nLab
double field theory

Contents

Context

String theory

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”.

Proposal of formalization in para-Hermitian geometry

Doubled space as para-Hermitian manifold

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 MM equipped with a vector bundle endomorphism FEnd(TM)F\in\mathrm{End}(T M) such that F 2=1F^2=1 and its ±1\pm 1 eigenbundles T ±MT^\pm M have same rank.

Definition

The para-complex projectors are the canonical projectors onto T ±MT^\pm M defined by P ±=12(1±F)P_\pm = \frac{1}{2}(1\pm F)

Definition

A ±\pm-para-complex manifold is an almost para-complex manifold (M,F)(M,F) such that T ±MT^\pm M of FF 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 MM equipped with the O(d,d)O(d,d)-structure η\eta carries a natural almost para-hermitian structure. On patches UU with coordinates (x μ,x˜ μ)(x^\mu,\tilde{x}_\mu) we have the canonical para-complex structure

Fx μ=x μ,Fx˜ μ=x˜ μF\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=x μT^+U=\big\langle\frac{\partial}{\partial x^\mu}\big\rangle,

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

with associated foliations

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

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

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

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

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

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

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

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

with properties:

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

  • (d ±) 2=0(\mathrm{d}^\pm)^2= 0,

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

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

X ± ±ξ=(d ±ι X ±+ι X ±d ±)ξ \mathcal{L}^\pm_{X_\pm}\xi = (\mathrm{d}^\pm\iota_{X_\pm} + \iota_{X_\pm}\mathrm{d}^\pm)\xi

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

Definition

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

Remark

The contraction with the metric η\eta defines two isomorphisms

ϕ ±:T ±M(T M) * \phi^\pm : T^\pm M \rightarrow (T^\mp M)^\ast

by

  • ϕ ±(X ±)=X ± \phi^\pm(X_\pm) = X_\pm^\flat and

  • (ϕ ±) 1(ξ ±)=ξ (\phi^\pm)^{-1}(\xi_\pm)=\xi^\sharp

that map a vector in T ±MT^\pm M to a 11-form in (T M) *(T^\mp M)^\ast and vice-versa. We used the notation ,\flat,\sharp for the musical isomorphisms induced by the metric η\eta between TMT M and T *MT^\ast M.

This can be used to define a new couple of isomorphisms

Φ ±:TMT ±M(T ±M) * \Phi^\pm : T M \longrightarrow T^\pm M\oplus(T^\pm M)^\ast

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

Definition

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

Foliated Courant algebroid and spacetime

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

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

  • the anchor 1 T ±M:=1 TM| T ±M1_{T^\pm M}:=1_{T M}|_{T^\pm M} are the restrictions of Lie bracket and identity of TMT M.

Since MM is ++-para-hermitian we have that T +MT^+M is integrable and therefore it is the tangent bundle T +M=T ±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}_+:

(T +M,[,] +,1 T +M)=(T +,[,] T +,1 T +). \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(T +M) *T^+ M\oplus(T^+ M)^\ast with skew-symmetric pairing

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

and symmetric pairing

X+α,Y+β +=ι Xβ+ι Yα. \langle X+\alpha,Y+\beta\rangle_+ = \iota_X\beta + \iota_Y\alpha.

The isomorphism Φ +:T +M(T +M) *T +MT M=TM\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 TMT M. This induces a metric η\eta on MM and a skew-symmetric pairing on TMT M by

[[X ++X ,Y ++Y ]] +:=[X +,Y +]+[ X + +Y Y + +X +d +(η(X ,Y +))] , [[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 XTMX\in T M is split in X ±=P ±XX_\pm=P_\pm X.

Remark

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

Remark

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

C-bracket

In previous section we assumed that the +1+1-eigenbundle T +MT^+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+α,Y+β]] C=[X,Y]+ Xβ Yα+d(η(X+α,Y+β))+[α,β] *+ α *Y β *Xd *(η(X+α,Y+β)) [[ 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\,], X\mathcal{L}_X and d\mathrm{d} are usual Lie bracket, Lie derivative and differential on T +T\mathcal{F}_+. On the other hand [,] *[\,\cdot\,,\,\cdot\,]^\ast, α *\mathcal{L}^\ast_\alpha and d *\mathrm{d}^\ast are Lie bracket, Lie derivative and differential on T + *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,ω)(M,\omega) there exist couples of lagrangian foliations +, \mathcal{F}_+,\mathcal{F}_- of MM defined by

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

For example for a symplectic space ( 2d,ω=dx μdp μ)(\mathbb{R}^{2d},\omega=\mathrm{d}x^\mu\wedge\mathrm{d}p_\mu) we can have +={x μ=const}\mathcal{F}_+ = \{x^\mu = \mathrm{const} \} and ={p μ=const}\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 MM.

Similarly in DFT, when MM 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 MM.

Relation with non-geometry

Double field theory is supposed to formalize the non-geometric backgrounds of type II string theory.

Examples

References

Foundational papers

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

Formalization in para-Hermitian geometry

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)

Formalization in higher geometry

Discussion of double field theory using higher differential geometry:

Discussion in the context of L-infinity algebra includes

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

Comprehensive discussion in higher differential geometry:

Last revised on July 10, 2020 at 05:00:36. See the history of this page for a list of all contributions to it.