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, ; F invariant under : φ 7! φ+" 7! +6@φε+F " F 7! F +"6@ on shell : φ = 0 6@ = 0 F = 0 idea: consider Φ = (φ, ; F ) as a single \superfield” on \superspace" 3j2 α R 3 (x ; θa) 1 Φ(x; θ) = φ(x) + (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+"a 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] mjk mjk • 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 1 smooth m-manifold M ' locally ringed space (M; CM ) with structure sheaf 1 1 CM : M ⊃ U 7! C (U; R) smooth mjk-supermanifold M := locally superringed space M = (M; C) with structure sheaf 1 • k C : U 7! C(U) ' C (U; R) ⊗R Λ R k ! f 2 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 )2Z2 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 1 • 1 • 4 f 2 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)2Z4 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 1 • 1 • k C(M) := C (M; Λ DM) ' C (M; R) ⊗R Λ R =: C0 defines a 4j4 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 2 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 + θ + 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 2 Der(G0) of G0, e.g. homogeneous derivation acting on f ; h 2 G0, f homogeneous X (fh) = X (f )h + (−1)jX jjf jfX (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 2 Der(G0), f 2 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 mjk supermanifolds M1 and M2, \superdiffeomorphisms" χ 2 HomSMan(M1; M2) can be abstractly defined as morphisms of locally superringed spaces, but a general explicit characterisation seems unavailable to date 1 1 however, just as HomMan(M1; M2) ' HomAlg(C (M2); C (M1)) ' 1 m 1 1 m fsmooth functions y ; ··· ; y 2 C (M1) s.t. (y (M1); ··· ; y (M2)) ⊂ M2g m for Mi ⊂ R ... ... one can show HomSMan(M1; M2) ' HomSAlg(G(M2); G(M1)) ' 1 m fodd and even functions y ; ··· ; y ; ξ1; ··· ; ξk ··· ; 2 G(M1) s.t. 1 m m (β(y (M1)); ··· ; β(y (M1))) ⊂ M2g 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 2 G(M1), y,ξ 2 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 φ + θ + F θ2 := φ(aα) + @ φ(aα)bβ θ2 + ··· + ambθ + ··· y,ξ 2 β a b α but y,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 + " @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 ! and dual vielbein eα into a \Cartan connection" βγ α Γ := ! + e = ! Lβγ + e Pα 2 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 ! 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 = !b + be + e = !b Lβγ + be Pα + e Qa 2 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; a; 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 !b = ! = Levi-Civita connection gα specified in terms of ! α a define derivations beα, ea 2 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 = fM = (M; E) j M = (M; G0); (M; e ) glob. hyp.; M ⊂ R g HomSLoc(M1; M2) = fE − preserving SMan morphisms which induce α α c.c. isom. embed. (M1; e1 ) ! (M2; e2 )g 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 4j4 \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