Musings

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

where the normal bundle $N_{S|B}=K_S^p$, 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 $\{f=0\}\subset Tot(V\to S)$

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

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 $$\varphi \in \Omega^{(2,0)}(S,ad(P)),\quad \overline{\varphi} \in \Omega^{(0,2)}(S,ad(P))$$ where $P\to S$ is a principal $G_S$-bundle. The left-handed fermions are $$\begin{aligned} \eta_\alpha &\in \Gamma(ad(P))\\ \psi_\alpha &\in \Omega^{(0,1)}(S,ad(P))\\ \chi_\alpha &\in \Omega^{(2,0)}(S,ad(P)) \end{aligned}$$ and their right-handed conjugates are $$\begin{aligned} \overline{\eta}_{\dot{\alpha}} &\in \Gamma(ad(P))\\ \overline{\psi}_{\dot{\alpha}} &\in \Omega^{(1,0)}(S,ad(P))\\ \overline{\chi}_{\dot{\alpha}} &\in \Omega^{(0,2)}(S,ad(P)) \end{aligned}$$ The conditions for a supersymmetric solution are

Here, $$F_S^{(2,0)} = {\left(D^{(1,0)}_S\right)}^2,\quad F_S^{(0,2)} = {\left(D^{(0,1)}_S\right)}^2,\quad F_S= \{D^{(1,0)}_S,D^{(0,1)}_S\}$$ 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 $\varphi=0$ (or, more generally, $[\varphi, \overline{\varphi}]=0$), (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 $$ad(P) \simeq \oplus_i (R_i \otimes \mathcal{T}_i)$$ where $R_i$ are irreps of $\Gamma_i$ and $\mathcal{T}_i$ 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 $S= \mathbb{F}_n$, the $n$^th Hirzebruch surface², $G_S= E_6$, $H_S = U(1)$ and the unbroken gauge group, $\Gamma_S= SO(10)$. That is, we have a nontrivial connection, satisfying the Donaldson-Uhlenbeck-Yau equation, for some line bundle $\mathcal{L}\to S$. Decomposing $$78 = 45_0 + 1_0 + 16_{-3} +\overline{16}_3$$ We get chiral multiplets in the $16$ from $H^1(S,\mathcal{L}^{-3})$, and $\overline{16}$s from $H^1(S,\mathcal{L}^{3})$. Their number is given by the index theorem. In the notation of the footnote, $$\begin{aligned} n_{16} &= (3b-1)\left[(3b+1) -\tfrac{3}{2}(2a+(2-n) b)\right]\\ n_{\overline{16}} &= (3b+1)\left[(3b-1) -\tfrac{3}{2}(2a+(2-n) b)\right] \end{aligned}$$

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

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 $R^{3,1}\times S'$, where $S'=Tot(K_S^p\to \Sigma)$.

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

Local model for intersecting ADE 7-branes

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

Over $\Sigma$, there’s a $G_S\times G_{S'}$ principal bundle, obtained by restricting the respective principal bundles over $S$ and $S'$. Decomposing $$ad(G_\Sigma) = ad(G_S) \oplus ad(G_{S'}) \oplus \left[\bigoplus_a U_a \otimes U'_a\right]$$ The twisted theory on $\mathbb{R}^{3,1}\times\Sigma$ has complex bosons, $\sigma_a$, and left-handed fermions, $\lambda_{a\alpha}$, transforming as³ sections of $K_\Sigma^{1/2}\otimes \mathcal{U}_a\otimes \mathcal{U}_a'$, 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 $H_{S'}\subset G_{S'}$. Under $G_S = \Gamma_S\times H_S$ and $G_{S'}= \Gamma_{S'}\times H_{S'}$, we decompose $$\begin{aligned} \mathcal{U}_a &= \bigoplus_n R_{n} \otimes \mathcal{V}_{a n}\\ \mathcal{U}'_a &= \bigoplus_n R'_{n} \otimes \mathcal{V}'_{a n}\\ \end{aligned}$$ So we obtain 4D massless chiral multiplets, $$(\sigma,\lambda_\alpha) \in H^0(\Sigma, K_\Sigma^{1/2} \otimes \mathcal{V}_{a n}\otimes \mathcal{V}'_{a n})$$ in the representation $(R_{n}, R'_{n})$ of $Gamma_S\times Gamma_{S'}$.

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 $$G_S = SO(12),\, H_S=U(1),\quad G_{S'}= SU(2),\, H_{S'}=U(1),\quad G_\Sigma = E_7$$ Under $E_7\supset SO(12)\times SU(2)$, the adjoint decomposes as $$133 = (66,1)\oplus (1,3)\oplus (32,2)$$ and decomposing further under $SO(12)\supset SO(10)\times U(1)$ $$\begin{aligned} 66 = 45_0 + 1_0 + {10}_2 + {10}_{-2}\\ 32 &= {16}_1 +\overline{16}_{-1} \end{aligned}$$ So we have 4D chiral multiplets in representations of $SO(10)$

where the $10$s are actually the restrictions to $\Sigma$ of “bulk” modes on the 7-brane wrapped on $S$ and $\mathcal{L},\mathcal{L}'$ 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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