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IPMU11-0147 A strange relationship between 2d CFT and 4d gauge theory Yuji Tachikawa IPMU, University of Tokyo, 5-1-5 Kashiwa-no-ha, Kashiwa, Chiba, 277-8583 Japan abstract * A relationship between 4d gauge theory and 2d CFT will be reviewed from the very basics. We will first cover the introductory material on the 2d CFT and on the instantons of 4d gauge theory. Next we will explicitly calculate and check the agreement of the norm of a coherent state on the 2d side and the instanton partition function on the 4d side. We will then see how this agreement can be understood from the perspective of string and M theory. to appear on the proceedings of the “Summer School on Mathematical Physics 2011”, Komaba * The review is prepared in Japanese as is customary for the proceedings of this summer school series, which has more than 20 years of history. An interested reader can find how to post T E X files written in CJK(Chinese-Japanese-Korean) languages to the arXiv by downloading the source code. arXiv:1108.5632v1 [hep-th] 29 Aug 2011

arXiv:1108.5632v1 [hep-th] 29 Aug 2011 · IPMU11-0147 ÛÛÛ!!!CCC†††üüü‚‚‚ ÖÖÖhhh„„„!!!CCCqqqbbb444 ÖÖÖnnn ˛˛˛pppjjj¢¢¢´´´ ¸Ý Õ„ 277-8583

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Page 1: arXiv:1108.5632v1 [hep-th] 29 Aug 2011 · IPMU11-0147 ÛÛÛ!!!CCC†††üüü‚‚‚ ÖÖÖhhh„„„!!!CCCqqqbbb444 ÖÖÖnnn ˛˛˛pppjjj¢¢¢´´´ ¸Ý Õ„ 277-8583

IPMU11-0147

A strange relationship between 2d CFT and 4d gauge theory

Yuji Tachikawa

IPMU, University of Tokyo,

5-1-5 Kashiwa-no-ha, Kashiwa, Chiba, 277-8583 Japan

abstract∗

A relationship between 4d gauge theory and 2d CFT will be reviewed from the very basics.

We will first cover the introductory material on the 2d CFT and on the instantons of 4d gauge

theory. Next we will explicitly calculate and check the agreement of the norm of a coherent

state on the 2d side and the instanton partition function on the 4d side. We will then see

how this agreement can be understood from the perspective of string and M theory.

to appear on the proceedings of the

“Summer School on Mathematical Physics 2011”, Komaba

∗The review is prepared in Japanese as is customary for the proceedings of this summer school series,

which has more than 20 years of history. An interested reader can find how to post TEX files written in

CJK(Chinese-Japanese-Korean) languages to the arXiv by downloading the source code.

arX

iv:1

108.

5632

v1 [

hep-

th]

29

Aug

201

1

Page 2: arXiv:1108.5632v1 [hep-th] 29 Aug 2011 · IPMU11-0147 ÛÛÛ!!!CCC†††üüü‚‚‚ ÖÖÖhhh„„„!!!CCCqqqbbb444 ÖÖÖnnn ˛˛˛pppjjj¢¢¢´´´ ¸Ý Õ„ 277-8583

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Page 3: arXiv:1108.5632v1 [hep-th] 29 Aug 2011 · IPMU11-0147 ÛÛÛ!!!CCC†††üüü‚‚‚ ÖÖÖhhh„„„!!!CCCqqqbbb444 ÖÖÖnnn ˛˛˛pppjjj¢¢¢´´´ ¸Ý Õ„ 277-8583

îîî!!!

0 oooXXX���kkk 1

1 ���!!!CCCqqqbbb444���ÖÖÖhhh]]]nnn³³³ÒÒÒüüüìììóóóÈÈȶ¶¶KKK 2

1.1 �!C�Ön� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Óé½íãpnhþ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Óé½íãpn³ÒüìóȶK . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 ÛÛÛ!!!CCC²²²üüü¸���ÖÖÖhhh¤¤¤óóó¹¹¹¿¿¿óóóÈÈÈóóónnnqqq������fff 11

2.1 ^ïÛ²ü¸�Ö . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 ¤ó¹¿óÈó . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.1 1-¤ó¹¿óÈóã . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.2 �ͤó¹¿óÈóã . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3 ¤ó¹¿óÈónq��f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.4 M�n@@� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.5 �ͤó¹¿óÈó�� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3 ���!!!CCChhhÛÛÛ!!!CCCnnn¢¢¢Â 25

3.1 þܢ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 ~Z�!Cx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3 U�km!Cx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.4 á5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

0 oooXXX���kkk

�&�Öo�þB¹go³Ækpf�kËÉU�f�D~[�W����ki��kº�U�_�Qg�B�~[��WKW�D�D�j¶Á<àK�$­Y�k��&�ÖLêñÛþn!Dpf�þahWfX(Y�no�UDjDh���~Y1��&�Öo�pf�ko^8k'Ñ�ko�10!Cn�ØS M kþWf�M¢ph|p

��� p Z(M) �þÜU[�K�MgY�SnK�Mo�hko*`�©U�fD~[�ng�&�ÖKo�S��~ZUhKpf�kMa�h�©U����gM�Ï~gbW~Y�WpWp�Z(M) ���gM�ÏkY�¹ÕoU���B�~Y�]�L�dB�hW�ZA� ZB h|s~W�F�Xk�X�n����nÕ¹g��W_`QgYK�� ZA h ZB oIWDoZgY(ó1)�ZA � ZB �pf�kMa�h�©gM�ÏgYK��ZA = ZB oþ(npfgqH�;5gY�WKW��&�Ö Z(M) o*`�³DO�Fk�©U�fD~[�K��WpWpZA = ZB opf�koUE�ËY�nKÄko$�jD;5hWfþ��Shkj�~Y�Sn�Fjþan�n��W g�^8kñD�nhWf90tã�-kz�U�_ßéüþ

ð'LBR��~Y�~_�Donaldson Ïh Seiberg-Witten Ïn�ôLi�g�FSeiberg-Witten �ÖK��Fn�Snþan��h�FShLgM~Y�U�k�²ü¸ûÍ�þÜ

1��L�~�fO�nL�~tEQ�p�n�kYÑøk�©LcfDf �pgojD`�F�hDFShgY�ã�f`cf Newton, Leibnitz nS�oD�D�OLLB�~W_�

1

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Z(M)弦理論(いいかげん)

数学(厳密) ZA ZB?

ó 1: &�ÖK�npf�þan�ú

K���nË hP�(¢¤ó·å¿¤ó�ØSn~ULþÜY�jipf�þa�Ö�`YShLúe~Y�ÊÞn�©go�å�R_�kÔy�hKj�EDgYL�K�ÕKWf��W�YD

ZA =�!Cqb4�Ön³ÒüìóȶKnÎëà� (0.1)

ZB =Û!C¤ó¹¿óÈón�M¢p (0.2)

n�dL�ôY�hDF���¬�W~Y�Z(M) o�m!Cn N = (2, 0) �Öh|p���nkj�~Y�wS�ko�ZA o c �-Ãûw�∆ � L0 nú$�λ �³ÒüìóȶKnÑéá¿hWf

ZA = 〈∆, λ|∆, λ〉 = 1 +λ2

2∆+

λ4(c+ 8∆)

4∆((1 + ∆)c− 10∆ + 16∆2)+ · · · � (0.3)

ZB o q, ε1,2, a �]�^�¤ó¹¿óÈóp�ÒKÕÏ�²ü¸wkþY��fÝÆó·ãëhWf�

ZB = Z instantonε1,ε2,a = 1 +

q

ε1ε2

2

(ε1 + ε2)2 − 4a2

+q2

ε21ε22

(8(ε1 + ε2)2 + ε1ε2 − 8a2)

((ε1 + ε2)2 − 4a2)((2ε1 + ε2)2 − 4a2)((ε1 + 2ε2)2 − 4a2)+ · · · (0.4)

hj�~Y�S��L

λ2 =q

(ε1ε2)2� ∆ =

1

ε1ε2((ε1 + ε2)2

4− a2), c = 1 + 6

(ε1 + ε2)2

ε1ε2(0.5)

hþÜØQ�h�hk�ôY��hDF�ngY��©n��o�~Zo�&�ÖnShoYcK�Ø�f�ZA J�s ZB �Kj�ú��jhS

�K�¬�W�gdKc_(��~Z¬�W~Y�!k�ZA h ZB �wS�k��Wf�ôY���º�W_na�]�Lin�Fkê6k�&�ÖK��ãgM�K�¬�Y��hDF�FkW_Dh�D~Y� SnÎüÈo(tøM~W_ [1] n�ÜHkjcfJ�~Yng��chíD�n�J}�n¹o]a��TÂgO`UD�

1 ���!!!CCCqqqbbb444���ÖÖÖhhh]]]nnn³³³ÒÒÒüüüìììóóóÈÈȶ¶¶KKK

1.1 ���!!!CCC���ÖÖÖnnn���

~Zo�!Cnp�i�K�oX�~W�F�(SnÀns0o��HpYÑø [2, 3, 4, 5] �ÂgnSh�)

2

Page 5: arXiv:1108.5632v1 [hep-th] 29 Aug 2011 · IPMU11-0147 ÛÛÛ!!!CCC†††üüü‚‚‚ ÖÖÖhhh„„„!!!CCCqqqbbb444 ÖÖÖnnn ˛˛˛pppjjj¢¢¢´´´ ¸Ý Õ„ 277-8583

Ø)ø èL N)ø

ó 2: ¤¸ó°!�n)¦�X'

−→ −→

ó 3: èL)¦gn¤¸ó°!�

�!Cni�ûn!Xj�hWf�¤¸ó°!���H~Y�Sn!�o��!Cbn�<P¹ (i, j) kê1¦ σi,j = ±1 LB��B�MMn�þU��º�o T �cnphWf

1

Zexp

1

T

∑i,j

(σi,jσi+1,j + σi,jσi,j+1)

(1.1)

gB��hDF�ngY�Z ohSnº�L1kj�_�n�<�n�pgY���Kk�

Z =∑σi,j

exp

1

T

∑i,j

(σi,jσi+1,j + σi,jσi,j+1)

(1.2)

gY�Z ��M¢ph|s~Y� σ �¹Ôóh|s�σ = +1 ��M�σ = −1 ���MhDD~W�F�Y�h�£��c_<P¹ p h q kþWf�¹Ôó σp h σq LÃcfD�{FLÃcfDjD4��� exp(2/T ) `Qïý'LØD�QgY� T �)¦h|s~W�F�Sn(�o!n�Fk�H��LgM~Y�T = 0 go�£��c_<P¹n¹ÔóoÃFº�LÃ�jDº�kþWf!Pkïý'LØD�gYK��hfn<P¹n¹Ôó σ L�ôW~Y�S�o�¹Ôó��<P¹kB�®�jÁón�M`h�Fh�)¦L^8kNDhMoûhSLêz�k�¹�kÁ�WfD��h�H~Y��¹�T = ∞ go��<P¹oìËgYK��ûhShWfo�hkqÑkjcfD~Y�S�oØ)goÁónÁ�L�H_�h�F�LgM~Y�T � 0 K� ∞ ~g��k�U[_4���SiSgÁ�L�H�n`�FK? Á�L�H�¬�oin�Fkj�nK�hDFSh��H~W�F�Sn!�o³ÆkãK�fD�ng�]��ã¬Y�Sh�úe~YL��Ñoѽ³óL^

8k7�jng�¹ÔóMM�Snº�k�cf��Wf�ULwScfD�K���n�ªOB�

3

Page 6: arXiv:1108.5632v1 [hep-th] 29 Aug 2011 · IPMU11-0147 ÛÛÛ!!!CCC†††üüü‚‚‚ ÖÖÖhhh„„„!!!CCCqqqbbb444 ÖÖÖnnn ˛˛˛pppjjj¢¢¢´´´ ¸Ý Õ„ 277-8583

5 10 20 50 100 200

1.00

0.50

0.20

0.30

0.15

0.70

ó 4: èL)¦gn 〈σ0,0σx,0〉 n/��

-4 -2 0 2 4

-4

-2

0

2

4

-4 -2 0 2 4

-4

-2

0

2

4

z 7→ eiθ · z z 7→ z + az2

ó 5: qbÛn�

~[��Wp�OJvh�T = 2.2 B_�gÁ�L�ÅY�ShL�K�~Y (ó2)�³Æãgo�

Tc =2

log(1 +√

2)(1.3)

hj�ShLå��fD~Y�Uf��¦)¦�SnèL¹ Tc kY�h�¹ÔónÃcfD�Jn'MUkD�D�j�nLB�ShL$�~Y�rhdMM�Öcf�� �� há'Wf��h�`D_D�_îL��gB�ShL$�~Y (ó3)��ch�Ï�ko�〈σ0,0σx,0〉 �,�Y�h�!þp°éÕgôÚkn�ShL$�~Y (ó4)�³Æãgo

