The Sony Vaio touchpad is what they call a 'ClickPad'. There are no buttons and it's one big touch surface. Linux Mint 13 isn't setup to handle this out of the box so you have to do the following to fix the problem.

Create directory

  sudo mkdir /etc/X11/xorg.conf.d

Create file /etc/X11/xorg.conf.d/synaptics.conf

  sudo gedit /etc/X11/xorg.conf.d/synaptics.conf

Add the following to that file

 Section "InputClass"
        Identifier "touchpad catchall"
        Driver "synaptics"
        MatchIsTouchpad "on"
        MatchDevicePath "/dev/input/event*"
        Option "LeftEdge"               "2900"
        Option "RightEdge"              "4000"
        Option "TopEdge"                "1620"
        Option "BottomEdge"             "3800"
        Option "TapButton1"             "1"
        Option "TapButton2"             "3"
        Option "TabButton3"             "2"
        Option "FastTaps"               "1"
        Option "CornerCoasting"         "1"
        Option "CoastingSpeed"          "0.30"
        Option "PalmDetect"             "true"
        Option "VertTwoFingerScroll"    "true"
        Option "HorizTwoFingerScroll"   "true"
        Option "SHMConfig"              "on"
        Option "SendCoreEvents"
        Option "MaxDoubleTapTime"       "50"
        Option "LBCornerButton"         "1"
        Option "RBCornerButton"         "3"
        Option "FingerLow"              "7"
        Option "FingerHigh"             "8"
        Option "MaxTapTime"             "180"
        Option "MaxTapMove"             "110"
        Option "VertScrollDelta"        "20"
        Option "HorizScrollDelta"       "20"
        Option "MinSpeed"               "0.60"
        Option "MaxSpeed"               "1.10"
        Option "AccelFactor"            "0.030"
        Option "EdgeMotionMinpeed"      "200"
        Option "EdgeMotionMaxSpeed"     "200"
        Option "SoftButtonAreas"        "4000 0 3800 0 2000 3999 3800 0"
 EndSection

Save & reboot your computer.

Now you should have a left, middle & right button at the bottom of your touchpad and you should be able to drag and drop.