Supersymmetry on Curved Spaces and Holography

Supersymmetry on Curved Spaces and Holography based on: 1205.1062 C.K.-Tomasiello-Zaffaroni 1207.2181 Cassani-C.K.-Martelli-Tomasiello-Zaffaroni Claudius Klare Università di Milano-Bicocca

Outline Motivations Basic Strategy & Results Examples Future Directions

Outline Motivations Basic Strategy & Results Examples Future Directions

Exact results in SUSY QFT's Localization reduces the path integral to a matrix model

Exact results in SUSY QFT's Localization reduces the path integral to a matrix model on on on on [Pestun '07] [Kapustin-Willett-Yaakov '09, Jafferis, Hama-Hosomichi-Lee '10] [Benini-Cremonesi, Doroud-Gomis-LeFloch-Lee '12] [Hosomichi-Seong-Terashima, Källén-Qiu-Zabzine '12] Similar: SCFT index on [Römelsberger, Kinney-Maldacena-Minwalla-Raju '05]

Exact results in SUSY QFT's Localization reduces the path integral to a matrix model yielding new observables & exact results Check of Seiberg dualities AGT -, - scaling Z-minimization F-theorem 4d, 3d 4d/2d 3d, 5/6d 3d 3d

Exact results in SUSY QFT's Localization reduces the path integral to a matrix model Typically this requires curved spaces (e.g. ) Question: Which spaces are allowed?

Exact results in SUSY QFT's Localization reduces the path integral to a matrix model Typically this requires curved spaces (e.g. ) Question: Which spaces are allowed?...and what happens to AdS/CFT?

Outline Motivations Basic Strategy & Results Examples Future Directions

Related Works 1105.0689 Festuccia-Seiberg 1109.5421 Jia-Sharpe 1203.3420 Samtleben-Tsimpis 1205.1062 C.K.-Tomasiello-Zaffaroni 1205.1115 Dumitrescu-Festuccia-Seiberg 1207.2181 Cassani-C.K.-Tomasiello-Zaffaroni 1207.2785 Liu-Pando Zayas-Reichmann 1209.4043 de Medeiros 1209.5408 Dumitrescu-Festuccia 1211.1367 Kehagias-Russo

Problem Curvature breaks SUSY Complicated construction of and following the Noether procedure

Problem Curvature breaks SUSY Complicated construction of and following the Noether procedure Elegant Solution Couple the QFT to supergravity Freeze the gravity fields to (complex) background value, s.t. Get and for free [Karlhede-Rocek '88] [Adams-Jockers-Kumar-Lapan '11] [Festuccia-Seiberg '11]

If the QFT is a CFT, naturally: couple to conformal SUGRA!

If the QFT is a CFT, naturally: couple to conformal SUGRA! Recall: in conformal SUGRA the whole superconformal group is gauged vs. Poincaré SUGRA only usual SUSY group is local

If the QFT is a CFT, naturally: couple to conformal SUGRA! Superconformal theories can also be understood via their gravity dual. The curved manifold arises then as the boundary of a bulk geometry.

If the QFT is a CFT, naturally: couple to conformal SUGRA! Superconformal theories can also be understood via their gravity dual. The curved manifold arises then as the boundary of a bulk geometry. Relation between the two!?

The bulk theory: Start with minimal gauged SUGRA.

The bulk theory: Consider asymptotically locally AdS (ALAdS) as the solution of a gauged SUGRA. minimal:

The bulk theory: ALAdS in gauged SUGRA - expansion On the boundary: Conformal SUGRA!

The bulk theory: ALAdS in gauged SUGRA - expansion On the boundary: Conformal SUGRA!

The bulk theory: ALAdS in gauged SUGRA - expansion On the boundary: Conformal SUGRA! -trace

The bulk theory: ALAdS in gauged SUGRA - expansion On the boundary: Conformal Killing spinor (CKS) with

Manifolds admitting a conformal Killing spinor, admit SUSY.