〈σ(x)σ(0)〉 ∝ 1/x1/4� (1.4)

gYK��¹±üënô ~x→ α~x k4cf

σ → α1/8σ (1.5)

hWf��pûo gY� S���ûn¹±üëÛ ~x→ α~x n�hgn 'h�D~Y�B�éDaö��_Y�!Cûgo��WûL¹±üëÛg j�p�êÕ�k�ch�

,n@@�jÒ¦�ÝdÛn�hg�ÖL kj�ShLå��fD~Y��!Cn§��z = x+ iy høOh��,kcG¢p f(z) �(Df

z 7→ z′ = f(z) (1.6)

hDFÍ\L@@�jÒ¦�Ýa~Y (ó5)�®�Ûo

z 7→ z′ = z +∑n

εnzn+1 (1.7)

4

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Nx

Nt

ó 6: �ñn¤¸ó°!�

gYK��!P�nÙ¯Èë4 ξn = zn+1∂z gûL gB�Shkj�~Y� � qynξn = zn+1∂z �B�~Y�S��n¤Û¢Âo���Y�h

[ξm, ξn] = (m− n)ξm+n� [ξm, ξn] = (m− n)ξm+n� [ξm, ξn] = 0 (1.8)

hj�~Y�Uf�YSW�¹�Hf�Sn�!CûnFa x ¹�oz�g�y ¹�oB�`h�D~W�

F: t = y� !Pk�Dh��SWDng�x ¹�o Nx ��t ¹�o Nt �<P¹LB�hW~Y(ó6)� B�B; t �ú�Y�h�¹Ôó σi (−∞ < i <∞) LB�~YL��d�dnMMkþWfÙ¯Èëz�nú� |σx〉 ��H~W�F�Snz� H ko�� 2Nx �ú�LB�~Y�H k\(Y�L���d�H~Y:

A|σx〉 = exp

[1

T

∑x

σxσx+1

]|σx〉� (1.9)

B|σx〉 =∑(σ′x)

exp

[1

T

∑x

σxσ′x

]|σ′x〉 (1.10)

Y�h��M¢p (1.2) o

Z = trH(AB)Ny = trH e−NyH FW H = log(AB) (1.11)

hj��B� Nt `QÏP�f�ÏßëÈË¢ó H gûLzUW_h�FShLúe~Y� DDKH�p�B�¹�kÕKYÙ¯Èë4 ∂t L�H k\(Y�L� H k�c_�QgY�

Nx �^8k'MOÖcfJDf�SnøMÛH�èL¹ T = Tc nhMkLFh�B�&2`QgjO�qbÛ ξn� ξn hSL H k\(Y�L�kj�~Y�]�� Ln� Ln høM~W�F��,k�äx�jÙ¯Èë4�ÏP�f�jL�k<MôYh�ÜcLe��~Y�!P�B� ξnn¤Û¢Â (1.8) �Ûþ!OôY�ko�!n�FkY�WKjDhå��fD~Y:

[Lm, Ln] = (m− n)Lm+n +c

12(m3 −m)δn,−m (1.12)

FW −∞ < n,m < ∞ otpg c ocn�p�S�LÓé½íãpgY� ξn �hf��P Lnkj���X¤Û¢Â��_W~Y� Ln n¨ëßüÈqyo L−n = L†n hW~Y�å�!Xn_��8 Ln oØ��ShkW~Y�

5

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1.2 ÓÓÓééé½½½íííãããpppnnnhhhþþþ

Óé½íãpnhþ�¿y�Mk�¿�/ÕPnÏP��©ÒW~W�F� KÕÏ��P p hMn��P q L [q, p] = i hDF¤Û¢Â��_Y�k�ÏßëÈË¢ó H = (p2 + q2)/2 �¿y_D�ÿÖ p = ∂/∂q hWf�®���P ∂2/∂q2 + q2 nú¢p�y�¢phWf¿yf�oDgYL�ãp�k�H~W�F�]n_�k a = p+ iq�a† = p− iq ��©Y�h�

H = a†a+1

2(1.13)

høMô[~Y�[a, a†] = −1 gYK��[H, a] = −a gY�Uf�H kú$ E nú¶K |E〉LBc_hW~W�F: H|E〉 = E|E〉�Y�h�

Ha|E〉 = (aH − a)|E〉 = (E − 1)a|E〉� (1.14)

Yj�a�a|E〉 o ú$ E − 1 núÙ¯ÈëgY� SnÍ\oU¦g�p�Ô[~Y� �¹g�¶Kz�Lc$�Yj�aÝKj¶K |ψ〉 kþWf ‖|ψ〉‖2 = 〈ψ|ψ〉 ≥ 0 hY�h�

E‖|E〉‖2 = 〈E|H|E〉 = ‖p|E〉‖2 + ‖q|E〉‖2 ≥ 0 (1.15)

gYK��ú$ E o^ gY��cf�an|E〉 oDZ��ÅWjDhDQjD�Yj�aUK¶K |vac〉 LBcf

a|vac〉 = 0� (1.16)

Y�h

H|vac〉 =1

2|vac〉� (1.17)

~_��Økú$ E n¶Kk a† ��Q�hú$ E + 1 n¶Kkj�n�:[~Y��cf�

|n+1

2〉 = (a†)n|vac〉 (1.18)

Lú$ 12 + n nú¶Kkj�~Y�

gYK��ÏßëÈË¢ó H kþWf�����P a† o¨Íë®ü� 1 BR��Å��P a

o¨Íë®ü� 1 �R��Å��P a g�U��¶KL�N¨Íë®ü¶K |vac〉 gY�Óé½íãpn4�o�L0 �ÏßëÈË¢óh�FnLý�LoDgY: H = L0�Y�h�

[L0, Ln] = −nLn (1.19)

gYK��L−n L¨Íë®ü� n BR�����Pg�Ln = L†−n L¨Íë®ü� n �R��Å��PgY�hfn�Å��Pg�U��¶KL�N¨Íë®ügYL�]Sgn H = L0 nú$� ∆ hW~W�F:

L0|∆〉 = ∆|∆〉� Ln|∆〉 = 0 (n > 0)� (1.20)

¿�/ÕPn4�hpj��∆ oS�`Qgo�~�~[���,k�L0 nú$L ∆ +N gB�¶Ko L−n1L−n2 · · ·L−nk |∆〉 g

∑ni = N hj��

Fj�ngY(ó7)� S��!ek

|∆;N ;n1, . . . , nk〉 = L−n1L−n2 · · ·L−nk |∆〉 (1.21)

6

Page 9: arXiv:1108.5632v1 [hep-th] 29 Aug 2011 · IPMU11-0147 ÛÛÛ!!!CCC†††üüü‚‚‚ ÖÖÖhhh„„„!!!CCCqqqbbb444 ÖÖÖnnn ˛˛˛pppjjj¢¢¢´´´ ¸Ý Õ„ 277-8583

L1

L1

L1

L�1

L�1

L�1

L�2

L�2

L�3L3

L2

L2

...

...

L0 = ! + 3

L0 = ! + 2

L0 = ! + 1

L0 = !

L3!1|!!

L2!1|!!

L!1|!!

L!2|!!

L!3|!!L!2L!1|!!

|!!

ó 7: Óé½íãpnhþ

høM~W�F�FW n1 ≥ n2 ≥ · · · ≥ nk hWfJM~Y� S��LhfÚbìËj4��Snhþ�Óé½íãpn Verma hþh�D~Y�

N o�8!p (grade) h|p�~Y�!pL 1 n¶Ko L−1|∆〉 n�gY�S�nÎëào�¤Û¢Â�dKFh

〈∆|L1L−1|∆〉 = 2∆〈∆|∆〉 (1.22)

hj�~YK��∆ ocgY�!k!pL 2 n¶Ko�,k

|ψ〉 = c11|∆; 2; 1, 1〉+ c2|∆; 2; 2〉 (1.23)

høQ~YL�S�nÎëào¤Û¢Â (1.12) �dKcf��Y�h

〈ψ|ψ〉 = (c11, c2)MN=2

(c11

c2

)FW M2 =

(4∆(2∆ + 1) 6∆

6∆ 4∆ + c/2

)(1.24)

hj�~Y�S�L kj�jD_�ko�

4∆(2∆ + 1)(4∆ + c/2) ≥ (6∆)2 (1.25)

gjQ�pDQ~[��øMÛH�h�

∆(∆−∆1,2)(∆−∆2,1) ≥ 0 (1.26)

gY�_`W

∆r,s =c− 1

24+

1

4(rα+ + sα−)2� α± =

√1− c±

√25− c√

24� (1.27)

c < 1 gYh ∆1,2� ∆2,1 o�gYK��∆1,2 < ∆ < ∆2,1 `hÄîj�QgY��,k!p N nhS�kB�¶Knpo�N �cntpn�hWføO4�np`QB�~

Y�]�� pN høO�kW~Yh� MN o pN × pN L�kj�~Y�¶Kz�Lc�$gB�_�ko�hfn N kþWf detMN ≥ 0 gjDhDQ~[��SnL��o Kac k�cf��U�fJ��

detMN ∝∏

r,s≥1; rs≤N(∆−∆r,s)

pN−rs (1.28)

7

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hj�~Y�c ≥ 1 nhMo{|êÕ�kS�ockj�~YL�c < 1 nhMo ∆ L ∆r,s ni�KgjDP��DZ�U�Kn ∆r,s� ∆s,r k�~�fÄîkjcfW~D~Y� S��naö��çk¿y�h�¶Kz�Lc�$gB�_�ko�

c ≥ 1 �WOo m ≥ 2 j�cntp�Öcfc = 1− 6

m(m+ 1)(1.29)

hj�ShLå��fD~Y�U�k���n4�o ∆ n$o r, s � 1 ≤ s ≤ r < m j�cntphWf∆ = ∆r,s kP��~Y�

α+ =m+ 1√m(m+ 1)

� α− =−m√

m(m+ 1)(1.30)

gYK��

∆r,s =((m+ 1)r −ms)2 − 1

4m(m+ 1)(1.31)

hj�~Y� m = 2 nhMo c = 0� 1U�� ∆ o ∆ = 0 n�gb}OB�~[�� !n m = 3

n4�o�c = 1/2� 1U�� ∆ o

∆1,1 = 0� ∆2,2 =1

16� ∆1,2 =

1

2(1.32)

n.^gY�¤¸ó°!�oèL¹go�¦Sn c = 1/2 nÓé½íãpL ξn K�e� Ln h ξn K�e

� Ln hLB�~Y� ¹±üëÛ (x, y)→ α(x, y) o®�Û α = 1 + ε kþWfo z = x+ iy

gøDf

z∂

∂z+ z

∂z= ξ0 + ξ0 (1.33)

g�H��~Y�Yj�a L0 + L0 gY� ¹Ôó��PnÛ'� (1.5) køM~W_L�®�Û��H�h L0 + L0 nú$L 1/8 gB�Shkj�~Y�S�o (1.32) g L0 Ês L0 nú$L!¹h� 1/16 kjcfD�ShkþÜW~Y�

1.3 ÓÓÓééé½½½íííãããpppnnn³³³ÒÒÒüüüìììóóóÈÈȶ¶¶KKK

Uf�q�~_¿�/ÕPk;Wf�³ÒüìóȶK��H~W�F� p h q o¤ÛW~[�K���Bú¶K�h�ShogM~[�� B�¶K |ψ〉 kþWf�〈O〉 = 〈ψ|O|ψ〉 he�Y�ShkY�h�p h q n�L�o

(δp)2 = 〈(p− 〈p〉)2〉� (δq)2 = 〈(q − 〈q〉)2〉 (1.34)

h�H~YK��

δp2δq2 = ‖(p− 〈p〉)|ψ〉‖2‖(q − 〈q〉)|ψ〉‖2 ≥ [Im〈ψ|(p− 〈p〉)(q − 〈q〉)|ψ〉]2 =1

4(1.35)

hj�ngW_� º�'���kY�¶KoïýjP�äx�j¶Kh�cf�oDgW�F�n�b�I�kY��dn¹Õo

i(p− 〈p〉)|ψ〉 = (q − 〈q〉)|ψ〉 (1.36)

8

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hY�p�DgY�λ = 〈p+ iq〉 hY�h�ψ L�Å��P a = p+ iq nú¶KgB�ShL�K�~Y:

a|ψ〉 = λ|ψ〉� (1.37)

S��³ÒüìóȶKh|vngW_�å�ú$ λ n³ÒüìóȶK� |λ〉 høO�kW~W�F�¿�/ÕPnú¶Ko³ÒüìóȶKn��gY: |vac〉 = |0〉� [a, a†] = 1 gB�Sh�

