Stacks

Prerequisites: Groupoids. Full path (3 pages)

Why is it interesting?
Intuitive
Concrete
Abstract
History

Why is it interesting?

The development of modern physics in the first half of the 20th century was closely related to the development of differential geometry, first via Riemannian geometry in Einstein’s theory of gravity and then later via Cartan geometry in Yang-Mills’s theory of gauge fields. But, as highlighted by Grothendieck in the second half of the 20th century and as witnessed by a multitude of modern developments, a more natural mathematical description of many phenomena in geometry is obtained by refining from traditional geometric spaces to more refined kinds of spaces known as “stacks”.

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Our main motivation to consider sheaves and stacks is to provide a nonperturbative framework in which we can do physics. Much of gauge theory is done in perturbation theory, but in fact non-perturbative effects such as Dirac monopoles and Yang-Mills instantons play a crucial role in fundamental physics [5]. The language of stacks is the natural language for these phenomena.

https://ncatlab.org/schreiber/files/Eggertsson2014.pdf

locality principle + gauge principle = stack principlehttps://ncatlab.org/schreiber/files/SchreiberTrento14.pdf

Intuitive

Many objects in physics are described patch by patch. On a curved space we use one coordinate chart here and another there; in electromagnetism we may need a different vector potential on different regions, related by gauge transformations where the regions overlap. To build a global object we glue the local pieces together along the overlaps.

For ordinary functions gluing is simple: two local functions either agree on the overlap or they do not. Gauge fields are different. Two local descriptions on an overlap are in general related by a specific gauge transformation, and that gauge transformation is part of the data. When three regions overlap, the three gauge transformations between them must fit together. A gauge field can also have symmetries of its own: gauge transformations that leave it unchanged.

A stack is the mathematical object that keeps track of all of this. It assigns to every region the collection of local objects together with the isomorphisms between them (for gauge fields: the gauge transformations), and it demands that compatible local pieces always glue to a global piece in an essentially unique way. In a quotient by a symmetry, an ordinary space forgets which points had extra symmetry; a stack remembers it. This is why stacks are the natural setting for the space of all gauge fields, and for topologically non-trivial configurations such as the Dirac monopole, where no single global vector potential exists.

Concrete

The simplest example is a $U(1)$ gauge field. We set $\hbar = c = 1$ and use the covariant derivative $D = d - ieA$ on a field of charge $e$ . Cover space-time (or space) $M$ by open sets $U_i$ . On each $U_i$ we have a vector potential $A_i$ , and on each overlap $U_i\cap U_j$ a transition function $g_{ij} = e^{ie\lambda_{ij}}$ with

$A_i = A_j + d\lambda_{ij}, \qquad \psi_i = g_{ij}\,\psi_j \quad \text{on } U_i\cap U_j .$

For this to be consistent on triple overlaps the transition functions have to satisfy the cocycle condition

$g_{ij}\, g_{jk}\, g_{ki} = 1 \quad\text{on } U_i\cap U_j\cap U_k .$

The data $(A_i, g_{ij})$ describe a principal $U(1)$ -bundle with connection. Two such data sets describe the same physical configuration if they are related by local gauge transformations $h_i: U_i\to U(1)$ , $A_i\mapsto A_i + \frac{1}{ie}h_i^{-1}dh_i$ , $g_{ij}\mapsto h_i g_{ij} h_j^{-1}$ .

Example: the Dirac monopole. On the sphere $S^2$ surrounding a magnetic charge $q_m$ take the two patches $U_N$ (sphere minus south pole) and $U_S$ (sphere minus north pole), with

$A_N = q_m(1-\cos\theta)\,d\phi, \qquad A_S = -q_m(1+\cos\theta)\,d\phi .$

Both give the field strength $F = dA = q_m\sin\theta\, d\theta\wedge d\phi$ with total flux $4\pi q_m$ , but neither is defined on the whole sphere. On the overlap $A_N - A_S = 2q_m\,d\phi = d\lambda_{NS}$ with $\lambda_{NS} = 2q_m\phi$ . The transition function $g_{NS} = e^{2ieq_m\phi}$ must be single valued on the equator, which requires $2eq_m\in\mathbb{Z}$ : the Dirac quantization condition.

