Introduction to a provisional mathematical definition of Coulomb
  branches of $3$-dimensional $\mathcal N=4$ gauge theories

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This is an introduction to a provisional mathematical definition of Coulomb branches of $3$ -dimensional $\mathcal N=4$ supersymmetric gauge theories, studied in arXiv:1503.03676, arXiv:1601.03586. This is an expanded version of an article arXiv:1612.09014 appeared in the 61st DAISUUGAKU symposium proceeding (2016), written originally in Japanese.

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Topological quantum field theories

Michael F. Atiyah (1988)

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A 'Darboux Theorem' for shifted symplectic structures on derived Artin stacks, with applications

Oren Ben-Bassat, Dominic Joyce, Vittoria Bussi … (2013)

This is the fifth in a series arXiv:1304.4508, arXiv:1305,6302, arXiv:1211.3259, arXiv:1305.6428 on the '

$k$ -shifted symplectic derived algebraic geometry' of Pantev, Toen, Vaquie and Vezzosi, arXiv:1111.3209. This paper extends the previous three from (derived) schemes to (derived) Artin stacks. We prove four main results: (a) If

$(X,\omega)$ is a

$k$ -shifted symplectic derived Artin stack for

$k<0$ in the sense of arXiv:1111.3209, then near each

$x\in X$ we can find a 'minimal' smooth atlas

$\varphi:U\to X$ with

$U$ an affine derived scheme, such that

$(U,\varphi^*(\omega))$ may be written explicitly in coordinates in a standard 'Darboux form'. (b) If

$(X,\omega)$ is a

$-1$ -shifted symplectic derived Artin stack and

$X'$ the underlying classical Artin stack, then

$X'$ extends naturally to a 'd-critical stack'

$(X',s)$ in the sense of arXiv:1304.4508. (c) If

$(X,s)$ is an oriented d-critical stack, we can define a natural perverse sheaf

$P^\bullet_{X,s}$ on

$X$ , such that whenever

$T$ is a scheme and

$t:T\to X$ is smooth of relative dimension

$n$ , then

$T$ is locally modelled on a critical locus Crit

$(f:U\to{\mathbb A}^1)$ for

$U$ smooth, and

$t^*(P^\bullet_{X,s})[n]$ is locally modelled on the perverse sheaf of vanishing cycles

$PV_{U,f}^\bullet$ of

$f$ . (d) If

$(X,s)$ is a finite type oriented d-critical stack, we can define a natural motive

$MF_{X,s}$ in a ring of motives

$\bar{\mathcal M}^{st,\hat\mu}_X$ on

$X$ , such that whenever

$T$ is a finite type scheme and

$t:T\to X$ is smooth of dimension

$n$ , then

$T$ is locally modelled on a critical locus Crit

$(f:U\to{\mathbb A}^1)$ for

$U$ smooth, and

${\mathbb L}^{-n/2}\odot t^*(MF_{X,s})$ is locally modelled on the motivic vanishing cycle

$MF^{mot,\phi}_{U,f}$ of

$f$ in

$\bar{\mathcal M}^{st,\hat\mu}_T$ . Our results have applications to categorified and motivic extensions of Donaldson-Thomas theory of Calabi-Yau 3-folds

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Publication date Created: 2017-06-16

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ArXiV ID: 1706.05154

SO-VID: 5b8284fd-a9c8-4378-baaf-8696c1e80eed

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http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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Categories math.RT hep-th math-ph math.AG math.DG math.MP

ScienceOpen disciplines: Mathematical physics,High energy & Particle physics,Mathematical & Computational physics,Geometry & Topology,Algebra

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Introduction to a provisional mathematical definition of Coulomb branches of $3$ -dimensional $\mathcal N=4$ gauge theories

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An Introduction to SAXS

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Topological quantum field theories

A 'Darboux Theorem' for shifted symplectic structures on derived Artin stacks, with applications

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Introduction to a provisional mathematical definition of Coulomb branches of 33-dimensional N=4\mathcal N=4 gauge theories

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An Introduction to SAXS

Topological quantum field theories

A 'Darboux Theorem' for shifted symplectic structures on derived Artin stacks, with applications

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Introduction to a provisional mathematical definition of Coulomb branches of $3$ -dimensional $\mathcal N=4$ gauge theories