)(Wf�aeλa

† |0〉 = λeλa† |0〉 (1.38)

Yj�a |λ〉 = eλa† |0〉 gY�¶K��<�Y�ko�

〈λ|λ〉 = 〈0|eλaeλa† |0〉 = eλλ〈0|0〉 (1.39)

hY�p�DgY�¿�/ÕPn³ÒüìóȶKoD�D�jÜ(LB�~Y�Óé½íãp�Í�gY�gY

K��Óé½íãpn³ÒüìóȶK��H����sL!OojDgW�F�|∆〉 g��U��Verma hþn-nÙ¯Èë |ψ〉 g��Å��P Ln (n > 0) Lú$��d�n��H~Y:

Ln|ψ〉 = λn|ψ〉� (1.40)

¤Û¢ÂK��YPk n ≥ 3 j� λn = 0 h�K�~Y�!Xn_�k λ2 �¼íhWfW~cf�λ ≡ λ1 g��U��¶K |∆, λ〉 ��H~W�F:

L1|∆, λ〉 = λ|∆, λ〉, L2|∆, λ〉 = 0� (1.41)

Y�h n > 2 kdDf Ln|∆, λ〉 = 0 oêÕ�k�D~Y� ¿�/ÕPkj�cf�eλL−1 |∆〉 ��H_DhS�gYL�Óé½íãpn¤Û¢Âo]�{i!XgjD_��]�go�ôYN~Y�~Z�ôk��Wf�~W�F�2WD¶Ko�j�k[�U�gM�HgYK��

|∆, λ〉 = |∆〉+ c1L−1|∆〉+ c11L2−1|∆〉+ c2L−2|∆〉+ · · · (1.42)

høM~Y�Y�h�L1c1L−1|∆〉 = λ|∆〉 (1.43)

K� c1 = λ/(2∆) h�~��

L1(c11L2−1|∆〉+ c2L−2|∆〉) = λc1L−1|∆〉, L2(c11L

2−1|∆〉+ c2L−2|∆〉) = 0 (1.44)

K� c11� c2 L�~�~Y�wUn�Wo�S���

〈∆, λ|∆, λ〉 = 1 +λ2

2∆+

λ4(c+ 8∆)

4∆((1 + ∆)c− 10∆ + 16∆2)+ · · · (1.45)

hj�~Y�Uf�Sn¿Pg�K�ÂpoYyfz����ngW�FK? !p 2 go�*åpL c11 h

c2 L�¦�d�¹��o!p 1 k¶KL 1 d�!p 0 k�¶KL 1 dBc_ng�Sa�� 2 d

9

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境界条件 境界条件|!, 1!!!, 1|

λ2 log

ó 8: Ï 〈∆, λ|∆, λ〉 o�wU 2 log λ n�ñn!tk¶K |∆, 1〉 L�LaöhWf�H��fD��nh�H��

g!�ãOShLgM~W_�S�o p2 = p1 + p0 `c_hDFv6kúeO�ng��,ko N

L'MOj�h pN � pN−1 + pN−2 hj��¹��npLNpkjcfãOnL��ðãkj�~Y���k���Wf��h�]�g�!�kãO�LgM�ShL�K�~Y�]n�1oSFgY[6]�2WD³ÒüìóȶKLBc_hWf�]��!pÎk~h�f

|∆, λ〉 = |ψ0〉+ λ|ψ1〉+ λ2|ψ2〉+ · · · (1.46)

hW~Y�Y�h�L1|ψN 〉 = |ψN−1〉� L2|ψN 〉 = 0 (1.47)

hjcfD�oZgY�Y�h�!Xk�K��Fk�

〈∆|(L1)N |ψN 〉 = 1� (1.48)

~_�]�å�nD���[go

〈∆|(L1)N−2L2|ψN 〉 = 0 (1.49)

Ihj�~Y�Yj�a�|ψN 〉 o�!p N n¶Kn-g� LN−1|∆〉 hn��M��cf�Önú�hoô¤WfD��QgY�]Sg�

|ψN 〉 =∑

i1+···+ik=N

(M−1N )11···1,i1i2···ikL−i1L−i2 · · ·L−ik |∆〉 (1.50)

hÖ�p�DShL�K�~W_� S�K��|∆, λ〉 n�M�YP��gM�

〈∆, λ|∆, λ〉 = 1 + λ2(M−11 )1,1 + λ4(M−1

2 )11,11 + λ6(M−13 )111,111 + · · · (1.51)

hj�~Y� YP�K��Fk etL0 |∆, λ〉 = et∆|∆, etλ〉 gYK��

〈∆, λ|∆, λ〉 = 〈∆, 1|λ2(L0−∆)|∆, 1〉 (1.52)

høMô[~Y� L0 L�ñk�!Cqb4�Ö�nD_hMnB�zUn��P`c_Sh��DúW~Yh�SnÀg��W_Ïo�ó 8 n�Fkó:Y�ShLgM~Y�

10

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2 ÛÛÛ!!!CCC²²²üüü¸���ÖÖÖhhh¤¤¤óóó¹¹¹¿¿¿óóóÈÈÈóóónnnqqq������fff

2.1 ^ïïïÛÛÛ²²²üüü¸���ÖÖÖ

Uf�YcK�q�Hf�SnÀgoÛ!Cn²ü¸�Önq�W~Y�(SnÀn�¹ns0o��HpYÑø [7] ���©2[8]�ÂgnSh�) ²ü¸�Ön�j!Xj�o Maxwell nûÁ�fgY�û4 ~E hÁ4 ~B oøþÖ�b�go Fµν = −Fνµ k~h~�ngW_:

F0i = Ei� F12 = B3� F23 = B1� F31 = B2� (2.1)

Y�h�¹��o∂µFµν = 0� ∂µFνρ + ∂νFρµ + ∂ρFµν = 0 (2.2)

høQ~Y�FWM�goá5U�_ Einstein n���Yj�a�XûWL�¦þ��hi�k�Ï ηµν �e�f³WR�hDF����D~W_:

∂µFµν =∑ρ,µ

∂ρηρµFµν� (2.3)

(2.2)n��oÛCÙ¯ÈëÝÆó·ãë Aµ �(Df

Fµν = ∂µAν − ∂νAµ (2.4)

hY�hêÕ�kãOShLúe~Y�FW�Fµν K� Aµ o���ko�~�Z�χ �ÝKj¹«éü¢phWf

Aµ 7→ Aµ + ∂µχ (2.5)

hWf� Fµν o��~[��S��²ü¸Ûh�D~Y� �FrhdÍ�j¹o� Aµ �ú,�j�fp`h�Fh\(�øO�LgM~Y�Yj�a

S =

∫d4x

1

4FµνFµν (2.6)

��Y�Shk�cf (2.2) Lúfe~Y� Maxwell ¹��n!XgF�WDhS�o�]nÚb'gY�Yj�a�Aµ h A′µ � 2 dnãhY�h�Aµ +A′µ �êÕ�kãkj�~Y�åoäxÖgW_L�S��ÏPÖkY�ko� Feynman nLïM��Y�Å�LB�~

Y�Yj�a

Z =

∫[DAµ]eiS (2.7)

hWf�ïýjYyfnÙ¯ÈëÝÆó·ãë Aµ nMMkþWf�Mø eiS �dQfM�[��hDFÍ\gY� åo�ÏL η = diag(−1,+1,+1,+1) nßó³Õ¹­z�gnpÖgW_L�å�!Xn_��ÏL δ = diag(+1,+1,+1,+1) næü¯êÃÉz�kq�H�ShkW~Y�Y�hLïM�o

Z =

∫[DAµ]e−S (2.8)

hjcf�*`!P!CnM�gYL�WoqD�YOj�~Y�20���Jni�n'Mjz�nrhdo�ûÁ�å�kB�Sn�nÖn2dn���7D

� h�1D� Lia��Sn Maxwell �Öná5gB�^ïÛ²ü¸4n�ÖgøK��h

11

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DFShgW_� ~Z�^ïÛ¤ SU(N) �JU�DW~W�F�g ∈ SU(N) o� N ×N L�g�æË¿ê g†g = 1� U�k det g = 1 hW_�ngY�SU(2) oyk

g =

(z −ww z

)(2.9)

g�d |z|2 + |w|2 = 1 hDF�ngYK��SU(2) ' S3 gB�ShL�K�~Y�g LXMCkÑDhWf� g = 1 + ε + · · · høM~Yh�g†g = 1 K� ε + ε† = 0� ôk

det g = 1 K� tr ε = 0 hj�~Y� hDF�Qg�ͨëßüÈgÈìü¹L¼ínL�hS�SU(N) nêüãph�D~Y�S��dKcf�²ü¸ÝÆó·ãë Aµ L µ = 1, 2, 3, 4 kþWf SU(N) nêüãpkec

fD�hW~W�F� ²ü¸4n7U�

Fµν = ∂µAν − ∂νAµ + [Aµ, Aν ] (2.10)

h���h�KÕ¹��o∂µFµν + [Aµ, Fµν ] = 0 (2.11)

hj�~Y�S�L Yang-Mills 4n¹��gY�S���H�\(o

S = − 1

2g2

∫trFµνFµν (2.12)

gY���k�Wf�fO`UD�N × N L�n4�k�S�� SU(N) ²ü¸�Öh|s~Y�g oP��ph|p�~Y��1D� oS�g N = 2 hW_�n��7D� o N = 3 hWf�U�kÏPÖkW_�ng

øK��ShL�KcfD~Y� ��k��Y�ko��o�LïM��W~Y:

Z =

∫[DAµ]e−S (2.13)

ÿÖ�S�o!P!CM�kjcf�³Æj�©Lúe�h Clay ÞnJ��Öc_�Fj�ngYL�pf�³ÆUk]�{iS`��jQ�p��HpBz R4 �^8k0Kj<P Z4 gÑ<Wf��è'¹üÑü³óÔåü¿ügM����Wf��ShLgM~Y�]FY�h�_hHpD�D�jÏÉíónêÏÔL��¤înÄògMa�h�þgM�ShL�KcfD~Y�Uf�Maxwell 4n\(L²ü¸Û (2.5) g `c_�Fk�^ïÛ²ü¸4n\(�

²ü¸Ûg gY�~Z�Aµ kþY�²ü¸ÛoBzK� SU(N) xn�Ï g(x) ��cf

Aµ 7→ gAµg−1 + g∂µg

−1 (2.14)

hY��nhW~Y�Y�h�Fµν kþWfo

Fµν 7→ gFµνg−1 (2.15)

hj��\(n«M�¢po

trFµνFµν 7→ tr gFµνg−1gFµνg

−1 = trFµνFµν (2.16)

12

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k=0 の配位

k=1 の配位k=−1 の配位

k=2 の配位

E−B=0E+B=0

E+B=0

ó 9: !P!CnMMz�o�¤ó¹¿óÈóp k géÙëU��è�z�k�K���]n]�^�n-g�(Í)êñÌþão\(����Y��

hj�ShL:[~Y� gYK���8

lim|x|→∞

g(x) = 1 (2.17)

hj� g(x) gn²ü¸ÛgPsdO�dn Yang-Mills ¹��não���W~Y�S���@@²ü¸Ûgn���h|s~W�F� �¹g� g(x) ≡ g h4@k��jD4�o (2.14)

gn®��L=a~Y� S��'ß²ü¸Ûh|s�'ß²ü¸ÛgPsdO�dnão���WjDShkW~Y�

Yang-Mills 4nKÕ¹�� (2.11) ko A kdDf!n�LB�~YK��^ÚbgY�ã Aµ h A′µ hL�d���f��Aµ +A′µ oãkoj�~[��gYK��Õüê¨ÛY�pãLB~�hDF�Qg�B�~[��WKW�¤ó¹¿óÈóh|p���#nwS�jãLå��fJ��]��o^8ks0k¿y��LgM~Y�]Sg�]�kdDf!k¬�W~W�F�

2.2 ¤¤¤óóó¹¹¹¿¿¿óóóÈÈÈóóó

Yang-Mills 4nLïM� (2.13) �Y�Sh��H~W�F�S L�UD{FLÄ�L'MDgYK��S ����Y�MMLÍ�]FgY� ]��¿y�ºk�~ZÌþ4

Fµν = εµνρσFµν/2 (2.18)

