+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### String theory +-- {: .hide} [[!include string theory - contents]] =-- =-- =-- \tableofcontents ## Idea 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 brane|wrapped]] the cycle along which the KK-reduction takes place. ## Definition {#Definition} ### 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 [here](fiber+integration+in+ordinary+differential+cohomology#ViaSmoothHomotopyType)): +-- {: .num_prop} ###### Proposition 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 $$ \array{ [\Sigma, \mathbf{B}^{p+1}U(1)_{conn}] &\stackrel{\int_\Sigma}{\longrightarrow}& \mathbf{B}^{p+1-k}U(1)_{conn} \\ \downarrow^{[\Sigma,curv]} && \downarrow^{curv} \\ [\Sigma,\mathbf{\Omega}^{p+2}] &\stackrel{\int_\Sigma}{\longrightarrow}& \mathbf{\Omega}^{p+2-k} } \,, $$ where the vertical morphisms are the [[curvature]] maps and the bottom morphisms reflects ordinary [[fiber integration]] of [[differential forms]]. =-- +-- {: .num_defn #DoubleDimensionalReductionForDifferentialCocycles} ###### Definition Given a [[cocycle]] $$ \nabla \;\colon\; X \times \Sigma \longrightarrow \mathbf{B}^{p+1}U(1)_{conn} \,, $$ 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 $$ X \longrightarrow [\Sigma, X \times \Sigma] \stackrel{[\Sigma,\nabla]}{\longrightarrow} [\Sigma,\mathbf{B}^{p+1}U(1)_{conn}] \stackrel{\int_\Sigma}{\longrightarrow} \mathbf{B}^{p+1-k}U(1)_{conn} \,, $$ where the first morphism is the [[unit of an adjunction|unit]] of the ([[Cartesian product]] $\dashv$ [[internal hom]])-[[adjunction]]. =-- ### Via cyclic loop spaces {#ViaCyclicLoopSpaces} We discuss here a formalization of double dimensional reduction via cyclification adjunction ([FSS 16, section 3](#FSS16), [BMSS 18, section 2.2](#BMSS18)). For more see at _[[geometry of physics -- fundamental super p-branes]]_ the _[section on double dimensional reduction](geometry+of+physics+--+fundamental+super+p-branes#DoubleDimensionalReduction)_. +-- {: .num_prop #GeneralReduction} ###### Proposition 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 $$ \mathbf{H} \underoverset {\underset{[G,-]/G}{\longrightarrow}} {\overset{hofib}{\longleftarrow}} {\bot} \mathbf{H}_{/\mathbf{B}G} \,, $$ where * $[G,-]$ denotes the [[internal hom]] in $\mathbf{H}$, * $[G,-]/G$ 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). Hence for * $\hat X \to X$ a $G$ [[principal ∞-bundle]] * $A$ a [[coefficient]] object, such as for some [[differential cohomology|differential]] [[generalized cohomology theory]] then there is a [[natural equivalence]] $$ \underset{ \text{original} \atop \text{fluxes} }{ \underbrace{ \mathbf{H}(\hat X\;,\; A) } } \;\; \underoverset {\underset{oxidation}{\longleftarrow}} {\overset{reduction}{\longrightarrow}} {\simeq} \;\; \underset{ \text{doubly} \atop { \text{dimensionally reduced} \atop \text{fluxes} } }{ \underbrace{ \mathbf{H}_{/\mathbf{B}G}(X \;,\; [G,A]/G) } } $$ given by $$ \left( \hat X \longrightarrow A \right) \;\;\; \leftrightarrow \;\;\; \left( \array{ X && \longrightarrow && [G,A]/G \\ & \searrow && \swarrow \\ && \mathbf{B}G } \right) $$ =-- +-- {: .proof #DimensionalReductionAbstractly} ###### 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]] $$ Act_G(\mathbf{H}) \simeq \mathbf{H}_{/\mathbf{B}G} $$ (from [NSS 12](geometry+of+physics+-+fundamental+super+p-branes#NSS12)) 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 $$ [G,X]/G \simeq [\ast, X \times \mathbf{B}G]_{\mathbf{B}G} \;\;\;\;\; \in \mathbf{H}_{/\mathbf{B}G} \,. $$ 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 $$ E \times_{\mathbf{B}G}(-) \;\;\; \dashv \;\;\; [E,-]_{\mathbf{B}G} $$ it is immediate that there is the following sequence of [[natural equivalences]] $$ \begin{aligned} \mathbf{H}_{/\mathbf{B}G}(Y, [G,X]/G) & \simeq \mathbf{H}_{/\mathbf{B}G}(Y, [\ast, X \times \mathbf{B}G]_{\mathbf{B}G}) \\ & \simeq \mathbf{H}_{/\mathbf{B}G}( Y \times_{\mathbf{B}G} \ast, \underset{p^\ast X}{\underbrace{X \times \mathbf{B}G }} ) \\ & \simeq \mathbf{H}( \underset{hofib(Y)}{\underbrace{p_!