supersymmetry on curved spaces and holography - ntuamib_infn_it_01.pdf · supersymmetry on curved...
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![Page 1: Supersymmetry on Curved Spaces and Holography - NTUAmib_infn_it_01.pdf · Supersymmetry on Curved Spaces and Holography Claudius Klare Università di Milano-Bicocca 20/9/12 - Eurostrings](https://reader031.vdocuments.site/reader031/viewer/2022022110/5c0ce1e009d3f213228cbefc/html5/thumbnails/1.jpg)
Supersymmetry on Curved Spaces and
Holography
Claudius KlareUniversità di Milano-Bicocca
20/9/12 - Eurostrings Corfu
based on:
1205.1062 CK-Tomasiello-Zaffaroni 1207.2181 Cassani-CK-Martelli-Tomasiello-Zaffaroni
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Outline●
Motivations
Basic Strategy
Results
Future Directions
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Outline●
Motivations
Basic Strategy
Results
Future Directions
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Exact results in SUSY QFT's
● Localization reduces the path integral to a matrix model yielding new observables & exact results
→ imagine talk by D. Jafferis
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Exact results in SUSY QFT's
● Check of Seiberg dualities ● AGT● -, - scaling● Z-minimization & F-theorem● SCFT index● …
← 4d, 3d
← 4d/2d
← 3d, 5/6d
← 3d
← 4d
● Localization reduces the path integral to a matrix model yielding new observables & exact results
→ imagine talk by D. Jafferis
![Page 6: Supersymmetry on Curved Spaces and Holography - NTUAmib_infn_it_01.pdf · Supersymmetry on Curved Spaces and Holography Claudius Klare Università di Milano-Bicocca 20/9/12 - Eurostrings](https://reader031.vdocuments.site/reader031/viewer/2022022110/5c0ce1e009d3f213228cbefc/html5/thumbnails/6.jpg)
● Localization reduces the path integral to a matrix model
● Typically this requires curved spaces (e.g. )
Exact results in SUSY QFT's
Question: Which spaces are allowed?
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Outline●
Motivations
Basic Strategy
Results
Future Directions
![Page 8: Supersymmetry on Curved Spaces and Holography - NTUAmib_infn_it_01.pdf · Supersymmetry on Curved Spaces and Holography Claudius Klare Università di Milano-Bicocca 20/9/12 - Eurostrings](https://reader031.vdocuments.site/reader031/viewer/2022022110/5c0ce1e009d3f213228cbefc/html5/thumbnails/8.jpg)
Problem● Curvature breaks SUSY● Complicated construction of and
following the “Noether procedure”
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Elegant Solution● Couple the QFT to supergravity ● Freeze the gravity fields to (complex)
background value, s.t.● Get and “for free”
Problem
[Festuccia-Seiberg '11]
● Curvature breaks SUSY● Complicated construction of and
following the “Noether procedure”
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● If the QFT is a CFT, naturally: couple to “conformal SUGRA”!
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Recall: in conformal SUGRA the whole superconformal group is gaugedvs. “Poincaré SUGRA” only usual SUSY ↔group is local
● If the QFT is a CFT, naturally: couple to “conformal SUGRA”!
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● 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.
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● 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!?
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The bulk theory: gauged SUGRA.
minimal:
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The bulk theory:
Conformal SUGRA!
- expansion
On the boundary:
Minimal gauged SUGRA
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Conformal Killing spinor (CKS)
On the boundary:
with
The bulk theory:
- expansion
Minimal gauged SUGRA
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Manifolds admitting a conformal Killing spinor (CKS), admit SUSY.