��eW~W�F�S�o�^øþÖ�k���û4 ~E hÁ4 ~B k�Q�p�a�Fi ~E h ~B �e�ÿH_�ngY�ÝKjMM Aµ L�H��_��\(LzcWjD�Fk!P`gE��O Fµν → 0 hjc

fD�hW~Y�SnhM�k �tphWf

−∫d4x trFµνFµν = −4

∫d4x tr ~E · ~B = 16π2k (2.19)

hj�ShLå��fD~Y� gYK��Aµ nMMnz�o!P!CgYL�]�otp k géÙëU��è�z�k�K�fD��QgY (ó9)� S�o�gYPk:W~Yng�]�~g��f�M~W�F�Sn k oi�go¤ó¹¿óÈóp�pfgo,�Áãüóph|p�~Y�

13

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Uf�

− trFµνFµν = −2 tr( ~E2 + ~B2) (2.20)

= −2 tr( ~E ± ~B)2 ± 4 tr ~E · ~B (2.21)

≥ ±4 tr ~E · ~B = ± trFµνFµν (2.22)

gYK�� ∫d4x trFµνFµν ≥

∣∣∣∣∫ d4x trFµνFµν

∣∣∣∣ = 16π2|k|� (2.23)

SnI÷��_Yko k Lcj�p

~B + ~E = 0, �WOo Fµν + Fµν = 0 (2.24)

gB�p�DShL�K�~Y(�¦ó9�Âg�UD)�S��ÍêñÌþ¹��hDD~Y� kL n4�o Fµν = Fµν hDFêñÌþ¹����H�poDgYL�,ê�k�XgYng�Ê�o k o^ hW~W�F�Uf�ÍêñÌþ¹���¿y�Mk�k Ltpkj�Sh�ºK�~W�F�~Z�trFµνFµν

Lh®�gB�Shkè�W~Y:

trFµνFµν = ∂µεµνρσ tr(AνFρσ −1

3AνAρAσ)� (2.25)

�cf�

−∫d4x trFµνFµν =

∫S3

dnµεµνρσ tr(−AνFρσ +1

3AνAρAσ) (2.26)

gY�!P`g Fµν = 0 hW~W_K��²ü¸Û (2.14) � Fµν = 0� Aµ = 0 K��k�cf�

=1

3

∫S3

dnµεµνρσ(g−1∂νg)(g−1∂ρg)(g−1∂σg)� (2.27)

_`W�g o!P`n S3 K�¤ SU(N) xn�ÏgY��ch�!Xj N = 2 nhM��H~Yh�SU(2) ∼ S3 gYK��g o

g : S3 → S3 (2.28)

h�FShLgM��M�oSn�ÏLUÞûMØDfD�K�,c_�nkj�~Y�Ô�Âp���Y�h�~MdMp� k hWf

= 16π2k� (2.29)

ÍêñÌþ¹��noDhS�o�~Z�S���_[pêÕ�k Yang-Mills ¹����_YShLB�~Y:

∂µFµν + [Aµ, Fµν ] = −∂µFµν − [Aµ, Fµν ] (2.30)

gYL�óºk Fµν n�©�ãeY�hêÕ�k¼íkj�ShL�K�~Y� ~_�Yang-Mills

¹��o��n®�¹��gYL�ÍêñÌþ¹��o��gY�U�k�^Úb��!gjOf�!gJU~�~Y�

14

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2.2.1 1-¤¤¤óóó¹¹¹¿¿¿óóóÈÈÈóóóããã

go�¤ó¹¿óÈópL k nÍêñÌþãoi�jb�WfD�ngW�FK? ~Z�j!Xjk = 1 n4���~W�F�S�o�,k

Aµ(x) =Hµν(x− x0)ν|x− x0|2 + ρ2

(2.31)

hDFb�WfD�hå��fD~Y�SSg x0 o¤ó¹¿óÈón-Ã�ρ o¤ó¹¿óÈón'MU�z�~Y� Hµν oiFÖ�pDDgW�FK? TH�ÍêñÌþkW_Dng�Hµν = −Hνµ�Hµν = −Hµν hDF N ×N L��h�ShkW~Y�Snb�ÍêñÌþ¹��kãeY�h�H01� H02� H03 L

H01 = [H02, H03]� H02 = [H03, H01]� H03 = [H01, H02] (2.32)

h�SO(3) n¤Û¢Â��_YyMShL�K�~Y�U�k�¤ó¹¿óÈóp���Y�h

k = 2 trH032 (2.33)

hj�Sh��K�~Y�gYK���j!Xjão�

H0i =1

2iσi ⊕ON−2 (2.34)

hY�ShgY�FW σ1,2,3 o�8nѦêL�

σ1 =

(0 1

1 0

), σ2 =

(0 i

i 0

), σ3 =

(1 0

0 −1

)(2.35)

g�S�o 2× 2 L�gYK��ON−2 o(N − 2)× (N − 2) L�ghf¼íj�nhWf�]��ØQ³Wf N ×N L�kY�ShkW~Y�⊕ oL��ÖíïþÒkj�y�Í\gY�rhdãLgM�h�ÝKj SU(N) L� g �Öcf

H0i = g−1(1

2iσi ⊕ON−2)g (2.36)

hWf�S6ãkj�~Y�g o (h Id2×2)⊕ g(N−2)×(N−2) hDFbnL�`hzÞ�W~Yng�S�g

N2 − 1− (N − 2)2 = 4N − 5 (2.37)

ê1¦`QãL���_Shkj�~Y�x0 kB�Ûdnê1¦J�s ρ kB��dnê1¦�³Yh��� 4N ê1¦B�ShL�K�~W_� S��nê1¦nSh�¤ó¹¿óÈónâ¸åé¤h|s~Y�yk N = 2 n4�o�4 · 2− 5 = 3 ê1¦o SU(2) L� g g~�Yê1¦]n�ngY�

Snê1¦o SU(2) ' S3 `QB�~Y�_`W�

g =

(−1 0

0 −1

)(2.38)

n4�o¤ÛWfW~cf Hµν kqÿLjDng���k�sLB�no SU(2)/{±1} ' S3/Z2

`QgY� PÖhWf�SU(2) n1-¤ó¹¿óÈóãnâ¸åé¤nz�M2,1 o

M2,1 ' R4 × R+ × S3/Z2 ' R4 × R4/Z2 (2.39)

gB_H���ShL�K�~Y�FW�óºgo'MUnÑéá¿ ρ h²ü¸n�MnÑéá¿S3/Z2 �O�B�[f R4/Z2 hW~W_�

15

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ó 10: 1-¤ó¹¿óÈóãn\(Ʀ

↓ ↓

ó 11: 2-¤ó¹¿óÈóãn\(Ʀ��dn¤ó¹¿óÈó�ÑeQ���²ü¸4n�MLiFjcfD�Kk�cfP�oWWOpj��

2.2.2 ���ÍÍͤ¤¤óóó¹¹¹¿¿¿óóóÈÈÈóóóããã

Uf�MÀgo 1-¤ó¹¿óÈóãko-ÃnMnk 4 Ñéá¿�µ¤ºk 1 Ñéá¿�²ü¸n�Mk 4N − 5 Ñéá¿B�Sh�fs~W_�wS�k\(Ʀ trF 2

µν �ó:Y�hó 10 n�Fkj�~Y�S�K���K��Fk�-ÃK�â��h²ü¸4n7U Fµν oA�k�UOj�~Y�gYK��1-¤ó¹¿óÈóã�u_d�G¹ A′µ o-ÃL x′0 µ¤ºL ρ′� �Frhd A′′µo-

ÃL x′′0 µ¤ºL ρ′′ gB��FkÖ�h� |x′0 − x′′0| � ρ′ + ρ′′ gB�p�!�L�Bk'Mj$koj�~[�K��A′µ + A′′µ �{h�iÍêñÌþ¹��nãkj�~Y� ÿÖ A′µA

′′µ K�O�

ÜcLB�ng�]��îcWf��jDhDQ~[�:

Aµ = A′µ +A′′µ +�UjÜc� (2.40)

Sn�ßgo�2-¤ó¹¿óÈóãko 8N Ñéá¿LB�ShL�K�~Y�¤ó¹¿óÈóL�dÑeD_4�o�Sn�Fj�ôjã�goDQ~[�L�]n4�g�¹���Ma�hãQ�ShLå��fD~Y(ó11)�

16

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�,k¤ó¹¿óÈópL k n4�o��ØkWfhfn¤ó¹¿óÈón-ÃLâ�fD�h�Ím��[�Shk�cf 4Nk�â¸åé¤��c_ãLdO�~Y� DDKH�h�Snâ¸åé¤z��MN,k høOh�â¸åé¤z�n�n¹goJJ�]

MN,k ∼ (MN,1)k/Sk (2.41)

hjcfD��Yj�a 1-¤ó¹¿óÈónâ¸åé¤n³ÔüL k �Bcf]��nÛ¤ Sk g���W_�nkjcfD~YL�-Ãèo�ch�ÑkjcfD~Y�MN,k �wS�køM�Y¹Õ� Atiyah-Drinfeld-Hitchin-Manin [9] k�cf:U�fD~

Y�]��Ma�h¬�Y�B�o0�B�~[�L�ðò�`Qo¬�W_Dh�D~Y(SnÀns0oìÓåü [10] ��©2 [11] I�ÂgnSh�)� ~Z�k = 1 n4�k;�~Yh�ãn®�jè�o H0i g�H��fD~W_�H0i o�X = H01 + iH02 g K�©CgM~YL�aöX2 = 0 h tr |X|2 = 1 LÅ�gY�SSg� ρ nê1¦� X k+�fW~Hp�tr |X|2k¢Y�aöo=hYShLgM~Y�]Sg�X2 = 0 k`QèîW~W�F�Y�h�X n�po 1 jng�

Xij = BiAj (2.42)

høO�LgM~Y�FW i, j = 1, . . . , N� X2 = 0 ��³U[�_�k

AiBi = 0 (2.43)

��BWf�U�kÝKj c ∈ C kþWf (Ai, Bi)→ (cAi, c−1Bi) hWf� X L��jDng�

AiAi −BjBj = 0 (2.44)

hWf[�fc ∈ R nê1¦o�YShkW~W�F�JW~Dk�-Ãnê1¦ x0 ∈ R4 �(z, w) ∈ C2 høOhY�h�P@ 1-¤ó¹¿óÈóão (z, w,Ai, B

i) g (2.43)� (2.44) ��_W�U�k

(Ai, Bi)→ (eiθAi, e

−iθBi) (2.45)

hDF\(kþWf����W_�n�høO�LgM~Y� Atiyah-Drinfeld-Hitchin-Manin o�S�nê6já5L k-¤ó¹¿óÈóãnâ¸åé¤��ðY����D`W~W_� TH`QøM~Yh�1, . . . , k �p�ûW a, b �(�W��n z, w,A,B kûW�ý Wf zab , wab , Aai , B

ja

hW~Y�]FWf�(2.43) ná5hWf

AaiBib + [z, w]ab = 0� (2.46)

(2.44) ná5hWfAaiA

ib −Bi

bBai + [z, z†]ab + [w,w†]ab = 0 (2.47)

�²W� (2.45) ná5hWf�

(Aai , Bib, z, w)→ (gabA

bi , B

ia(g−1)ab , gzg

−1, gwg−1) (2.48)

hDF k × k æË¿êL� gab n\(g���Y�Shk[��hDFnL|�n�dQ_h:gY�â¸åé¤np�Ø�W~W�F�A� B ko�� 4Nk ê1¦LB��z� w ko 4k2 �ê1

¦LB�~Y� (2.46) g 2k2 �aö�²W�(2.47) g k2 ��U�k (2.48) g k2 �ê1¦�Ö�dOng�P@ 4Nk �ê1¦LB�Shkj�~Y� 1-¤ó¹¿óÈóã� k �hcfM_4���þÜY� (Ai, B

i, z, w) � k DhcfO�p�(2.46)� (2.48) �JJ�]ãOL��\�noÖíïþÒk&y�pDDgYL�[z, w] n¤ÛPnB_�K�¤ó¹¿óÈó�nø�\(LúfO��QgY�

17

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インスタントン数が k の配位

E+B=0Aμ

AμASD

δAμ

ó 12: �,nMM Aµ�ÍêñÌþè� AASDµ h]SK�nZ� δAµ k�ãY��

2.3 ¤¤¤óóó¹¹¹¿¿¿óóóÈÈÈóóónnnqqq������fff

Uf�¤ó¹¿óÈóãkdDf��f�`hS�g�LïM�nU¡k�i�~W�F�\(o¤ó¹¿óÈópL k nMMn-goÍêñÌþjMMg��kj��LïM�k�ch�Ä�Y�ngYK���,nMM Aµ �

Aµ = AASDµ + δAµ (2.49)

n�FkÍêñÌþè� AASDµ h]SK�nZ� δAµ k�ãY�Sh��H~Y(ó12)�

δAµ L�UQ�p�

S =8π2k

g2+

∫d4x(δAµn�!�) + (Ø!�) (2.50)

hj��LïM�nM�p��ãY�h

Z =

∫[DAµ]e−S =

∑k

∫[DAASD

µ ]

