The Geometry of Non-Projected Supermanifolds through Simple Examples Simone Noja Universit` a del Piemonte Orientale simone.noja@uniupo.it 15 May 2018 Simone Noja Supergeometry and Non-Projected Supermanifolds 15 May 2018 1 / 38
Scheme of the Talk WHY SUPERGEOMETRY? Motivations and Premises from Theoretical Physics: Simone Noja Supergeometry and Non-Projected Supermanifolds 15 May 2018 2 / 38
Scheme of the Talk WHY SUPERGEOMETRY? Motivations and Premises from Theoretical Physics: issues in Superstring Perturbation Theory a result due to Donagi and Witten (2013) Simone Noja Supergeometry and Non-Projected Supermanifolds 15 May 2018 2 / 38
Scheme of the Talk WHY SUPERGEOMETRY? Motivations and Premises from Theoretical Physics: issues in Superstring Perturbation Theory a result due to Donagi and Witten (2013) HOW SUPERGEOMETRY? Supergeometry in Action: Simone Noja Supergeometry and Non-Projected Supermanifolds 15 May 2018 2 / 38
Scheme of the Talk WHY SUPERGEOMETRY? Motivations and Premises from Theoretical Physics: issues in Superstring Perturbation Theory a result due to Donagi and Witten (2013) HOW SUPERGEOMETRY? Supergeometry in Action: Instruments and Methods from Algebraic Geometry: sheaves / vector bundles, exact sequences, ˇ Cech cohomology and invariants... Simone Noja Supergeometry and Non-Projected Supermanifolds 15 May 2018 2 / 38
Scheme of the Talk WHY SUPERGEOMETRY? Motivations and Premises from Theoretical Physics: issues in Superstring Perturbation Theory a result due to Donagi and Witten (2013) HOW SUPERGEOMETRY? Supergeometry in Action: Instruments and Methods from Algebraic Geometry: sheaves / vector bundles, exact sequences, ˇ Cech cohomology and invariants... Supermanifolds and Non-Projected Supermanifolds A lot of examples: Supermanifolds over Projective Spaces CP n and Grassmannians G ( k ; C n ) . Simone Noja Supergeometry and Non-Projected Supermanifolds 15 May 2018 2 / 38
Bibliography S.N., S.L. Cacciatori, F. Dalla Piazza, A. Marrani, R. Re, One-Dimensional Super Calabi-Yau Manifolds and their Mirrors , JHEP 04 (2017) 094 S.N., Supergeometry of Π -Projective Spaces , J.Geom.Phys., 124 , 286 (2018) S.L. Cacciatori, S.N, Projective Superspaces in Practice , J.Geom.Phys., 130 , 40 (2018) S.L. Cacciatori, S.N., R. Re, Non-Projected Calabi-Yau Supermanifolds over P 2 , arXiv:1706.01354 S.N., Topics in Algebraic Supergeometry over Projective Spaces , Ph.D. Thesis, Universit` a degli Studi di Milano (2018) Simone Noja Supergeometry and Non-Projected Supermanifolds 15 May 2018 3 / 38
Motivation and Premises from Physics What is Supergeometry Supergeometry is the study of varieties characterized by sheaves of Z 2 -graded algebras O M = O M , 0 ⊕ O M , 1 , whose even elements commute odd elements anti-commute (...and as such, they are nilpotent!) Such algebras are called superalgebras . Simone Noja Supergeometry and Non-Projected Supermanifolds 15 May 2018 4 / 38
Motivation and Premises from Physics What is Supergeometry Supergeometry is the study of varieties characterized by sheaves of Z 2 -graded algebras O M = O M , 0 ⊕ O M , 1 , whose even elements commute odd elements anti-commute (...and as such, they are nilpotent!) Such algebras are called superalgebras . (Non-Trivial) Supergeometry and Physics There are some scenarios in which supersymmetry does not boil down to make something either commute or anticommute! Simone Noja Supergeometry and Non-Projected Supermanifolds 15 May 2018 4 / 38
Motivation and Premises from Physics What is Supergeometry Supergeometry is the study of varieties characterized by sheaves of Z 2 -graded algebras O M = O M , 0 ⊕ O M , 1 , whose even elements commute odd elements anti-commute (...and as such, they are nilpotent!) Such algebras are called superalgebras . (Non-Trivial) Supergeometry and Physics There are some scenarios in which supersymmetry does not boil down to make something either commute or anticommute! Superstring Field Theory ← → De Rham Theory on Supermanifolds ; Simone Noja Supergeometry and Non-Projected Supermanifolds 15 May 2018 4 / 38
Motivation and Premises from Physics What is Supergeometry Supergeometry is the study of varieties characterized by sheaves of Z 2 -graded algebras O M = O M , 0 ⊕ O M , 1 , whose even elements commute odd elements anti-commute (...and as such, they are nilpotent!) Such algebras are called superalgebras . (Non-Trivial) Supergeometry and Physics There are some scenarios in which supersymmetry does not boil down to make something either commute or anticommute! Superstring Field Theory ← → De Rham Theory on Supermanifolds ; Superstring Perturbation Theory ← → Non-Projected Supermanifolds . Simone Noja Supergeometry and Non-Projected Supermanifolds 15 May 2018 4 / 38
