Algebra-geometry equivalence in string theory

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The discussion centers on the equivalence between geometric structures, specifically manifolds, and algebraic structures related to associative algebras in the context of string theory. Mirror symmetry is highlighted as a key concept, representing a relationship between these structures, although it traditionally pertains to manifolds alone. The conversation also touches on the connection to category theory through the definition of composition in one of the structures. The original poster seeks literature to better understand these concepts, particularly in relation to geometric Langlands. Overall, the thread emphasizes the intricate relationships between algebra, geometry, and string theory.
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I am looking for literature on a certain topic in mathematics inspired by string theory of which I have heard bits and pieces. Since I am not at all familiar with string theory and haven't found anything online, I was hoping someone more knowledgeable might recognize some of the keywords I remember.

The most important point was that there was a certain equivalence between geometric structures and algebraic ones. The geometric structures were manifolds and the algebraic structures were related to associative algebras, if I'm not mistaken. The manifolds could be used to represent some state of a string. The term "mirror symmetry" was also mentioned. From what I gathered mirror symmetry was this equivalence between geometric and algebraic structures, but from Wikipedia I understand that in the context of string theory "mirror symmetry" refers to a relation between manifolds only. For what its worth: there was also a connection to category theory by defining some kind of composition on one of the structures.

I apologize for the vagueness. Hopefully, my description will ring a bell for someone.
 
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