Introduction Supermanifolds DiffGeo on supermanifolds Wess-Zumino Model
(Work in progress on)
Quantum Field Theory on Curved Supermanifolds
Paul-Thomas Hack
Universität Hamburg
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
spacetime R3 : φ, ψ, F
invariant under :
φ 7→ φ+εψ
on shell :
ψ 7→ ψ+6∂ φε+F ε
φ = 0
6∂ ψ = 0
F 7→ F +ε6∂ ψ
F =0
idea: consider Φ = (φ, ψ, F ) as a single “superfield” on “superspace”
R3|2 3 (x α , θa )
Φ(x, θ) = φ(x) + ψ(x)θ +
1
F (x)θ2
2
(θ2 = θθ)
Φ0 (x, θ) = Φ(x 0 , θ0 )
“special supertranslations”
Paul-Thomas Hack
x α 7→ x 0α = x α +εγ α θ
θa 7→ θa0 = θa +εa
(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
Rm|k → RL
• L k
' (Λ• RL )m
0 × (Λ R )1
arbitrarily large, possibly infinite, number L of Grassmann parameters
needed /
2
indirect definition by defining functions on CSM
[Berezin, Leı̆tes, 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
C : U 7→ C(U) ' C ∞ (U, R) ⊗R Λ• Rk
→ f ∈ C(M), i.e. “functions on M” are locally (θa a basis of Rk )
f '
X
i
fi1 ,··· ,ik (x) ⊗R θ1i1 ∧ · · · ∧ θkk =:
X
i
fi1 ,··· ,ik θ1i1 · · · θkk
(i1 ,··· ,ik )∈Zk2
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
f ∈ C ∞ (M, Λ• DM) ' C ∞ (M, R) ⊗R Λ• R4 can be written as
X
X
fi1 ,··· ,if θ1i1 · · · θ4i4
f =
fi1 ,··· ,i4 (x) E1i1 (x) ∧ · · · ∧ E4i4 (x) '
(i1 ,··· ,i4 )∈Z24
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
C(M) := C ∞ (M, Λ• DM) ' C ∞ (M, R) ⊗R Λ• Rk =: 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)
Φ = φ + ψθ +
Paul-Thomas Hack
1 2
Fθ
2
(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
Z-grading of Λ• Rk
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
∂
(e0 , · · · , e4 , ∂ 1 , · · · , ∂ 4 ) where eα vierbein and ∂ a = ∂θ
a
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
Ω(G0 ) dual of Der (G0 ) with basis (e α , dθa )
heα , e β i = δαβ
h∂ a , e α i = 0
heα , dθa i = 0
h∂ b , dθa i = δab
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 )) '
{smooth functions y 1 , · · · , y m ∈ C ∞ (M1 ) s.t. (y 1 (M1 ), · · · , y m (M2 )) ⊂ M2 }
for Mi ⊂ Rm ...
... one can show HomSMan (M1 , M2 ) ' HomSAlg (G(M2 ), G(M1 )) '
{odd and even functions y 1 , · · · , y m , ξ1 , · · · , ξk · · · , ∈ G(M1 ) s.t.
(β(y 1 (M1 )), · · · , β(y m (M1 ))) ⊂ M2 } for Mi = (Mi , G), Mi ⊂ Rm
Paul-Thomas Hack
(Work in progress on) Quantum Field Theory on Curved Supermanifolds
Introduction Supermanifolds DiffGeo on supermanifolds Wess-Zumino Model
Superdiffeomorphisms II
given y α , ξa ∈ G(M1 ), ψy ,ξ ∈ HomSAlg (G(M2 ), G(M1 )) is defined by a
“pullback”, e.g. (in 3d)
y α = aα (x) + b α (x)θ2
ξa = mab (x)θb
1
ψy ,ξ φ + ψθ + F θ2 := φ(aα ) + ∂β φ(aα )b β θ2 + · · · + ψ a mab θb + · · ·
2
but ψy ,ξ must preserve parity ⇒ only parity preserving y α , ξb allowed,
SUSY supertranslations y α = x α + εγ α θ disallowed unless one introduces
“flesh”!
