Supersymmetry: Mathematical Framework — Mathematics Knowledge System

Overview

Supersymmetry (SUSY) is a symmetry that interchanges bosons (integer spin) and fermions (half-integer spin). Mathematically, it is formalized via superalgebras (Z_2-graded algebras) and supergeometry (geometry on supermanifolds where coordinates include anticommuting Grassmann variables). The super-Poincare algebra extends the Poincare algebra with fermionic generators Q (supercharges). Mathematically, SUSY has been productive in pure mathematics: Witten's proof of the Morse inequalities via SUSY quantum mechanics, the positive energy theorem in GR, and mirror symmetry (which relates topologically distinct Calabi-Yau manifolds via string theory duality).

Intuition

A superalgebra has two types of generators: bosonic (commuting, even graded) and fermionic (anticommuting, odd graded). Grassmann variables theta satisfy theta^2 = 0 (and theta_i * theta_j = -theta_j * theta_i for i != j). Superspace adds these anticommuting directions to ordinary spacetime. A superfield F(x, theta) = f(x) + theta*psi(x) + theta^2*F(x) encodes bosonic (f, F) and fermionic (psi) components in one object. SUSY quantum mechanics: the Hamiltonian H = Q*Q^+ + Q^+*Q with supercharge Q makes the ground state energy non-negative, and zero-energy states are topologically robust (Morse theory).

Formal Definition

Definition

A Lie superalgebra g = g_0 + g_1 (Z_2-graded vector space) with a Z_2-graded bracket [,] satisfying: [a,b] = -(-1)^{|a||b|}[b,a] and the super Jacobi identity. The super-Poincare algebra adds supercharges Q_alpha (alpha=1,2 for 4D N=1 SUSY) to the Poincare algebra: {Q_alpha, Q-bar_{dot{beta}}} = 2*sigma^mu_{alpha dot{beta}} * P_mu, {Q_alpha, Q_beta} = 0. A supermanifold M = (|M|, O_M) is a manifold |M| with a sheaf of supercommutative algebras O_M = functions in x_1,...,x_n, theta_1,...,theta_m with theta_i*theta_j = -theta_j*theta_i.

\{Q_\alpha, \bar{Q}_{\dot{\beta}}\} = 2\sigma^\mu_{\alpha\dot{\beta}}P_\mu

Super-Poincare algebra (SUSY anticommutator)

\{Q_\alpha, Q_\beta\} = \{\bar{Q}_{\dot{\alpha}}, \bar{Q}_{\dot{\beta}}\} = 0

Supercharge nilpotency conditions

H = \{Q, Q^\dagger\} = QQ^\dagger + Q^\dagger Q \ge 0

SUSY Hamiltonian (non-negative)

\Phi(x,\theta,\bar{\theta}) = \phi(x) + \theta\psi(x) + \theta^2 F(x) + \cdots

Chiral superfield expansion

Notation

Notation	Meaning
$Q_\alpha$	Supercharge (fermionic generator)
$\theta_\alpha$	Grassmann (anticommuting) coordinate on superspace
$\mathfrak{g}_0, \mathfrak{g}_1$	Even and odd parts of a Lie superalgebra
$H = \{Q,Q^\dagger\}$	Supersymmetric Hamiltonian

Theorems

Theorem 1: Haag-Lopuszanski-Sohnius Theorem

Any Lie superalgebra containing the Poincare algebra as its even part, consistent with analyticity of the S-matrix and with finite-dimensional representations for massive particles, must be a super-Poincare algebra (possibly with extended SUSY N > 1 and central charges). This 'no-go theorem' says SUSY is essentially the only non-trivial extension of the Poincare group.

Theorem 2: Witten's SUSY Proof of Morse Inequalities

For a smooth function f on a Riemannian manifold M, define the deformed exterior derivative d_t = e^{-tf} d e^{tf} and the Witten Laplacian H_t = d_t d_t^* + d_t^* d_t. The number of critical points of f of index k equals the number of small eigenvalues of H_t^{(k)} in the limit t -> inf. This gives: (number of index-k critical points) >= b_k (Morse inequalities), and the equality constraints (strong Morse inequalities). The proof uses the SUSY quantum mechanics Hamiltonian H = d_t^* d_t.

