Musings

In the local model, $B$ is noncompact, and is the total space of the line bundle

where the normal bundle $N_{S\mid B}=K_S^p$, a power of the canonical bundle of the surface, $S$, on which the 7-brane is wrapped. Away from the zero section, we have an elliptic curve (with affine coordinates $x,y$) fibered over the base. Over the zero section, the curve degenerates, and the total space of $X\to S$ looks like an isolated ADE surface singularity fibered over $S$. Denoting the fiber coordinate of $B\to S$ as $z$, the total space of $X\to S$ is given as the locus $\{f(x,y,z)=0\}\subset V$, with

ADE 7-branes as the locus $\{f=0\}\subset \mathrm{Tot}(V\to S)$

	$f(x,y,z)$	$V$	$N_{S\mid B}$	$K_B$	$\Delta \in \Gamma \left(K_B^{-12}\right)$
$A_n$	$y^2-x^2-z^{n+1}$	$K_S^{(n+1)/2}\oplus K_S^{(n+1)/2}\oplus K_S$	$K_S$	$𝒪$	?
$D_n$	$y^2-x^{2}z-z^{n-1}$	$K_S^{n-2}\oplus K_S^{n-1}\oplus K_S^2$	$K_S^2$	${\pi }^{*}\left(K_S^{-1}\right)$	?
$E_6$	$y^2-x^3-z^4$	$K_S^4\oplus K_S^6\oplus K_S^3$	$K_S^3$	${\pi }^{*}\left(K_S^{-2}\right)$	$-2^4{3}^3{z}^8$
$E_7$	$y^2-x^3-x{z}^3$	$K_S^6\oplus K_S^9\oplus K_S^4$	$K_S^4$	${\pi }^{*}\left(K_S^{-3}\right)$	$-2^6{z}^9$
$E_8$	$y^2-x^3-z^5$	$K_S^{10}\oplus K_S^{15}\oplus K_S^6$	$K_S^6$	${\pi }^{*}\left(K_S^{-5}\right)$	$-2^4{3}^3{z}^{10}$

Now, I’m a little confused by the $A_n$ and $D_n$ cases. As written, these are not elliptically-fibered; away from $z=0$, the fiber of $X\to B$ looks to me like a smooth quadric. But the main focus of attention is on the exceptional cases where, indeed, we have a Weierstrass form for the equation of the elliptic fiber.

Now, the idea is that the local physics is captured by the 8D twisted SYM theory on $R^{3,1}\times S$. The super Yang-Mills multiplet, in 8 dimensions, consists of a $G_S$-connection, a complex scalar in the adjoint representation, and some fermions, also in the adjoint. After twisting, the scalar becomes a 2-form $$\phi \in {\Omega }^{(2,0)}(S,\mathrm{ad}(P)),\overline{\phi }\in {\Omega }^{(0,2)}(S,\mathrm{ad}(P))$$ where $P\to S$ is a principal $G_S$-bundle. The left-handed fermions are $$\begin{array}{rl}{\eta }_{\alpha }& \in \Gamma (\mathrm{ad}(P))\\ {\psi }_{\alpha }& \in {\Omega }^{(0,1)}(S,\mathrm{ad}(P))\\ {\chi }_{\alpha }& \in {\Omega }^{(2,0)}(S,\mathrm{ad}(P))\end{array}$$ and their right-handed conjugates are $$\begin{array}{rl}{\overline{\eta }}_{\dot{\alpha }}& \in \Gamma (\mathrm{ad}(P))\\ {\overline{\psi }}_{\dot{\alpha }}& \in {\Omega }^{(1,0)}(S,\mathrm{ad}(P))\\ {\overline{\chi }}_{\dot{\alpha }}& \in {\Omega }^{(0,2)}(S,\mathrm{ad}(P))\end{array}$$ The conditions for a supersymmetric solution are

Here, $$F_S^{(2,0)}={\left(D_S^{(1,0)}\right)}^2,F_S^{(0,2)}={\left(D_S^{(0,1)}\right)}^2,F_S=\{D_S^{(1,0)},D_S^{(0,1)}\}$$ are, respectively, the $(2,0)$, $(0,2)$ and $(1,1)$ parts of the field strength on $S$, and $\omega$ is the Kähler form.

