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nLab > Latest Changes: D-branes

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    • CommentRowNumber1.
    • CommentAuthorzskoda
    • CommentTimeSep 21st 2011

    There are now two different pages: old D-branes and new one created by Urs today D-brane. I found D-branes as I recalled we had something and I looked under nLab search.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeSep 21st 2011

    I was just about to announce that I have split off D-brane from brane.

    I had not been aware of the entry D-branes. I’ll merge what is there to the singular-case entry in a moment.

    But then I have to quit. I need to take care of another urgent task.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeSep 21st 2011

    Okay, I have merged the information that used to be in the entry “D-branes” and which wasn’t yet at D-brane into the latter (namely the two comments on D-branes of LG models and the pointer to Kapustin’s article).

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeSep 22nd 2011

    Thanks for hard work :)

    I had not been aware of the entry D-branes

    Surely we all sometimes created the doubles :) Nobody does this regularily, including of course myself, but somehow we should in principle do some nnLab search when creating a page for a very common/probable concept. Here the bad circumstance was that the name was in plural, hence more likely to escape the attention.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeSep 22nd 2011

    I am inclined to say that, apart from the plural, here the problem was that the page “D-branes” was not linked to from the relevant entries. Otherwise I would have seen it.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeMay 28th 2013
    • (edited May 28th 2013)

    added to D-brane a brief paragraph on D-brane charge in terms of an index in KK-theory.

    Should be further expanded later…

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeMay 29th 2013

    I have cleaned it up a bit more.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeDec 23rd 2016
    • (edited Dec 27th 2016)

    It should be true that, in full beauty: “the WZW-term” of the D-brane action functional (sometimes called “the Chern-Simons term”) is nothing but the Kronecker pairing in (twisted) differential K-theory.

    In def. 1.6 of arXiv:hep-th/0607020 the shadow of this statement in ordinary cohomology is given. Is there any reference that mentiones the full statement?

    Here is what I mean: Write KU∈Stab(Smooth∞Grpd)\mathbf{KU} \in Stab(Smooth \infty Grpd) for (some version) of multiplicative differential complex K-theory. Let X∈Smooth∞GrpdX \in Smooth \infty Grpd be a smooth manifold. With tt denoting the “type of string theory” (t=0t = 0 for type IIA string theory, and t=1t = 1 for type IIB string theory) then

    1. for p∈2ℤp \in 2\mathbb{Z} a D p+tD_{p+t}-brane in XX is a cycle in differential K-homology ϕ:S p+t+1→KU∧Σ + ∞X\phi \colon S^{p+t+1} \to \mathbf{KU} \wedge \Sigma^\infty_+ X;

    2. an RR-field configuration on XX is a cocycle in differential KU-cohomology c:Σ + ∞X→Σ tKUc \colon \Sigma_+^\infty X \to \Sigma^t\mathbf{KU}

    and the value of the WZW-type action functional of that D-brane in that RR-background should be the Kronecker pairing

    exp(iℏS WZW):𝕊⟶ϕΩ p+t+1KU∧Σ + ∞X⟶id∧cΩ p+t+1KU∧Σ tKU⟶multiplyΩ 1KU \exp(\tfrac{i}{\hbar} S_{WZW}) \;\colon\; \mathbb{S} \stackrel{\phi}{\longrightarrow} \Omega^{p+t+1} \mathbf{KU} \wedge \Sigma_+^\infty X \stackrel{id \wedge c }{\longrightarrow} \Omega^{p+t+1}\mathbf{KU} \wedge \Sigma^{t}\mathbf{KU} \stackrel{multiply}{\longrightarrow} \Omega^{1} \mathbf{KU}

    in π 1(KU(*))≃π 1(Kℂ)→ℂ ×\pi_1(\mathbf{KU}(\ast)) \simeq \pi_1(K \mathbb{C}) \to \mathbb{C}^\times.

    • CommentRowNumber9.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 23rd 2016

    Why write the exponent as p+(1−t)+2p+(1-t)+2?

