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

Superconformal indices for Sasaki-Einstein backgrounds

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

Introduction and overview

(Very similar results hold for d=4, N=1 and d=10 type IIB, AdS5 x SE5.)

• Osp(2|4) multiplets
• Unitarity bounds
• Short multiplets
• Superconformal index
• Kaluza-Klein spectrum
• Laplace operator
• Kohn-Rossi cohomology
• Sum over cohomologies

d=3 SCFT, N=2, k=1 d=11 SUGRA, AdS4 x SE7

Sasaki-Einstein geometry

η 2J = dη J2 = −1 + η ⊗ η T ∗SE = Ω1,0 ⊕ Ω0,1 ⊕ Cη d = ∂B + ¯ ∂B + η ∧ £η

Kohn, Rossi; Yau; Gauntlett, Martelli, Sparks, Waldram; Boyer, Galicki

Decomposition of (co)tangent bundle Kohn-Rossi cohomology . . .

¯ ∂B

− − → Ωp,q−1

¯ ∂B

− − → Ωp,q

¯ ∂B

− − → Ωp,q+1

¯ ∂B

− − → . . . Hp,q

¯ ∂B (SE)

Osp(2|4) multiplets from the Kaluza-Klein spectrum

Ceresole, Dall’Agata, D’Auria, Ferrara, Fre, Gualtieri, Merlatti, Termonia

• Spectrum of ∆ is difficult beyond coset case G/H.
• Reproduces multiplet structure in supergravity.
• Possible due to SUSY.

∆f = δf, {∂Bf, ¯ ∂Bf, η ∧ f, J ∧ f, η ∧ ¯ ∂Bf, . . . } conformal energy ↔ AdS4 mass ↔ spectrum of ∆ on SE Recall:

• Approach: Generic SE manifolds.

Pope; Richard Eager, J.S., Yuji Tachikawa

Example: The graviton multiplet

Spin Energy Charge Mass2 Name Wave-f. 2 E0 + 1 y 4(E0 2)(E0 + 1) h f [0;q] c, ?

3 2

E0 + 1

2

y + 1 E0 2 + f [3/2]

∗

c, ? 1 E0 + 2 y 4E0(E0 + 1) W f [1;q;−] 1 E0 + 1 y 2 4E0(E0 1) Z f [2;q−4] ? 1 E0 + 1 y + 2 4E0(E0 1) Z f [2;q+4] 1 E0 + 1 y 4E0(E0 1) Z f [2;q;a,b] ? 1 E0 + 1 y 4E0(E0 1) Z f [2;q;a,b] 1 E0 y 4(E0 2)(E0 1) A f [1;q;+] c, ?, p E0 + 1 y 4E0(E0 1)

• f [2s;q]

∗

Unitarity bounds from supergravity

∆ = 2∆¯

∂B − £2 η − 2ı(n − d0)£η + 2LΛ + 2(n − d0)LηΛη + 2ı(Lη ¯

∂∗

B − ¯

∂BΛη) [Λ, ¯ ∂B] = −ı∂∗

B + ıLηΛ + (n − d0)Λη

The unitarity bounds: ✏ j3 + y + 1|j3 6= 0; ✏ y + 1 _ ✏ = y|j3 = 0 The Laplace operator on SE2n+1 horizontal degree Lefschetz and adjoint Reeb and adjoint Proof via Kähler-like identities

J.S.

Short multiplets and cohomology

Short graviton Short gravitino Short vector Z/Betti Short vector A Hyper Hyper Hyper H3,0

¯ ∂B (SE)

H1,0

¯ ∂B (SE)

H1,1

¯ ∂B (SE)

H2,0

¯ ∂B (SE)

H2,1

¯ ∂B (SE)

H0,0

¯ ∂B (SE)

H1,2

¯ ∂B (SE)

Richard Eager, J.S

The superconformal index

Is.t.(t) = Trs.t.[(−1)F t✏+j3], ✏ = j3 + y

Bhattacharya, Bhattacharyya, Kinney, Maldacena, Minwalla, Raju; Romelsberger; Gadde, Rastelli, Razamat, Yan

1 + Is.t. = X Tr t£η|H0,0

¯ ∂B (SE) H2,0 ¯ ∂B (SE) H2,1 ¯ ∂B (SE)

t2H1,0

¯ ∂B (SE) t2H1,1 ¯ ∂B (SE) t2H1,2 ¯ ∂B (SE) t2H3,0 ¯ ∂B (SE)

Supergravity - matches known duals Only these contribute.

Richard Eager, J.S.

Summary

Future directions:

• Revisit gauge duals.
• A complete Kaluza-Klein analysis.
• Decomposition of the Laplacian - mathematics and applications to physics.
• Beyond Sasaki-Einstein.

Thank you.

• Osp(2|4) multiplets
• Unitarity bounds
• Short multiplets
• Superconformal index
• Kaluza-Klein spectrum
• Laplace operator
• Kohn-Rossi cohomology
• Sum over cohomologies

d=3 SCFT, N=2, k=1 d=11 SUGRA, AdS4 x SE7