12.2 Lie superalgebras and supersymmetry

Lie superalgebras extend Lie algebras by incorporating both bosonic and fermionic elements. This powerful mathematical framework unifies the description of particles with different spins, providing a foundation for supersymmetry in physics.

Supersymmetry, a key application of Lie superalgebras, proposes a symmetry between bosons and fermions. This concept has far-reaching implications in particle physics and quantum field theory, potentially addressing fundamental questions about the nature of our universe.

Z2-grading in Lie superalgebras

Definition and properties of Z2-grading

Top images from around the web for Definition and properties of Z2-grading

• 

  A Class of Lie 2-Algebras in Higher-Order Courant Algebroids View original

  Is this image relevant?

• 

  Frontier Orbitals, Combustion and Redox Transfer from a Fermionic-Bosonic Orbital Perspective View original

  Is this image relevant?

• 

  Construction of bosonic symmetry-protected-trivial states and their topological terms via $G ... View original

  Is this image relevant?

• 

  A Class of Lie 2-Algebras in Higher-Order Courant Algebroids View original

  Is this image relevant?

• 

  Frontier Orbitals, Combustion and Redox Transfer from a Fermionic-Bosonic Orbital Perspective View original

  Is this image relevant?

1 of 3

Top images from around the web for Definition and properties of Z2-grading

• 

  A Class of Lie 2-Algebras in Higher-Order Courant Algebroids View original

  Is this image relevant?

• 

  Frontier Orbitals, Combustion and Redox Transfer from a Fermionic-Bosonic Orbital Perspective View original

  Is this image relevant?

• 

  Construction of bosonic symmetry-protected-trivial states and their topological terms via $G ... View original

  Is this image relevant?

• 

  A Class of Lie 2-Algebras in Higher-Order Courant Algebroids View original

  Is this image relevant?

• 

  Frontier Orbitals, Combustion and Redox Transfer from a Fermionic-Bosonic Orbital Perspective View original

  Is this image relevant?

1 of 3

• A Lie superalgebra is a Z2-graded vector space $g = g_0 \oplus g_1$
  • $g_0$ represents the even part (bosonic)
  • $g_1$ represents the odd part (fermionic)
• The Z2-grading is compatible with the Lie bracket $[g_i, g_j] \subseteq g_{i+j} \text{ (mod 2)}$ for $i, j \in \{0, 1\}$
  • Ensures the closure of the Lie bracket within the superalgebra
• Elements in $g_0$ are called even (bosonic) elements, while elements in $g_1$ are called odd (fermionic) elements
  • Bosonic elements follow commutation relations (like in ordinary Lie algebras)
  • Fermionic elements follow anticommutation relations

Graded versions of anticommutativity and Jacobi identity

• The Lie bracket in a Lie superalgebra satisfies graded versions of anticommutativity and the Jacobi identity
• Graded anticommutativity: $[x, y] = -(-1)^{|x||y|}[y, x]$
  • $|x|$ and $|y|$ represent the Z2-grading of elements $x$ and $y$ (0 for even, 1 for odd)
  • Reduces to ordinary anticommutativity for even elements and commutation for odd elements
• Graded Jacobi identity: $(-1)^{|x||z|}[[x, y], z] + (-1)^{|y||x|}[[y, z], x] + (-1)^{|z||y|}[[z, x], y] = 0$
  • Ensures the consistency of the Lie bracket with the Z2-grading
  • Reduces to the ordinary Jacobi identity for even elements

Universal enveloping algebra

• The universal enveloping algebra of a Lie superalgebra is a Z2-graded associative algebra
  • Extends the Lie superalgebra to an associative algebra while preserving the Z2-grading
• The multiplication in the universal enveloping algebra respects the grading
  • Multiplying an even element with any element preserves the grading
  • Multiplying two odd elements results in an even element

