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Infinite Analysis Seminar Tokyo
Seminar information archive ～09/07｜Next seminar｜Future seminars 09/08～
| Date, time & place | Saturday 13:30 - 16:00 117Room #117 (Graduate School of Math. Sci. Bldg.) |
|---|
2026/09/10
14:00-16:00 Room #123 (Graduate School of Math. Sci. Bldg.)
Motohico Mulase (UC Davis)
The SL(2) that relates Gaiotto's Conformal Limit and the Geometric Langlands Correspondence (English)
Motohico Mulase (UC Davis)
The SL(2) that relates Gaiotto's Conformal Limit and the Geometric Langlands Correspondence (English)
[ Abstract ]
The purpose of this talk is to show that Gaiotto's "conformal limit" construction of opers via the non-Abelian Hodge correspondence gives the same map that appears in the Geometric Langlands Correspondence for the case of opers. The resulting map is a unique canonical biholomorphic map from the moduli space of G-spectral curves to the moduli space of LG-opers, both defined over a smooth complex projective curve X with a complex simple algebraic group G and its Langlands dual LG.
The Structure of the Talk:
In the first hour of the talk I will present the main theorem and explain motivational materials, including:
・The differential equations interpreting the stability conditions in algebraic geometry of moduli spaces;
・Simpson's NAH map between the moduli spaces of Higgs bundles and connections on X;
・The moduli space of opers that forms a Lagrangian in the moduli space of local systems on X;
・From a spectral curve to an oper as a quantization procedure; and
・The idea of Gaiotto on taking a scaling limit of NAH that changes a transcendental topological map into a calculable algebraic map.
In the second hour, I will talk about the following subjects:
・The Geometric Langlands Correspondence for the case of opers due to Beilinson-Drinfeld;
・Kostant's TDS;
・The proof of Gaiotto's conformal limit for the case of sl(2);
・GLC for sl(2);
・The parallelism of these proofs going from sl(2) to the general case through Kostant's TDS.
The purpose of this talk is to show that Gaiotto's "conformal limit" construction of opers via the non-Abelian Hodge correspondence gives the same map that appears in the Geometric Langlands Correspondence for the case of opers. The resulting map is a unique canonical biholomorphic map from the moduli space of G-spectral curves to the moduli space of LG-opers, both defined over a smooth complex projective curve X with a complex simple algebraic group G and its Langlands dual LG.
The Structure of the Talk:
In the first hour of the talk I will present the main theorem and explain motivational materials, including:
・The differential equations interpreting the stability conditions in algebraic geometry of moduli spaces;
・Simpson's NAH map between the moduli spaces of Higgs bundles and connections on X;
・The moduli space of opers that forms a Lagrangian in the moduli space of local systems on X;
・From a spectral curve to an oper as a quantization procedure; and
・The idea of Gaiotto on taking a scaling limit of NAH that changes a transcendental topological map into a calculable algebraic map.
In the second hour, I will talk about the following subjects:
・The Geometric Langlands Correspondence for the case of opers due to Beilinson-Drinfeld;
・Kostant's TDS;
・The proof of Gaiotto's conformal limit for the case of sl(2);
・GLC for sl(2);
・The parallelism of these proofs going from sl(2) to the general case through Kostant's TDS.
