Chern characters and Hirzebruch-Riemann-Roch formula for matrix factorizations (Q442756)

A matrix factorization of an element \(w\) in a commutative ring \(R\) is a \(\mathbb Z/2\)-graded finitely generated projective \(R\)-module \(W=E_0\oplus E_1\) together with an odd endomorphism \(\delta_E\) such that \(\delta_E^2=w\cdot\text{id}_E\). Matrix factorizations are a classical tool in the study of hypersurface singularity algebras, which is said to measure geometrically the failure of every coherent sheaf on the hypersurface \(w=0\) to have a locally free resolution. Also, they are used in link homology and in mathematical physics suggested by Kontsevich to describe D-branes in topological Landau-Ginzburg models.NEWLINENEWLINEThe present paper is motivated by the rich structure arising on the Hochschild homology of the category \(\text{MF}(w)\) of matrix factorizations of an isolated singularity \(w=0\) where \(w(x_1,\dots,x_n)\) is a formal power series. The Hochschild (co)homology of this category has an interpretation as the state space for the closed string sector of the open-closed topological string theory associated with the Landau-Ginzburg model of the potensial \(w\). This article gives explicit formulas for some of the natural structures on this state space using the tools of the theory of differential graded categories (dg-categories), and these formulas are used in another text to construct a purely algebraic version of the Fan-Jarvis-Ruan theory.NEWLINENEWLINEThe category \(\text{MF}(w)\) for an isolated singularity fits naturally into the framework of noncommutative geometry developed from the point of view of dg-categories, or \(A_\infty\)-categories. It provides an example of a smooth and proper noncommutative space. Recently, Shklyarov showed that the Hochschild homology \(HH_\ast(\mathcal C)\) of a smooth proper dg-category \(\mathcal C\) is equipped with a canonical nondegenerate bilinear form \(\langle\cdot,\cdot\rangle\). His categorical Hirzebruch-Riemann-Roch formula expresses the Euler characteristic of the \(\text{Hom}\)-spaces between two objects in the derived category of \(\mathcal C\) in terms of Chern characters taking values in \(HH_\ast(\mathcal C)\) and the form \(\langle\cdot,\cdot\rangle\). There exists an even more general formula computing the traces of certain endomorphisms of the \(\text{Hom}\)-spaces between two objects. In the case of a Calabi-Yau category \(\mathcal C\) this generalized formula is equivalent to the Cardy condition for the corresponding open-closed two-dimensional topological quantum field theory.NEWLINENEWLINEIn concrete situations the difficulty shifts to calculating explicitly the Hochschild homology of the corresponding category along with the Chern character map and the canonical bilinear form. In this article, these ingredients are worked out in the case of the (\(\mathbb Z/2\)-graded) dg-category of matrix factorizations of an isolated singularity \(w\), including the \(G\)-equivariant version, where \(G\) is a finite group of symmetries of \(w\). This leads to a concrete Hirzebruch-Riemann-Roch formula for matrix factorizations.NEWLINENEWLINEIn the classical (nonequivariant) setting the formulation of the results are as follows: For an isolated singularity \(w\in k[[x_1,\dots,x_n]]\) the Hochschild homology of the (\(\mathbb Z/2\)-graded dg-) category \(\text{MF}(w)\) of matrix factorizations of \(w\) is isomorphic as \(\mathbb Z/2\)-graded vector space