Simple Lie superalgebras and representations

Classification of simple Lie superalgebras

• Simple Lie superalgebras are classified into two types: classical and Cartan type
• Classical Lie superalgebras include:
  • The special linear Lie superalgebra $\mathfrak{sl}(m|n)$
    • Generalizes the special linear Lie algebra $\mathfrak{sl}(n)$
  • The orthosymplectic Lie superalgebra $\mathfrak{osp}(m|2n)$
    • Combines orthogonal and symplectic Lie algebras
  • The periplectic Lie superalgebra $\mathfrak{p}(n)$
  • The queer Lie superalgebra $\mathfrak{q}(n)$
• Cartan type Lie superalgebras are denoted by $W(n)$, $S(n)$, $H(n)$, and $K(n)$
  • Generalizations of the Witt, special, Hamiltonian, and contact Lie algebras

Representations of Lie superalgebras

• Representations of Lie superalgebras are Z2-graded modules $V = V_0 \oplus V_1$
  • $V_0$ represents the even (bosonic) subspace
  • $V_1$ represents the odd (fermionic) subspace
• The action of the Lie superalgebra $g$ on the module $V$ respects the Z2-grading
  • Even elements of $g$ map even (odd) elements of $V$ to even (odd) elements
  • Odd elements of $g$ map even (odd) elements of $V$ to odd (even) elements
• Irreducible representations of classical Lie superalgebras can be classified using highest weight theory
  • Similar to the classification of representations for ordinary Lie algebras
  • Highest weight vectors and Verma modules play a crucial role in the classification

Lie superalgebras and supersymmetry

Supersymmetry in particle physics

• Supersymmetry is a proposed symmetry between bosons and fermions in particle physics
  • Extends the Poincaré algebra (symmetries of special relativity) to a Lie superalgebra
  • Introduces superpartners for each particle: bosons have fermionic partners, and fermions have bosonic partners
• The generators of supersymmetry, called supercharges, are odd elements of the Lie superalgebra
  • Satisfy anticommutation relations, unlike the commutation relations of the Poincaré generators
• The even part of the supersymmetry algebra contains the generators of the Poincaré algebra
  • Translations, rotations, and boosts
• The odd part of the supersymmetry algebra contains the supercharges
  • Relate bosonic and fermionic states within a supermultiplet

Supersymmetric theories and supermultiplets

• Representations of the supersymmetry algebra lead to supermultiplets
  • Contain both bosonic and fermionic states related by the action of supercharges
  • Examples include the chiral supermultiplet (scalar boson + Weyl fermion) and the vector supermultiplet (vector boson + Majorana fermion)
• Supersymmetric quantum field theories (QFTs) and supergravity theories are constructed using Lie superalgebras and their representations
  • Supersymmetric QFTs extend the Standard Model by including superpartners for each particle
  • Supergravity theories combine supersymmetry with general relativity, aiming to unify all fundamental interactions

Structure and applications of Lie superalgebras

Problem-solving involving Lie superalgebras

• Determine the Z2-grading and Lie bracket relations for given examples of Lie superalgebras
  • Identify the even and odd subspaces and their dimensions
  • Compute the Lie bracket between basis elements and verify the graded Jacobi identity
• Construct the root system and Dynkin diagram for classical Lie superalgebras
  • Extend the concept of roots and simple roots from ordinary Lie algebras
  • Use the Cartan matrix and Dynkin diagram to classify classical Lie superalgebras
• Classify irreducible representations of classical Lie superalgebras using highest weight theory
  • Determine the highest weight vectors and construct Verma modules
  • Identify the irreducible quotients of Verma modules and their characters

Applications in physics and mathematics

• Apply Lie superalgebras to solve problems in supersymmetric quantum mechanics and quantum field theory
  • Construct supersymmetric Hamiltonians and study their spectra
  • Analyze the structure of supersymmetric vacua and spontaneous supersymmetry breaking
• Use the representation theory of Lie superalgebras to study the structure of supermultiplets in supersymmetric theories
  • Classify supermultiplets based on their highest weights and R-symmetry representations
  • Investigate the relations between different supermultiplets using tensor product decompositions
• Apply Lie superalgebras in mathematical areas such as topology, geometry, and representation theory
  • Study the cohomology of Lie superalgebras and their representations
  • Investigate supergeometry and supermanifolds using Lie supergroups and superalgebras