨ξ, Ldf X⟩ + ρ(X)ρ∗ (ξ)f − ρ∗ (LX ξ)f(A.79)

Ͱ͋Δ͜ͱΛ༻͍Δͱɼ(A.78) ͸͞Βʹ੔ཧ͢Δ͜ͱ͕Ͱ͖ͯɼ

ρV ([e1 , e2 ]V )f = [ρ(X1 ), ρ(X2 )]f + {ρ(Lξ1 X2 ) − ρ∗ (LX2 ξ1 )}f
1− ρρ∗∗ d0 (⟨ξ1 , X2 ⟩ − ⟨ξ2 , X1 ⟩)f
2+ [ρ∗ (ξ1 ), ρ∗ (ξ2 )]f − {ρ(Lξ2 X1 ) − ρ∗ (LX1 ξ2 )}f
1+ ρ∗ ρ∗ d0 (⟨ξ1 , X2 ⟩ − ⟨ξ2 , X1 ⟩)f
2= [ρ(X1 ), ρ(X2 )]f + {ρ(Lξ1 X2 ) − ρ∗ (LX2 ξ1 ) − ρρ∗∗ d0 ⟨ξ1 , X2 ⟩}f
1+ ρρ∗∗ d0 (⟨ξ1 , X2 ⟩ − ⟨ξ2 , X1 ⟩)f
2+ [ρ∗ (ξ1 ), ρ∗ (ξ2 )]f − {ρ(Lξ2 X1 ) − ρ∗ (LX1 ξ2 ) − ρρ∗∗ d0 ⟨ξ2 , X1 ⟩}f
1(A.80)+ ρ∗ ρ∗ d0 (⟨ξ1 , X2 ⟩ − ⟨ξ2 , X1 ⟩)f
2͔͜͜Βɼ[ρ(X), ρ∗ (ξ)]f Λ·ͱΊΔɽෆ଍͍ͯ͠Δ߲͸଍ͯ͠Ҿ͖ɼா৲Λ߹ΘͤΔͱҎԼͷΑ

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58
͏ʹͳΔɽ

ρV ([e1 , e2 ]V )f = [ρ(X1 ), ρ(X2 )]f − [ρ(X2 ), ρ∗ (ξ1 )]f− ⟨ξ1 , Ldf X2 ⟩ + ρ(X2 )ρ∗ (ξ1 )f − ρ∗ (LX2 )ξ1 f
1+ ρρ∗∗ d0 (⟨ξ1 , X2 ⟩ − ⟨ξ2 , X1 ⟩)f
2+ [ρ∗ (ξ1 ), ρ∗ (ξ2 )]f + [ρ(X1 ), ρ∗ (ξ2 )]f+ ⟨ξ2 , Ldf X1 ⟩ − ρ(X1 )ρ∗ (ξ2 )f + ρ∗ (LX1 ξ2 )f
1+ ρ∗ ρ∗ d0 (⟨ξ1 , X2 ⟩ − ⟨ξ2 , X1 ⟩)f
2= [ρ(X1 ), ρ(X2 )]f + [ρ∗ (ξ1 ), ρ(X2 )]f + [ρ∗ (ξ1 ), ρ∗ (ξ2 )]f + [ρ(X1 ), ρ∗ (ξ2 )]f− ⟨ξ1 , Ldf X2 − [X2 , d∗ f ]E ⟩ + ⟨ξ2 , Ldf X1 − [X1 , d∗ f ]E ⟩(A.81)

͞Βʹɼρ ͷఆ͔ٛΒ (A.81) ্ஈͷ 4 ߲͸·ͱΊΔ͜ͱ͕Ͱ͖ͯɼ

ρV ([e1 , e2 ]V )f = [ρV (e1 ), ρV (e2 )] − ⟨ξ1 , Ldf X2 − [X2 , d∗ f ]E ⟩ + ⟨ξ2 , Ldf X1 − [X1 , d∗ f ]E ⟩11
+ ρρ∗∗ d0 (⟨ξ1 , X2 ⟩ − ⟨ξ2 , X1 ⟩)f + ρ∗ ρ∗ d0 (⟨ξ1 , X2 ⟩ − ⟨ξ2 , X1 ⟩)f (A.82)22Ұൠʹɼ೚ҙͷ Xi , ξ1 ʹର͠ӈล͸ 0 ʹͳΒͳ͍ɽΑͬͯɼ(V, [·, ·]V , ρV , (·, ·)+ ) ʹ͓͍ͯɼ

Axiom C2 ͸ഁΕ͍ͯΔɽ

A.9 Axiom C3 ͷ֬ೝ࣮ࡍʹɼࠨล [e1 , f e2 ]V Λల։͠ɼӈลʹߦ͖ண͘͜ͱΛࣔ͢ɽρ ͷఆ͔ٛΒɼ[e1 , f e2 ]V = [X1 + f ξ1 , X2 , +f ξ2 ]V= [X1 , f X2 ]V + [X1 , f ξ2 ]V + [ξ1 , f X2 ]V + [ξ1 , f ξ2 ]V .

(A.83)

͜͜Ͱɼ[·, ·]V ͷఆ͔ٛΒɼ

1
1[X1 , f ξ2 ]V = −Lf ξ2 X1 + d∗ (f ⟨ξ2 , X1 ⟩) + LX1 (f ξ2 ) − d(f ⟨ξ2 , X1 ⟩)2
2
1
1= −f Lξ2 X1 − (d∗ f )⟨ξ2 , X1 ⟩ + (d∗ f )⟨ξ2 , X1 ⟩ + f d∗ ⟨ξ2 , X1 ⟩2
2
1
1
+ f d⟨ξ2 , X1 ⟩ + ιX df ξ2 + f ιX dξ2 − (df )⟨ξ2 , X1 ⟩ − f d⟨ξ2 , X1 ⟩2
2
1
(A.84)= f [X1 , ξ2 ]V + (ρ(X1 )f )ξ2 − Df ⟨ξ2 , X1 ⟩.
2[ξ1 , f X2 ]V ΋ಉ༷ʹɼ[·, ·]V ͷఆ͔ٛΒɼ
1[ξ1 , f X2 ]V = f [ξ1 , X2 ]V + (ρ∗ (ξ1 )f )X2 − Df ⟨ξ1 , X2 ⟩.
2(A.85)·ͨɼL ͱ L∗ ͕ͦΕͧΕ Lie ѥ୅਺Ͱ͋Δ͜ͱ͔Βɼ

[X1 , f X2 ]V = [X1 , f X2 ]E = f [X1 , X2 ]V + (ρ(X1 )f )X2 ,(A.86)[ξ1 , f ξ2 ]V = [ξ1 , f ξ2 ]E ∗ = f [ξ1 , ξ2 ]V + (ρ∗ (ξ1 )f )ξ2 .

