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The base of any algebraically completely integrable system has canonically defined special geometry:
I asked GPT, and according to GPT, special Kaehler manifolds (which correspond to the definition in the entry) are just a particular class of special manifolds. Let me reproduce the answer here
(GPT said)
Special geometry is a concept related to complex and symplectic geometry, often used in the context of supergravity and string theory in mathematical physics. Special geometry manifolds are not characterized by their holonomy group but by the geometric structures they possess, particularly Kähler and symplectic structures. These structures are related to the underlying mathematical structure of certain physical theories. In the context of supergravity theories and string theory, special geometry manifolds are associated with scalar manifolds that appear in the formulation of the theory. The moduli space of scalar fields in supergravity theories is often a special geometry manifold, where the metric encodes the kinetic terms for these scalar fields. The most well-known example of special geometry is the concept of a special Kähler manifold, which is used to describe the moduli space of scalar fields in N=2 supergravity theories. Special Kähler manifolds have specific constraints on their metric and connection that arise from the supersymmetric structure of the theory.
Special geometry manifolds are associated with the geometric structures, often Kähler and symplectic, used to describe the moduli spaces of scalar fields in certain physical theories, particularly in the context of supergravity and string theory.
I am not feeling competent to reconcile this with the current entry entirely at this moment, maybe somebody can give an advice.
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Conversely, any special Kähler manifold is so related to an integrable system. (just a note, as I am leaving offline now)
Thanks for bringing this up. In Strominger’s paper, what is currently known as special Kähler manifold is referred to as special manifold. Nowadays the “special” condition is taken to be a (weaker) structure on a manifold, so that special Kähler is a Kähler structure compatible in some way with the special structure. Phrased in this way, one can then consider making other kinds of structures like para-Kahler or hyperKahler compatible with special condition, and the implications of taking these spaces as target spaces. Seems 1909.06240 explains this in detail, will try to update when I get the chance.
I am not feeling competent to reconcile this with the current entry entirely at this moment, maybe somebody can give an advice.
Don’t use LLMs. They’re perfectly fine for producing for producing entirely grammatical, convincing sounding synthetic text which is at best meaningless and at worst completely false (and this is well-documented by now; you can get the same quality output with a lower carbon footprint from snarXiv).
We show equivalences of some definitions and give examples which show that earlier definitions are not equivalent, and are not sufficient to restrict the Kähler metric to one that occurs in N = 2 supersymmetry.
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