nLab Hanany-Witten effect




In string theory, what has come to be known as the Hanany-Witten-effect (or -process, -tansition, or -move) is an expected phenomenon exhibited by branes in type II string theory which is roughly described as follows:

Moving an NS5-brane across a D ( 2 p + σ ) D (2p+\sigma) -brane creates a D ( 2 p 2 + σ ) D (2p-2 + \sigma) -brane stretching between the two”.

(Here σ=0\sigma = 0 in type IIA string theory and σ=1\sigma=1 in type IIB string theory.)

Something like this was informally argued in Hanany & Witten (1997). Later authors have tried to offer explanation and illustration of what is meant to be happening here, as follows:

Nakatsu et al (1998):

Marolf (2001):

Eto et al (2005):

Pelc (2008):

Gang et al (2011):

Marolf (2001) suggested that the Hanany-Witten effect may be understood entirely as a consequence of the famous (pregeometric) Bianchi identity for the RR-field-fluxes F 82pσF_{8-2p-\sigma} sourced by the D-branes, which is twisted by the B-field-flux H 3H_3 sourced by NS5-branes:

(1)dF 82pσ=H 3F 8(2p+2)σ \mathrm{d} F_{8-2p-\sigma} \;=\; H_3 \wedge F_{8-(2p+2)-\sigma}

and offered an argument in prose [Marolf (2001) pp. 3].

Below we spell out this approach in more detail.


D6 stretching between D8 and NS5

(cf. D6-D8 brane bound state and Bergshoeff, Lozano & Ortin (1998))

We explain the “Hanany-Witten effect” in detail for the case of “flat” NS5D6D8-configurations, as a consequence of the pregeometric Bianchi identities (1) satisfied by the flux densities sourced by these branes.

In this case we are dealing with

  • Minkowski spacetimes\; 1,p\mathbb{R}^{1,p} which we may identify with their canonical coordinate charts (x 0,x)(x 0,x 1,,x p)(x^0, \vec x) \,\coloneqq\, (x^0, x^1, \cdots, x^p).

  • A closed differential 3-form

    H 3kvol S 3Ω dR 3(S 3) cl(pr S 3) *Ω dR 3( 1,5× +×S 3) clΩ dR 3( 1,9 1,5) cl H_3 \,\coloneqq\, k vol_{S^3} \,\in\, \Omega^3_{dR}\big( S^3 \big)_{cl} \xrightarrow{(pr_{S^3})^\ast} \Omega^3_{dR} \big( \mathbb{R}^{1,5} \times \mathbb{R}_{+} \times S^3 \big)_{cl} \,\simeq\, \Omega^3_{dR} \big( \mathbb{R}^{1,9} \setminus \mathbb{R}^{1,5} \big)_{cl}

    being the B-field flux density sourced by an NS5-brane with worldvolume

    5,1x(x,0) 9,1. \mathbb{R}^{5,1} \xhookrightarrow{ \vec x \,\mapsto\, (\vec x, \vec 0) } \mathbb{R}^{9,1} \,.
  • A closed differential 0-form (the “Romans mass”)

    F 0Ω dR 0( 9,1 8,1) F_0 \,\in\, \Omega^0_{dR} \Big( \mathbb{R}^{9,1} \setminus \mathbb{R}^{8,1} \Big)

    being the RR-field flux density sourced by a D8-brane whose worldvolume

    1,8x(x,d) 1,9 \mathbb{R}^{1,8} \overset{ \vec x \,\mapsto\, (\vec x, d) }{ \hookrightarrow } \mathbb{R}^{1,9}

    we take to be located at x 9=dx^9=d for some d +d \in \mathbb{R}_+.

  • A differential 2-form

    F 2Ω dR 3( 9,1( 5,1 6,1)) F_2 \,\in\, \Omega^3_{dR} \Big( \mathbb{R}^{9,1} \setminus \big( \mathbb{R}^{5,1} \cup \mathbb{R}^{6,1} \big) \Big)

    being the RR-field flux density sourced by a D6-brane with potential worldvolume (“potential” because the D6 may have \sim vanishing charge carried by parts of its worldvolume, hence may “disappear” from its would-be worldvolume locus in parts, as we will see):

    (2) 6,1(x 0,x)(x 0,0,x) 9,1 \mathbb{R}^{6,1} \xhookrightarrow{ (x^0, \vec x) \,\mapsto\, \big(x^0, \vec 0, \vec x\big) } \mathbb{R}^{9,1}

    transverse to that of both the NS5-brane and the D8-brane and satisying the equation

    (3)dF 2=H 3F 0Ω dR 3( 9,1( 5,1 8,1 6,1)), \mathrm{d} F_{2} \;=\; H_3 \wedge F_{0} \;\;\;\;\; \in \; \Omega^3_{dR} \Big( \mathbb{R}^{9,1} \setminus \big( \mathbb{R}^{5,1} \cup \mathbb{R}^{8,1} \cup \mathbb{R}^{6,1} \big) \Big) \,,

    where now H 3H_3 and F 0F_0 are understood of the pullback (restriction) of the above differential forms of these names to the complement of the D6-brane worldvolume (2).

