This document provides an overview of doubled geometry and double field theory. It introduces Lie and Courant algebroids, which are structures that underlie doubled geometry. Doubled geometry is defined on a manifold with fibers that transform as modules under the O(D,D) group. This allows for the description of duality-covariant and non-geometric backgrounds. Generalized tensors, connections, torsion, and curvature are defined on the doubled space. Double field theory is a field theory formulated on this doubled space.

1. Doubled Geometry and Double Field Theory
An overview
Luigi Alfonsi
2016
Luigi Alfonsi Doubled Geometry and Double Field Theory 2016 1 / 30

2. Outline
1 Introduction
2 Lie and Courant algebroids
3 Doubled Geometry
O(D, D) action
Generalised Lie derivative
Generalised Tensors
Generalized Connection and Torsion
Generalized Curvature
Generalised Scalar Curvature
4 Double Field Theory
Luigi Alfonsi Doubled Geometry and Double Field Theory 2016 2 / 30

3. Outline
1 Introduction
2 Lie and Courant algebroids
3 Doubled Geometry
O(D, D) action
Generalised Lie derivative
Generalised Tensors
Generalized Connection and Torsion
Generalized Curvature
Generalised Scalar Curvature
4 Double Field Theory
Luigi Alfonsi Doubled Geometry and Double Field Theory 2016 3 / 30

4. Introduction
Stringy geometry: beyond Riemann? We would like:
Duality-covariance geometrically realised
Description of non-geometric backgrounds
Background fields (g, b, φ) in a single object
Double Field Theory: field theory on a spacetime patched by U × T2D
gives a
O(D, D)-covariant field theory on a spacetime patched by U × TD
.
Luigi Alfonsi Doubled Geometry and Double Field Theory 2016 4 / 30

5. Outline
1 Introduction
2 Lie and Courant algebroids
3 Doubled Geometry
O(D, D) action
Generalised Lie derivative
Generalised Tensors
Generalized Connection and Torsion
Generalized Curvature
Generalised Scalar Curvature
4 Double Field Theory
Luigi Alfonsi Doubled Geometry and Double Field Theory 2016 5 / 30

6. Lie and Courant algebroids
Definition
A Lie algebroid over a manifold M is a vector bundle (L, M, π) equipped with Lie
bracket
[·, ·] : Γ(L) × Γ(L) −→ Γ(L) (1)
and with a morphism ρ of vector bundles (called anchor) ρ : L −→ TM whose
tangent map dρ preserves the bracket
dρ([X, Y ]) = [dρ(X), dρ(Y )] (2)
and such that it holds the following Leibniz rule:
[X, fY ] = f [X, Y ] + ρ(X)[f ]Y (3)
for all X, Y ∈ Γ(L) and f ∈ C∞
(M).
Luigi Alfonsi Doubled Geometry and Double Field Theory 2016 6 / 30

7. Definition
A Lie bialgebroid over a manifold M is a couple (L, L∗
) where L and its dual
bundle L∗
are Lie algebroids such that satisfy
dL[X, Y ] = [dLX, Y ] + [X, dLY ]. (4)
Example
The tangent bundle TM is a Lie algebroid with the commutator [·, ·] as Lie
bracket and ρ = 1TM . The couple (TM, T∗
M) is a bialgebroid.
Definition 2.1
The Jacobiator of a bilinear skew-symmetric operator [·, ·] on a vector space V ,
given ei ∈ V , is
Jac(e1, e2, e3) = [[e1, e2], e3] + [[e2, e3], e1] + [[e3, e1], e2]. (5)
Luigi Alfonsi Doubled Geometry and Double Field Theory 2016 7 / 30

8. If (L, L∗
) is a Lie bialgebroid, L ⊕ L∗
has a new structure:
Definition
A Courant algebroid is a vector bundle E over M equipped with
1 a skew-symmentric bracket ·, · : Γ(E) × Γ(E) → Γ(E),
2 a non-degenerate bilinear form ·, · : E × E → R,
3 a bundle map π : E → TM (”anchor”),
such that the following conditions are satisfied ∀ei ∈ Γ(E), ∀f , g ∈ C∞
(M):
π( e1, e2 ) = [π(e1), π(e2)],
Jac(e1, e2, e3) = 1
3 d e1, e2 , e3 + c.p. ,
e1, fe2 = f e1, e2 + π(e1)[f ]e2 − e1, e2 df ,
π ◦ d = 0 ⇒ df , dg = 0,
π(e1)[ e2, e3 ] = e1 • e2, e3 + e2, e1 • e3 ,
where we defined e1 • e2 ≡ e1, e2 + d e1, e2 .
TM ⊕ T∗
M is naturally a Courant algebroid with
X + ξ, Y + η = [X, Y ] + LX η − LY ξ +
1
2
(ıY ξ − ıX η), (6)
Luigi Alfonsi Doubled Geometry and Double Field Theory 2016 8 / 30

