Rinse the mint and thyme and pat dry. Rinse 3 limes and cut into thin slices.
Add the lime slices and the herbs to a bowl. Cover them with the rum and the Gewürztraminier to the bowl. Cover and chill in the refrigerator for 1 hour.
Pass the herb mixture through a fine strainer into a clean bowl, then fill with the sparkling wine. Cut the remaining limes into slices, then add to the punch. Remove the leaves from the lemon balm, add to the punch, and serve.