Superconformal indices for Sasaki-Einstein backgrounds Johannes - PowerPoint PPT Presentation

Superconformal indices for Sasaki-Einstein backgrounds Johannes Schmude RIKEN (until tomorrow), Oviedo (from Friday) Gauge/gravity duality 2013, Munich Based on work with Richard Eager and Yuji Tachikawa: arXiv:1207.0573, 1305.3547, 1307.xxxx

Introduction and overview d=3 SCFT, N =2, k=1 d=11 SUGRA, AdS 4 x SE 7 • Osp(2|4) multiplets • Kaluza-Klein spectrum • Unitarity bounds • Laplace operator • Short multiplets • Kohn-Rossi cohomology • Superconformal index • Sum over cohomologies (Very similar results hold for d=4, N =1 and d=10 type IIB, AdS 5 x SE 5 .)

Sasaki-Einstein geometry 2 J = d η η Decomposition of (co)tangent bundle J 2 = − 1 + η ⊗ η T ∗ SE = Ω 1 , 0 ⊕ Ω 0 , 1 ⊕ C η d = ∂ B + ¯ ∂ B + η ∧ £ η Kohn-Rossi cohomology ¯ ¯ ¯ ¯ ∂ B ∂ B ∂ B ∂ B → Ω p,q − 1 → Ω p,q +1 H p,q ∂ B ( SE ) → Ω p,q − − − − − − − − → . . . ¯ . . . Kohn, Rossi; Yau; Gauntlett, Martelli, Sparks, Waldram; Boyer, Galicki

Osp(2|4) multiplets from the Kaluza-Klein spectrum Recall: conformal energy ↔ AdS 4 mass ↔ spectrum of ∆ on SE • Spectrum of ∆ is difficult beyond coset case G/H. Ceresole, Dall’Agata, D’Auria, Ferrara, Fre, Gualtieri, Merlatti, Termonia • Approach: Generic SE manifolds. { ∂ B f, ¯ ∂ B f, η ∧ f, J ∧ f, η ∧ ¯ ∂ B f, . . . } ∆ f = δ f, • Reproduces multiplet structure in supergravity. • Possible due to SUSY. Pope; Richard Eager, J.S., Yuji Tachikawa

Example: The graviton multiplet Mass 2 Spin Energy Charge Name Wave-f. f [0; q ] 2 E 0 + 1 y 4( E 0 � 2)( E 0 + 1) h c, ? f [3 / 2] 3 E 0 + 1 � + y + 1 E 0 � 2 c, ? 2 2 ∗ f [1; q ; − ] 1 E 0 + 2 4 E 0 ( E 0 + 1) y W f [2; q − 4] 1 E 0 + 1 y � 2 4 E 0 ( E 0 � 1) Z ? f [2; q +4] 1 E 0 + 1 y + 2 4 E 0 ( E 0 � 1) Z f [2; q ; a,b ] 1 E 0 + 1 4 E 0 ( E 0 � 1) y Z ? f [2; q ; a,b ] 1 E 0 + 1 4 E 0 ( E 0 � 1) y Z f [1; q ;+] 1 4( E 0 � 2)( E 0 � 1) E 0 y A c, ? , p f [2 s ; q ] 0 E 0 + 1 4 E 0 ( E 0 � 1) y � ∗

Unitarity bounds from supergravity The unitarity bounds: ✏ � j 3 + y + 1 | j 3 6 = 0; ✏ � y + 1 _ ✏ = y | j 3 = 0 The Laplace operator on SE 2n+1 η − 2 ı ( n − d 0 ) £ η + 2 L Λ + 2( n − d 0 ) L η Λ η + 2 ı ( L η ¯ B − ¯ ∂ B − £ 2 ∆ = 2 ∆ ¯ ∂ ∗ ∂ B Λ η ) horizontal degree Reeb and adjoint Lefschetz and adjoint Proof via Kähler-like identities [ Λ , ¯ B + ıL η Λ + ( n − d 0 ) Λ η ∂ B ] = − ı ∂ ∗ J.S.

Short multiplets and cohomology H 3 , 0 Short graviton ∂ B ( SE ) ¯ H 1 , 0 Short gravitino ∂ B ( SE ) ¯ H 1 , 1 Short vector Z/Betti ∂ B ( SE ) ¯ H 2 , 0 Short vector A ∂ B ( SE ) ¯ H 2 , 1 Hyper ∂ B ( SE ) ¯ H 1 , 2 Hyper ∂ B ( SE ) ¯ H 0 , 0 Hyper ∂ B ( SE ) ¯ Richard Eager, J.S

The superconformal index Only these contribute. I s.t. ( t ) = Tr s.t. [( − 1) F t ✏ + j 3 ] , ✏ = j 3 + y Bhattacharya, Bhattacharyya, Kinney, Maldacena, Minwalla, Raju; Romelsberger; Gadde, Rastelli, Razamat, Yan Supergravity - matches known duals X Tr t £ η | H 0 , 0 ∂ B ( SE ) H 2 , 0 ∂ B ( SE ) � H 2 , 1 1 + I s.t. = ∂ B ( SE ) ¯ ¯ ¯ � t 2 H 1 , 0 ∂ B ( SE ) t 2 H 1 , 1 ∂ B ( SE ) � t 2 H 1 , 2 ∂ B ( SE ) t 2 H 3 , 0 ∂ B ( SE ) ¯ ¯ ¯ ¯ Richard Eager, J.S.

Summary d=3 SCFT, N =2, k=1 d=11 SUGRA, AdS 4 x SE 7 • Osp(2|4) multiplets • Kaluza-Klein spectrum • Unitarity bounds • Laplace operator • Short multiplets • Kohn-Rossi cohomology • Superconformal index • Sum over cohomologies Future directions: • Revisit gauge duals. • A complete Kaluza-Klein analysis. • Decomposition of the Laplacian - mathematics and applications to physics. • Beyond Sasaki-Einstein. Thank you.