Determinant line of an analytic family of worldsheet operators

Let $\pi :\mathcal{C}\to B$ be an analytic family of pointed compact worldsheets in the sense of Pointed analytic worldsheet and family, with $B$ a complex manifold, and let $E\to \mathcal{C}$ be a holomorphic vector bundle of finite rank. Fix an integer Sobolev order $s\ge 1$. For each $b\in B$, let ${\mathbb{E}}_b^s=H^s(C_b,E{\mathrm{\mid }}_{C_b}),{\mathbb{F}}_b^{s-1}=H^{s-1}(C_b,{\mathrm{\Lambda }}^{0,1}{T}^{\ast }{C}_b\otimes E{\mathrm{\mid }}_{C_b})\mathrm{.}$ Here $H^k(C_b,G)$ is the completion of smooth sections of $G$ in an integer-order Sobolev norm formed from supplied smooth metrics and a smooth connection on the compact fiber $C_b$. Require these completions to be locally trivial complex Hilbert bundles ${\mathbb{E}}^s,{\mathbb{F}}^{s-1}\to B$; in the holomorphic case their transition maps are holomorphic in operator norm. This Hilbert-bundle structure, including the local identifications used to compare fibers, is part of the family data.

A continuous, respectively holomorphic, operator family $D:{\mathbb{E}}^s\to {\mathbb{F}}^{s-1}$ is a bundle map whose local representatives $b\mapsto D_b:{\mathbb{E}}_b^s\to {\mathbb{F}}_b^{s-1}$ are norm-continuous, respectively holomorphic in operator norm, bounded complex-linear maps. Assume every $D_b$ is Fredholm in the sense of Fredholm operator cokernel and index: it has closed image and finite-dimensional kernel and cokernel ${\mathbb{F}}_b^{s-1}\mathrm{/}\mathrm{im}{D}_b$. The determinant-line construction is applied locally on $B$ using Finite stabilization defines the analytic Fredholm determinant line, which proves the Fredholm index is locally constant; no common index is imposed across distinct connected components. Write $\mathrm{coker}{D}_b={\mathbb{F}}_b^{s-1}\mathrm{/}\mathrm{im}{D}_b,\det (D_b)={\mathrm{\Lambda }}^{\max }\ker D_b\otimes ({\mathrm{\Lambda }}^{\max }\mathrm{coker}{D}_b{)}^{\ast }\mathrm{.}$

For $b_0\in B$, a finite-dimensional subspace $W\subset {\mathbb{F}}_{b_0}^{s-1}$ is a stabilizer if $\mathrm{im}{D}_{b_0}+W={\mathbb{F}}_{b_0}^{s-1}$. In a local trivialization of ${\mathbb{F}}^{s-1}$, regard $W$ as a constant finite-dimensional subbundle and set $K_W(b)=\{u\in {\mathbb{E}}_b^s:D_{b}u\in W\}$. On the neighborhood where $\mathrm{im}{D}_b+W={\mathbb{F}}_b^{s-1}$, the local stabilization chart is ${\mathrm{\Lambda }}^{\max }{K}_W(b)\otimes ({\mathrm{\Lambda }}^{\max }W{)}^{\ast }$. A determinant line bundle for this family means the line bundle with fibers $\det (D_b)$ and local charts given by these stabilization lines; the finite-stabilization theorem referenced below proves their canonical, choice-independent gluing.

Pointed analytic worldsheet and family

Finite stabilization defines the analytic Fredholm determinant line

Fredholm operator cokernel and index

Zinger, The Determinant Line Bundle for Fredholm Operators, §§3.1, 3.3, 4.3, 5.1

This convention is the dual of Quillen's displayed convention ${\mathrm{\Lambda }}^{\max }(\ker D{)}^{\ast }\otimes {\mathrm{\Lambda }}^{\max }(\mathrm{coker}D)$; Zinger §3.3 compares these conventions. The two lines are naturally dual, not silently identified. The usual index-zero determinant section belongs to the dual convention, so this item does not include that section. Kernel and cokernel dimensions may jump while stabilization keeps the line rank one. A Quillen metric, connection, anomaly formula, sewing isomorphism, and identification with a functional-integral determinant are additional structures and are not included in this definition.

In the worldsheet application, the Dolbeault operator ${\bar{\mathrm{\partial }}}_E$ is the operator encoding the given holomorphic structure on $E$, characterized by ${\bar{\mathrm{\partial }}}_E(fs)=\bar{\mathrm{\partial }}f\otimes s+f{\bar{\mathrm{\partial }}}_{E}s$ in a local holomorphic frame. A Cauchy–Riemann-type operator is a first-order operator with the same principal symbol. For a first-order operator $P$, its principal symbol is the bundle map ${\sigma }_P(x,\xi )$ determined by $P(fs)-fP(s)={\sigma }_P(x,df)s$ for smooth scalar $f$ and section $s$; $P$ is elliptic when ${\sigma }_P(x,\xi )$ is an isomorphism for every nonzero real cotangent vector $\xi$. The Dolbeault symbol is $s\mapsto {\xi }^{0,1}\otimes s$, which is an isomorphism on a complex curve for real $\xi \mathrm{\ne }0$. This contextual example does not establish that any particular family is bounded or Fredholm on the stated completions; those are explicit hypotheses above.