    Notice that even if/though the original forms H 3H_3 is not exact, after this pullback/restriction to this it does become exact, so that the equation (3) is meaningful.

In order to model the Hanany-Witten effect proper as considered by previous authors, we declare the following boundary conditions:

  • The Romans vacuum is to the “right” of the D8-brane

    (4)x 9>dF 0(x 9)=0. x^9 \gt d \;\;\;\;\;\;\; \Rightarrow \;\;\;\;\;\;\; F_0(x^9) = 0 \,.

    Notice that the Bianchi identity for F 0F_0 is dF 0=0\mathrm{d} F_0 = 0, which means that it is a locally constant function whose jump across the D8-brane worldvolume measures the charge/number NN of these D8-branes (cf. eg. Fazzi (2017) p. 40). Hence

    (5)x 9<dF 0(x 9)=N. x^9 \lt d \;\;\;\;\;\;\; \Rightarrow \;\;\;\;\;\;\; F_0(x^9) = N \,.
  • There is no D6-brane to the far “left” of the NS5/D8,

    (6)limx 9F 2(x 9)=0, \underset{\underset{x^9 \to -\infty}{\longrightarrow}}{\lim} F_2(x^9) \;\; = \;\; 0 \,,

    which just means to disregard D6-brane whose presence is not due to the Hanany-Witten effect.

With these boundary conditions set, the differential equation (3) may be solved for F 2F_2, and hence for the density of D6-brane charge/number which is “created” by the Hanany-Witten effect: At given x 9x^9 the number/charge of D6-branes seen is the integral of F 2F^2 around the 2-sphere in

9,1 6,1 6,1× +×S 2 \mathbb{R}^{9,1} \setminus \mathbb{R}^{6,1} \;\simeq\; \mathbb{R}^{6,1} \times \mathbb{R}_+ \times S^2

over (0,x 9) 6,1(\vec 0, x^9) \,\in\, \mathbb{R}^{6,1}. Now there are two cases:

  1. d<0d \lt 0 – indicated in the top part of figure (7)

    In this case the NS5-brane is to the “right” of the D8-brane and hence in the Romans vacuum where F 0=0F_0 = 0. Here the differential equation (3) becomes

    dF 2 = F 0H 3 = 0 \begin{array}{c} \mathrm{d} F_2 &=& F_0 \wedge H_3 \\ &=& 0 \end{array}

    This means that F 2F_2 does not change in the Romans vacuum, and hence if it vanishes to the left, as the boundary condition assumes, then it vanishes throughout.

    Since the integral of F 2F_2 counts the number/charge of D6-brane, the conclsusion is that no D6-branes are created in this situation.

  2. d>0d \gt 0 – indicated in the bottom part of figure (7)

    In this case the NS5-brane is to the “left” of the D8-brane and hence in the region where F 0=NF_0 = N equals the number of D8-branes. Here the differential equation (3) becomes

    dF 2 = F 0H 3 = NH 3 \begin{array}{c} \mathrm{d} F_2 &=& F_0 \wedge H_3 \\ &=& N \, H_3 \end{array}

    This means now that as approach the NS5/D8-branes from the “left”, F 2F_2 is continuously increasing by a rate measured by the density H 3H_3 of NS5-brane flux. While this is everywhere non-vanishing, it is concentrated around the NS5-brane worldvolume locus (falling off by a power law from there). Therefore once we have “crossed the NS5-brane” and thus picked up most of its flux, S 2F 2\int_{S^2} F_2 will have increased from its asympotically vanishing value to N S 2 dH 3N S 3H 3NkN \int_{S^2} \int_{-\infty}^d H_3 \,\sim\, N \int_{S^3} H_3 \,\sim\, N k, the product of the number of NS5-branes and D8-branes. This is hence the number of D6-branes “created” by the Hanany-Witten effect.

D3 stretching between D5 and NS5

(cf. D3-D5 brane bound state)

We discuss the “creation” of D3-brane flux “in between” a background of D5-brane- and NS5-brane flux as seen from (just) the pregeometric flux-equations.

If we assume that there are no D7-branes present in that the RR-field flux density which they source vanishes, F 1=0F_1 = 0, then the Bianchi identity for the D5-brane-flux is just

dF 3 = H 3F 1 = 0 \begin{array}{rcl} \mathrm{d} \, F_3 &=& H_3 \wedge F_1 \\ &=& 0 \end{array}

just as for the B-field flux density

dH 3=0. \mathrm{d}\, H_3 \;=\; 0 \,.

