Mega-Space Current Algebra and Green-Schwarz Geometry in Heterotic String Theory

We study the Lorentz and gauge connections in heterotic string theories within a framework of manifest T-duality. A central motivation is the Bergshoeff--de Roo (BdR) type parallelism between the torsionful Lorentz and Yang-Mills connections in heterotic $\alpha'$-corrections. We formulate worldsheet current algebras in a mega-space that incorporates the Lorentz and gauge sectors simultaneously. By generalizing the Pol\'a\v{c}ek-Siegel scheme, we evaluate the commutators of the extended generators and determine the generalized connections. We show that this mega-space current algebra geometrically embeds the heterotic Chern--Simons structures into the generalized vielbeins and curvatures. We also show that the Green-Schwarz anomaly cancellation condition follows from the Jacobi identity for the algebra of stringy covariant derivatives, the ``nachos'' operators $\triangleright_{\mathcal{M}}$.