9. Outline
1 Introduction
2 Lie and Courant algebroids
3 Doubled Geometry
O(D, D) action
Generalised Lie derivative
Generalised Tensors
Generalized Connection and Torsion
Generalized Curvature
Generalised Scalar Curvature
4 Double Field Theory
Luigi Alfonsi Doubled Geometry and Double Field Theory 2016 9 / 30

10. O(D, D) action
Definition
A left G-module is an abelian group (A, +) with a left group action · : G × A → A
such that g · (a + b) = g · a + g · b.
Let M be a fibre bundle over the base manifold N where the fibre F is an
O(D, D)-module. Consider torus fibers F = T2D
with the action



x 1
...
x 2D


 = h



x1
...
x2D


 , h ∈ O(D, D) (7)
From String Theory (Level Matching Condition) we have the axiom that any
couple of fields A, B satisfies ∂M A∂M
B = 0 on T2D
.
This means that, given coordinates (x1
, x2
, . . . , x2D
) on T2D
, every field depends
only on x1
, . . . , xD
, i.e on a torus TD
⊂ T2D
.
Luigi Alfonsi Doubled Geometry and Double Field Theory 2016 10 / 30

11. From each patch U × T2D
we can choose a subpatch U × TD
.
Subpatches can glue together only with diffeomorphisms to form a manifold.
Subpatches can glue togheter with diffeomorphisms and
O(D, D)-transormations to form a T-fold (non-geometric background).
Figure: T-fold [credits: Falk Hassler]
Luigi Alfonsi Doubled Geometry and Double Field Theory 2016 11 / 30

12. Generalised Lie derivative
A generalised vector W is an element of a vector bundle E(M), transforming as
δW = LV W (8)
Definition
The generalised Lie derivative is defined as:
LV W ≡ [V , W ] + V , ∂M
W eM (9)
for a basis {eM }M=1,...,2D of Γ(E).
where
Definition
Metric on E defined by V , W ≡ 1
2 V T
ηW , with η =
0 1
1 0
singlet of O(D, D).
Luigi Alfonsi Doubled Geometry and Double Field Theory 2016 12 / 30

13. The fibres of E are assumed to be O(D, D)-modules, so that, given h ∈ O(D, D),
we have W (hx) = hW (x).
Definition
The D-bracket:
[·, ·]D : Γ(E) × Γ(E) −→ Γ(E)
(V , W ) −→ [V , W ]D ≡ LV W
(10)
Definition
The C-bracket:
·, · C : Γ(E) × Γ(E) −→ Γ(E)
(V , W ) −→ V , W C ≡ 1
2 ([V , W ]D − [W , V ]D)
(11)
We have the identity:
[LV , LW ] = L V ,W C
(12)
The C-bracket closes the algebra of generalised Lie derivatives.
Luigi Alfonsi Doubled Geometry and Double Field Theory 2016 13 / 30

14. Given a patch U ⊂ M, since a generalised vector W ∈ E(U) depends only on the
first x1
, . . . , xD
coordinates, it can be thought as an element of a vector bundle
E(U) over a subpatch U ⊂ U.
On E(U):
·, · C reduces to a Courant bracket ·, ·
Hence E(U) is a Courant algebroid.
We can say that E(M) is ”locally” a Courant algebroid.
We can show that E(U) is a deformation of the Courant algebroid TU ⊕ T∗
U.
Luigi Alfonsi Doubled Geometry and Double Field Theory 2016 14 / 30

15. There exists bundle isomorphism L : E(U) → TU ⊕ T∗
U defined by:
W = LW , L =
1 0
−b 1
. (13)
The vector W ∈ TU ⊕ T∗
U transforms under Diff(U) as
W (X ) = ΛW (X) (14)
where we defined
Λ =
J 0
0 (JT
)−1 , Jµ
ν ≡
∂x µ
∂xν
.
From (13) and (14) W ∈ E(U) transforms under Diff(U) as:
W (X ) = ΛW (X), (15)
where we defined
Λ ≡ L −1
ΛL(X), L (X ) =
1 0
−b (x ) 1
(16)
Luigi Alfonsi Doubled Geometry and Double Field Theory 2016 15 / 30

