Motivation and Overview Overview Spaces of Superconformal field - PowerPoint PPT Presentation

2D CFTs for a class of 4D N = 1 theories Vladimir Mitev PRISMA Cluster of Excellence, Institut f¨ ur Physik, THEP , Johannes Gutenberg Universit¨ at Mainz IGST Paris, July 18 2017 [V. Mitev, E. Pomoni, arXiv:1703.00736]

Motivation and Overview

Overview Spaces of Superconformal field theories in 4D • N = 4 SYM is unique up to the choice of a gauge group Integrable in the planar limit • N = 3: not known to exist until recently [Garc´ ıa-Etxebarria, Regalado, 2015] • The space of N = 2 SCFTs is rich Theories of class S [Gaiotto, 2009] led to AGT

Overview Spaces of Superconformal field theories in 4D • N = 4 SYM is unique up to the choice of a gauge group Integrable in the planar limit • N = 3: not known to exist until recently [Garc´ ıa-Etxebarria, Regalado, 2015] • The space of N = 2 SCFTs is rich Theories of class S [Gaiotto, 2009] led to AGT • The space of N = 1 SCFTs much less understood Let’s see which techniques carry over from N = 2!

Exact results for N = 2 SCFTs • Seiberg-Witten: complete low-energy effective action (IR)

Exact results for N = 2 SCFTs • Seiberg-Witten: complete low-energy effective action (IR) • Nekrasov: instanton partition functions

Exact results for N = 2 SCFTs • Seiberg-Witten: complete low-energy effective action (IR) • Nekrasov: instanton partition functions • Pestun: UV observables thanks to localization on S 4

Exact results for N = 2 SCFTs • Seiberg-Witten: complete low-energy effective action (IR) • Nekrasov: instanton partition functions • Pestun: UV observables thanks to localization on S 4 • Gaiotto: 4D N = 2 class S , 6D (2,0) compactified on a Riemann surface

Exact results for N = 2 SCFTs • Seiberg-Witten: complete low-energy effective action (IR) • Nekrasov: instanton partition functions • Pestun: UV observables thanks to localization on S 4 • Gaiotto: 4D N = 2 class S , 6D (2,0) compactified on a Riemann surface • 4D superconformal index = 2D correlation function of a TFT

Exact results for N = 2 SCFTs • Seiberg-Witten: complete low-energy effective action (IR) • Nekrasov: instanton partition functions • Pestun: UV observables thanks to localization on S 4 • Gaiotto: 4D N = 2 class S , 6D (2,0) compactified on a Riemann surface • 4D superconformal index = 2D correlation function of a TFT • AGT: 4D partition functions = 2D CFT correlators

Exact results for N = 1 SCFTs • Superconformal index • Intriligator and Seiberg: generalized SW techniques • Witten: IIA/M-theory approach to the SW curve

Beware! • for N = 2 the SW curve fixes the prepotential (complete IR) • for N = 1 the SW curve fixes the superpotential BUT there are also K¨ ahler terms

Beware! • for N = 2 the SW curve fixes the prepotential (complete IR) • for N = 1 the SW curve fixes the superpotential BUT there are also K¨ ahler terms • No localization yet for N = 1 • S 4 partition function Z plagued with scheme ambiguities [Gerchkovitz, Gomis, Komargodski, 2014] • BUT derivatives of log Z can be scheme independent [Bobev, Elvang, Kol, Olson, Pufu, 2014]

A class of 4D N = 1 theories

The S k theories Nice Class of N = 1 theories • N = 1 superconformal • Constructed by orbifolding N = 2 theories (inheritance) • Labeled by a punctured Riemann surface Σ g , n • Its index is a correlation function of a 2D TFT [Gaiotto, Razamat, 2015] • SW curves known [Coman, Pomoni, Taki, Yagi, 2015]

The quiver construction • Circles are SU ( N ) gauge groups ( N = 2 vector multiplets) • Squares are SU ( N ) flavor symmetry • Line segments are N = 2 hypermultiplets

The quiver construction • N = 2 vector ⇒ N = 1 vector and N = 1 chiral (blue) • N = 2 hyper ⇒ N = 1 chiral (red) and N = 1 chiral (green)

The quiver construction

The SW curves

Seiberg-Witten theory SU ( N ) M −→ U ( 1 ) M ( N − 1 ) The Coulomb branch low energy effective action is given by • an auxiliary algebraic curve given by G ( x , t ) = 0 • the meromorphic SW differential λ SW = x dt

Seiberg-Witten theory SU ( N ) M −→ U ( 1 ) M ( N − 1 ) The Coulomb branch low energy effective action is given by • an auxiliary algebraic curve given by G ( x , t ) = 0 • the meromorphic SW differential λ SW = x dt � � λ SW = ∂ F a = a D = λ SW ∂ a α β τ IR = ∂ a D ∂ a

Brane contruction Gauge theory as the world-volume theory on D4 branes ending on NS5 branes

Brane contruction Gauge theory as the world-volume theory on D4 branes ending on NS5 branes

Brane contruction Gauge theory as the world-volume theory on D4 branes ending on NS5 branes

Uplift to M-theory

Uplift to M-theory

The brane configuration for S k 2 π i k v for k = 1 , 2 , 3 , . . . Orbifold identification: v ∼ e