to the Milnor ring of \(w\) (up to a shift of grading): NEWLINE\[NEWLINEHH_\ast(\text{MF}(w))\simeq\mathcal A_w\cdot d\mathbf{x}[n],NEWLINE\]NEWLINE where \(\mathcal A_w=k[[x_1,\dots,x_n]]/\mathcal I_w\) with \(\mathcal I_w=(\partial_1 w,\dots,\partial_n w)\) and \(d\mathbf{x}=dx_1\wedge\dots\wedge dx_n\). The following formula for the Chern character \(\text{ch}(\bar{E})\in HH_0(\text{MF}(w))\) of a matrix factorization \(\bar{E}=(E,\delta_E)\) is derived: NEWLINE\[NEWLINE\text{ch}(\bar{E})=\text{str}_R(\partial_n\delta_E\cdot\dots\cdot\partial_1\delta_E)\cdot d\mathbf{x},NEWLINE\]NEWLINE where \(\text{str}_R\) is the supertrace of a matrix with entries in \(R\). More generally, there is a canonical map NEWLINE\[NEWLINE\tau^{\bar{E}}(\alpha):\text{Hom}^\ast(\bar{E},\bar{E})\rightarrow HH_\ast(\text{MF}(w))NEWLINE\]NEWLINE called the boundary bulk map such that \(\text{ch}(\bar{E})=\tau^{\bar{E}}(\text{id}_E)\) and the formula for the Chern character generalizes to NEWLINE\[NEWLINE\tau^{\bar{E}}(\alpha)=\text{str}_R(\partial_n\delta_E\cdot\dots\cdot\partial_1\delta_E\circ\alpha)\cdot d\mathbf{x}\text{ mod }\mathcal I_w\cdot d\mathbf{x},NEWLINE\]NEWLINE where \(\alpha\) is an arbitrary endomorphism of \(\bar{E}\). The authors show that the formula leads to the identification of the canonical bilinear form on \(HH_\ast(\text{MF}(w))\) with the form NEWLINE\((f\otimes d\mathbf{x},g\otimes d\mathbf{x}) = (-1)^{\binom{n}{2}}\text{Res}(f\cdot g)\), where \(\text{Res}\) is the linear functional on the Milnor ring \(\mathcal A_w\) given by the generalized residue \(\text{Res}(f)=\text{Res}_{k[x]/k}\left[\begin{matrix} f(x)\cdot dx_1\wedge\cdots\wedge dx_n\\ \partial_1w,\dots,\partial_n w\end{matrix}\right].\) A consequence is an analogue of the Hirzebruch-Riemann-Roch formula for the Euler characteristic of the \(\mathbb Z/2\)-graded space NEWLINE\[NEWLINE\text{Hom}^\ast(\bar{E},\bar{F}) : \chi(\bar{E},\bar{F})=\langle\text{ch}(\bar{E}),\text{ch}(\bar{F})\rangle.NEWLINE\]NEWLINE More generally, for \(\alpha\in\text{Hom}^\ast(\bar{E},\bar{E})\) and \(\beta\in\text{Hom}^\ast(\bar{F},\bar{F})\), NEWLINE\[NEWLINE\text{str}_k(m_{\alpha,\beta})=\langle\tau^{\bar{E}}(\alpha),\tau^{\bar{E}}(\beta)\rangle,NEWLINE\]NEWLINE where \(m_{\alpha,\beta}\) is the endomorphism of \(\text{Hom}^\ast(\bar{E},\bar{F})\) sending \(f\) to NEWLINE\[NEWLINE(-1)^{|\alpha|\cdot|\beta|+|\alpha|\cdot|f|}\beta\circ f\circ\alpha.NEWLINE\]NEWLINE Combining these results, the authors obtain explicit formulas, and all the way they prove the \(G\)-equivariant versions. Finally, in the case of a quasi-homogenous singularity one can also consider \(\mathbb Z\)-graded versions of the categories of matrix factorizations. An analogue of the Hirzebruch-Riemann-Roch formula for these categories follows from this formula for the category of \(G\)-equivariant matrix factorizations, where \(G\) is an appropriate cyclic group.NEWLINENEWLINEThe paper contains a detailed exposition on Hochschild homology for smooth proper dg-categories, including perfect derived categories. Then the category of matrix factorizations in most generality is considered, leading up to the definition of the Chern character and the boundary-bulk maps. This is necessary to be able to study the Hirzebruch-Riemann-Roch formula, both classical and equivariant. The article is very long, but very interesting and well written, and with a lot of interesting results coming from matrix factorizations in general.