Introduction: SUSY and LQG
Supermanifolds and super Lie groups
Gravity as Cartan geometry
Supergravity
Application for loop quantum supergravity
Summary

Towards a manifestly supersymmetric formulationof loop quantum supergravity theories
Konstantin Eder
FAU Erlangen-Nürnberg

June 19, 2020

Konstantin Eder

Towards a manifestly supersymmetric formulation of loop quantum supergra

Introduction: SUSY and LQG
Supermanifolds and super Lie groups
Gravity as Cartan geometry
Supergravity
Application for loop quantum supergravity
Summary

Section 1
Introduction: SUSY and LQG

Konstantin Eder

Towards a manifestly supersymmetric formulation of loop quantum supergra

The SUSY algebra

Coleman-Mandula-Theorem (CM)In the presence of a mass gap, the only possible (Lie) algebra ofsymmetries of the S-matrix for an interacting QFT is given byiso(R1,3 ) ⊕ internal sym.
Haag-Łopuszański-Sohnius theorem(CM) does not apply in case of super Lie algebras, i.e. Z2 -gradedalgebras (g, [·, ·]) of the formg = g0 ⊕ g1

with [·, ·] : (anti) commutator on g0 (g1 )such that [gi , gj ] ⊆ gi+j(+ graded Jacobi identity)

The SUSY algebra⇒ smallest possible super Lie algebra of symmetries is given by thesuper Poincaré algebra (N = 1 SUSY-algebra)
iso(R1,3|4 ) = R1,3 n so(1, 3) ⊕ SR{z} |{z}|g0αgenerators: PI , MIJ , Q = (Q[PI , Q α ] = 0−A

g1, QA0 )T (Majorana spinor)

1Q β (γI )β α
4L

1Q β (γIJ )β α211[Q α , Q β ] = (C γI )αβ P I + (C γ IJ )αβ MIJ28L[MIJ , Q α ] =AdS4 := {x ∈ R5 | − (x 0 )2 + (x 1 )2 + . . . + (x 3 )2 − (x 4 )2 = −L2 }LQG
nonperturbative and manifestly background independent
quantum theory of gravity
starting point: reformulation of GR in terms of Ashtekar-Barbero
variables
gives canonical GR the structure of a SU(2) Yang-Mills theory
Hilbert space: projective limit of L2 -functions on group-valued
holonomies (parallel transport map) along one-dimensional paths
Peter-Weyl theory ⇒ basis: spin networks (→ string-like
excitations of the gravitational field)

j1

j2
j3SUSY and LQG

SUSY and LQG:
SUGRA in Ashtekar variables? [Füllöp, Jacobson, Pullin et al]
Quantization [Smolin, Bodendorfer et al]

But:
Canonical treatment of SUGRA theories generically suffer fromcomplicated constraints
SUSY hidden in the formalism → keep SUSY manifest (at leastpartially)?
role of the Barbero-Immirzi parameter?

SUSY and LQG

What is needed for LQSG:
super Cartan geometry
super analog of Ashtekar’s connection
anticommuting classical fermionssuper holonomies
generalization of Peter-Weyl theory to super Lie groups

Introduction: SUSY and LQG
Supermanifolds and super Lie groups
Gravity as Cartan geometry
Supergravity
Application for loop quantum supergravity
Summary

Section 2
Supermanifolds and super Lie groups

Konstantin EderTowards a manifestly supersymmetric formulation of loop quantum supergra

Supermanifolds
Various different approaches:
Berezin-Kostant-Leites(algebro-geometric): "Definition like in NC
geometry". But: what are the points?
Rogers-DeWitt(concrete): "Start with topological space of points".
But: Too many points, ambiguities"
Molotkov ’84 and Sachse ’08(categorial): Shows BKL and RdWare two sides of the same coin.
algebro-geom.: Observation: Structure of a smooth manifold M
completely encoded in a suitable ring of functions on it→ describe M as locally ringed space (|M|, OM ) s.t.
|M| paracompact topological Hausdorff space
OM abstract sheaf of local rings on |M|
M is locally Euclidean (Rn , CR∞n )Supermanifolds

