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\section*{double dimensional reduction}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{string_theory}{}\paragraph*{{String theory}}\label{string_theory}

[[!include string theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{Definition}{Definition}\dotfill \pageref*{Definition} \linebreak
\noindent\hyperlink{via_fiber_integration_in_ordinary_differential_cohomology}{Via fiber integration in ordinary differential cohomology}\dotfill \pageref*{via_fiber_integration_in_ordinary_differential_cohomology} \linebreak
\noindent\hyperlink{ViaCyclicLoopSpaces}{Via cyclic loop spaces}\dotfill \pageref*{ViaCyclicLoopSpaces} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{ReductionOnTheCircle}{Reduction on the circle}\dotfill \pageref*{ReductionOnTheCircle} \linebreak
\noindent\hyperlink{for_general_super_branes}{For general super $p$-branes}\dotfill \pageref*{for_general_super_branes} \linebreak
\noindent\hyperlink{from_mbranes_to_fbranes}{From M-branes to F-branes}\dotfill \pageref*{from_mbranes_to_fbranes} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak
\noindent\hyperlink{reduction_of_membrane_to_string}{Reduction of membrane to string}\dotfill \pageref*{reduction_of_membrane_to_string} \linebreak
\noindent\hyperlink{reduction_of_m5brane_to_d4brane}{Reduction of M5-brane to D4-brane}\dotfill \pageref*{reduction_of_m5brane_to_d4brane} \linebreak
\noindent\hyperlink{reduction_of_mwaves_and_mk6s}{Reduction of M-Waves and MK6s}\dotfill \pageref*{reduction_of_mwaves_and_mk6s} \linebreak
\noindent\hyperlink{reduction_of_black_m2s_and_black_m5s}{Reduction of black M2s and black M5s}\dotfill \pageref*{reduction_of_black_m2s_and_black_m5s} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

What is called \emph{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 brane|wrapped]] the cycle along which the KK-reduction takes place.

\hypertarget{Definition}{}\subsection*{{Definition}}\label{Definition}

\hypertarget{via_fiber_integration_in_ordinary_differential_cohomology}{}\subsubsection*{{Via fiber integration in ordinary differential cohomology}}\label{via_fiber_integration_in_ordinary_differential_cohomology}

Let $\mathbf{H}$ be the [[smooth infinity-groupoid|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 \href{fiber+integration+in+ordinary+differential+cohomology#ViaSmoothHomotopyType}{here}):

\begin{prop}
\label{}\hypertarget{}{}
For $\Sigma$ an [[orientation|oriented]] [[closed manifold]] of [[dimension]] $k \leq p+1$, then [[fiber integration]] in [[ordinary differential cohomology]] is reflected by a morphism of the form

\begin{displaymath}
\itexarray{
    [\Sigma, \mathbf{B}^{p+1}U(1)_{conn}]
    &\stackrel{\int_\Sigma}{\longrightarrow}&
    \mathbf{B}^{p+1-k}U(1)_{conn}
    \\
    \downarrow^{[\Sigma,curv]} && \downarrow^{curv}
    \\
    [\Sigma,\mathbf{\Omega}^{p+2}]
    &\stackrel{\int_\Sigma}{\longrightarrow}&
    \mathbf{\Omega}^{p+2-k}
  }
  \,,
\end{displaymath}
where the vertical morphisms are the [[curvature]] maps and the bottom morphims reflects ordinary [[fiber integration]] of [[differential forms]].

