Orientifold Singularities and Duality

Ben Heidenreich (Harvard)

with Iñaki García Etxebarria (MPI Munich)and Timm Wrase (Stanford SITP)

1210.7799, 1307.1701and forthcoming

SO(2k) SO(2k + 1)

C2

B2

O3� :

O3+ :

O3

Review (N = 4)

(Witten '98)AdS/CFT description:

Montonen-OliveDualityUSp(2k)

N D3s+

1/6

H3(S5/Z2, Z) ⇠= Z2

⌧ ! �1/⌧

⌧ ! ⌧ + 1

Orbifolds (N = 1) (BH, García-Etxebarria,Wrase '12, '13)

C3/Z3 : zi ! e2⇡i/3zi

OR

SO(N � 4)

SU(N) SU(N)

USp(N + 4)

C2

B2

SO(2k � 1)⇥ SU(2k + 3)SO(2k)⇥ SU(2k + 4)

USp(2k + 4)⇥ SU(2k){Novel duality

� : zi ! �zi

2/6

H3(S5/Z6, Z) ⇠= Z2

Beyond Orbifolds dP1 : x y z w

C⇤ 2 2 �1 �3Z2 � � � +

⌥

⌥

⌥

±

SU(N)SU(N � 4)

SU(N + 4) SU(N)

3/6

IA� IA� IA+ IA+

IB+ IB+IB�IB�

Discrete Torsion & DualitydP1

H3

✓S3 ⇥ S2

Z2, Z

◆⇠= Z2 � Z2

+

C2

B2⌧ ! �1/⌧

⌧ ! ⌧ + 1?

4/6

New Involutions from Deconfinement

SeibergDuality:

Deconfinement:

( )

+ flavors(Berkooz '95,Pouliot '95)

New Involution!

5/6

III+III+

II�� II+�II++

New OrientifoldsdP1

“Phase II”:

SU(N + F � 4)

SO(N � 4)

SU(N)

SU(F )

IA� IA� IA+ IA+

IB+ IB+IB�IB�

II�+

III� III�

(�1)N

(�1)F

+

��

�

Anomalies, SCI,Low N examplesmatch!

6/6

⌥

⌥

⌥

±

IA� IA� IA+ IA+

IB+ IB+IB�IB�

?

+

��

�

covariance requires additional orientifoldsdP1SL(2,Z)

Constructed by deconfinement of parent theory

III+III+

II�� II+�II++

IA� IA� IA+ IA+

IB+ IB+IB�IB�

II�+

III� III�Verified by strong duality checks