Counting instantons in N=1 theories of class S k Elli Pomoni - PowerPoint PPT Presentation

���������������������������������� ������������������������ Counting instantons in N=1 theories of class S k Elli Pomoni [1512.06079 Coman,EP,Taki,Yagi] [1703.00736 Mitev,EP] [1712.01288 Bourton,EP]

Motivation: N=2 exact results Seiberg-Witten theory: effective theory in the IR Nekrasov: instanton partition function Pestun: observables in the UV (path integral on the sphere localizes) String/M-/F-theory realizations Gaiotto: 4D N=2 class S : 6D (2,0) on Riemann surface � ��� AGT: 4D partition functions = 2D CFT correlators 2D/ 4D relations 4D SC Index = 2D correlation function of a TFT

What can we do for N=1 theories? [Romelsberge 2005] Superconformal Index [Kinney,Maldacena,Minwalla,Raju 2005] Intriligator and Seiberg: generalized SW technology Witten: IIA/M-theory approach to curves Holomorphy fixes: N=2 theories: prepotential (that’ s all in the IR) N=1 theories: superpotential (there are also Kähler terms) No Localization (No Nekrasov, no Pestun) An S 4 partition function plagued with scheme ambiguities. [Gerchkovitz, Gomis, Komargodski 2014] Derivatives of the free energy scheme independent. [Bobev, Elvang, Kol, Olson, Pufu 2014]

What can we do for N=1 theories? [Leigh,Strassler 1995] Can construct conformal N=1 theories. [Kachru,Silverstein 1998] AdS/CFT natural route to several examples. [Lawrence,Nekrasov,Vafa1998] 6D (1,0) on a Riemann Surface. [Gaiotto,Razamat 2015] [Heckman,Vafa….] Conformal Class S k ( S Γ ): Obtained by orbifolding N=2 (inheritance) [Gaiotto,Razamat 2015] Labeled by punctured Riemann Surface 2D/ 4D relation Index = 2D correlation function of a TFT

Plan Is there AGT k ? 4D partition functions = 2D CFT correlators Introduce N=1 theories in class S k Spectral curves for N=1 theories in class S k From the curves: 2D symmetry algebra and representations Conformal Blocks Instanton partition function Instanton partition function from Dp/D(p-4) branes on orbifold Free trinion partition functions on S 4 3pt functions

Class S k

������������ � �� � �������������������������������������������� Class S and S k N M5 branes on X 4 x C g,n [Gaiotto 2009] 4D/2D 2D theory on C g,n SU(N) theory on X 4 relation 6D (2,0) SCFT on Riemann surface: 4D N=2 theories of class S 6D (1,0) SCFT on Riemann surface: 4D N=1 theories of class S k x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 N M5-branes . . . . . − − − − − − A k − 1 orbifold . . . . . . . − − − − [Gaiotto,Razamat 2015]

������������� ��� � Class S k [Gaiotto,Razamat 2015] U(1) r SU(2) R of N=2 x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 ( x 10 ) Type IIA M NS5 branes . . . . . − − − − − − N D4-branes . . . . . − − − − − − A k − 1 orbifold . . . . . . . _____ _____ − − − − U(1) R x 4 , x 5 NS 5 NS 5 NS 5 NS 5 m 3 m 1 a 1 D 4 m 3 − m 2 m 1 a 1 D 4 − m 4 − a 2 4 / Z 2 R a 2 m 4 a 2 m 2 m 4 m 2 − a 1 D 4 − m 1 D 4 − m 3 1/g 2 1/g 2 x 6 ✏ = Γ 0 Γ 1 Γ 2 Γ 3 Γ 4 Γ 5 ✏ = Γ 0 Γ 1 Γ 2 Γ 3 Γ 6 ✏ = Γ 4 Γ 5 Γ 7 Γ 8 ✏ ______

Class S k x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 Type IIB __________ A M − 1 orbifold . . . . . . − − − − N D3-branes . . . . . . − − − − _____ _____ A k − 1 orbifold . . . . . . − − − − Γ = Z k × Z M N=1 orbifold daughter of N=4 SYM _ _ AdS 5 × S 5 / ( Z k × Z M ) Useful for AdS/CFT ( orbifold inheritance ) [Bershadsky, Kakushadze,Vafa 1998] String theory technics to calculate instantons [Dorey, Hollowood, Khoze, Mattis,…] [Lerda,…]

����������� � ���������������� ������������������������� Class S k [Gaiotto,Razamat 2015] 4D field theory point of view U (1) t U (1) α c U (1) β i +1 − c U (1) γ i V ( i,c ) 0 0 0 0 � 1 � 1 +1 Φ ( i,c ) 0 Q ( i,c − 1) +1 / 2 � 1 +1 0 e Q ( i,c − 1) +1 / 2 � 1 +1 0 Large global symmetry group

������������������������� ����������� � �� ��������������� Class S k Begin with N=2 class S with SU(kN) gauge groups: M − 1 ⇣ ⌘ Q ( c − 1) Φ ( c ) ˜ Q ( c − 1) − ˜ X W S = Q ( c ) Φ ( c ) Q ( c ) c =1 0 1 Orbifold projection: [Douglas,Moore 1996] Q (1 ,c ) N B C B Q (2 ,c ) C B C Q ( c ) = ... B C 0 1 Φ (1 ,c ) @ A Q ( k,c ) Φ (2 ,c ) B C B C ... B C Φ ( c ) = 0 1 B C e Q ( k,c ) B C kN x kN B C B Φ ( k − 1 ,c ) C e B Q (1 ,c ) C @ A e B C Q ( c ) = Φ ( k,c ) ... B C @ A N x N e Q ( k − 1 ,c ) M − 1 ⇣ ⌘ k X X Q ( i,c − 1) Φ ( i,c ) ˜ Q ( i,c − 1) � ˜ W S k = Q ( i,c ) Φ ( i,c ) Q ( i +1 ,c ) i =1 c =1

