Finite deformations from a heterotic superpotential: holomorphic Chern-Simons and an L∞ algebra

Abstract

We consider finite deformations of the Hull-Strominger system. Starting from the heterotic superpotential, we identify complex coordinates on the off-shell parameter space. Expanding the superpotential around a supersymmetric vacuum leads to a third-order Maurer-Cartan equation that controls the moduli. The resulting complex effective action generalises that of both Kodaira-Spencer and holomorphic Chern-Simons theory. The supersymmetric locus of this action is described by an L 3 algebra. C The off-shell N = 1 parameter space and holomorphicity of Ω 41 C.1 SU(3) × SO(6) structures in the NS-NS sector 41 C.2 SU(3) × SO(6 + n) structures in heterotic supergravity 44 C.3 The off-shell hermitian structure on V 45 D Comments on D-terms 48 D.1 Massless deformations 48 D.2 Including bundle moduli 52 D.3 Polystable bundles 54 D.4 Full Maurer-Cartan equations 56 E Massless moduli 571 More precisely, it is the complex vector bundle V C (defined in appendix C.3) that is a holomorphic bundle. 2 The curvature R in the Bianchi identity is the curvature of a connection on T X, satisfying its own hermitian Yang-Mills conditions in order for the equations of motion to be fulfilled [33]. To O(α ′ ), this connection is ∇ − , given by taking the connection in (A.7) with the opposite sign for H.

Superpotential of three dimensional $$ \mathcal{N} $$ = 1 heterotic supergravity

“…Another interesting direction is to view ∆W as an action which can be quantised. Indeed, the famous Donaldson-Thomas invariants of Calabi-Yau manifolds equipped with holomorphic sheaves can be interpreted as correlation functions within holomorphic Chern-Simons theory [1][2][3], and it has been suggested that similar invariants may be defined for the Hull-Strominger system using the corresponding superpotential [11]. Donaldson and Segal have suggested that similar invariants could be defined for G 2 manifolds [4], though it has been debated to what extent such invariants depend on the given G 2 structure.…”

“…As required by N = 1 supersymmetry in four dimensions, it is a holomorphic functional on the off-shell parameter space of the system, and its critical locus corresponds to the infinitesimal moduli space. Adopting this perspective leads to an understanding of the structure of the finite deformations of these configurations, and the holomorphicity of the superpotential provides constraints enough to determine a third-order Maurer-Cartan equation for finite moduli [11].…”

Supergravity Fluxes and Generalised Geometry

“…In light of this, it is natural to speculate that a generalised geometry based on this bundle could be a useful tool in studying these systems. In [], the finite deformations of the system were studied both as an interesting problem in their own right, and as a means to uncover the relevant generalised geometry of .…”

“…This is equivalent to a smoothly varying element of the homogeneous space One can then perform a decomposition of the adjoint representation of to identify complex coordinates on this space, which when allowed to vary smoothly over the manifold X provide us with complex coordinates on the space of structures. In [] it is argued that the relevant coordinates on the space of structures can be identified with (which is exactly as for the almost complex structure), and and which are the (1,1) and (0,2) parts of the holomorphic variation .…”

Nongeometric heterotic strings and dual F-theory with enhanced gauge groups

“…See, e.g.,[6,7,8,9,10,11,12,13,14,15] for recent studies of heterotic strings 2. See, for example,[17,18,19,20,21] for recent studies on the stable degenerations in F-theory/heterotic duality.…”