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The case is studied:
Manifolds with a CKS / with SUSY
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Manifolds with a CKS / with SUSY
● Euclidean Case →
Dimension
Manifold
Cone
The case is studied:
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● Euclidean Case →
● Lorentzian Case either → Fefferman (in 4d) ) or pp-wave metrics
Manifolds with a CKS / with SUSY
The case is studied:
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Manifolds with a CKS / with SUSY
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Manifolds with a CKS / with SUSY
1. Bispinors forms defining -structure
General strategy:
![Page 23: Supersymmetry on Curved Spaces and Holography - NTUAmib_infn_it_01.pdf · Supersymmetry on Curved Spaces and Holography Claudius Klare Università di Milano-Bicocca 20/9/12 - Eurostrings](https://reader031.vdocuments.site/reader031/viewer/2022022110/5c0ce1e009d3f213228cbefc/html5/thumbnails/23.jpg)
Manifolds with a CKS / with SUSY
1. Bispinors forms defining -structure
Example (4d Euclidean):
General strategy:
-structure
![Page 24: Supersymmetry on Curved Spaces and Holography - NTUAmib_infn_it_01.pdf · Supersymmetry on Curved Spaces and Holography Claudius Klare Università di Milano-Bicocca 20/9/12 - Eurostrings](https://reader031.vdocuments.site/reader031/viewer/2022022110/5c0ce1e009d3f213228cbefc/html5/thumbnails/24.jpg)
Manifolds with a CKS / with SUSY
1. Bispinors forms defining -structure
2. CKS equation conditions on the forms
General strategy:
![Page 25: Supersymmetry on Curved Spaces and Holography - NTUAmib_infn_it_01.pdf · Supersymmetry on Curved Spaces and Holography Claudius Klare Università di Milano-Bicocca 20/9/12 - Eurostrings](https://reader031.vdocuments.site/reader031/viewer/2022022110/5c0ce1e009d3f213228cbefc/html5/thumbnails/25.jpg)
Manifolds with a CKS / with SUSY
1. Bispinors forms defining -structure
2. CKS equation conditions on the forms
4d Euclidean:
General strategy:
is complex!
[CK-Tomasiello-ZaffaroniDumitrescu-Festuccia-Seiberg]
![Page 26: Supersymmetry on Curved Spaces and Holography - NTUAmib_infn_it_01.pdf · Supersymmetry on Curved Spaces and Holography Claudius Klare Università di Milano-Bicocca 20/9/12 - Eurostrings](https://reader031.vdocuments.site/reader031/viewer/2022022110/5c0ce1e009d3f213228cbefc/html5/thumbnails/26.jpg)
Manifolds with a CKS / with SUSY
1. Bispinors forms defining -structure
2. CKS equation conditions on the forms
4d Euclidean:
General strategy:
is complex!
determines+ (roughly):
![Page 27: Supersymmetry on Curved Spaces and Holography - NTUAmib_infn_it_01.pdf · Supersymmetry on Curved Spaces and Holography Claudius Klare Università di Milano-Bicocca 20/9/12 - Eurostrings](https://reader031.vdocuments.site/reader031/viewer/2022022110/5c0ce1e009d3f213228cbefc/html5/thumbnails/27.jpg)
Manifolds with a CKS / with SUSY
1. forms defining -structure
2. CKS equation conditions on the forms
4d Euclidean:
is complex!