∫[δAµ]e

− 8π2kg2

+···(2.51)

hj�~Y�SnU��)(WfÏP Yang-Mills �Ö�¿y�FhoX�kL�pc_nL ’t

Hooft nÖ�[12]gYL�ú�N δAµ næ�o^8k'jng�ÊÞo]��YcK��eWf�å�nȤâÇë��H~W�F:

Z instantontoy =

∑k

qk∫

[DASDµ ] =

∑k

qk∫MN,k

dvol� (2.52)

FW dvol MN,k nnê6jSMb�gY� q = exp(−8π/g2) o¤ó¹¿óÈópn�fÝÆó·ãë�Yj�a�¤ó¹¿óÈó��dûk�eY��n³¹È`h�F�LgM~Y� Bhoê6jSMb�å�oU�M�WfD~[�K��inÑéá¿nÍêñÌþMM��Xº�gwS�F�hDF¶Á��HfD~Y�ÿÖSn~~go Z oBznM�n_�kzcW~Y�Yj�a�d4Nks n-ko�¤ó¹¿ó

Èón-ÃnMn R4 k¢Y�M�LB�ng�]n�nzcLB�~Y�q��f�fs~Yh��8SnOLoBz��º L n'Mj�˹Sn±ke�f��Wf�logZ ∼ L4 hj�]nÔ��p�Ö�`YShgæ�W~YL�¤ó¹¿óÈóo±kD��hôkã�LãWOj�hDF'êLB�~Yng�a�ch%nSh�Wf�~Y�]n_�k��ch!Xj¹�Pn!���H~W�F�Xk�PL (x, y) ∈ R2 nin�@k

��Xº�gX(WF�hW~Y�Y�h�ÿÖ�M¢po

Z =

∫ ∞−∞

∫ ∞−∞

dxdy =∞ (2.53)

18

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x

y

ó 13: J = x2 + y2 oøz�nÞâ��O�

hjcfzcW~Y�]nK��k�¬¦¹�nàP�KgD�fÎ_U[~W�F:

Zε =

∫ ∞−∞

∫ ∞−∞

e−πε(x2+y2)dxdy =

1

ε� (2.54)

ε → 0 hY�hÎ_àP�Ö�UFShkjcf�]�kh�jcf Z �zcW~Y�gYK� 1/ε

oÎ_àP�e�_�n�¹�j R2 nbMh�FShLúe~Y�S�oiF�f�Kg����Î_U[_�Fj�XgYL��FYSW�sØQ�Y�ShLg

M~Y�(x, y) oz�n4@hW~W_L�Ý¢½óì'�e�f�fûnøz�n§�h�FShkW~Y:

{x, y}P.B = 1� (2.55)

]FY�h�J = (x2 + y2)/2 høOh�

{H,x}P.B. = y� {H, y}P.B. = −x (2.56)

gYK��J o (x, y) sbnÞânÏßëÈË¢ógYm(ó13)�gYK���n Zε o�

Zε =

∫ ∞−∞

∫ ∞−∞

e−2πεJdxdy (2.57)

h�ÞânÏßëÈË¢ó J L'MOj�h ε nÍ�g �Y��FkM�WjUD�hDFSh`h�H~Y�ÊÞoSn¹Õgû�±ke�~W�F�¤ó¹¿óÈókq�;W~Yh�ÊoBzoÛ!Cg x1, x2, x3, x4 hB�~YK��Ý¢½ó

ì'�{x1, x2}P.B. = 1� {x3, x4}P.B. = 1 (2.58)

]nÖo¼í�hWfMøz�h�FShkW~W�F�Sn�Fk R4 kMøz�nË �D��h�ê6k¤ó¹¿óÈónâ¸åé¤z�MN,k k�øz�nË Le�ShLå��fD~Y�MN,k n�¹�h�~W�F�Y�h�rhd Aµ(x) hDF¤ó¹¿óÈóMML�~�~YK�� R4

nÞâ�Y�h�A′µ(x) hDF%�n¤ó¹¿óÈóMML�~��MN,k n%n�¹kj�~Y�gYK��R4 nÞâoê6kMN,k kÍM~Y� ]Sg�R4 n (x1, x2) sb�ÞYÞâLMN,k

19

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k�MwSY\(��H�S�nÏßëÈË¢ó� J1, R4 n (x3, x4) �ÞYÞâkþÜY�ÏßëÈË¢ó� J2 h|vShkW~W�F�]�^�kþY�Í�� ε1,2 h|vh�

e−2π(ε1J1+ε2J2) (2.59)

hDFÍ��dQ�ShkW~Y�~_�'ß²ü¸Û�MN,k nÛkj�~Y�!Xn_� SU(2) ��H�h�σ3 g��U

��²ü¸ÛkþÜY�ÏßëÈË¢ó K ��H�Í� a �dQ~W�F�Y�h��p¢pn©k aK hDFàP�D��Shkj�~Y��,n SU(N) go�þÒL�n²ü¸Û diag(a1, . . . , aN ) kþWf�ÏßëÈË¢ó

K1, . . . ,KN ��H��p¢pn©k∑

i aiKi hDFàP�D��ShkW~Y�hDF�Qg���L�H_D¤ó¹¿óÈónq��f!�o!n�Fj�ngY:

Z instantonε1,ε2;ai =

∑k

qkZN,k� FW ZN,k =

∫MN,k

e−2π(ε1J1+ε2J2+∑i aiKi)dvol� (2.60)

����SWDSh��D~W_L�P@o�¤ó¹¿óÈó�¢qD��h q `Q �Y��(x1, x2) sb�nÒKÕÏL'MDh ε1J1 `Q �Y�� (x3, x4) sb�nÒKÕÏL'MDh ε2J2 `Q �Y�� ²ü¸Û'L'MDh Ñéá¿ ai kÜXf

∑i aiKi `Q �Y��hD

F�¤ó¹¿óÈóL¢qBc_pBDkq��f�kqJFhY��k�H�Shnúe��ch�!Xj!�kjcfD~Y�J1� J2� Ki �wSboP@oXk¿�ÝÆó·ãëkj�`QgY�]����k SU(2) n 1-¤ó¹¿óÈón4�kºK�f�~W�F�â¸åé¤z�oHk

R4 × R4/Z2 `hDD~W_�rhd�n R4 o¤ó¹¿óÈón-à (x1, x2, x3, x4) �Ñéá¿WfDf�u_d�n R4 o²ü¸n�M�B��YL� H0i h ¤ó¹¿óÈónµ¤º ρ �B�[_�ngW_� u_d�n R4/Z2 �� pu_d (z, w) � (z, w)→ −(z, w)g���W_�nh�FShkW~W�F�²ü¸ÞâoXk (z, w) LSU(2) n�!ChþgYK�

(z, w) 7→ (eiaz, e−iaw) (2.61)

hÍM~Y��¹� x1, x2 sbn ε1 ÞâJ�s x3, x4 sbn ε2 Þâo

(x1 + ix2, x3 + ix4, z, w) 7→ (eiε1(x1 + ix2), eiε2(x3 + ix4), ei(ε1+ε2)/2z, ei(ε1+ε2)/2w) (2.62)

h\(W~Y�SSg�(z, w) xn\(o�BznÞâL Hµν �÷\�ShK��� (2.36) kB���� g �ÛY�Å�LB��g L (z, w) ∈ C2 nÒ¦è�gBc_ShK��X~Y�åK��þÜY�ÏßëÈË¢óo

ε1J1 + ε2J2 + aK =ε12

(x21 + x2

2) +ε22

(x23 + x2

4)

+(ε1 + ε2)/2 + a

2|z|2 +

(ε1 + ε2)/2− a2

|w|2 (2.63)

gYm� Y�h

Z2,1 =1

2

1

ε1

1

ε2

1

(ε1 + ε2)/2 + a

1

(ε1 + ε2)/2− a (2.64)

20

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hj�~Y�H-n 1/2 o R4/Z2 n /Z2 K�e~Y�YSW`QSn��P�nãÈ�WfJM~W�F�1/(ε1ε2) oBzn±n'MUgYK��±

�'MOY�_�k ε1,2 → 0 hW~Y�Y�h¬¦¹M� (2.64) nÎ_'LªOj�~YK��a o�ZhWf��LOLjD�FkW~Y� Y�h�Bzn±L'MDuPgo�¤ó¹¿óÈóL��B�Shk��¹�oXMBzSMB_�

ε1ε2 logZ2,1 ∼1

2

1

(ε1 + ε2)/2 + a

1

(ε1 + ε2)/2− a ∼ −1

2

1

a2(2.65)

hj��QgY�_`W |a| � |ε1,2| hW~W_�

2.4 MMM���nnn@@@@@@���

Uf�SU(2) n4�n Z instanton ���Y�ko�Sn¿PgM2,2, M2,3, . . .nM����gM�pDDngYL�]��ô¥LFnojKjK'gY�]Sg�Duistermaat-Heckman n@@�l�hDF�n�dKD~Y�(SnÀn�¹ns0o�YÑø[13]�ÂgnSh�)Snl�o��¦��L��W_D�Fj�ÏßëÈË¢ó��p¢pn©kn[_�nn

M��_aiS�kúefW~F�ngY��j!Xj�hWf��!C�b��H~Y�ï¦�−π

2 < θ < π2 � L¦� 0 < ψ < 2π hWf�b� � cos θdθdψ hW~Y�L¦¹�ÞânÏßëÈ

Ë¢óo H = sin θgY�]Sg

Z =

∫∫e−2πεH cos θdθdψ (2.66)

��H~W�F�S�o�gMf�

=e−H(θ=π/2)

ε− e−H(θ=−π/2)

ε(2.67)

hj�~YL�S��!n�FkøM~Y2 :

=∑

p=�u,Wu

e−H(p)

(p gnÞâÒ)� (2.68)

Duistermaat-Heckman nl�o�S�L�,kj�_d�hDF�ng�M L 2n!CnÑ�Kjøz�g�H LÏßëÈË¢óhWf]nkA���MwSW�A�nú�¹ p LdËWfD�hY�h�

Z =

∫e−2πεHdvol =

∑p

e−H(p)∏ni=1 θi,p

(2.69)

L��Ëd�hDF�ngY�_`W��ú�¹ p n~��go�A�o¹ p ~��nÞâkj�~YK��R2n � n �n R2 k�Qf]�^�nbLÒ¦ θi,p (i = 1, . . . , n)gÞâWfD��hDF�FkW~W_��Frhd����~W�F� M = R4 hWf�(x1, x2) �Ò�¦ ε1 g�(x3, x4) �Ò�¦ ε2 g

ÞYSh��H�h�ÏßëÈË¢óo

J =ε12

(x21 + x2

2) +ε22

(x23 + x2

4) (2.70)

2�p¢pn�M�nl�Lú�¹��n�.gB�hDFno�� [14] K�fs~W_�

21

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u v

u=v′=0 u′=v=0v′ u′

↓u v

u=v=0

ó 14: R4/Z2 nÖíü¢Ã×�u = v = 0 nhS�k�u′ = 1/v′ gÑéá¿ØQU�_ S2 �?eW_�

gY�l��i(Y�ko�ú�¹�¢Umpj�~[�L�ÿÖ]�o�¹`Qg�]SgnÞâÒo ε1 h ε2 gYm��cf ∫ ∞

−∞

∫ ∞−∞

∫ ∞−∞

∫ ∞−∞

e−2πJd4x =1

ε1ε2(2.71)

hj�oZgYL�S�oXk¬¦¹M�gY�S�oH{i�c_M�n R4 è�gY�Sn��� R4/Z2 ki(Y�koiFY�p�DgW�FK? rhdOLo���oÑ�Kj

�ØSkþWfn���ËdngYL�R4/Z2 oHïLhLcfD~Y� hL��ã�Y�º�Öíü¢Ã×hDFÍ\�W~Y�~Z�(z, w) ∈ C2 ' R4 �§�hWf�(z, w) 7→ −(z, w) hDF Z2

\(grcfD~YK��u = z2, v = w2, t = zw L Z2 \(g gY�S��o uv = t2 ��_W~Y�u = v = t = 0 nB_�L^8k�cfD~Y�]Sg��c_hS�k S2 �îW¼�gÑ�KkW~Y(ó14)� S2 n�ugo@@�ko

(u, v′) FW uv′ = t� S2 nWugo@@�ko (u′, v) FW u′v = t LoD§�kjcfD~Y�z nÞâÒL (ε1 + ε2)/2 + a� w nÞâÒL (ε1 + ε2)/2− a gW_K�� �ugnÞâÒo

