(Work in Progress On) Quantum Field Theory on Curved Supermanifolds

Introduction Supermanifolds DiﬀGeo on supermanifolds Wess-Zumino Model

(Work in progress on) Quantum Field Theory on Curved Supermanifolds

Paul-Thomas Hack

Universit¨atHamburg

LQP 31, Leipzig, November 24, 2012

in collaboration with Andreas Schenkel

Paul-Thomas Hack (Work in progress on) Quantum Field Theory on Curved Supermanifolds Introduction Supermanifolds DiﬀGeo on supermanifolds Wess-Zumino Model Motivation

supergravity theories can be described as ﬁeld theories on supermanifolds = “curved superspace”, invariant under “superdiﬀeomorphisms”

want to obtain a systematic treatment of supergravity theories as “locally superinvariant” algebraic QFT on curved supermanifolds (CSM)

long term goal: understand SUSY non-renormalisation theorems in the algebraic language and extend them to curved spacetimes

ﬁrst: understand Wess-Zumino model as QFT on CSM covariant under “superdiﬀeomorphisms” → locally supercovariant QFT

Paul-Thomas Hack (Work in progress on) Quantum Field Theory on Curved Supermanifolds Introduction Supermanifolds DiﬀGeo on supermanifolds Wess-Zumino Model Supersymmetry and superspace for pedestrians

a simple SUSY model (Wess-Zumino model) in d = 2 + 1 Minkowski 3 spacetime R : φ, ψ, F invariant under : φ 7→ φ+εψ ψ 7→ ψ+6∂φε+F ε F 7→ F +ε6∂ψ

on shell : φ = 0 6∂ψ = 0 F = 0

idea: consider Φ = (φ, ψ, F ) as a single “superﬁeld” on “superspace” 3|2 α R 3 (x , θa) 1 Φ(x, θ) = φ(x) + ψ(x)θ + F (x)θ2 (θ2 = θθ) 2 Φ0(x, θ) = Φ(x 0, θ0)

α 0α α α 0 “special supertranslations” x 7→ x = x +εγ θ θa 7→ θa = θa+εa

Paul-Thomas Hack (Work in progress on) Quantum Field Theory on Curved Supermanifolds Introduction Supermanifolds DiﬀGeo on supermanifolds Wess-Zumino Model

Supermanifolds

Paul-Thomas Hack (Work in progress on) Quantum Field Theory on Curved Supermanifolds Introduction Supermanifolds DiﬀGeo on supermanifolds Wess-Zumino Model Supermanifolds: the good, the bad and the ugly

two mathematical deﬁnitions of supermanifolds:

1 sets with a suitable topology [DeWitt, Rogers]

m|k m|k • L m • L k R → RL ' (Λ R )0 × (Λ R )1 arbitrarily large, possibly infinite, number L of Grassmann parameters needed / 2 indirect definition by defining functions on CSM [Berezin, Le˘ıtes,Manin, ... ] finite number of Grassmann parameters sufficient ,

Paul-Thomas Hack (Work in progress on) Quantum Field Theory on Curved Supermanifolds Introduction Supermanifolds DiﬀGeo on supermanifolds Wess-Zumino Model Supermanifolds: the good

∞ smooth m-manifold M ' locally ringed space (M, CM ) with structure sheaf

∞ ∞ CM : M ⊃ U 7→ C (U, R)

smooth m|k-supermanifold M := locally superringed space M = (M, C) with structure sheaf

∞ • k C : U 7→ C(U) ' C (U, R) ⊗R Λ R

k → f ∈ C(M), i.e. “functions on M” are locally (θa a basis of R )

X i1 ik X i1 ik f ' fi1,··· ,ik (x) ⊗R θ1 ∧ · · · ∧ θk =: fi1,··· ,ik θ1 ··· θk k (i1,··· ,ik )∈Z2

Paul-Thomas Hack (Work in progress on) Quantum Field Theory on Curved Supermanifolds Introduction Supermanifolds DiﬀGeo on supermanifolds Wess-Zumino Model From spin manifolds to supermanifolds I

1 consider globally hyperbolic smooth manifold 4-dim (M, g) ⇒ trivial Lorentz frame bundle FM ' M × SO0(1, 3)

2 choose trivial spin structure SM ' M × Spin0(1, 3)

