what are the halachic implications of forgetting to put a talis katan when getting an aliyah during a fast day? when it should have been done and was forgotten? I am sorry to ask this question since it agonizes me very much. How can do teshuvah for this mistake.

There are no particular connections between a fast and talit katan, nor between TK and an aliya. It is forbidden for a person not fasting to get an aliyah on afast day, even if thwe person is allowed to eat (e.g. for health reasons). Wearing a talit katan is not an absolute obligation, though there are opinions that there is a positive rabbinic commandment to wear a TK. From the Torah it is an obligation to place tzitzit on a garment if it has 4 corners and you are wearing it. I suggest that you resolve to study the laws of tzitzit snd become familiar with different circumstances, but certainly there is no need to feel distress over the fast.