determines+ (roughly):
● This determines and
Comments:
*
● is a local statement*
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Outline●
Motivations
Basic Strategy
Results
Future Directions
![Page 29: Supersymmetry on Curved Spaces and Holography - NTUAmib_infn_it_01.pdf · Supersymmetry on Curved Spaces and Holography Claudius Klare Università di Milano-Bicocca 20/9/12 - Eurostrings](https://reader031.vdocuments.site/reader031/viewer/2022022110/5c0ce1e009d3f213228cbefc/html5/thumbnails/29.jpg)
conformal Killing
Manifolds with a CKS / with SUSY- Results:
Dimension Forms Condition
4d (–+++)
complex 4d (++++)
3d (+++)
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Comment: The non-conformal case
● (Non-conformal) SUSY Theories with an -symmetry are automatically included in the analysis as the special case where
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Comment: The non-conformal case
● (Non-conformal) SUSY Theories with an -symmetry are automatically included in the analysis as the special case where
● Get the same manifolds as before! ...with subtleties:
● Local global→● In 4d Lorentz, CKV KV→
![Page 32: Supersymmetry on Curved Spaces and Holography - NTUAmib_infn_it_01.pdf · Supersymmetry on Curved Spaces and Holography Claudius Klare Università di Milano-Bicocca 20/9/12 - Eurostrings](https://reader031.vdocuments.site/reader031/viewer/2022022110/5c0ce1e009d3f213228cbefc/html5/thumbnails/32.jpg)
SUSY Theories with an -symmetry(the non-conformal case)
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SUSY Theories with an -symmetry
● Naturally: couple to “new minimal SUGRA”
constraint:
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SUSY Theories with an -symmetry
● Naturally: couple to “new minimal SUGRA”
(3d follows by dimensional reduction... )
constraint:
→ see also: talk by G. Tartaglino-Mazzucchelli
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SUSY Theories with an -symmetry
● Naturally: couple to “new minimal SUGRA”
● Fact: This is a special case of CKS equation
![Page 36: Supersymmetry on Curved Spaces and Holography - NTUAmib_infn_it_01.pdf · Supersymmetry on Curved Spaces and Holography Claudius Klare Università di Milano-Bicocca 20/9/12 - Eurostrings](https://reader031.vdocuments.site/reader031/viewer/2022022110/5c0ce1e009d3f213228cbefc/html5/thumbnails/36.jpg)
where and
SUSY Theories with an -symmetry
● Naturally: couple to “new minimal SUGRA”
● Fact: This is the special case of CKS equation for ,
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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
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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)
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SUSY Theories with an -symmetry
Comment: No surprise!
Recall: Superconformal tensor calculus …
New minimal SUGRA
Conformal SUGRA
gauge fixing the conformal symmetries (upon a “compensator
multiplet” )
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Future Directions
● Regularity in bulk Imprint on boundary↔(via perturbative construction in )
● Other dimensions:
interesting in the context of the M5 →and AGT.New feature: Non-abelian!
● in 4d
→ see also: A. Passias' talk
→ see also: N. Lambert's talk
D. Jafferis' talk
F. Bonetti's talk
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Thank You...
...Questions?
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Outline●
Motivations
Basic Strategy & Results
Examples
Future Directions
![Page 43: Supersymmetry on Curved Spaces and Holography - NTUAmib_infn_it_01.pdf · Supersymmetry on Curved Spaces and Holography Claudius Klare Università di Milano-Bicocca 20/9/12 - Eurostrings](https://reader031.vdocuments.site/reader031/viewer/2022022110/5c0ce1e009d3f213228cbefc/html5/thumbnails/43.jpg)
More explicitly: 4d Euclidean
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More explicitly: 4d Euclidean
● Extra Detail:
● Equations for -structure :
is complex!
![Page 45: Supersymmetry on Curved Spaces and Holography - NTUAmib_infn_it_01.pdf · Supersymmetry on Curved Spaces and Holography Claudius Klare Università di Milano-Bicocca 20/9/12 - Eurostrings](https://reader031.vdocuments.site/reader031/viewer/2022022110/5c0ce1e009d3f213228cbefc/html5/thumbnails/45.jpg)
More explicitly: 4d Euclidean
Example: Kähler manifolds
( is real.)
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More explicitly: 4d Euclidean
Example: Kähler manifolds
Note: in new minimal set-up:
Characterises Kähler manifolds
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More explicitly: 4d Euclidean
Example: complex but non-Kähler?
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More explicitly: 4d Euclidean
Example:
( is imaginary!)