Manifolds with a CKS / with SUSY Note: Spin - part of Conformally covariant

Manifolds with a CKS / with SUSY The case is studied: ( twistor equation )

Manifolds with a CKS / with SUSY The case is studied: Euclidean Case Dimension Manifold Cone

Manifolds with a CKS / with SUSY The case is studied: Euclidean Case Lorentzian Case either Fefferman (in 4d) ) or pp-wave metrics

Manifolds with a CKS / with SUSY

Manifolds with a CKS / with SUSY General strategy: 1. Bispinors forms defining -structure

Manifolds with a CKS / with SUSY General strategy: 1. Bispinors forms defining -structure Example (4d Euclidean): -structure

Manifolds with a CKS / with SUSY General strategy: 1. Bispinors forms defining -structure 2. Define intrinsic torsions from the forms

Manifolds with a CKS / with SUSY General strategy: 1. Bispinors forms defining -structure 2. Define intrinsic torsions from the forms Example:

Manifolds with a CKS / with SUSY General strategy: 1. Bispinors forms defining -structure 2. Define intrinsic torsions from the forms 3. CKS equation condition on intrinsic torsions

Manifolds with a CKS / with SUSY General strategy: 1. Bispinors forms defining -structure 2. Define intrinsic torsions from the forms 3. CKS equation condition on intrinsic torsions Example: etc.

Manifolds with a CKS / with SUSY General strategy: 1. Bispinors forms defining -structure 2. Define intrinsic torsions from the forms 3. CKS equation condition on intrinsic torsions Example: etc. computation computation

Manifolds with a CKS / with SUSY General strategy: 1. Bispinors forms defining -structure 2. Define intrinsic torsions from the forms 3. CKS equation condition on intrinsic torsions result: ( roughly!)

Manifolds with a CKS / with SUSY General strategy: 1. Bispinors forms defining -structure 2. Define intrinsic torsions from the forms 3. CKS equation condition on intrinsic torsions result: (and is complex! determined) [CK-Tomasiello-Zaffaroni Dumitrescu-Festuccia-Seiberg]

Comments: This determines and through the coupling (here: linearised) * is a local statement result: (and is complex! * determined)

Manifolds with a CKS / with SUSY - Results: [C.K.-Tomasiello-Zaffaroni, Cassani-C.K.-Martelli-Tomasiello-Zaffaroni,...] Dimension Forms Condition 4d (++++) complex 4d ( +++) conformal Killing 3d (+++)

SUSY Theories with an -symmetry (the non-conformal case)

SUSY Theories with an -symmetry Naturally: couple to new minimal SUGRA constraint:

SUSY Theories with an -symmetry Naturally: couple to new minimal SUGRA constraint: (3d follows by dimensional reduction... )

SUSY Theories with an -symmetry Naturally: couple to new minimal SUGRA Fact: This is a special case of CKS equation

SUSY Theories with an -symmetry Naturally: couple to new minimal SUGRA Fact: This is the special case of CKS equation for, where and

SUSY Theories with an -symmetry Naturally: couple to new minimal SUGRA Fact: This is the special case of CKS equation for The constraint can be imposed by fixing an ambiguity in the definition of

SUSY Theories with an -symmetry Naturally: couple to new minimal SUGRA Fact: This is the special case of CKS equation for Lorentzian case: reality of Weyl rescaling ( becomes Killing vector)

SUSY Theories with an -symmetry Comment: No surprise! Recall: Superconformal tensor calculus Conformal SUGRA gauge fixing the conformal symmetries (upon a compensator multiplet ) New minimal SUGRA

Outline Motivations Basic Strategy & Results Examples Future Directions

More explicitly: 4d Euclidean

More explicitly: 4d Euclidean Extra Detail: Equations for -structure : is complex!

More explicitly: 4d Euclidean Example: Kähler manifolds ( is real.)

More explicitly: 4d Euclidean Example: Kähler manifolds Note: in new minimal set-up: Characterises Kähler manifolds

More explicitly: 4d Euclidean Example: complex but non-kähler?

More explicitly: 4d Euclidean Example: ( is imaginary!)

