Sony Vaio Duo 13
Legacy BIOS Boot Mode
The Duo 13 does not support legacy BIOS boot mode.
UEFI Boot Mode
The Duo 13 only supports UEFI boot.
You may have noticed that the Arch install media won't boot at all, and that even if you try to add a "nomodeset" parameter to the boot line it still won't work. Well it turns out that gummiboot won't work at all with the Duo 13, period. You'll have to create a custom install media using GRUB as your UEFI boot loader. Once you've installed Arch Linux onto the SSD drive, you'll have to chainload the SSD from the USB stick (which means you'll have to have your USB "key" plugged in when you power on in order to boot the main OS on the SSD drive).
The wireless WiFi card and the touchpad are connected to the motherboard via the newer SDIO interface (not USB or PCI), so the lspci and lsusb commands will not show you that those two pieces of hardware they exist.
This isn't solved yet, so you're best bet right now is to use an external USB mouse. If you figure it out, please update the Wiki here (and many thanks for doing so!).
This isn't solved yet either, so a recommendation would be to run to your local store (e.g. Walmart) and buy a cheap USB wifi dongle. The Netgear N150 works right out of the box with stock kernel drivers and no need to install any drivers manually.