The brane configuration for S k 2 π i k v for k = 1 , 2 , 3 , . . . Orbifold identification: v ∼ e

The brane configuration for S k 2 π i k v for k = 1 , 2 , 3 , . . . Orbifold identification: v ∼ e

The S k SW curve The SW curve for one SU ( N ) gauge group � � N � � v k − m k t 2 − ( 1 + q ) v Nk + t L , i i = 1 u k ℓ v ( N − ℓ ) k � � � N N � � v k − m k + + q = 0 R , i ℓ = 1 i = 1 [Coman, Pomoni, Taki, Yagi, 2015] q = e 2 π i τ = coupling u s = Coulomb moduli

Set v = xt and rewrite: � N k ℓ ( t ) x k ( N − ℓ ) = 0 φ ( 4 ) ℓ = 0 ( − 1 ) ℓ c ( ℓ, k ) t 2 + u k ℓ t + ( − 1 ) ℓ c ( ℓ, k ) q φ ( 4 ) L R k ℓ ( t ) = for ℓ = 1 , . . . , N t k ℓ ( t − 1 )( t − q ) • Casimirs c ( ℓ, k ) = � i 1 < ··· < i ℓ m k i 1 · · · m k i ℓ • φ ( 4 ) because there are four poles in t : two full ⊙ at t = 0 , t = ∞ and two simple • at t = 1 , t = q

Set v = xt and rewrite: � N k ℓ ( t ) x k ( N − ℓ ) = 0 φ ( 4 ) ℓ = 0 ( − 1 ) ℓ c ( ℓ, k ) t 2 + u k ℓ t + ( − 1 ) ℓ c ( ℓ, k ) q φ ( 4 ) L R k ℓ ( t ) = for ℓ = 1 , . . . , N t k ℓ ( t − 1 )( t − q ) • Casimirs c ( ℓ, k ) = � i 1 < ··· < i ℓ m k i 1 · · · m k i ℓ • φ ( 4 ) because there are four poles in t : two full ⊙ at t = 0 , t = ∞ and two simple • at t = 1 , t = q The 2D CFT duals of φ ( n ) are blocks with insertions of currents

Trinion curve Weak coupling limit q → 0 gives the trinion curve It describes a bunch of free chiral hypermultiplets

Observations SW curve for N = 1 S k and SU ( N ) is the SW curve of N = 2 S with SU ( kN ) with 2 π i k s m L , j + Ns �−→ m L , j e 2 π i k s m R , j + Ns �−→ m R , j e 2 π i k s a j + Ns �−→ a j e � u s if s mod k = 0 u s �−→ 0 otherwise

The AGT correspondence

� � 1 • Z inst = exp + · · · − F ���� ǫ 1 ǫ 2 SW prepotential from SW curve

� � 1 • Z inst = exp + · · · − F ���� ǫ 1 ǫ 2 SW prepotential from SW curve �� W s ( t ) �� 2D CFT Blocks with currents • lim ǫ i → 0 = SW curve coefficients φ s ( t ) .

� � 1 • Z inst = exp + · · · − F ���� ǫ 1 ǫ 2 SW prepotential from SW curve �� W s ( t ) �� 2D CFT Blocks with currents • lim ǫ i → 0 = SW curve coefficients φ s ( t ) . The SW curve knows a lot about the 2D CFT: type of algebra and nature of the primary fields

Repeat • The standard AGT correspondence is well established and has given many new results • It can be generalized in many directions (parafermionic Toda, etc...) • We want to generalize to the N = 1 SCFTs • The class S k SCFTs are obtained by orbifolding the usual N = 2 class S SCFTs with a Z k • Their SW curves are known. What do they say about the 2D CFT?

The 2D CFT side

W N Toda overview N − 1 free bosons with an exponential potential One coupling b 2 = ǫ 1 ǫ 2

W N Toda overview N − 1 free bosons with an exponential potential One coupling b 2 = ǫ 1 ǫ 2 • N = 2 ⇒ Liouville CFT is the simplest non-rational 2D CFT � ∞ z − n − 2 L n stress-tensor: W 2 ( z ) = T ( z ) = n = −∞ c 12 m ( m 2 − 1 ) δ m , − n + ( m − n ) L m + n [ L m , L n ] =

W N Toda overview N − 1 free bosons with an exponential potential One coupling b 2 = ǫ 1 ǫ 2 • N = 2 ⇒ Liouville CFT is the simplest non-rational 2D CFT � ∞ z − n − 2 L n stress-tensor: W 2 ( z ) = T ( z ) = n = −∞ c 12 m ( m 2 − 1 ) δ m , − n + ( m − n ) L m + n [ L m , L n ] = • Toda: Virasoro → W N with higher spin currents W 2 , . . . , W N The algebra becomes non-linear

Decomposition of the correlation function � � V 1 ( ∞ ) V 2 ( 1 ) V 3 ( q ) V 4 ( 0 ) = � �������� �� �������� � OPE

Decomposition of the correlation function � � V 1 ( ∞ ) V 2 ( 1 ) V 3 ( q ) V 4 ( 0 ) = � �������� �� �������� � OPE � � � C ( L − Y α ) 34 � V 1 ( ∞ ) V 2 ( 1 )( L − Y V α )( 0 ) � = d α q ∆ 3 +∆ 4 − ∆( L − Y α ) ���� Young diagram Y � ���������� �� ���������� � primary descendants