Sigma Model Geometry
U. Lindström1
1Department of Physics and Astronomy Division of Theoretical Physics University of Uppsala
November 20 ASC workshop "Geometry and Physics"
U. Lindström Superspace is smarter Outline
Formulations of Generalized Kähler Geometry The corresponding Sigma Models
M. Göteman, C. Hull, M. Rocek,ˇ I. Ryb, R. von Unge and M. Zabzine.
U. Lindström Superspace is smarter Outline II: Sigma Model geometry
Sigma models SUSY sigma models and geometry Complex geometry Kähler Bihermitean geometry Generalized complex geometry Generalized Kähler geometry Superspace descriptions
U. Lindström Superspace is smarter Outline II: Sigma Model geometry
Sigma models SUSY sigma models and geometry Complex geometry Kähler Bihermitean geometry Generalized complex geometry Generalized Kähler geometry Superspace descriptions
U. Lindström Superspace is smarter Outline II: Sigma Model geometry
Sigma models SUSY sigma models and geometry Complex geometry Kähler Bihermitean geometry Generalized complex geometry Generalized Kähler geometry Superspace descriptions
U. Lindström Superspace is smarter Outline II: Sigma Model geometry
Sigma models SUSY sigma models and geometry Complex geometry Kähler Bihermitean geometry Generalized complex geometry Generalized Kähler geometry Superspace descriptions
U. Lindström Superspace is smarter Outline II: Sigma Model geometry
Sigma models SUSY sigma models and geometry Complex geometry Kähler Bihermitean geometry Generalized complex geometry Generalized Kähler geometry Superspace descriptions
U. Lindström Superspace is smarter Outline II: Sigma Model geometry
Sigma models SUSY sigma models and geometry Complex geometry Kähler Bihermitean geometry Generalized complex geometry Generalized Kähler geometry Superspace descriptions
U. Lindström Superspace is smarter Outline II: Sigma Model geometry
Sigma models SUSY sigma models and geometry Complex geometry Kähler Bihermitean geometry Generalized complex geometry Generalized Kähler geometry Superspace descriptions
U. Lindström Superspace is smarter Outline II: Sigma Model geometry
Sigma models SUSY sigma models and geometry Complex geometry Kähler Bihermitean geometry Generalized complex geometry Generalized Kähler geometry Superspace descriptions
U. Lindström Superspace is smarter Outline III: Relation to supergravity
Pure Spinors Generalized Calabi Yau Supergravity Relation to the Sigma Model formulation
U. Lindström Superspace is smarter Outline III: Relation to supergravity
Pure Spinors Generalized Calabi Yau Supergravity Relation to the Sigma Model formulation
U. Lindström Superspace is smarter Outline III: Relation to supergravity
Pure Spinors Generalized Calabi Yau Supergravity Relation to the Sigma Model formulation
U. Lindström Superspace is smarter Outline III: Relation to supergravity
Pure Spinors Generalized Calabi Yau Supergravity Relation to the Sigma Model formulation
U. Lindström Superspace is smarter Sigma Models
φi :Σ → T
ξ 7→ φi (ξ) Z i j S = dφ gij (φ) ? dφ Σ 2 i 2 i j i k ∇ φ := ∂ φ + ∂φ Γjk ∂φ = 0 Z n µν i j o S = dξ η ∂µX gij (X)∂νX + ... ΣB
U. Lindström Superspace is smarter Susy Sigma Models
Susy σ models ⇐⇒ Geometry of T
d= 6 4 2 Geometry N= 1 2 4 Hyperkähler N= 1 2 Kähler N= 1 Riemannian
(Odd dimensions have the same structure as the even dimension lower.)
U. Lindström Superspace is smarter Sigma models in d=2
The (1,1) analysis by Gates Hull and Rocekˇ gives:
Susy (0,0) (1,1) (2,2) (2,2) (4,4) (4,4) Bgd G, B G G, B G G, B Geom Riem. Kähler biherm. hyperk. bihyperc.
U. Lindström Superspace is smarter Sigma models in d=2
The (1,1) analysis by Gates Hull and Rocekˇ gives:
Susy (0,0) (1,1) (2,2) (2,2) (4,4) (4,4) Bgd G, B G G, B G G, B Geom Riem. Kähler biherm. hyperk. bihyperc.
