I read a little bit the survey: http://www.claymath.org/sites/default/files/hodge.pdf
Well I understand its a conjecture in Algebraic Geometry, so it seems there should be some consequences of proving this conjecture in mathematical physics.
So are there any? and what are they?
Thanks.
There are connections to string theory as explained in Open Strings and Extended Mirror Symmetry, by Johannes Walcher (2007). In one sentence: The Gromov-Witten theory of a Calabi-Yau manifold is solved by the Hodge theory of its mirror symmetric manifold.
A more elaborate quote from this 2011 conference:
The mathematical theory that describes how integrals and differential equations control the shape of algebraic spaces in various dimensions is known as Hodge theory. The most important conjecture in algebraic geometry -- the Hodge Conjecture -- can be thought of as "a metaphor for transforming transcendental computations into algebraic ones." The physical theory able to describe the universe at both micro- (quantum mechanics) and macro- (general relativity) scales, and at the same time thought to be a suitable candidate for unifying all known forces of nature, is string theory. There are several variants of this "theory of everything," linked by dualities which can radically alter mathematical formulations while preserving physical predictions. String dualities thus imply conjectures: seemingly unrelated pieces of mathematics must be related since they offer different descriptions of the same physical world. Although a role for Hodge theory in string theory has been hinted at for some time, only very recently has the depth and precision of this relationship begun to emerge. Recent results suggest that a mathematical "grand unification" relating arithmetic geometry and symplectic geometry is taking shape.
