Highest-Weight Representations of Affine Lie Algebras and the Level k

1. Highest-Weight Representations of Affine Algebras

The concept of a highest-weight representation is a unifying principle in the representation theory of many Lie algebras you have encountered. Let's briefly recall two key examples:

• For su(2): A highest-weight state |j,j⟩ is annihilated by the raising operator J+ and is an eigenstate of J_z. The full (finite-dimensional) representation is generated by acting with the lowering operator J-.
• For the Virasoro algebra: A highest-weight state |h⟩ is annihilated by all L_n with n > 0 and is an eigenstate of L_0 with eigenvalue h. The (infinite-dimensional) representation is generated by acting with L_n for n < 0.

We will now apply this same logic to the affine algebra $\hat{\mathfrak{g}}$.

A highest-weight state |λ⟩ in a representation of an affine algebra is defined by three conditions:

1. It is annihilated by all "raising" operators, which we define as the modes with positive mode number:
   $J_n^a|\lambda ⟩=0\text{for all }n>0\text{ and for all }a$
2. The zero-modes $\{J_0^a\}$ form the horizontal subalgebra, which is isomorphic to the original Lie algebra $\mathfrak{g}$. The state |λ⟩ must be a highest-weight state of a representation of this subalgebra $\mathfrak{g}$. This representation of $\mathfrak{g}$ is labeled by its highest weight, which we denote as λ.
3. As an irreducible representation, the central element K must act as a multiple of the identity. We call this eigenvalue the level k. In our physics notation where the central term is written as k n \delta_{n+m,0}, this is already encoded.

The entire representation space, called a Verma module, is then generated by acting on this highest-weight state |λ⟩ with all possible combinations of the "lowering" operators J^a_n (where n < 0) and the lowering operators of the horizontal subalgebra $\mathfrak{g}$.

The resource states that for unitary representations of affine SU(2), both k and 2j (where j is the spin of the horizontal SU(2)_0 representation) must be integers satisfying 0 ≤ 2j ≤ k. This is a profound constraint. It implies that for a given integer level k, there are only k+1 possible representations of the horizontal subalgebra that can serve as the "ground state" of a unitary affine representation.

Let's investigate where this constraint comes from.

2. Unitarity, Null States, and the Quantization of k

In a quantum theory, the space of states is a Hilbert space, which requires a positive-definite inner product. This property is called unitarity. For an affine algebra based on a compact Lie algebra (like su(N)), the inner product requires the hermitian conjugation property (J^a_n)† = J^a_{-n}.

With this structure, we can compute the norm of any state in the Verma module. A crucial issue arises: for certain values of k and λ, we can construct non-zero states that have zero norm. These are called null states or null vectors.

The existence of a null state |ψ⟩ ≠ 0 with ⟨ψ|ψ⟩ = 0 signals two things:

1. The Verma module is reducible; the set of all descendants of the null state forms a proper submodule.
2. If we simply work with the Verma module, the Hilbert space structure breaks down because it contains states with non-positive norms (descendants of the null state can have negative norm).

To obtain a unitary, irreducible representation, we must quotient the Verma module by all its null states and their descendants. The condition for a null state to appear becomes a constraint on the parameters k and λ.

To see this in action, let's analyze the su(2) case in detail.

As the resource demonstrates, for a highest-weight state |ℓ⟩ (where ℓ is the spin, j in our previous notation), the state (J^+_{-1})^{N}|ℓ⟩ has a norm proportional to N! ∏_{n=1}^{N} (k+1-n-2ℓ).

For this norm to be non-negative for all N, we must avoid the factors (k+1-n-2ℓ) becoming negative. This leads directly to the constraints:

1. The product becomes zero when n = k+1-2ℓ. For this to happen at an integer N=n, k must be an integer (since 2ℓ is an integer).
2. For the norm to be positive for N < k+1-2ℓ, we require k+1-n-2ℓ > 0 for these n. The most restrictive case is the largest n, i.e., n=N. This requires k+1-N-2ℓ > 0. A unitary representation must not have negative-norm states at any level. The appearance of a zero-norm state at level N = k+1-2ℓ prevents further states from being generated, effectively truncating the representation in that direction. For this to define a consistent unitary theory, we must have N ≥ 1, which means k+1-2ℓ ≥ 1, or k ≥ 2ℓ.

So, for the affine su(2) algebra (denoted ${\hat{A}}_1$), unitary highest-weight representations are labeled by a pair (k, ℓ) where:

• k is a non-negative integer (the level).
• ℓ is the su(2)_0 spin, which can take values 0, 1/2, 1, ... , k/2.

This is a remarkable result. For any given integer level k, there are only a finite number (k+1) of allowed primary fields. This property, known as rationality, makes these theories highly constrained and often exactly solvable. This generalises to other compact Lie groups, where for a given integer level k, only a finite number of highest-weight representations of $\mathfrak{g}$ are "admissible".

Conclusion

In this lesson, we have defined the highest-weight representations that form the building blocks of CFTs with current algebras. We saw that the abstract algebraic structure we defined previously has a concrete realization in terms of states in a Hilbert space, organized by a highest weight.

Key Takeaways:

• A highest-weight representation of an affine algebra $\hat{\mathfrak{g}}$ is specified by the level k (the eigenvalue of the central element) and the highest weight λ of the horizontal subalgebra $\mathfrak{g}$.
• The representation space (Verma module) is generated by acting on the highest-weight state |λ⟩ with the negative modes J^a_{n<0}.
• The requirement of unitarity (positive-definite norm) is a powerful physical constraint. It leads to the existence of null states, which forces the parameters k and λ to take specific, constrained values.
• For affine algebras based on compact Lie groups, the level k must be a non-negative integer, and for a given k, there is only a finite set of allowed representations λ of the horizontal subalgebra.

We have now established both the algebra and its representations. The next natural question is: where do these structures appear? In our next lesson, we will introduce Wess-Zumino-Witten (WZW) models, which are concrete CFTs whose symmetry currents realize this affine algebra structure. We will see how the level k appears directly in the action of the theory and how the primary fields of the WZW model correspond precisely to the unitary highest-weight representations we studied today.