Disclaimer: Nothing in this note is original.

A short introduction about sheaf can be found here.

Gauged linear sigma model

A gauged linear sigma model (GLSM) is a type of quantum field theory that incorporates both gauge symmetry and a set of scalar fields which interact with each other through a potential that is usually taken to be a function of the sum of the squares of the scalar fields. These models are a generalization of the linear sigma models that were originally introduced to describe the dynamics of pions in particle physics.

The term “linear” in this context refers to the fact that the potential energy function for the scalar fields is quadratic, at least around the vacuum of the theory. This means that the interactions can be written in terms of fields and their derivatives in a polynomial form that does not exceed second degree when expanded about the vacuum expectation value.

Often, GLSMs are discussed in the context of supersymmetry. This means that for every boson (like the scalar fields and gauge fields in the model), there is a corresponding fermion. Supersymmetric GLSMs are particularly interesting for string theory and the study of Calabi-Yau manifolds.

In essence, the gauged linear sigma model allows for the study of a wide array of phenomena including spontaneous symmetry breaking, phase transitions, and the dynamics of topological defects. By gauging a symmetry, you add a rich structure to the model that includes interactions mediated by gauge bosons and the possibility of incorporating richer topological properties.

A key feature of GLSMs is that they can have a “Higgs phase,” where the gauge symmetry is spontaneously broken and the gauge bosons acquire mass via the Higgs mechanism, and a “Coulomb phase,” where the gauge symmetry is unbroken and the gauge bosons remain massless. Transitions between these phases can often be studied using GLSMs, providing insights into non-perturbative aspects of quantum field theories.

Toric variety

We all know what a variety is. So what is a toric variety?

A toric variety is a type of variety that is built from combinatorial objects known as fans, which are collections of cones. Toric varieties are particularly nice because they provide a bridge between algebraic geometry and combinatorics, and they are also rich in geometric properties.

Algebraic Torus.

Start with an algebraic torus $( \mathbb{C}^* )^n$, which is the product of $n$ copies of the multiplicative group $\mathbb{C}^* = \mathbb{C} \setminus {0}$. This group acts on itself by multiplication, and this action can be extended to act on other algebraic varieties.

Fans and Cones.

A fan is a set of cones (not in the usual sense of solid geometry, but in the sense of vector spaces). These cones are generated by a set of vectors in $\mathbb{R}^n$ that satisfy certain combinatorial conditions – they must be strongly convex (no line can be contained in a cone), and the intersection of any two cones in the fan must be a face of each. The dimension of the toric variety is equal to the dimension of the space where the fan lives.

Construction of Toric Varieties.

Each cone in the fan corresponds to an affine variety that is invariant under the action of a subtorus of $( \mathbb{C}^* )^n$. The toric variety is constructed by “gluing” these affine varieties together in a way that corresponds to how their cones intersect.

Divisors and Line Bundles. Toric varieties are also interesting because there is a correspondence between the combinatorial data of the fan and the divisor class group of the variety. This makes the calculation of certain cohomological and geometric properties much simpler than in general varieties.

A key feature of toric varieties is that they come with a lot of symmetry – namely, the action of the torus. This symmetry makes them easier to study and gives them a rich structure. Toric varieties have applications in many areas of mathematics, including combinatorics, symplectic geometry, and mirror symmetry in string theory. They serve as local models in the Minimal Model Program and are also used in the study of singularities.

For a concrete example, consider the complex projective space $\mathbb{CP}^n$, which is a toric variety. The fan for $\mathbb{CP}^n$ consists of cones that correspond to the origin and the standard basis vectors in $\mathbb{R}^n$, as well as all their faces. The rich interplay between the algebraic and combinatorial structures of toric varieties makes them a fascinating subject of study in modern algebraic geometry.

Stacks

Let $\mathcal{T}$ be the category of topological spaces, where the objects are topological spaces and the morphisms are continuous maps.

A groupoid fibration, or a category fibered in groupoids, over $\mathcal{T}$ is another category (which is a groupoid) $\mathcal{X}$, together with a functor

\[F: \mathcal{X} \to \mathcal{T}\]

such that there exists a pullback and the pull back is unique. First, some terminologies. If the functor maps $X \in \mathcal{X}$ to $T \in \mathcal{T}$ then we say that $X$ lies over $T$, or that $X$ is a $\mathcal{X}$-family parametrized by $T$, and we write $X / T$. If the morphism $\eta: X \to Y$ is mapped by $F$ to $f: T\to S$ then we say that $\eta$ lies over $f$, or $\eta$ covers $f$. The requirement of pullback essentially says that for every $\mathcal{X}$-family $X / T$ and a morphism $T’ \to T$ there exists another $\mathcal{X}$-family $X’ / T’$ such and such. And this pullback is essentially unique. $X’ / T’$ is said to be the pullback of $X / T$ via the continuous map $f$. We use notations

\[X' = f^{\ast } X \text{ or } X' = X \mid _ {T'}.\]

Groupoid fibration captures two notions at once:

  • isomorphism of families. By the arrows in $\mathcal{X}$ (Since $\mathcal{X}$ is a groupoid).
  • pullback.