Definition Supermanifold
Supermanifold M is a locally ringed space (M, OM ) s.t.
M paracompact topological Hausdorff space
OM abstract sheaf of local super rings on M
M locally looks like flat superspace Rm|n = (Rm , CR∞m ⊗VRn )→ locally, functions of the form:f (x , θ) = f0 (x ) + fi (x )θi + . . . + fn (x )θ1 · · · θn2

J := O1 + hO1 i (nilpotent sub ideal) → (M, OM /J ) ordinary
manifold (body)super Lie groups as group objects in this category SMan.

Supermanifolds

Supermanifold M yields functor of points M : SManop → Set
T 7→ M(T ) := HomSMan (T , M)

(T -point)(f : T → S) 7→ (M(f ) : g 7→ g ◦ f )restrict on superpoints T ∼= ({∗}, Λ) (→ Grassmann algebras)
M(Λ) ∼= HomSAlg (O(M), Λ)
contains real spectrum SpecR (O(M)) = HomSAlg (O(M), R)
topological space via Zariski or Gelfand topology
equip M(Λ) with coarsest topology s.t. M(Λ) → SpecR (O(M)) is
continuous → DeWitt-topology

Supermanifolds

M(ΛN ) structure of topological manifold →Rogers-DeWitt
supermanifold⇒ yields functor
Gr → Man, Λ 7→ M(Λ)→ Supermanifold in the sense of Molotkov and Sachse
starting point for the construction of infinite-dimensionalsupermanifolds requiring M(Λ) to be a Banach [M ’84, S ’08] orFréchet supermanifold [Schütt ’19]
→ groups of super diffeomorphisms and supersymmetrytransformations

Introduction: SUSY and LQG
Supermanifolds and super Lie groups
Gravity as Cartan geometry
Supergravity
Application for loop quantum supergravity
Summary

Section 3
Gravity as Cartan geometry

Konstantin Eder

Towards a manifestly supersymmetric formulation of loop quantum supergra

Klein geometry

F. Klein: ”Classify geometry of space via group symmetries”.
Example: Minkowski spacetime M = (R1,3 , η)isometry group ISO(R1,3 ) = R1,3 n SO0 (1, 3)
event p ∈ M: Gp = SO0 (1, 3) (isotropy subgroup)
ISO(R1,3 )/SO0 (1, 3) ∼=MDefinition
A Klein geometry is a pair (G, H) where G is a Lie group and H ⊆ G aclosed subgroup such that G/H is connected.

Cartan geometry
flat spacetime ↔ metric Klein geometry⇒ Cartan geometry as deformed Klein geometry
metric Cartan geometry
A metric Cartan geometry modeled on a metric Klein geometry (G, H)
is a principal H-bundlerHPoπM

together with a Cartan connection A ∈ Ω1 (P, g) s.t.
IIIIIIrg∗ A = Ad(g −1 )A ∀g ∈ He ) = X , ∀X ∈ h X
e := (1 ⊗ Xe ) ◦ r ∗A(XA : Tp P → g is an isomorphism ∀p ∈ Pwithout (III): generalized Cartan geometry

Cartan geometryπGravity as metric Cartan geometry (M ← P, A) modeled on(ISO(R1,3 ), SO0 (1, 3))
Decomposition
A = prR1,3 ◦ A + prso(1,3) ◦ A =: ξ + ωω Lorentz-connectionξ ∈ Ω1hor (P, R1,3 ) soldering form, induces isomorphismΞ : P ×(H,Ad) R1,3 ∼= TM ⇒ metric g := η ◦ (Ξ−1 × Ξ−1 ) on MAction
ZS[A] =Ms ∗ ξ =: e I PI ⇒ {e I } coframes ∗ (R[A]IJ ∧ ξ K ∧ ξ L )IJKLIntroduction: SUSY and LQG
Supermanifolds and super Lie groups
Gravity as Cartan geometry
Supergravity
Application for loop quantum supergravity
Summary