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeDec 27th 2016
    • (edited Dec 27th 2016)

    Sorry, it should have been p+t+1p+t+1 (fixed now). So for pp even then the D-branes of IIA are p-branes (in every even degree) and those of IIB are p+1p+1-branes (every odd degree). (The worldvolume is is of dimension one higher.)

    • CommentRowNumber11.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 28th 2016

    Does #8 have something to do with KK-theory then?

    the coupling of D-branes and their Chan-Paton bundles in twisted K-theory with RR-charge in string theory is naturally captured by the coupling between K-homology and K-cohomology in KK-theory

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeDec 28th 2016
    • (edited Dec 28th 2016)

    KK-theory is one model for KU-(co-)homology. The references cited in the entry on KK-theory discuss mainly the push-forward maps in KU-theory in terms of pairings in KK-theory.

    However what I am after in #8 is the WZW-term of the D-branes (e.g. def. 4.4 in arXiv:1606.03206). I am thinking this should be equivalently the (Kronecker) pairing, whith relative degree 1 of the homology cycle of a D-brane against the cohomology cocycle of the background RR-field, in differential KU-theory. This is a statement that I don’t see in existing literature. In def. 1.6 of arXiv:hep-th/0607020 the explicit formula is given in K-theoretic terms, but not as an intrinsically differential K-theoretic pairing construction.

    What I am really after is this: the brane bouquet gives a derivation from first principles of Dp-branes with (twisted) complex line bundles on their worldvolume and equipped with that WZW-functional

    S WZW=∫ Σ^exp(F 2)∧ϕ^ *C S_{WZW} = \int_{\hat \Sigma } \exp(F_2) \wedge \hat \phi^\ast C

    where Σ^\hat \Sigma is manifold of dimension p+2p+2 whose boundary ∂Σ^=Σ\partial \hat \Sigma = \Sigma is the given Dp-brane wordvolume, where ϕ:Σ^→X\phi \colon \hat \Sigma \to X is the way that the D-brane sits in spacetime, and where =C 2+C 4+⋯= C_2 + C_4 + \cdots denote the backrground RR-field strengths.

    One observes that, rationally, this functional is invariant under forming the “external tensor product” with the canonical line bundle on S 2S^2 with curvature F 2 βF_2^\beta, in that for Σ=Σ′×S 2\Sigma = \Sigma' \times S^2 then the above functional becomes

    ∫ Σ^′×S 2exp(F′ 2+kF 2 β)∧ϕ^ *C=k∫ Σ^′exp(F′ 2)∧ϕ^ *C. \int_{\hat \Sigma' \times S^2 } \exp(F'_2 + k F_2^\beta) \wedge \hat \phi^\ast C = k \int_{\hat \Sigma' } \exp(F'_2) \wedge \hat \phi^\ast C \,.

    Hence it makes sense to consider the “quotient” of the moduli of complex line bundles on the D-branes by this relation, since the functional that we are interested in descends to this “quotient”. More precisely, this means saying that multiplying with the “Bott generator” should be invertible. Now Snaith’s theorem says that the result of starting with complex line bundles and then inverting the multiplication with the Bott generator is complex K-theory

    KU≃𝕊[BU(1)][β −1]. KU \simeq \mathbb{S}[BU(1)][\beta^{-1}] \,.

    I’d like to turn this argument into an alternative “derivation”, starting from just the brane bouquet, that D-brane charge is in K-theory.

    • CommentRowNumber13.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 29th 2016

    And you have a parallel for much of this for M-branes?

    Returning to the more ambitious route via spectral supergeometry, there would need to be a spectral group scheme to play the role of ℝ 0|1\mathbb{R}^{0|1}? Then successive extensions of this group scheme would lead inexorably to KUKU and the M-brane spectrum?

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeDec 29th 2016

    And you have a parallel for much of this for M-branes?

    I have the rational story for the M-branes and for the D-branes and we know the non-rational answer for the D-branes. If I knew a systematic way to deduce the non-rational answer for the D-branes from the rational version (some general principle), then the analogous systematics applied to the rational M-branes should give the non-rational M-branes, which is currently an open problem.