3 construct Majorana spinor bundle DM

4 pick global frame Ea of DM, any section ∞ • ∞ • 4 f ∈ C (M, Λ DM) ' C (M, R) ⊗R Λ R can be written as

X i1 i4 X i1 i4 f = fi1,··· ,i4 (x) E1 (x) ∧ · · · ∧ E4 (x) ' fi1,··· ,if θ1 ··· θ4 2 (i1,··· ,i4)∈Z4

Paul-Thomas Hack (Work in progress on) Quantum Field Theory on Curved Supermanifolds Introduction Supermanifolds DiﬀGeo on supermanifolds Wess-Zumino Model From spin manifolds to supermanifolds II

⇒ M = (M, C) with

∞ • ∞ • k C(M) := C (M, Λ DM) ' C (M, R) ⊗R Λ R =: C0 deﬁnes a 4|4 supermanifold

we have “global Fermionic coordinates” θa (corresponding to a global vielbein of DM) and thus avoid sheaf-theoretic complications

by construction the coefficients fi1,··· ,i4 of f ∈ C0 transform under antisymmetrised products of the Majorana representation and can be interpreted as fields of different spin

⇒ rigorous understanding of (in 3d) 1 Φ = φ + ψθ + F θ2 2

Paul-Thomas Hack (Work in progress on) Quantum Field Theory on Curved Supermanifolds Introduction Supermanifolds DiﬀGeo on supermanifolds Wess-Zumino Model

Diﬀerential Geometry on Supermanifolds

Paul-Thomas Hack (Work in progress on) Quantum Field Theory on Curved Supermanifolds Introduction Supermanifolds DiﬀGeo on supermanifolds Wess-Zumino Model Vector ﬁelds on supermanifolds

G0 is a supercommutative superalgebra with a Z2-grading induced by the • k Z-grading of Λ R

vector ﬁelds on the supermanifold M = (M, G) are derivations X ∈ Der(G0) of G0, e.g. homogeneous derivation acting on f , h ∈ G0, f homogeneous X (fh) = X (f )h + (−1)|X ||f |fX (h)

a (G0-left supermodule) basis of Der(G0) is given by 1 4 a ∂ (e0, ··· , e4, ∂ , ··· , ∂ ) where eα vierbein and ∂ = ∂θa

Paul-Thomas Hack (Work in progress on) Quantum Field Theory on Curved Supermanifolds Introduction Supermanifolds DiﬀGeo on supermanifolds Wess-Zumino Model Forms and diﬀerentials on supermanifolds

α Ω(G0) dual of Der(G0) with basis (e , dθa)

β β a α b b heα, e i = δα h∂ , e i = 0 heα, dθai = 0 h∂ , dθai = δa

diﬀerential d : G0 → Ω(G0) deﬁned by hX , df i := X (f ) for all X ∈ Der(G0), f ∈ G0

higher forms + extension of d can be introduced

Paul-Thomas Hack (Work in progress on) Quantum Field Theory on Curved Supermanifolds Introduction Supermanifolds DiﬀGeo on supermanifolds Wess-Zumino Model Superdiﬀeomorphisms I

given two m|k supermanifolds M1 and M2, “superdiﬀeomorphisms” χ ∈ HomSMan(M1, M2) can be abstractly deﬁned as morphisms of locally superringed spaces, but a general explicit characterisation seems unavailable to date

∞ ∞ however, just as HomMan(M1, M2) ' HomAlg(C (M2), C (M1)) ' 1 m ∞ 1 m {smooth functions y , ··· , y ∈ C (M1) s.t. (y (M1), ··· , y (M2)) ⊂ M2} m for Mi ⊂ R ...

... one can show HomSMan(M1, M2) ' HomSAlg(G(M2), G(M1)) ' 1 m {odd and even functions y , ··· , y , ξ1, ··· , ξk ··· , ∈ G(M1) s.t. 1 m m (β(y (M1)), ··· , β(y (M1))) ⊂ M2} for Mi = (Mi , G), Mi ⊂ R

Paul-Thomas Hack (Work in progress on) Quantum Field Theory on Curved Supermanifolds Introduction Supermanifolds DiﬀGeo on supermanifolds Wess-Zumino Model Superdiﬀeomorphisms II

α given y , ξa ∈ G(M1), ψy,ξ ∈ HomSAlg(G(M2), G(M1)) is deﬁned by a “pullback”, e.g. (in 3d)