(A.87)

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59
Αͬͯɼ

[e1 , f e2 ]V = (A.83)= (A.86) + (A.84) + (A.85) + (A.87)= f [X1 , X2 ]V + (ρ(X1 )f )X2
1+ f [X1 , ξ2 ]V + (ρ(X1 )f )ξ2 − Dg⟨ξ2 , X1 ⟩2
1
+ f [ξ1 , X2 ]V + (ρ∗ (ξ1 )f )X2 − Df ⟨ξ1 , X2 ⟩2+ f [ξ1 , ξ2 ]V + (ρ∗ (ξ1 )f )ξ2= f [e1 , e2 ]V + (ρV (e1 )f )e2 − Df (e1 , e2 )− .

(A.88)

Αͬͯɼ(V, [·, ·]V , ρV , (·, ·)+ ) ʹ͓͍ͯɼAxiom C3 ͸੒ཱ͢Δɽ

A.10 Axiom C4 ͷ֬ೝD ͷఆٛΛ࢖͏ͱɼ(3.24) ͷࠨล͸ҎԼͷΑ͏ʹม‫͖Ͱܗ‬Δɽͨͩ͠ɼΓ(T ∗ M ) ্ͷ֎ඍ෼ԋࢉࢠΛ d0 ͱͨ͠ɽ(Df, Dg)+ = (df + d∗ f, dg + d∗ g)+1= (⟨df, d∗ g⟩ + ⟨dg, d∗ f ⟩)2
1
= (ρ∗ (df )g + ρ(d∗ f )g)2
1= (ρ∗ ρ∗ d0 f + ρρ∗∗ d0 f )g.
2(A.89)

͕ͨͬͯ͠ɼ(A.89) ͕ 0 ʹͳΓɼ(3.24) ͕੒ཱ͢Δʹ͸ ρ∗ ρ∗ = −ρρ∗∗ ͕੒Γཱ͍ͬͯΕ͹Α͍ɽॳΊʹɼderivation ৚݅Λ՝ͤ͹ඞͣ anchor ρ ͕൓ରশ ρρ∗∗ = −ρ∗ ρ∗ ʹͳΔ͜ͱΛࣔ͢ɽͨͩ͠ɼຊจதͰ΋ड़΂ͨ௨Γ anchor ʹ෇͍͍ͯΔ্෇͖ͷ ∗ ͸ɼadjoint operator ͷҙຯͰ͋Γɼ಺ੵΛ௨ͯ͠‫ݩ‬ͷԋࢉࢠͷసஔͰఆٛ͞ΕΔɽρ : E → TM

ρ∗ : E ∗ → T M

ρ∗ : T ∗ M → E ∗ρ∗∗ : T ∗ M → E(A.90)

͕ͨͬͯ͠ɼρρ∗∗ : T ∗ M → T M ɼρ∗ ρ∗ : T ∗ M → T M Ͱ͋ΔɽɽҰൠʹɼʮ͋Δԋࢉࢠ O ɿ T ∗ M → T M ͕൓ରশͰ͋Δʯͱ͖ɼҎԼͷ͕ࣜ੒ཱ͢Δɽશͯͷx ∈ Γ(T ∗ M ) ʹରͯ͠ɼ⟨Ox, x⟩ = 0.

(A.91)

ԋࢉࢠ O Λ ρρ∗∗ ɼx Λ d0 f ∈ Γ(T M ) ʹஔ͖‫͑׵‬Δ͜ͱͰҎԼͷؔ܎͕ಘΒΕΔɽͨͩ͠ɼ

f ∈ Γ(T M ), d0 ͸ T ∗ M ্ͷ֎ඍ෼࡞༻ૉͰ͋Δɽ⟨ρρ∗∗ (d0 f ), d0 f ⟩ = ρρ∗∗ (d0 f ) · f = 0.

(A.92)

Αͬͯɼ(A.92) ͕੒ཱ͢ΔͳΒ͹ ρ∗ ρ∗ = −ρρ∗∗ ͕੒ཱ͢ΔɽԾʹɼҎԼͷ͕ࣜ੒Γཱͭ΋ͷͱ͠Α͏ɽ

Ldf X + [d∗ f, X]E = 0(3.60)

Soryushiron Kenkyu

60͜ͷࣜ͸ɼจ‫[ ݙ‬44] ͷ Proposition 3.4 ͔Βɼdarivation ৚͕݅੒ཱ͍ͯ͠Ε͹෇ਵͯ͠੒ΓཱͭࣜͰ͋ΔɽX Λ d∗ f Ͱஔ͖‫͑׵‬ΔͱɼҎԼͷؔ܎͕੒ཱ͢Δɽ"!
d∗ ρρ∗∗ (d0 f ) · f = 0.

(A.93)!
"d∗ ρρ∗∗ (d0 f 2 ) · f 2 = 0.

(A.94)

ρρ∗∗ (d0 f 2 ) · f 2 = ⟨d0 f 2 , ρρ∗∗ d0 f 2 ⟩ = ⟨df 2 , d∗ f 2 ⟩ = 4f 2 ⟨df, d∗ f ⟩,(A.95)f Λ f 2 ʹஔ͖‫͑׵‬Ε͹ɼ

ҰํͰɼ

ͳͷͰɼ

!

