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  Instanton Moduli Space with a Surface Operator

+ 12 like - 0 dislike

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 ${\bf C}\subset{\bf S}$.

[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].)

This post has been migrated from (A51.SE)

asked Oct 7, 2011 in Theoretical Physics by Satoshi Nawata (345 points) [ revision history ]
edited Dec 31, 2015 by Dilaton
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

This post has been migrated from (A51.SE)
Oh, I didn't know that. Thanks for enlightening me, Yuji.

This post has been migrated from (A51.SE)

1 Answer

+ 8 like - 0 dislike

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.

This post has been migrated from (A51.SE)
answered Oct 7, 2011 by Alexander Braverman (580 points) [ no revision ]
Thank you very much. This is exactly the answer I wanted. It is such an honor to have your response.

This post has been migrated from (A51.SE)
You are welcome. If you have any further questions, I'll be happy to try answer.

This post has been migrated from (A51.SE)

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