α α α 2 b y = a (x) + b (x)θ ξa = ma (x)θb 1 ψ φ + ψθ + F θ2 := φ(aα) + ∂ φ(aα)bβ θ2 + ··· + ψambθ + ··· y,ξ 2 β a b

α but ψy,ξ must preserve parity ⇒ only parity preserving y , ξb allowed, SUSY supertranslations y α = x α + εγαθ disallowed unless one introduces “ﬂesh”!

however, inﬁnitesimal SUSY supertranslations are available as odd derivations on a ﬁxed supermanifold

α a X = εγ θeα + ε ∂a

Paul-Thomas Hack (Work in progress on) Quantum Field Theory on Curved Supermanifolds Introduction Supermanifolds DiﬀGeo on supermanifolds Wess-Zumino Model Superconnection and supervielbein I

on ordinary manifolds, we can combine a connection ω and dual vielbein eα into a “Cartan connection”

βγ α Γ := ω + e = ω Lβγ + e Pα ∈ Ω(M) ⊗R poin(1, 3)

4 Lβγ , Pα generators of poin(1, 3), Lie algebra of Poin(1, 3) := Spin0(1, 3) n R

Γ can be interpreted as induced by a connection on a Poin(1, 3)-bundle after reducing the latter to a Spin0(1, 3)-bundle

Levi-Civita connection ω can be speciﬁed by a suitable torsion constraint

Paul-Thomas Hack (Work in progress on) Quantum Field Theory on Curved Supermanifolds Introduction Supermanifolds DiﬀGeo on supermanifolds Wess-Zumino Model Superconnection and supervielbein II

on the supermanifold M = (M, G0), we want to consider a “super Cartan connection”

βγ α a suppoin bΓ = ωb + be + e = ωb Lβγ + be Pα + e Qa ∈ Ω(G0) ⊗R (1, 2)

coming from a reduction Suppoin(1, 3) → Spin0(1, 3)

L/R suppoin(1, 3) super Lie algebra generated by even Lβγ , Pα, odd Qa

L/R b L/R L/R [Lαβ , Qa ] = ([γα, γβ ])a Qb [Pα, Qb ] = 0 1 [QL, QR ] = (π γα) P [QL, QL] = 0 π = 1 ∓ iγ5 a b L ab α a b L/R 2

Paul-Thomas Hack (Work in progress on) Quantum Field Theory on Curved Supermanifolds Introduction Supermanifolds DiﬀGeo on supermanifolds Wess-Zumino Model Superconnection and supervielbein III

textbooks: suitable “supertorsion constraints” + “superdiffeomorphism gauge fixing”: bΓ is specified in terms of (eα, ψa, a, bα), we choose “metric backgrounds” (eα, 0, 0, 0) ⇒

α α α ab a a α ab be = e − dθa(γ ) θb e = dθ + e gα θb

ab ωb = ω = Levi-Civita connection gα speciﬁed in terms of ω

α a deﬁne derivations beα, ea ∈ Der(G0) as inverses of be , e

A A α E := be + e = E (Q ⊕ P)A ⇒ sdet E = det(e )

Paul-Thomas Hack (Work in progress on) Quantum Field Theory on Curved Supermanifolds Introduction Supermanifolds DiﬀGeo on supermanifolds Wess-Zumino Model Category of CSM for QFT

⇒ “best” category SLoc of backgrounds for locally supercovariant QFT so far:

α 4 ObjSLoc = {M = (M, E) | M = (M, G0), (M, e ) glob. hyp., M ⊂ R }

HomSLoc(M1, M2) = {E − preserving SMan morphisms which induce α α c.c. isom. embed. (M1, e1 ) → (M2, e2 )}

Paul-Thomas Hack (Work in progress on) Quantum Field Theory on Curved Supermanifolds Introduction Supermanifolds DiﬀGeo on supermanifolds Wess-Zumino Model

