1 Introduction↩︎

1.1 Motivation and main results↩︎

The theory of factorization algebras arises when dealing with families of local operators over an algebraic variety or a smooth manifold. Its origin is the theory of vertex operator algebras — a mathematical structure describing the chiral part of quantum field operators in a two-dimensional conformal field theory [1], [2]. In their original context, which is the subject of this paper, factorization algebras were introduced by Beilinson and Drinfeld [3] for families of local operator over finite subsets of a smooth algebraic curve, arising in the study of the geometric Langlands correspondence. They were later generalized to higher dimensional smooth schemes [4], as well as smooth manifolds [5], [6], and have been used to describe the structure of observables in a general quantum field theory [7].

Heuristically, a factorization algebra over an algebraic curve \(X\) is an assignment of a vector space (or a chain complex) \(\cA_{(x_1,\dots,x_n)}\) for each finite subset \(\{x_1,\dots,x_n\} \subset X\), and an isomorphism \[\cA_{(x_1,\dots,x_n)} \simeq \cA_{x_1} \otimes \dotsb \otimes \cA_{x_n}\] whenever \(x_1,\dots,x_n\) are disjoint. The assignment \[(x_1,\dots,x_n) \mapsto \cA_{(x_1,\dots,x_n)}\] should have a flat connection along each variable \(x_i\), and assemble to a collection of D-modules \(\cA_I\) over \(X^I\) for each finite set \(I\), compatible with diagonal embeddings. A compatible collection of D-modules over powers of a curve \(X\) can be equivalently defined as a D-module over \(\Ran X\) — the moduli space of finite subsets of \(X\). The structure of a factorization algebra then translates to an equivariance condition with respect to the (disjoint) union operation. If \(X\) is proper, one then have a de Rham global sections functor over \(\Ran X\) — the chiral (or factorization) homology of \(X\) with coefficients in \(\cA\), which we denote by \(\int_{X} \cA\).

Any vertex operator algebra \(V\) defines a factorization algebra \(\cA_{V,X}\) over any smooth curve [2]. Factorization algebras arising in this way have a special property — they are defined over any smooth curve in a compatible way, and assemble to a family of factorization algebras over the universal curve \(\cX_g \to \cM_g\) of the moduli of genus-\(g\) smooth curves. Forgetting the factorization structure, one gets the structure of a universal D-module. The precise structure of the letter is defined in [8], using the stack of étale germs of \(n\)-dimensional varieties. We give here a factorization version of this construction, which generalizes the definition in [3] to the derived setting, using the moduli space \(\MDisk_{\Ran}\) of pointed multidisks:

In the case where a universal factorization algebra arises from a vertex operator algebra, this construction reduces to that of [2]. In particular, we have an isomorphism \(H^0\int_{X} \cA_X \simeq \VV(V)_{[X]}\), where \(\VV(V)_{[X]}\) denotes the space of coinvariants (with vacuum coefficients), dual to the space of conformal blocks, which is central in the construction of correlation functions in conformal field theory.

If we start with a family of curves \(X/S\), the resulted chiral homology would be a sheaf over \(S\). Furthermore, using the action of the Virasoro algebra, we get a structure of a twisted D-module [9]. In particular, taking \(S\) to be the universal base \(\cM_{g}\), one gets a twisted D-module of (complexes of) vector spaces over \(\cM_g\), whose zeroth homology is the sheaf of coinvariants. An important question in the study of vertex operator algebras is the extension of the sheaf of coinvariants to the boundary, namely to the Deligne-Mumford compactification \(\bcM_g\), together with a gluing formula, relating the space of coinvariants for a nodal curve with that of its normalization. The motivation is twofold:

The first is computational: Under certain finiteness assumptions, the sheaf of coinvariants is a vector bundle, and in particular, its rank depends on the genus only. By extending to the boundary, one reduces the computation to the case of a nodal curve. The gluing formula then expresses the genus-\(g\) rank in terms of lower genera. This is known as the Verline formula, which was conjectured in [10], and proven in [11] for the case of the integrable quotient vertex algebra, and more generally in [12].

