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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 13th 2015
    • (edited Nov 13th 2015)

    I am maybe beginning to see how to obtain the F-theory picture from super-cohesion. Here is some brainstorming.


    Let’s consider the M-theory bouquet, but starting down in IIA:

    𝔪5𝔟𝔯𝔞𝔫𝔢 𝔪2𝔟𝔯𝔞𝔫𝔢 [6] 𝔇0𝔟𝔯𝔞𝔫𝔢= 10,1|32 [3] 9,1|16+16¯ [1] \array{ \mathfrak{m}5\mathfrak{brane} \\ \downarrow \\ \mathfrak{m}2\mathfrak{brane} &\longrightarrow& \mathbb{R}[6] \\ \downarrow \\ {{\mathfrak{D}0\mathfrak{brane}} \atop {= \mathbb{R}^{10,1\vert \mathbf{32}}}} &\longrightarrow& \mathbb{R}[3] \\ \downarrow \\ \mathbb{R}^{9,1\vert \mathbf{16}+ \overline{\mathbf{16}}} &\stackrel{}{\longrightarrow}& \mathbb{R}[1] }

    In Lie integrating and globalizing this, we should really be complexifying the coefficients.

    Recall the reason for this for the M2-brane (from Structure Theory for Higher WZW Terms (schreiber)):

    Considering its WZW term as complexified L WZW M2: 10,1|32B 3(/) conn\mathbf{L}_{WZW}^{M2} \colon \mathbb{R}^{10,1\vert \mathbf{32}} \longrightarrow \mathbf{B}^3(\mathbb{C}/\mathbb{Z})_{conn} gives us the freedom to add a closed 3-form α\alpha, and if we take α\alpha to be the associative 3-form on 7 10,1|32\mathbb{R}^7 \hookrightarrow \mathbb{R}^{10,1\vert \mathbf{32}} then we get the neat statement that

    1. globalizing L WZW M2+iα\mathbf{L}_{WZW}^{M2} + i \alpha over an 11-dimensional supermanifold XX is precisely the structure of making XX a solution to the equations of motion of 11d supergravity and fibered by G 2G_2-manifolds together with the classical anomaly cancellation data that makes the M2 be globally defined on this background, hence is precisely the data of M-theory on G2-manifolds;

    2. Since the thus globalized α\alpha is a calibration on XX, it follows that the volume holonomy of the globalized L WZW M2+iα\mathbf{L}_{WZW}^{M2} + i \alpha gives precisely the membrane instanton contributions.

    So let’s think about what it means to complexify the coefficients of the other stages in the bouquet.

    Consider the D0-brane. Its WZW term is the RR-field which in the literature is often written C 1C_1. Under T-duality, this is identified with the WZW-term C 0C_0 of the D(-1)-brane in type IIB. Now, the complexification of C 0C_0 is well-known and famous: this is the axio-dilaton τC 0+ig 2\tau \coloneqq C_0 + i g_2, where g sg_s is the string coupling constant. The proposal of F-theory is to think of τ\tau as varying over spacetime, defining an elliptic fibration over 10d spacetime.

    But we may be more systematic about this: passing back via T-duality, we find that the complexified D0-brane WZW term is identified with the axio-dilaton. But now we know that this WZW term induces an extension of spacetime by Lie extension/D0-brane condensation.

    To summarize this: we know

    1. from first principles that the complexified D0 WZW term induces an extension of 10d spacetime by fibers locally modeled on \mathbb{C};

    2. from comparison with the string literature we see that this is nothing but that type IIA pre-image of the F-theory elliptic fibration.

    So we may conclude that the integrated version of the D0-brane cocycle is to be identified with the map

    L WZW D0+ig˜ s:X//SL(2,)¯ \mathbf{L}_{WZW}^{D0} + i \tilde g_s \colon X \longrightarrow \overline{\mathbb{H}//SL(2,\mathbb{Z})}

    to the compactified complex moduli stack of elliptic curves which classifies the elliptic fibration.

    (Maybe we may fill in more details in this step. What if we used //GL(2,)\mathbb{C}//GL(2,\mathbb{Z}) instead? which is more natural from the perspective of integrating our [1]\mathbb{C}[1]-valued Lie algebra cocycle)

    Once at this point, it makes sense to rewrite this as

    L WZW D0+ig˜ s:X ell¯(). \mathbf{L}_{WZW}^{D0} + i \tilde g_s \colon X \longrightarrow \overline{\mathcal{M}_{ell}}(\mathbb{C}) \,.

