A lax monoidal topological quantum field theory for representation varieties

González-Prieto, Ángel; Logares, Marina; Muñoz, Vicente

doi:10.1016/j.bulsci.2020.102871

Cited by 16 publications

(35 citation statements)

References 35 publications

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“…after taking the stacky quotient under the conjugacy action, giving rise directly to the right answer. This is in sharp contrast with the TQFT of [18,15], which was only able to compute the virtual class of the representation variety R G (W ). To obtain the virtual class of the GIT quotient χ G (W ) = R G (W ) G from that TQFT, an additional calculation was needed by means of a thorough study of the Luna stratification of the adjoint action [13].…”

Section: Introductionmentioning

confidence: 87%

“…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%

“…holds for any special algebraic group G. The latter approach has a crucial short-coming, it forgets the group action of G on the representation variety. On the other hand, we will use (13) to compare the virtual class of the character stack in K(Stck/BG) with the virtual class of the representation variety in K(Var k ) in the case of AGL 1 (C) ( [18,20]).…”

Section: Field Theory In Dimensionmentioning

confidence: 99%

“…Nevertheless, despite its success, eventually, the geometric approach strongly depends on some clever stratifications which are not clear how to generalize to other situations. To address this issue, González-Prieto, Logares, and Muñoz developed the quantum method [18]. The key result in this approach is the construction of a Topological Quantum Field Theory (TQFT) computing the Epolynomial of the character variety.…”

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: Introductionmentioning

confidence: 87%

Section: Constructing the Stacky Tqftmentioning

confidence: 99%

Section: Field Theory In Dimensionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Virtual Classes of Character Stacks

González-Prieto¹,

Hablicsek²,

Vogel³

2022

Preprint

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“…It suggests that some type of recursion formalism, in the spirit of a Topological Quantum Field Theory (TQFT for short), must hold. 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.…”

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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