A brief survey of FJRW theory

Abstract

In this paper we describe some of the constructions of FJRW theory. We also briefly describe its relation to Saito-Givental theory via Landau-Ginzburg mirror symmetry and its relation to Gromov-Witten theory via the Landau-Ginzburg/Calabi-Yau correspondence. We conclude with a discussion of some of the recent results in the field.

D-brane central charge and Landau-Ginzburg orbifolds

“…These authors have also shown that the FJRW invariants contain the same information as the Gromov-Witten invariants, as is expected from by Landau-Ginzburg/Calabi-Yau correspondence [9]. For reviews of FJRW theory see also [70,71].…”

“…It satisfies certain key properties and axioms such that the set of correlations functions defines a cohomological field theory, called FJRW theory, in the sense of Kontsevich and Manin [77] (cf. also [71]), on the space H FJRW (W, G) of chiral primary fields of the Landau-Ginzburg orbifold (W, G). FJRW theory is then intersection theory on W g,n (W, G), generalizing the case of topological gravity [74].…”

Mirror Symmetry for Nonabelian Landau-Ginzburg Models

“…This definition is the standard definition of diagonal symmetries (see, e.g., [17]). Note that G diag W can be viewed as a subgroup of G max W via diagonal matrices.…”

“…While the state spaces have similar definitions, the grading and product structures are very differentfor example, the multiplicative identity on the B-side lies in the untwisted sector, whereas the multiplicative identity on the A-side lies in the sector indexed by j W . We won't discuss the product structure any further here, but the interested reader can find the definition of the B-model multiplication in [2] or [20], and the definition of the A-model product in [14] or [17], when G and G * are groups of diagonal symmetries. When G and G * are not diagonal, the B-model product has not yet been defined, as far as we know, whereas the A-model structure comes from invariants of the associated Gauged Linear Sigma Model (see, e.g., [15], though we are unaware of the invariants having been computed for any examples).…”

“…In the past, mathematicians have primarily studied LG models of pairs (W, G) where G is an abelian group comprised of so-called diagonal symmetries (see [17]). Mirror theorems at various levels of structure have been proved in this setting.…”