Is there timezone data for any Indian cities such as Mumbai, Dehli, &c.?

I looked and couldn't find any names of Indian cities that I recognized:
Dehli, New Dehli, Mumbai, Kolkata Bangalore, Madras, Chandigarh, Amritsar,
&c., &c., &c.  None of the timezones I found beginning with "Indian/" are
actually in India.

I have built an infrastructure for a web application that makes extensive
use of Datetime, and especially the Timezone module, so I now I have a
problem in that my colleague pointed out to me he couldn't find any Indian
timezone data.  But we now have clients in India and we have to support
them (and that means that my system for handling timezones is broken unless
I find a solution to properly support Indian timezones - i don't know - is
there more than one Indian timezone).  How do I do that without breaking,
or having to spend a lot of time finding a replacement for, my use of
Datetime::Timezone?

So, are Indian timezones supported, using a name I am not aware of, or is
there a way to add support for Indian timezones without breaking anything?

Thanks

Ted