This is a bit of a repackaging of the same info in the other answer, but maybe it will be more clear.

The short answer is (almost) both: A fermion is a section of the parity shifted spinor bundle on a manifold. As such, you can't have a fermion without a spin structure.

Each aspect of this can be considered separately: there is no classical reason that an anti-commutative field has to be a section of the spinor bundle, and there is no reason that a section of the spinor bundle has to be anti-commutative. However, in physics the Spin-Statistics Theorem says that to have a consistent, Lorentz-invariant theory in >2 spatial dimensions, all anti-commutative fields must be spinors (have half-integral spin).

However, you only need the parity shifted bundle here. The full formalism of supermanifolds is for when you have supersymmetry, which is an odd (ie, anti-commuting) symmetry that relates bosons and fermions.

You can look at this in two ways. The first is as supersymmetric quantum mechanics, where you have maps from, say, the super manifold $\mathbb{R}^{1|1}$ to a Riemannian manifold. Here, the need for a spin-structure arises when you try to quantize theory in order to patch together the Clifford algebras that arise on each local chart.

The second way to look at this is to have your fields be functions on a supermanifold. Here, the supermanifold is modeled on super-Minkowski space, which is acted on by the super-Poincare group. In super-Minkowski space, the odd part is (some number of copies of) the parity shifted the spinor bundle, so the need for the spin-structure is part of the definition.

Dan Freed's notes on this stuff are very good.

This is a bit of a repackaging of the same info in the other answer, but maybe it will be more clear.

The short answer is (almost) both: A fermion is a section of the parity shifted spinor bundle on a manifold. As such, you can't have a fermion without a spin structure.

Each aspect of this can be considered separately: there is no classical reason that an anti-commutative field has to be a section of the spinor bundle, and there is no reason that a section of the spinor bundle has to be anti-commutative. However, in physics the Spin-Statistics Theorem says that to have a consistent, Lorentz-invariant theory in >2 spatial dimensions, all anti-commutative fields must be spinors (have half-integral spin).

However, you only need the parity shifted bundle here. The full formalism of supermanifolds is for when you have supersymmetry, which is an odd (ie, anti-commuting) symmetry that relates bosons and fermions.

You can look at this in two ways. The first is as supersymmetric quantum mechanics, where you have maps from, say, the super manifold $\mathbb{R}^{1|1}$ to a Riemannian manifold. Here, the need for a spin-structure arises when you try to quantize theory in order to patch together the Clifford algebras that arise on each local chart.

The second way to look at this is to have your fields be functions on a supermanifold. Here, the supermanifold is modeled on super-Minkowski space, which is acted on by the super-Poincare group. In super-Minkowski space, the odd part is (some number of copies of) the parity shifted the spinor bundle, so the need for the spin-structure is part of the definition.

Dan Freed's notes Classical field theory and supersymmetry on this stuff are very good.

This is a bit of a repackaging of the same info in the other answer, but maybe it will be more clear.

The short answer is (almost) both: A fermion is a section of the parity shifted spinor bundle on a manifold. As such, you can't have a fermion without a spin structure.

Each aspect of this can be considered separately -- there is no classical reason that a anti-commutative field has to be a section of the spinor bundle and there is no reason that a section of the spinor bundle has to be anti-commutative. However, in physics the Spin-Statistics Theorem says that to have a consistent, Lorentz-invariant theory in >2 spatial dimensions, all anti-commutative fields must be spinors (have half-integral spin).

However, you only need the parity shifted bundle here. The full formalism of supermanifolds is for when you have supersymmetry, which is an odd (ie, anti-commuting) symmetry that relates bosons and fermions.

You can look at this in two ways. The first is as supersymmetric quantum mechanics, where you have maps from, say, the super manifold $\mathbb{R}^{1|1}$ to a Riemannian manifold. Here, the need for a spin-structure arises when you try to quantize theory in order to patch together the Clifford algebras that arise on each local chart.

The second way to look at this is to have your fields be functions on a supermanifold. Here, the supermanifold is modeled on super-Minkowski space, which is acted on by the super-Poincare group. In super-Minkowski space, the odd part is (some number of copies of) the parity shifted the spinor bundle, so the need for the spin-structure is part of the definition.

Dan Freed's notes on this stuff are very good.

This is a bit of a repackaging of the same info in the other answer, but maybe it will be more clear.

The short answer is (almost) both: A fermion is a section of the parity shifted spinor bundle on a manifold. As such, you can't have a fermion without a spin structure.

Each aspect of this can be considered separately: there is no classical reason that an anti-commutative field has to be a section of the spinor bundle, and there is no reason that a section of the spinor bundle has to be anti-commutative. However, in physics the Spin-Statistics Theorem says that to have a consistent, Lorentz-invariant theory in >2 spatial dimensions, all anti-commutative fields must be spinors (have half-integral spin).

However, you only need the parity shifted bundle here. The full formalism of supermanifolds is for when you have supersymmetry, which is an odd (ie, anti-commuting) symmetry that relates bosons and fermions.

You can look at this in two ways. The first is as supersymmetric quantum mechanics, where you have maps from, say, the super manifold $\mathbb{R}^{1|1}$ to a Riemannian manifold. Here, the need for a spin-structure arises when you try to quantize theory in order to patch together the Clifford algebras that arise on each local chart.

The second way to look at this is to have your fields be functions on a supermanifold. Here, the supermanifold is modeled on super-Minkowski space, which is acted on by the super-Poincare group. In super-Minkowski space, the odd part is (some number of copies of) the parity shifted the spinor bundle, so the need for the spin-structure is part of the definition.

Dan Freed's notes on this stuff are very good.