.'
!
"
"
( /
ρρ∗∗ (d0 f ) · f d∗ f 2 = d∗ ρρ∗∗ (d0 f ) · f f 2 − d∗ ρρ∗∗ (d0 f ) · f f 2
"
!
"
1 !
= d∗ ρρ∗∗ (d0 f 2 ) · f 2 − d∗ ρρ∗∗ (d0 f ) · f f 2 .
4(A.96)

͜͜Ͱɼӈลʹ (A.93), (A.94) ͔Βফ͑Δ߲͕͋ΔͷͰɼ!

"ρρ∗∗ (d0 f ) · f ρ∗∗ d0 f 2 = 0.

(A.97)!
"2
2f ρρ∗∗ (d0 f ) · f = 0.

(A.98)

͞Βʹɼ͜Ε͸ d0 f ͱͷ಺ੵͩͱࢥ͏ͱ

ͱͳΔɽ͕ͨͬͯ͠ɼderivation ৚͕݅՝͞ΕΕ͹ 0 = ρρ∗∗ (d0 f ) · f = ⟨ρρ∗∗ (d0 f ) ͱͳΔɽρρ∗∗͕൓ରশͰ͋Δʹ͸ɼderivation ৚݅Λ՝͢ඞཁ͕͋ΔɽҎԼͷ‫ࢉܭ‬͸্‫ه‬ͷ݁Ռͷิ଍Ͱ͋Δɽderivation ৚݅Λ՝͞ͳ͍Ұൠͷ৔߹͸ɼρ∗ ρ∗ = −ρρ∗∗͕੒ཱ͠ͳ͍৔߹ΛഉআͰ͖ͳ͍͜ͱΛࣔ͢ɽ۩ମతʹ͸ɼ(V, [·, ·]V , ρV , (·, ·)+ ) ͸ɼؔ܎ࣜ(3.24) ຬͨ͞ͳ͍͜ͱΛ֬ೝ͢Δɽ۩ମతʹ͸ɼ(A.92) ͷࠨลΛผͷํ๏Ͱಋग़͠ɼҰൠʹ 0 ʹͳΒͳ͍͜ͱΛࣔ͢ɽ[·, ·]E ͷੑ࣭͔ΒɼX, Y ∈ Γ(E) ͱ͢Δͱd∗ [X, f Y ]E = d∗ (f [X, Y ]E + (ρ(X) · f )Y )= d∗ (f [X, Y ]E ) + d∗ ((ρ(X) · f )Y )= d∗ f ∧ [X, Y ]E + f [X, Y ]E + d∗ ρ(X) · f ∧ Y + ρ(X) · f d∗ Y

(A.99)

Ͱ͋Δ͜ͱ͔ΒɼҎԼͷؔ܎͕ࣜ੒ཱ͢Δɽ

[X, d∗ f ]E ∧ Y = Ldf X ∧ Y − d∗ [X, f Y ]E + f d∗ [X, Y ]E

+ d∗ f ∧ LX Y + LX d∗ f ∧ Y + (ρ(X) · f )d∗ Y + f LY d∗ X − f Lf Y d∗ X
(A.100)X Λ d∗ f ʹஔ͖‫ͯ͑׵‬੔ཧ͢ΔͱɼҎԼͷؔ܎͕ࣜಘΒΕΔɽ[d∗ f, d∗ f ]E ∧ Y = Ldf d∗ f ∧ Y − d∗ [d∗ f, f Y ]E + f d∗ [d∗ f, Y ]E
+ d∗ f ∧ Ld∗ f Y + (ρ(d∗ f ) · f )d∗ Y=0(A.101)

Soryushiron Kenkyu

61ΑͬͯɼҎԼͷ͕ࣜ੒ཱ͢Δɽ

Ldf d∗ f ∧ Y = d∗ [d∗ f, f Y ]E − f d∗ [d∗ f, Y ]E − d∗ f ∧ Ld∗ f Y − (ρ(d∗ f ) · f )d∗ Y(A.102)·ͨɼ(A.102) ͸ɼf Λ f 2 ʹஔ͖‫ͯ͑׵‬΋੒ཱ͢Δɽ

Ldf 2 d∗ f 2 ∧Y = d∗ [d∗ f 2 , f 2 Y ]E −f 2 d∗ [d∗ f 2 , Y ]E −d∗ f 2 ∧Ld∗ f 2 Y −(ρ(d∗ f 2 )·f 2 )d∗ Y (A.102′ )͞Βʹɼ(A.92) ͔Βల։͍ͯ͘͠ͱɼ

ρρ∗∗ (d0 f 2 ) · f 2 = ⟨ρρ∗∗ (d0 f 2 ), d0 f 2 ⟩= ⟨d∗ f 2 , df 2 ⟩= 4⟨f d∗ f, f df ⟩= 4f 2 ⟨d∗ f, df ⟩= 4(ρρ∗∗ (d0 f ) · f )f 2

(A.103)ͳͷͰɼ͜ΕΛ྆ล d∗ Ͱୟ͘ͱd∗ (ρρ∗∗ (d0 f 2 ) · f 2 ) = 4d∗ ((ρρ∗∗ (d0 f ) · f )f 2 )= 4f 2 d∗ (ρρ∗∗ (d0 f ) · f ) + 4(ρρ∗∗ (d0 f ) · f )d∗ f 2(A.104)

Ͱ͋Δ͜ͱ͔Βɼ(A.100) ࣜ͸࣍ͷΑ͏ʹॻ͖‫͑׵‬Δ͜ͱ͕Ͱ͖Δɽ

4f 2 d∗ (ρρ∗∗ (d0 f ) · f ) ∧ Y + 4(ρρ∗∗ (d0 f ) · f )d∗ f 2 ∧ Y'(= d∗ [d∗ f 2 , f 2 Y ]E + −(d∗ f 2 ) ∧ Ld∗ f 2 Y − f Ld∗ f 2 d∗ Y − ρ(d∗ f 2 )[f 2 ]d∗ Y .