(ε1 + ε2) + 2a,−2a (2.72)

WugnÞâÒo2a, (ε1 + ε2)− 2a (2.73)

hj�~Y�]Sg�Duistermaat-Heckman nl��dKFh�∫e−2πHdvol =

e−H�u

((ε1 + ε2) + 2a)(−2a)+

e−HWu

(2a)((ε1 + ε2)− 2a)(2.74)

gY�Öíü¢Ã×Y�~Hn R4/Z2 nM��å�ko�S2 L�UOj�uP�h�~YL�]FY�h�Pn H oUk[��¹kLcf¼íkj�~Yng�

→ 1

((ε1 + ε2) + 2a)(−2a)+

1

(2a)((ε1 + ε2)− 2a)(2.75)

=2

((ε1 + ε2) + 2a)((ε1 + ε2)− 2a)(2.76)

=1

2

1

((ε1 + ε2)/2 + a)((ε1 + ε2)/2− a)(2.77)

22

Page 25: arXiv:1108.5632v1 [hep-th] 29 Aug 2011 · IPMU11-0147 ÛÛÛ!!!CCC†††üüü‚‚‚ ÖÖÖhhh„„„!!!CCCqqqbbb444 ÖÖÖnnn ˛˛˛pppjjj¢¢¢´´´ ¸Ý Õ„ 277-8583

hj��H{in����þW~W_�M2,2 hSnM�kY�ko�R4 nM���Q�p�DgY� Y�h�(2.64) go��`c_

�n��

Z2,1 =1

ε1ε2

1

((ε1 + ε2) + 2a)(−2a)+

1

ε1ε2

1

(2a)((ε1 + ε2)− 2a)(2.78)

h��k�rW_Shkj�~Y�

2.5 ���ÍÍͤ¤¤óóó¹¹¹¿¿¿óóóÈÈÈóóó������

åMÀgoR4 × R4/Z2n¬¦¹M��^8k~��OiOLc_�QgYL�Sn¹ÕnoDhS�o�MÀg9ËW_ ADHM Ë�hD���[�h��,n N , k kþWf��Lúe�hS�gY[15, 16, 17, 18]� �úoãWDngMa�hogM~[�L�ðò�`Q¬�W~W�F�SU(2) 1-¤ó¹¿óÈón4��Öíü¢Ã×o uv = t2 n�cfD�hS�k S2 �Ë�¼�~W_�S�o�ADHM Ë�gDFh��cfD�¶Áo

A1B1 +A2B2 = 0�∑i=1,2

|Ai|2 − |Bi|2 = 0 (2.79)

g (Ai, Bi) 7→ (eiθAi, e−iθBi) hDF����WfD_�n��

A1B1 +A2B2 = 0�∑i=1,2

|Ai|2 − |Bi|2 = r2 (2.80)

hH�ShkøSW~Y�Bi = 0 hY�h�

|A1|2 + |A2|2 = r2 (2.81)

hDF S3 LB�~YL�S�� Ai 7→ eiθAi grc_nL�o�¼�` S2 kjcfD_�QgY�S2n�uWuo�

(A1, A2) = (r, 0)� (A1, A2) = (0, r) (2.82)

k]�^�þÜW~Y�M��h�~Yh�SU(2) Û�Y�hÿÖÞcfW~D~Y

(A1, A2) = (r, 0) 7→ (eiar, 0) (2.83)

L�S�oiF[ (A1, A2) ' eiθ(A1, A2) h���Y�ngW_K��

' (r, 0) (2.84)

hjcf�eiθ nÄògú�U�fD�Shkj�~Y�(A1, A2) = (0, r) n4���ØgYm�WKW�A1, A2 n!¹L 0 gjDh�SU(2) ÞâY�h

(A1, A2) 7→ (eiaA1, e−iaA2) (2.85)

hÛW~YK��eiθ gCk;YShLúe~[�� gYK���o�SU(2)²ü¸ÞâJ�s�R4

n (x1, x2) sbh (x3, x4)sbnÞâLin�Fk eiθ ÞâgøºgM�KhDFShgY��,ko�(2.46)�Ýc_~~�(2.47) nóº� r2 kY�ShLMN,k �Ñ�KkY�Shkj

�~Y�ú�¹o k = 1 nhMh�Ø�Bi = 0 nhS�kB�~Y� ú�¹��ØkSU(N) ²ü

23

Page 26: arXiv:1108.5632v1 [hep-th] 29 Aug 2011 · IPMU11-0147 ÛÛÛ!!!CCC†††üüü‚‚‚ ÖÖÖhhh„„„!!!CCCqqqbbb444 ÖÖÖnnn ˛˛˛pppjjj¢¢¢´´´ ¸Ý Õ„ 277-8583

s

ó 15: äó°ó Y n± s k¢Y�Uw AY (s)�³wLY (s)�ón4�o AY (s) = 2� LY (s) = 3

hj��s L Y n�kB�pBDoUw�³wo kj��

¸ÞâJ�s�R4 n (x1, x2) sbh (x3, x4)sbnÞâLin�Fk U(k) ÞâgøºgM�Kgz~cfD~Y� SU(N) ²ü¸Þâ� diag(a1, . . . , aN ) g�R4 nÞâ� ε1,2 gY�ShkY�h�ú�¹oäó°ón N �D Y1, . . . , YN g�±npL�� k gB��Fj�ngéÙëØQLjU�~Y�]n��øºk(D�U(k) Þâo�þÒ��L

ai + jε1 + kε2 (2.86)

FW 0 ≤ j < (Yin�np), 0 ≤ k < (Yin j �înØU) hj�~Y�gYK��MN,k �Öíü¢Ã×W_�ngnM�o�l� (2.69) k��ú�¹ p =

(Y1, . . . , YN ) n³WRkj�~Y�CnMN,k gnM�o��Pn e−H n�Lhf 1 kjcf�

ZN,k =

∫MN,k

e−2π(ε1J1+ε2J2+aiKi)dvol =∑

p=(Y1,...,YN )

∏i

1

θi,p(2.87)

hj�~Y��ú�¹gnÒ¦n�� θi,p ��LY�h�

ZN,k =∑

Y1,...,YN

N∏i,j=1

∏s∈Yi

(−LYj (s)ε1 + (AYi(s) + 1)ε2 + aj − ai)−1

×∏t∈Yj

((LYi(t) + 1)ε1 −AYj (s)ε2 + aj − ai)−1 (2.88)

hDFwS�j�g�H��~Y� FW�∑ai = 0hW� Y1, . . . , YN oäó°óg�±npL��

k �gB��FkW~Y�AY (s)�LY (s) oäó°ó Y n± s nUw�³wh����ó 15 n�Fkz�~Y�MN,k o 4Nk !CgYK���Íko 2Nk �ÞâÒL&�gD�oZgYng�]��º�Wf�fO`UD� N = 2 n4�o (a1, a2) = (a,−a) hY�ShkW~Y� k = 1 hY�h�(Y1, Y2) = ( , 0) K = (0, )gY�l�nD���[Ö�j��U�Y�h�(2.78) n���]�^��þY�nL�K�h�D~Y�

k = 2 hY�h�(Y1, Y2) o ( , 0), ( , 0), ( , ), (0, ), (0, ) n���B�~Y��Hp( , 0) K�n�o[

−4ε21ε2(ε1 − ε2)a(2a+ ε1)(2a+ ε1 + ε2)(2a+ 2ε1 + ε2)]−1

(2.89)

hj�~Y�Ön���5cf��W~Yh�

Z2,2 =(8(ε1 + ε2)2 + ε1ε2 − 8a2)

ε21ε22((ε1 + ε2)2 − 4a2)((2ε1 + ε2)2 − 4a2)((ε1 + 2ε2)2 − 4a2)

(2.90)

24

Page 27: arXiv:1108.5632v1 [hep-th] 29 Aug 2011 · IPMU11-0147 ÛÛÛ!!!CCC†††üüü‚‚‚ ÖÖÖhhh„„„!!!CCCqqqbbb444 ÖÖÖnnn ˛˛˛pppjjj¢¢¢´´´ ¸Ý Õ„ 277-8583

hj�~Y�j�`K��SWDP�kj�~W_L��FYSW`Qi���¹�h�`Wf�~W�F�¤ó

¹¿óÈó�dn4�o (2.78) gW_�¤ó¹¿óÈóu_dL`Oâ�fD�p�1-¤ó¹¿óÈóã��dÍm��[�p2-¤ó¹¿óÈóãLgM~Y�gYK�� â¸åé¤z�n�ègo�JJ�] M2,2 ∼ (M2,1)2/Z2 gW_� gYK��M2,2nM�o�JJ�]M2,1nM�n{|�Wg�H��f�îo¤ó¹¿óÈó�dL�DkÑeD_�nø�\(K�O�oZgY�]����kºK�~W�F�|a| � ε1,2 hW_hW~Y�Y�h� (2.90) kB� Z2,2 � (2.78)2 g�H��� Z2,1

2�]�^�(ε1ε2)−2 nàPLB�~Y�Yj�aBzn±nµ¤ºn�WnàPLB�~Y�S�o�Bz�u_diSLÕDfD�M�K�e~Y��¹�ø�\(n¹�o

Z2,2 −1

2Z2,1

2 =20a2 + 7ε21 + 16ε1ε2 + 7ε22

ε1ε2((ε1 + ε2)2 − 4a2)((2ε1 + ε2)2 − 4a2)((ε1 + 2ε2)2 − 4a2)(2.91)

hjcf� (ε1ε2)−1 n�LøºW� (ε1ε2)−1 nÄ�WKB�~[��S�o�Bzn�¹�ø�\(¹hWf�]Sk�dn¤ó¹¿óÈóLÑeDfM_hMkJS�Ä�LB��hDFShkjcfD~Y�%n�D¹�W~Yh�

Z instantonε1,ε2;a = 1 + qZ2,1 + q2Z2,2 + · · · (2.92)

hW~Yh� qk n�o¤ó¹¿óÈóL k �B�~YK� (ε1ε2)−k nàPLB�~YL�þp�hcf��h

logZ instantonε1,ε2;a =

q

ε1ε2

1

2

1

((ε1 + ε2)/2 + a)((ε1 + ε2)/2− a)(2.93)

+q2

ε1ε2

20a2 + 7ε21 + 16ε1ε2 + 7ε22((ε1 + ε2)2 − 4a2)((2ε1 + ε2)2 − 4a2)((ε1 + 2ε2)2 − 4a2)

+ · · · (2.94)

→ q

ε1ε2

−1

a2+

q2

ε1ε2

−5

16a4+ · · · (2.95)

hjcf�q nyMkKK��ZBzSMnàP (ε1ε2)−1 LÎÞg�Shkj�~Y�S�o�k¤ó¹¿óÈóLø�\(Y�no�Bz¹kJDfgB��hDFØP�IHfD��QgY�(2.95) go |a| � ε1,2 hDFuP�h�~W_�

3 ���!!!CCChhhÛÛÛ!!!CCCnnn¢¢¢ÂÂÂ

3.1 þþþÜÜÜ¢¢¢ÂÂÂ

Uf��F]�]�oX�kU��c_nKJØ�gojDKh�D~YL�M�Àgo��!Cqb4�Ög�¶K |∆〉 K���U�� Verma hþ��Hf�]n-g³ÒüìóȶK

L1|∆, λ〉 = λ|∆, λ〉� L2|∆, λ〉 = 0 (3.1)

���

〈∆, λ|∆, λ〉 = 〈∆, 1|λ2(L0−∆)|∆, 1〉 = 1 +λ2

2∆+

λ4(c+ 8∆)

4∆((1 + ∆)c− 10∆ + 16∆2)+ · · · (3.2)

25

Page 28: arXiv:1108.5632v1 [hep-th] 29 Aug 2011 · IPMU11-0147 ÛÛÛ!!!CCC†††üüü‚‚‚ ÖÖÖhhh„„„!!!CCCqqqbbb444 ÖÖÖnnn ˛˛˛pppjjj¢¢¢´´´ ¸Ý Õ„ 277-8583

���W~W_�~_MÀgo�¤ó¹¿óÈónq��f��H~W_�Bz��˹Sn±kD��ÿ��

k�BznÞâJ�s²ü¸nÞâkþWf�fÝÆó·ãë�e��

Z instantonε1,ε2,a =

∑k

qk∫M2,k

e−2π(ε1J1+ε2J2+aK)dvol� (3.3)

��H~W_�FW N = 2 n SU(2) n4���H~Y�S���5cf��Y�h�

= 1 +q

ε1ε2

2

(ε1 + ε2)2 − 4a2

+q2

ε21ε22

(8(ε1 + ε2)2 + ε1ε2 − 8a2)

