arXiv:1108.5632v1 [hep-th] 29 Aug 2011 · IPMU11-0147 ÛÛÛ!!!CCC†††üüü‚‚‚...

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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 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

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

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

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1

Z(M)弦理論(いいかげん)

数学(厳密) ZA ZB?

ó 1: &�ÖK�npf�þan�ú

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ZA =�!Cqb4�Ön³ÒüìóȶKnÎëà� (0.1)

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ZA = 〈∆, λ|∆, λ〉 = 1 +λ2

2∆+

λ4(c+ 8∆)

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

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hj�~Y�S��L

λ2 =q

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1

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4− a2), c = 1 + 6

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ε1ε2(0.5)

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1

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1

T

∑i,j

(σi,jσi+1,j + σi,jσi,j+1)

(1.1)

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Z =∑σi,j

exp

1

T

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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^

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3

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

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0

2

4

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-4

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0

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4

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ó 5: qbÛn�

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Tc =2

log(1 +√

2)(1.3)

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〈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�

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z 7→ z′ = f(z) (1.6)

hDFÍ\L@@�jÒ¦�Ýa~Y (ó5)�®�Ûo

z 7→ z′ = z +∑n

εnzn+1 (1.7)

4

Nx

Nt

ó 6: �ñn¤¸ó°!�

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[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)

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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�

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[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

1.2 ÓÓÓééé½½½íííãããpppnnnhhhþþþ

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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)

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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

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

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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

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

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

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

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

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

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

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

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

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

�,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

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

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

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

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

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

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

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

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

���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

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

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

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

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

!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

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

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

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������...

[1] ËÝÕ�, “qb4�Öh¤ó¹¿óÈónq��f,” i�f�� 65(9) (2010) 703–707.

[2] C. Itzykson and J.-M. Druffe, Statistical Field Theory. Cambridge University Press,

1991.

[3] ÝGÄû��, qb4�Öh�!CÏPû. ©âø�, 1997.

[4] q0ðf, qb4�Öe�. ù¨(, 2006.

[5] äKø, qb4�Ö. µ¤¨ó¹>, 2011.

[6] A. Marshakov, A. Mironov, and A. Morozov, “On Non-Conformal Limit of the AGT

Relations,” Phys. Lett. B682 (2009) 125–129, arXiv:0909.2052 [hep-th].

[7] S. Coleman, Aspects of Symmetry. Cambridge University Press, 1988.

[8] G. ’t Hooft, “Monopoles, Instantons and Confinement,” arXiv:hep-th/0010225.

[9] M. F. Atiyah, N. J. Hitchin, V. G. Drinfeld, and Y. I. Manin, “Construction of

Instantons,” Phys. Lett. A65 (1978) 185–187.

[10] N. Dorey, T. J. Hollowood, V. V. Khoze, and M. P. Mattis, “The Calculus of Many

Instantons,” Phys. Rept. 371 (2002) 231–459, arXiv:hep-th/0206063.

34

[11] äKø, “N = 2 �þð²ü¸�Öh¤ó¹¿óÈó,” �PÖ�v 116(4) (2008)

5–107.

[12] G. ’t Hooft, “Computation of the Quantum Effects Due to a Four- Dimensional

Pseudoparticle,” Phys. Rev. D14 (1976) 3432–3450.

[13] V. Guillemin and S. Sternberg, Symplectic Techniques in Physics. Cambridge

University Press, 1990.

[14] -öS, “¤ó¹¿óÈónpHRhÉÊëɽó Ï,” å,pf�t�¢Ö¹Èé¯ÈÆ(2011) .

[15] G. W. Moore, N. Nekrasov, and S. Shatashvili, “Integrating over Higgs Branches,”

Commun. Math. Phys. 209 (2000) 97–121, arXiv:hep-th/9712241.

[16] N. A. Nekrasov, “Seiberg-Witten Prepotential from Instanton Counting,” Adv. Theor.

Math. Phys. 7 (2004) 831–864, arXiv:hep-th/0206161.

[17] R. Flume and R. Poghossian, “An Algorithm for the Microscopic Evaluation of the

Coefficients of the Seiberg-Witten Prepotential,” Int. J. Mod. Phys. A18 (2003) 2541,

arXiv:hep-th/0208176.

[18] H. Nakajima and K. Yoshioka, “Lectures on Instanton Counting,”

arXiv:math/0311058.

