Theorem. Ordered exterior operators span the endomorphisms [fdip-0092]
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  • November 2, 2026

On every square torus the kernel of \(d+d^\dagger \) has dimension \(4\), by the Hodge theorem. For periods at least \(3\), the centered Dirac operator has kernel dimension \(2^{2+e}\). That dimension equals \(4\) when both periods are odd, and it is strictly larger when either period is even.

Let \(k\) be a commutative ring and let \(n\) be a positive integer. Write \(\bigwedge \) for the exterior algebra of the free module on basis vectors \(e_1,\ldots ,e_n\), with basis the monomials \(e_I\) indexed by subsets \(I\), written in increasing order, and with \(e_\varnothing =1\). Exterior multiplication and contraction along the basis are the linear operators \[\epsilon _i(e_I)=e_i\wedge e_I,\] \[\iota _i(e_I)=0\] when \(i\notin I\), and, when \(i\) occupies position \(p\) in \(I\), \[\iota _i(e_I)=(-1)^p e_{I\setminus \{i\}}.\] For subsets \(S\) and \(T\), write \(\epsilon _S\) and \(\iota _T\) for the compositions of these operators in increasing index order, the largest index acting first, and write \(M_{S,T}=\epsilon _S\circ \iota _T\). There are \(2^n\) choices for each subset, hence \(4^n\) such operators, and they are a basis of the endomorphism ring of \(\bigwedge \). In particular \(\gamma _i=\epsilon _i+\iota _i\) satisfies \[\gamma _i\gamma _j+\gamma _j\gamma _i=2\delta _{ij}I.\] The configuration Clifford algebra is the square quotient of a supplied quadratic form. The operators here are defined before any such form, and the displayed relation is the square relation of the standard dot product. The note does not claim that the resulting Clifford representation is injective.

Proof. 

  • November 2, 2026

The dimension \(4\) is the kernel dimension in the Hodge theorem. The centered count is the cited formula at \(d=2\). It equals \(4\) when \(e=0\). When \(e\ge 1\) the factor \(2^e\) makes it strictly larger. The two statements have the scopes already named.

The monomials are a basis, so operator identities may be checked on \(e_I\). If \(i\in I\), then \(e_i\wedge e_I=0\). If \(i\notin I\) and \(q\) indices of \(I\) are smaller than \(i\), then \(e_i\wedge e_I=(-1)^q e_{I\cup \{i\}}\). Thus \(\epsilon _i^2=0\). For \(i\ne j\) the wedge of the two basis vectors changes sign when their order is exchanged, so \(\epsilon _i\epsilon _j+\epsilon _j\epsilon _i=0\). A monomial contains \(i\) at most once, so \(\iota _i^2=0\).

For the contractions, take \(i\ne j\) and suppose both lie in \(I\), with positions \(p\) and \(r\) and with \(i<j\), so \(p<r\). Removing \(j\) first contributes \((-1)^r\), and \(i\) is still at position \(p\), so the second removal contributes \((-1)^p\). Removing \(i\) first contributes \((-1)^p\), after which \(j\) stands one place earlier, and the second removal contributes \((-1)^{r-1}\). The two signs are opposite, and the remaining monomial is the same. If either index is missing, both double contractions vanish. Hence \(\iota _i\iota _j+\iota _j\iota _i=0\).

Now compare \(\iota _j\epsilon _i\) with \(\epsilon _i\iota _j\). If \(i\ne j\) and \(j\notin I\), the contraction of the input is zero. Exterior multiplication either vanishes or produces a monomial that still omits \(j\), so the later contraction vanishes as well. If \(j\in I\) and \(i\in I\), exterior multiplication of the input vanishes, and the contraction still leaves \(i\) in place, so the later exterior multiplication vanishes. The remaining pattern is \(j\in I\) and \(i\notin I\). Let \(p\) be the position of \(j\) and let \(q\) be the number of indices of \(I\) smaller than \(i\). Removing \(j\) contributes \((-1)^p\). If \(i<j\), wedging \(e_i\) back contributes the same sign \((-1)^q\) that \(e_i\wedge e_I\) carries, while the later removal of \(j\) from \(I\cup \{i\}\) contributes \((-1)^{p+1}\), because \(i\) stands before \(j\). If \(i>j\), wedging \(e_i\) after the removal contributes \((-1)^{q-1}\), because \(j\) was one of the indices counted by \(q\), while the removal of \(j\) from \(I\cup \{i\}\) still contributes \((-1)^p\). In both subcases the two compositions land on the same monomial with opposite signs. If \(i=j\) and \(i\notin I\), the contraction of the input vanishes and \(\iota _i\epsilon _i\) returns \(e_I\), and the product of the two signs \((-1)^q\) is \(1\). If \(i\in I\), exterior multiplication of the input vanishes and \(\epsilon _i\iota _i\) returns \(e_I\), and the product of the two signs \((-1)^p\) is \(1\). Therefore \(\epsilon _i\iota _j+\iota _j\epsilon _i=\delta _{ij}I\).

The square of \(\gamma _i\) is \(\epsilon _i\iota _i+\iota _i\epsilon _i=I\). For \(i\ne j\) each of the 4 pairs in the expansion of \(\gamma _i\gamma _j+\gamma _j\gamma _i\) anticommutes, so the sum is zero. This is the displayed Clifford relation.

The monomials are a free basis, so the endomorphism ring is free on the operators that send one monomial to one monomial and kill the rest. Order the pairs of subsets so that the cardinality of \(T\) increases, and index both the operators \(M_{S,T}\) and these monomial matrix units by that order. The coefficient of \(e_S\) in \(M_{S',T'}e_T\) can be nonzero only when \(T'\subseteq T\). A column with a strictly larger second set therefore does not meet this row. When the cardinalities are equal, necessarily \(T'=T\). The contraction \(\iota _T\) removes every factor of \(e_T\). Each removal meets a present factor, so the image of the contraction is \((\pm 1)\,1\). The following exterior product inserts the factors in \(S'\), each into a monomial that does not yet contain it, and the image is \((\pm 1)\,e_{S'}\). The coefficient of \(e_S\) is therefore zero unless \(S'=S\), in which case it is \(\pm 1\). Within one cardinality the block is diagonal. A strictly smaller \(T'\) meets the row in an earlier column. The matrix is consequently lower triangular, with units \(\pm 1\) on the diagonal. Given a target endomorphism, its coefficients against the \(M_{S,T}\) are the unique solution of that system: start from the empty second set, invert the diagonal unit, and at each later row use the coefficients already found. The operators are therefore a basis.

For a single basis vector the 4 operators are the identity, exterior multiplication, contraction, and their composite. On a space of dimension \(2\) any 4 independent endomorphisms are a basis, and the triangular count supplies the independence.