More explicitly: 3d Euclidean

More explicitly: 3d Euclidean Get by dimensional reduction: Extra Detail: Equations for -structure :

More explicitly: 3d Euclidean Ex.: (Squashed) 3-sphere

More explicitly: 3d Euclidean Ex.: (Squashed) 3-sphere Choose vielbein (I): Determine background fields: see also: [Hama-Hosomichi-Lee]

More explicitly: 3d Euclidean Ex.: (Squashed) 3-sphere Choose vielbein (II): Determine background fields: see also: [Imamura-Yokoyama], [Martelli-Sparks]

More explicitly: 4d Lorentzian [Cassani-CK-Martelli-Tomasiello-Zaffaroni]

More explicitly: 4d Lorentzian The bilinears define a vielbein......up to an ambiguity: ( -structure )

More explicitly: 4d Lorentzian The bilinears define the vielbein: SUSY z Killing special coordinates: Parametrise most general metric: for some independent of

More explicitly: 4d Lorentzian We have explicit coordinate expressions for the background fields: etc....

More explicitly: 4d Lorentzian Note: in Lorentzian case, we have a complete classification of the gravity duals! [Gauntlett-Gutowski '03 Caldarelli-Klemm '03] Reduction to boundary reproduces our CKS results from the full bulk solutions Hard but interesting: Which of our boundary solutions extent to regular ones in the bulk?

Future Directions Regularity in bulk Imprint on boundary (via perturbative construction in )

Future Directions Other dimensions, supersymmetries: and AGT. New feature: Non-abelian! interesting in the context of the M5 in 4d [Lambert-Papageorgakis- Schmidt Sommerfeld, Douglas, '10] [Hosomichi-Seong-Terashima, Källén-Qiu-Zabzine, '12]

Thank You......Questions?

New Minimal in 3d: Disposal

Disposal The other vector in 4d Lorentzian:

Disposal

Detour: the free energy in The matrix model of the CFT on to be a very deep object. turns out

Detour: the free energy in The matrix model of the CFT on to be a very deep object. turns out It can be solved at large N applying a saddle point approximation as [Herzog, Klebanov, Pufu, Tesileanu] [Martelli-Sparks, Jafferis-Klebanov-Pufu-Safdi] Extremum of

Detour: the free energy in The matrix model of the CFT on to be a very deep object. turns out Actually, one finds R-charges of the fields Note: R-symmetry is not uniquely defined...

Detour: the free energy in Reproduces scaling [Drukker-Marino-Putrov '10,...] is extremised for the exact R-charge [Jafferis '10] F seems to obey an F-theorem [Jafferis-Klebanov-Pufu-Safdi '11] [ ] [Amariti-CK-Siani '11] [...] reproduces nicely the AdS expectations of Z-minimisation [Herzog-Klebanov-Pufu-Tesileanu '10] [Martelli-Sparks '11] [ ] [Amariti-CK-Siani '11] [...]

The matrix model [Kapustin-Willett-Yaakov '10] [Jafferis, Hama-Hosomichi-Lee '10] The partition function of the field theory on S^3 localizes in field space: all other fields Extra adjoint scalar in 3d from dim. red.:

The matrix model [Kapustin-Willett-Yaakov '10] [Jafferis, Hama-Hosomichi-Lee '10] The partition function of the field theory on S^3 localizes to a matrix model Cartan of G

The matrix model [Kapustin-Willett-Yaakov '10] [Jafferis, Hama-Hosomichi-Lee '10] The partition function of the field theory on S^3 localizes to a matrix model Cartan of G, typically: when For U(N), the roots are and, etc...

The large N saddle point [Herzog, Klebanov, Pufu, Tesileanu] [Martelli-Sparks, Jafferis-Klebanov-Pufu-Safdi] This can be solved at large N applying a saddle point approximation as Extremum of

The large N saddle point [Herzog, Klebanov, Pufu, Tesileanu] [Martelli-Sparks, Jafferis-Klebanov-Pufu-Safdi] This can be solved * at large N applying a saddle point approximation as Extremum of * Note: Only true for vector-like theories (i.e., wih matter ~ )

The large N saddle point [Herzog, Klebanov, Pufu, Tesileanu] [Martelli-Sparks, Jafferis-Klebanov-Pufu-Safdi] This can be solved at large N applying a saddle point approximation as Extremum of Compare with the AdS dual : [ ],[Amariti-CK-Siani],[...]

F-maximisation [Jafferis] Actually, one finds with ( ) maximised exact R-symmetry

F-theorem? F decreases along RG flow [Jafferis-Klebanov-Pufu-Safdi] [Amariti-CK-Siani]