(A.102”)ࠨลʹ (A.102) Λ୅ೖ͢ΔͱɼҎԼͷΑ͏ʹͳΔɽ

4f 2 d∗ [d∗ f, f Y ]E − 4f 2 (d∗ f ) ∧ Ld∗ f Y − 4f 2 Ld∗ f d∗ Y − 4f 2 (ρ(d∗ f ) · f )d∗ Y + 4(ρρ∗∗ (d0 f ) · f )d∗ f 2 ∧ Y= d∗ [d∗ f 2 , f 2 Y ]E − (d∗ f 2 ) ∧ Ld∗ f 2 Y − f Ld∗ f 2 d∗ Y − (ρ(d∗ f 2 ) · f 2 )d∗ Y.

͜ΕΛ ρρ∗∗ (d0 f ) · f d∗ f 2 ͷ͋Δ߲Ͱ·ͱΊΔͱɼ

4ρρ∗∗ (d0 f ) · f d∗ f 2 ∧ Y = −4f 2 d∗ [d∗ f, f Y ]E + 4f 2 (d∗ f ) ∧ Ld∗ f Y+ 4f 2 Ld∗ f d∗ Y + 4f 2 ρ(d∗ f ) · f d∗ Y

+ d∗ [d∗ f 2 , f 2 Y ]E − (d∗ f 2 ) ∧ Ld∗ f 2 Y− f Ld∗ f 2 d∗ Y − (ρ(d∗ f 2 ) · f 2 )d∗ Y.

(A.105)(A.105) ͷ྆ลʹ ρ Λ࡞༻ͤ͞Δͱɼࠨล͸
ρ((ρρ∗∗ (d0 f ) · f )(d∗ f 2 )) = (ρρ∗∗ (d0 f ) · f )ρ(d∗ f 2 )= (ρρ∗∗ (d0 f ) · f )ρρ∗∗ (d0 f 2 )= 2f ((ρρ∗∗ (d0 f ) · f )ρρ∗∗ d0 f )Ͱ͋Δ͜ͱ͔Βɼ

4ρρ∗∗ (d0 f ) · f d∗ f 2 ∧ Y = 8f ((ρρ∗∗ (d0 f ) · f )ρρ∗∗ d0 f ) ∧ Y(A.106)

Soryushiron Kenkyu

62ͱͳΓɼ(A.105) ͸ (ρ(X ∧ Y ) = ρ(X) ∧ ρ(Y ) Ͱ͋ΔͳΒ͹) ҎԼͷΑ͏ʹॻ͘͜ͱ͕Ͱ͖Δɽ

8f (ρρ∗∗ (d0 f ) · f ρρ∗∗ d0 f ) ∧ ρ(Y ) = −4f 2 ρ(d∗ [d∗ f, f Y ]E ) + 4f 2 ρ(d∗ f ) ∧ ρ(Ld∗ f Y )+ 4f 2 ρ(Ld∗ f d∗ Y ) + 4f 2 ρ(d∗ f ) · f ρ(d∗ Y )+ ρ(d∗ [d∗ f 2 , f 2 Y ]E ) − ρ((d∗ f 2 )) ∧ ρ(Ld∗ f 2 Y )− f ρ(Ld∗ f 2 d∗ Y ) − (ρ(d∗ f 2 ) · f 2 )ρ(d∗ Y ).

(A.107)͞Βʹɼ྆ลͰ d0 f ͱ಺ੵΛͱΔͱɼࠨล͸8f ιd0 f (((ρρ∗∗ (d0 f ) · f ), ρρ∗∗ d0 f ) ∧ ρ(Y ))= 8f ιd0 f ((ρρ∗∗ (d0 f ) · f ), ρρ∗∗ d0 f ) ∧ ρ(Y ) − 8f ((ρρ∗∗ (d0 f ) · f ), ρρ∗∗ d0 f ) ∧ ιd0 f ρ(Y )= 8f (ρρ∗∗ (d0 f ) · f )2 ∧ Y − 8f ((ρρ∗∗ (d0 f ) · f ), ρρ∗∗ d0 f ) ∧ ιd0 f ρ(Y )(A.108)ͱͳΔ͜ͱ͔Βɼ

8f (ρρ∗∗ (d0 f ) · f )2 ∧ Y= 8f ((ρρ∗∗ (d0 f ) · f )ρρ∗∗ d0 f ) ∧ id0 f ρ(Y )− 4f 2 ρ(d∗ [d∗ f, f Y ]E ) + 4f 2 ρ(d∗ f ) ∧ ρ(Ld∗ f Y )+ 4f 2 ρ(Ld∗ f d∗ Y ) + 4f 2 ρ(d∗ f ) · f ρ(d∗ Y )+ ρ(d∗ [d∗ f 2 , f 2 Y ]E ) − ρ((d∗ f 2 )) ∧ ρ(Ld∗ f 2 Y )− f ρ(Ld∗ f 2 d∗ Y ) − (ρ(d∗ f 2 ) · f 2 )ρ(d∗ Y ).