Hence assume that there is a pair of flat parallel D5-branes and NS5-branes at a positive distance 2d +2 d \in \mathbb{R}_+ from each other

NS5 5,1 10,1 (t,x) (t,x,d,0) D5 5,1 10,1 (t,x) (t,x,+d,0) \array{ \mathbb{R}^{5,1}_{NS5} &\xhookrightarrow{\phantom{--}}& \mathbb{R}^{10,1} \\ (t,\vec x) &\mapsto& (t,\vec x, -d , \vec 0) } \;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\;\; \array{ \mathbb{R}^{5,1}_{D5} &\xhookrightarrow{\phantom{--}}& \mathbb{R}^{10,1} \\ (t,\vec x) &\mapsto& (t,\vec x, +d , \vec 0) }

in that the respective flux densities are

H 3vol S 3Ω dR 3(S 3)pr S 3 *Ω dR 3( NS5 5,1× +×S 3)Ω dR 3( 9,1 D5 5,1) H_3 \;\coloneqq\; vol_{S^3} \;\in\; \Omega^3_{dR}(S^3) \xrightarrow{ pr_{S^3}^\ast } \Omega^3_{dR}\big( \mathbb{R}^{5,1}_{NS5} \times \mathbb{R}_+ \times S^3 \big) \;\simeq\; \Omega^3_{dR}\big( \mathbb{R}^{9,1} \setminus \mathbb{R}^{5,1}_{D5} \big)


F 3vol S 3Ω dR 3(S 3)pr S 3 *Ω dR 3( D5 5,1× +×S 3)Ω dR 3( 9,1 D5 5,1) F_3 \;\coloneqq\; vol_{S^3} \;\in\; \Omega^3_{dR}(S^3) \xrightarrow{ pr_{S^3}^\ast } \Omega^3_{dR}\big( \mathbb{R}^{5,1}_{D5} \times \mathbb{R}_+ \times S^3 \big) \;\simeq\; \Omega^3_{dR}\big( \mathbb{R}^{9,1} \setminus \mathbb{R}^{5,1}_{D5} \big)

In the following we regard both of these as further pulled back/restricted to the joint complement 9,1( NS5 5,1 D5 5,1 D3 3,1)\mathbb{R}^{9,1} \setminus \big( \mathbb{R}^{5,1}_{NS5} \cup \mathbb{R}^{5,1}_{D5} \cup \mathbb{R}^{3,1}_{D3}\big). Notice that as such they differ, one being the translate of the other by 2d2d along one of the axes.

Now the Bianchi identity for the RR-field flux density F 5F_5 sourced by any D3-branes is

(8)dF 5=H 3F 3, \mathrm{d} \, F_{5} \;=\; H_3 \wedge F_3 \,,

Imposing the boundary condition that F 5F_5 vanishes at infinity, the differential equation will imply the presence of non-vanishing F 5F_5-flux and hence of D3D3-branes “in between” the D5/NS5 iff the wedge product H 3F 3H_3 \wedge F_3 has the form of a “multipole” roughly supported in between these branes.

The following graphics, which shows a 2-dimensional shadow of the situation, qualitatively indicates that and why this is the case:


  • the strength of the circular lines indicates the absolute value of the flux densities H 3H_3, F 3F_3 (vanishing at infinity and spiked at the locus of the sourcing 5-brane),

  • the little arrows indicate the orientation? of the two flux densities, projected to the plane that is being shown

  • the little parallelograms indicate the orientation of the wedge product H 3F 3H_3 \wedge F_3 in the projection to the plane being shown.

(Hence what the graphics is really showing, approximately, is the wedge product of a pair of translates of dvol S 1dvol_{S^1} pulled back to around either puncture of the 2-punctured plane.)

The point is to observe that — besides the abolute value of the wedge product F 3H 3F_3 \wedge H_3 evidently being concentrated near the branes, hence particularly between them when they are close — the sign of the orientation of the wedge product changes as the transverse coordinate passes through the axis between the branes.

Hence as we solve the differential equation dF 5=H 3F 3\mathrm{d} F_5 \,=\, H_3 \wedge F_3 incrementally by running, say, from the top to the bottom in the above graphics, we see that the absolute value of F 5F_5 — indicated in red shading\color{red}\text{red shading} above — fist incrementally picks up positive contributions as we approach the area between the branes, and then dimishies again as we leave that area. In total, the absolute value of the solution F 5F_5 will be supported roughly in between the two 5-branes.

Since F 5F_5 is the flux sourced by D3-branes, this means that D3-branes are seen to be “created” in between the 5-branes.

However, from this analysis of the fluxes alone, it does not follow that the D3-branes would disappear as the NS5- and D5-brane “swap positions”, as Hanany-Witten would suggest. In fact there is no intrinsic such directedness in the problem in the first place with respect to which one could even speak about the branes “swapping positions”.

Hence to see the full Hanany-Witten effect in this case, if it exists, one may have to look beyond the bare fluxes and include also the field of gravity. A suggestion for how this might work is given, for the analogous situation of M2-M5 brane intersections, in Hosomichi (2000), §5.

brane intersections/bound states/wrapped branes/polarized branes

S-duality\,bound states:



The original article:

Further discussion:

See also:

The case of D6-D8 brane bound states:

Discussion for M2-M5 brane intersections:

Last revised on January 27, 2024 at 16:58:28. See the history of this page for a list of all contributions to it.