[Paper Review] On uniformization of N=2 superconformal and N=1 superanalytic DeWitt super-Riemann surfaces

Katrina Barron|arXiv (Cornell University)|Jul 17, 2008

Black Holes and Theoretical Physics47 references3 citations

TL;DR

This paper establishes a uniformization theorem for N=2 superconformal and N=1 superanalytic DeWitt super-Riemann surfaces, proving that such surfaces are equivalent to ringed-space supermanifolds (i.e., with no odd functions of even variables in transition functions) if and only if a specific first Čech cohomology group vanishes. The key result classifies these super-Riemann surfaces via holomorphic line bundles and theta functions, particularly showing that genus-zero N=2 superconformal DeWitt super-Riemann surfaces form a countably infinite family, while simply connected ones are classified by conformal equivalence of line bundles over the body.

ABSTRACT

We prove a general uniformization theorem for N=2 superconformal and N=1 superanalytic DeWitt super-Riemann surfaces, showing that in general an N=2 superconformal (resp. N=1 superanalytic) DeWitt super-Riemann surface is N=2 superconformally (resp., N=1 superanalytically) equivalent to a manifold with transition functions containing no odd functions of the even variable if and only if a certain cohomology group is trivial, namely the first Cech cohomology group of the body Riemann surface with coefficients in the sheaf consisting of the reciprocal of a line bundle tensor the holomorphic vector fields over the body. In particular, this gives a general criteria for when a DeWitt N=1 superanalytic super-Riemann surface is N=1 superanalytically equivalent to a ringed-space (1,1)-supermanifold, as studied in the algebro-geometric setting. This general classification result implies there is a countably infinite family of N=2 superconformal equivalence classes of N=2 superconformal DeWitt super-Riemann surfaces with genus-zero compact body, and N=2 superconformal DeWitt super-Riemann surfaces with simply connected body are classified up to N=2 superconformal equivalence by conformal equivalence classes of holomorphic line bundles over the underlying body Riemann surface. In addition, N=2 superconformal DeWitt super-Riemann surfaces with compact genus-one body and transition functions which correspond to the trivial cocycle in the first Cech cohomology group of the body Riemann surface with coefficients in the reciprocal of a line bundle tensor the sheaf of holomorphic vector fields over the body are classified up to N=2 superconformal equivalence by holomorphic line bundles over the torus modulo conformal equivalence. The corresponding results for the uniformization of N=1 superanalytic DeWitt super-Riemann surfaces of genus zero or one are presented.

Motivation & Objective

• To establish a general uniformization theorem for N=2 superconformal and N=1 superanalytic DeWitt super-Riemann surfaces.
• To determine the conditions under which such super-Riemann surfaces are equivalent to ringed-space supermanifolds (i.e., with transition functions free of odd functions of even variables).
• To classify N=2 superconformal DeWitt super-Riemann surfaces with compact genus-zero or genus-one bodies via holomorphic line bundles and theta functions.
• To extend the classification to N=1 superanalytic super-Riemann surfaces, particularly in genus zero and one.
• To provide a cohomological criterion for the existence of such uniformizations, linking superconformal geometry to algebraic geometry via sheaf cohomology.

Proposed method

• The paper uses the DeWitt supermanifold approach, allowing general transition functions that include odd functions of even variables, to model superconformal field theory worldsheets.
• It employs the Čech cohomology group $\check{H}^1(M_B, \mathcal{L}^{-1} \otimes TM_B)$ as the central cohomological invariant, where $M_B$ is the body Riemann surface and $\mathcal{L}$ is a holomorphic line bundle over $M_B$.
• The main result is derived by analyzing consistency conditions on triple overlaps in the homogeneous coordinate setting, which makes the cohomological dependence transparent.
• The classification of N=2 superconformal super-Riemann surfaces with genus-zero body is shown to be countably infinite, parameterized by holomorphic line bundles over $\mathbb{P}^1$.
• For genus-one bodies, the classification is given by theta functions modulo trivial ones, or equivalently, by holomorphic line bundles modulo conformal equivalence.
• The paper also uses the N=2 to N=1 equivalence established in [DRS] to extend results to N=1 superanalytic super-Riemann surfaces.

Experimental results

Research questions

• RQ1 Under what cohomological condition is an N=2 superconformal DeWitt super-Riemann surface N=2 superconformally equivalent to a ringed-space supermanifold?
• RQ2 How are N=2 superconformal DeWitt super-Riemann surfaces with genus-zero compact body classified up to equivalence?
• RQ3 What is the classification of N=2 superconformal DeWitt super-Riemann surfaces with simply connected body?
• RQ4 How are N=2 superconformal DeWitt super-Riemann surfaces with genus-one compact body and trivial Čech cocycle classified?
• RQ5 What is the corresponding classification for N=1 superanalytic DeWitt super-Riemann surfaces in genus zero and one?

Key findings

• An N=2 superconformal DeWitt super-Riemann surface is N=2 superconformally equivalent to a ringed-space supermanifold if and only if $\check{H}^1(M_B, \mathcal{L}^{-1} \otimes TM_B) = 0$.
• There exists a countably infinite family of N=2 superconformal equivalence classes of N=2 superconformal DeWitt super-Riemann surfaces with genus-zero compact body.
• N=2 superconformal DeWitt super-Riemann surfaces with simply connected body are classified up to N=2 superconformal equivalence by the conformal equivalence classes of holomorphic line bundles over the body Riemann surface.
• N=2 superconformal DeWitt super-Riemann surfaces with compact genus-one body and trivial Čech cocycle are classified by theta functions modulo trivial ones, or equivalently, by holomorphic line bundles over the torus modulo conformal equivalence.
• For genus-zero N=1 superconformal DeWitt super-Riemann surfaces, the classification depends on the spin structure: trivial spin structure is parameterized by $b_S \in (\bigwedge_{*-1}^0)_S$ and $\delta \in \bigwedge_{*-1}^1 / \langle \pm 1 \rangle$, while nontrivial spin structure gives one class per $b \in \bigwedge_{*-1}^0$ with $b_B = \tau$.
• For genus-one N=1 superconformal DeWitt super-Riemann surfaces, the moduli space is determined by holomorphic $GL(1)$-bundles over the torus, with classification via exponentiation of the $\mathfrak{u}(1)$ affine Lie algebra over the body manifold.