The second comes from the idea of an algebraic quantum field theory [13], [14]. In its simplest form, it describes a structure of a topological quantum field theory arising from a conformal field theory which is equipped with strong equivariance with respect to deformations. In that case, the resulted structure is that of a functor out of a \([1,2]\)-cobordism category [15]. In the case of a vertex operator algebra \(V\), it assigns the category \(V\mh\Mod\) to the circle, and the functor \[\VV(V,-)_{[\overline{\Sigma},\{x_i\}]} : V\mh\Mod^{\otimes n} \to \Vect\] to a Riemann surface \(\Sigma\) with boundary \((S^1)^{\sqcup n}\), where \(\VV(V,{M_1,\dots,M_n})_{[\overline{\Sigma},\{x_i\}]}\) denotes the spaces of coinvariants corresponding to \(M_i\) assigned to \(x_i\). Here \(\overline{\Sigma} = \Sigma \underset{(S^1)^{\sqcup n}}{\cup} D^{\sqcup n}\), and \(x_i\) the center of the \(i\)-th disk. The gluing law then translate to a composition law — the behavior of the spaces \(\VV(V,\{M_i\})_{[\overline{\Sigma},\{x_i\}]}\) with respect to compositions of cobordisms. The precise structure is described in [16] under strong finiteness assumptions on the category of modules. In general, however, one cannot expect a completely topological structure: even of we have a flat connection on \(\cM_g\), this connection may fail to be integrable. Instead, one only expects a composition law to hold for nodal curves.

Our main result is an extension of chiral homology to the boundary, and a gluing formula for nodal curves.

Here the tensor product of two modules \(M,N\) over a non-unital associative algebra \(A\) is given by the non-unital bar complex \[M \underset{A}{\otimes} N = \colim(\dotsb \rrrarrow M \otimes A \otimes N \rrarrow M \otimes N)~.\]

The fibers \(\int_{X \backslash \{x_1,\dots,x_n\}} \cA\) are isomorphic to the chiral homology of the proper curve \(X\) with coefficients in a certain \(\cA\)-module \(\fZ_{\cA}^+\) supported at each marked point \(x_i\), which we regard as a puncture. Similarly, the fibers at a nodal curve \(X\) can be interpreted as factorization homology for \(X\) with coefficients in a chiral bimodule \(\fZ_{\cA}^{\nd}\). However, the latter description does not work in families — in a nodal degeneration with smooth general fiber, the node does not extend to a section over the smooth locus. Instead, we interpret it as chiral homology for a module \(\Vac(\cA)^{\epsilon}_{(X,x_1,\dots,x_n)}\) whose support is the entire space \(\Ran X\).

In [17], we show that for a vertex operator algebra \(V\) with associated universal factorization algebra \(\cA_V\), the associative algebra \(H^0\fZ_{\cA}^0\) is isomorphic to Zhu’s associative algebra \(A(V)\) defined in [18], and the chiral module \(H^0\fZ_{\cA}^+\) is the vertex algebra module associated to \(A(V)\). Assume now \(A(V)\) is semisimple, namely decomposes as \(A(V) \simeq \bigoplus_{i=1}^{\ell} M_i \otimes M_i^{\lor}\), where \(M_1,\dots,M_n\) are the simple \(A(V)\)-modules. In particular, \(H^0\fZ_{\cA}^+\) decomposes as \(H^0\fZ_{\cA}^+ \simeq \bigoplus_{i=1}^{\ell} \cM_i \otimes \cM_i^{\lor}\). Then we show that at the level of \(H^0\), the formula ?? reduces to \[\VV(V)_{[X \underset{x \sim y}{\cup} Y]} \simeq \bigoplus_{i=1}^{\ell} \VV(V, \cM_i)_{[X,x]} \otimes \VV(V, \cM_i^{\lor})_{[Y,y]}\] This is the usual form of the Verlinde formula, which is proven in [11], [12].