(A.109)܎਺ఔ౓ͷζϨ͸͋Δ͕ɼࠨล͸ (A.92) ࣜͷதԝͷ߲͕‫ݱ‬Ε͍ͯΔɽҰൠʹ೚ҙͷ f, Y ʹରͯ͠ӈล͸໌Β͔ʹ 0 Ͱ͸ͳ͍ͷͰɼρρ∗∗ (d0 f ) · f ͕ 0 ʹͳΒͳ͍ՄೳੑΛഉআͰ͖ͳ͍ɽ

A.11 Axiom C5 ͷ֬ೝρ(e)(e1 , e2 ) = ([e, e1 ]V + D(e, e1 ), e2 ) + (e1 , [e, e2 ]V + D(e, e2 ))(3.25)

͕੒ཱ͍ͯ͠Δ͔Ͳ͏͔Λ͔֬ΊΔɽ͜Ε͸ɼ(A.32) Λ࢖͏͜ͱͰ௚ͪʹࣔͤΔɽ(A.32) ͔Βɼ11([e, e1 ]V , e2 )+ = T (e, e1 , e2 ) + ρV (e)(e1 , e2 )+ − ρV (e1 )(e, e2 )+2211(e1 , [e, e2 ]V )+ = T (e, e2 , e1 ) + ρV (e)(e2 , e1 )+ − ρV (e2 )(e, e1 )+22
͜ͷ 2 ࣜΛ଍͢ͱɼҎԼͷؔ܎͕ࣜಘΒΕΔɽT (e, e2 , e1 ) ͸‫׬‬શ൓ରশͳͷͰଧͪফ͋͠͏ɽ

([e, e1 ]V , e2 )+ + (e1 , [e, e2 ]V )+1
1= T (e, e1 , e2 ) + T (e, e2 , e1 ) + ρV (e)(e1 , e2 )+ − ρ(e1 )(e, e2 )+ − ρ(e2 )(e, e1 )+22
11= ρV (e)(e1 , e2 )+ − ρ(e1 )(e, e2 )+ − ρ(e2 )(e, e1 )+(A.110)2
2(A.110) Λ ρ(e)(e1 , e2 )+ ʹ͍ͭͯղ͘ͱɼҎԼͷΑ͏ʹͳΔɽ
11
ρ(e)(e1 , e2 )+ = ([e, e1 ]V , e2 )+ + (e1 , [e, e2 ]V )+ + ρV (e1 )(e, e2 )+ + ρV (e2 )(e, e1 )+ (A.111)
2
2

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63͜͜Ͱɼρ ͱ D ͷఆ͔ٛΒɼei = Xi + ξi ͳͷͰɼ111ρV (e1 )(e, e2 )+ = ρ(X1 )(e, e2 )+ + ρ∗ (ξ1 )(e, e2 )+22
2
1= (⟨X1 , d(e, e2 )+ ⟩ + ⟨ξ1 , d∗ (e, e2 )+ ⟩)2= (d(e, e2 )+ + d∗ (e, e2 )+ , X1 + ξ1 )+= (D(e, e2 )+ , e1 )+(A.112)ಉ༷ʹɼ1ρV (e2 )(e, e1 )+ = (D(e, e1 )+ , e2 )+2(A.113)͜ΕΒΛ (A.111) ʹ୅ೖͯ͠ɼρV (e)(e1 , e2 )+ = ([e, e1 ]V , e2 )+ + (e1 , [e, e2 ]V )+ + (D(e, e2 )+ , e1 )+ + (D(e, e1 )+ , e2 )+= ([e, e1 ]V + D(e, e1 ), e2 ) + (e1 , [e, e2 ]V + D(e, e2 ))(A.114)(A.114) ͸·͞ʹ (3.25) ͦͷ΋ͷͰ͋ΔɽΑͬͯɼ(V, [·, ·]V , ρV , (·, ·)+ ) ʹ͓͍ͯɼAxiom C5 ͸

੒ཱ͢Δɽ

A.12 NK = NP + NP̃ Ͱ͋Δ͜ͱͷ֬ೝNK ͸ɼ(4.1) Ͱ༩͑ͨɼ֓ύϥෳૉߏ଄ͷՄੵ෼ੑΛධՁ͢ΔͨΊͷྔͰ͋ΔɽNP (X, Y ),NP̃ (X, Y ) Λ
NP (X, Y ) = P̃ [P (X), P (Y )], NP̃ (X, Y ) = P [P̃ (X), P̃ (Y )](A.115)ͱͨ͠ͱ͖ɼNK (X, Y ) = NP (X, Y ) + NP̃ (X, Y ) Ͱ͋Δ͜ͱΛ͔֬ΊΔɽ͜͜Ͱɼ(4.2) ࣜͷ P, P̃ ͷఆ͔ٛΒɼNP (X, Y ), NP̃ (X, Y ) Λ K ͕͋ΒΘʹͳΔΑ͏ʹॻ͖Լ͢ͱɼҎԼͷΑ͏ʹͳΔɽ

NP (X, Y ) = P̃ [P (X), P (Y )]
0
1
1
1
1
= (1 − K) (1 + K)X, (1 + K)Y
2
2
2
1
1
= (1 − K) [X + K(X), Y + K(Y )]2
41'
1= (1 − K) XY + XK(Y ) + K(X)Y + K(X)K(Y )24(− Y X − Y K(X) − K(Y )X − K(Y )K(X)1
1= (1 − K) ([X, Y ] + [X, K(Y )] + [K(X), Y ] + [K(X), K(Y )])2
41= ([X, Y ] + [X, K(Y )] + [K(X), Y ] + [K(X), K(Y )])81+ (−K[X, Y ] − K[X, K(Y )] − K[K(X), Y ] − K[K(X), K(Y )])8

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64
ಉ༷ʹɼ

NP̃ (X, Y ) = P [P̃ (X), P̃ (Y )]
0
1
1
1
1
= (1 + K) (1 − K)X, (1 − K)Y
2
2
2
1
1
= (1 + K) [X − K(X), Y − K(Y )]2
41'
1= (1 + K) XY − XK(Y ) − K(X)Y + K(X)K(Y )24(− Y X + Y K(X) + K(Y )X − K(Y )K(X)1
1= (1 + K) ([X, Y ] − [X, K(Y )] − [K(X), Y ] + [K(X), K(Y )])2
41= ([X, Y ] − [X, K(Y )] − [K(X), Y ] + [K(X), K(Y )])81+ (K[X, Y ] − K[X, K(Y )] − K[K(X), Y ] + K[K(X), K(Y )])8͕ͨͬͯ͠ɼ

