Spin(8, ℂ)-Higgs bundles fixed points through spectral data

Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>X</m:mi> </m:math> X be a compact Riemann surface of genus <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>g</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:math> g≥ 2 . The geometry of the moduli space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi class="MJX-tex-caligraphic" mathvariant="script">ℳ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi mathvariant="normal">Spin</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>8</m:mn> <m:mo>,</m:mo> <m:mi mathvariant="double-struck">C</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {{ { M} }}({Spin}(8,{C})) of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="normal">Spin</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>8</m:mn> <m:mo>,</m:mo> <m:mi mathvariant="double-struck">C</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {Spin}(8,{C}) -Higgs bundles over <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>X</m:mi> </m:math> X is of great interest both in algebraic geometry and mathematical physics. Consequently, several works have studied subvarieties of fixed points of this moduli space, especially those arising from the automorphism induced by the action of triality. In this work, fixed points of automorphisms of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi class="MJX-tex-caligraphic" mathvariant="script">ℳ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi mathvariant="normal">Spin</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>8</m:mn> <m:mo>,</m:mo> <m:mi mathvariant="double-struck">C</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {{ { M} }}({Spin}(8,{C})) induced by outer automorphisms of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="normal">Spin</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>8</m:mn> <m:mo>,</m:mo> <m:mi mathvariant="double-struck">C</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {Spin}(8,{C}) are studied. This is done by giving explicit descriptions of the spectral data of these fixed points induced by the Hitchin fibration, under certain technical conditions that must be required. Specifically, it is proved that stable <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="normal">Spin</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>8</m:mn> <m:mo>,</m:mo> <m:mi mathvariant="double-struck">C</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {Spin}(8,{C}) -Higgs bundles that admit nontrivial automorphisms reduce their structure group to a subgroup isomorphic to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="normal">SL</m:mi> <m:msup> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>2</m:mn> <m:mo>,</m:mo> <m:mi mathvariant="double-struck">C</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mn>4</m:mn> </m:mrow> </m:msup> <m:mo>=</m:mo> <m:mi mathvariant="normal">SL</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>2</m:mn> <m:mo>,</m:mo> <m:mi mathvariant="double-struck">C</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>×</m:mo> <m:mi mathvariant="normal">SL</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>2</m:mn> <m:mo>,</m:mo> <m:mi mathvariant="double-struck">C</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>×</m:mo> <m:mi mathvariant="normal">SL</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>2</m:mn> <m:mo>,</m:mo> <m:mi mathvariant="double-struck">C</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>×</m:mo> <m:mi mathvariant="normal">SL</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>2</m:mn> <m:mo>,</m:mo> <m:mi mathvariant="double-struck">C</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {SL}{(2,{C})}⁴={SL}(2,{C})× {SL}(2,{C})× {SL}(2,{C})× {SL}(2,{C}) . Subsequently, the spectral data of these reductions are described and the manner in which the outer automorphisms of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="normal">Spin</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>8</m:mn> <m:mo>,</m:mo> <m:mi mathvariant="double-struck">C</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {Spin}(8,{C}) act on them is analyzed. The final application of these results allows a description of the mentioned fixed points through their spectral data.