Xingdu is an emerging mathematical game about searching for a corresponding polyline for a given polyline in a 2-dimensional or 3-dimensional mesh grid, meanwhile the searched polyline should have the same start and end points as those of the given polyline and the corresponding line segments on the searched and given polylines are perpendicular each other respectively. Studying from mathematical perspective the problem solving and designing of Xingdu is helpful of finding a general method for Xingdu problem solving, but still not attract much more attentions due to the relatively short history of Xingdu. Referring to the achievements in mathematical methods for Sudoku, we investigated preliminarily the possibility of using optimization theory to solve the problem of Xingdu. According to the definition of Xingdu, the points on the given polyline and the solution polyline have to be the grid nodes in the mesh grid so that their coordinates are integer. In addition, there should not be any points appearing repeatedly and any three sequential points on the solution polyline cannot be located on a same line. Due to these constraints, the optimization problem related to Xingdu belongs to integer programming with nonlinear constraints, which is very difficult and usually can only be solved by enumerating. For this reason, we solve the related problem first by neglecting the nonlinear constraints so that it can be converted as an integer programming with linear constraints only. Afterwards, we can check whether it fulfill the related nonlinear constraints if a feasible solution is obtained. Based on that, we proposed two integer programming models for Xingdu problem, which are the model using only line segments perpendicular property as constraints and the binary integer programming model using line segments perpendicular property together with no repeated points as constraints. The test results show that the proposed binary integer programming model has a very good performance in finding the solution of Xingdu problem. We also discussed the possibility of using the proposed binary integer programming model together with random simulation to design a Xingdu problem. Due to the challenge of discriminating the uniqueness of Xingdu solution, we finally presented an approach, inspired from the branch and bound method for integer programming, which solves and designs the Xingdu problem by constructing and pruning a search tree.

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In this paper we compute spaces of chiral operators in general two-dimensional (0,2) nonlinear sigma models, both in theories twistable to the A/2 or B/2 model, as well as in non-twistable theories, and apply them to check recent duality conjectures. The fact that in a nonlinear sigma model, the Fock vacuum can act as a section of a line bundle on the target space plays a crucial role in our (0,2) computations, so we begin with a review of this property. We also take this opportunity to show how even in (2,2) theories, the Fock vacuum encodes in this way choices of target space spin structures, and discuss how such choices enter the A and B model topological field theories. We then compute chiral operators in general (0,2) nonlinear sigma models, and apply them to the recent Gadde-Gukov-Putrov triality proposal, which says that certain triples of (0,2) GLSMs should RG flow to nontrivial IR fixed points. We find that different UV theories in the same proposed universality class do not necessarily have the same space of chiral operators – but, the mismatched operators do not contribute to elliptic genera and are in non-integrable representations of the proposed IR affine symmetry groups, suggesting that the mismatched states become massive along RG flow. We find this state matching in examples not only of different geometric phases of the same GLSMs, but also in phases of different GLSMs, indirectly verifying the triality proposal, and giving a clean demonstration that (0,2) chiral rings are not topologically protected. We also check proposals for enhanced IR affine E6 symmetries in one such model, verifying that (matching) chiral states in phases of corresponding GLSMs transform as 27’s, \bar{27}’s.

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We calculate the dimension of the locus of Jacobian elliptic surfaces over P1 with a given Picard number, in the corresponding moduli space.