Superstring Perturbation Theory - RNS Action RNS Action - The Superstring 1 � � Ψ µ / a ∂ α Φ µ e a β ∂ β Φ µ − i ¯ d 2 x e e α S RNS ( e , χ, Φ , Ψ) = − D Ψ µ + 4 πα ′ χ α γ β γ α Ψ µ ∂ β Φ µ + 1 � ¯ Ψ µ Ψ µ ¯ χ α γ β γ α χ β + 2¯ 2 Simone Noja Supergeometry and Non-Projected Supermanifolds 15 May 2018 5 / 38
Superstring Perturbation Theory - RNS Action RNS Action - The Superstring 1 � � Ψ µ / a ∂ α Φ µ e a β ∂ β Φ µ − i ¯ d 2 x e e α S RNS ( e , χ, Φ , Ψ) = − D Ψ µ + 4 πα ′ χ α γ β γ α Ψ µ ∂ β Φ µ + 1 � ¯ Ψ µ Ψ µ ¯ χ α γ β γ α χ β + 2¯ 2 (a lot of) Symmetries! local supersymmetry : A ǫ B ( ∂ α Φ µ − ¯ δ susy Φ µ = ¯ ǫ A Ψ µ δ susy Ψ µ A = − i ( γ α ) B Ψ µ B χ α B ) A δ susy e a ǫ B ( γ a ) A α = − 2 i ¯ B χ α A δ susy χ α A = D α ǫ A ; Diffeomorphisms + Weyl on the worldsheet; Lorentz on the worldsheet; Poincar´ e in the spacetime. Simone Noja Supergeometry and Non-Projected Supermanifolds 15 May 2018 5 / 38
Superstring Perturbation Theory - RNS Action RNS Action - On Super Riemann Surface (Witten 2014) i � S RNS = − D [ dz L , dz R | d θ L , d θ R ] D θ L X ( z L | θ L ; z R | θ R ) · D θ R X ( z L | θ L ; z R | θ R ) 2 πα S Σ g (a lot of) Symmetries! local supersymmetry : δ susy Φ µ = ¯ A ǫ B ( ∂ α Φ µ − ¯ ǫ A Ψ µ δ susy Ψ µ A = − i ( γ α ) B Ψ µ B χ α B ) A δ susy e a ǫ B ( γ a ) A α = − 2 i ¯ B χ α A δ susy χ α A = D α ǫ A ; Diffeomorphisms + Weyl on the worldsheet; Lorentz on the worldsheet; Poincar´ e in the spacetime. Simone Noja Supergeometry and Non-Projected Supermanifolds 15 May 2018 6 / 38
Superstring Perturbation Theory - The Path Integral Path Integral Quantization � Z vac = [ D Fields] exp ( − S TOT ) where S TOT = S RNS + “Topological Term” Simone Noja Supergeometry and Non-Projected Supermanifolds 15 May 2018 7 / 38
Superstring Perturbation Theory - The Path Integral Path Integral Quantization � Z vac = [ D Fields] exp ( − S TOT ) where S TOT = S RNS + “Topological Term” Faddeev-Popov Procedure and its Geometry Need to carry out a F-P procedure for G = SWeyl ⋉ SDiff × U (1) S Σ Simone Noja Supergeometry and Non-Projected Supermanifolds 15 May 2018 7 / 38
Superstring Perturbation Theory - The Path Integral Path Integral Quantization � Z vac = [ D Fields] exp ( − S TOT ) where S TOT = S RNS + “Topological Term” Faddeev-Popov Procedure and its Geometry Need to carry out a F-P procedure for G = SWeyl ⋉ SDiff × U (1) S Σ Faddeev-Popov Procedure ⇐ ⇒ Reduction to Supermoduli Space M g Simone Noja Supergeometry and Non-Projected Supermanifolds 15 May 2018 7 / 38
Superstring Perturbation Theory - The Path Integral Path Integral Quantization � Z vac = [ D Fields] exp ( − S TOT ) where S TOT = S RNS + “Topological Term” Faddeev-Popov Procedure and its Geometry Need to carry out a F-P procedure for G = SWeyl ⋉ SDiff × U (1) S Σ Faddeev-Popov Procedure ⇐ ⇒ Reduction to Supermoduli Space M g M g = { isomorphy classes of super Riemann surfaces S Σ g of genus g } 0 | 0 g = 0 dim C M g = 1 | 0 e 1 | 1 o g = 1 3 g − 3 | 2 g − 2 g ≥ 2 . Simone Noja Supergeometry and Non-Projected Supermanifolds 15 May 2018 7 / 38
Superstring Perturbation Theory - Partition Function Superstring Partition Function = Sum Over Topologies + ∞ � � � � e λ (1 − g ) Z vac = d µ g M g g =0 Simone Noja Supergeometry and Non-Projected Supermanifolds 15 May 2018 8 / 38
Superstring Perturbation Theory - Partition Function Superstring Partition Function = Sum Over Topologies + ∞ � � � � e λ (1 − g ) Z vac = d µ g M g g =0 + + + Simone Noja Supergeometry and Non-Projected Supermanifolds 15 May 2018 8 / 38
Superstring and Supermoduli Space Integral over the Supermoduli Space Superstring Interactions = ⇒ Measure for Supermoduli Space M g Simone Noja Supergeometry and Non-Projected Supermanifolds 15 May 2018 9 / 38
Superstring and Supermoduli Space Integral over the Supermoduli Space Superstring Interactions = ⇒ Measure for Supermoduli Space M g The Idea : get rid of the fermionic part of M g ! integrate the fermionic fibers out; deal with M spin instead; g M g M g Simone Noja Supergeometry and Non-Projected Supermanifolds 15 May 2018 9 / 38
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