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Chern-Weil theory for ∞-local systems

Acceso Cerrado

Idioma: Inglés

Publicado: 25/01/2024

APC (est): No disponible

Abstract:

Let <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> be a compact connected Lie group with Lie algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German g"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">g</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {g}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We show that the category <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper L times bold o times bold c Subscript normal infinity Baseline left-parenthesis upper B upper G right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">L</mml:mi> <mml:mi mathvariant="bold">o</mml:mi> <mml:mi mathvariant="bold">c</mml:mi> </mml:mrow> </mml:mrow> <mml:mi mathvariant="normal">∞</mml:mi> </mml:msub> <mml:mo>⁡</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>B</mml:mi> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {\mathbf {Loc}} _\infty (BG)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal infinity"> <mml:semantics> <mml:mi mathvariant="normal">∞</mml:mi> <mml:annotation encoding="application/x-tex">\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-local systems on the classifying space 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> can be described infinitesimally as the category <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper I bold n bold f bold upper L bold o bold c Subscript normal infinity Baseline left-parenthesis German g right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">I</mml:mi> <mml:mi mathvariant="bold">n</mml:mi> <mml:mi mathvariant="bold">f</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">L</mml:mi> <mml:mi mathvariant="bold">o</mml:mi> <mml:mi mathvariant="bold">c</mml:mi> </mml:mrow> </mml:mrow> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">g</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{\operatorname {\mathbf {Inf}\mathbf {Loc}}} _{\infty }(\mathfrak {g})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of basic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German g"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">g</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {g}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Subscript normal infinity"> <mml:semantics> <mml:msub> <mml:mi>L</mml:mi> <mml:mi mathvariant="normal">∞</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">L_\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> spaces. Moreover, we show that, given a principal bundle <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi colon upper P right-arrow upper X"> <mml:semantics> <mml:mrow> <mml:mi>π</mml:mi> <mml:mo>:</mml:mo> <mml:mi>P</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\pi \colon P \to X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with structure group <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> and any connection <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="theta"> <mml:semantics> <mml:mi>θ</mml:mi> <mml:annotation encoding="application/x-tex">\theta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P"> <mml:semantics> <mml:mi>P</mml:mi> <mml:annotation encoding="application/x-tex">P</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, there is a differntial graded (DG) functor <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper C script upper W Subscript theta Baseline colon bold upper I bold n bold f bold upper L bold o bold c Subscript normal infinity Baseline left-parenthesis German g right-parenthesis long right-arrow bold upper L bold o bold c Subscript normal infinity Baseline left-parenthesis upper X right-parenthesis comma"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="script">C</mml:mi> <mml:mi mathvariant="script">W</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>θ</mml:mi> </mml:mrow> </mml:msub> <mml:mo>:</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">I</mml:mi> <mml:mi mathvariant="bold">n</mml:mi> <mml:mi mathvariant="bold">f</mml:mi> </mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">L</mml:mi> <mml:mi mathvariant="bold">o</mml:mi> <mml:mi mathvariant="bold">c</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">g</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">⟶</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">L</mml:mi> <mml:mi mathvariant="bold">o</mml:mi> <mml:mi mathvariant="bold">c</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\begin{equation*} \mathscr {CW}_{\theta } \colon \mathbf {Inf}\mathbf {Loc}_{\infty }(\mathfrak {g}) \longrightarrow \mathbf {Loc}_{\infty }(X), \end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> which corresponds to the pullback functor by the classifying map of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P"> <mml:semantics> <mml:mi>P</mml:mi> <mml:annotation encoding="application/x-tex">P</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The DG functors associated to different connections are related by an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Subscript normal infinity"> <mml:semantics> <mml:msub> <mml:mi>A</mml:mi> <mml:mi mathvariant="normal">∞</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">A_\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-natural isomorphism. This construction provides a categorification of the Chern-Weil homomorphism, which is recovered by applying the functor <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper C script upper W Subscript theta"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="script">C</mml:mi> <mml:mi mathvariant="script">W</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>θ</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mathscr {CW}_{\theta }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to the endomorphisms of the constant <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal infinity"> <mml:semantics> <mml:mi mathvariant="normal">∞</mml:mi> <mml:annotation encoding="application/x-tex">\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-local system.

Tópico:

Quantum chaos and dynamical systems

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Información de la Fuente:

FuenteTransactions of the American Mathematical Society	Cuartil año de publicaciónNo disponible	VolumenNo disponible
IssueNo disponible	PáginasNo disponible	pISSNNo disponible
ISSN1088-6850	Perfil OpenAlexhttps://openalex.org/S60030702

Enlaces e Identificadores:

Scienti ID	0000154881-24	Scienti URL	https://www.ams.org/journals/tran/2024-377-05/S0002-9947-2024-09068-2/home.html	Doi URL	https://doi.org/10.1090/tran/9068
Openalex URL	https://openalex.org/W4391231476

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