Complementary Information For Advanced String Theory

Representations of SUSY Algebra

In string theory, different particles are associated with different representations of the supersymmetry algebra. It then makes sense to gain an understanding of the Poincare algebra. Poincare algebra are characterized by the inclusion of two Casimir-projectors P, W, so that W is the Pauli-Lubanski vector. For massive particles in rest frame, it's attractive to choose P to be the 4-momentum vector, which will cause W² to include terms for the spin.

Massless particles see P = (E, 0, 0, E) and W = MP, with helicity M = ±s. Within these representations, the spin, helicity and energy states act as the regular quantum-mechanical bases that should be familiar to the reader. The superparticle then is an irreducible representation of the SUSY algebra, and with that a representation of the Poincare algebra by group relation. A superparticle corresponds to a collection of particles, related by the action of the supersymmetry generators and with a spin difference by one. It's often referred to as a "supermultiplet" for that reason. In every representation of supersymmetry algebra, W² loses its Casimir property, since helicity doesn't commute with supersymmetry generators. The particles of a supermultiplet share the same mass, but have different spin (due to non-conservation). Since this mass degeneracy is not in general observed in particle physics, SUSY can be assumed to be broken in nature (or at the natural energy scales. The energy of all states is assumed to be positive definite. The supermultiplet contains an equal number of bosonic and fermionic degrees of freedoms. For this, define a fermion number operator for both kinds of state, so that

At massless representations, all central charges vanish (why is covered elsewhere). Since all supersymmetry generators commute, at rest frame,

This means that the generators are trivialized. The creation/annihilation operators function as non-trivial generators, defined by said generators.

Constructing a representation is best approached from choosing the Clifford vacuum, a state annihilated by all annihilators. It carries some helicity, which will find itself in the quantum state. The Clifford vacuum is exclusively defined by these quantum numbers. The full supermultiplet is obtained by acting on this vacuum with creation operators, iteratively. To complete the representation, include the states with flipped helicity. The specific versions of supermultiplet of an N=1 SUSY can be easily obtained by setting the helicity, and constructing the CPT-reflected superposition.

Anti-Self-Dual Tensor Fields & Type IIB Action

Define the Hodge Star operator on a p-form, so that it squares on

It's constructed to transform p-forms to (d-p)-forms. The anti-self-dual p-form is supposed to function as

Fₚ = ±*Fₚ, making the Hodge-star automorphic. This fixes p to the middle-dimensions, and since p = d/2, where d is even. Direct application gives

The condition for the squared Hodge star is trivial for an odd d²/4, so the anti-self-dual forms in Minkowski signature will require it to be even, specifically d = 4k + 2. As the C-form differential, the field strength contribution to the action is defined through

meaning that the action contribution is null. This is implausible.

The (anti-)chiral boson in ℝ¹ ¹ is a scalar field, so that it's F-form (meaning the field differential) is (anti-)self-dual. It doesn't have a Lagrangian description precisely because of the above reasons, and has to be studied through the equivalent theory of free fermions.

For the type IIB SuGra action in Einstein frame, get the differential of the different parts.

Cohomology Classes & Kaehler Manifold

A space of differential forms A of degree r on M are sections of the vector bundle Ωʳ(M). The metric on M defines that of L² as differential forms

The exterior differential d: Aʳ(M) → Aʳ⁺¹(M), which satisifies d² = 0. The Rahm cohomology H is defined as usual, and given d, the adjoint can be defined so that

Δ is an elliptic self-adjoint differential operator on M, and futher fixes H(M) as r-forms in the kernel of Δ. This defines harmonic forms. ( , ) then is a positive definite inner product. Every form is closed. The (unique) Hodge decomposition gives an harmonic hʳ

For all closed ωʳ exist harmonic hʳ in the same cohomology class, implying surjectivity. Since every harmonic form hʳ is closed, the map is also injective, and we get full isomorphism, and so every cohomology class has a harmonic representation

A Kaehler metric ω is a closed, positive (1, 1)-form. If dω = 0, then it's locally d-exact, and since

With this set of non-zero components of the affine correction, the standard structure is covariantly constant. Since U(N) is the maximal subgroup of SO(2N) preserving the structure, the holonomy is also already maximal.