An important ingredient, which is missing from the current paper, is that of sewing, namely the extension of a gluing formula to the formal neighborhood of the boundary \(\partial \bcM_{g}\). We hope to address that in future work.

1.2 Overview↩︎

In Section 2, we review the definitions of chiral and factorization algebras, chiral homology and chiral modules, and write down the corresponding definitions in the relative case of families of smooth curves. We formalize the notion of a universal factorization algebra as a certain commutative coalgebra in some symmetric monoidal category, and define a localization functor from the category of universal factorization algebras to the category of families of factorization algebras over any family of smooth curves.

In Section 3, we define the moduli space of semistable modifications of a curve \(\fM_{X}^{\sst}\), and construct various Ran spaces (namely, moduli spaces of finite subsets) over it. In particular, we define the pseudo-proper prestack \(\bcM_{g,n,\Ran} \to \fM^{\sst}_{g,n}\) of stable configurations over semistable modifications, which serves as our maximal compactification for the Ran space of the punctured, smooth locus \(\mathring{\Ran}_{g,n}\), and which maps to the minimal compactification \(\Ran_{g,n}\), given by Ran space of the universal curve.

In Section 4, we define chiral monoidal structures for the spaces of semistable modifications, as well as a chiral module space structure on the space \(\bcM_{g,n,\Ran}\) of stable configurations. We then construct, for any universal factorization algebra, a factorization \(\cA\)-module \(\Vac(\cA)_{g,n}^{\epsilon}\) over \(\bcM_{g,n,\Ran}\). By pushing forward along \(\bcM_{g,n,\Ran} \to \Ran_{g,n}\), we get a factorization \(\cA\)-module \(\Vac(\cA)^0_{g,n}\) over \(\Ran_{g,n}\), whose fibers over unmarked smooth points are the vacuum module. We prove that in the case of two-pointed genus zero curves \(\bcM_{0,2,\Ran}\), we have an additional (non-symmetric) monoidal structure, which we call the join monoidal structure, and which is given by concatenation of semistable modifications, and show that \(\Vac(\cA)_{g,n}^{\epsilon}\) are equivariant with respect to this monoidal structure.

Finally, in Section 5, we prove the main result of this paper. We define the associative algebra \(\fZ_{\cA}^0\) for any universal factorization algebra \(\cA\) as global sections over \(\bcM_{0,2,\Ran}\), with multiplication induced by the join product. We also define \(\fZ_{\cA}^0\)-modules \(\fZ_{\cA}^+,\fZ_{\cA}^-\), which admit an additional structure of chiral modules supported at a point. We define the chiral bimodule \(\fZ_{\cA}\), which is the value of our extension at singular points. We then prove a gluing formula, relating factorization homology with coefficients in \(\fZ_{\cA}\) at a node to factorization homology with coefficients in \(\fZ_{\cA}^{\pm}\) over its normalization, and in particular proving that \(\fZ_{\cA} \simeq \fZ_{\cA}^+ \underset{\fZ_{\cA}^0}{\otimes} \fZ_{\cA}^-\).

1.3 Notations and conventions↩︎

Throughout this paper, and unless specified otherwise, all categories are assumed to be \((\infty,1)\)-categories. In particular, for a field \(k\), \(\Vect_k\) will refer to the \((\infty,1)\)-category of \(k\)-chain complexes. We use the theory of ind-coherent sheaf as developed in [19]. We use the following notations:

• \(k\) denotes a fixed algebraically closed field of characteristic zero.

• For a base scheme or stack \(S\), \(\PreSt_S\) refers to the category of \(S\)-prestacks, namely presheaves of groupoids over the category of affine schemes over \(S\). We will write \(\PreSt = \PreSt_k\).

• \(\PreSt^{\aft} \subset \PreSt\) denotes the full subcategory of prestacks locally almost of finite type.

