Yang-Mills connections on orientable and nonorientable surfaces

Abstract

In “ <italic>The Yang-Mills equations over Riemann surfaces</italic> ”, Atiyah and Bott studied Yang-Mills functional over a Riemann surface from the point of view of Morse theory. In “ <italic>Yang-Mills Connections on Nonorientable Surfaces</italic> ”, we study Yang-Mills functional on the space of connections on a principal <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G Subscript double-struck upper R"> <mml:semantics> <mml:msub> <mml:mi>G</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">G_{\mathbb {R}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -bundle over a closed, connected, nonorientable surface, where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G Subscript double-struck upper R"> <mml:semantics> <mml:msub> <mml:mi>G</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">G_{\mathbb {R}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is any compact connected Lie group. In this monograph, we generalize the discussion in “ <italic>The Yang-Mills equations over Riemann surfaces</italic> ” and “ <italic>Yang-Mills Connections on Nonorientable Surfaces</italic> ”. We obtain explicit descriptions of equivariant Morse stratification of Yang-Mills functional on orientable and nonorientable surfaces for non-unitary classical groups <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S upper O left-parenthesis n right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">SO(n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S p left-parenthesis n right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>p</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">Sp(n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . When the surface is orientable, we use Laumon and Rapoport’s method in “ <italic> The Langlands lemma and the betti numbers of stacks of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -bundle on a curve </italic> ” to invert the Atiyah-Bott recursion relation, and write down explicit formulas of rational equivariant Poincaré series of the semistable stratum of the space of holomorphic structures on a principal <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S upper O left-parenthesis n comma double-struck upper C right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">SO(n,\mathbb {C})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -bundle or a principal <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S p left-parenthesis n comma double-struck upper C right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>p</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">Sp(n,\mathbb {C})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -bundle.