    Returning to the more ambitious route via spectral supergeometry, there would need to be a spectral group scheme to play the role of ℝ 0|1\mathbb{R}^{0|1}? Then successive extensions of this group scheme would lead inexorably to KU$ and the M-brane spectrum?

    That’s a guess I have expressed, yes. But for the moment it’s just that, a guess.

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeDec 30th 2016
    • (edited Dec 30th 2016)

    Here is a more polished way to state the observation in #12.

    So consider the D-brane cocycles, say in type IIA (just for definiteness of notation):

    𝔰𝔱𝔯𝔦𝔫𝔤 IIA ⟶⊕ 2p[exp(f 2)∧C] p+2 ⊕ 2pb 2p+1ℝ ↓ ℝ 9,1|16+16¯ μ F1↘ b 2ℝ \array{ \mathfrak{string}_{IIA} &\overset{\oplus_{2p} [\exp(f_2) \wedge C]_{p+2} }{\longrightarrow}& \oplus_{2p} b^{2p+1} \mathbb{R} \\ \downarrow \\ \mathbb{R}^{9,1\vert \mathbf{16} + \overline{\mathbf{16}}} \\ & {}_{\mathllap{\mu_{F1}}}\searrow \\ && b^2 \mathbb{R} }

    (with notation and conventions from here).

    By the process of differential Lie integration, the cocoycles

    μ Dp≔[exp(f 2)∧C] p+2 \mu_{Dp} \coloneqq [exp(f_2) \wedge C]_{p+2}

    yield global WZW terms (here)

    L Dp:String˜ IIA⟶B 2p+1U(1) conn \mathbf{L}_{Dp} \;\colon\; \widetilde{String}_{IIA} \longrightarrow \mathbf{B}^{2p+1}U(1)_{conn}

    where String˜ IIA\widetilde{String}_{IIA} is a super-2-group extension of the super-Minkowski group ℝ 9,1|16+16¯\mathbb{R}^{9,1\vert \mathbf{16}+ \overline{16}} by BU(1) conn\mathbf{B}U(1)_{conn}, fitting into a homotopy fiber sequence of the form

    BU(1) conn ⟶ String IIA˜ ↓ ℝ 9,1|16+16¯ ⟶L F1 IIA B(BU(1) conn) \array{ \mathbf{B}U(1)_{conn} &\longrightarrow& \widetilde{String_{IIA}} \\ && \downarrow \\ && \mathbb{R}^{9,1\vert \mathbf{16}+ \overline{\mathbf{16}}} &\overset{\mathbf{L}_{F1}^{IIA}}{\longrightarrow}& \mathbf{B}(\mathbf{B}U(1)_{conn}) }

    (this example).

    Now consider the “differential Bott generator” in this context, the map

    String IIA˜×S 2⟶id×1String IIA˜×BU(1) conn⟶String IIA˜×String IIA˜⟶⋅String IIA˜, \widetilde{String_{IIA}} \times S^2 \overset{id \times 1}{\longrightarrow} \widetilde{String_{IIA}} \times \mathbf{B}U(1)_{conn} \longrightarrow \widetilde{String_{IIA}} \times \widetilde{String_{IIA}} \overset{\cdot}{\longrightarrow} \widetilde{String_{IIA}} \,,

    where “1” is the map that modulates the circle-principal connection on the sphere with unit first Chern class and curvature 2-form the volume form, where the second morphism is the inclusion from the above extension diagram, and where the last morphism is the product in the 2-group.

    The hom-adjunct of this morphism is a map

    β:String IIA˜⟶[S 2,String IIA˜]. \beta \;\colon\; \widetilde{String_{IIA}} \longrightarrow [S^2, \widetilde{String_{IIA}}] \,.

