Related papers: Quantum Master Equation and Open Gromov-Witten the…
Long ago, in math.AG/0112004, we pledged more details on the algebraic version of Chen-Ruan's math.AG/0103156. This is it.
We define Gromov-Witten classes and invariants of smooth projective schemes of finite presentation over a Dedekind domain. We prove that they are deformation invariants and verify the fundamental axioms. For a smooth projective scheme over…
We study the coarse-graining approach to derive a generator for the evolution of an open quantum system over a finite time interval. The approach does not require a secular approximation but nevertheless generally leads to a…
The master equation is quantized. This is an example of quantization of a gauge theory with nilpotent generators. No ghosts are needed for a generation of the gauge algebra. The point about the nilpotent generators is that one can't write…
We prove a quantum version of Kalkman's wall-crossing formula comparing Gromov-Witten invariants on geometric invariant theory (git) quotients related by a change in polarization. The wall-crossing terms are gauged Gromov-Witten invariants…
This expository article introduces the Kapustin-Witten equations to mathematicians. We discuss the connections between the Complex Yang-Mills equations and the Kapustin-Witten equations. In addition, we show the relation between the…
We reframe a collection of well-known comparison results in genus zero Gromov-Witten theory in order to relate these to integral transforms between derived categories. This implies that various comparisons among Gromov-Witten theories and…
We compute the quantum cohomology relative to a Lagrangian submanifold in some complete intersections. For quadric hypersurfaces, we also give a full computation of the genus zero open Gromov-Witten invariants.
A systematic first-order correction to the standard Markov master equation for open quantum systems interacting with a bosonic bath is presented. It extends the Markov Lindblad master equation to the more general case of non-Markovian…
A huge family of solvable potentials can be generated by systematically exploiting the factorization (Darboux) method. Starting from the free case, a large class of the known solvable families is thus reproduced, together with new ones. We…
In this paper we demonstrate how to generate the strong-coupling master equations for open quantum systems of continuous variables. These are the dissipative master equations of quantum Brownian particles for which the environmental noise…
We introduce the most general to date version of the permutation-equivariant quantum K-theory, and express its total descendant potential in terms of cohomological Gromov-Witten invariants. This is the higher-genus analogue of adelic…
We prove a `Whitney' presentation, and a `Coulomb branch' presentation, for the torus equivariant quantum K theory of the Grassmann manifold $\mathrm{Gr}(k;n)$, inspired from physics, and stated in an earlier paper. The first presentation…
Virasoro constraints for orbifold Gromov-Witten theory are described. These constraints are applied to the degree zreo, genus zero orbifold Gromov-Witten potentials of the weighted projective stacks $\mathbb{P}(1,N)$, $\mathbb{P}(1,1,N)$…
We propose two conjectural relationships between the equivariant Gromov-Witten invariants of the resolved conifold under diagonal and anti-diagonal actions and the Gromov-Witten invariants of $\mathbb{P}^1$, and verify their validity in…
We define a perturbative approximation for the solution of Lindblad master equations with time-dependent generators that satisfies the fundamental property of complete positivity, as essential for quantum simulations and optimal control.…
Motivated by algorithmic problems arising in quantum field theories whose dynamical variables are geometric in nature, we provide a quantum algorithm that efficiently approximates the colored Jones polynomial. The construction is based on…
A formulation of quantum mechanics based on replacing the general unitary group by finite groups is considered. To solve problems arising in the context of this formulation, we use computer algebra and computational group theory methods.
Complete sets of commutation relations for arbitrary pairs of quantum minors are computed, with explicit coefficients in closed form.
We consider an open quantum many-particle system in which there are dissipative processes. The evolution of this system is described by a kinetic equation for the density matrix. From the equation describing a random Markov process in this…