double dimensional reduction




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+1d+1 in which a p+1p+1-brane propagates, its KK-reduction results in a dd-dimensional effective spacetime containing a p+1p+1-brane together with a “doubly reduced” pp-brane, which is the reduction of those original (p+1)(p+1)-brane configurations that wrapped the cycle along which the KK-reduction takes place.


Via fiber integration in ordinary differential cohomology

Let H\mathbf{H} be the smooth topos. For p+1p+1 \in \mathbb{N} write B p+1U(1) connH\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 Σ\Sigma an oriented closed manifold of dimension kp+1k \leq p+1, then fiber integration in ordinary differential cohomology is reflected by a morphism of the form

[Σ,B p+1U(1) conn] Σ B p+1kU(1) conn [Σ,curv] curv [Σ,Ω p+2] Σ Ω p+2k, \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.


Given a cocycle

:X×ΣB p+1U(1) conn, \nabla \;\colon\; X \times \Sigma \longrightarrow \mathbf{B}^{p+1}U(1)_{conn} \,,

on the Cartesian product of some smooth space XX with Σ\Sigma, then its double dimensional reduction is the cocycle on XX which is given by the composite

X[Σ,X×Σ][Σ,][Σ,B p+1U(1) conn] ΣB p+1kU(1) conn, 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.


Let H\mathbf{H} be any (∞,1)-topos and let GG be an ∞-group in H\mathbf{H}. Then the right base change/dependent product along the canonical point inclusion *BG\ast \to \mathbf{B}G into the delooping of GG takes the following form: There is a pair of adjoint ∞-functors of the form

H[G,]/GhofibH /BG, \mathbf{H} \underoverset {\underset{[G,-]/G}{\longrightarrow}} {\overset{hofib}{\longleftarrow}} {\bot} \mathbf{H}_{/\mathbf{B}G} \,,


Hence for

then there is a natural equivalence

H(X^,A)originalfluxesoxidationreductionH /BG(X,[G,A]/G)doublydimensionally reducedfluxes \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

(X^A)(X [G,A]/G BG) \left( \hat X \longrightarrow A \right) \;\;\; \leftrightarrow \;\;\; \left( \array{ X && \longrightarrow && [G,A]/G \\ & \searrow && \swarrow \\ && \mathbf{B}G } \right)

First observe that the conjugation action on [G,X][G,X] is the internal hom in the (∞,1)-category of GG-∞-actions Act G(H)Act_G(\mathbf{H}). Under the equivalence of (∞,1)-categories

Act G(H)H /BG Act_G(\mathbf{H}) \simeq \mathbf{H}_{/\mathbf{B}G}

(from NSS 12) then GG with its canonical ∞-action is (*BG)(\ast \to \mathbf{B}G) and XX with the trivial action is (X×BGBG)(X \times \mathbf{B}G \to \mathbf{B}G).


[G,X]/G[*,X×BG] BGH /BG. [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]/GH /BG[G,X]/G \in \mathbf{H}_{/\mathbf{B}G} is to mean in the first place, abstractly.

But now since the slice (∞,1)-topos H /BG\mathbf{H}_{/\mathbf{B}G} is itself cartesian closed, via

E× BG()[E,] BG E \times_{\mathbf{B}G}(-) \;\;\; \dashv \;\;\; [E,-]_{\mathbf{B}G}

it is immediate that there is the following sequence of natural equivalences

H /BG(Y,[G,X]/G) H /BG(Y,[*,X×BG] BG) H /BG(Y× BG*,X×BGp *X) H(p !(Y× BG*)hofib(Y),X) H(hofib(Y),X) \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:BG*p \colon \mathbf{B}G \to \ast denotes the terminal morphism and p !p *p_! \dashv p^\ast denotes the base change along it.


Reduction on the circle


When Σ=S 1\Sigma = S^1 is the circle, and we think of X×S 1X \times S^1 as a spacetime of 11-dimensional supergravity, then :X×S 1B 3U(1) conn\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 XX, in the sense of string theory.


For Σ=S 1\Sigma = S^1 a circle as in example , then the morphism X[Σ,X×Σ]X \longrightarrow [\Sigma, X \times \Sigma] in def. sends each point of XX to the loop in X×S 1X\times S^1 that winds identically around the copy of S 1S^1 at that point. Hence in this case it would make sense to consider, more generally, for each pp \in \mathbb{Z} the “order pp” double dimensional reduction, given by the operation where one instead considers the map that lets the loop wind pp times around the S 1S^1.

The resulting double dimensional reduction is just pp-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 pC_p-fixed points of the free loop space (for C pC_p the cyclic group of order pp). 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 pp-branes

Double dimensional reduction for the super-pp-branes in DD 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:

The brane scan

In particular

From M-branes to F-branes

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

M-theory on S A 1×S B 1S^1_A \times S^1_B-elliptic fibrationKK-compactification on S A 1S^1_Atype IIA string theoryT-dual KK-compactification on S B 1S^1_Btype 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 1S_A^1double dimensional reduction \mapstotype IIA superstring\mapstotype IIB superstring\mapsto\mapstoheterotic superstring
M2-brane wrapping S B 1S_B^1\mapstoD2-brane\mapstoD1-brane\mapsto
M2-brane wrapping pp times around S A 1S_A^1 and qq times around S B 1S_B^1\mapstopp strings and qq D2-branes\mapsto(p,q)-string\mapsto
M5-brane wrapping S A 1S_A^1double dimensional reduction \mapstoD4-brane\mapstoD5-brane\mapsto
M5-brane wrapping S B 1S_B^1\mapstoNS5-brane\mapstoNS5-brane\mapsto\mapstoNS5-brane
M5-brane wrapping pp times around S A 1S_A^1 and qq times around S B 1S_B^1\mapstopp D4-brane and qq NS5-branes\mapsto(p,q)5-brane\mapsto
M5-brane wrapping S A 1×S B 1S_A^1 \times S_B^1\mapsto\mapstoD3-brane\mapsto
KK-monopole/A-type ADE singularity (degeneration locus of S A 1S^1_A-circle fibration, Sen limit of S A 1×S B 1S^1_A \times S^1_B elliptic fibration)\mapstoD6-brane\mapstoD7-branes\mapstoA-type nodal curve cycle degeneration locus of elliptic fibration ADE 2Cycle (Sen 97, section 2)SU-gauge enhancement
KK-monopole orientifold/D-type ADE singularity\mapstoD6-brane with O6-planes\mapstoD7-branes with O7-planes\mapstoD-type nodal curve cycle degeneration locus of elliptic fibration ADE 2Cycle (Sen 97, section 3)SO-gauge enhancement
exceptional ADE-singularity\mapsto\mapsto\mapstoexceptional ADE-singularity of elliptic fibration\mapstoE6-, E7-, E8-gauge enhancement

(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

and further discussion 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-pp-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 M-Waves and MK6s

Reduction of black M2s and black M5s

Last revised on July 10, 2020 at 04:52:32. See the history of this page for a list of all contributions to it.