Quantising Chiral Bosons On Riemann Surfaces

Technical Summary: Quantising Chiral Bosons On Riemann Surfaces

Problem Statement
The quantisation of chiral bosons (self-dual fields) in two dimensions presents significant challenges, particularly regarding the formulation of a consistent action on arbitrary Riemann surfaces and the achievement of modular invariance. While Sen's action [1, 2] provides a framework for chiral gauge fields in $4k+2$ dimensions using a physical metric $g$ and a flat auxiliary metric $\eta$, this formulation is limited to manifolds admitting a flat metric. Furthermore, the partition function for a single chiral boson is generally not modular invariant, and standard approaches often require imposing self-duality constraints by hand rather than deriving them from the action. The authors aim to generalise Sen's action to a bi-metric formulation where the auxiliary field couples to an arbitrary second metric $\bar{g}$, allowing for the quantisation of chiral scalars on any two-dimensional manifold, specifically Riemann surfaces, and to investigate the resulting modular properties and anomalies.

Methodology
The paper employs the bi-metric generalisation of Sen's action [3], adapted for two-dimensional chiral scalars ($k=0$). The methodology proceeds through the following steps:

1. Action Formulation: The authors utilise an action involving a physical scalar $P$ coupled to a physical metric $g$ and a "shadow" scalar $C$ coupled to a second metric $\bar{g}$. The metrics are parameterised using complex variables $\tau, \tilde{\tau}$ for $g$ and $\rho, \tilde{\rho}$ for $\bar{g}$. The tensor $M$, which encodes the dependence on the metrics, is explicitly evaluated to ensure the action is well-defined.
2. Analytic Continuation: To define the path integral, the authors analytically continue the real Lorentzian and Euclidean metrics to complex metrics. They identify regions in the parameter space of $\tau, \tilde{\tau}, \rho, \tilde{\rho}$ where the functional integral is convergent (specifically requiring $\text{Im}(\rho) < 0$ and $\text{Im}(\tau) > 0$ for certain conditions). The results are then analytically continued to the full parameter space.
3. Path Integral and Canonical Quantisation: The partition function is calculated via two methods:
   • Path Integral: The auxiliary field is integrated out, reducing the system to a non-chiral scalar coupled to a complex metric $h = e^{2\sigma'}(dx + \rho dy)(dx + \tau dy)$. The path integral is evaluated on a torus by separating zero-modes (winding and momentum) and non-zero oscillator modes. Zeta-function regularisation is used to handle divergent products.
   • Canonical Quantisation: The system is quantised on a spatial circle, treating $y$ as time. The Hilbert space is shown to factorise into left-moving ($\tau$) and right-moving ($\rho$) sectors, allowing the partition function to be computed as a trace over the decoupled Hamiltonians.
4. Modular Analysis: The transformation properties of the partition function under modular transformations of the moduli $\tau$ and $\rho$ are examined. The authors distinguish between diagonal modular transformations (where $\tau$ and $\rho$ transform identically) and independent transformations.
5. Extension to Higher Genus: The path integral is generalised to arbitrary Riemann surfaces by analysing the Laplacian associated with the complex metric and evaluating contributions from zero-modes (periods over 1-cycles) and non-zero modes.

Key Contributions and Results

• Effective Action and Complex Metric: The authors demonstrate that the bi-metric action for chiral bosons reduces to an effective action for a non-chiral scalar coupled to a complex metric $h$. This allows the use of standard techniques for non-chiral scalars to study chiral systems.
• Partition Function Structure:
  • For a general chiral boson on a circle or torus, the partition function $Z(\tau, \rho)$ depends on both moduli. In the general case, it is a sum of terms that do not factorise.
  • For a rational torus (where the compactification radius squared is rational), the partition function becomes a finite sum of products of holomorphic functions: $Z(\tau, \rho) = \sum C_{a\bar{b}} \chi_a(\tau) \chi_{\bar{b}}(\rho)$.
  • Crucially, for $8k$ chiral bosons associated with an even self-dual lattice (such as the root lattice of $E_8$ or the weight lattice of $Spin(32)/\mathbb{Z}_2$), the partition function factorises completely into a single product: $Z(\tau, \rho) = F(\tau) \tilde{F}(\rho)$.
• Modular Invariance and Anomalies:
  • The partition function is shown to be invariant under the diagonal subgroup of the modular group $SL(2, \mathbb{Z})_\tau \times SL(2, \mathbb{Z})_\rho$, corresponding to coordinate transformations.
  • The authors identify gravitational anomalies associated with both the physical metric $g$ and the shadow metric $\bar{g}$.
  • They propose a mechanism to achieve full modular invariance and anomaly cancellation by introducing a "shadow" sector. Specifically, for the heterotic string (16 chiral bosons), the physical sector cancels the $g$-anomaly, while a second set of fields coupling to $\bar{g}$ cancels the $\bar{g}$-anomaly. This results in a total partition function that is invariant under independent modular transformations of $\tau$ and $\rho$.
• Heterotic String Application: The formalism is applied to the heterotic string. By coupling the physical heterotic string fields to $g$ and a shadow heterotic string to $\bar{g}$, the authors construct a world-sheet action whose quantisation is modular invariant and free from gravitational anomalies in both sectors. This mirrors structures found in Sen's and other heterotic string field theories involving shadow sectors.
• Higher Genus: The path integral is successfully extended to general Riemann surfaces, with the partition function expressed in terms of the determinant of the Laplacian on the complex metric and theta-functions involving the period matrices of the surface.

Significance
The paper claims that the bi-metric formulation provides a robust framework for quantising chiral bosons on arbitrary Riemann surfaces without imposing self-duality constraints by hand. By analytically continuing to complex metrics, the authors derive a well-defined partition function that naturally incorporates the shadow sector. The primary significance lies in the construction of a consistent, modular-invariant theory for chiral bosons on even self-dual lattices, which is directly applicable to the heterotic string. The work suggests that the "shadow" sector, often viewed as unphysical, plays a necessary role in cancelling gravitational anomalies and ensuring modular invariance when the theory is formulated on a general two-dimensional manifold with two metrics. The results confirm that for specific lattice choices (even self-dual), the partition function factorises elegantly, facilitating the construction of anomaly-free string theories.