(Y \times_{\mathbf{B}G} \ast)}}, X ) \\ & \simeq \mathbf{H}(hofib(Y),X) \end{aligned} $$ Here $p \colon \mathbf{B}G \to \ast$ denotes the terminal morphism and $p_! \dashv p^\ast$ denotes the [[base change]] along it. =-- ## Examples ### Reduction on the circle {#ReductionOnTheCircle} +-- {: .num_example #ReductionOnTheCircle} ###### Example 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]]. =-- +-- {: .num_remark #Cyclotomic} ###### Remark 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]]. =-- ### For general super $p$-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 * 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 * [[duality between M-theory and type IIA string theory]] [[!include F-branes -- table]] ## Related concepts * [[cyclic loop space]], [[cyclic loop stack]] * [[caloron correspondence]] ## References {#References} ### General Formalization of double dimensional reduction via [[cyclic loop space|cyclification]] of [[classifying spaces]]/characteristic [[L-infinity algebras|$L_\infty$-algebras]]: in [[rational homotopy theory]]: * {#FSS16} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], Section 3 of _[[schreiber:T-Duality from super Lie n-algebra cocycles for super p-branes]]_, ATMP Volume 22 (2018) Number 5 ([arXiv:1611.06536](https://arxiv.org/abs/1611.06536)) and in full [[homotopy theory]]: * {#BMSS18} [[Vincent Braunack-Mayer]], [[Hisham Sati]], [[Urs Schreiber]], Section 2.2 of: _[[schreiber:Gauge enhancement of Super M-Branes|Gauge enhancement of Super M-Branes via rational parameterized stable homotopy theory]]_, Communications in Mathematical Physics 371: 197 (2019) ([arXiv:1806.01115](https://arxiv.org/abs/1806.01115), [doi:10.1007/s00220-019-03441-4](https://doi.org/10.1007/s00220-019-03441-4)) * [[Hisham Sati]], [[Urs Schreiber]]: *[[schreiber:Cyclification of Orbifolds]]*, Comm. Math. Phys. **405** 67 (2024) [[doi:10.1007/s00220-023-04929-w](https://doi.org/10.1007/s00220-023-04929-w), [arXiv:2212.13836](https://arxiv.org/abs/2212.13836), [[schreiber:cyclic loop spaces 2022|talk]]] Exposition is in * {#Schreiber16} [[Urs Schreiber]], [Section 4](https://ncatlab.org/schreiber/show/Super+Lie+n-algebra+of+Super+p-branes#DoubleDimensionalReduction) of _[[schreiber:Super Lie n-algebra of Super p-branes]]_, talks at [Fields, Strings, and Geometry Seminar](http://www.surrey.ac.uk/maths/news/seminars/fsg/), Surrey, 5th-9th Dec. 2016 and further discussion in * {#Alfonsi19} [[Luigi Alfonsi]], _Global Double Field Theory is Higher Kaluza-Klein Theory_, Fortsch. d. Phys. **68** 3-4 (2020) [[arXiv:1912.07089](https://arxiv.org/abs/1912.07089), [doi:10.1002/prop.202000010](https://doi.org/10.1002/prop.202000010)] * [[Luigi Alfonsi]]: *The puzzle of global Double Field Theory: open problems and the case for a Higher Kaluza-Klein perspective*, Fortschr. Phys. **69** 7 (2021) [[arXiv:2007.04969](https://arxiv.org/abs/2007.04969), [doi:10.1002/prop.202000102](https://doi.org/10.1002/prop.202000102)] The case of [[duality between 11D and 10D IIA supergravity]]: * [[Grigorios Giotopoulos]], [[Hisham Sati]]: *Flux Quantization on 10D Type IIA Superspace via Cyclification from 11D* \[<a href="https://arxiv.org/abs/2606.16857">arXiv:2606.16857</a>\] * [[Grigorios Giotopoulos]], [[Hisham Sati]], [[Urs Schreiber]]; §3.2 in: *[[schreiber:Nesselwang 2026|Higher Superspace Supergravity & its