((ε1 + ε2)2 − 4a2)((2ε1 + ε2)2 − 4a2)((ε1 + 2ε2)2 − 4a2)+ · · · (3.4)

hj�~W_�u_dnP�(3.2), (3.4) ��Ôy~Yh�,��~go

λ2 =q

(ε1ε2)2� ∆ =

1

ε1ε2((ε1 + ε2)2

4− a2) (3.5)

hY�p�ôW~Y�,�o�U�k

c = 1 + 6(ε1 + ε2)2

ε1ε2(3.6)

hY�h�D~Y�SS~gooX�n���cf3dnp�¢ÂØQ_`Qn�Fk�H�K�W�~[�L�,Û��,��hi�i���Y�h��npnþÜg (3.2) h (3.4) LDd~g��ôY�hDFShL�K�~Y[19, 20]�MÀg¬�W_ (3.2) h (3.4) ���Y�¢ë´êºàoMÀhM�ÀkøM~W_ng�/^!n����Wf�fO`UD� K��g��h 3-¤ó¹¿óÈónB_�K��6b�kj�~Yng�×í°éà�øD_{FLDDgW�F�Mathematica g�ÅW_�nL ×ì×êóÈÚü¸n ancillary files nhS�knDfB�~Yng�]��T§O`UD�Ï 〈∆, λ|∆, λ〉 o�!Cni�K�úfe_�ngY�Ï Z instanton

ε1,ε2,a oÛ!Cni�K�úfe_�ngY�S��L�ôY�hDFþa��in�Fk�ãY�poDgW�FK� ̹h���¹Õo$cfD~Y�³ÒüìóȶKnwUn�Wol� (1.51) gwS�k�H��fD~YW�¤ó¹¿óÈón�M¢pol� (2.88) gwS�k�H��fD~Y� gYK��ÌooØ�f�S��nl��h�g<�W�F�hY�ShLúe���]n�Fj<�Lâkå��fD~Y[21, 22, 23]� ~_�â¸åé¤z�k\(Y�!P!Cãp�Ë�Y�hDFno�pfgo~Uf�hþÖhDF��Î�jY{igY�_hHp�C2/Γ FW Γ ⊂ SU(2) hDFz�gn¤ó¹¿óÈónâ¸åé¤z���H�h�Γ �n Kac-Moody ãpnhþLB����hDFnL [24] g:U�fD~Y� gYK���I�oΓ gr�ÿ��k ε1,2 �e�_ê6já5kjcfDf�~Uf�hþÖK�n<�L]�]�zhU��hDFBgY[25]� S���dnKÕo�ó16 g�Hp�pfn-kY~cf¢Â��ãW�F�hDF�ngY� WKW�Õn��opfgojO&�ÖgYng�³ÆgojD&�Ön�L��NWf�in�FkSn¢ÂL�ãgM�K�hDFn�¬�W_Dh�D~Y�

26

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6次元 N=(2,0) 理論弦理論(いいかげん)

数学(厳密) ?!!, !|!, !" Z instanton

!1,!2,a

2次元の量 4次元の量

ó 16: �!CnÏhÛ!CnÏhn¢Â

3.2 ~~~ZZZ���!!!CCCxxx

��n�ãW_D�o

〈∆, 1|λ2(L0−∆0)|∆, 1〉 =∑k

qk∫M2,k

e−2π(ε1J1+ε2J2+aK)dvol (3.7)

hDF�ngW_��!ChÛ!CnÏ�Ô�WfD�hDFåMk� æºoâÕ¢pn�Wnb�WfJ��óºoXj�M�gY�S��iFÔ�Y�p�DgW�F? ]n_�k�óº�å�n�Fk�HôWf�~W�F�ψk(t) ��M2,k n¢pg

ψk(t) = e−π(ε1J1+ε2J2+aK) (3.8)

g��~Y�wS�ko�k = 1 go

ψ1(x1, x2, x3, x4, z, w) = e−π(ε1(x21+x22)+ε2(x23+x24)+(ε1+ε2

2+a)|z|2+(

ε1+ε22−a)|w|2) (3.9)

g�¬¦¹�n¢pgY�Y�h�óºo

=∑k

qk∫M2,k

|ψk(t)|2dvol (3.10)

gY� φk(t)��M2,k �ÕDfD�ÏP�f��PnâÕ¢p`h�cf |φk〉 høM~W�F:

|φk〉 ∈ H(M2,k) (3.11)

_`W H(M2,k) oM2,k nâÕ¢pnjYÒëÙëÈz�gY�Y�h�óºoU�k

=∑k

qk〈φk|φk〉 (3.12)

høQ~Y�]Sg�U�k

|φ〉 = |φ0〉 ⊕ |φ1〉 ⊕ · · · ∈ H(M2,0)⊕H(M2,1)⊕ · · · (3.13)

hDFÙ¯Èë��H�ÏßëÈË¢ó H � H(M2,k) gú$ k ��d�Fj��P`hY�h�¢Â�o

〈∆, 1|λ2L0−∆|∆, 1〉 = 〈φ|qH |φ〉 (3.14)

27

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hj�~Y�|∆, 1〉 �

|∆, 1〉 = |ψ0〉 ⊕ |ψ1〉 ⊕ |ψ2〉 ⊕ · · · ∈ V0 ⊕ V1 ⊕ V2 ⊕ · · · (3.15)

hDFU�LB�~W_�FW Vk o!pL k n��g�]Sgo L0 −∆ = k jngW_�Vk oP!CgW_: dimVk = pk _`W pk o k �ctpn�hWføO¹Õnp��¹g

H(M2,k) oS6!P!CgY�WKW�U�Kn�sg

Vk ⊂ H(M2,k) (3.16)

hê6kË�¼~�fJ��]nË�¼�n�hg

|ψk〉 = |φk〉 (3.17)

hjcfD�j�p���n¢Â�oê6k�D~Y�WKW�¤ó¹¿óÈónâ¸åé¤z��ÕO�PhDFnoiFDFShgW�FK? S�

o�Û!Cn²ü¸�ÖgjO��!Cn²ü¸�Ö��H�hê6kþ�~Y� äóßëº4� R4 gjO R5 = R4 × Rt g�H~W�F�ØQ H_�¹��B�`h�FShkW~Y:

(x1, x2, x3, x4, t) ∈ R5� \(o ∫1

2g25d

trFµνFµνdtd4x (3.18)

gY� �!CnMM�rhd Aµ(x1, x2, x3, x4, t) �h�h��B; t Îk R4 nMM

Aµ(xi; t) = Aµ(x1, x2, x3, x4, t) (3.19)

L�~cfD�h�H~Y�B; t = t1 gn Aµ(xi; t1)¤ó¹¿óÈóp� k hY�h�¤ó¹¿óÈópotpgYK��ÖnDdnB; t = t2 g�¤ó¹¿óÈópo k kj�~Y� ÏP�fkY�_�kLïM��Y�Sh��H~Yh��B;g¨Íë®ü�u�kY�_�k��B; t gAµ(xi; t) LÍêñÌþgB��FjMML�ch�Ä�L'MOj�~Y(ó17)� �B;gn¨Íë®üo ∫

1

2g25d

trFµνFµνd4x ∼ 8π2k

g25d

(3.20)

hj�~Y�¨Íë®ühêÏoI¡gYK�� ¤ó¹¿óÈó�dLêÏ 8π2/g25d n�Pk�H

��QgY�Uf�ÝKj¤ó¹¿óÈóp k nÍêñÌþão 4Nk �nÑéá¿ si gz~cfD�oZ

gYK��B; t k�XWf si(t) Lz~�~W_:

si(t) : Rt →MN,k� (3.21)

gYK���!Cn²ü¸�Ög�LïM�k��Ä�n'MDè�o�¤ó¹¿óÈónâ¸åé¤z��ÕOÏP�f��PnKÕgIH���ShL�K�~W_�WKW���oÛ!Cn²ü¸�Ö��HfD~W_��!Cn²ü¸�ÖK�Û!Cn²ü

¸�Ö�dO�!Xj¹Õo��dn¹���³óѯÈ� Y�ShgY��Hp�t ¹�� [0, L]

nÚ�kWfW~D~W�F�Y�h��Ö� L ��^8kJJMj¹±üëg��P�o�t ¹��

28

Page 31: arXiv:1108.5632v1 [hep-th] 29 Aug 2011 · IPMU11-0147 ÛÛÛ!!!CCC†††üüü‚‚‚ ÖÖÖhhh„„„!!!CCCqqqbbb444 ÖÖÖnnn ˛˛˛pppjjj¢¢¢´´´ ¸Ý Õ„ 277-8583

t = 0 t = 1 t = 2 t = 3

ó 17: �!C²ü¸�ÖhÍêñÌþãn�

:%Y�ShogMZ��êÛ!Cn�Ökj�~Y�^8k�ôko��!Cn\((3.18) kJDf�Fµν L t ¹�k�WjQ�p�dt M��WfW~cf∫

1

2g24d

trFµνFµνd4x (3.22)

hgM�hDF�QgY�FW�1

g24d

=L

g25d

� (3.23)

�,k�Sn�Fj³óѯÈ��Y�h Kaluza-Klein �PhDF�nLþ�~Y��!CnBzkêÏnjD�PLBc_hW~W�F�Y�h�¨Íë®ühKÕÏo

E2 = ~p 2 + p25 (3.24)

��_W~Y�_`W ~p o R3 ¹�nKÕÏg�p5 o,�¹�nKÕÏhW~Y�,�¹�� L

k³óѯÈ�Y�h�ÏP�f�koKÕÏoâÕ¢pnMø ei~p·~x gYK��2πp5L otpgjDhDQ~[��]�� k hY�h�p5 = 2πk/L hj�~Y�Y�h�Û!CnË4K�o�

E2 − ~p 2 =

(2πk

L

)2

(3.25)

hjcf�êÏL 2πk/L n�PLþ�~Y�S�o L L�UQ�pi�i�ÍOj�ng�L L�UD{i,�kOOj��QgY�²ü¸�Ökq�;W~Yh���oÛ!CnP��pL g4d n²ü¸�Ö��H_Dng�

�!Cn²ü¸�Ö� (3.23) g�~�wU L nÚ�k³óѯÈ�W~Y ó18� Y�h�Snûn�M¢po�

Z = 〈Φ|e−LH |Φ〉 (3.26)

g�H���Shkj�~Y�FW Φ o t = 0 J�s t = L gn�LaöK��~�¶Kg�H ot ¹�xnB�zUn��PgY�

3.3 UUU���kkkmmm!!!CCCxxx

SS~go^8k�,�j�ßgW_L��!Cn²ü¸�ÖhWf�Xk²ü¸4`Qn�ÖgjO��'�þð SU(N) ²ü¸�Öh�p���n�Ö���LaögJ�n�þð�Ýd�n

29

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L

1,2,3,4t 境界条件 境界条件

!!, 1|

λ2 logR4 R4

|!, 1!x

ó 18: æ: �!C²ü¸�ÖK�Û!C²ü¸�Ö�dO��ó: ó8��²��!Cqb4�Ön³ÒüìóȶKnÎëà�

��Fh�Û!Cn�ÖhWf� N = 2 SU(N)²ü¸�ÖhDF�nkj�~Y�Sn�ÖkôkNekrasov n ε1,2 bhDF�n� H�h��M¢pL��nqcfM_q��f!�h�ôY�ShLå��fD~Y�D~��o N = 2 ��HfD~Y�Y�h�� |Φ〉 o |φ〉 h�ôW~Y� Uf�S�oôkqb4�Ön³ÒüìóȶK |∆, 1〉 h�ôY�ngYL�¶Á�Ôy�h�!C�Ú�kJDfÛ!C�Ö�dO�nh� ³ÒüìóȶKnÎëà���Y�¶Áo{h�i�XgYm(ó18)� �!C¹�nwU L o

L ∼ 1

g24d

∼ log q (3.27)

gW_L��ñn*Eo ∼ log λ g�(3.5) g�_þÜ¢ÂK���U����gY�gYK���!C�'�þð SU(2) ²ü¸�Ön t ¹�nzUn��P H o��!Cn

�Ön��P L0 h���YyMgY��XShgYL�ó 18 n,�¹��Yj�a L ¹�h�ó 8 n�ñn log λ ¹�o���YyMgY� �!Cn�Ön�¹�oYyf SO(5) ÞâgI¡gY:

R4n�¹�SO(5) Þâ−−−−−−−→,�¹� (3.28)

�¹g�qbÛo L0 `QgjO�Ln �+��yk L±1 o�ñn R ¹���ñnS1¹�kÞYÛ�+�gD~Y�i��ko��ñn-n�¹�oI¡j�QgY�

