Gauged Linear Sigma-Models

Among the general supersymmetric sigma models, a particular type of models [see "Phases…" by E.Witten] has been studied extensively. As sigma-models, they all exhibit the following properties:

1. The domain (base space) is any Riemann surface (as needed for string theory) with at least (0,2)-supersymmetry (so the effective spacetime theory would end up with at least N=1 supersymmetry).
2. The target space is the (complex) affine space, C, in which the actually desired target space, X, is found as an intersection of hypersurfaces and gauge-quotient slices.
3. The immersion map is provided by (lowest component fields of) the coordinate (super)fields.
4. The Hamiltonian action functional, S, (often, mistakenly, called the ‘energy function’) has four terms:
   a. the flat kinetic term for coordinate (super)fields, i.e., flat metric for C;
   b. the standard kinetic terms, and
   c. a Fayet-Illiopoulos term (with a ‘radial’ parameter) for each of the several U(1)-symmetries, G, which together create the Toric variety as the quotient T=C/G;
   d. a homogeneity-0 function of the coordinate (super)fields for the superpotential, of the form W(p,z)=pG(z), where G(z) is homogeneous of degree -deg(p).

In the simplest case, this constrains the ground state configurations to the target space, X, found to be (in the limit of infinite ‘radii’) the union of the:
  - (intersection of) hypersurfaces {p=0, z><0, G(z)=0}: Calabi-Yau varieties, and
  - the point(s) {p><0, z=0}: Landau-Ginzburg orbifolds.
These are the limiting ‘corners’ of the ‘phases’ of the gauged linear sigma model; the finite values of the ‘radii’ mapping out the entire phase space.

In this way, the gauged linear sigma model is a perfect analogue of the original Landau-Ginzburg models, the ‘radii’ serving, instead of temperature, as the parameter(s) interpolating between the Landau-Ginzburg orbifold and the Calabi-Yau compactifications.

In all of these, of particular interest are wave-functions of lowest possible energy, as the entire Hilbert space (of wave functions) may then be reconstructed from these ‘ground states’, or ‘vacua’.

In supersymmetric theories, energy is proportional to the trace of the matrix of anticommutators of the supercharges, H = Tr{Q,Q⁺}, which is positive definite. That is, the Hamiltonian always turns out to be a sum of nonnegative terms. Lowest energy therefore is zero, and it is attained by states which are annihilated by the supercharges, Q, and so are invariant under supersymmetry: supersymmetric vacua. The Hamiltonian being a sum of nonnegative terms, each must vanish separately. In particular, the vanishing of the potential energy (non-derivative ) terms provides an array of simultaneous algebraic equations over the fields space - whence the ground state is described as a complete intersection of algebraic equations in T=C/G.

The parameters in the action functional, S, are allowed to vary - so as to ‘feel out’ all possible choices of their values. Ultimately, some generalized principle of lowest energy will then determine which particular values of these parameters represents the stablest candidate for the effective “Theory of Everything”. These parameters are called ‘moduli’ (of the target space, X), and their variations correspond to light particles in the effective ‘particle’ theory. Moduli space geometry and parameter space dynamics are therefor, in many ways, merely two different aspects of the same thing; the latter also determines the dynamics of the light particles. This is how the geometry of the moduli space of the target space, X, ends up determining the dynamics of the light particles from which the observed matter and interactions are forged.

The underlying gauged linear sigma model turns out to admit nontrivial transformations which however leave the dynamics of the light particles unchanged. The latter constituting the ‘real physics’, such transformations are deemed symmetries. Other nontrivial transformations on the other hand turn out to relate the dynamics of two or more models with different light particle dynamics. These are the mirror and the duality transforms.

© Tristan HŸbsch, 2008