• For a prestack \(\cX\), we write \(\cX_{\dR}\) for its de Rham prestack, whose functor of points is given by \(\cX_{\dR}(R) = \cX(R^{\red})\). For a base stack \(S\) and \(\cX \in \PreSt_S\), we write \[\cX_{S\mh\dR} = \cX_{\dR} \times_{S_{\dR}} S\] the relative de Rham prestack. If the base stack is understood implicitly, such as in the case \(S = \fM_{g,n}^{\sst}\), we will write \[\cX_{S\mh\dR} = \cX_{\dR/}\]

• For a category \(\cC\), we let \(\cC^{\corr}\) be the \(1\)-category of correspondences in \(\cC\), namely the category with same objects as in \(\cC\), and with morphisms \(X \to Y\) given by spans \(X \from Z \to Y\) in \(\cC\). Compositions are defined via pullback.

• We will denote by \(\Cat_k = \Vect_k\mh\Mod(\Cat)\) the symmetric monoidal category of \(k\)-linear categories, namely stable categories whose hom spaces have a structure of \(k\)-vector spaces in a compatible way. The monoidal strcture is given by Lurie’s tensor product [5].

• We denote by \(\hat{\cO} = k[[t]]\) the algebra of Taylor series over \(k\), and by \(\hat{D} = \Spf k[[t]]\) its formal spectrum. We denote by \[\Aut(\hat{\cO}) \simeq k[[a_0]][a_1^{\pm 1},a_2,a_3,\dots]\] the group indscheme of continuous automorphisms of \(\hat{\cO}\), and \[\Aut_*(\hat{\cO}) \simeq k[a_1^{\pm 1},a_2,a_3,\dots]\] its subgroup of automorphisms preserving the closed point \(\Spec k \to \Spf k[[t]]\).

• By a flat family of curves \(X \to S\) over a base stack \(S\) we mean a flat morphism, of finite presentation, of relative dimension \(\leq 1\), with \(X\) an algebraic space.

• The category \(\fSet^{\surj}\) is the category of non-empty finite sets and surjections between them. The category \(\fSet^{\surj,\subset}\) refers to the category of non-empty finite sets, together with a choice of a non-empty subset \(I \subset J\). Morphisms are given by surjective morphisms of sets which induce surjective morphisms on the chosen subsets.

• The category \(\Delta\) is the simplex category of finite non-empty ordered sets \([n] = \{0 < 1 < \dotsb < n\}\) and order preserving morphisms between them. The subcategories \(\Delta^{\surj},\Delta^{\inj}\) are the subcategories with same objects, and with morphisms given by surjective and injective maps resp.

For the most part, we will only deal with the subcategories \(\PreSt^{\aft}_S\) over an algebraic stack \(S\) locally of finite type. We will use the symmetric monoidal structure on \(\PreSt_S^{\corr}\) induced by the Cartesian symmetric monoidal structure \((-) \times_S (-)\) on \(\PreSt_S\). We then have a lax-monoidal functor \[\IndCoh : \PreSt^{\aft,\corr} \to \Cat_k\] sending a prestack \(\cX\) to the category \[\IndCoh(\cX) = \lim_{\Spec R \to \cX} \IndCoh(\Spec R)\] where the limit is taken with respect to \(!\)-pullback, and a morphism given by a span \(\cX \xleftarrow{f} \cZ \xrightarrow{g} \cY\) is sent to the functor \[g_*f^! : \IndCoh(\cX) \to \IndCoh(\cY)\] The lax-monoidal structure is given by the exterior tensor product \[\IndCoh(\cX) \otimes \IndCoh(\cY) \xrightarrow{\boxtimes} \IndCoh(\cX \times \cY)~.\]

As a consequence, for any base stack \(S \in \PreSt^{\aft}\), the category \(\IndCoh(S)\) is a commutative algebra in \(\Cat_k\), with multiplication given by \[\IndCoh(S) \otimes \IndCoh(S) \xrightarrow{\boxtimes} \IndCoh(S \times S) \xrightarrow{\Delta^!} \IndCoh(S)\] where \(\Delta : S \to S \times S\) is the diagonal map. We let \[\Cat_S \mathrel{\vcenter{:}}= \IndCoh(S)\mh\Mod(\Cat_k)~.\]

