# Supersymmetry v.s. BRST symmetry: QFT examples

Questions: Can any expert contrast the differences and similarities of

Supersymmetry (SUSY) v.s. BRST (global) symmetry?

(Question 1) What are the RULES and CRITERIA that having one symmetry implies that having the other symmetry? SUSY v.s. BRST (global) symmetry.

(Question 2) Is it true that we can only have a BRST charge $$Q$$? But we may have many SUSY charge $$Q_i$$. Are the two $$Q$$ charges related in some way?

Here let me give two QFT elementary examples and list some properties.

Ex 1. SUSY example below (P&S QFT p.74): A complex $$\mathbb{C}$$ Lorentz scalar field $$\phi$$, a complex $$\mathbb{C}$$ Lorentz Weyl spinor field $$\chi$$, a complex $$\mathbb{C}$$ Lorentz scalar auxiliary field $$F$$. And an anti-commuting complex $$\mathbb{C}$$ Lorentz spinor field $$\epsilon$$ as a SUSY global transformation parameter.

Ex 2. BRST (global) symmetry example below (P&S QFT p.517): A real $$\mathbb{R}$$ Lorentz 4-vector gauge field $$A^a$$, a real $$\mathbb{R}$$ Lorentz scalar auxiliary field $$B^a$$, a complex $$\mathbb{C}$$ Lorentz Weyl spinor field $$\psi$$. And an anti-commuting complex $$\mathbb{C}$$ Lorentz scalar field $$\epsilon$$ as a BRST global transformation parameter.

• 1. Both have auxiliary field without their kinetic terms (so no dynamics?):

Ex 1 has complex $$\mathbb{C}$$ Lorentz scalar field $$F$$,

Ex 2 has a real $$\mathbb{R}$$ Lorentz scalar field $$B^a$$. (Just like $$A^a$$ is a real $$\mathbb{R}$$ field, but $$A^a$$ is a Lorentz 4-vector gauge field.)

• 2. Both have anti-commuting Grassman number $$\epsilon$$, but Ex 1 spinor and Ex 2 scalar:

Ex 1 has anti-commuting complex $$\mathbb{C}$$ Lorentz spinor field $$\epsilon$$ .

But Ex 2 has anti-commuting complex $$\mathbb{C}$$ Lorentz scalar field $$\epsilon$$.

• 3. Ex 2 has ghost field, but Ex 1 does not require ghost field:

Ex 2 ghost field has anti-commuting complex $$\mathbb{C}$$ Lorentz scalar field $$c$$.

• 4. Ex 1 needs not to be gauge theory, but Ex 2 is a (gauge-fixed constraint) gauge theory.

See how the BRST symmetry relates to gauge symmetry transformation.

• 5. Ex 1 has SUSY charge $$Q$$, while Ex 2 has BRST charge $$Q$$.

Are two $$Q$$ charges related in some way?

# Ex 2. BRST (global) symmetry example:

A supersymmetry is a Grassmann-odd symmetry that takes Grassmann-even objects into Grassmann-odd objects, and vice-versa.

Main examples of supersymmetry:

1. Poincare supersymmetry (often abbreviated as SUSY). The Poincare superalgebra is a $$\mathbb{Z}_2$$ graded extension of the Poincare algebra. The number of supercharges $$Q_A$$ are labelled by an integer $${\cal N}$$ times the number of components of the appropriate spinor (Dirac, Majorana, ...). The anticommutators of supercharges are proportional to momenta.

2. BRST supersymmetry (often called BRST symmetry). This encodes the gauge symmetry. There is only 1 BRST charge $$Q$$. It is Grassmann-odd, nilpotent and has ghostnumber 1.

The two above examples typically have nothing to do with each other per se, but they are allowed to co-exist (at least in component-form -- superfield formulation is more tricky). Both examples are $$x$$-independent/global symmetries, although global SUSY can sometimes be gauged into local SUSY aka. SUGRA.

• thanks +1 --- "SUSY is a Grassmann-odd symmetry that takes Grassmann-even objects into Grassmann-odd objects, and vice-versa." but isnt that BRST also does that? Oct 7, 2020 at 23:54
• BRST takes Grassmann-even objects into Grassmann-odd objects, and vice-versa, since both have anti-commuting Grassman number 𝜖, but Ex 1 spinor and Ex 2 scalar Oct 7, 2020 at 23:55
• Does my Ex 1 count as Poincare supersymmetry? Oct 8, 2020 at 3:17