Sphere Partition Functions, the Zamolodchikov Metric and Surface Operators Jaume Gomis Princeton, Strings 2014 with Gerchkovitz, Komargodski, arXiv:1405.7271 with Le Floch, to appear
Introduction • Recent years have seen dramatic progress in the exact computation of partition functions of supersymmetric field theories on curved spaces • In geometries S 1 × M d , the partition function has a standard Hilbert space interpretation as a sum over states � ( − 1) F e − βH � Z [ S 1 × M d ] = Tr H 1) What does the partition function of a (S)CFT on S d compute? • Physical Interpretation • Ambiguities of Z S d 2) Sphere partition function = ⇒ M2 ⊂ M5-brane surface operators
Sphere Partition Function in Conformal Manifold � d d xλ i O i define a family of CFTs spanning the • Exactly marginal operators conformal manifold S : λ i are coordinates and O i are vectors fields in S • Conformal manifold S admits Riemannian metric: Zamolodchikov metric � O i ( x ) O j (0) � p = G ij ( p ) p ∈ S x 2 d • CFT can be canonically put on sphere for any p ∈ S • Sphere partition function is an infrared finite observable • Z S d is a probe of the conformal manifold S
• Observable �O� λ defined by expansion around reference CFT � �� � k � � d d x √ gλ i O i ( x ) 1 �O� λ = O k ! k • Integrated correlation functions have ultraviolet divergences • Need to renormalize so that �O� λ has a continuum limit • The structure of divergences of sphere partition function is log Z S 2 n = A 1 [ λ i ]( r Λ UV ) 2 n ... + A n [ λ i ]( r Λ UV ) 2 + A [ λ i ] log( r Λ UV ) + F 2 n [ λ i ] log Z S 2 n +1 = B 1 [ λ i ]( r Λ UV ) 2 n +1 ... + B n +1 [ λ i ]( r Λ UV ) + F 2 n +1 [ λ i ] • Different renormalization schemes differ by diffeomorphism invariant local terms with ∆ ≤ d constructed from background fields g mn ( x ) and λ i → λ i ( x ) L ( g mn , λ i )
• All power-law divergences can be tuned by appropriate counterterms • In even dimensions: • The finite piece F 2 n [ λ i ] is ambiguous, there is a finite counterterm � d 2 n x √ gF 2 n [ λ i ] E 2 n • There is no local counterterm for the A [ λ i ] log( r Λ UV ) term • Consistency requires that A [ λ i ] = A , the A-type anomaly • In odd dimensions: • There is no finite counterterm for Re( F 2 n +1 [ λ i ]) • Consistency requires that Re( F 2 n +1 [ λ i ]) = Re( F 2 n +1 )
• All power-law divergences can be tuned by appropriate counterterms • In even dimensions: • The finite piece F 2 n [ λ i ] is ambiguous, there is a finite counterterm � d 2 n x √ gF 2 n [ λ i ] E 2 n • There is no local counterterm for the A [ λ i ] log( r Λ UV ) term • Consistency requires that A [ λ i ] = A , the A-type anomaly • In odd dimensions: • There is no finite counterterm for Re( F 2 n +1 [ λ i ]) • Consistency requires that Re( F 2 n +1 [ λ i ]) = Re( F 2 n +1 ) Summary • Unambiguous quantities A and Re( F 2 n +1 ) are constant along S • A and Re( F 2 n +1 ) measure entanglement entropy across a sphere in the CFT Casini,Huerta,Myers
SCFT Sphere Partition Functions • Regulate the divergences in a supersymmetric way • Preserve a “massive” subalgebra of superconformal algebra { Q, Q } = SO ( d + 1) ⊕ R-symmetry This is the general supersymmetry algebra of a massive theory on S d • Counterterms are diffeomorphism and supersymmetric invariant = ⇒ supergravity counterterms • Realize S 2 n as supersymmetric background in a supergravity theory Festuccia,Seiberg supergravity multiplet: g mn , ψ m , . . . • Represent λ i as bottom component of a superfield Φ i ( x, Θ) | = λ i ( x ) • Supergravity invariant constructed from supergravity multiplet and Φ i L ( g mn , ψ m , . . . ; λ i , . . . )
Two Dimensional N = (2 , 2) SCFTs • Includes worldsheet description of string theory on Calabi-Yau manifolds • Conformal manifold S is K¨ ahler and locally S c × S tc • ∄ an N = (2 , 2) superconformal invariant regulator. ∃ two massive N = (2 , 2) subalgebras on S 2 mirror SU (2 | 1) A ← − − − → SU (2 | 1) B • Defines partition functions Z A and Z B Benini,Cremonesi; Doroud,J.G,Le Floch,Lee Doroud,J.G • Compute the exact K¨ ahler potential K on the conformal manifold Jockers,Kumar,Lapan,Morrison,Romo J.G,Lee Z A = e − K tc Z B = e − K c • Partition function subject to ambiguity under Kahler transformations ¯ K → K + F ( λ i ) + ¯ F (¯ i ) λ F is a holomorpic function instead of an arbitrary real function of the moduli
