What is called double dimensional reduction is a variant of Kaluza-Klein mechanism combined with fiber integration in the presence of branes: given a spacetime of dimension $d+1$ in which a $p+1$-brane propagates, its KK-reduction results in a $d$-dimensional effective spacetime containing a $p+1$-brane together with a “doubly reduced” $p$-brane, which is the reduction of those original $(p+1)$-brane configurations that wrapped the cycle along which the KK-reduction takes place.
Let $\mathbf{H}$ be the smooth topos. For $p+1 \in \mathbb{N}$ write $\mathbf{B}^{p+1}U(1)_{conn} \in \mathbf{H}$ for the universal moduli stack of circle n-bundles with connection (given by the Deligne complex).
Notice that fiber integration in ordinary differential cohomology has the following stacky incarnation (see here):
For $\Sigma$ an oriented closed manifold of dimension $k \leq p+1$, then fiber integration in ordinary differential cohomology is reflected by a morphism of the form
where the vertical morphisms are the curvature maps and the bottom morphims reflects ordinary fiber integration of differential forms.
Given a cocycle
on the Cartesian product of some smooth space $X$ with $\Sigma$, then its double dimensional reduction is the cocycle on $X$ which is given by the composite
where the first morphism is the unit of the (Cartesian product $\dashv$ internal hom)-adjunction.
We discuss here a formalization of double dimensional reduction via cyclification adjunction (FSS 16, section 3, BMSS 18, section 2.2). For more see at geometry of physics – fundamental super p-branes the section on double dimensional reduction.
Let $\mathbf{H}$ be any (∞,1)-topos and let $G$ be an ∞-group in $\mathbf{H}$. Then the right base change/dependent product along the canonical point inclusion $\ast \to \mathbf{B}G$ into the delooping of $G$ takes the following form: There is a pair of adjoint ∞-functors of the form
where
$[G,-]$ denotes the internal hom in $\mathbf{H}$,
$[G,-]/G$ denotes the homotopy quotient by the conjugation ∞-action for $G$ equipped with its canonical ∞-action by left multiplication and the argument regarded as equipped with its trivial $G$-$\infty$-action (for $G = S^1$ the circle group this is the cyclic loop space construction).
Hence for
$\hat X \to X$ a $G$ principal ∞-bundle
$A$ a coefficient object, such as for some differential generalized cohomology theory
then there is a natural equivalence
given by
First observe that the conjugation action on $[G,X]$ is the internal hom in the (∞,1)-category of $G$-∞-actions $Act_G(\mathbf{H})$. Under the equivalence of (∞,1)-categories
(from NSS 12) then $G$ with its canonical ∞-action is $(\ast \to \mathbf{B}G)$ and $X$ with the trivial action is $(X \times \mathbf{B}G \to \mathbf{B}G)$.
Hence
Actually, this is the very definition of what $[G,X]/G \in \mathbf{H}_{/\mathbf{B}G}$ is to mean in the first place, abstractly.
But now since the slice (∞,1)-topos $\mathbf{H}_{/\mathbf{B}G}$ is itself cartesian closed, via
it is immediate that there is the following sequence of natural equivalences
Here $p \colon \mathbf{B}G \to \ast$ denotes the terminal morphism and $p_! \dashv p^\ast$ denotes the base change along it.
When $\Sigma = S^1$ is the circle, and we think of $X \times S^1$ as a spacetime of 11-dimensional supergravity, then $\nabla \colon X \times S^1 \to \mathbf{B}^3 U(1)_{conn}$ may represent the supergravity C-field as a cocycle in ordinary differential cohomology. Then its double dimensional reduction in the sense of def. is the differential cocycle representing the B-field on $X$, in the sense of string theory.
For $\Sigma = S^1$ a circle as in example , then the morphism $X \longrightarrow [\Sigma, X \times \Sigma]$ in def. sends each point of $X$ to the loop in $X\times S^1$ that winds identically around the copy of $S^1$ at that point. Hence in this case it would make sense to consider, more generally, for each $p \in \mathbb{Z}$ the “order $p$” double dimensional reduction, given by the operation where one instead considers the map that lets the loop wind $p$ times around the $S^1$.
The resulting double dimensional reduction is just $p$-times the original one, so in a sense nothing much is changed, but maybe it is suggestive that now we are looking at the space of $C_p$-fixed points of the free loop space (for $C_p$ the cyclic group of order $p$). In E-infinity geometry this fixed-point structure on the free loop spaces makes the derived function algebras – the topological Hochschild homology of the original function algebras – be cyclotomic spectra.
Double dimensional reduction for the super-$p$-branes in $D$ dimensions which are described by the Green-Schwarz action functional corresponds to moving down and left the diagonals in the brane scan table of consistent such branes:
In particular
the superstring of type IIA string theory appears as the double dimensional reduction of the M2-brane in the KK-compactification from 11-dimensional supergravity/M-theory down to 10-dimensional type II supergravity/type II string theory.
the D4-brane appears as the double dimensional reduction of the M5-brane under this process;
the double dimensional reduction of the super 2-brane in 4d is super 1-brane in 3d (see there).
from M-branes to F-branes: superstrings, D-branes and NS5-branes
(e.g. Johnson 97, Blumenhagen 10)
Formalization of double dimensional reduction is discussed in rational homotopy theory in
and in full homotopy theory in
Exposition is in
The concept of double dimensional reduction was introduced, for the case of the reduction of the supermembrane in 11d to the Green-Schwarz superstring in 10d, in
The above “brane scan” table showing the double dimensional reduciton pattern of the super-$p$-branes given by the Green-Schwarz action functional (see there for more references on this) is taken from
Paul Townsend, D-branes from M-branes, Phys. Lett. B373 (1996) 68-75 (arXiv:hep-th/9512062)
Malcolm Perry, John Schwarz, Interacting Chiral Gauge Fields in Six Dimensions and Born-Infeld Theory, Nucl. Phys. B489 (1997) 47-64 (arXiv:hep-th/9611065)
Neil Lambert, Constantinos Papageorgakis, Maximilian Schmidt-Sommerfeld, M5-Branes, D4-Branes and Quantum 5D super-Yang-Mills, JHEP 1101:083 (2011) (arXiv:1012.2882)
Edward Witten, Fivebranes and Knots (arXiv:1101.3216)
Last revised on January 20, 2019 at 08:30:33. See the history of this page for a list of all contributions to it.