\end{prop}
\begin{defn}
\label{DoubleDimensionalReductionForDifferentialCocycles}\hypertarget{DoubleDimensionalReductionForDifferentialCocycles}{}
Given a [[cocycle]]

\begin{displaymath}
\nabla \;\colon\; X \times \Sigma \longrightarrow \mathbf{B}^{p+1}U(1)_{conn}
  \,,
\end{displaymath}
on the [[Cartesian product]] of some [[smooth space]] $X$ with $\Sigma$, then its \emph{double dimensional reduction} is the cocycle on $X$ which is given by the composite

\begin{displaymath}
X \longrightarrow [\Sigma, X \times \Sigma]
  \stackrel{[\Sigma,\nabla]}{\longrightarrow}
  [\Sigma,\mathbf{B}^{p+1}U(1)_{conn}]
  \stackrel{\int_\Sigma}{\longrightarrow}
  \mathbf{B}^{p+1-k}U(1)_{conn}
  \,,
\end{displaymath}
where the first morphism is the [[unit of an adjunction|unit]] of the ([[Cartesian product]] $\dashv$ [[internal hom]])-[[adjunction]].

\end{defn}
\hypertarget{ViaCyclicLoopSpaces}{}\subsubsection*{{Via cyclic loop spaces}}\label{ViaCyclicLoopSpaces}

We discuss here a formalization of double dimensional reduction via cyclification adjunction (\hyperlink{FSS16}{FSS 16, section 3}, \hyperlink{BMSS18}{BMSS 18, section 2.2}). For more see at \emph{[[geometry of physics -- fundamental super p-branes]]} the \emph{\href{geometry+of+physics+--+fundamental+super+p-branes#DoubleDimensionalReduction}{section on double dimensional reduction}}.

\begin{prop}
\label{GeneralReduction}\hypertarget{GeneralReduction}{}
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

\begin{displaymath}
\mathbf{H}
    \underoverset
      {\underset{[G,-]/G}{\longrightarrow}}
      {\overset{hofib}{\longleftarrow}}
      {\bot}
  \mathbf{H}_{/\mathbf{B}G}
  \,,
\end{displaymath}
where

\begin{itemize}%
\item $[G,-]$ denotes the [[internal hom]] in $\mathbf{H}$,


\item $[G,-]/G$ denotes the [[homotopy quotient]] by the [[conjugation action|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).



\end{itemize}
Hence for

\begin{itemize}%
\item $\hat X \to X$ a $G$ [[principal ∞-bundle]]


\item $A$ a [[coefficient]] object, such as for some [[differential cohomology|differential]] [[generalized cohomology theory]]



\end{itemize}
then there is a [[natural equivalence]]

\begin{displaymath}
\underset{
    \text{original} \atop \text{fluxes}
  }{
  \underbrace{
    \mathbf{H}(\hat X\;,\; A)
  }
  }
    \;\;
    \underoverset
      {\underset{oxidation}{\longleftarrow}}
      {\overset{reduction}{\longrightarrow}}
      {\simeq}
    \;\;
  \underset{
     \text{doubly} \atop { \text{dimensionally reduced} \atop \text{fluxes} }
  }{
  \underbrace{
    \mathbf{H}_{/\mathbf{B}G}(X \;,\; [G,A]/G)
  }
  }
\end{displaymath}
given by

\begin{displaymath}
\left(
     \hat X \longrightarrow A
  \right)
  \;\;\;
    \leftrightarrow
  \;\;\;
  \left(
    \itexarray{
       X && \longrightarrow && [G,A]/G
       \\
       & \searrow && \swarrow
       \\
       && \mathbf{B}G
    }
  \right)
\end{displaymath}
\end{prop}
\begin{proof}
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]]

\begin{displaymath}
Act_G(\mathbf{H}) \simeq \mathbf{H}_{/\mathbf{B}G}
\end{displaymath}
(from \href{geometry+of+physics+-+fundamental+super+p-branes#NSS12}{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

\begin{displaymath}
[G,X]/G
    \simeq
  [\ast, X \times \mathbf{B}G]_{\mathbf{B}G}
  \;\;\;\;\;
  \in
   \mathbf{H}_{/\mathbf{B}G}
  \,.
\end{displaymath}
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 (infinity,1)-category|cartesian closed]], via

\begin{displaymath}
E \times_{\mathbf{B}G}(-)
  \;\;\;
   \dashv
  \;\;\;
  [E,-]_{\mathbf{B}G}
\end{displaymath}
it is immediate that there is the following sequence of [[natural equivalences]]