The composition \[\IndCoh_S : \PreSt_S^{\aft,\corr} \to \PreSt^{\aft,\corr} \to \Cat_k\] then upgrades to a lax-monoidal functor \[\IndCoh_S : \PreSt_S^{\aft,\corr} \to \Cat_S\] where the lax structure is given by the composition \[\begin{align} \IndCoh(\cX) \underset{\IndCoh(S)}{\otimes} \IndCoh(\cY) & \simeq \IndCoh(S) \underset{\IndCoh(S) \otimes \IndCoh(S)}{\otimes} \IndCoh(\cX) \otimes \IndCoh(\cY) \\ & \xrightarrow{\id \otimes \boxtimes} \IndCoh(S) \underset{\IndCoh(S \times S)}{\otimes} \IndCoh(\cX \times \cY) \\ & \to \IndCoh(S \times_{S \times S} \cX \times \cY) \simeq \IndCoh(\cX \times_S \cY)~. \end{align}\] Denote this composition by \[\otimes_S^! : \IndCoh(\cX) \otimes_{\IndCoh(S)} \IndCoh(\cY) \to \IndCoh(\cX \times_S \cY)~.\]

1.4 Ind-coherent sheaves over infinite type stacks↩︎

Since the group indscheme \(\Aut(\hat{\cO})\) has infinite type, we will also need some extensions of the theory of ind-coherent sheaves to the infinite context. We will follow the formalism in [20], [21]. Unlike the finite type case, in the infinite case we gnerally do not have both \(!\)-pullback and \(*\)-pushforward.

One can always define a functor \[\IndCoh^! : \PreSt^{\op} \to \Cat_k\] by right Kan extension, so that we have \(!\)-pullback along any map \(f : \cX \to \cY\). However, not much can be said about the category \(\IndCoh^!(\cX)\) without further assumptions. One can also define a \(*\)-theory for a larger class of prestacks:

For any renormalizable prestack \(\cX\), one can attach a compactly generated category \(\IndCoh_*^{\ren}(\cX)\), and this construction defines a functor \[\IndCoh_*^{\ren} : \PreSt_{\operatorname{w.ren}} \to \Cat_k~.\]

We will need the following properties of this functor:

1.5 Relation to previous work↩︎

For a vertex operator algebra \(V\) and a curve \(X\), Frenkel and Ben-Zvi [2] constructed a family of D-modules \(\cV_{g}\) over the universal curve \(\cX_{g}\), using Gelfand-Kazhdan descent. This can be interpreted as a classifying map \(X \to B\Aut_*\hat{\cO}\) for any family of smooth curves \(X/S\), which agrees with our map \(\pi^{X/S}_{1}\) from Example 7, and our construction reduces to their construction in the case where a universal factorization algebra comes from a vertex operator algebra.

Our definition of the space of \(I\)-pointed multidisks is borrowed from [22], and provides a (one-dimensional) multi-point version of the stack of étale germs defined in [8].

In [23], the sheaf \(\cV_g\) is extended to a sheaf \(\overline{\cV}_g\) over the universal curve \(\bcX_g \to \bcM_{g}\), using the isomorphism \(\bcX_{g} \simeq \bcM_{g,1}\). This allows one to identify a node \(p \in X \subset \bcX_g\) with a smooth point in the stable model for \((X,p)\), given by \(1 \in (X \underset{\{p\} \sim \{0,\infty\}}{\cup} \PP^1) / \GG_m \subset \bcM_{g,1}\). Since the vertex algebra bundle can be defined over any smooth point, one gets the desired extension. Inspired by this construction, we define the extension of a universal factorization algebra using the moduli of semistable modifications at \(p\), with a sequence of curves \(\PP^1 \cup \dotsb \cup \PP^1\) of arbitrary length inserted at \(p\), and with an arbitrary number of smooth points over it. This provides a multi-points extension for the above construction. However, we do not know whether the resulted spaces of conformal blocks agree with \(H^0\) of our extension, beyond the rational case. In [24], the above results are generalized using the mode transition associative algebra, which we identify with the zeroth homology of \(\fZ_{\cA}\) in [17].

