Contents

duality

# Contents

## Idea

The term double field theory has come to be used for field theory (prequantum field theory/quantum field theory) on spacetimes which are T-folds (doubled geometries) hence for “T-duality-equivariant field theory”.

## T-duality-covariant formalism

### Para-hermitian geometry

The use of para-hermitian geometry in Double Field Theory was introduced by Izu Vaisman in (I.Vaisman 2012), then by David Svoboda (D.Svoboda 2018).

###### Definition

An almost para-complex manifold is a manifold $M$ equipped with a vector bundle endomorphism $F\in\mathrm{End}(T M)$ such that $F^2=1$ and its $\pm 1$ eigenbundles $T^\pm M$ have same rank.

###### Definition

The para-complex projectors are the canonical projectors onto $T^\pm M$ defined by $P_\pm = \frac{1}{2}(1\pm F)$

###### Definition

A $\pm$-para-complex manifold is an almost para-complex manifold $(M,F)$ such that $T^\pm M$ of $F$ is Frobenius integrable as a distribution. A para-complex manifold is a manifold that is both $+$-para-complex and $-$-para-complex.

###### Remark

A doubled manifold $M$ equipped with the $O(d,d)$-structure $\eta$ carries a natural almost para-hermitian structure. On patches $U$ with coordinates $(x^\mu,\tilde{x}_\mu)$ we have the canonical para-complex structure

$F\frac{\partial}{\partial x^\mu} = \frac{\partial}{\partial x^\mu},\,\, F\frac{\partial}{\partial \tilde{x}_\mu} = - \frac{\partial}{\partial \tilde{x}_\mu}$

with eigenbundles

• $T^+U=\big\langle\frac{\partial}{\partial x^\mu}\big\rangle$,

• $T^-U=\big\langle\frac{\partial}{\partial \tilde{x}_\mu}\big\rangle$,

with associated foliations

• $\mathcal{F}_+=\{x^\mu = \mathrm{const}\}$,

• $\mathcal{F}_-=\{\tilde{x}_\mu = \mathrm{const}\}$.

In analogy with complex geometry, if we define $\Omega^{r,s}(M)$ as the space of sections of $\Lambda^r(T^+M)\wedge\Lambda^s(T^-M)$, we have the decomposition

$\Omega^k(M)=\bigoplus_{k=r+s}\Omega^{r,s}(M)$

Let us call $\pi^{r,s}:\Omega^{r+s}(M)\rightarrow\Omega^{r,s}(M)$ the canonical projector induced by $P_\pm$.

In analogy with complex geometry, there are para-Dolbeault operators:

• $\mathrm{d}^+=\pi^{r+1,s}\circ\mathrm{d} : \Omega^{r,s}(M)\longrightarrow \Omega^{r+1,s}(M)$,

• $\mathrm{d}^-=\pi^{r,s+1}\circ\mathrm{d} : \Omega^{r,s}(M)\longrightarrow \Omega^{r,s+1}(M)$.

with properties:

• $\mathrm{d} = \mathrm{d}^+ + \mathrm{d}^-$,

• $(\mathrm{d}^\pm)^2= 0$,

• $\mathrm{d}^+\mathrm{d}^-+\mathrm{d}^-\mathrm{d}^+ =0$.

We can also define Lie derivatives on the eigenbundles $T^\pm M$ by

$\mathcal{L}^\pm_{X_\pm}\xi = (\mathrm{d}^\pm\iota_{X_\pm} + \iota_{X_\pm}\mathrm{d}^\pm)\xi$

for any vector $X_\pm\in\Gamma(T^\pm M)$ and $\xi\in\Omega^{r,s}(M)$.

###### Definition

An almost para-hermitian manifold is an almost para-complex manifold $M$ equipped with a compatible metric, i.e a symmetric tensor $\eta\in\mathrm{Sym}^2(T M)$ such that $\eta(F\cdot\,,F\cdot\,)=-\eta(\,\cdot\,,\,\cdot\,)$.

###### Remark

The contraction with the metric $\eta$ defines two isomorphisms

$\phi^\pm : T^\pm M \rightarrow (T^\mp M)^\ast$

by

• $\phi^\pm(X_\pm) = X_\pm^\flat$ and

• $(\phi^\pm)^{-1}(\xi_\pm)=\xi^\sharp$

that map a vector in $T^\pm M$ to a $1$-form in $(T^\mp M)^\ast$ and vice-versa. We used the notation $\flat,\sharp$ for the musical isomorphisms induced by the metric $\eta$ between $T M$ and $T^\ast M$.

This can be used to define a new couple of isomorphisms

$\Phi^\pm : T M \longrightarrow T^\pm M\oplus(T^\pm M)^\ast$

that maps a vector a vector $X=X_++X_-$ with $X_\pm\in T^\pm M$ into $X_+ + X_-^\flat$.

###### Definition

A para-hermitian manifold is an almost para-hermitian manifold $(M,\eta,F)$ such that $(M,F)$ is a para-complex manifold.

### Foliated Courant algebroid and spacetime

Given a $+$-para-hermitian manifold $(M,\eta,F)$, consider the triple $\big(T^+ M,[\,\cdot\,,\,\cdot\,]_+,1_{T^+M}\big)$ where

• the skew-symmetric bracket $[\,\cdot\,,\,\cdot\,]_+:=[\,\cdot\,,\,\cdot\,]|_{T^+ M}$

• the anchor $1_{T^\pm M}:=1_{T M}|_{T^\pm M}$ are the restrictions of Lie bracket and identity of $T M$.

