Line defects and 5d instanton partition functions

We consider certain line defect operators in five-dimensional SUSY gauge theories, whose interaction with the self-dual instantons is described by 1d ADHM-like gauged quantum mechanics constructed by Tong and Wong. The partition function in the presence of these operators is known to be a generating function of BPS Wilson loops in skew symmetric tensor representations of the gauge group. We calculate the partition function and explicitly prove that it is a finite polynomial of the defect mass parameter $x$, which is an essential property of the defect operator and the Wilson loop generating function. The relation between the line defect partition function and the qq-character defined by N. Nekrasov is briefly discussed.Comment: 17 pages, 1 figure; typos fixed, references corrected; version to be published in JHE

Weak type estimates on certain Hardy spaces for smooth cone type multipliers

Let $\varrho\in C^{\infty} ({\Bbb R}^d\setminus\{0\})$ be a non-radial homogeneous distance function satisfying $\varrho(t\xi)=t\varrho(\xi)$. For $f\in\frak S ({\Bbb R}^{d+1})$ and $\delta>0$, we consider convolution operator {\Cal T}^{\delta} associated with the smooth cone type multipliers defined by \hat {{\Cal T}^{\delta} f}(\xi,\tau)= (1-\frac{\varrho(\xi)}{|\tau|} )^{\delta}_+\hat f (\xi,\tau), (\xi,\tau)\in {\Bbb R}^d \times \Bbb R. If the unit sphere $\Sigma_{\varrho}\fallingdotseq\{\xi\in {\Bbb R}^d : \varrho(\xi)=1\}$ is a convex hypersurface of finite type and $\varrho$ is not radial, then we prove that {\Cal T}^{\delta(p)} maps from $H^p({\Bbb R}^{d+1})$, $0<p<1$, into weak-$L^p(\Gamma_{\gamma})$ for the critical index $\delta(p)=d(1/p -1/2)-1/2$, where $\Gamma_{\gamma}=\{(x,t)\in {\Bbb R}^d\times\Bbb R : |t|\geq\gamma |x|\}$ for $\gamma=\max\{\sup_{\varrho(\xi)\leq 1}|\xi|,1\}$. Moreover, we furnish a function $f\in\frak S({\Bbb R}^{d+1})$ such that \sup_{\lambda>0} \lambda^p|\{(x,t)\in \bar{{\Bbb R}^{d+1}\setminus\Gamma_{\gamma}} : |{\Cal T}_{\varrho}^{\delta(p)}f(x,t)|>\lambda\}|=\infty.Comment: 13 page