[19] L. F. Alday, D. Gaiotto, and Y. Tachikawa, “Liouville Correlation Functions from

Four-Dimensional Gauge Theories,” Lett. Math. Phys. 91 (2010) 167–197,

arXiv:0906.3219 [hep-th].

[20] D. Gaiotto, “Asymptotically Free N = 2 Theories and Irregular Conformal Blocks,”

arXiv:0908.0307 [hep-th].

[21] R. Poghossian, “Recursion Relations in CFT and N = 2 SYM Theory,” JHEP 12

(2009) 038, arXiv:0909.3412 [hep-th].

[22] V. A. Fateev and A. V. Litvinov, “On AGT Conjecture,” JHEP 02 (2010) 014,

arXiv:0912.0504 [hep-th].

[23] L. Hadasz, Z. Jaskolski, and P. Suchanek, “Proving the AGT Relation for NF = 0, 1, 2

Antifundamentals,” arXiv:1004.1841 [hep-th].

[24] H. Nakajima, “Instantons on ALE Spaces, Quiver Varieties and Kac-Moody Algebras,”

Duke Math. Journal 76 (1994) 365.

[25] D. Maulik and A. Okounkov, unpublished. 2011?

[26] E. Witten, “Geometric Langlands from Six Dimensions,” arXiv:0905.2720 [hep-th].

[27] E. Witten, “Geometric Langlands and the Equations of Nahm and Bogomolny,”

arXiv:0905.4795 [hep-th].

35

[28] R. Dijkgraaf, E. P. Verlinde, and H. L. Verlinde, “Matrix String Theory,” Nucl. Phys.

B500 (1997) 43–61, arXiv:hep-th/9703030.

[29] M. Berkooz, M. Rozali, and N. Seiberg, “Matrix Description of M Theory on T 4 and

T 5,” Phys. Lett. B408 (1997) 105–110, arXiv:hep-th/9704089.

[30] J. Polchinski, String Theory. Cambridge University Press, 1998.

[31] P. Bouwknegt and K. Schoutens, “W Symmetry in Conformal Field Theory,” Phys.

Rept. 223 (1993) 183–276, arXiv:hep-th/9210010.

[32] �,�, “7ãpnq ∼ ¹üÑüû³óÕ©üÞëãpnhþÖx,” in �p�i�2004 �?Æ. 2004.

[33] N. Wyllard, “AN−1 Conformal Toda Field Theory Correlation Functions from

Conformal N = 2 SU(N) Quiver Gauge Theories,” JHEP 11 (2009) 002,

arXiv:0907.2189 [hep-th].

[34] A. Mironov and A. Morozov, “On AGT Relation in the Case of U(3),”

arXiv:0908.2569 [hep-th].

[35] M. Taki, “On AGT Conjecture for Pure Super Yang-Mills and W- algebra,” JHEP 05

(2011) 038, arXiv:0912.4789 [hep-th].

[36] A. Braverman, B. Feigin, M. Finkelberg, and L. Rybnikov, “A Finite Analog of the

AGT Relation I: Finite W-Algebras and Quasimaps’ Spaces,” arXiv:1008.3655

[math.AG].

[37] N. Wyllard, “Instanton Partition Functions in N = 2 SU(N) Gauge Theories with a

General Surface Operator, and Their W-Algebra Duals,” JHEP 02 (2011) 114,

arXiv:1012.1355 [hep-th].

[38] H. Kanno and Y. Tachikawa, “Instanton Counting with a Surface Operator and the

Chain- Saw Quiver,” JHEP 06 (2011) 119, arXiv:1105.0357 [hep-th].

[39] T. Nishioka and Y. Tachikawa, “Para-Liouville/Toda Central Charges from

M5-Branes,” arXiv:1106.1172 [hep-th].

[40] V. Belavin and B. Feigin, “Super Liouville Conformal Blocks from N = 2 SU(2) Quiver

Gauge Theories,” JHEP 07 (2011) 079, arXiv:1105.5800 [hep-th].

[41] G. Bonelli, K. Maruyoshi, and A. Tanzini, “Instantons on ALE Spaces and Super

Liouville Conformal Field Theories,” arXiv:1106.2505 [hep-th].

[42] G. Bonelli, K. Maruyoshi, and A. Tanzini, “Gauge Theories on ALE Space and Super

Liouville Correlation Functions,” arXiv:1107.4609 [hep-th].

36

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