Welcome back! In our last lesson, we constructed the 4D N=1 superconformal algebra, psu(2,2|1), identifying its bosonic and fermionic generators and the crucial role of the U(1)_R symmetry. This algebra defines the "rules of the game" for a superconformal field theory. Today, we start playing that game by studying the allowed states and fields, which must form unitary representations of this algebra.
Our goal is to define the BPS bound and the principle of representation shortening in superconformal theories. We will discover that representations of the supersymmetry algebra fall into two main classes: generic "long" multiplets and special "short" multiplets. Short multiplets satisfy a specific bound on their quantum numbers, known as the BPS bound. This property makes them incredibly powerful tools in modern theoretical physics because they are often "protected" from quantum corrections, allowing for exact calculations even at strong coupling. This is a cornerstone of the methods used in studying AdS/CFT and string theory.
1. The BPS Bound: A Warm-up with Poincaré Supersymmetry
Before tackling the superconformal case, let's build our intuition by examining the simpler, non-conformal Poincaré superalgebra. To see the key structure, we need an algebra with a central charge. The simplest example is the N=2 super-Poincaré algebra.
PiTP Lectures on BPS States and Wall-Crossing in d = 4, N ...
To understand the origin of the BPS bound, we'll start with the N=2 super-Poincaré algebra. This is a non-conformal algebra that extends the familiar Poincaré group with two sets of supercharges and, crucially, a central charge Z. We will use a classic set of lecture notes by Gregory Moore, 'PiTP Lectures on BPS States and Wall-Crossing'.
Please read the following parts of the lecture notes: Section 2.1 (page 5) to see the structure of the N=2 supersymmetry algebra. Note the anticommutators in equation (2.5), especially the appearance of the central charge Z. Section 2.2 (pages 5-6), focusing on the setup for particle representations at rest. The derivation leading to the BPS bound, equations (2.15) through (2.17). The key step is realizing that the square of a Hermitian operator must be positive semi-definite.
The crucial insight from the reading comes from analyzing the supersymmetry algebra in the rest frame of a massive particle, where P^\mu = (M, 0, 0, 0). The anticommutator for the supercharges Q and their Hermitian conjugates $\bar{Q}$ schematically takes the form:
In the N=2 algebra, this is enriched by the central charge Z, which commutes with all other generators. A careful combination of the supercharges (as shown in eq. 2.12 in the text) leads to a relation like:
where $\mathcal{Q}$ is a specific linear combination of the original supercharges.
Since we are working in a unitary theory, the Hilbert space has a positive-definite norm. The operator { \mathcal{Q}, \mathcal{Q}^\dagger } is manifestly positive semi-definite because for any state $|\psi\rangle$:
Applying this to the relation $\{ \mathcal{Q}, \mathcal{Q}^\dagger \} = M - |Z|$, we find that the eigenvalue of this operator on any physical state must be non-negative. This forces the famous BPS bound:
This is a profound constraint: for any single-particle state in a theory with N=2 supersymmetry, its mass must be greater than or equal to the magnitude of its central charge.
2. Representation Shortening
The BPS bound immediately partitions the representations of the algebra into two categories. This is the principle of representation shortening.
-
Long Multiplets (
$M > |Z|$):
When the inequality is strict, all supercharges act non-trivially to create or annihilate states within the multiplet. This gives rise to a "long" or generic representation, which has the maximum number of states (forN=2, this is typically 16 states for a given momentum). -
Short (BPS) Multiplets (
$M = |Z|$):
When the bound is saturated, something special happens. The operator$\{ \mathcal{Q}, \mathcal{Q}^\dagger \}$must have an eigenvalue of zero. Since it is a sum of squared norms, this can only happen if the constituent norms are themselves zero. This implies that some of the supercharges must annihilate the state:This means that some supersymmetry is "preserved" by the state. When a generator acts as zero, it cannot create a new, independent state from
$|\psi_{BPS}\rangle$. Consequently, the resulting multiplet has fewer states than a long multiplet. The representation is "shorter".
PiTP Lectures on BPS States and Wall-Crossing in d = 4, N ...
The same lecture notes by Moore provide a clear distinction between these two types of representations.
Please read sections 2.2.1 (Long representations) and 2.2.2 (Short representations) on pages 7-9. Focus on the conceptual difference: In long representations, all supercharges are represented non-trivially. In short representations, the saturation of the bound forces some supercharges (the 'preserved supersymmetries') to be represented by zero, thus shortening the multiplet.
These short, or BPS, multiplets are the "rigid" states mentioned in the introduction of Moore's notes. Because their mass is fixed by their charge, their properties are often protected against quantum corrections, making them invaluable for exact calculations.
3. The Superconformal BPS Bound
Now we are ready to apply this logic to superconformal theories. The principles are the same, but the algebra and the resulting bound are different.
In a CFT, we classify states not by their mass but by their scaling dimension $\Delta$ and spin $j, \bar{\jmath}$. Representations are built from a superconformal primary state $|\phi\rangle$, which is the state in the multiplet with the lowest scaling dimension. By definition, it is annihilated by all generators that lower the scaling dimension:
The $S$ charges have $\Delta_S = -1/2$, so they indeed lower the dimension.
The BPS bound for SCFTs comes from considering the norm of a state created by acting with the $Q$ supercharge, $Q_\alpha|\phi\rangle$. This state has dimension $\Delta + 1/2$. The norm \(\|Q_\alpha|\phi\rangle\|^2 must be non-negative.