\begin{displaymath}
\begin{aligned}
   \mathbf{H}_{/\mathbf{B}G}(Y, [G,X]/G)
   & \simeq
   \mathbf{H}_{/\mathbf{B}G}(Y, [\ast, X \times \mathbf{B}G]_{\mathbf{B}G})
   \\
   & \simeq
   \mathbf{H}_{/\mathbf{B}G}(
     Y \times_{\mathbf{B}G} \ast,
     \underset{p^\ast X}{\underbrace{X \times \mathbf{B}G }}
  )
  \\
  & \simeq
  \mathbf{H}(
     \underset{hofib(Y)}{\underbrace{p_!(Y \times_{\mathbf{B}G} \ast)}},
     X
  )
  \\
  & \simeq
  \mathbf{H}(hofib(Y),X)
  \end{aligned}
\end{displaymath}
Here $p \colon \mathbf{B}G \to \ast$ denotes the terminal morphism and $p_! \dashv p^\ast$ denotes the [[base change]] along it.

\end{proof}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{ReductionOnTheCircle}{}\subsubsection*{{Reduction on the circle}}\label{ReductionOnTheCircle}

\begin{example}
\label{ReductionOnTheCircle}\hypertarget{ReductionOnTheCircle}{}
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. \ref{DoubleDimensionalReductionForDifferentialCocycles} is the differential cocycle representing the [[B-field]] on $X$, in the sense of [[string theory]].

\end{example}
\begin{remark}
\label{Cyclotomic}\hypertarget{Cyclotomic}{}
For $\Sigma = S^1$ a circle as in example \ref{ReductionOnTheCircle}, then the morphism $X \longrightarrow [\Sigma, X \times \Sigma]$ in def. \ref{DoubleDimensionalReductionForDifferentialCocycles} 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]].

\end{remark}
\hypertarget{for_general_super_branes}{}\subsubsection*{{For general super $p$-branes}}\label{for_general_super_branes}

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:

[[branescan.gif:pic]]

In particular

\begin{itemize}%
\item 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]].


\item the [[D4-brane]] appears as the double dimensional reduction of the [[M5-brane]] under this process;


\item the double dimensional reduction of the [[super 2-brane in 4d]] is [[super 1-brane in 3d]] (see there).



\end{itemize}
\hypertarget{from_mbranes_to_fbranes}{}\subsubsection*{{From M-branes to F-branes}}\label{from_mbranes_to_fbranes}

\begin{itemize}%
\item [[duality between M-theory and type IIA string theory]]

\end{itemize}
[[!include F-branes -- table]]

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[caloron correspondence]]

\end{itemize}
\hypertarget{References}{}\subsection*{{References}}\label{References}

Formalization of double dimensional reduction is discussed in [[rational homotopy theory]] in

\begin{itemize}%
\item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], Section 3 of \emph{[[schreiber:T-Duality from super Lie n-algebra cocycles for super p-branes]]}, ATMP Volume 22 (2018) Number 5 (\href{https://arxiv.org/abs/1611.06536}{arXiv:1611.06536})

\end{itemize}
and in full [[homotopy theory]] in

\begin{itemize}%
\item [[Vincent Braunack-Mayer]], [[Hisham Sati]], [[Urs Schreiber]], Section 2.2 of \emph{[[schreiber:Gauge enhancement of Super M-Branes|Gauge enhancement of Super M-Branes via rational parameterized stable homotopy theory]]} (\href{https://arxiv.org/abs/1806.01115}{arXiv:1806.01115})

\end{itemize}
Exposition is in

\begin{itemize}%
\item [[Urs Schreiber]], \href{https://ncatlab.org/schreiber/show/Super+Lie+n-algebra+of+Super+p-branes#DoubleDimensionalReduction}{Section 4} of \emph{[[schreiber:Super Lie n-algebra of Super p-branes]]}, talks at \href{http://www.surrey.ac.uk/maths/news/seminars/fsg/}{Fields, Strings, and Geometry Seminar}, Surrey, 5th-9th Dec. 2016