This example shows the two features that make a stack. First, the global object exists only as local data glued by gauge transformations. Second, to decide whether two configurations are "the same" we have to specify a gauge transformation between them, and a configuration can have non-trivial automorphisms (here the constant gauge transformations $h_i = h\in U(1)$ , which leave every $(A_i,g_{ij})$ unchanged). The rule "to each open set $U$ assign the groupoid whose objects are $U(1)$ -bundles with connection on $U$ and whose morphisms are gauge transformations" is the stack of $U(1)$ gauge fields. The gluing property above (local data plus compatible transition functions determine a global object, unique up to unique isomorphism) is what makes it a stack and not only a groupoid-valued assignment.

Abstract

Fix a site, for example the category of smooth manifolds with open covers, or of schemes with the étale topology. A prestack is a pseudofunctor $\mathcal{X}: \mathcal{C}^{op}\to\mathbf{Grpd}$ (equivalently, a category fibred in groupoids over $\mathcal{C}$ ). For a cover $\{U_i\to U\}$ the category of descent data has as objects families $x_i\in\mathcal{X}(U_i)$ with isomorphisms $\phi_{ij}: x_j|_{U_{ij}}\to x_i|_{U_{ij}}$ satisfying $\phi_{ij}\circ\phi_{jk} = \phi_{ik}$ on $U_{ijk}$ . $\mathcal{X}$ is a stack if for every cover the restriction functor

$\mathcal{X}(U)\longrightarrow \mathrm{Desc}(\{U_i\to U\})$

is an equivalence of categories. Fully faithful means isomorphisms glue uniquely (a sheaf condition on morphisms); essentially surjective means objects glue. A sheaf of sets is a stack whose groupoids have only identity morphisms, and every manifold or scheme $X$ defines a stack through $U\mapsto\mathrm{Hom}(U,X)$ , since maps glue along covers (the topology is subcanonical) and by the Yoneda lemma no information is lost.

Standard examples:

For a Lie group $G$ , $\mathbf{B}G$ assigns to $U$ the groupoid of principal $G$ -bundles on $U$ . Maps $M\to\mathbf{B}G$ are principal $G$ -bundles on $M$ , so $\mathbf{B}G$ is a moduli stack of bundles. With connections one obtains $\mathbf{B}G_{\rm conn}$ , the stack of gauge fields.
For a $G$ -action on $X$ , the quotient stack $[X/G]$ assigns to $U$ principal $G$ -bundles $P\to U$ with a $G$ -equivariant map $P\to X$ . When $G$ acts freely and properly, $[X/G]\simeq X/G$ ; otherwise the stack retains the stabilizer groups, and $[\mathrm{pt}/G] = \mathbf{B}G$ . Orbifolds are differentiable stacks that are locally of the form $[V/\Gamma]$ with $\Gamma$ a finite group.
A differentiable stack is one equivalent to $[G_1\rightrightarrows G_0]$ , the stack associated with a Lie groupoid; two Lie groupoids present equivalent stacks if and only if they are Morita equivalent. In algebraic geometry the analogous notions are Deligne-Mumford and Artin stacks, introduced to handle moduli problems such as the moduli of curves $\mathcal{M}_g$ , whose objects have automorphisms.

Replacing groupoids by higher groupoids gives higher stacks. Bundle gerbes with connection, which describe the $B$ -field of string theory, form the 2-stack $\mathbf{B}^2U(1)_{\rm conn}$ (see gerbes), and in general $\mathbf{B}^nU(1)_{\rm conn}$ classifies degree $n+1$ differential cohomology. In this language a gauge field configuration on $M$ is a map $M\to\mathbf{B}G_{\rm conn}$ , and the gauge redundancy is built into the morphisms instead of being divided out. Global effects like monopole charges and instanton numbers appear as the topological classes of these maps, i.e. characteristic classes of the underlying bundles.

See section 6 in Mirror Symmetry, Hitchin's Equations, And Langlands Duality by Edward Witten

History

Stacks (champs) were introduced by Alexander Grothendieck and developed by Jean Giraud in the 1960s (Giraud, Cohomologie non abélienne, 1971); Deligne and Mumford introduced algebraic stacks in 1969. Grothendieck's 1983 manuscript Pursuing Stacks is about higher stacks. For some more information, see https://ncatlab.org/nlab/show/Pursuing+Stacks