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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeNov 13th 2015
   • (edited Nov 13th 2015)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI 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 [[schreiber:The brane bouquet|bouquet]], but starting down in IIA: $$ \array{ \mathfrak{m}5\mathfrak{brane} \\ \downarrow \\ \mathfrak{m}2\mathfrak{brane} &\longrightarrow& \mathbb{R}[6] \\ \downarrow \\ {{\mathfrak{D}0\mathfrak{brane}} \atop {= \mathbb{R}^{10,1\vert \mathbf{32}}}} &\longrightarrow& \mathbb{R}[3] \\ \downarrow \\ \mathbb{R}^{9,1\vert \mathbf{16}+ \overline{\mathbf{16}}} &\stackrel{}{\longrightarrow}& \mathbb{R}[1] } $$ In Lie integrating and [[definite globalization of WZW term|globalizing]] this, we should really be complexifying the coefficients. Recall the reason for this for the M2-brane (from _[[schreiber:Structure Theory for Higher WZW Terms]]_): Considering its WZW term as complexified $\mathbf{L}_{WZW}^{M2} \colon \mathbb{R}^{10,1\vert \mathbf{32}} \longrightarrow \mathbf{B}^3(\mathbb{C}/\mathbb{Z})_{conn}$ gives us the freedom to add a closed 3-form $\alpha$, and if we take $\alpha$ to be the [[associative 3-form]] on $\mathbb{R}^7 \hookrightarrow \mathbb{R}^{10,1\vert \mathbf{32}}$ then we get the neat statement that 1. globalizing $\mathbf{L}_{WZW}^{M2} + i \alpha$ over an 11-dimensional supermanifold $X$ is precisely the structure of making $X$ a solution to the equations of motion of 11d supergravity and fibered by $G_2$-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]]; 1. Since the thus globalized $\alpha$ is a [[calibration]] on $X$, it follows that the volume holonomy of the globalized $\mathbf{L}_{WZW}^{M2} + i \alpha$ 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 $C_1$. Under T-duality, this is identified with the WZW-term $C_0$ of the D(-1)-brane in type IIB. Now, the complexification of $C_0$ is well-known and famous: this is the [[axio-dilaton]] $\tau \coloneqq C_0 + i g_2$, where $g_s$ is the string coupling constant. The proposal of [[F-theory]] is to think of $\tau$ 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 $\mathbb{C}$; 1. 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 $$ \mathbf{L}_{WZW}^{D0} + i \tilde g_s \colon X \longrightarrow \overline{\mathbb{H}//SL(2,\mathbb{Z})} $$ 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 $\mathbb{C}//GL(2,\mathbb{Z})$ instead? which is more natural from the perspective of integrating our $\mathbb{C}[1]$-valued Lie algebra cocycle) Once at this point, it makes sense to rewrite this as $$ \mathbf{L}_{WZW}^{D0} + i \tilde g_s \colon X \longrightarrow \overline{\mathcal{M}_{ell}}(\mathbb{C}) \,. $$ Now by GHL this moduli stack of course carries the bundle $\mathcal{O}^{top}_{ell}$ of elliptic spectra, and so we may say that the D0-brane cocycle twists elliptic cohomology on $X$. 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; 1. mathematically, the restriction of $\mathcal{O}^{top}_{ell}$ to the nodal curve is $KU$ with its $\mathbb{Z}_2$-action sitting over $\ast/\mathbb{Z}_2$ (see the table [here](http://ncatlab.org/nlab/show/modular+equivariant+elliptic+cohomology#General)), hence mathematically this gives indeed [[nLab:KR-theory]] cohomology in exactly the right way as needed for general type II backgrounds (see at [[nLab:orientifold]]). *** [continued in next comment]

   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:

   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 gives us the freedom to add a closed 3-form , and if we take to be the associative 3-form on then we get the neat statement that

