Flat F-Manifolds, F-CohFTs, and Integrable Hierarchies

We define the double ramification hierarchy associated to an F-cohomological field theory and use this construction to prove that the principal hierarchy of any semisimple (homogeneous) flat F-manifold possesses a (homogeneous) integrable dispersive deformation at all orders in the dispersion parameter. The proof is based on the reconstruction of an F-CohFT starting from a semisimple flat F-manifold and additional data in genus 1, obtained in our previous work. Our construction of these dispersive deformations is quite explicit and we compute several examples. In particular, we provide a complete classification of rank 1 hierarchies of DR type at the order 9 approximation in the dispersion parameter and of homogeneous DR hierarchies associated with all 2-dimensional homogeneous flat F-manifolds at genus 1 approximation.

Towards formalization of the soliton counting technique for the Khovanov–Rozansky invariants in the deformed ℛ-matrix approach

We consider recently developed Cohomological Field Theory (CohFT) soliton counting diagram technique for Khovanov (Kh) and Khovanov–Rozansky (KhR) invariants.[Formula: see text] Although, the expectation to obtain a new way for computing the invariants has not yet come true, we demonstrate that soliton counting technique can be totally formalized at an intermediate stage, at least in particular cases. We present the corresponding algorithm, based on the approach involving deformed [Formula: see text]-matrix and minimal positive division, developed previously in Ref. 3. We start from a detailed review of the minimal positive division approach, comparing it with other methods, including the rigorous mathematical treatment.4 Pieces of data obtained within our approach are presented in the Appendices.

Gromov–Witten theory and cycle-valued modular forms

In this paper, we review Teleman’s work on lifting Givental’s quantization of{\mathcal{L}^{(2)}_{+}{\rm GL}(H)}action for semisimple formal Gromov–Witten potential into cohomological field theory level. We apply this to obtain a global cohomological field theory for simple elliptic singularities. The extension of those cohomological field theories over large complex structure limit are mirror to cohomological field theories from elliptic orbifold projective lines of weight(3,3,3),(2,4,4),(2,3,6). Via mirror symmetry, we prove generating functions of Gromov–Witten cycles for those orbifolds are cycle-valued (quasi)-modular forms.

DUBROVIN’S SUPERPOTENTIAL AS A GLOBAL SPECTRAL CURVE

We apply the spectral curve topological recursion to Dubrovin’s universal Landau–Ginzburg superpotential associated to a semi-simple point of any conformal Frobenius manifold. We show that under some conditions the expansion of the correlation differentials reproduces the cohomological field theory associated with the same point of the initial Frobenius manifold.

Matrix factorizations and cohomological field theories

We give a purely algebraic construction of a cohomological field theory associated with a quasihomogeneous isolated hypersurface singularity

Invariants of four-manifolds with flows via cohomological field theory

This chapter argues that the Jones–Witten invariants can be generalized for smooth, nonsingular vector fields with invariant probability measure on three-manifolds, thus giving rise to new invariants of dynamical systems. After a short survey of cohomological field theory for Yang–Mills fields, Donaldson–Witten invariants are generalized to four-dimensional manifolds with non-singular smooth flows generated by homologically non-trivial p-vector fields. The chapter studies the case of Kähler manifolds by using the Witten's consideration of the strong coupling dynamics of N = 1 supersymmetric Yang–Mills theories. The whole construction is performed by implementing the notion of higher-dimensional asymptotic cycles. In the process Seiberg–Witten invariants are also described within this context. Finally, the chapter gives an interpretation of the asymptotic observables of four-manifolds in the context of string theory with flows.