\end{itemize}
\hypertarget{reduction_of_membrane_to_string}{}\subsubsection*{{Reduction of membrane to string}}\label{reduction_of_membrane_to_string}

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

\begin{itemize}%
\item [[Michael Duff]], [[Paul Howe]], T. Inami, [[Kellogg Stelle]], \emph{Superstrings in $D=10$ from Supermembranes in $D=11$}, Phys. Lett. B191 (1987) 70 and in [[Michael Duff]] (ed.) \emph{[[The World in Eleven Dimensions]]} 205-206 (1987) (\href{http://inspirehep.net/record/245249}{spire:245249})

\end{itemize}
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

\begin{itemize}%
\item [[Michael Duff]], \emph{Supermembranes: the first fifteen weeks} CERN-TH.4797/87 (1987) (\href{http://ccdb4fs.kek.jp/cgi-bin/img/allpdf?198708425}{scan})

\end{itemize}
\hypertarget{reduction_of_m5brane_to_d4brane}{}\subsubsection*{{Reduction of M5-brane to D4-brane}}\label{reduction_of_m5brane_to_d4brane}

The [[double dimensional reduction]] of the [[M5-brane]] to the [[D4-brane]]:

\begin{itemize}%
\item [[Paul Townsend]], \emph{D-branes from M-branes}, Phys. Lett. B373 (1996) 68-75 (\href{https://arxiv.org/abs/hep-th/9512062}{arXiv:hep-th/9512062})


\item [[Mina Aganagic]], Jaemo Park, Costin Popescu, [[John Schwarz]], Section 6 of \emph{World-Volume Action of the M Theory Five-Brane}, Nucl.Phys. B496 (1997) 191-214 (\href{http://arxiv.org/abs/hep-th/9701166}{arXiv:hep-th/9701166})


\item [[Mina Aganagic]], Jaemo Park, Costin Popescu, [[John Schwarz]], Section 6 of \emph{Dual D-Brane Actions}, Nucl. Phys. B496 (1997) 215-230 (\href{https://arxiv.org/abs/hep-th/9702133}{arXiv:hep-th/9702133})


\item [[Neil Lambert]], Constantinos Papageorgakis, Maximilian Schmidt-Sommerfeld, \emph{M5-Branes, D4-Branes and Quantum 5D super-Yang-Mills}, JHEP 1101:083 (2011) (\href{http://arxiv.org/abs/1012.2882}{arXiv:1012.2882})


\item [[Edward Witten]], \emph{Fivebranes and Knots} (\href{http://arxiv.org/abs/1101.3216}{arXiv:1101.3216})



\end{itemize}
\hypertarget{reduction_of_mwaves_and_mk6s}{}\subsubsection*{{Reduction of M-Waves and MK6s}}\label{reduction_of_mwaves_and_mk6s}

\begin{itemize}%
\item [[José Figueroa-O'Farrill]], [[Joan Simón]], \emph{Supersymmetric Kaluza-Klein reductions of M-waves and MKK-monopoles}, Class. Quant. Grav.19:6147-6174, 2002 (\href{https://arxiv.org/abs/hep-th/0208108}{arXiv:hep-th/0208108})

\end{itemize}
\hypertarget{reduction_of_black_m2s_and_black_m5s}{}\subsubsection*{{Reduction of black M2s and black M5s}}\label{reduction_of_black_m2s_and_black_m5s}

\begin{itemize}%
\item [[José Figueroa-O'Farrill]], [[Joan Simón]], \emph{Supersymmetric Kaluza-Klein reductions of M2 and M5-branes}, Adv. Theor. Math. Phys. 6:703-793, 2003 (\href{https://arxiv.org/abs/hep-th/0208107}{arXiv:hep-th/0208107})

\end{itemize}
[[!redirects double dimensional reductions]]

[[!redirects double dimensional reduction]]



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