����������� � ���������������� ������������������������� Coulomb and Higgs branch u ` = h tr φ ` i h Q i = 0 h � i = diag ( a 1 , . . . , a N ) Coulomb Branch: with and E = r Higgs Branch: with operators e.g. h φ i = a = 0 E = 2 R m i = 0 Q {I ¯ � � µ IJ = h tr i Q J } � � � � ` i , Φ (1) · · · Φ ( k ) u ` k = h tr Coulomb Branch: parameterised by h Q i = 0 � � ⇣ ⌘ 2 ⇡ i ( k − 1) 2 ⇡ i 4 ⇡ i E = r h U − 1 Φ U i = diag ( a 1 , a 2 , · · · , a N ) ⌦ diag k · · · e k , e 1 , e k Higgs Branch: similar w/ mother theory, operators charged under new beta and gamma symmetries. CB and HB do not mix (no relations): charged under different charges! [Bourton,Pini,EP to appear]

Curves [1512.06079 Coman,EP,Taki,Yagi]

Generic N=1 C urves The spectral curve computes the effective YM coupling constants. [Intriligator,Seiberg] R 3 , 1 × CY 3 × R 1 , M-theory on [Bah, Beem, Bobev, Wecht] CY 3 locally two holomorphic line bundles on the curve C g,n N=1 spectral curve is an overdetermined algebraic system of eqns. [Bonelli,Giacomelli,Maruyoshi,Tanzini] [Xie… ] For class S k on the Coulomb Branch ( ) only one equation h Q i = 0 exactly like for N=2 theories. [Coman,EP,Taki,Yagi]

� � ���������������������������������� ���������������������������������� S k curves [Coman,EP,Taki,Yagi] U(1) r SU(2) R of N=2 v = x 4 + ix 5 x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 ( x 10 ) w = x 7 + ix 8 M NS5 branes . . . . . − − − − − − N D4-branes . . . . . − − − − − − t = e − x 6+ ix 10 R 10 A k − 1 orbifold . . . . . . . _____ _____ − − − − U(1) R ⇣ ⌘ 2 π i k v , e − 2 π i k w ( v , w ) ∼ e NS 5 NS 5 m 3 m 1 a 1 D 4 − m 2 − m 4 [Lykken,Poppitz,Trivedi 97] − a 2 4 / Z 2 R a 2 h Q i = 0 m 4 m 2 − a 1 D 4 − m 1 − m 3 Zero vevs for Higgs branch operators!

������������������������������������������� � � C urves from M-theory [Witten 1997] v = x 4 + ix 5 NS 5 NS 5 m 3 m 1 a 1 D 4 a 2 m 4 m 2 D 4 q 2D surface F(t,v)=0 in the 4D t = e − x 6+ ix 10 R 10 space {x 4 , x 5 , x 6 , x 10 }={v,t}. ��������������������������������������������������������������� M-theory: a single M5 brane with non trivial topology ( t − 1)( t − q ) v 2 − P 1 ( t ) v + P 2 ( t ) = 0 ( v − m 1 )( v − m 2 ) t 2 + − (1 + q ) v 2 + qMv + u � � t + q ( v − m 3 )( v − m 4 ) = 0 _ _ - coupling constant q=e 2 π i � �� u = tr � 2 M=m 1 +m 2 +m 3 +m 4

� � NS 5 NS 5 Class S Curve m 3 m 1 a 1 D 4 SU(2) with 4 flavors a 2 m 4 m 2 D 4 ( t − 1)( t − q ) v 2 − P 1 ( t ) v + P 2 ( t ) = 0 q ( v − m 1 )( v − m 2 ) t 2 + − (1 + q ) v 2 + qMv + u � � t + q ( v − m 3 )( v − m 4 ) = 0 _ _ - q=e 2 π i � �� u = tr � 2 M=m 1 +m 2 +m 3 +m 4 t = e Class S k Curve 2 π i k v v ∼ e 2 ) t 2 + P ( v ) t + q ( v k − m k ( v k − m k 1 )( v k − m k 3 )( v k − m k 4 ) = 0 NS 5 NS 5 P ( v ) = − (1 + q ) v 2 k + u k v k + u 2 k m 3 m 1 _ _ a 1 D 4 − m 2 − m 4 − a 2 vevs of gauge invariant operators: � � 4 / Z 2 � � R a 2 parameterize the Coulomb branch m 4 m 2 D 4 − a 1 � � � � 2 i ⇠ u 2 k − m 1 − m 3 h tr Φ (1) · · · Φ ( k ) i ⇠ u k Φ (1) · · · Φ ( k ) h tr _ _

S k curves SW or IR curve � Gaiotto or UV curve � ��� X of g=kN-1 a sphere with n punctures N x kN = − � (4) X k ` ( t ) x k ( N − ` ) N c ( s,k ) = X m k i 1 · · · m k i s ` =1 i 1 < ··· <i s =1 t 2 + u k ` t + ( − 1) ` c ( ` ,k ) k ` ( t ) = ( − 1) ` c ( ` ,k ) q � (4) L R t k ` ( t − 1)( t − q ) Full/maximal puncture : k images of N mass parameters Simple puncture : No further parameters k ` ( t ) = ( � 1) ` c ( ` ,k ) t � c ( ` ,k ) � (3) L R t k ` ( t � 1)