I figured out the problem. We were not pulling in the jQuery UI CSS file, and I had a global transition on "a" elements that was clashing with the UI handle styles.
We have installed the Ajax Filters extension on a site we are developing, and most of it seems to be working fine, but we are having trouble with the price range slider.
As you can see, there seem to be some styles missing to make the slider handles look right. Based on your demo site (http://demos.nop-templates.com/shoes?demo=filters), the .ui-slider parent element is missing a position: relative style, and the .ui-slider-handle elements are missing position: absolute. The behavior of our slider is not working correctly, either. If I drag the handles the range bar anchors to the left instead of centering between the handles, and the handles don't move smoothly as I drag them.
I'm guessing a script is not getting applied somehow, but I can't figure out what it is. I can't give out our client's development link in a public forum, so I hope this question provides enough information.