however, infinitesimal SUSY supertranslations are available as odd
derivations on a fixed supermanifold
X = εγ α θeα + εa ∂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
e α into a “Cartan connection”
Γ := ω + e = ω βγ Lβγ + e α Pα ∈ Ω(M) ⊗R poin(1, 3)
Lβγ , Pα generators of poin(1, 3), Lie algebra of Poin(1, 3) := Spin0 (1, 3) n R4
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”
b
Γ=ω
b +b
e +e
e=ω
b βγ Lβγ + b
e α Pα + e
e a Qa ∈ Ω(G0 ) ⊗R suppoin(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
[Lαβ , QaL/R ] = ([γα , γβ ])ab Qb
[QaL , QbR ] = (πL γ α )ab Pα
Paul-Thomas Hack
[QaL , QbL ] = 0
L/R
[Pα , Qb
πL/R
]=0
1
1 ∓ iγ 5
=
2
(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”: b
Γ is specified in terms of (e α , ψ a , a, b α ), we choose
“metric backgrounds” (e α , 0, 0, 0) ⇒
b
e α = e α − dθa (γ α )ab θb
ω
b = ω = Levi-Civita connection
e
e a = dθa + e α gαab θb
gαab specified in terms of ω
define derivations b
eα , e
ea ∈ Der (G0 ) as inverses of b
eα, e
ea
E := b
e +e
e = E A (Q ⊕ P)A ⇒ sdet E A = det(e α )
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:
ObjSLoc = {M = (M, E ) | M = (M, G0 ), (M, e α ) glob. hyp., M ⊂ R4 }
HomSLoc (M1 , M2 ) =
{E − preserving SMan morphisms which induce
c.c. isom. embed. (M1 , e1α ) → (M2 , e2α )}
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
Paul-Thomas Hack
1
a vector bundle with non-degenerate bilinear form
2
an equation of motion
(Work in progress on) Quantum Field Theory on Curved Supermanifolds
Introduction Supermanifolds DiffGeo on supermanifolds Wess-Zumino Model
Chiral superfields
L/R
chiral coordinates θa
b
:= π L/R
θb
a
2
θL/R
:= θL/R θL/R
(left) chiral superfields Φ ∈ CL ⊂ GC0 := G0 ⊗R C
DaR Φ := hπ R e
ea , dΦi = 0
⇒Φ=φ+
√
1
1
2 ψθL + F θL2 + θR 6∂ φθL + φθL2 θR2 − √ θR 6∇ψθL2
4
2
Φ ∈ GL ⇔ Φ∗ ∈ GR
define G⊕ 3 Φ⊕ = (Φ, Φ∗ )T , 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” G⊕
0 3 F = (f , f ) , f has smooth and
compactly supported expansion coefficients
non-degenerate bilinear form on G⊕
0
Z
Z
Z
Z
hF1 , F2 i := <
dx dθL2 sdet(E A )f1 f2 = <
dVol dθL2 f1 f2
M
Z
Paul-Thomas Hack
dθL2 θL2 := 1 ,
M
integrals of linear θL polynomials=0
(Work in progress on) Quantum Field Theory on Curved Supermanifolds
Introduction Supermanifolds DiffGeo on supermanifolds Wess-Zumino Model
Equation of motion
define “chiral projector” PL : GC → GC by
1
1
PL Φ := − C ab hπ L e
eb , d(DaL Φ)+ωa c DcL Φi+ RθR2 Φ =
4
6
1
1
− DLa DaL + RθR2 Φ
4
6
⇒ PL : GL 7→ GR and for Φ ∈ GL
√
1
1
1
PL Φ = F − 2 θR 6∇ψ+ + R φθR2 + √ 6∇2 ψθL θR2 + F θL2 θR2 +θR 6∂ F θL
6
4
2
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
operators GP± , GP := GP− − GPR
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
Φ(f1 )Φ(f2 ) − (−1)|f1 ||f2 | Φ(f2 )Φ(f1 ) = ihf1⊕ , GP f2⊕ i1
¯
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!
φ(h) ' Φ(f )
ψ(p) ' Φ(f )
f =
f = (h − ih)θL2
√
1
2pθL − √ θR 6∇pθL2
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