U. Lindström Superspace is smarter Complex Geometry
Manifold (M2d , J)
Complex structure: J ∈ End(TM) J2 = −1
1 Projectors: π± := 2 (11 ± iJ) These define an involutive distribution if
π∓[π±u, π±v] = 0 ⇐⇒ N (J) = 0 (Nijenhuis)
Here
N (J)(X, Y ) := [X, Y ] − [JX, JY ] + J[JX, Y ] + J[X, JY ]
or, in a local coordinates in a basis ∂µ,
µ λ µ µ λ N νρ = J [νJ ρ],λ + J λJ [ν,ρ]. This is integrability of J.
U. Lindström Superspace is smarter Hermitean
In local holomorphic coordinates, M2d ⊃ O ≈ C2k , and the transition functions are holomorphic.
Hermitean Metric: Jt gJ = g
(g → g = g + Jt gJ)
Symplectic 2-form: ω := gJ
Kähler: dω = 0, ∇J = 0, gzz¯ = ∂z ∂z¯K (z, z¯)
Hyperkähler: JA, A = 1, 2, 3 JAJB = −δAB + ABCJC
U. Lindström Superspace is smarter Kähler Geometry
Erich Kähler 1906-2000
K 3
U. Lindström Superspace is smarter Gates-Hull-Rocekˇ Bihermitean.
(M, g, J(±), H)
2 t (±) J(±) = −11 , J(±)gJ(±) = g , ∇ J(±) = 0
(±) 0 1 −1 Γ = Γ ± 2 g H , H = dB N (J±) = 0.
E := g + B
U. Lindström Superspace is smarter GKG II
(M, g, J(±))
2 t J(±) = −11 , J(±)gJ(±) = g , ω(±) := gJ(±)
c c c d(+)ω(+) + d(−)ω(−) = 0 , dd(±)ω(±) = 0 ,
c c H := d(+)ω(+) = −d(−)ω(−)
U. Lindström Superspace is smarter Generalized Complex Geometry
Complex structure:
J ∈ End(TM ⊕ T ∗M), J 2 = −1 1 Π± := 2 (11 ± J ) Involutivity
Π∓[Π±U, Π±V ]H = 0
where
U = (u, ξ) , V = (v, ρ) 1 [U, V ]H := [u, v] + Luρ − Lv ξ − 2 d(ıuρ − ıv ξ) + ıuıv H
=: [U, V ]C + ıuıv H .
U. Lindström Superspace is smarter It follows that the “Nijenhuis” tensor vanishes:
NC(J )(U, V ) := [U, V ]C − [J U, J V ]C + J [J U, V ]C + J [U, J V ]C .
µ In a coordinate basis (∂µ, du ), where we define
JP J = L −Jt the corresponding local conditions read
µ µν σ N ρλ(J) + P L[ρλ,ν] + J [ρHλ]σν = 0
σ[ν ρλ] P P σ = 0 . (0.1) and two more relations. This is integrability of J .
U. Lindström Superspace is smarter "Newlander-Nirenberg"
When J is integrable there are local holomorphic and Darboux − coordinates such that M2d looks like Ck × R2d k .
i, i
a
U. Lindström Superspace is smarter The automorphisms of this courant bracket are diffeomorphisms and b-transforms:
b e (u, ξ) = (u, ξ + ıub) , db = 0 .
ν In the coordinate basis (∂µ, du ) a b-transform acts on J as follows:
1 0 1 0 J , b 1 −b 1
U. Lindström Superspace is smarter In such a basis, the natural pairing
< (u, ξ), (v, ρ) >= ıuρ + ıv ξ is represented by the matrix
0 1 I = 1 0
An additional requirement of GCG (used above) is that
J t IJ = I
U. Lindström Superspace is smarter Generalized Kähler Geometry
Two commuting generalized complex structures
2 J(1,2) = −11 , [J(1), J(2)] = 0 ,
t J(1,2)IJ(1,2) = I , G := −J(1)J(2)
Ex. Kähler (ω = gJ):
J 0 0 −ω−1 0 g−1 J = J = G = 1 0 −Jt 2 ω 0 g 0
U. Lindström Superspace is smarter The G-map
GKG ↔ Bi-Hermitean:
J(1,2) =
−1 −1 1 0 J(+) ± J(−) −(ω(+) ∓ ω(−)) 1 0 t t B 1 ω(+) ∓ ω(−) −(J(+) ± J(−)) −B 1
U. Lindström Superspace is smarter GKG IV Local Symplectic Description
(M, J(±))
Locally, ∃ “symplectic” two-forms F(±) such that
t F(±)(v, J(±)v) > 0 , d(F(+)J(+) − J(−)F(−)) = 0 .
