Virtual classes of parabolic <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msub><mml:mrow><mml:mi mathvariant="normal">SL</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="double-struck">C</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>-character varieties

González-Prieto, Ángel

doi:10.1016/j.aim.2020.107148

Cited by 12 publications

(16 citation statements)

References 28 publications

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“…In this section, we follow [14,16,18,45,20] to construct a Topological Quantum Field Theory (TQFT) which computes the classes of character stacks in the Grothendieck ring of stacks K(RStck/BG).…”

Section: Constructing the Stacky Tqftmentioning

confidence: 99%

“…Hence, this map is actually the multiplication map by a polynomial of Z[q] and this polynomial turns out to be precisely the E-polynomial of the representation variety e(R G (W )). This work has been substantially extended in [15] and [16], where the TQFT was adapted to work also in the parabolic setting, even for non-generic parabolic structures, in [17] to surfaces with conic singularities, in [46] for non-orientable surfaces and in [20] for G the group of upper triangular matrices of rank ≤ 4.…”

Section: Introductionmentioning

confidence: 99%

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Virtual Classes of Character Stacks

González-Prieto¹,

Hablicsek²,

Vogel³

2022

Preprint

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In this paper, we extend the Topological Quantum Field Theory developed by González-Prieto, Logares and Muñoz for computing virtual classes of representation varieties of closed orientable surfaces in the Grothendieck ring of varieties to the setting of the character stacks.To this aim, we define a suitable Grothendieck ring of representable stacks, over which this Topological Quantum Field Theory is defined. We apply this theory to the case of the affine linear group of rank 1, providing an explicit expression for the virtual class of the character stack of closed orientable surfaces of arbitrary genus. This virtual class remembers the natural adjoint action, and in particular from this we can derive the virtual class of the character variety.For this reason, the algebraic structure of the character variety χ G (Σ g ) has been deeply studied. For instance, when G is a complex group, the character variety itself is a complex algebraic variety so its cohomology is naturally endowed with a mixed Hodge structure. In this way, a natural question is to compute the E-polynomialwhich encodes the information of the Hodge numbers h k;p,q c (χ G (Σ g )) = dim C H k;p,q c (χ G (Σ g )) on the compactly supported cohomology of χ G (Σ g ). Three main approaches coexist in the literature to this aim: the so-called arithmetic, the geometric and quantum methods.

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Section: Constructing the Stacky Tqftmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Virtual Classes of Character Stacks

González-Prieto¹,

Hablicsek²,

Vogel³

2022

Preprint

Self Cite

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“…Therefore, it offers a systematic method that can be applied to more general contexts in which geometric or arithmetic methods fail. For instance, in [15], it is computed the virtual classes of SL 2 (C)-parabolic representation varieties in the general case by means of the quantum method. This result is unavailable using the geometric or the arithmetic approach due to very subtle interaction between the monodromies of the punctures that cannot be captured with the classical methods.…”

Section: 3mentioning

confidence: 99%

“…This leads to the third computational method, the quantum method, introduced in [12], that formalizes this set up and provides a powerful machinery to compute E-polynomials of character varieties. Moreover, this technique allows us to keep track of the classes in the Grothendieck ring of varieties (also known as virtual classes, as defined in section 2.4) of the representation varieties and had been successfully used in [14,15] in the parabolic context, in which we deal with punctured surfaces with prescribed monodromy around the puctures. This paper applies the geometric, arithmetic and quantum methods to the group of affine transformation of the line, G = AGL 1 (C).…”

Section: Introductionmentioning

confidence: 99%

Representation variety for the rank one affine group

González-Prieto¹,

Logares²,

Muñoz³

2020

Preprint

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The aim of this paper is to study the virtual classes of representation varieties of surface groups onto the rank one affine group. We perform this calculation by three different approaches: the geometric method, based on stratifying the representation variety into simpler pieces; the arithmetic method, focused on counting their number of points over finite fields; and the quantum method, which performs the computation by means of a Topological Quantum Field Theory. We also discuss the corresponding moduli spaces of representations and character varieties, which turn out to be non-equivalent due to the non-reductiveness of the underlying group.

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“…The TQFT that we will use associates to such M the class [X G (M )] in R = K(Var k ). This method was used in [3] to compute the virtual class of the (parabolic) SL 2 (C)-representation and character varieties of Σ g . It was also used in [7,5] to compute the virtual class of X G (Σ g ) for G the groups of upper triangular matrices of rank 2, 3 and 4.…”

Section: Introductionmentioning

confidence: 99%

Representation Varieties of Non-orientable Surfaces via Topological Quantum Field Theories

Vogel¹

2020

Preprint

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We study the G-representation varieties of non-orientable surfaces. By a geometric method using Topological Quantum Field Theories (TQFTs), we compute virtual classes of these Grepresentation varieties in the Grothendieck ring of varieties, for G the groups of complex upper triangular matrices of rank 2 and 3. We discuss various ways to use conjugacy invariance to simplify the computations. Finally, we give a number of remarks on the resulting classes, and observe and explain the zero eigenvalues that appear in the TQFT.

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Virtual classes of parabolic SL2(C)-character varieties

Cited by 12 publications

References 28 publications

Virtual Classes of Character Stacks

Virtual Classes of Character Stacks

Representation variety for the rank one affine group

Representation Varieties of Non-orientable Surfaces via Topological Quantum Field Theories

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