We can compute this norm using the superconformal algebra we derived in the last lesson. Schematically:
(This isn't quite right, we need $Q^\dagger$ which is $\bar{Q}$ or $\bar{S}$, but the logic is similar). The key anticommutator we derived was:
When we evaluate the expectation value of this on the primary state $|\phi\rangle$, the $D$ operator becomes the scaling dimension $\Delta$, $J$ becomes the spin, and $R$ becomes the R-charge. The requirement of a non-negative norm then translates into an inequality involving $\Delta$, spin, and R-charge.
This leads to the superconformal unitarity bounds, which are the BPS bounds for SCFTs.
Superconformal symmetry and representations
Let's now study this in detail using the lecture notes 'Superconformal symmetry and representations' by Leonardo Rastelli. This text provides a concise and clear derivation of these bounds and the structure of the resulting multiplets.
Please read the following sections: Section 5.1 (Structure of superconformal representations): This introduces the idea of a superconformal primary and how a superconformal multiplet contains several conformal multiplets. Section 5.2 (Unitarity bounds): This is the core of our lesson. It explains how unitarity (norm \ge 0) leads to bounds on $\Delta$. Pay attention to Figure 2, which illustrates the general structure of long (green region) and short (lines) multiplets. Glance at Table 1 and Table 2 for 4D N=1 SCFTs. You don't need to memorize them, but appreciate how they explicitly state the bound ($\Delta > ...$ for long multiplets) and the exact value ($\Delta = ...$ for short multiplets).
As the reading explains, the structure is richer than in the Poincaré case. For 4D N=1 theories, we can have different types of shortening conditions, leading to various families of short multiplets (denoted A1, A2, B1, etc. in Table 1).
The general principle remains the same:
- Long Multiplets: Have a scaling dimension
$\Delta$that strictly satisfies the unitarity bound (e.g.,$\Delta > 2 + j - \frac{3}{2}r$). They have no null states at the level of conformal primaries. - Short (BPS) Multiplets: Have a scaling dimension
$\Delta$that saturates the bound (e.g.,$\Delta = 2 + j - \frac{3}{2}r$). This saturation implies that a descendant state, like$Q|\phi\rangle$has zero norm and is a null vector. This null vector must be removed from the Hilbert space, "shortening" the representation.
Test your understanding!
A certain N=1 superconformal primary operator has spin j=0, $\bar{\jmath}=0$ and R-charge r=2. It is observed to have a scaling dimension $\Delta=1$ and is part of a multiplet denoted B1 $\bar{L}$. Looking at Table 2 from the LINK resource, we see that for a B1 $\bar{L}$ multiplet with j=0, the dimension is $\Delta = -3r/2$. In our case, this would mean $\Delta = -3(2)/2 = -3$, which contradicts $\Delta=1$.
What is the flaw in this reasoning? Hint: The tables list conditions on the quantum numbers.
Show answer
The flaw is in assuming that a B1 $\bar{L}$ multiplet can exist for any r. If you look at the header for that column in Table 2, you'll see the condition r < -(\bar{\jmath}+2)/3.
For our operator with $\bar{\jmath}=0$, this condition becomes r < -2/3. Our operator has r=2, which violates this condition. Therefore, an operator with (j=0, $\bar{\jmath}=0$, r=2) cannot belong to a B1 $\bar{L}$ multiplet.
Instead, let's look for a multiplet that can accommodate (j=0, $\bar{\jmath}=0$, r=2). The A2 $\bar{B1}$ multiplet requires r=2/3. The A2 $\bar{A2}$ multiplet requires r=0. The A2 $\bar{A1}$ multiplet requires r = - \bar{\jmath}/3 = 0. This exercise shows that the existence of short multiplets is highly constrained by the quantum numbers of the primary operator. The operator in question would likely belong to a long multiplet, as it doesn't fit the strict criteria for these short multiplets.
Conclusion
In this lesson, we have explored one of the most powerful consequences of supersymmetry: the existence of BPS bounds and the corresponding shortened representations.
Key Takeaways:
- The BPS Bound: Unitarity (the positivity of the Hilbert space norm) imposes a lower bound on the mass or scaling dimension of a state in terms of its charges.
- In Poincaré SUSY:
$M \ge |Z|$. - In Superconformal SUSY:
$\Delta \ge f(j, R)$, where$f$is a function of spin and R-charge.
- In Poincaré SUSY:
- Representation Shortening: States that saturate the BPS bound belong to "short" multiplets. The saturation implies that some supercharges annihilate the state, leading to null vectors in the representation space. Quotienting out these null states reduces the total number of states in the multiplet.
- Long vs. Short: States that strictly obey the inequality (
$M > |Z|$or$\Delta > f(j, R)$) belong to "long" multiplets, which have a generic number of states. - Physical Importance: Short (BPS) multiplets are "rigid". Their properties are often protected from quantum corrections, making them essential for obtaining exact, non-perturbative results in QFT and for making precise statements in the AdS/CFT correspondence.
Next Lesson Preview:
We have now defined the abstract conditions for shortening. In the next lesson, we will make this concrete. Our goal will be to identify BPS operators in simple SCFTs by applying these shortening conditions. We will see how to check if a given operator, like a chiral primary in N=2 theory, satisfies the $\Delta = f(R)$ relation and is therefore a BPS operator. This will be our first step towards performing practical computations in SCFTs.