�ñn R ¹� L1 Þâ−−−−−→�ñnS1¹� (3.29)

WKW��!C�Ön,�¹� =�ñn R ¹� (3.30)

gY�S���D���[�h��!C�Ön R4 ¹�o��ñnz�¹�xÞâU[�ShLúe�hDF;5kó�~Y�]n_�ko�m!Cn�ÖLÅ�gY� ��k�!C�'�þðSU(2) ²ü¸�Ö��HfD�d��`c_L�]�om!CnB��Ö��h R n�hk³óѯÈ�W_�ngB�h�FyMgB��hDFShgY�m!Cn�Ö��hk³óѯÈ�W_ngB�p�YPMkJU�DW_�Fk�tp k kþ

WfÍU 2πk/R n Kaluza-Klein �PLg�oZgYL�ºKk��!C�Öko¤ó¹¿óÈó�PLB��]nÍUo 8π2k/g5d gW_�gYK���!C�Ön¤ó¹¿óÈó�Po��om

30

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!C�Ön Kaluza-Klein �PgB��

R =g2

5d

4π(3.31)

gB��hDFShL�K�~Y�gYK����om!Cn�Ö��S1 × [0, L]× R4 g�HfD_ngY�Snû�ã�Y�

�k�

• SSg�S1 L^8k�UDhY�h�[0, L] × R4 g�!C²ü¸�Ö��U�k L ��UDhWf�P@ R4 gÛ!C²ü¸�Ö��H�hDF¹ÕLB�~Y�

• �¹� R4 ¹�koÝÆó·ãë ε1(x21 + x2

2) + ε2(x23 + x2

4) �e�fD�ng�ε1,2 L'MDh�R4 ¹�n±�^8k�UOY�ShLúe~Y�Y�h�S1 × [0, L] g�!C�Ö��H��hDFShkj�~Y�

Sn���nU¡Õ�Ô�Y�Shk���

Z instantonε1,ε2,a = 〈∆, λ|∆, λ〉 (3.32)

L�����QgY�go�Snm!C�Öoj�gW�FK? H{i��!C²ü¸�Ö�wU L nÚ�k³óÑ

¯È�Y�p�JJ�]Û!C²ü¸�Ökj�Sh�¬�W~W_K���Øk�m!C²ü¸�Ö��h R n�hk³óѯÈ�WfD�ngojDK�h�ôko�H~Y�WKW�]FY�h(3.23) h�ØkWf�

1

g25d

∝ R (3.33)

hjcfW~D�(3.31) ho~c_O�kjcfW~D~Y�Ma�h (3.31) �úY�Fjm!C�Öo���nmLgo�m!C N = (2, 0) �Ö

h|p�fD~Y�S�n'êoocM�ho�KcfD~[�L�²ü¸�Ön Fµν =

∂µAν − ∂νAµ + [Aµ, Aν ] nÿ��k�Fµνρ = ∂µBνρ + · · · hDF4LB��

Fµνρ = Fµνρ FW Fµνρ =1

6εµνραβγFαβγ (3.34)

L��ËcfD���Fj�n `h���fD~Y�Snm!C�ÖoD�D�j!Cgn�þð²ü¸�Önª�`h���fJ���Ñ;zk�vU�fD~Y (pf��Qn~h�o[26, 27] I�ÂgnSh)��!Cn�'�þð²ü¸�Öo�P��p g5d �'MOY�h�¤ó¹¿óÈó�PLýO

j���h R ∼ g25d n�h���Wfm!Cn�ÖkjcfW~F�QgYL�Sn��o [28] k

:�U�f 1997 tk [29] LoX�k�XW~W_� S�o 1995 tK��ãU�Ë�_��&�Öo M �ÖgB� hDF��n�°gY� ���Type IIA �&�Öo10!Cn�ÖjngYL�&�ÖnP��p�'MOY�h�D0-ÖìüóhDF�PLýOj���h���Wf11!Cn�ÖkjcfW~D~Y� Type IIA �&ko D4-ÖìüóhDF�!Ckr�Lc_iSLB��SnÍ\kh�jcf�M�Ön M5-ÖìüóhDF6!Ckr�Lc_iSL��U�_�hkûMØDfD�¶Kkj�~Y(SnB_�ns0o�YÑø [30] I�ÂgnSh)� D4-ÖìüóL N �ÍjcfD�h�]nko�!C�'�þð SU(N) ²ü¸�ÖLO�~Y�S��

31

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7P�kY�h�M5-ÖìüóL N �ÍjcfD�Shkj�~Y�SnkO�gD�nL���nå�_D6!CSU(N)� N = (2, 0) �ÖgY�Snm!C�ÖoocM�ho�KcfD~[�L�S���h R n�hk³óѯÈ�W�

U�kS��wU L nÚ�k³óѯÈ�W_ûn�M¢p����n¹Õg��W�FhW_hS��ÕhqW�orhdn¹Õgo 〈∆, λ|∆, λ〉 ���Frhdn¹Õgo Z instanton

ε1,ε2,a kj�ShL�Kc_ng�S��oIWDoZ`�hDF��k¿�@D_ngW_ [19, 20]3�

3.4 ááá555

Uf�SS~go〈∆, λ|∆, λ〉 = Z instanton

ε1,ε2,a (3.35)

hDF_`�dn¢Â�k^cfj�yOwS�k¬��WfM_d��gYL� MÀgn¬�K��YSW&�Ön-��KH�h�~�g�¢#Y�¢Â�L����ShL�K�gW�F�]��DOdKðyfJW~DkW_Dh�D~Y�~Z�SnI�goæºk�!CnÓé½íãpn³ÒüìóȶKLB�� óºgo SU(2)

n¤ó¹¿óÈón�M¢pLB�~Y� SU(2) nK��k SU(N) kY�hiFj�gW�FK?

óº���Y�Sho!Xg�l�oâk (2.88) køM~W_� æºoiFY�p�DgW�F? 1980tãEOK�1990tã�-kKQf�Óé½íãpo WN ãph�p��!P!Cãpn¯é¹n-g N = 2 n�j!Xj�n�Yj�a W2 ãpgB�hDFShL�XU�~W_�(WãpkdDfo�ìÓåü[31] ���p�i�2004 n�,H�n�©2 [32] I�Âg�)gYK�����,�Wf

WN ãpn³ÒüìóȶKnÎëà = SU(N)¤ó¹¿óÈón�M¢p (3.36)

hDFSh��H�noê6gY[33, 34]��Hp�N = 3 n4�o�Ln k Hf Wn hDF��PLB��¤Û¢Âo

[Ln, Lm] = (n−m)Ln+m +c

12(n3 − n)δn,−m� (3.37)

[Ln,Wm] = (2n−m)Wn+m� (3.38)

[Wn,Wm] =1

48

[c(22 + 5c)

3 · 5!n(n2 − 1)(n2 − 4)δn,−m + 16(n−m)Λn+m

+ (22 + 5c)(n−m)

((n+m+ 2)(n+m+ 3)

15− (n+ 2)(m+ 2)

6

)Ln+m

]�

(3.39)

g�H��~Y�FW Λn o

Λn =∑m≤−2

LmLn−m +∑m≥−1

Ln−mLm −3

10(n+ 2)(n+ 3)Ln (3.40)

3côjSh��D~YhÕL��k¿�@D_hDFno�DNNgY�Õo¤ó¹¿óÈón�M¢pn���îëÖ�kWfD_ngYL�]��åcfD_ D. Gaiotto LB�å� F. Alday hnq��vnN�g�Z instanton

L�!Cqb4�ÖnSFDFÏg�Q�H`hjc_�`LºK�fO�jDK�hÕk�D~W_�]Sg�ptMkøD_ Mathematica ×í°éà�ѽ³óK���úWf�Q��K�Z��Wf��hºKk�ôWfD_ngZ�W_�hDFnL [19] n�øgY�]n��U�k��!X�W_nL [20] g�Sn�©go!X�W_�n`Q�¬�W~W_�

32

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L

x1,2,3,4t クォーク2重項が居る

境界条件 境界条件!!, 1|

λ2 log

|!, 1, m!

R4 R4

ó 19: 2Í�¯©ü¯��de�_¶Á�æ: �!Cn²ü¸�Ö�Ú�knD_�ó: þÜY��!Cqb4�Ön¶Á�

g�©U�~Y�SSg�Wm n�<�o�8n�nn√

(22 + 5c)/216 kW~W_�ê14h:K�oSa�n{FLê6gY�W3ãpnú,�jhþo

L0|∆, w〉 = ∆|∆, w〉, W0|∆, w〉 = w|∆, w〉 (3.41)

g n > 0 j�Ln, Wn ��Q�h�H��Fj¶KK���U�~Y�]��nÚbP�K��³ÒüìóȶK�

W1|∆, w, λ〉 = λ|∆, w, λ〉 (3.42)

g���h�]nÎëàL�(2.88)g N = 3 hW_ Z instanton h�ôY�ShLå��fD~Y[34, 35]�SSg�Ñéá¿nþÜo

c = 2 + 24(b+1

b)2, ∆ = (b+

1

b)2 − a2

1 − a1a2 − a22, w = ia1a2(a1 + a2) (3.43)

hW~Y�FW�!Xn_� ε1 = b, ε2 = 1/b hWU�k(a1, a2, a3) = (a1, a2,−a1 − a2) hÖ�~W_�~_�SU(2) n~~g�²ü¸�Ökiê4�³YSh�úe~Y��Hp�þ�n�1D� n

�hgo¯©ü¯o SU(2) n�Í�gYK��]�k#cf�Í�n¯©ü¯nÄ��¤ó¹¿óÈó�M¢pk H�ShLgM~Y�Y�h�(2.88) n�Pk�D�D��L ��ShLå��fD~Y���n-�g¯©ü¯�rhd³Y!Xj¹Õo�G¹n�Lk`Q¯©ü¯�³YShgY�Y�h�þÜWf�!Cnqb4�Ögo�Gtn�LaöLôU�~Y(ó19)�Y�h�

〈∆, λ|∆, λ,m〉 = Z instantonwith quark (3.44)

hDFI�L��Ëa~Y[20]�_`W�|∆, λ,m〉 o L1 � L2 �Îó¼ínú$��d�Fj³ÒüìóȶKgY:

L1|∆, λ,m〉 = λ|∆, λ,m〉� L2|∆, λ,m〉 =√λm|∆, λ,m〉� (3.45)

oX�nÖ� [19] gq��_no�U�k�Ñk¯©ü¯�Í��Ûd H_4�gW_�%ná5hWf�WN ãpo¢Õ¡¤ó SU(N) ãpK�ÏP Drinfeld-Sokolov �ChDFK

Õg\�ShLúe~YL��CkoÇü¿hWf N n�r ρ = (N1, . . . , Nk) ���Wf��ShLgM~Y�WN oyk ρ = (N) hDF4�gYL��,n ρ kþWf W (SU(N), ρ) ãphD

33

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F�nLB�~Y� S��úYko²ü¸�Ötkin�Fjô� H�poDKhDFn�å��fJ��R4 n¤ó¹¿óÈó��H��k�x1 = x2 = 0 nsbk¿cf²ü¸4kyp'

Aµdxµ ∼ diag(α(1), . . . , α(1)︸ ︷︷ ︸

n1 times

, α(2), . . . , α(2)︸ ︷︷ ︸n2 times

, . . . , α(k), . . . , α(k)︸ ︷︷ ︸nk times

)idθ, (3.46)

�e��poDgY[36, 37, 38]�~_�R4 gpK�¤ó¹¿óÈó��HfM~W_L�]nK��k R4/Γ g¤ó¹¿óÈó

��H�piFj�gW�FK�_`W�Γ ⊂ SU(2) hW~Y�ε1,2 �e�jD¶Áo [24] k�cf¿y���Γ �nKac-Moody ãpLg�ShLå��fD~W_�Γ = Zm n4�oU�k ε1, ε2 k��b�e��ShLúe�m-!Ñé WN ãphDFP�WDãpLúfO�h���fD~Y[39]�yk m = N = 2 n4�o�2-! ÑéW2 ãphDFno�8n�þðÓé½íãpkj�ngwS�jº��DOd�Y�ShLúe��ÑD�D�hÖ�LúfD~Y[40, 41, 42]�

������

~Zo�nf!�p�i�2011 n�qºn�Øk�Sn�Fj_���HfO`Uc_����W_Dh�D~Y� W�nÕ�oè��k¢áê«NSFn°éóÈj÷ PHY-0969448 ÊsØI�v@n Marvin L. Goldberger membership K�nô©�×Q~W_�~_�pi#:��_Ë��Xfå,ý�èÑf��LÈÃ×ìÙë�và¹×í°éàK�nÜ©�×QfD~Y�

ÂÂÂ������...

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36