Hochschild cohomology of Fermat type polynomials with non–abelian symmetries

Basalaev, Alexey; Ionov, Andrei

doi:10.1016/j.geomphys.2021.104450

Cited by 4 publications

(4 citation statements)

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“…[4,5,6,14,19,20,21]). Also some work was done for the symmetry groups G = S ⋉ G d with G d acting diagonally and S ⊆ S N -a subgroup of a symmetric group on N elements [2,3,10,11,12,18,24,27,32]. We relax both conditions in this paper.…”

Section: Landau-ginzburg Orbifoldsmentioning

confidence: 99%

“…, x N ] and g ∈ GL f we have Remark 4.5. In the literature (see, for example, [25]) a different definition could be found where the sum is taken over the representatives of the conjugacy classes of G and the invariants in each sector are taken with respect to the centralizer of the corresponding g. The two definitions are in fact equivalent in the same way as in [3,Proposition 42].…”

Section: B-model Group Actionmentioning

confidence: 99%

“…Pick ξ (1,2,3) and ξ (1,3,2) in such a way that (1, 2)(ξ (1,2,3) ) = ξ (1,3,2) ; ξ (1,2,3)j f and ξ (1,3,2)j f in such a way that (1, 2)(ξ (1,2,3)j f ) = ξ (1,3,2)j f and so on. Then the basis of B(f, G) is ξ id and x 3 1 x 3 2 x 3 3 x 3 4 x 3 5 ξ id in bidegrees (0, 0) and (3,3), respectively;…”

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confidence: 99%

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Hodge Diamonds of the Landau-Ginzburg Orbifolds

Basalaev,

Ionov

2024

SIGMA

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Consider the pairs $(f,G)$ with $f = f(x_1,\dots,x_N)$ being a polynomial defining a quasihomogeneous singularity and $G$ being a subgroup of ${\rm SL}(N,\mathbb{C})$, preserving $f$. In particular, $G$ is not necessary abelian. Assume further that $G$ contains the grading operator $j_f$ and $f$ satisfies the Calabi-Yau condition. We prove that the nonvanishing bigraded pieces of the B-model state space of $(f,G)$ form a diamond. We identify its topmost, bottommost, leftmost and rightmost entries as one-dimensional and show that this diamond enjoys the essential horizontal and vertical isomorphisms.

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Section: Landau-ginzburg Orbifoldsmentioning

confidence: 99%

Section: B-model Group Actionmentioning

confidence: 99%

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Hodge Diamonds of the Landau-Ginzburg Orbifolds

Basalaev,

Ionov

2024

SIGMA

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“…Finally, we would like to mention that in the diagonal case, much more is known beyond an isomorphism of state spaces. Both the LG A-model and LG B-model state spaces can be given the structure of a Frobenius algebra (see [14], [23], or [17] for the A-model, and [3] for the B-model), and in [17], it was shown that for a large class of LG models, the two state spaces are also isomorphic as Frobenius algebras. This is conjectured to be true in general, but the general case remains open.…”

Section: Introductionmentioning

confidence: 99%

Mirror Map for Landau-Ginzburg models with nonabelian groups

Clawson¹,

Johnson²,

Duncan³

et al. 2023

Preprint

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BHK mirror symmetry as introduced by Berglund-Hübsch and Marc Krawitz between Landau-Ginzburg (LG) models has been the topic of much study in recent years. An LG model is determined by a potential function and a group of symmetries. BHK mirror symmetry is only valid when the group of symmetries is comprised of the so-called diagonal symmetries. Recently, an extension to BHK mirror symmetry to include nonabelian symmetry groups has been conjectured. In this article, we provide a mirror map at the level of state spaces between the LG A-model state space and the LG B-model state space for the mirror model predicted by the BHK mirror symmetry extension for nonabelian LG models. We introduce two technical conditions, the Diagonal Scaling Condition, and the Equivariant Φ condition, under which a bi-degree preserving isomorphism of state spaces (the mirror map) is guaranteed to exist, and we prove that the condition is always satisfied if the permutation part of the group is cyclic of prime order.

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