Week 8

作者: 悟空金月饺子 | 来源:发表于2021-04-26 01:58 被阅读0次

K. Costello, D. Gaiotto and J. Yagi, "Q-operator are 't Hooft lines"

还是这个文章，有一些想看懂的地方，但是还是有很多不懂。

We define a 't Hooft line such the the field strength on the $S^2$ surronding this line appraoches Dirac monopole which describes a G-bundle on $S^2$ of first Chern class k, up to some appropriate class of gauge transformation. That means the field strength takes the form
$F\sim |x|^{-3} \epsilon_{ijk}x_i dx_j dx_k \rho+\text{regular temrs},$
where $\rho$ is an element of $\mathfrak{g}$ .

The upshot is that 't Hooft line describes some singluar field configuratation. To describe the Dirac monopole more conveneintly in the 4DCS theory instead of considering the unit two-sphere we consider a a region which is the boundary of the solid cylinder: $|z|\leq\epsilon$ and $|y|\leq \epsilon$ . (So we assume the line is along the $x$ --direction.)

The boundary can be divided into three regions (From the point of view the 2-sphere, we introduce two pathes around the N pole and S pole and there is an overlapping region around the equator):

$y=-\epsilon,\quad |z|\leq \epsilon,$
$-\epsilon\leq y\leq \epsilon,\quad z=\epsilon,$
$y=\epsilon,\quad |z|\leq \epsilon.$

A solution to the equation of motion (modulo the gauge transformation)\footnote{Recall that the equation of motion is simply the flatness condition.} for the 4DCS theory on this region is described by:

A holomorphic bundle on the discs $|z|\leq \epsilon$ at $y=\pm\epsilon$
An isomorphism between these two bundles restricted to $|z|=\epsilon$ , The isomorphism is provided by parallel transport in the $y$ --direction.

Trivializing the holomorphic bundles at $y=\pm \epsilon$ (Just like we trivialize the bundles at the disorder defects), the parallel transport (which will form the moduli space after remove the gauge symmtry.) is an element of the loop group $LG$ of the complex group $G$ . The gauge transoformation at the top and bottom boundaries is not $LG$ because the gauge transformation should be able to extend to the center of the disc therefore the guage transformation is $L_+G$ . The moduli space of solution is then to be
$L_+G\ G / L_+G \sim G((z))/G[[z]],$
where $G((z))$ and $G[[z]]$ are Laurent series and Taylor series valued in $G$ , respectivly. The coset space $G((z))/G[[z]]$ is known as the affine Grassmannian.

Therefore, we can think of the 't Hooft line is associated with a singular guage transformation $z^\mu$ , where $\mu$ is a coweight.

't Hooft line from surface defects

The idea is similar to engineer 2d integrable field theories by inserting defects. First let us define the topological surface defect.\par
Let $X$ be a complex manifold with a $G$ --action. We consider the analytically-continued version of Poisson $\sigma$ --model with target space $X$ . The fields of the theory are a map $\sigma:\mathbb{R}^2\rightarrow X$ and a one-form $\eta\in \Omega^1(\mathbb{R}^2,\sigma^\star TX)$ . The Lagrangian is

$\int_{\mathbb{R}^2\times 0} \eta d\sigma$

The field $\eta$ has a guage transformation
$\eta \mapsto \eta+d\chi,\quad \chi\in \Omega^0(\mathbb{R}^2,\sigma^\star TX).$
This defect theory can be minimal coupled 4DCS by promoting the derivative to covariant deriavative. For example, if we insert the surface defect at $z=0$ , the Lagrangian for the coupled theory is simply
$\int_{\mathbb{R}^2\times 0}\eta\wedge \text{d}_A \sigma.$
Let $V_i^a$ be the holomorphic vector fields on $X$ corresponding to the action of $\mathfrak{g}$ then the Lagrangian is
$\int_{\mathbb{R}^2\times 0}( \eta d\sigma+\eta^i(\sigma^\star V_i^a)A_a )$
The gauge transformation for $\eta$ is now given the covariant derivative
$\delta\eta=\eta+\text{d}_A\chi.$
There is also bulk gauge transformation？(Not clear the meaning "the bulk gauge transformation act on the fields in the obvious way.")

But we can use this gauge symmetry to fix $\sigma$ to be a constant and we also assume that the stabilizer of apoint in $X$ is a parabolic subgroup $P \subset G$ then the tangent space of $X$ at x is the quotient $\mathfrak{g}/\mathfrak{p}$ . If the algebra has a triangular decomposition as
$\mathfrak{g}=\mathfrak{n}^{-}\oplus \mathfrak{l}\oplus \mathfrak{n}^+$
we can veiw the field $\eta$ as a one-form valued in $\mathfrak{n}^{-}$ . In this gauge, the Lagrangian takes the form
$\int_{\mathbb{R}^2\times0} \eta_i A_i^++\int_{\mathbb{R}^2\times \mathbb{C}}dz CS(A)$

Varing $A$ we get the equation of motion
$\eta_i\delta_{z=0}+dz F(A)_i^-=0.$

Choosing the gauge such that $A_{\bar{z}}=0$ , these equations are given by
$\p_z A_x^{-}+\eta_x=\p_z A_y^-+\eta_y=0.$
Thus on the surface defects the gauge fields satisfy: $A^+$ has a zero and $A^{-}$ has a pole at $z=0$ .

Week 8

't Hooft line from surface defects

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