1.6 Acknowledgements↩︎

I am deeply grateful to my advisor, John Francis, for his invaluable guidance and support throughout the writing of this paper. I would like to thank Alexander Beilinson for generously sharing his insights on the problem and for his meticulous feedback, which helped correct several errors in my arguments. I also thank Sam Raskin, David Nadler, Chiara Damiolini, Kevin Lin, and Justin Campbell for many helpful discussions.

2 Chiral and factorization algebras↩︎

2.1 Chiral monoidal structure↩︎

A factorization algebra \(\cA_X\) over a smooth curve \(X\) is given roughly by the following data: a collection of D-modules \(\cA_{I}\) over \(X^I\) for each finite set \(I\), compatible with restrictions along diagonal embeddings \(X^I \to X^J\), together with a collection of isomorphism \[\cA_I \mid X^I \backslash \Delta \simeq \cA_1^{\boxtimes I} \mid X^I \backslash \Delta\] where \(\Delta \subset X^I\) is the union of all the diagonals subspaces. In particular, both the values over the diagonal and away from the diagonal are determined by \(\cA_1\), and the only information missing is the gluing rules between \(\cA_1^{\boxtimes I}\) over \(X^I \backslash \Delta\) and \(\cA_1\) over \(\Delta\). The latter is captured precisely by the structure of a chiral algebra: A D-module \(\cB\) over \(X\), and a collection of gluing rules \[\cB^{\boxtimes I} \mid X^I \backslash \Delta \to \Delta_!\cB~.\] These gluing rules are then expected to satisfy certain compatibility conditions. In [3], Beilinson and Drinfeld formulated the structures of chiral and factorization algebras as certain cocommutative and Lie algebras in the appropriate pseudo-tensor category (namely, a \(k\)-linear colored operad), and proved that they are in fact equivalent. This was later reinterpreted in [4] as Koszul duality between Lie and cocommutative algebras in a certain symmetric monoidal category. One possible way of implementing this symmetric monoidal category is via the Ran space — the moduli space of finite nonempty subsets of \(X\). We will describe this idea, in the more general context of families of smooth curves, following the formalism in [4].

Throughout this section, we fix an algebraic stack of finite type \(S\). We will only discuss here the case of curves.

Let \(\fSet^{\surj}\) be the category of finite sets and surjective morphisms between them. Define a functor \[(X/S)^{\fSet}_{\dR} : \fSet^{\op} \to \PreSt_{S}^{\aft}\] by \[I \mapsto X^{\times_S I}_{S\mh\dR}, \quad (I \xrightarrow{\alpha} J) \mapsto (X^{\times_S J}_{S\mh\dR} \xrightarrow{\Delta_{(\alpha)}} X^{\times_S I}_{S\mh\dR})~,\] where \(\Delta_{(\alpha)}\) is the diagonal map corresponding to \(\alpha\). Then we have a presentation of \(\Ran (X/S)\) as a colimit:

The \(S\)-prestack \(\Ran (X/S)\) admits a symmetric monoidal structure given by the union of finite subsets. If we restrict the union operation to disjoint subsets only, we get merely an operad structure, which can be implemented by a symmetric monoidal structure in the category \(\PreSt^{\corr}_S\) of correspondences in \(\PreSt_S\), as we describe below.

