Rozansky–Witten Theory, Localised Then Tilted

The paper has two parts, in the first part, we apply the localisation technique to the Rozansky–Witten theory on compact hyperkähler targets. We do so via first reformulating the theory as some supersymmetric sigma-model. We obtain the exact formula for the partition function with Wilson loops on $$S^1\times \Sigma _g$$ S 1 × Σ g and the lens spaces, the results match with earlier computations using Feynman diagrams on K3. The second part is motivated by a very curious paper (Gukov in J Geom Phys 168, 104311, 2021), where the equivariant index formula for the dimension of the Hilbert space of the Rozansky–Witten theory is interpreted as a kind of Verlinde formula. In this interpretation, the fixed points of the target hyperkähler geometry correspond to certain ‘states’. We extend the formalism of part one to incorporate equivariance on the target geometry. For certain non-compact hyperkähler geometry, we can apply the tilting theory to the derived category of coherent sheaves, whose objects label the Wilson loops, allowing us to pick a basis for the latter. We can then compute the fusion products in this basis and we show that the objects that have diagonal fusion rules are intimately related to the fixed points of the geometry. Using these objects as basis to compute the dimension of the Hilbert space leads back to the Verlinde formula, thus answering the question that motivated the paper.

Tilting separation simulation and theory verification of mask projection stereolithography process

Purpose This study aims to accurately simulate the tilting separation process of mask projection stereolithography (MPSL) and verify the tilting theory. Design/methodology/approach The finite element separation models of MPSL 3D printing process were established. The established models simulated both tilting and pulling-up separation process by changing the constraints and boundary conditions. The bilinear cohesive curves were used to define the separation interface. The stress distribution of the cured part and FEP film at different times during the whole separation process was extracted. Different orientations of pulling-up and tilting were also compared for stress distribution. The stress change was analyzed for the center and edge points of the upper surface of cured part. Findings The results showed that the stress increased with the separation speed, and the stress at the edge position of exposure area was greater than the internal position. The tilting traction stress distribution was affected by the exposure area function and the velocity distribution. Alternation of the exposure area function changed the cohesive stiffness. The non-coincidence of the calculated traction stress with the input bilinear cohesive curve reflected the influence of the material properties and the separation methods. The high-speed side of tilting had fast separation and high traction stress. Originality/value This study proposes a technical method for simulation tilting separation and verified the tilting theory. The cohesive zone model was proved applicable to the tilting traction stress calculation.

TILTING THEORY FOR GORENSTEIN RINGS IN DIMENSION ONE

In representation theory, commutative algebra and algebraic geometry, it is an important problem to understand when the triangulated category $\mathsf{D}_{\operatorname{sg}}^{\mathbb{Z}}(R)=\text{}\underline{\mathsf{CM}}_{0}^{\mathbb{Z}}R$ admits a tilting (respectively, silting) object for a $\mathbb{Z}$ -graded commutative Gorenstein ring $R=\bigoplus _{i\geqslant 0}R_{i}$ . Here $\mathsf{D}_{\operatorname{sg}}^{\mathbb{Z}}(R)$ is the singularity category, and $\text{}\underline{\mathsf{CM}}_{0}^{\mathbb{Z}}R$ is the stable category of $\mathbb{Z}$ -graded Cohen–Macaulay (CM) $R$ -modules, which are locally free at all nonmaximal prime ideals of $R$ . In this paper, we give a complete answer to this problem in the case where $\dim R=1$ and $R_{0}$ is a field. We prove that $\text{}\underline{\mathsf{CM}}_{0}^{\mathbb{Z}}R$ always admits a silting object, and that $\text{}\underline{\mathsf{CM}}_{0}^{\mathbb{Z}}R$ admits a tilting object if and only if either $R$ is regular or the $a$ -invariant of $R$ is nonnegative. Our silting/tilting object will be given explicitly. We also show that if $R$ is reduced and nonregular, then its $a$ -invariant is nonnegative and the above tilting object gives a full strong exceptional collection in $\text{}\underline{\mathsf{CM}}_{0}^{\mathbb{Z}}R=\text{}\underline{\mathsf{CM}}^{\mathbb{Z}}R$ .

REPETITIVE EQUIVALENCES AND TILTING THEORY

Let $R$ be a ring and $T$ be a good Wakamatsu-tilting module with $S=\text{End}(T_{R})^{op}$ . We prove that $T$ induces an equivalence between stable repetitive categories of $R$ and $S$ (i.e., stable module categories of repetitive algebras $\hat{R}$ and ${\hat{S}}$ ). This shows that good Wakamatsu-tilting modules seem to behave in Morita theory of stable repetitive categories as that tilting modules of finite projective dimension behave in Morita theory of derived categories.