16. Generalised Tensors
Definition
A (p, q)-generalised tensor is an element of Γ(E⊗p
⊗ E∗⊗q
).
Like for vectors
Untwisted tensor:
T
M1···Mp
N1···Nq
= LM1
A1
· · · L
Mp
Ap
T
A1···Ap
B1···Bq
(L−1
)B1
N1
· · · (L−1
)
Bq
Nq
.
Transformation:
T
M1···Mp
N1···Nq
(X ) = ΛM1
A1
· · · Λ
Mp
Ap
T
A1···Ap
B1···Bq
(X)(Λ−1
)B1
N1
· · · (Λ−1
)
Bq
Nq
.
Luigi Alfonsi Doubled Geometry and Double Field Theory 2016 16 / 30

17. A (0, 2)-generalised tensor H such that H−1
= ηT
Hη can decompose as:
HMN =
gµν − bµαgαβ
bβν bµαgαν
−gµα
bαν gµν (17)
where gµν is symmetric and bµν is antisymmetric.
From transformations δH = LV H we get
gµν(x ) = gαβ(x)
∂xα
∂x µ
∂xβ
∂x ν
, bµν(x ) = bαβ(x) + ∂α ˜vβ − ∂β ˜vα
∂xα
∂x µ
∂xβ
∂x ν
.
where we decompose V = (v, ˜v)T
.
⇒ g, b are metric and 2-form field on spacetime manifold.
LV on E(U) reduce to Supergravity gauge transformations.
From h ∈ O(D, D) transformations H (X ) = hT
H(X)h we get the Buscher
rules for T-duality.
Luigi Alfonsi Doubled Geometry and Double Field Theory 2016 17 / 30

18. Doubled Manifolds
Definition
A doubled manifold (M, H, d) is a fibre bundle M over the base manifold N where
the fibre T2D
is an O(D, D)-module, equipped with
1 a (0, 2) generalised tensor H ∈ Γ(E∗
⊗ E∗
) such that H−1
= ηT
Hη, called
generalised metric,
2 a scalar density d, called dilaton density,
3 a volume form Vol = e−2d
VolN ∧ d2D
x with VolN volume form of N.
Luigi Alfonsi Doubled Geometry and Double Field Theory 2016 18 / 30

19. Generalized Connection and Torsion
Definition
Generalised connection is a map
: Γ(E) × Γ(E) −→ Γ(E)
(V , W ) −→ V W
(18)
such that it satisfies the usual properties:
(V +W )Y = V Y + W Y ,
V (W + Y ) = V W + V Y ,
fV W = f V W ,
V (fW ) = V M
(∂M f )W + V W .
Linearity ⇒ connection determined by its components
M eN = ΩK
MN eK (19)
where {eM } is a coordinate basis. Then:
V W = V M
(∂M W N
+ ΩN
MK W K
)eN , (20)
Luigi Alfonsi Doubled Geometry and Double Field Theory 2016 19 / 30

20. Definition
The generalised torsion T is the a generalised tensor defined by
LV − LV WM = TMNK V N
W K
, (21)
where LV is the generalized Lie derivative with ∂M replaced by M .
In terms of connection components:
TMNK = ΩMNK − ΩNMK + ΩKMN . (22)
Luigi Alfonsi Doubled Geometry and Double Field Theory 2016 20 / 30

21. Definition
A generalised Levi-Civita connection is a generalised connection such that:
1 it preserves the η metric: M ηNK = 0,
2 it preserves the generalised metric: M HNK = 0,
3 the generalised torsion vanishes: TMNK = 0,
4 integration by parts is V M
N W N
Vol = − W N
N V M
Vol.
Solve first condition:
M ηNK = ∂M ηNK − ΩP
MN ηPK − ΩP
MK ηNP = 0 ⇒ ΩM[NK] = 0 (23)
Luigi Alfonsi Doubled Geometry and Double Field Theory 2016 21 / 30

22. Generalized Curvature
Define R by the commutator:
[ M , N ]AK = −R L
MNK AL − T L
MN LAK . (24)
In terms of connection components:
RMNKL = ∂M ΩNKL − ∂N ΩMKL
+ ΩMPLΩP
NK − ΓNPLΓP
MK . (25)
Hence R[MN]KL = 0.
But R is not a generalised tensor.
Definition
The generalised curvature is the generalised tensor
RMNKL ≡ RMNKL + RKLMN + ΩPMN ΩP
KL (26)
where R is determined by the expression (24).
Luigi Alfonsi Doubled Geometry and Double Field Theory 2016 22 / 30

23. Symmetries:
By definition: RMNKL = −RKLMN .
R[MN]KL = 0 ⇒ R[MN]KL = 0.
ΩM[NK] = 0 ⇒ RMN[KL] = 0.
Analogous of Bianchi identities: Ω[MNK] = 0 ⇒ R[MNK]L = 0.
Luigi Alfonsi Doubled Geometry and Double Field Theory 2016 23 / 30