1([X, Y ] + [X, K(Y )] + [K(X), Y ] + [K(X), K(Y )])81+ (−K[X, Y ] − K[X, K(Y )] − K[K(X), Y ] − K[K(X), K(Y )])81+ ([X, Y ] − [X, K(Y )] − [K(X), Y ] + [K(X), K(Y )])81+ (K[X, Y ] − K[X, K(Y )] − K[K(X), Y ] + K[K(X), K(Y )])81= ([X, Y ] − K[X, K(Y )] − K[K(X), Y ] + [K(X), K(Y )])4= NK (X, Y )(A.116)NP (X, Y ) + NP̃ (X, Y ) =ΑͬͯɼNP (X, Y ), NP̃ (X, Y ) Λ

NP (X, Y ) = P̃ [P (X), P (Y )], NP̃ (X, Y ) = P [P̃ (X), P̃ (Y )].
ͱͨ͠ͱ͖ɼNK = NP + NP̃ ͕੒ཱ͢Δɽ

(A.117)

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65ࢀߟจ‫ݙ‬[1] T. Kaluza, “On the Problem of Unity in Physic”, Sitzungsber. Preuss. Akad. Wiss. Berlin(Math. Phys.) (1921) 66.
[2] O. Klein, “Quantum Theory and Five-Dimensional Theory of Relativity. (In German
and English),” Z. Phys. 37 (1926) 895, High. Energ. Phys. 5 (1986) 241.
[3] C. G. Callan, Jr., E. J. Martinec, M. J. Perry and D. Friedan, “Strings in Background
Fields,” Nucl. Phys. B 262 (1985) 593.
[4] J. M. Maldacena, “The Large N limit of superconformal field theories and supergravity,” Int. J. Theor. Phys. 38 (1999) 1113 [Adv. Theor. Math. Phys. 2 (1998) 231] [hepth/9711200].
[5] A. Sen, “Strong - weak coupling duality in four-dimensional string theory,” Int. J. Mod.
Phys. A 9 (1994) 3707 [hep-th/9402002].
[6] N. A. Obers and B. Pioline, “U duality and M theory,” Phys. Rept. 318 (1999) 113[hep-th/9809039].
[7] A. Giveon, M. Porrati and E. Rabinovici, “Target space duality in string theory,” Phys.
Rept. 244 (1994) 77 [hep-th/9401139].
[8] C. Hull and B. Zwiebach, “Double Field Theory,” JHEP 0909 (2009) 099[arXiv:0904.4664 [hep-th]].
[9] C. Hull and B. Zwiebach, “The Gauge algebra of double field theory and Courant brackets,” JHEP 0909 (2009) 090 [arXiv:0908.1792 [hep-th]].
[10] A. Einstein, “The Foundation of the General Theory of Relativity,” Annalen Phys. 49(1916) no.7, 769 [Annalen Phys. 14 (2005) 517] [Annalen Phys. 354 (1916) no.7, 769].
[11] N. Hitchin, “Generalized Calabi-Yau manifolds,” Quart. J. Math. 54 (2003) 281[arXiv:math/0209099 [math-dg]].
[12] C. M. Hull, “Doubled Geometry and T-Folds,” JHEP 0707 (2007) 080 [arXiv:hepth/0605149].
[13] H. Mori, S. Sasaki and K. Shiozawa, “Doubled Aspects of Vaisman Algebroid and Gauge
Symmetry in Double Field Theory,” J. Math. Phys. 61 (2020) 013505 [arXiv:1901.04777[hep-th]].
[14] I. Vaisman, “On the geometry of double field theory,” J. Math. Phys. 53 (2012) 033509[arXiv:1203.0836 [math.DG]].
[15] I. Vaisman, “Towards a double field theory on para-Hermitian manifolds,” J. Math. Phys.

Soryushiron Kenkyu

66
54 (2013) 123507 [arXiv:1209.0152 [math.DG]].
[16] M. Gualtieri, “Generalized complex geometry,” Oxford University Ph.D. thesis.
[arXiv:math/0401221 [math.DG]].
[17] T. J. Courant, “Dirac Manifolds,” Transactions of the American Mathematical Society,vol. 319 (1990) 631.
[18] O. Hohm, C. Hull and B. Zwiebach, “Generalized metric formulation of double field
theory,” JHEP 1008 (2010) 008 [arXiv:1006.4823 [hep-th]].
[19] O. Hohm and B. Zwiebach, “Towards an invariant geometry of double field theory,” J.
Math. Phys. 54 (2013) 032303 [arXiv:1212.1736 [hep-th]].
[20] A. Deser and C. Sämann, “Extended Riemannian Geometry I: Local Double Field Theory,” arXiv:1611.02772 [hep-th].
[21] A. Deser and C. Sämann, “Derived Brackets and Symmetries in Generalized Geometryand Double Field Theory,” PoS CORFU 2017 (2018) 141 [arXiv:1803.01659 [hep-th]].
[22] K. Lee, “Towards Weakly Constrained Double Field Theory,” Nucl. Phys. B 909 (2016)
429 [arXiv:1509.06973 [hep-th]].
[23] O. Hohm, C. Hull and B. Zwiebach, “Background independent action for double field
theory,” JHEP 1007 (2010) 016 [arXiv:1003.5027 [hep-th]].
[24] C. M. Hull, “A Geometry for non-geometric string backgrounds,” JHEP 0510 (2005)
065 [hep-th/0406102].
[25] D. Svoboda, “Algebroid Structures on Para-Hermitian Manifolds,” J. Math. Phys. 59(2018) no.12, 122302 [arXiv:1802.08180 [math.DG]].
[26] L. Freidel, F. J. Rudolph and D. Svoboda, “Generalised Kinematics for Double Field
Theory,” JHEP 1711 (2017) 175 [arXiv:1706.07089 [hep-th]].
[27] L. Freidel, F. J. Rudolph and D. Svoboda, “A Unique Connection for Born Geometry,”
Commun. Math. Phys. 372 (2019) no.1, 119 [arXiv:1806.05992 [hep-th]].
[28] V. E. Marotta and R. J. Szabo, “Para ʖ Hermitian Geometry, Dualities and Generalized
Flux Backgrounds,” Fortsch. Phys. 67 (2019) no.3, 1800093 [arXiv:1810.03953 [hep-th]].
[29] W. Siegel, “Superspace duality in low-energy superstrings,” Phys. Rev. D 48 (1993) 2826
[hep-th/9305073].
[30] W. Siegel, “Two vierbein formalism for string inspired axionic gravity,” Phys. Rev. D
47 (1993) 5453 [hep-th/9302036].
[31] M. Gualtieri, “Generalized Complex Geometry,” Ann. of Math. 174 (2011) 75[arXiv:math/0703298].
[32] M. Grana, R. Minasian, M. Petrini and D. Waldram, “T-duality, Generalized Geometry
and Non-Geometric Backgrounds,” JHEP 0904 (2009) 075 [arXiv:0807.4527 [hep-th]].
[33] I. Y. Dorfmann, “Dirac Structures of Integrable Evolution Equations”, Phys. Lett.
A125(5) (1987) 240.
[34] D. Roytenberg, “Courant algebroids, derived brackets and even symplectic supermanifolds,” math/9910078 [math.DG].