In the particular case¹ of $\phi =0$ (or, more generally, $[\phi ,\overline{\phi }]=0$), (2), these equations imply the Donaldson-Uhlenbeck-Yau equation, whose solution is an anti-self-dual connection, with field strength $F_S$, for some subgroup $H_S\subset G_S$. Correspondingly, there’s a reduction of the structure group of $P$ from $G_S$ to $H_S$, Denoting by ${\Gamma }_S$, the commutant of $H_S$ in $G_S$, we decompose $$\mathrm{ad}(P)\simeq {\oplus }_i(R_i\otimes {𝒯}_i)$$ where $R_i$ are irreps of ${\Gamma }_i$ and ${𝒯}_i$ are vector bundles associated to our $H_S$ principal bundle.

One obtains massless 4D chiral multiplets, in representation $R_i$ from the Dolbeault cohomology groups

As a toy example, take $S={𝔽}_n$, the $n$^th Hirzebruch surface², $G_S=E_6$, $H_S=U(1)$ and the unbroken gauge group, ${\Gamma }_S=\mathrm{SO}(10)$. That is, we have a nontrivial connection, satisfying the Donaldson-Uhlenbeck-Yau equation, for some line bundle $ℒ\to S$. Decomposing $$78={45}_0+1_0+{16}_{-3}+{\overline{16}}_3$$ We get chiral multiplets in the $16$ from $H^1(S,{ℒ}^{-3})$, and $\overline{16}$s from $H^1(S,{ℒ}^3)$. Their number is given by the index theorem. In the notation of the footnote, $$\begin{array}{rl}{n}_{16}& =(3b-1)[(3b+1)-\frac{3}{2}(2a+(2-n)b)]\\ n_{\overline{16}}& =(3b+1)[(3b-1)-\frac{3}{2}(2a+(2-n)b)]\end{array}$$

The Yukawa couplings of the chiral multiplets (3) are obtained from the trilinear form $$c_{ijk}:H^0(S,K_S\otimes {𝒯}_i)\otimes H^1(S,{𝒯}_j)\otimes H^1(S,{𝒯}_i)\to ℂ$$

Unfortunately, for del Pezzo and Hirzebruch surfaces, $H^0(S,K_S\otimes E)$ vanishes, for any vector bundle, $E$, admitting a connection satisfying Donaldson-Uhlenbeck-Yau. So there are no nontrivial Yukawa couplings among these “bulk modes.”

Intersecting Branes

To get nontrivial Yukawa couplings, we need to consider intersecting branes. Let $\Sigma \subset S$ be a smooth, irreducible curve. We can modify our setup to include 7-branes wrapping $R^{3,1}\times S\prime$, where $S\prime =\mathrm{Tot}(K_S^p\to \Sigma )$.

Since $S\prime$ is noncompact, the $G_{S\prime }$ gauge theory on it is decoupled. But there are new degrees of freedom localized on $\Sigma =S\cap S\prime$, given essentially by 6 dimensional twisted gauge theory, for the group $G_{\Sigma }\supset G_S\times G_{S\prime }$. Let $\alpha \in H^0(S,{𝒪}_S(\Sigma ))$. As before, we want the fiber over a generic point in $S$ to be an isolated $\mathrm{ADE}$ surface singularity, corresponding to $G_S$. But, over $S\prime$ (roughly, taking $z\ne 0$, and allowing $\alpha$ to vary), we want a $G_{S\prime }$ surface singularity.