    Finally consider the transgression of the D(p+2)-cocycle on the 2-sphere, given by the composite

    τ S 2L D(p+2):[S 2,String IIA˜]⟶[S 2,L Dp][S 2,B p+3U(1) conn]⟶∫ S 2B p+1U(1) conn. \tau_{S^2} \mathbf{L}_{D(p+2)} \;\colon\; [S^2, \widetilde{String_{IIA}}] \overset{[S^2, \mathbf{L}_{D p}]}{\longrightarrow} [S^2, \mathbf{B}^{p+3}U(1)_{conn}] \overset{\int_{S^2}}{\longrightarrow} \mathbf{B}^{p+1}U(1)_{conn} \,.

    Using this we may now state a “Bott periodicity” among the Dp-brane cocycles: I think it is true that L Dp\mathbf{L}_{D p} coincides with the 2-sphere transgression of L D(p+2)\mathbf{L}_{D(p+2)} composed with the “differential Bott generator”, i.e. that

    L Dp:String IIA˜⟶β[S 2,String IIA˜]⟶τ S 2L D(p+2)B p+1U(1) conn. \mathbf{L}_{D p} \;\colon\; \widetilde{String_{IIA}} \overset{\beta}{\longrightarrow} [S^2, \widetilde{String_{IIA}}] \overset{\tau_{S^2} \mathbf{L}_{D(p+2)}}{\longrightarrow} \mathbf{B}^{p+1}U(1)_{conn} \,.

    The argument for why this is true is that L Dp\mathbf{L}_{D p} is given by the Lie integration of the cocycle [exp(f 2)∧C] p+2[\exp(f_2) \wedge C]_{p+2}, with local data given by integrals of the differential form obtained from this by substituting differential forms for the CE-generators, in particular by substituting the curvature 2-form F 2F_2 for f 2f_2. Now for maps in the image of the above composite with β\beta, then

    F 2=F 2 Σ+F 2 S 2 F_2 = F_2^{\Sigma} + F_2^{S^2}

    where the first summand is some contribution and the second summand is the volume form on the 2-sphere. Under the integral ∫ S 2\int_{S^2} involved in the transgression map, the component in [exp(F 2)∧C] p+4[\exp(F_2) \wedge C]_{p+4} which is linear in that form F 2 S 2F_2^{S^2} is picked out, and that component is simply [exp(F 2 Σ)∧C] p+2[\exp(F_2^{\Sigma}) \wedge C]_{p+2}.

    I should check carefully that this arguments holds water, but I think it looks good.

    So this is then the idea that I am after:

    Observe that the Dp-brane WZW terms obtained from the brane bouquet (hence “from first principles”) enjoy this extra “differential Bott periodicity” condition

    L Dp≃τ S 2L D(p+2)∘β. \mathbf{L}_{D p} \simeq \tau_{S^2} \mathbf{L}_{D (p+2)}\circ \beta \,.

    Then somehow argue that, therefore, we should “quotient out” this “symmetry” of the joint WZW term (it’s standard to pass a Lagrangian to the “reduced phase space” obtained by quotienting out all its symmetries from the original phase space, one just needs to be careful to read this Bott periodicity as a symmetry in this sense, but I suppose it looks like it should make sense). In conclusion this discovers essentially the quotient operation that Snaith’s theorem shows turns BU(1)BU(1) into KUKU. Well, up to some handwaving for the moment. I should still say how the passage to spectra via Σ + ∞\Sigma_+^\infty before “quotienting” is motivated from te brane bouquet story.

    • CommentRowNumber16.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 2nd 2017

    Is that systematic enough to help with

    If I knew a systematic way to deduce the non-rational answer for the D-branes from the rational version (some general principle), then the analogous systematics applied to the rational M-branes should give the non-rational M-branes ?

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeJan 2nd 2017
    • (edited Jan 2nd 2017)

    Not yet. There is the open issue that I mentioned at the end. And, worse, I realize now that there is a gap even in the argment that I sketched: the second clause of the condition here is not satisfied, unless one makes some adjustments. Not sure. For that purpose it might be better to consider the cocycles on AdS and then take the contraction limit. Unfortunately in this case I am not sure exactly how the D-brane cocycles behave.

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