There is no one agreed upon definition of giftedness in academic fields. The State of North Carolina defines giftedness as:
Academically or intellectually gifted students perform or show the potential to perform at substantially high levels of accomplishment when compared with others of their age, experience, or environment. Academically or intellectually gifted students exhibit high performance capability in intellectual areas, specific academic fields, or in both the intellectual areas and specific academic fields. Academically or intellectually gifted students require differentiated educational services beyond those ordinarily provided by the regular educational program. Outstanding abilities are present in students from all cultural groups, across all economic strata, and in all areas of human endeavor.
--NC Article 9B (Legal definition of giftedness in NC) https://www.ncleg.gov/EnactedLegislation/Statutes/PDF/ByArticle/Chapter_115C/Article_9B.pdf
Is It a Cheetah?--essay by Stephanie Tolan about identifying and serving gifted students
"Bright Child" vs. "Gifted Learner" What's the Difference--Article contrasting the two
10 Myths & Facts About Gifted Students and Gifted Education
Neuroscience of Giftedness--Summaries of neuroscience research on giftedness