An extended version of additive <i>K</i>-theory

Garoufalidis, Stavros

doi:10.1017/is008007022jkt059

J K-Theor

2008

DOI: 10.1017/is008007022jkt059

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An extended version of additive K-theory

Stavros Garoufalidis¹

Abstract: Abstract. There are two infinitesimal (i.e., additive) versions of the K-theory of a field F : one introduced by Cathelineau, which is an F -module, and another one introduced by Bloch-Esnault, which is an F * -module. Both versions are equipped with a regulator map, when F is the field of complex numbers.In our short paper we will introduce an extended version of Cathelineau's group, and a complex-valued regulator map given by the entropy. We will also give a comparison map between our extended version and Ca… Show more

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Cited by 2 publications

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“…The path-integral (13) can be defined for arbitrary coupling constants ,˜ by enlarging the domain of integration and deforming the integration contour [17], as we will comment more in [1]. The existing literature has focused on the case of maximal puncture, see in particular [18][19][20][21][22] for the case N > 2.…”

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Taming supersymmetric defects in 3d–3d correspondence

Gang¹,

Kim²,

Romo³

et al. 2016

J. Phys. A: Math. Theor.

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We study knots in 3d Chern-Simons theory with complex gauge group SL(N, C), in the context of its relation with 3d N = 2 theory (the so-called 3d-3d correspondence). The defect has either co-dimension 2 or co-dimension 4 inside the 6d (2, 0) theory, which is compactified on a 3-manifold M . We identify such defects in various corners of the 3d-3d correspondence, namely in 3d SL(N, C) Chern-Simons theory, in 3d N = 2 theory, in 5d N = 2 super Yang-Mills theory, and in the M-theory holographic dual. We can make quantitative checks of the 3d-3d correspondence by computing partition functions at each of these theories. This Letter is a companion to a longer paper [1], which contains more details and more results.Introduction.-One lesson from history is that physics and mathematics often develop hand in hand; the development on one side facilitates development in the other, creating a virtuous cycle of feedback. The recently-discovered 3d-3d correspondence [2-7] is a perfect example for this interplay. The correspondence states that there exists a surprising connection between 3d SL(N, C) Chern-Simons (CS) theory defined on a 3-manifold M on the one hand, and a 3d N = 2 supersymmetric gauge theory (which we call T N [M ]) on the other. Being a topological field theory, the SL(N, C) CS theory provides a functor equipped with a Hilbert space. For the N = 2 case, we have the SL(2, C) CS theory and the relevant Hilbert space corresponds to a quantization of the moduli space of SL(2, C)-flat connections on M , and it contains the space of hyperbolic structures on M when M is a hyperbolic manifold. The 3d-3d relation thus unifies and enriches the mathematics of both knot theory and 3d hyperbolic geometry.The physical origin of the 3d-3d correspondence is the compactification of the 6d (2, 0) theory on a closed 3-manifoldM , along which the theory is partially topologically twisted:

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mentioning

confidence: 99%

Taming supersymmetric defects in 3d–3d correspondence

Gang¹,

Kim²,

Romo³

et al. 2016

J. Phys. A: Math. Theor.

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An Ansatz for the asymptotics of hypergeometric multisums

Garoufalidis

2008

Advances in Applied Mathematics

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Sequences that are defined by multisums of hypergeometric terms with compact support occur frequently in enumeration problems of combinatorics, algebraic geometry and perturbative quantum field theory. The standard recipe to study the asymptotic expansion of such sequences is to find a recurrence satisfied by them, convert it into a differential equation satisfied by their generating series, and analyze the singulatiries in the complex plane. We propose a shortcut by constructing directly from the structure of the hypergeometric term a finite set, for which we conjecture (and in some cases prove) that it contains all the singularities of the generating series. Our construction of this finite set is given by the solution set of a balanced system of polynomial equations of a rather special form, reminiscent of the Bethe ansatz. The finite set can also be identified with the set of critical values of a potential function, as well as with the evaluation of elements of an additive K-theory group by a regulator function. We give a proof of our conjecture in some special cases, and we illustrate our results with numerous examples.Contents 1991 Mathematics Classification. Primary 57N10. Secondary 57M25.

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An extended version of additive K-theory

Cited by 2 publications

References 10 publications

Taming supersymmetric defects in 3d–3d correspondence

Taming supersymmetric defects in 3d–3d correspondence

An Ansatz for the asymptotics of hypergeometric multisums

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