Apple has acknowledged a bug with Mavericks and Apple Mail, where for some email providers, new incoming email messages only come in when Mail is first launched… and no new mail is received until Mail is quit and relaunched again. Gmail has been reported to be one of these providers.
Apple’s 10.9.2 update claims to “resolve an issue that prevented Mail from receiving new messages from certain providers”, however people that have updated have reported on the Apple Discussion forums that they are still experiencing this problem.
Apple does not have a fix for this issue, but they do provide a manual workaround for receiving new mail without quitting and relaunching the Mail app. There workaround is detailed in this Knowledgebase article: http://support.apple.com/kb/TS5367