Contents

# Contents

## 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 the cycle along which the KK-reduction takes place.

## Definition

### Via fiber integration in ordinary differential cohomology

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):

###### Proposition

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

$\array{ [\Sigma, \mathbf{B}^{p+1}U(1)_{conn}] &\stackrel{\int_\Sigma}{\longrightarrow}& \mathbf{B}^{p+1-k}U(1)_{conn} \\ \downarrow^{[\Sigma,curv]} && \downarrow^{curv} \\ [\Sigma,\mathbf{\Omega}^{p+2}] &\stackrel{\int_\Sigma}{\longrightarrow}& \mathbf{\Omega}^{p+2-k} } \,,$

where the vertical morphisms are the curvature maps and the bottom morphims reflects ordinary fiber integration of differential forms.

###### 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 the (Cartesian product $\dashv$ internal hom)-adjunction.

### Via cyclic loop spaces

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.

###### 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 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

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

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) 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, 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

###### 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. is the differential cocycle representing the B-field on $X$, in the sense of string theory.

###### Remark

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.

### 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:

In particular

### From M-branes to F-branes

from M-branes to F-branes: superstrings, D-branes and NS5-branes

M-theory on $S^1_A \times S^1_B$-elliptic fibrationKK-compactification on $S^1_A$type IIA string theoryT-dual KK-compactification on $S^1_B$type IIB string theorygeometrize the axio-dilatonF-theory on elliptically fibered-K3 fibrationduality between F-theory and heterotic string theoryheterotic string theory on elliptic fibration
M2-brane wrapping $S_A^1$double dimensional reduction $\mapsto$type IIA superstring$\mapsto$type IIB superstring$\mapsto$$\mapsto$heterotic superstring
M2-brane wrapping $S_B^1$$\mapsto$D2-brane$\mapsto$D1-brane$\mapsto$
M2-brane wrapping $p$ times around $S_A^1$ and $q$ times around $S_B^1$$\mapsto$$p$ strings and $q$ D2-branes$\mapsto$(p,q)-string$\mapsto$
M5-brane wrapping $S_A^1$double dimensional reduction $\mapsto$D4-brane$\mapsto$D5-brane$\mapsto$
M5-brane wrapping $S_B^1$$\mapsto$NS5-brane$\mapsto$NS5-brane$\mapsto$$\mapsto$NS5-brane
M5-brane wrapping $p$ times around $S_A^1$ and $q$ times around $S_B^1$$\mapsto$$p$ D4-brane and $q$ NS5-branes$\mapsto$(p,q)5-brane$\mapsto$
M5-brane wrapping $S_A^1 \times S_B^1$$\mapsto$$\mapsto$D3-brane$\mapsto$
KK-monopole/A-type ADE singularity (degeneration locus of $S^1_A$-circle fibration, Sen limit of $S^1_A \times S^1_B$ elliptic fibration)$\mapsto$D6-brane$\mapsto$D7-branes$\mapsto$A-type nodal curve cycle degeneration locus of elliptic fibration (Sen 97, section 2)SU-gauge enhancement
KK-monopole orientifold/D-type ADE singularity$\mapsto$D6-brane with O6-planes$\mapsto$D7-branes with O7-planes$\mapsto$D-type nodal curve cycle degeneration locus of elliptic fibration (Sen 97, section 3)SO-gauge enhancement
exceptional ADE-singularity$\mapsto$$\mapsto$$\mapsto$exceptional ADE-singularity of elliptic fibration$\mapsto$E6-, E7-, E8-gauge enhancement

(e.g. Johnson 97, Blumenhagen 10)

## References

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

and in full homotopy theory in

Exposition is in

### 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

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

• Michael Duff, Supermembranes: the first fifteen weeks CERN-TH.4797/87 (1987) (scan)

### Reduction of M5-brane to D4-brane

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

### Reduction of black M2s and black M5s

Last revised on July 23, 2019 at 05:01:01. See the history of this page for a list of all contributions to it.