Local model for intersecting ADE 7-branes

$G_S$	$G_{S\prime }$	$G_{\Sigma }$	$f(x,y,z)$	$V$	$L$	$N_{S\mid B}$	$K_B$
$A_n$	$A_m$	$A_{n+m}$	$y^2-x^2-{\alpha }^{m+1}{z}^{n+1}$	$L\oplus L\oplus K_S$	${𝒪}_S\left(\frac{m+1}{2}\Sigma \right)\otimes K_S^{(n+1)/2}$	$K_S$	$𝒪$
$D_n$	$U(1)$	$D_{n+1}$	$y^2+x^{2}z-{\alpha }^2{z}^{n-1}$	$(L\otimes K_S^{-1})\oplus L\oplus K_S^2$	${𝒪}_S(\Sigma )\otimes K_S^{n-1}$	$K_S^2$	${\pi }^{*}\left(K_S^{-1}\right)$
$E_6$	$A_2$	$E_8$	$y^2-x^3-{\alpha }^2{z}^4$	$L^2\oplus L^3\oplus (L\otimes K_S)$	${𝒪}_S(\Sigma )\otimes K_S^2$	$L\otimes K_S$	${\pi }^{*}\left(L^{-1}\right)$
$E_7$	$A_1$	$E_8$	$y^2+x^3-16\alpha x{z}^3$	$L^2\oplus L^3\oplus (L\otimes K_S)$	${𝒪}_S(\Sigma )\otimes K_S^3$	$L\otimes K_S$	${\pi }^{*}\left(L^{-1}\right)$
$E_8$	—	$E_8$	$y^2-x^3-\alpha z^5$	$L^2\oplus L^3\oplus (L\otimes K_S)$	${𝒪}_S(\Sigma )\otimes K_S^5$	$L\otimes K_S$	${\pi }^{*}\left(L^{-1}\right)$

Over $\Sigma$, there’s a $G_S\times G_{S\prime }$ principal bundle, obtained by restricting the respective principal bundles over $S$ and $S\prime$. Decomposing $$\mathrm{ad}(G_{\Sigma })=\mathrm{ad}(G_S)\oplus \mathrm{ad}(G_{S\prime })\oplus [\underset{a}{⨁}{U}_a\otimes U{\prime }_a]$$ The twisted theory on ${ℝ}^{3,1}\times \Sigma$ has complex bosons, ${\sigma }_a$, and left-handed fermions, ${\lambda }_{a\alpha }$, transforming as³ sections of $K_{\Sigma }^{1/2}\otimes {𝒰}_a\otimes {𝒰}_a\prime$, and their Hermitian conjugates. The equations for a supersymmetric background are $$D^{(0,1)}\sigma =D^{(0,1)}{\lambda }_{\alpha }=0$$ As before, we take the background gauge field on each 7-brane to be an anti-self dual connection for subgroups $H_S\subset G_S$ and $H_{S\prime }\subset G_{S\prime }$. Under $G_S={\Gamma }_S\times H_S$ and $G_{S\prime }={\Gamma }_{S\prime }\times H_{S\prime }$, we decompose $$\begin{array}{rl}{𝒰}_a& =\underset{n}{⨁}{R}_n\otimes {𝒱}_{an}\\ 𝒰{\prime }_a& =\underset{n}{⨁}R{\prime }_n\otimes 𝒱{\prime }_{an}\\ \end{array}$$ So we obtain 4D massless chiral multiplets, $$(\sigma ,{\lambda }_{\alpha })\in H^0(\Sigma ,K_{\Sigma }^{1/2}\otimes {𝒱}_{an}\otimes 𝒱{\prime }_{an})$$ in the representation $(R_n,R{\prime }_n)$ of ${\mathrm{Gamma}}_S\times {\mathrm{Gamma}}_{S\prime }$.

These have Yukawa couplings with the “bulk” fields discussed previously, given by the trilinear form

As a more refined version of the previous example, consider intersecting 7-branes, with $$G_S=\mathrm{SO}(12),H_S=U(1),G_{S\prime }=\mathrm{SU}(2),H_{S\prime }=U(1),G_{\Sigma }=E_7$$ Under $E_7\supset \mathrm{SO}(12)\times \mathrm{SU}(2)$, the adjoint decomposes as $$133=(66,1)\oplus (1,3)\oplus (32,2)$$ and decomposing further under $\mathrm{SO}(12)\supset \mathrm{SO}(10)\times U(1)$ $$\begin{array}{r}66={45}_0+1_0+{10}_2+{10}_{-2}\\ 32& ={16}_1+{\overline{16}}_{-1}\end{array}$$ So we have 4D chiral multiplets in representations of $\mathrm{SO}(10)$

where the $10$s are actually the restrictions to $\Sigma$ of “bulk” modes on the 7-brane wrapped on $S$ and $ℒ,ℒ\prime$ are appropriately-chosen line bundles. The Yukawa couplings are obtained by applying the trilinear form (4) to the modes in (5).

Yet more interesting structure arises when $\Sigma$ is reducible, $\Sigma ={\bigcup }_i{\Sigma }_i$, and one gets further contributions from the intersections, ${\Sigma }_i\cap {\Sigma }_j$.

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