based on the Mishneh Torah instead of the Shulchan Aruch. They are not affected by chasidut and their main rabbi, Rabbi Yihhyah Qafahh, has written against Kabbalah and the Zohar.
What is wrong with Shulhan Aruch? And what is wrong with the Zohar? I thought these were part of the Jewish tradition. Didn't Chaim talk positively about the Kabbalah?
I don't think there is necessarily anything wrong with the Shulchan Aruch, but it was based largely on the Mishneh Torah and the author of the Shulchan Aruch said he did not intend to stop people from practicing according to the Mishneh Torah (written by Rambam), which many people used to do and the Yemenites still do. Mishneh Torah also encompasses all of Jewish law whereas the Shulchan Aruch only concentrates on laws applicable during the exile. I feel the less influence by the modern movements the better, but I respect the Shulchan Aruch. I love studying the Mishneh Torah and my interest in it is second only to the Tanach.
I personally don't agree with the Zohar or what is commonly known as Kabbalah. There is apparently a secret knowledge ("sod") involved with Torah ("ma'ase breshit", etc.), but this is not available to the layman and certainly isn't the new age type stuff you see when you pick up a book on Kabbalah. Zohar authorship is extrapolated back onto ancient authors and was actually written much later than it claims to be written--kind of like there is one Mormon book that Mormon's claim Abraham wrote. I'm not sure if Chaim discussed Kabbalah at length, but I would agree with the followers of Rabbi Yihhyah Qafahh on this issue.
I do not know if you saw Rabbi Bar Hayyim's interview with Voz Iz Neias, but he does a great job explaining the deficiencies of the Shulchan Aruch, as well as its true intentions.
He received Semicha from Rabbi Kapach, and is very knowledgable in the practices and educational traditions of Yemenite Jews. He also talks of the primacy of Maimonides.
Nevertheless, Rabbi Kapach was not so much against Kabbalah as he was against the fact that so many people were trying to learn it. Kabbalah was ruining the Jewish community in Yemen because people tried to read it, yet lacked any of the prerequisites for genuine comprehension of the material. Rabbi Kapach himself did not promote Kabbalah, but I would guess he would not mind that someone who is very well-learned such as Rabbi Elyashiv studies Kabbalah.
On the other hand, for the same reason that Kabbalah is only meant for those who are incredibly well-learned (My guess is that maybe 20 or 30 people in the entire world have a firm grip on the teachings of Kabbalah), I cannot pass judgment on its merits for the fact that I am not a Torah Scholar. Unless one is a Gadol BTorah, I don't give any credence to what they say about Kabbalah, and that includes myself. Any judgment or opinion I could give regarding the Zohar and Kabbalah is meaningless and bears now weight.
For this reason I cannot formulate any opinion regarding something I completely do not understand.