based on - 京都大学soken.editorial/sokendenshi/...applications explicit checks of 3d duality...

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

http://arxiv.org/find/hep-th/1/au:+Dimofte_T/0/1/0/all/0/1

http://arxiv.org/find/hep-th/1/au:+Dimofte_T/0/1/0/all/0/1

http://arxiv.org/find/hep-th/1/au:+Gaiotto_D/0/1/0/all/0/1

http://arxiv.org/find/hep-th/1/au:+Gaiotto_D/0/1/0/all/0/1

http://arxiv.org/find/hep-th/1/au:+Gukov_S/0/1/0/all/0/1

http://arxiv.org/find/hep-th/1/au:+Gukov_S/0/1/0/all/0/1

http://arxiv.org/find/hep-th/1/au:+Gang_D/0/1/0/all/0/1

http://arxiv.org/find/hep-th/1/au:+Gang_D/0/1/0/all/0/1

http://arxiv.org/find/hep-th/1/au:+Koh_E/0/1/0/all/0/1

http://arxiv.org/find/hep-th/1/au:+Koh_E/0/1/0/all/0/1

http://arxiv.org/find/hep-th/1/au:+Lee_S/0/1/0/all/0/1

http://arxiv.org/find/hep-th/1/au:+Lee_S/0/1/0/all/0/1

http://arxiv.org/find/hep-th/1/au:+Park_J/0/1/0/all/0/1

http://arxiv.org/find/hep-th/1/au:+Park_J/0/1/0/all/0/1

Today’s talk

・ index formulaRP2

・ Role of in 3d dualityZ2

・ 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 + . . .


Localization formula (2-steps)


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




ExampleI = TrH(�1)F q

12 (H+j3)

+ �

=1X

2s=�1

1

2⇡

Z 2⇡

0d✓ s, ✓


=⇣q

1��2

⌘s (e�iQ✓q|Qs|+ 2��2 ; q)1

(e+iQ✓q|Qs|+�2 ; q)1

s, ✓




12 (H+j3)

+ �

=1X

2s=�1

1

2⇡

Z 2⇡

0d✓ s, ✓

⇣ ⇣





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✓




Check: 3d mirror symmetryI = TrH(�1)F q

12 (H+j3)

=⇣ ⇣ ⇣ ⇣




Check: 3d mirror symmetryI = TrH(�1)F q

12 (H+j3)

=⇣ ⇣ ⇣ ⇣

q-binomial theorem+

Ramanujan’s sum formulaKrattenthaler, Spiridonov, Vartanov (2011), Kapustin, Willett (2011)


Today’s talk



=X

s

Zd✓ ・ Index formula (review)✔


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



�

Aµ

+ :=Aµ :=

Coupling

PCP P! CP

PP !CPCP !

!!



P

CP

P

8<

:

8<

:

P

CP

P

8<

:!8<

:

�

Aµ

+ :=Aµ :=

Coupling PP

CPCP

!!!!



I = TrH(�1)F q12 (H+j3)H/Z2

(�1)F q12 (H+j3)

⇥


STEP1


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, ✓


STEP2


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+�


= �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 ?/?


=⇣ ⇣ ⇣ ⇣


I = TrH(�1)F q12 (H+j3)H/Z2

(�1)F q12 (H+j3)

P⇣ ⇣ ⇣ ⇣

=?


=⇣ ⇣ ⇣ ⇣


I = TrH(�1)F q12 (H+j3)H/Z2

(�1)F q12 (H+j3)

=⇣⇣

P⇣ ⇣A.T, H. Mori, T. Morita (2014)


=⇣ ⇣ ⇣ ⇣


I = TrH(�1)F q12 (H+j3)H/Z2

(�1)F q12 (H+j3)

A.T, H. Mori, T. Morita (2015)

CP

⇣⇣=

⇣ ⇣


P⇣ ⇣

=⇣⇣

=⇣ ⇣ ⇣ ⇣Mirror symmetry eq decomposition



P⇣ ⇣

=⇣⇣


CP

⇣⇣=

⇣ ⇣

q-binomial theorem+

Ramanujan’s sum formula

q-binomial theorem

Ramanujan’ssum formula

Today’s talk


CP =X

s

X

✓2Z2

・ Index formula (review) =X

s

Zd✓

P =X

s2Z2

Zd✓

8<

:


✔

✔


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)


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


● 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



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+�


= +1,⇣q

��18 e

iQ✓4

⌘�1(e�iQ✓q

4��2 ;q2)1

(e+iQ✓q2+�


= �1,

Unless , they are anomalous.Q 2 4Z

● CS term?→Generically, no... But (AdA - BdB) type is OK.

Thank you !

based on - 京都大学soken.editorial/sokendenshi/...applications explicit checks of 3d duality...

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