   1. globalizing over an 11-dimensional supermanifold is precisely the structure of making a solution to the equations of motion of 11d supergravity and fibered by -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 is a calibration on , it follows that the volume holonomy of the globalized 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 . Under T-duality, this is identified with the WZW-term of the D(-1)-brane in type IIB. Now, the complexification of is well-known and famous: this is the axio-dilaton , where is the string coupling constant. The proposal of F-theory is to think of 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 ;

   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

   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 instead? which is more natural from the perspective of integrating our -valued Lie algebra cocycle)

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

   Now by GHL this moduli stack of course carries the bundle of elliptic spectra, and so we may say that the D0-brane cocycle twists elliptic cohomology on . 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 to the nodal curve is with its -action sitting over (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]

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeNov 13th 2015
   • (edited Nov 13th 2015)
   • PermaLink
   Author: Urs
   Format: MarkdownItex[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: $$ \array{ \vdots \\ \downarrow \\ X \times_{\tilde \tau} E &\stackrel{\mathbf{L}_{WZW}^{M2} + i \alpha}{\longrightarrow}& \mathbf{B}^3 (\mathbb{C}/\mathbb{Z})_{conn} \\ \downarrow \\ X &\stackrel{\tilde \tau}{\longrightarrow}& \overline{\mathcal{M}_{ell}} } $$ This D0-M2 stage is directly analogous to the M2-M5 stage which we analyzed in [[schreiber:The WZW term of the M5-brane]]. If we are to proceed as we did there, then we have to take the above diagram and rewrite it as $$ \array{ \vdots \\ \downarrow \\ X \times_{\tilde \tau} E && \stackrel{\mathbf{L}_{WZW}^{M2} + i \alpha}{\longrightarrow}&& \mathbf{B}^3 (\mathbb{C}/\mathbb{Z})_{conn} \\ \downarrow && && \downarrow \\ X && \longrightarrow && Q \\ & {}_{\mathllap{\tilde \tau}}\searrow && \swarrow \\ && \overline{\mathcal{M}_{ell}}(\mathbb{C}) } $$ with $Q$ something like a $K(\mathbb{Z},4)$-fibration over $\overline{\mathcal{M}_{ell}}$. And then we should check if $\mathbf{L}_{WZW}^{M2} + i \alpha$ descends after transgression ([[double dimensional reduction]]) along the fiber to give a map $$ X \longrightarrow \overline{\mathcal{M}_{ell}} \tilde \times K(\mathbb{Z},3) $$ 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. ***

   [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:

   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

   with something like a -fibration over . And then we should check if descends after transgression (double dimensional reduction) along the fiber to give a map

   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.

   ---

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeNov 13th 2015
   • (edited Nov 13th 2015)
   • PermaLink
   Author: Urs
   Format: MarkdownItexTo see what this ought to be, restrict attention to $X = Y \times (*/\mathbb{Z}_2)$ and the ordinary "non-elliptic" case where $\tilde \tau$ is constant on the inclusion of the nodal curve. The fiber of $\mathcal{O}^{top}_{ell}$ over $\tilde \tau$ then is just $ko \simeq ku^{\mathbb{Z}_2}$ And so in this case the charges of the M2 are found to sit in $ko$ $$ \array{ X \times E && \stackrel{L_{M2}}{\longrightarrow} && ko \\ & {}_{\mathllap{\tilde \tau}}\searrow && \swarrow \\ && \{nodal\} } $$ Accordingly, after double dimensional reduction/transgression along the circle, the string will take charges in $\Sigma^{-1} ko$. And this is indeed the right answer for type II backgrounds, that's the point around p. 13 in [Freed 12](http://ncatlab.org/nlab/show/orientifold#Freed).

   To see what this ought to be, restrict attention to and the ordinary “non-elliptic” case where is constant on the inclusion of the nodal curve. The fiber of over then is just And so in this case the charges of the M2 are found to sit in

   Accordingly, after double dimensional reduction/transgression along the circle, the string will take charges in .

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