1 (2,0) (0,2) F(±) = 2 i(B(±) − B(±) ) ∓ ω(±) 1 t 1 t t F(+) = − 2 E(+)J(+) , F(−) = − 2 J(−)E(−)
U. Lindström Superspace is smarter GKG V Generalized Kähler Potential
Geometric data: (M, g, H, J(±)) or (M, g, J(±)) or (M, F(±), J(±)). In each case, there is a complete description in terms of a Generalized Kähler potential K . Unlike the Kähler case, the expressions are non-linear in second derivatives of K . E.g.,
J 0 J(+) = −1 −1 (KLR) [J, KLL](KLR) JKLR
g = Ω[J(+), J(−)]
−1 F(+) = dλ(+) , λ(+)` = iKRJ(KLR) KL` , ...
U. Lindström Superspace is smarter Generating function
There are two special sets of Darboux coordinates for the L symplectic form Ω. One set, (X , YL), is also canonical R coordinates for J(+) and the other set, (X , YR) is canonical coordinates for J(−). The symplectomorphism that relates the two sets of coordinates has thus a generating function. This generating function is in fact the generalized Kähler-potential K (XL, XR).
L L R R (X , YL) ← K (X , X ) → (X , YR) i 0 i 0 J = J = (+) 0 −i (−) 0 −i ` r dΩ = dX ∧ dY` + c.c. dΩ = X ∧ Yr + c.c
This fact is a key ingredient in the proof that we have a complete description or GKG.
U. Lindström Superspace is smarter Superspaces
d = 2 , N = (2, 2)
Algebra: ¯ {D±, D±} = i∂++ = Constrained superfields:
¯ a D±φ = 0 , ¯ a0 a0 D+χ = D−χ = 0 , ¯ ` D+X = 0 , ¯ r D−X = 0 .
Notation: c := a, a¯ , t := a0, a¯0 , L := `, `¯ , R := r, ¯r .
U. Lindström Superspace is smarter Superspace I
The (2, 2) formulation uses the generalized Kähler Potential. Z ¯ ¯ c t L R S = D+D+D−D−K (φ , χ , X , X )
K → K (XL, XR)
Reduction to (2, 1) superspace
L L D− =: D− − iQ− , X| =: X , Q−X | =: Ψ− Z ¯ L R S = D+D+D− KLΨ− + KRJD−X Z ¯ α S = i D+D+D−(λ(+)αD−ϕ + c.c.)
which uses the “Liouville form” (F(+) = dλ(+))
U. Lindström Superspace is smarter Superspace II
Reduction to (1, 1) finally yields Z S = D+D− (D+XED−X) .
R R The reduction goes via D+ =: D+ − iQ+ , Q+X | =: Ψ+ and L R both the auxiliary spinors Ψ− and Ψ+ have been eliminated.
The (1, 1) formulation uses E = g + B directly.
U. Lindström Superspace is smarter Summary
Superspace encodes and dictates all the geometric formulations of Generalized Kähler Geometry.
U. Lindström Superspace is smarter Pure Spinors
Multi forms ρ on M are spinors of T ⊕ T ∗.
U = (u, ξ) ∈ T ⊕ T ∗ acts on a form ρ according to
U · ρ = ıuρ + ξ ∧ ρ
This satisfies the Clifford algebra identity for the indefinite metric I:
{U, V }· ρ = (U · V + V · U) · ρ = 2I(U, V )ρ
U. Lindström Superspace is smarter The null space of a spinor ρ
∗ Lρ = {U ∈ TM ⊕ T M|U · ρ = 0} is isotropic. A spinor ρ is pure if its null space is maximally isotropic, rank d.
A GCS J may alternatively be defined via decomposition
∗ (T ⊕ T ) ⊗ C = L + L¯ where L is the +i eigenbundle of J .