Section 4
Supergravity

Konstantin Eder

Towards a manifestly supersymmetric formulation of loop quantum supergra

Supergravity
Supergravity as super Cartan geometry modeled on super Klein
geometry (ISO(R1,3|4 ), Spin(1, 3)) [D’Auria-Fré-Regge ’80, D’Auria-Fré’80, Castellani-D’Auria-Fré ’91]

PoSpin(1, 3)πMsuper Cartan connection A ∈ Ω1 (P, iso(R1,3|4 ))Decompose
A = prg1 ◦ A + prR1,3 ◦ A + prspin(1,3) ◦ A| {z } | {z } |{z}ψξ

ψ = ψ α Q α (spin-3/2) Rarita-Schwinger fieldωSupergravity

SUGRA action (D = 4, N = 1)
Z
S[A] =s ∗ hR[A] ∧ σ ∧ σi”=”SP [e, ω] +MZi µνρσψ̄µ γσ γ∗ Dνω ψρ2M

R[A] := dA + 21 [A ∧ A]gσ := ψ + ξ ∈ Ω1 (P, t1,3|4 ) super soldering form (super vielbein)s : U ⊂ M → P local section

Introduction: SUSY and LQG
Supermanifolds and super Lie groups
Gravity as Cartan geometry
Supergravity
Application for loop quantum supergravity
Summary

Section 5Application for loop quantum supergravity

Konstantin EderTowards a manifestly supersymmetric formulation of loop quantum supergra

Super Ashtekar connection
self-dual Ashtekar connection:
i1e i )T +IJ
ω IJ − IJ KL ω KL = (Γi − i KA+IJ =i
22
Ti± = 21 (Ji ± iKi ), Ji = − 12 i jk Mjk , Ki = M0i
Proposition: (Ti+ , Q A ) generate sub super Lie algebra[Ti+ , Tj+ ] = ij k Tk+i
[Ti+ , Q A ] = − Q B (σi )B A
2
1
[Q A , Q B ] = 0+ (σ i )AB Ti+2Lgenerate generalized Takiff Lie superalgebra sl(2, C) o C2
(orthosymplectic Lie superalgebra osp(1|2)C )Super Ashtekar connection

Definition
A+ := ψ A Q A + A+i Ti+⇒ Defines 1-form A+ ∈ Ω1 (P, osp(1|2)C )Propertiesrg∗ A+ = Ad(g −1 )A+ , ∀g ∈ SL(2, C)e ) = X , ∀X ∈ sl(2, C)
A+ (XCondition (III) not satisfied ⇒ generalized super Cartan connection.

Super Ashtekar connection
Proposition:[KE ’20]
Consider associated OSp(1|2)C -principal bundle
P ×SL(2,C) OSp(1|2)C oOSp(1|2)CπM

lift of A+ gives principal connection →super Ashtekar connection
even exists for N > 1 (extended SUSY)
what about real β?
⇒ both chiral components of Q in Aβ
A0

But: [Q A , Q ] ∝ P → no proper sub super algebra!
→ no super holonomies!

Consequences

A+ is the right starting point for LQSG
fundamental description via super Cartan geometry→ resolves confusions in literature:
existence with and without cosmological constantrequires chiral description of SUGRA (no real β)contains both gravity and matter d.o.f. → unified description, more
fundamental way of quantizing fermions→ can give hint how to quantize fermions in spin foam approach[Livine+Oeckl ’03]

substantially simplifies constraints (canonical form of Einsteinequations): partial solution via gauge invariance[Fülöp ’93,Ling+Smolin ’00, Tsuda ’00, Livine+Oeckl ’03, KE+Sahlmann ’20]