Since $M$ is $+$-para-hermitian we have that $T^+M$ is integrable and therefore it is the tangent bundle $T^+M = T\mathcal{F}_\pm$ of a foliation $\mathcal{F}_+$. This means that this triple is just the tangent Lie algebroid of the foliation $\mathcal{F}_+$:

$\big(T^+ M,[\,\cdot\,,\,\cdot\,]_+,1_{T^+ M}\big) = \big(T\mathcal{F}_+,[\,\cdot\,,\,\cdot\,]_{T\mathcal{F}_+},1_{T\mathcal{F}_+}\big).$

There is a natural Courant algebroid structure on the bundle $T^+ M\oplus(T^+ M)^\ast$ with skew-symmetric pairing

$[X+\alpha,Y+\beta]_{+} = [X,Y]_+ + \mathcal{L}_X^+\beta -\mathcal{L}_Y^+\alpha +\mathrm{d}_+(\iota_Y\alpha)$

and symmetric pairing

$\langle X+\alpha,Y+\beta\rangle_+ = \iota_X\beta + \iota_Y\alpha.$

The isomorphism $\Phi^+:T^+ M\oplus (T^+ M)^\ast\rightarrow T^+ M\oplus T^- M = T M$ previously defined induces a Courant algebroid isomorphism and hence a Courant algebroid structure on $T M$. This induces a metric $\eta$ on $M$ and a skew-symmetric pairing on $T M$ by

$[[X_++X_-,Y_++Y_-]]_+ := [X_+,Y_+] + \Big[\mathcal{L}_{X_+}^+ Y_-^\flat -\mathcal{L}_{Y_+}^+ X_-^\flat +\mathrm{d}_+\big(\eta(X_-,Y_+)\big)\Big]^\sharp,$

where $X\in T M$ is split in $X_\pm=P_\pm X$.

###### Remark

Since $M$ is assumed $+$-para-hermitian, $T^+ M\oplus (T^+ M)^\ast$ can be written as $T\mathcal{F}_+\oplus T^\ast\mathcal{F}_+$. Therefore we constructed an isomorphism between the Courant algebroid on the whole $T M$ and the generalized tangent bundle $T\mathcal{F}_+\oplus T^\ast\mathcal{F}_+$ of the foliation. In other terms para-hermitian geometry of the doubled manifold $M$ reduces to Generalized Geometry of physical spacetime.

###### Remark

The same argument can be clearly applied to $T^-M$ too.

### C-bracket

In previous section we assumed that the $+1$-eigenbundle $T^+M$ is integrable. This is equivalent to assuming that there exists a well defined foliation that can be interpreted as the physical spacetime. However it is possible to construct a more general bracket that does not require such an assumption, but only an almost para-complex structure. Therefore it works even when a global physical spacetime foliation is not defined. This is achieved by Vaisman with the definition of C-bracket by using a generalization of the notion of Levi-Civita connection (look (I.Vaisman 2012)).

In the special case of an (inetgrable) para-hermitian manifold C-bracket is given by

$[[ X+\alpha,Y+\beta ]]_{\mathrm{C}} = [X,Y] + \mathcal{L}_X\beta -\mathcal{L}_Y\alpha + \mathrm{d}\big(\eta(X+\alpha,Y+\beta)\big) + [\alpha,\beta]^\ast + \mathcal{L}^\ast_\alpha Y -\mathcal{L}^\ast_\beta X - \mathrm{d}^\ast\big(\eta(X+\alpha,Y+\beta)\big)$

where $[\,\cdot\,,\,\cdot\,]$, $\mathcal{L}_X$ and $\mathrm{d}$ are usual Lie bracket, Lie derivative and differential on $T\mathcal{F}_+$. On the other hand $[\,\cdot\,,\,\cdot\,]^\ast$, $\mathcal{L}^\ast_\alpha$ and $\mathrm{d}^\ast$ are Lie bracket, Lie derivative and differential on $T\mathcal{F}_+^\ast$ induced by the former ones.

### Analogy with geometric quantization

The analogy between geometric quantization and DFT was firstly noticed by David Berman.

Given a symplectic manifold $(M,\omega)$ there exist couples of lagrangian foliations $\mathcal{F}_+,\mathcal{F}_-$ of $M$ defined by

$\forall X,Y\in T\mathcal{F}_\pm, \: \omega(X,Y)=0.$

For example for a symplectic space $(\mathbb{R}^{2d},\omega=\mathrm{d}x^\mu\wedge\mathrm{d}p_\mu)$ we can have $\mathcal{F}_+ = \{x^\mu = \mathrm{const} \}$ and $\mathcal{F}_-= \{p_\mu = \mathrm{const} \}$. But notice that any symplectic rotation of this choice is a couple of lagrangian foliations that works fine.

Heuristically, in geometrical quantization we make a choice of a couple of lagrangian foliations $\mathcal{F}_\pm$ to “select” a physical spacetime $\mathcal{F}_+$ from the whole symplectic-covariant theory on $M$.

Similarly in DFT, when $M$ is an (integrable) para-hermitian manifold we make a choice of a couple of lagrangian foliations $\mathcal{F}_\pm$ to “select” a physical spacetime $\mathcal{F}_+$ from the whole T-duality-covariant theory on $M$.

The idea of “doubled spacetime geometry” is a variant of the idea of T-folds, due to

The coinage of the term “double field theory” for field theory on such doubled geometry goes back to

Discussion in the context of L-infinity algebra includes

Comprehensive discussion in higher differential geometry:

Discussion of an extended version of Riemannian geometry suitable for the description of double field theory

Discussion about para-Hermitian formalism started in

• Izu Vaisman, On the geometry of double field theory Journal of Mathematical Physics 53, 033509 (2012) (arXiv:1203.0836)

Para-Hermitian formalism further developed and generalized in

• David Svoboda, Algebroid structures on para-Hermitian manifolds Journal of Mathematical Physics 59, 122302 (2018) (arxiv:1802.08180)