I would like to understand the mathematical language which is relevant to instanton moduli space with a surface operator.

Alday and Tachikawa stated in 1005.4469 that the following moduli spaces are isomorphic.

- the moduli space of ASD connections on $\mathbb{R}^4$ which are smooth away from $z_2=0$ and with the behavior $A\sim (\alpha_1,\cdots,\alpha_N)id\theta$ close to $r\sim 0$ where the $\alpha_i$ are all distinct and $z_2=r\exp(i\theta)$. (Instanton moduli space with a full surface operator)
- the moduli space of stable rank-$N$ locally-free sheaves on $\mathbb{P}^1\times \mathbb{P}^1$ with a parabolic structure $P\subset G$ at $\{z_2=0\}$ and with a framing at infinities, $\{z_1=\infty\}\cup\{z_2=\infty\}$. (Affine Laumon space)

I thought the moduli space ${\rm Bun}_{G,P}({\bf S}, {\bf D}_\infty)$ in [B] also corresponds to the instanton moduli space with a surface operator. Note that ${\rm Bun}_{G,P}({\bf S}, {\bf D}_\infty)$ is the moduli space of principal $G$-bundle on ${\bf S}=\mathbb{P}^2$ of second Chern class $-d$ endowed with a trivialization on ${\bf D}_\infty$ and a parabolic structure $P$ on the horizontal line ${\bf C}\subset{\bf S}$.

[B] http://arxiv.org/abs/math/0401409

However, [B] considers the moduli space of parabolic sheaves on $\mathbb{P}^2$ instead of $\mathbb{P}^1\times \mathbb{P}^1$. What in physics does ${\rm Bun}_{G,P}({\bf S}, {\bf D}_\infty)$ correspond to? Is it different from the affine Laumon space?

In addition, I would like to know the relation between [B] and [FFNR].

[FFNR] http://arxiv.org/abs/0812.4656

Do \mathfrak{Q}*{\underline d} and $\mathcal{Q}_{\underline d}$ in [FFNR] correspond to $\mathcal{M}_{G,P}$ and $\mathcal{QM}_{G,P}$ in the section 1.4 of [B]? (Sorry, this does not show \mathfrak properly. \mathfrak{Q}*{\underline d} is the one which appears the first line of the section 1.1 in [FFNR].)

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