I am a Physics undergrad who is interesting in pursuing a PhD in pure maths in the future (algebraic geometry/topology) but I am a bit unsure. My question is quite general, and I don't wish to provide more background for fear of bias in the answers. My question is it is advisable/possible/unfavorable/favorable to apply for a PhD in a field, different from that in which you have done most of your undergraduate research. My research (includes just reading and understanding papers, writing summaries until now, I havent published anything) mostly includes Quantum field theory, and gauge theory. Would the selection committe turn down an application to a pure maths field, if I have no research experience whatsoever?
I would also like to ask the question other way round. What if I concentrate my undergraduate research SOLELY on topics in pure maths such as algebraic geometry/topology and take other physics courses, would I be able to apply to a string theory phd with a high chance of success?
Should I consider spending time on both of these (which is almost an impossible task), to improve my chances in both the areas or would research in one area, and grad level courses in the other suffice?
In brief, Should the PhD field you are applying to be the same as your undergraduate research area?