# Instanton Moduli Space with a Surface Operator

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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.

1. 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)
2. 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 ${\bf C}\subset{\bf S}$.

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].

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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edited Dec 31, 2015
To the readers who are interested in this subject, I would recommend to watch the following videos delivered by Braverman and Finkelberg. http://media.scgp.stonybrook.edu/video/video.php?f=20110321_4_branes_qtp.mp4 http://www.sms.cam.ac.uk/media/538617;jsessionid=9540827CB40AC9F1E61BF944127EBAF4

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Oh, I didn't know that. Thanks for enlightening me, Yuji.

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Let me try to answer. For your first question the statement is that you can work with either ${\mathbb P}^2$ or ${\mathbb P}^1\times {\mathbb P}^1$ - the moduli space is the same. More generally, if $S$ is any surface which contains ${\mathbb A}^2$ as an open subset and $D_{\infty}$ is the divisor at $\infty$ then $Bun_G(S,D_{\infty})$ is independent of $S$.

For the second question: it is true that ${\mathfrak Q}={\mathcal M}_{G,P}$ (for $P$ being the Borel subgroup and $G=SL(n)$) but it is not true that $Q={\mathcal QM}_{G,P}$. The point is that the quasi-maps' space ${\mathcal QM}_{G,P}$ is defined for any $G$ and it is singular; for $G=SL(n)$ (and only in that case) it has a nice resolution of singularities which is given by the Laumon space. If you are interested to know more, you can read my 2006 ICM talk ("Spaces of quasi-maps into the flag varieties and their applications") - the above questions are discussed there.

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answered Oct 7, 2011 by (570 points)
Thank you very much. This is exactly the answer I wanted. It is such an honor to have your response.

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You are welcome. If you have any further questions, I'll be happy to try answer.

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