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M2 Determinant Lines and Twisted Curve Euler Characteristics

Physical scope

Pure mathematical example of information loss under the determinant map. The bundles are not competing physical compactifications or membrane vacua.

Statement

Exhibit trivial rank-one and rank-two bundles over a curve with the same trivial determinant, but twisted holomorphic Euler characteristics two and four after tensoring with the same fixed line. This is information lost by the determinant line, not evidence for competing physical compactifications.

Construction

Let $C = {C P}^{1}$ with its standard complex structure, choose a point $p \in C$ , let $D = [p]$ be the degree-one divisor, and put $L = O_{C} (D) ≅ O (1)$ . Set $E_{1} = O_{C}, E_{2} = O_{C} \oplus O_{C} .$ Give each trivial summand its constant frame. Their determinants are both the same trivial line, while their ranks are one and two. Here the topological genus convention is $χ_{t o p} (C) = 2 - 2 g$ for a compact connected oriented surface; ${C P}^{1}$ is the sphere and has genus zero.

For $G = L$ or $L \oplus L$ , let $O (G)$ be its sheaf of holomorphic sections, put $h^{q} (C, G) : = \dim_{C} H^{q} (C, O (G))$ , and use the curve convention $χ (C, G) : = h^{0} (C, G) - h^{1} (C, G)$ . Finiteness for these particular bundles is established below; no rank-two Riemann–Roch theorem is presumed. The determinant of a rank- $r$ bundle is its top exterior power $Λ^{r} G$ .

Facts & Assumptions

Given: Full AC and the bundles in the Construction.

[F1]

The Riemann-Roch theorem on a compact Riemann surface: assuming full AC, for a compact Riemann surface of genus $g$ and divisor $D$ , $χ (O (D)) = \deg D + 1 - g$ .

[F2]

The holomorphic line bundle associated to a divisor: a divisor $D$ determines the holomorphic line bundle $O (D)$ , and $O (0)$ is the trivial line.

[F3]

Divisors, principal divisors and canonical divisors on a Riemann surface: on a compact Riemann surface a divisor has finite support and degree equal to the sum of its coefficients.

[F4]

Riemann surfaces and holomorphic atlases: a Riemann surface is a nonempty connected Hausdorff second-countable complex one-manifold; the standard complex structure on ${C P}^{1}$ is such a surface.

[F5]

M2 Combined Fermion Cfield Descent Data: the Euclidean descent package is mathematical data and does not assert physical realization or supply a physical vacuum multiplicity.

[F6]

Sheaf cohomology as right derived global sections: under AC, sheaf cohomology is computed by global sections of an injective resolution, independently of the resolution up to natural isomorphism. Scalar multiplication induces the complex vector-space structure for sheaves of complex vector spaces.

Proof

1.1givenF2algebra

The determinant of the rank-one line $E_{1}$ is $O_{C}$ . The determinant of $E_{2}$ is $O_{C} \otimes O_{C} ≅ O_{C}$ , with the tensor of the constant frames. Thus the determinant lines agree, while $rank E_{1} = 1$ and $rank E_{2} = 2$ .

1.2F1F2F3F4algebra

The standard charts $z$ on $C$ and $w = 1 / z$ near infinity have holomorphic transition on $C^{*}$ , so they give the Riemann-surface structure in [F4]. The one-point compactification of $C$ is homeomorphic to $S^{2}$ by stereographic projection; its cell structure has one zero-cell and one two-cell, hence Euler characteristic $2 = 2 - 2 g$ and genus $g = 0$ . The single point divisor has $\deg D = 1$ by [F3]. Since $L = O_{C} (D)$ by [F2], Riemann–Roch [F1] gives $χ (C, L) = 1 + 1 - 0 = 2$ .

2.1F1F5F6step 1.1step 1.2algebra∎

To verify finite direct-sum additivity, choose an injective resolution $L \to I^{∙}$ as in [F6]. A finite direct sum of injective sheaves is injective: extension of a map along a monomorphism can be performed separately in each component. Hence $L \oplus L \to I^{∙} \oplus I^{∙}$ is an injective resolution. Global sections split componentwise, and the kernels and images of the componentwise differential split. Thus $H^{q} (C, L \oplus L) = H^{q} (C, L) \oplus H^{q} (C, L)$ for each $q$ , so the degree-zero and degree-one groups are finite by [F1], and $χ (C, L \oplus L) = χ (C, L) + χ (C, L) = 4$ . Therefore the twisted rank-one and rank-two bundles have holomorphic Euler characteristics two and four despite having the same untwisted determinant line. The determinant calculation is a mathematical information-loss example; by [F5] it constructs no physical vacuum.

Sources

Freed–Moore, Setting the Quantum Integrand of M-Theory, §1.3: determinant/Pfaffian line context only; the rank and curve-index calculation is local and uses the pinned Riemann–Roch supplier.

Depends on

Dependency tree · 2 levels

Sources

Daniel S. Freed and Gregory W. Moore, Setting the Quantum Integrand of M-Theory