This week I have been reading Frank Hellmann’s PhD thesis on The geometry of state sums. Here I’ll review the section on the geometric states arising in the representation theory of SU(2), Spin(4) and SL(2,C).

**The Geometry of Representations**

We scan construct a state sum invariant using the representation theory of SU(2). A similar construction can be given for Spin(4),

the covering group of SO(4), and SL(2,C), the covering group of SO(3, 1)+ the identity connected component of SO(3, 1).

**The Geometry of SU(2)**

One way to the representation theory of SU(2) is by the quantization of the sphere

**Coherent States**

Coherent states are defined as the eigenstates of Lie algebra elements. Given a 3-dimensional unit vector n ∈ S² the associated coherent states αj(n) are defined by:

This fixes the states j(n) up to a phase. Every state in the fundamental representation is proportional to a coherent state.

Every normalized state α’ is a coherent state α’(n(α’)).

In particular we have that

The anti-linear map J transforms a coherent state to one associated

to the opposite direction:

The coherent states satisfy an exponential property for every αj(n) there is an α(n) in the fundamental representation such that

Lie algebra elements transform under the vector representation of SU(2), their transformation behaviour under rotations that stabilize n is,

The modulus square of the inner product between two coherent states in arbitrary representations is

This shows that in the large quantum number limit j → ∞ coherent states become orthogonal.

A resolution of the identity is given by integrating over

the sphere:

Coherent states give the geometry of a sphere associated to particular representations j. Extending this we can give a geometric interpretation to the invariant subspace of the tensor product of representations Inv(j1⊗j2. . .⊗jn)

Next we look at three-valent intertwiners which are associated to triangles and four-valent intertwiners and their relationship with tetrahedra.

**Coherent Triangles**

The shape space of triangles can be described as a constrained space of three spheres of radius j_{i}. This space has the product symplectic structure of that of the sphere used in geometric quantisation. Given three vectors j_{a}n_{a} of length j_{a} that are on the spheres the constraint

forces them to describe the edge vectors of a triangle with edge lengths j_{a}.

To quantize this state space we take the quantized unconstrained state space

** Coherent Tetrahedra**

Just as three-valent intertwiners can be interpreted as quantized triangles we can interpret four-valent intertwiners as quantized tetrahedra.

Consider a set of four vectors in directions n_{a} of lengths j_{a} satisfying closure

The space of such closing vectors that are non-degenerate is the shape space of non-degenerate tetrahedra.

* Tetrahedra from Closing Vectors*

*Four vectors j*_{a}n_{a} of length j_{a} that span 3-dimensional space and satisfy are the outward normals of a non-degenerate tetrahedron with areas j_{a} embedded in R³ which is unique up to translation.

Again the constraint generates rotations of the tetrahedron and the non-degenerate sector of the reduced phase space is the shape space of tetrahedra. Implementing the constraint quantum mechanically again we obtain the over-parametrisation of the space of four-valent intertwiners by:

**The Geometry of Spin(4)**

For the 4-dimensional models need to understand the geometry of the representation theory of Spin(4). To do so we will consider the

Lie algebra spin(4) which is isomorphic to so(4), the Lie algebra of 4-dimensional rotations. We can understand this Lie algebra as arising from bivectors. It decomposes into a left and right sector under the action of the Hodge star. This allows us to give the representation theory of Spin(4) in terms of SU(2) representations and define coherent bivectors. We will look at a necessary and sufficient set of conditions for a set of bivectors to define a geometric 4-simplex σ^{4}.

Bivectors in R^{4 }are elements of Λ²(R^{4}), that is, 2-dimensional antisymmetric tensors B^{IJ} = −B^{JI} , I, J = 0, . . . , 3. Λ²(R^{4}) is 6-dimensional.

We define the norm of a bivector by |B|² = ½B^{IJ} B_{IJ} . The Lie algebra so(4) is the algebra of antisymmetric matrices with Lie product

given by the commutator. The Lie algebra structure constants are:

** Geometric Bivectors**

We call a set of bivectors geometric if they are the bivectors of the faces of a geometric 4-simplex σ^{4}_{i} in R^{4}. Using a, b = 1, . . . , 5 to denote the tetrahedral faces σ^{3}_{i} of the 4-simplex we denote its outward facing normal vectors by N_{a}. The bivectors can then be written as

The associated to the triangles of a non-degenerate geometric 4 simplex satisfy the geometricity conditions:

**The Geometry of SL(2,C)**

In order to work with theories with Lorentz symmetry we need to consider representations of SO(3, 1). More specifically the identity connected component SO(3, 1)+ and its double cover SL(2,C). That is, the part of SO(3, 1) that takes future pointing normals to future pointing normals.

The corresponding Lorentzian generators are then

the structure constants are given by the same calculation as in the spin(4) case:

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