IR Completions]]*, lecture notes [[arXiv:2607.20312](https://arxiv.org/abs/2607.20312)] ### 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 * {#DuffHoweInamiStelle87} [[Michael Duff]], [[Paul Howe]], T. Inami, [[Kellogg Stelle]], _Superstrings in $D=10$ from Supermembranes in $D=11$_, Phys. Lett. B191 (1987) 70 and in [[Michael Duff]] (ed.) _[[The World in Eleven Dimensions]]_ 205-206 (1987) [[spire:245249](http://inspirehep.net/record/245249), <a href="https://doi.org/10.1016/0370-2693(87)91323-2">doi:10.1016/0370-2693(87)91323-2</a>] The above "brane scan" table showing the double dimensional reduction pattern of the super-$p$-branes given by the [[Green-Schwarz action functional]] (see there for more references on this) is taken from * {#Duff} [[Michael Duff]]: *Supermembranes: the first fifteen weeks*, Class. Quantum Grav. **5** 189 (1988) [[doi:10.1088/0264-9381/5/1/023](https://iopscience.iop.org/article/10.1088/0264-9381/5/1/023)] ### Reduction of M5-brane to D4-brane The [[double dimensional reduction]] of the [[M5-brane]] to the [[D4-brane]]: * [[Paul Townsend]], _D-branes from M-branes_, Phys. Lett. B **373** (1996) 68--75 [<a href="https://doi.org/10.1016/0370-2693(96)00104-9">doi:10.1016/0370-2693(96)00104-9</a>, [arXiv:hep-th/9512062](https://arxiv.org/abs/hep-th/9512062)] * {#APPS97a} [[Mina Aganagic]], [[Jaemo Park]], Costin Popescu, [[John Schwarz]], Section 6 of: *World-Volume Action of the M Theory Five-Brane*, Nucl. Phys. B **496** (1997) 191-214 [[arXiv:hep-th/9701166](http://arxiv.org/abs/hep-th/9701166), <a href="https://doi.org/10.1016/S0550-3213(97)00227-7">doi:10.1016/S0550-3213(97)00227-7</a>] * {#APPS97b} [[Mina Aganagic]], [[Jaemo Park]], Costin Popescu, [[John Schwarz]], Section 6 of: _Dual D-Brane Actions_, Nucl. Phys. B **496** (1997) 215-230 [[arXiv:hep-th/9702133](https://arxiv.org/abs/hep-th/9702133), <a href="https://doi.org/10.1016/S0550-3213(97)00257-5">doi:10.1016/S0550-3213(97)00257-5</a>] * [[Neil Lambert]], Constantinos Papageorgakis, Maximilian Schmidt-Sommerfeld, _M5-Branes, D4-Branes and Quantum 5D super-Yang-Mills_, JHEP 1101:083 (2011) ([arXiv:1012.2882](http://arxiv.org/abs/1012.2882)) * {#Witten11} [[Edward Witten]], _Fivebranes and Knots_ ([arXiv:1101.3216](http://arxiv.org/abs/1101.3216)) * Anajak Phonchantuek, [[Pichet Vanichchapongjaroen]]: *Double dimensional reduction of M5-brane action in Sen formalism*, Eur. Phys. J. C **83** (2023) 721 [[arXiv:2305.04861](https://arxiv.org/abs/2305.04861), [doi:10.1140/epjc/s10052-023-11892-2](https://doi.org/10.1140/epjc/s10052-023-11892-2)] ### Reduction of black M2s and black M5s The [[differential geometry]] of the double dimensional reduction of the [[M2-brane]]- and [[M5-brane]]-charges was maybe first clearly written out in: * [[Varghese Mathai]], [[Hisham Sati]], §4 in: *Some Relations between Twisted K-theory and $E_8$ Gauge Theory*, J. High Energ. Phys. **2004** 03 (2004) 016 [[arXiv:hep-th/0312033](https://arxiv.org/abs/hep-th/0312033), [doi:10.1088/1126-6708/2004/03/016](https://doi.org/10.1088/1126-6708/2004/03/016)] For their [[black brane]]-reductions see: * [[José Figueroa-O'Farrill]], [[Joan Simón]], _Supersymmetric Kaluza-Klein reductions of M2 and M5-branes_, Adv. Theor. Math. Phys. 6:703-793, 2003 ([arXiv:hep-th/0208107](https://arxiv.org/abs/hep-th/0208107)) ### Reduction of M-Waves and MK6s * [[José Figueroa-O'Farrill]], [[Joan Simón]], _Supersymmetric Kaluza-Klein reductions of M-waves and MKK-monopoles_, Class. Quant. Grav.19:6147-6174, 2002 ([arXiv:hep-th/0208108](https://arxiv.org/abs/hep-th/0208108)) [[!redirects double dimensional reductions]] [[!redirects double-dimensional reduction]] [[!redirects double-dimensional reductions]]