• K¨ ahler ambiguity counterterm in Type A/B 2d N = (2 , 2) supergravity. Supergravities gauge either U (1) V or U (1) A R-symmetry • Coordinates in S c are bottom components of chiral multiplets Φ i • Coordinates in S tc are bottom components of twisted chiral multiplets Ω i • The SU (2 | 1) B K¨ ahler ambiguity is due to the supergravity coupling � � d 2 x √ g F ( λ i ) + c.c 1 d 2 xd 2 Θ εR F (Φ i ) + c.c ⊃ r 2 F : holomorphic function R : chiral superfield containing R as top component ε : chiral density superspace measure • The SU (2 | 1) A K¨ ahler ambiguity is parametrized by � d 2 xd Θ + d ˜ Θ − ˆ εF F (Ω i ) + c.c
4d N = 2 SCFTs • Conformal Manifold S of 4d N = 2 SCFTs is K¨ ahler • SCFT on S 4 can be deformed by exactly marginal operators � � � � d 4 x √ g i ¯ τ i C i + ¯ τ ¯ C ¯ i i C i : top component of 4d N = 2 chiral multiplet with bottom component A i τ i : coordinates on conformal manifold S • Regulate divergences of Z S 4 in an OSp (2 | 4) ⊂ SU (2 , 2 | 2) invariant way • Calculate by supersymmetric localization or using Ward identity �� � � S 4 d 4 x √ g C i ( x ) S 4 d 4 y √ g ¯ ∂ i ∂ ¯ j log Z S 4 = C ¯ j ( y ) � � A i ( N ) ¯ = A ¯ j ( S ) = G i ¯ j = ∂ i ∂ ¯ j K • Z S 4 of 4d N = 2 SCFTs computes the K¨ ahler potential on S Z S 4 = e K/ 12
• How about 4d N = 1 SCFTs? • Conformal Manifold S is K¨ ahler • Partition Function regulated in an OSp (1 | 4) ⊂ SU (2 , 2 | 1) invariant way • ∃ 4d N = 1 (old minimal) supergravity finite counterterm � � � d 4 x √ gF ( λ i , ¯ i ) ⊃ 1 D 2 − 8 R ) R ¯ ¯ ¯ d 2 Θ ε ( ¯ RF (Φ i , ¯ d 4 x i ) Φ λ r 4 F arbitrary = ⇒ Z S 4 for N = 1 SCFTs is ambiguous
• How about 4d N = 1 SCFTs? • Conformal Manifold S is K¨ ahler • Partition Function regulated in an OSp (1 | 4) ⊂ SU (2 , 2 | 1) invariant way • ∃ 4d N = 1 (old minimal) supergravity finite counterterm � � � d 4 x √ gF ( λ i , ¯ i ) ⊃ 1 D 2 − 8 R ) R ¯ ¯ ¯ d 2 Θ ε ( ¯ RF (Φ i , ¯ d 4 x i ) Φ λ r 4 F arbitrary = ⇒ Z S 4 for N = 1 SCFTs is ambiguous Summary • S 2 n partition function of SCFTs may have reduced space of ambiguities • Sphere partition functions of 2d N = (2 , 2) and 4d N = 2 SCFTs capture the exact K¨ ahler potential on their conformal manifold
Surface Operators and M2-branes to appear J.G, Le Floch • M2-branes ending on N f M5-branes M5 0 1 2 3 4 5 M2 0 1 6 insert a surface operator in the 6d N = (2 , 0) A N f − 1 SCFT • Surface operators labeled by a representation R of SU ( N f ) • M5-branes wrapping a punctured Riemann surface C realize a large class of 4d N = 2 theories (class S) Gaiotto • M2-branes ending on N f M5-branes insert a surface operator in the corresponding 4d N = 2 theory C M5 0 1 2 3 4 5 M2 0 1 6
• Surface operators in 4d gauge theories Gukov,Witten • Order parameters that go beyond the Wilson-’t Hooft criteria • Can be described by coupling 2d defect dof to the bulk gauge theory • Coupled 4d/2d system can exhibit new dynamics and dualities • M2-brane surface operators preserve a 2d N = (2 , 2) subalgebra of 4d N = 2
• Surface operators in 4d gauge theories Gukov,Witten • Order parameters that go beyond the Wilson-’t Hooft criteria • Can be described by coupling 2d defect dof to the bulk gauge theory • Coupled 4d/2d system can exhibit new dynamics and dualities • M2-brane surface operators preserve a 2d N = (2 , 2) subalgebra of 4d N = 2 • We have identified the 2d gauge theories corresponding to M2-branes · · · N N n f N n − 1 − N n ← → · · · N 1 N 2 N n N 2 − N 3 N f N 1 − N 2
• Surface operator obtained by identifying the SU ( N f ) × SU ( N f ) × U (1) symmetry of the 2d gauge theory with a corresponding gauge or global symmetry of 4d N = 2 theory 4d 2d 4d 2d N N f f N 1 · · · N n N N f f N 1 · · · N n N N f f • A superpotential on the defect couples 2d fields to 4d fields
• S 4 b partition function of T C is captured by Toda CFT correlator in C Pestun AGT Z [ T C ] ↔ • Conjecturally, a degenerate puncture describes a surface operator AGGTV
• S 4 b partition function of T C is captured by Toda CFT correlator in C Pestun AGT Z [ T C ] ↔ • Conjecturally, a degenerate puncture describes a surface operator AGGTV S 4 b partition function of T C Toda CFT correlator on C + our 2d gauge theory on S 2 • = + extra degenerate labelled by R (Ω) with momentum α = − b Ω Ω Z R [Ω] b ↔ S 2 ⊂ S 4
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