Soryushiron Kenkyu

67[35] Z.-J. Liu, A. Weinstein, P. Xu, “Manin Triples for Lie Bialgebroids,” J. Diﬀerential
Geom. 45 (1997) 547 [dg-ga/9508013].
[36] A. Chatzistavrakidis, L. Jonke, F. S. Khoo and R. J. Szabo, “Double Field Theory and
Membrane Sigma-Models,” JHEP 1807 (2018) 015 [arXiv:1802.07003 [hep-th]].
[37] Y. Kosmann-Schwarzbach, “Lie bialgebras, Poisson Lie groups and dressing transformations,” Integrability of Nonlinear Systems, Second edition, Lecture Notes in Physics
638, Springer-Verlag, (2004) 107.
[38] C. M. Marle, “The Schouten-Nijenhuis bracket and interior products,” Journal of Geometry and Physics. 23 (1997) 350.
[39] C. Chevalley and S. Eilenberg, “Cohomology Theory of Lie Groups and Lie Algebras,”
Trans. Am. Math. Soc. 63 (1948) 85.
[40] J. A. Schouten, “Über Diﬀerentialkonkomitanten zweier kontravarianten Grössen,”.
Indag. Math. 2 (1940) 449, “On the diﬀerential operators of the first order in tensor
calculus,” In Cremonese. Convegno Int. Geom. Diﬀ. Italia. (1953) 1,
A. Nijenhuis, “Jacobi-type identities for bilinear diﬀerential concomitants of certain tensor fields I,” Indagationes Math. 17 (1955) 390.
[41] M. Gerstenhaber, “The Cohomology Structure of an Associative Ring,” Ann. of Math.
78 (1963) 267.
[42] V. G. Drinfel’d, “Quantum groups,” Proc. Internat. Congr. Math. (1986) 789.
[43] J. Pradines, “Théorie de Lie pour les groupoı̈des diﬀérentiables. Relations entre propriétés locales et globales,” C. R. Acad. Sci. Paris, Sér. A-B 263 (1966) A907.
[44] K. C. H. Mackenzie and P. Xu, “Lie bialgebroids and Poisson groupoids,” Duke Math.
J. 73 (1994) 415.
[45] N. Hitchin, “Lectures on generalized geometry,” arXiv:1008.0973[math.DG].
[46] P. Koerber, “Lectures on Generalized Complex Geometry for Physicists,” Fortsch. Phys.
59 (2011) 169 [arXiv:1006.1536 [hep-th]].
[47] T. Asakawa, H. Muraki, S. Sasa and S. Watamura, “Poisson-generalized geometry andR-flux,” Int. J. Mod. Phys. A 30 (2015) no.17, 1550097 [arXiv:1408.2649 [hep-th]].
[48] T. Asakawa, H. Muraki and S. Watamura, “Topological T-duality via Lie algebroids andQ-flux in Poisson-generalized geometry,” Int. J. Mod. Phys. A 30 (2015) no.30, 1550182[arXiv:1503.05720 [hep-th]].
[49] T. Asakawa, H. Muraki and S. Watamura, “Gravity theory on Poisson manifold with
R-flux,” Fortsch. Phys. 63 (2015) 683 [arXiv:1508.05706 [hep-th]].
[50] K. Uchino, “Remarks on the Definition of a Courant Algebroid,” Lett. Math. Phys. 60(2002) 171 [math/0204010 [math.DG]].
[51] D. Roytenberg and A. Weinstein, “Courant algebroids and strongly homotopy Lie algebras.” Lett. Math. Phys. 46 (1998) 1 [math/9802118[math.QA]].
[52] I. Vaisman, “Generalized Sasaki metrics on tangent bundles,” arXiv:1312.4279[math.DG].

