To make the simple syrup, combine water sugar and the squeezed limes into a sauce pan. Bring to a boil and stir until the sugar is completely dissolved. Remove from heat and take out the lime halves. Take the juice from the lime and muddle with the mint leaves and add this to the simple syrup and stir.

To make the cocktail combine the simple syrup, pomegranate juice and vodka in a cocktail shaker and shake well. Pour over ice into two glasses and top with soda water. Garnish with mint sprigs and pomegranates.