24. Introduce the projectors to the GL(D)-subbundles:
Π ≡
1
2
(η − H), Π ≡
1
2
(η + H), (27)
A generalised vector can be decomposed as V = ΠV + ΠV .
If is generalised Levi-Civita we have conditions:
Π = Π = 0. (28)
Notation convention:
Π N
M VN ≡ VM , Π
N
M VN ≡ VM (29)
.
Luigi Alfonsi Doubled Geometry and Double Field Theory 2016 24 / 30

25. From ΩM[NK] = 0 and Ω[MNK] = 0 obtain:
ΩMNK = ΩMNK + ΩMNK +
+ ΩMNK + ΩMKN + ΩNKM + ΩKNM +
+ ΩMNK + ΩMKN + ΩKMN + ΩNMK
(30)
Thus we only need ΩMNK , ΩMNK , ΩMNK and ΩMNK to determine the connection.
For a generalised Levi-Civita connection we find:
ΩMNK = −Π P
M (Π∂P Π)KN
ΩMNK = −Π
P
M (Π∂P Π)KN
ΩMNK = 4
D−1 ΠM[N Π P
K] ∂P d + (Π∂Q
Π)[QP] + ΩMNK
ΩMNK = 4
D−1 ΠM[N Π
P
K] ∂P d + (Π∂Q
Π)[QP] + ΩMNK
(31)
where ΩMNK and ΩMNK are unfixed and constrained by:
ηMK
ΩMNK = 0, ηMK
ΩMNK = 0. (32)
Luigi Alfonsi Doubled Geometry and Double Field Theory 2016 25 / 30

26. Hence decomposition ΩMNK = ΩMNK + ΣMNK , where
ΩMNK is fixed,
ΣMNK = ΩMNK + ΩMNK is unfixed.
We can calculate the fixed part:
ΩMNK =
1
2
HKP ∂M HP
N +
1
2
δ P
[N H Q
K] + H P
[N δ Q
K] ∂P HQM +
+
2
D − 1
ηM[N δ Q
K] + HM[N H Q
K] ∂Qd +
1
4
HPM
∂M HPQ .
(33)
Also generalised curvature has unfixed terms:
RMNKL = RMNKL + 2 [M ΣN]KL + 2 [K ΣK]MN +
+ 2Σ[M|PLΣ P
N]K + 2Σ[K|PN Σ P
L]M + ΣPMN ΣP
KL,
(34)
Luigi Alfonsi Doubled Geometry and Double Field Theory 2016 26 / 30

27. Generalised Scalar Curvature
Definition
The generalised scalar curvature is
R ≡ R
MN
MN = −R MN
MN
(35)
Lemma
It can be proved that R is univocally fixed by H,d (i.e. R = R).
We can calculate:
R(H, d) = 4HMN
∂M ∂N d − ∂M ∂N HMN
+
+ 4HMN
∂M d∂N d + 4∂M HMN
∂N d+
+
1
8
HMN
∂M HKL
∂N HKL −
1
2
HMN
∂M HKL
∂K HNL.
(36)
Luigi Alfonsi Doubled Geometry and Double Field Theory 2016 27 / 30

28. Outline
1 Introduction
2 Lie and Courant algebroids
3 Doubled Geometry
O(D, D) action
Generalised Lie derivative
Generalised Tensors
Generalized Connection and Torsion
Generalized Curvature
Generalised Scalar Curvature
4 Double Field Theory
Luigi Alfonsi Doubled Geometry and Double Field Theory 2016 28 / 30

29. Free Double Field Theory
Free DFT action
The action funtional of DFT is
SDFT[H, d] =
M
R(H, d)Vol(d), (37)
where R(H, d) is the generalised scalar curvature.
If we vary the generalised metric:
δH = ΠδKΠ + ΠδKΠ ⇒ δSDFT =
M
δKMN
RMN Vol, (38)
where δK is a symmetric matrix.
Field equations:
δSDFT
δKNM
= RMN = 0. (39)
They give exactly the equations of Supergravity.
Luigi Alfonsi Doubled Geometry and Double Field Theory 2016 29 / 30

30. Summary
DFT describes T-duality covariant field theories.
DFT describes field theories on non-geometric backgrounds.
DFT describes geometric fields g, b in a single object H.
Further developments
Analogous construction for U-duality groups En.
Relaxing Strong Constraint
Rigorous mathematical foundations
Further reading:
Gerardo Aldazabal, Diego Marques, Carmen Nunez.
Double Field Theory: A Pedagogical Review. 2013.
Chris Hull, Barton Zwiebach.
Double Field Theory. 2009.
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