Soryushiron Kenkyu

68[53] P. Ševera, “Letters to Alan Weinstein about Courant algebroids,” arXiv:1707.00265[math.DG].
[54] Y. Kosmann-Schwarzbach, “Exact Gerstenharber algebras and Lie bialgebroids,” Acta
Appl. Math. 41 (1995) 153.
[55] Y. Kosmann-Schwarzbach, F. Magri, “Poisson-Nijenhuis structures,” Ann. de l’I.H.P.
Physique théorique 53 (1990) 35.
[56] ాଜ Ұ࿠, “༿૚ͷτϙϩδʔ,” ‫ؠ‬೾ॻళ (1976)[57] A. Newlander and L. Nirenberg, “Complex Analytic Coordinates in Almost Complex
Manifolds,” Ann. of Math. 65 (1957) 391.
[58] W. M. Tulczyjew, “The graded Lie algebra of multivector fields and the generalized Liederivative of forms,” Bull. Acad. Pol. Sci., Sér. Sci. Math. Astr. Phys. 22 (1974) 937.
[59] A. Vaintrob, “Lie algebroids and homological vector fields,” Uspekhi Mat. Nauk 52
no.2(314) (1997) 161 translation in Russian Math. Surveys 52 no.2 (1997) 428.
[60] T. Kimura and S. Sasaki, “Gauged Linear Sigma Model for Exotic Five-brane,” Nucl.
Phys. B 876 (2013) 493 [arXiv:1304.4061 [hep-th]].
[61] J. Berkeley, D. S. Berman and F. J. Rudolph, “Strings and Branes are Waves,” JHEP
1406 (2014) 006 [arXiv:1403.7198 [hep-th]].
[62] D. S. Berman and F. J. Rudolph, “Branes are Waves and Monopoles,” JHEP 1505
(2015) 015 [arXiv:1409.6314 [hep-th]].
[63] I. Bakhmatov, A. Kleinschmidt and E. T. Musaev, “Non-geometric branes are DFT
monopoles,” JHEP 1610 (2016) 076 [arXiv:1607.05450 [hep-th]].
[64] T. Kimura, S. Sasaki and K. Shiozawa, “Worldsheet Instanton Corrections to Five-branes
and Waves in Double Field Theory,” JHEP 1807 (2018) 001 [arXiv:1803.11087 [hep-th]].
[65] O. Hohm and B. Zwiebach, “Large Gauge Transformations in Double Field Theory,”
JHEP 1302 (2013) 075 [arXiv:1207.4198 [hep-th]].
[66] J. H. Park, “Comments on double field theory and diﬀeomorphisms,” JHEP 1306 (2013)098 [arXiv:1304.5946 [hep-th]].
[67] D. S. Berman, M. Cederwall and M. J. Perry, “Global aspects of double geometry,”
JHEP 1409 (2014) 066 [arXiv:1401.1311 [hep-th]].
[68] M. Cederwall, “The geometry behind double geometry,” JHEP 1409 (2014) 070[arXiv:1402.2513 [hep-th]].
[69] C. M. Hull, “Finite Gauge Transformations and Geometry in Double Field Theory,”
JHEP 1504 (2015) 109 [arXiv:1406.7794 [hep-th]].
[70] U. Naseer, “A note on large gauge transformations in double field theory,” JHEP 1506
(2015) 002 [arXiv:1504.05913 [hep-th]].
[71] S. J. Rey and Y. Sakatani, “Finite Transformations in Doubled and Exceptional Space,”
arXiv:1510.06735 [hep-th].
[72] V. G. Drinfel’d, “Hamiltonian structures on Lie groups, Lie bialgebras and the geometricmeaning of classical Yang-Baxter equations,” Dokl. Akad. Nauk SSSR 268(2) (1983) 285.

Soryushiron Kenkyu

69[73] R. Loja Fernandes and M. Crainic, “Lectures on Integrability of Lie Brackets,”math/0611259.
[74] P. Ševera and M. Širaň, “Integration of diﬀerential graded manifolds,” [arXiv:1506.04898[math.DG]].
[75] D. Li-Bland and P. Ševera, “Integration of Exact Courant Algebroids,” Electronic Research Announcements in Mathematical Sciences (ERA-MS) 19 (2012) 58[arXiv:1101.3996 [math.DG]].
[76] M. Grana and D. Marques, “Gauged Double Field Theory,” JHEP 1204 (2012) 020[arXiv:1201.2924 [hep-th]].
[77] A. Deser and J. Stasheﬀ, “Even symplectic supermanifolds and double field theory,”
Commun. Math. Phys. 339 (2015) no.3, 1003 [arXiv:1406.3601 [math-ph]].
[78] U. Carow-Watamura, N. Ikeda, T. Kaneko and S. Watamura, “DFT in supermanifold
formulation and group manifold as background geometry,” [arXiv:1812.03464 [hep-th]].
[79] D. Roytenberg, “On the structure of graded symplectic supermanifolds and Courant
algebroids,” [arXiv:math/0203110 [math-sg]].
[80] Y.

Kosmann-Schwarzbach, “Derived brackets,” Lett.

Math.

Phys. 69 (2004) 61,[arXiv:math/0312524 [math.DG]].
[81] C. Klimcik and P. Severa, “Dual nonAbelian duality and the Drinfeld double,” Phys.
Lett. B 351 (1995) 455 [arXiv:hep-th/9502122].
[82] R. Von Unge, “Poisson Lie T plurality,” JHEP 0207 (2002) 014 [arXiv:hep-th/0205245].
[83] D. Lüst and D. Osten, “Generalised fluxes, Yang-Baxter deformations and the O(d,d)
structure of non-abelian T-duality,” JHEP 1805 (2018) 165 [arXiv:1803.03971 [hep-th]].
[84] V. E. Marotta, F. Pezzella and P. Vitale, “Doubling, T-Duality and Generalized Geometry: a Simple Model,” JHEP 1808 (2018) 185 [arXiv:1804.00744 [hep-th]].
[85] P. Ševera and F. Valach, “Courant algebroids, Poisson-Lie T-duality, and type II supergravities,” [arXiv:1810.07763 [math.DG]].
[86] S. Demulder, F. Hassler and D. C. Thompson, “Doubled aspects of generalised dualities
and integrable deformations,” arXiv:1810.11446 [hep-th].
[87] O. Hohm and H. Samtleben, “Exceptional Field Theory I: E6(6) covariant Form of MTheory and Type IIB,” Phys. Rev. D 89 (2014) no.6, 066016 [arXiv:1312.0614 [hep-th]].
[88] O. Hohm and H. Samtleben, “Higher Gauge Structures in Double and Exceptional Field
Theory,” Fortsch. Phys. 67 (2019) no.8-9, 1910008 [arXiv:1903.02821 [hep-th]].