abstract duality: opposite category,
concrete duality: dual object, dualizable object, fully dualizable object, dualizing object
Examples
between higher geometry/higher algebra
Langlands duality, geometric Langlands duality, quantum geometric Langlands duality
In QFT and String theory
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”.
The most popular proposal of formalization of double field theory is in the context of para-Hermitian geometry.
The use of para-hermitian geometry in Double Field Theory was introduced by Izu Vaisman in (Vai 12), then by David Svoboda (Svo 18).
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.
The para-complex projectors are the canonical projectors onto $T^\pm M$ defined by $P_\pm = \frac{1}{2}(1\pm F)$
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.
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
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
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
for any vector $X_\pm\in\Gamma(T^\pm M)$ and $\xi\in\Omega^{r,s}(M)$.
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 $\eta(F\cdot\,,F\cdot\,)=-\eta(\,\cdot\,,\,\cdot\,)$.
The contraction with the metric $\eta$ defines two isomorphisms
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
that maps a vector a vector $X=X_++X_-$ with $X_\pm\in T^\pm M$ into $X_+ + X_-^\flat$.
A para-hermitian manifold is an almost para-hermitian manifold $(M,\eta,F)$ such that $(M,F)$ is a para-complex manifold.
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 $\mathcal{F}_+$:
There is a natural Courant algebroid structure on the bundle $T^+ M\oplus(T^+ M)^\ast$ with skew-symmetric pairing
and symmetric pairing
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
where $X\in T M$ is split in $X_\pm=P_\pm X$.
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.
The same argument can be clearly applied to $T^-M$ too.
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 (Vai 2012)).
In the special case of an (integrable) para-hermitian manifold C-bracket is given by
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.
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
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 $M$.
Let us start from a bundle gerbe $\pi:P\rightarrow M$ on a $d$-dimensional smooth manifold $M$. This is locally isomorphic to $P|_U \cong U\times \mathbf{B}U(1)$, where $\mathbf{B}U(1)$ is the circle 2-group and $U$ is an open set. If we consider an atlas $\mathbb{R}^d\rightarrow U$, we immediately have an atlas $\mathbb{R}^d\times \mathbf{B}U(1)\rightarrow U\times \mathbf{B}U(1)$ which extends the former. In other words the bundle gerbe is locally modelled on the Lie 2-group $\mathbb{R}^d\times \mathbf{B}U(1)$.
The Lie 2-algebra of $\mathbb{R}^d\times \mathbf{B}U(1)$ is $\mathbb{R}^d \oplus \mathbf{b}\mathfrak{u}(1)$. An atlas for this Lie 2-algebra will be just an ordinary Lie algebra $\mathfrak{a}$ and a homomorphism of Lie n-algebras
which is surjective in the lowest degree. What choice of $\mathfrak{a}$ can we make?
To find a working atlas we can consider the dual homomorphism of Chevalley-Eilenberg algebras, i.e. the injective homomorphism
The dg-algebra $\mathrm{CE}(\mathbb{R}^d \oplus \mathbf{b}\mathfrak{u}(1))$ has a degree $2$ generator $b$ which is annihilated by the differential, i.e. $\mathrm{d} b=0$. Therefore its image $\omega := f^\ast b$ in $\mathfrak{a}$ is a of degree $2$ element which satisfies the equation
Since $\mathbb{R}^d$ must be a subalgebra of $\mathfrak{a}$ and $\omega$ must be rotation-invariant, we have no other choice than $\mathfrak{a}=\mathbb{R}^d \oplus (\mathbb{R}^d)^\ast$ and
where $\mathrm{d}\tilde{x}_\mu$ and $\mathrm{d}x^\mu$ are respectively generators of $\mathrm{CE}((\mathbb{R}^d)^\ast)$ and $\mathrm{CE}(\mathbb{R}^d)$.
The vector space $\mathbb{R}^d \oplus (\mathbb{R}^d)^\ast$ equipped with $\omega = \mathrm{d}\tilde{x}_\mu \wedge \mathrm{d}x^\mu \in \Omega_{\mathrm{LI}}^1(\mathbb{R}^d \oplus (\mathbb{R}^d)^\ast)$ is nothing but the local space on which para-Hermitian geometry is modelled. On the other side of the atlas map $f: \mathfrak{a} \longrightarrow \mathbb{R}^d \oplus \mathbf{b}\mathfrak{u}(1)$ we have an extended Minkowski spacetime, on which bundle gerbes are modelled.
This relation is introduced in (Alf20).
Double field theory is supposed to formalize the non-geometric backgrounds of type II string theory.
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 about para-Hermitian formalism started in
Para-Hermitian formalism further developed and generalized in
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:
Luigi Alfonsi, Global Double Field Theory is Higher Kaluza-Klein Theory, Fortsch. d. Phys. 2020 [arXiv:1912.07089, doi:10.1002/prop.202000010]
(relating Kaluza-Klein compactification on principal ∞-bundles to double field theory, T-folds, non-abelian T-duality, type II geometry, exceptional geometry, …)
Luigi Alfonsi, The puzzle of global Double Field Theory: open problems and the case for a Higher Kaluza-Klein perspective (arXiv:2007.04969)
Luigi Alfonsi, Towards an extended/higher correspondence – Generalised geometry, bundle gerbes and global Double Field Theory (arXiv:2102.10970)
Last revised on December 8, 2022 at 08:08:58. See the history of this page for a list of all contributions to it.