Quantum theory
What do we need for quantum theory?
holonomies (parallel transport map)
Hilbert spaces
spin network basis
Holonomies[KE ’20 (in prep.)]:general problem: pullback of superfields to the body of asupermanifolds M are purely bosonic → no fermionic degrees of
freedom on the underlying spacetime manifold!
Resolution: Consider enriched category of supermanifolds [Schmitt’96, Deligne ’99, Sachse ’09, Groeger ’14, Hack-Hanisch-Schenkel ’15]

Choose parametrizing supermanifold S and consider S-relative
supermanifold M/S := (S × M, prS )Have to require that physical quantities behave natural underchange of parametrization λ : S → S 0

Holonomies→ Consider super connection 1-form A on S-relative principal super
fiber bundles
yields parallel transport map along γ : S × [0, 1] → MAPS,γ: Γ(γ0∗ P) → Γ(γ1∗ P)∗λ A∗natural under reparametrization: λ∗ ◦ PSA0 ,γ = PS,λ∗γ ◦ λSuper Wilson loop observable I(s̃)
Wγ [A] = str gγ [ω] · Pexp − Adgγ [ω]−1 ψ: S→Gγ

A = ω + ψ and γ : [0, 1] → M ⊂ M
gγ [ω]: parallel transport map associated to ω

Holonomies
Proposition
Wγ [A] : S → G element in G(S) (S-point of G)
natural under reparametrization λ∗ Wγ [A] = Wγ [λ∗ A]
fermionic component ψ ∈ Ω1 (M, E ) ⊗ O(S)1 describes odd
functional on supermanifold S
choice of S: S object in SPtop ∼= Gr
Wγ [A] ∈ G(Λ) (group element in Rogers-DeWitt supermanifold)Λ∞ universal property: ∃λN : ΛN ,→ Λ∞ ∀N ∈ N0Alternatively: S as configuration space (infinite-dimensional smf!)
S = Ω1 (M, Ad(P)) ⊕ Ω1 (M, E )⇒ ψ describes odd functional on configuration space[Schmitt ’96,Rejzner ’10]

Hilbert space

Super Peter Weyl:
use that super LieVgroups have relatively simple structure, i.e.
OG ∼= C ∞ (G) ⊗ g∗1 ⇒ invariant integrals of the formZZZf =dµHdθ f (θ)
GGB⇒ (OG , h·, ·i) structure of Krein spacefor U(1|1) and SU(1|1) (u(1|1)0 = u(1) ⊕ u(1)) [KE ’20 (in prep.)]
OGk·k∼=M(m,n)∈Z2π(m,n)

Introduction: SUSY and LQG
Supermanifolds and super Lie groups
Gravity as Cartan geometry
Supergravity
Application for loop quantum supergravity
Summary

Section 6Summary

Konstantin EderTowards a manifestly supersymmetric formulation of loop quantum supergra

Summary

Considered supergravity as super Cartan geometry
found that chiral generators (Ti+ , Q A ) generate sub super Liealgebra sl(2, C) o C2 (or osp(1|2)C ) of super Poincaré algebra
gives rise to super Ashtekar connection A+ as generalized superCartan connection (even for extended SUSY)
A+ can be lifted to a super principal connection (à la Ehresmann)on associated principal super fiber bundle → super holonomies
requires chiral description (no real β)considered parallel transport of super connection on parametrized
supermanifolds

Summaryyields description of fermionic component of super connection asfunctional on supermanifold ↔ description of fermions inpAQFT[Schmitt ’96, Rejzner ’10]
⇒ in LQG: holonomies encode both matter and gravity d.o.f.
Fermions: 1(+1) dimensional quantum excitations (as gravity).
Construction of Hilbert space → derived super Peter Weyl theory for
SU(1|1) and U(1|1).
Outlook
non-manifestly supersymmetric approach to SUGRA:
Quantization of the SUSY-constraint S (without K -term!)[KE+Sahlmann ’20 (in prep.)]

since {S, S} ∝ H ⇒ consistency condition for Hamilton
constraint
solutions of S turn out to be really supersymmetric (need tocontain both boson+fermion)
SUSY cosmology