Adjunctions and defects in Landau–Ginzburg models

Abstract

We study the bicategory of Landau-Ginzburg models, which has polynomials as objects and matrix factorisations as 1-morphisms. Our main result is the existence of adjoints in this bicategory and formulas for the evaluation and coevaluation maps in terms of Atiyah classes and homological perturbation. The bicategorical perspective offers a unified approach to Landau-Ginzburg models: we show how to compute arbitrary correlators and recover the full structure of open/closed TFT, including the Kapustin-Li disc correlator and a simple proof of the Cardy condition, in terms of defect operators which in turn are directly computable from the adjunctions. Nils Carqueville and Daniel Murfetboundary-bulk map (which reduces to the Chern character (−1) ( m+1 2 ) str(∂ z 1 D . . . ∂ zm D) for Φ = 1) and the Kapustin-Li disc correlator.Another application of our construction of adjunctions in LG k is a new proof of the Cardy condition (see Section 9 for its precise statement). This generalisation of the Hirzebruch-Riemann-Roch theorem is the most "quantum" among the axioms for open/closed TFTs (as it stems from a one-loop diagram) and may accordingly be viewed as a particularly deep structure. In the case of Landau-Ginzburg models it was proved only recently in [PV12] when k is a field of characteristic zero. Our proof works for any ring k and simply follows from the fact that the 2-morphism in LG k to be read off from the diagram(which is to be identified with an annulus correlator) can be evaluated in two ways: either by first contracting the inner X-loop and then contracting the outer Y -loop, or by first fusing X with Y and then contracting the fused (X ∨ ⊗ Y )-loop. Applying special cases of our evaluation and coevaluation maps then immediately produces the Cardy condition, see Theorem 9.1.Let us conclude with future applications of our results. One of the most intriguing properties of Landau-Ginzburg models is that they are on one side of the CFT/LG correspondence. This roughly states that many aspects of a large class of conformal field theories (CFTs) can be described in terms of (non-conformal) Landau-Ginzburg models, and one may wonder which structures encountered in rational CFT can also be found in the theory of Landau-Ginzburg models.One example is the generalised orbifold procedure of [FFRS09] which constructs all rational CFTs of fixed central charge and with identical left and right chiral algebras from any given single such CFT. Carried over to Landau-Ginzburg models this leads to the following picture: under the right circumstances a Landau-Ginzburg model with potential V can be obtained from a model with potential W by identifying an object A in the monoidal category LG k (W, W ) that can be equipped with the structure of a special symmetric Frobenius algebra (see e. g. [FRS02, Section 3]). Then the category of matrix factorisations of V is equivalent to the category of A-modules. The results of the present paper facilitate the construction of suitable algebras; the details appear in [CR].The rest of the prese...

Realizing IR theories by projections in the UV

“…12 Note that the Hom-complex is untwisted! Since the factorized polynomials add upon taking the tensor product, this is indeed a matrix factorization of ( 13 Adjunctions of B-type defects in LG models have been studied in [11,19] (see [20] for a nice review). Adjoints are given by…”

“…Following [4,11], we define λ −1 h I : h I → I ⊗ h I to be the natural junction of the identity defect with the symmetry defect h I. It is given by…”

Orbifold equivalence: structure and new examples

“…We summarise some abstract properties of the notions of orbifold equivalence and quantum dimensions in a theorem; all statements were proven before, see [11,14,13] and references therein:…”

“…matrix factorisations of the difference), and 2-morphisms by morphisms (as occurring in (1.7)) of those matrix factorisations. This bicategory is "graded pivotal" [11,14,8], in particular it has adjoints: for each 1-morphism Q, i.e. each defect between V 1 (x) and V 2 (y), the adjoint Q † , a defect between V 2 (y) and V 1 (x), is given by…”

“…(This equation holds if the number of y-variables is even, otherwise there is an additional exchange of E and J; see [11,14,12] for details, which will not play a role in what follows. )…”