On a Deformed Holomorphic Chern-Simons Theory

We deform classical holomorphic Chern--Simons theory on a Calabi--Yau three-fold $X$ by deforming the complex structure by a deformation parameter $h \in\mathscr{H}^{0,1}(T^{1,0}X)$. The corresponding equations of motion admit new "instanton solutions" which which are invariant under re-scalings of $h$, and are perhaps more reminiscent of $G_2$-instantons for $G_2$ manifolds. We give examples of such instantons. In particular, when $h$ has non-vanishing Yukawa coupling ${\rm Yuk}(h,h,h)\neq 0$, it may be used to define a connection on ${\rm End}(T^{1,0}X)$ solving the instanton constraint. Interestingly, this connection gives rise to a hermitian (self-adjoint) connection for a real gauge theory on the real bundle ${\rm End}(TX)$ for only specific directions in deformation space, which may be classified using Morse theory. We quantize the deformed theory around these instanton backgrounds, and derive explicit expressions for the partition function in the limit where the complex structure deformation is large. We study anomalies, and the $h$-dependece of the partition function. In particular, coupling the theory to additional gravitational degrees of freedom, we find that the special directions in deformation space give rise to novel anomaly free theories on ${\rm End}(T^{1,0}X)$.