    Now by GHL this moduli stack of course carries the bundle 𝒪 ell top\mathcal{O}^{top}_{ell} of elliptic spectra, and so we may say that the D0-brane cocycle twists elliptic cohomology on XX. This is pretty much the proposal by Kriz-Sati. Notice that when the cocycle map hits (only) the point of the nodal curve, then this statement comes out precisely matching the known data:

    1. physically (see at F-branes – table) this is the phase of F-theory with ordinary type IIB D-branes around which carry their ordinary D-brane charge in K-theory;

    2. mathematically, the restriction of 𝒪 ell top\mathcal{O}^{top}_{ell} to the nodal curve is KUKU with its 2\mathbb{Z}_2-action sitting over */ 2\ast/\mathbb{Z}_2 (see the table here), hence mathematically this gives indeed KR-theory (nlab) cohomology in exactly the right way as needed for general type II backgrounds (see at orientifold (nlab)).


    [continued in next comment]

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeNov 13th 2015
    • (edited Nov 13th 2015)

    [continued from previous comment]


    All this looks good. So we need to see how it all integrates into one integrated bouquet. At this point we have this:

    X× τ˜E L WZW M2+iα B 3(/) conn X τ˜ ell¯ \array{ \vdots \\ \downarrow \\ X \times_{\tilde \tau} E &\stackrel{\mathbf{L}_{WZW}^{M2} + i \alpha}{\longrightarrow}& \mathbf{B}^3 (\mathbb{C}/\mathbb{Z})_{conn} \\ \downarrow \\ X &\stackrel{\tilde \tau}{\longrightarrow}& \overline{\mathcal{M}_{ell}} }

    This D0-M2 stage is directly analogous to the M2-M5 stage which we analyzed in The WZW term of the M5-brane (schreiber). If we are to proceed as we did there, then we have to take the above diagram and rewrite it as

    X× τ˜E L WZW M2+iα B 3(/) conn X Q τ˜ ell¯() \array{ \vdots \\ \downarrow \\ X \times_{\tilde \tau} E && \stackrel{\mathbf{L}_{WZW}^{M2} + i \alpha}{\longrightarrow}&& \mathbf{B}^3 (\mathbb{C}/\mathbb{Z})_{conn} \\ \downarrow && && \downarrow \\ X && \longrightarrow && Q \\ & {}_{\mathllap{\tilde \tau}}\searrow && \swarrow \\ && \overline{\mathcal{M}_{ell}}(\mathbb{C}) }

    with QQ something like a K(,4)K(\mathbb{Z},4)-fibration over ell¯\overline{\mathcal{M}_{ell}}. And then we should check if L WZW M2+iα\mathbf{L}_{WZW}^{M2} + i \alpha descends after transgression (double dimensional reduction) along the fiber to give a map

    X ell¯×˜K(,3) X \longrightarrow \overline{\mathcal{M}_{ell}} \tilde \times K(\mathbb{Z},3)

    that map would then be the combined twisting class of F-theory, combining the elliptic parameterization with the B-field twist for K-theory.

    Something like this.


    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeNov 13th 2015
    • (edited Nov 13th 2015)

    To see what this ought to be, restrict attention to X=Y×(*/ 2)X = Y \times (*/\mathbb{Z}_2) and the ordinary “non-elliptic” case where τ˜\tilde \tau is constant on the inclusion of the nodal curve. The fiber of 𝒪 ell top\mathcal{O}^{top}_{ell} over τ˜\tilde \tau then is just koku 2ko \simeq ku^{\mathbb{Z}_2} And so in this case the charges of the M2 are found to sit in koko

    X×E L M2 ko τ˜ {nodal} \array{ X \times E && \stackrel{L_{M2}}{\longrightarrow} && ko \\ & {}_{\mathllap{\tilde \tau}}\searrow && \swarrow \\ && \{nodal\} }

    Accordingly, after double dimensional reduction/transgression along the circle, the string will take charges in Σ 1ko\Sigma^{-1} ko.

    And this is indeed the right answer for type II backgrounds, that’s the point around p. 13 in Freed 12.