Theorem 3: Positive Energy Theorem

In general relativity, the ADM mass (total energy) of an isolated gravitating system is non-negative: E_ADM >= 0, with equality iff the spacetime is Minkowski space. Witten's 1981 proof used the SUSY algebra structure: E_ADM = ||Q|psi>||^2 >= 0 for a suitable spinor field |psi>. This was the first clean proof of the positive energy theorem; previous proofs by Schoen-Yau used different methods.

Worked Examples

1
H = Q*Q† + Q†*Q. For any state |psi>: <psi|H|psi> = <psi|Q*Q†|psi> + <psi|Q†*Q|psi> = ||Q†|psi>||^2 + ||Q|psi>||^2 >= 0.
$\langle\psi|H|\psi\rangle = \|Q^\dagger|\psi\rangle\|^2 + \|Q|\psi\rangle\|^2 \ge 0$
2
If H|psi> = 0, then <psi|H|psi> = 0, so Q|psi> = 0 and Q†|psi> = 0.
3
Therefore the state is annihilated by both Q and Q† -- it is invariant under supersymmetry transformations generated by Q.
$H|\psi\rangle=0 \Rightarrow Q|\psi\rangle=0 \text{ and } Q^\dagger|\psi\rangle=0$
4
Such states are BPS (Bogomolny-Prasad-Sommerfield) or SUSY-invariant states. Their number (the Witten index Tr(-1)^F) is a topological invariant.

✓ Answer

H = {Q, Q†} >= 0 since H = ||Q|psi>||^2 + ||Q†|psi>||^2 >= 0. Zero-energy states satisfy Q|psi>=0 and Q†|psi>=0 (SUSY-invariant).

Practice Problems

Hardfree response

Explain the Witten index and why it is a topological invariant of the SUSY quantum mechanics system.

Common Mistakes

Common Mistake

Thinking Grassmann variables theta satisfy theta != 0 but theta^2 = 0 implies they are nilpotent matrices.

Grassmann variables are abstract generators of a supercommutative algebra, not matrices. theta_i * theta_j = -theta_j * theta_i for all i,j (including i=j, giving theta_i^2 = -theta_i^2 = 0). They are not nilpotent matrices -- they live in an abstract (noncommutative) algebra. When computing with them, one uses the rules of the exterior algebra (wedge product): a function f(theta_1,...,theta_n) is always a polynomial of degree <= n.

Quiz

The SUSY relation {Q_alpha, Q-bar_beta-dot} = 2 sigma^mu P_mu implies:

Supercharges are nilpotentThe square of a supercharge is the HamiltonianTranslations are generated by squares of superchargesSupercharges commute with the Hamiltonian

Historical Background

Gol'fand and Likhtman (1971) first wrote down the superalgebra extension of the Poincare algebra. Wess and Zumino (1974) wrote the first 4D supersymmetric field theory. Haag, Lopuszanski, and Sohnius (1975) proved the HLS theorem: SUSY is the only non-trivial extension of the Poincare algebra consistent with S-matrix theory. In mathematics, Witten (1982) used SUSY quantum mechanics to prove the Morse inequalities. The Mirror Symmetry conjecture (1991) and its proof for certain cases (Kontsevich's homological mirror symmetry, 1994) have driven major developments in algebraic geometry.

1971
Gol'fand-Likhtman first write the super-Poincare algebra
Yuri Gol'fand, Evgeny Likhtman
1974
Wess-Zumino write first 4D SUSY field theory
Julius Wess, Bruno Zumino
1982
Witten uses SUSY QM to prove Morse inequalities
Edward Witten
1994
Kontsevich proposes homological mirror symmetry connecting symplectic and algebraic geometry
Maxim Kontsevich

Summary

SUSY mathematically: Lie superalgebras with Z_2-graded bracket, supercharges Q with {Q,Q-bar}=2P_mu.
SUSY QM: H = {Q,Q†} >= 0; zero-energy states are topological (Witten index = Tr(-1)^F).
Witten's Morse inequality proof: deformed Laplacian H_t counts critical points via SUSY.
Mathematical applications: positive energy theorem, mirror symmetry, index theorems.

References

BookFreed, D.S. Five Lectures on Supersymmetry. AMS, 1999.
BookHori, K. et al. Mirror Symmetry. Clay Mathematics Monographs, 2003.
WebsiteWikipedia -- Supersymmetry