To every GCS J with +i eigenbundle LJ is associated a complex pure spinor ρJ :
LJ = LρJ
U. Lindström Superspace is smarter Generalized CY
A generalized Calabi-Yau structure:
dρ = 0 , (ρ, ρ¯) 6= 0
A generalized Calabi-Yau metric structure is defined as a pair of closed pure spinors ρ1 and ρ2 such that the corresponding generalized complex structures J1 and J2 give rise to a GKS and
(ρ1, ρ¯1) = α(ρ2, ρ¯2) 6= 0
Mukai pairing:
X j 2j d−2j 2j+1 d−2j−1 (ρ1, ρ2) = (−1) [ρ1 ∧ ρ2 + ρ1 ∧ ρ2 ] , j
U. Lindström Superspace is smarter Supergravity
The conditions for Type II supergravity solutions in
2 2A(y) 2 m n ds(10) = e ds(4) + gmndy dy
is
4A−Φ 4A ˜ dH (e <ρ1) = e F 3A−Φ dH (e ρ2) = 0 2A−Φ dH (e =ρ1) = 0
where F˜ is (part of) the polyform of RR fields.
U. Lindström Superspace is smarter The generalized CY metric structure defines a Type II supersymmetric supergravity solution. (No RR fluxes).
Use the Gualtieri map to find (gµν, Hµνρ, Φ).
−2Φ√ 1 D (ρ1, ρ¯1) = α(ρ2, ρ¯2) = e g dx ∧ .... ∧ dx
(+) (−) Rµν + 2∇µ ∂νΦ = 0 , automatically satisfied.
U. Lindström Superspace is smarter Construction from the sigma model
Ansatz:
R1,2+iS1,2 ρ1,2 = N1,2 ∧ e , (0.2)
where
f (φ) 1 dc N1 = e dφ ∧ ... ∧ dφ , = g(χ) χ1 ∧ ... ∧ χ , N2 e d d dt R1 = −d(KLdXL) ,
R2 = −d(KRdXR) ,
S1 = d(KT Jdχ + KLJdXL − KRJdXR) ,
S2 = −d(KCJdφ + KLJdXL + KRJdXR) ,
These are pure spinors with the correct properties.
U. Lindström Superspace is smarter The Generalized Monge-Ampère equation
(ρ1, ρ1) = α(ρ2, ρ2) =⇒
−K ¯ −Klr −K ¯ ¯ ll lt dsdc f (φ) f (φ¯) (−1) e e det −K¯r¯l −K¯rr −K¯r¯t (0.3) −Kt¯l −Ktr −Kt¯t Kl¯r Kl¯l Klc¯ g(χ) g¯(¯χ) = α e e det Kr¯r Kr¯l Krc¯ Kc¯r Kc¯l Kcc¯
−f (φ) −¯f (φ¯) −Kl¯l −Klr −Kl¯t 2Φ dsdc e e e = (−1) det −K¯r¯l −K¯rr −K¯r¯t (0.4) det KLR −Kt¯l −Ktr −Kt¯t
U. Lindström Superspace is smarter Outlook
Include RR fields in the geometry. See recent work by Waldram and collaborators. Understand reduction of GKG. Cavalcanti, Gualtieri,.... SKT (strong Kähler with Torsion). Cavalcanti,...,(2, 1) -models......
U. Lindström Superspace is smarter Outlook
Include RR fields in the geometry. See recent work by Waldram and collaborators. Understand reduction of GKG. Cavalcanti, Gualtieri,.... SKT (strong Kähler with Torsion). Cavalcanti,...,(2, 1) -models......
U. Lindström Superspace is smarter Outlook
Include RR fields in the geometry. See recent work by Waldram and collaborators. Understand reduction of GKG. Cavalcanti, Gualtieri,.... SKT (strong Kähler with Torsion). Cavalcanti,...,(2, 1) -models......
U. Lindström Superspace is smarter Outlook
Include RR fields in the geometry. See recent work by Waldram and collaborators. Understand reduction of GKG. Cavalcanti, Gualtieri,.... SKT (strong Kähler with Torsion). Cavalcanti,...,(2, 1) -models......
U. Lindström Superspace is smarter THANK YOU FOR YOUR ATTENTION!
U. Lindström Superspace is smarter