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More explicitly: 3d Euclidean
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More explicitly: 3d Euclidean
● Get by dimensional reduction:
● Extra Detail:
● Equations for -structure :
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More explicitly: 3d Euclidean
Ex.: (Squashed) 3-sphere
![Page 52: Supersymmetry on Curved Spaces and Holography - NTUAmib_infn_it_01.pdf · Supersymmetry on Curved Spaces and Holography Claudius Klare Università di Milano-Bicocca 20/9/12 - Eurostrings](https://reader031.vdocuments.site/reader031/viewer/2022022110/5c0ce1e009d3f213228cbefc/html5/thumbnails/52.jpg)
More explicitly: 3d Euclidean
Choose vielbein:
Determine background fields:
Ex.: (Squashed) 3-sphere
![Page 53: Supersymmetry on Curved Spaces and Holography - NTUAmib_infn_it_01.pdf · Supersymmetry on Curved Spaces and Holography Claudius Klare Università di Milano-Bicocca 20/9/12 - Eurostrings](https://reader031.vdocuments.site/reader031/viewer/2022022110/5c0ce1e009d3f213228cbefc/html5/thumbnails/53.jpg)
More explicitly: 4d Lorentzian[Cassani-CK-Martelli-Tomasiello-Zaffaroni]
![Page 54: Supersymmetry on Curved Spaces and Holography - NTUAmib_infn_it_01.pdf · Supersymmetry on Curved Spaces and Holography Claudius Klare Università di Milano-Bicocca 20/9/12 - Eurostrings](https://reader031.vdocuments.site/reader031/viewer/2022022110/5c0ce1e009d3f213228cbefc/html5/thumbnails/54.jpg)
More explicitly: 4d Lorentzian
● The bilinears define a vielbein...
(“ -structure”)
...up to an ambiguiety:
![Page 55: Supersymmetry on Curved Spaces and Holography - NTUAmib_infn_it_01.pdf · Supersymmetry on Curved Spaces and Holography Claudius Klare Università di Milano-Bicocca 20/9/12 - Eurostrings](https://reader031.vdocuments.site/reader031/viewer/2022022110/5c0ce1e009d3f213228cbefc/html5/thumbnails/55.jpg)
More explicitly: 4d Lorentzian
● The bilinears define the vielbein:
● SUSY z Killing special coordinates:
for some independent of
Parametrise most general metric:
![Page 56: Supersymmetry on Curved Spaces and Holography - NTUAmib_infn_it_01.pdf · Supersymmetry on Curved Spaces and Holography Claudius Klare Università di Milano-Bicocca 20/9/12 - Eurostrings](https://reader031.vdocuments.site/reader031/viewer/2022022110/5c0ce1e009d3f213228cbefc/html5/thumbnails/56.jpg)
More explicitly: 4d Lorentzian
We have explicit coordinate expressions for the background fields:
etc. ...
![Page 57: Supersymmetry on Curved Spaces and Holography - NTUAmib_infn_it_01.pdf · Supersymmetry on Curved Spaces and Holography Claudius Klare Università di Milano-Bicocca 20/9/12 - Eurostrings](https://reader031.vdocuments.site/reader031/viewer/2022022110/5c0ce1e009d3f213228cbefc/html5/thumbnails/57.jpg)
More explicitly: 4d Lorentzian
Note: in Lorentzian case, we have a complete classification of the gravity duals!
[Gauntlett-Gutowski '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?
![Page 58: Supersymmetry on Curved Spaces and Holography - NTUAmib_infn_it_01.pdf · Supersymmetry on Curved Spaces and Holography Claudius Klare Università di Milano-Bicocca 20/9/12 - Eurostrings](https://reader031.vdocuments.site/reader031/viewer/2022022110/5c0ce1e009d3f213228cbefc/html5/thumbnails/58.jpg)
Disposal
New Minimal in 3d:
![Page 59: Supersymmetry on Curved Spaces and Holography - NTUAmib_infn_it_01.pdf · Supersymmetry on Curved Spaces and Holography Claudius Klare Università di Milano-Bicocca 20/9/12 - Eurostrings](https://reader031.vdocuments.site/reader031/viewer/2022022110/5c0ce1e009d3f213228cbefc/html5/thumbnails/59.jpg)
Disposal
The other vector in 4d Lorentzian: