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index andits applications
AKINORI TANAKA (RIKEN)
Based on A. T, H. Mori, and T. Morita, Phys. Rev. D91 (2015) 105023, arXiv:1408.3371 [hep-th].
A. T, H. Mori, and T. Morita, JHEP 09 (2015) 154, arXiv:1505.07539 [hep-th].
H. Mori, A.T., arXiv:151x.xxxxx [hep-th].
RP2
(2+1)d IndexI = TrH(�1)F q
12 (H+j3)
Y. Imamura & S. Yokoyama(2011)
● Exactly calculable (Localization)
H �4-supercharges
orP
CP
A.T, H. Mori & T. Morita(2014-2015)H �4-supercharges/Z2
Applications● Explicit checks of 3d duality
● 3d-3d correspondence
Krattenthaler, Spiridonov, Vartanov (2011), Kapustin, Willett (2011)
Ramanujan’s Sum
q-binomial Formula+
0
@
0
@3d mirror symmetry(1-flavor)
0
@
0
@ =
,3d mirror symmetry(1-flavor)
0
@
0
@ =0
@
0
@
Dimofte, Gaiotto, Gukov (2011), Gang, Koh, Lee, Park (2013)
/Z2
Today’s talk
・ index formulaRP2
・ Role of in 3d dualityZ2
・ Index formula (review)
・ Index formula (review)
I = TrH(�1)F q12 (H+j3)
H
SUSY QFT on
j3
⇥
Definition
+ �
Our notation for quiver diagram
Aµ := :=Degrees of freedom
�
Aµ
+ :=Aµ :=
Coupling
� 1
4Fµ⌫F
µ⌫ + . . .
+ 1(@/� iA/) 1 + . . .
+ 2(@/+ iA/) 2 + . . .
・ Index formula (review)
Localization formula (2-steps)
・ Index formula (review)
STEP1=
1X
2s=�1
1
2⇡
Z 2⇡
0d✓ s, ✓
STEP2
I = TrH(�1)F q12 (H+j3)
s, ✓ =⇣q
1��2
⌘s (e�iQ✓q|Qs|+ 2��2 ; q)1
(e+iQ✓q|Qs|+�2 ; q)1
Y. Imamura & S. Yokoyama(2011)
Localization formula (2-steps)
・ Index formula (review)
ExampleI = TrH(�1)F q
12 (H+j3)
+ �
=1X
2s=�1
1
2⇡
Z 2⇡
0d✓ s, ✓
Y. Imamura & S. Yokoyama(2011)
=⇣q
1��2
⌘s (e�iQ✓q|Qs|+ 2��2 ; q)1
(e+iQ✓q|Qs|+�2 ; q)1
s, ✓
Localization formula (2-steps)
・ Index formula (review)
ExampleI = TrH(�1)F q
12 (H+j3)
+ �
=1X
2s=�1
1
2⇡
Z 2⇡
0d✓ s, ✓
⇣ ⇣
Y. Imamura & S. Yokoyama(2011)
Localization formula (2-steps)
・ Index formula (review)
ExampleI = TrH(�1)F q
12 (H+j3)
+ �
=1X
2s=�1
1
2⇡
Z 2⇡
0d✓ s, ✓
⇣ ⇣
=1X
2s=�1
Idz
2⇡iz
⇣q
1��2
⌘|2s| (z�1q|s|+2��
2 ; q)1
(zq|s|+0+�
2 ; q)1⇥ (zq|s|+
2��2 ; q)1
(z�1q|s|+0+�
2 ; q)1
z = ei✓
Y. Imamura & S. Yokoyama(2011)
Localization formula (2-steps)
・ Index formula (review)
Check: 3d mirror symmetryI = TrH(�1)F q
12 (H+j3)
=⇣ ⇣ ⇣ ⇣
Y. Imamura & S. Yokoyama(2011)
Localization formula (2-steps)
・ Index formula (review)
Check: 3d mirror symmetryI = TrH(�1)F q
12 (H+j3)
=⇣ ⇣ ⇣ ⇣
q-binomial theorem+
Ramanujan’s sum formulaKrattenthaler, Spiridonov, Vartanov (2011), Kapustin, Willett (2011)
Y. Imamura & S. Yokoyama(2011)
Today’s talk
・ index formulaRP2
・ Role of in 3d dualityZ2
=X
s
Zd✓ ・ Index formula (review)✔
・ index formulaRP2
I = TrH(�1)F q12 (H+j3)
H
SUSY QFT on
j3
⇥
Definition
H/Z2(�1)F q
12 (H+j3)
Aµ := :=Degrees of freedom
⇒ SUSY parity conds
CPCP !
PP ! !
!
(@µ � iAµ) ! ±(@ � iAµ)
(@µ � iAµ) ! ±(@ + iAµ)
� ! �
✓�1
�2
◆!
✓�2
�1
◆
µ
µ
・ index formulaRP2
Our notation for quiver diagram
�
Aµ
+ :=Aµ :=
Coupling
PCP P! CP
PP !CPCP !
!!
・ index formulaRP2
Our notation for quiver diagram
P
CP
P
8<
:
8<
:
P
CP
P
8<
:!8<
:
�
Aµ
+ :=Aµ :=
Coupling PP
CPCP
!!!!
・ index formulaRP2
Our notation for quiver diagram
I = TrH(�1)F q12 (H+j3)H/Z2
(�1)F q12 (H+j3)
⇥
Localization formula (2-steps)
STEP1
・ index formulaRP2
CP
P = q+18(q2; q2)1(q; q2)1
X
s±=0,1
1
2⇡
Z 2⇡
0d✓ s±, ✓
= q�18(q; q2)1(q2; q2)1
1X
s=�1
1
2
X
✓±=0,⇡
s, ✓±
=1X
2s=�1
1
2⇡
Z 2⇡
0d✓ s, ✓
Localization formula (2-steps)
STEP2
・ index formulaRP2
I = TrH(�1)F q12 (H+j3)H/Z2
(�1)F q12 (H+j3)
s±, ✓
s, ✓±
=
8><
>:
⇣q
��18 e
iQ✓4
⌘+1(e�iQ✓q
2��2 ;q2)1
(e+iQ✓q0+�
2 ;q2)1for ei⇡Qs±
= +1,⇣q
��18 e
iQ✓4
⌘�1(e�iQ✓q
4��2 ;q2)1
(e+iQ✓q2+�
2 ;q2)1for ei⇡Qs±
= �1,
=⇣q
1��2
⌘|Qs|
⇣e�iQ✓±q|Qs|+ (2��)
2 ; q⌘
1⇣e+iQ✓±q|Qs|+ (0+�)
2 ; q⌘
1
etc.
Check: 3d mirror symmetry
=⇣ ⇣ ⇣ ⇣
Localization formula (2-steps) ・ index formulaRP2
I = TrH(�1)F q12 (H+j3)H/Z2
(�1)F q12 (H+j3)
/P CP ?/?
Check: 3d mirror symmetry
=⇣ ⇣ ⇣ ⇣
Localization formula (2-steps) ・ index formulaRP2
I = TrH(�1)F q12 (H+j3)H/Z2
(�1)F q12 (H+j3)
P⇣ ⇣ ⇣ ⇣
=?
Check: 3d mirror symmetry
=⇣ ⇣ ⇣ ⇣
Localization formula (2-steps) ・ index formulaRP2
I = TrH(�1)F q12 (H+j3)H/Z2
(�1)F q12 (H+j3)
=⇣⇣
P⇣ ⇣A.T, H. Mori, T. Morita (2014)
Check: 3d mirror symmetry
=⇣ ⇣ ⇣ ⇣
Localization formula (2-steps) ・ index formulaRP2
I = TrH(�1)F q12 (H+j3)H/Z2
(�1)F q12 (H+j3)
A.T, H. Mori, T. Morita (2015)
CP
⇣⇣=
⇣ ⇣
A.T, H. Mori, T. Morita (2014)
P⇣ ⇣
=⇣⇣
=⇣ ⇣ ⇣ ⇣Mirror symmetry eq decomposition
・ index formulaRP2
A.T, H. Mori, T. Morita (2014)
P⇣ ⇣
=⇣⇣
A.T, H. Mori, T. Morita (2015)
CP
⇣⇣=
⇣ ⇣
q-binomial theorem+
Ramanujan’s sum formula
q-binomial theorem
Ramanujan’ssum formula
Today’s talk
・ index formulaRP2
CP =X
s
X
✓2Z2
・ Index formula (review) =X
s
Zd✓
P =X
s2Z2
Zd✓
8<
:
・ Role of in 3d dualityZ2
✔
✔
・ Role of in 3d dualityZ2
3d abelian mirror symmetry
=. . .
. . .
P CP CP . . .
. . .
CP= P . . .P
P , CP
NS5� NS5’�D5�
D3�
�� �
�
V
D5�
�� �
�
D5��� �
�
Q1, Q1
Q1,¯Q1 Q2,
¯Q2 Q3,¯Q3
Q2, Q2 Q3, Q3
D5� D5’�NS5�
D3�
NS5� NS5�
�� �
�
����
����
V1 V2 Z, Z
X, XY, Y b12, b12
b21, b21
b23, b23
b32, b32
H. Mori, A.T (To apper)
・ Role of in 3d dualityZ2
Duality between loop operators
Wilson loop :
⇥ S1Pon
CP
eieRdtAt
eieRdt�
9=
;
H. Mori, A.T (To apper)
Vortex loop :P
onCP
eieRdtAt
eieRdt�
9=
;S�1 S
SS�1⇥ S1
・ Role of in 3d dualityZ2
● Gauge WP
CP
8<
:
8<
: = Topological VCP
8<
:
8<
:P
● Gauge VP
CP
8<
:
8<
: = Topological WCP
8<
:
8<
:P
P , CPDuality between loop operators H. Mori, A.T (To apper)
Today’s talk
・ index formulaRP2
・ Role of in 3d dualityZ2
CP =X
s
X
✓2Z2
・ Index formula (review) =X
s
Zd✓
P =X
s2Z2
Zd✓
8<
:
P , CP
✔
✔
✔
・ Comments
● Parity anomaly = Gauge anomaly?
s±, ✓ =
8><
>:
⇣q
��18 e
iQ✓4
⌘+1(e�iQ✓q
2��2 ;q2)1
(e+iQ✓q0+�
2 ;q2)1for ei⇡Qs±
= +1,⇣q
��18 e
iQ✓4
⌘�1(e�iQ✓q
4��2 ;q2)1
(e+iQ✓q2+�
2 ;q2)1for ei⇡Qs±
= �1,
Unless , they are anomalous.Q 2 4Z
● CS term?→Generically, no... But (AdA - BdB) type is OK.
Thank you !
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