The free Wess-Zumino model as a QFT on CSM

Paul-Thomas Hack (Work in progress on) Quantum Field Theory on Curved Supermanifolds Introduction Supermanifolds DiﬀGeo on supermanifolds Wess-Zumino Model The free Wess-Zumino model as a QFT on CSM

plan: construct the free quantum Wess-Zumino model as an algebraic QFT on CSM (ﬁrst: arbitrary 4|4 “spin”-CSM)

a linear QFT on CST is speciﬁed by

1 a vector bundle with non-degenerate bilinear form

2 an equation of motion

Paul-Thomas Hack (Work in progress on) Quantum Field Theory on Curved Supermanifolds Introduction Supermanifolds DiﬀGeo on supermanifolds Wess-Zumino Model Chiral superﬁelds

b L/R L/R 2 L/R L/R chiral coordinates θa := π θb θL/R := θ θ a

C (left) chiral superﬁelds Φ ∈ CL ⊂ G0 := G0 ⊗R C

R R Da Φ := hπ ea, dΦi = 0 √ 2 1 2 2 1 2 ⇒ Φ = φ + 2 ψθL + F θL + θR 6∂φθL + φθLθR − √ θR 6∇ψθL 4 2

∗ Φ ∈ GL ⇔ Φ ∈ GR

⊕ ⊕ ∗ T deﬁne G 3 Φ = (Φ, Φ ) ,Φ ∈ GL

Paul-Thomas Hack (Work in progress on) Quantum Field Theory on Curved Supermanifolds Introduction Supermanifolds DiﬀGeo on supermanifolds Wess-Zumino Model Test superfunctions and bilinear form

⊕ ∗ T “chiral supertestfunctions” G0 3 F = (f , f ) , f has smooth and compactly supported expansion coeﬃcients

⊕ non-degenerate bilinear form on G0 Z Z Z Z 2 A 2 hF1, F2i := < dx dθL sdet(E )f1f2 = < dVol dθL f1f2 M M Z 2 2 dθLθL := 1 , integrals of linear θL polynomials=0

Paul-Thomas Hack (Work in progress on) Quantum Field Theory on Curved Supermanifolds Introduction Supermanifolds DiﬀGeo on supermanifolds Wess-Zumino Model Equation of motion

C C deﬁne “chiral projector” PL : G → G by 1 1 1 1 P Φ := − C abhπLe , d(DLΦ)+ω c DLΦi+ Rθ2 Φ = − DaDL + Rθ2 Φ L 4 eb a a c 6 R 4 L a 6 R

⇒ PL : GL 7→ GR and for Φ ∈ GL √ 1 2 1 2 2 1 2 2 PLΦ = F − 2 θR 6∇ψ+ + R φθR + √ 6∇ ψθLθR + F θLθR +θR 6∂F θL 6 2 4

deﬁne (massless) chiral superwave operator P : G⊕ → G⊕ 3 Φ⊕: by ⊕ Φ 0 PR Φ PΦ := P ∗ := ∗ Φ PL 0 Φ

⊕ ⊕ ⊕ ⊕ hPf1 , f2 i = hf1 , Pf2 i and P has unique advanced/retarded Green’s ± − R operators GP , GP := GP − GP

Paul-Thomas Hack (Work in progress on) Quantum Field Theory on Curved Supermanifolds Introduction Supermanifolds DiﬀGeo on supermanifolds Wess-Zumino Model Quantization I

ﬁeld algebra A(M) of the quantum Wess-Zumino ﬁeld on M is free tensor algebra generated by 1, Φ(f ) (the quantizations of the functionals hΦ⊕, f ⊕i) and relations ¯

∗ Φ(PR f ) = 0

|f1||f2| ⊕ ⊕ Φ(f1)Φ(f2) − (−1) Φ(f2)Φ(f1) = ihf , GP f i1 1 2 ¯

for fi homogeneous

Paul-Thomas Hack (Work in progress on) Quantum Field Theory on Curved Supermanifolds Introduction Supermanifolds DiﬀGeo on supermanifolds Wess-Zumino Model Quantization II

⇒ the canonical supercommutation relations combine the CCR for φ and the CAR for ψ in a nice way!

2 φ(h) ' Φ(f ) f = (h − ih)θL √ 1 2 ψ(p) ' Φ(f ) f = 2pθL − √ θR 6∇pθL 2

quantization can be formulated in terms of a functor A : SLoc → Alg

Paul-Thomas Hack (Work in progress on) Quantum Field Theory on Curved Supermanifolds Introduction Supermanifolds DiﬀGeo on supermanifolds Wess-Zumino Model

Thanks a lot for your attention!

Paul-Thomas Hack (Work in progress on) Quantum Field Theory on Curved Supermanifolds