From Lemma 3, we get \[\IndCoh(\Ran (X/S)) \simeq \lim_{I \in \fSet^{\surj}, (-)^!} \IndCoh(X^{\times_S I}_{S\mh\dR})~.\]

Furthermore, since all the transition maps \(\Delta_{(\alpha)}\) are closed embeddings, the corresponding pullbacks \(\Delta_{(\alpha)}^!\) admit left adjoints \(\Delta_{(\alpha),!} \simeq \Delta_{(\alpha),*}\). In particular, we get an isomorphism \[\label{eq:colim-present} \cF \simeq \colim_{\fSet^{\op}} \Delta_{I,!}\Delta_{I}^!\cF \eqqcolon \colim_{\fSet^{\op}}\Delta_{I,!}\cF_{I}\tag{1}\] for any \(\cF \in \IndCoh(\Ran (X/S))\). Here \(\Delta_I : X_{S\mh\dR}^{\times_S I} \to \Ran (X/S)\) is the diagonal embedding.

In terms of this presentation, an ind-coherent sheaf \(\cF\) over \(\Ran (X/S)\) can be described as a collection \(\cF_I \in \IndCoh(X_{S\mh\dR}^{\times_S I})\) for \(I \in \fSet\), together with isomorphisms \[\label{eq:Ran-iso} \cF_I \iso \Delta_{(\alpha)}^!\cF_J\tag{2}\] for any surjective \(\alpha : J \onto I\). The chiral monoidal structure can be then described by \[(\otimes^{\ch}_{S,i \in I} \cF_i)_J = \bigoplus_{\alpha : J \onto I} \jmath_{(\alpha),*}\jmath_{(\alpha)}^!\otimes^!_{S,i \in I}(\cF_i)_{J_i}~.\]

Here \(\jmath_{(\alpha)} : X^{(\alpha)}_{S\mh-\dR} \into X^{\times_S I}_{S\mh\dR}\), where \[X^{(\alpha)}_{S\mh\dR}(R) = \left\{(x_i)_{i \in I} \in X^{\times_SI}_{S\mh\dR}(R) :\: x_i \cap x_{i'} = \emptyset \text{ for }\alpha(i) \neq \alpha(i')\right\}~.\]

Now, since \(\sqcup\) is étale (see, e.g., [25]), \(\sqcup^! = \sqcup^*\) is left adjoint to \(\sqcup_*\).

Given a map \(f: T \to S\), we have a symmetric monoidal base-change functor \[(-)_T : \PreSt_S \to \PreSt_T\] right adjoint to the forgetful functor \[\Oblv_S^T : \PreSt_T \to \PreSt_S\] It induces a symmetric monoidal functor \[(-)_T : \PreSt_S^{\corr} \to \PreSt_T^{\corr}\] and therefore an induced map on commutative (co)algebra objects. Given a smooth family of curves \(X/S\), it sends the commutative algebra \(\Ran(X/S)\) to a commutative algebra \(\Ran(X/S)_T\). In general, this commutative algebra is different from the commutative algebra \(\Ran(X_T/T)\): While the underlying spaces are the same, the notions of disjointness are different, namely \(\Ran(X/S)^{\times_SI}_{\disj} \times_S T\) is different from \(\Ran(X_T/T)^{\times_TI}\). We always have a map \[\label{eq:bc-Ran-spaces} \Ran(X/S)^{\times_SI}_{\disj} \times_S T \to \Ran(X_T/T)^{\times_TI}_{\disj}~,\tag{3}\] but this map may not be an isomorphism even if \(u_f\) is étale.

We have a unit map \[u_f : \Oblv^T_S\Ran(X/S)_T \to \Ran(X/S)\] in \(\PreSt_S\), which induces maps \[(\id : u_f) : \Ran(X/S)_T \xleftarrow{\id} \Ran(X/S)_T \xrightarrow{u_f} \Ran(X/S)\] and \[(u_f : \id) : \Ran(X/S) \xleftarrow{u_f} \Ran(X/S)_T \xrightarrow{\id} \Ran(X/S)\] in \(\PreSt_S^{\corr}\).

Any factorization algebra defines a chiral algebra in the following way: Given a factorization algebra \(\cA\) over a smooth curve \(X_0 / \Spec k\), the D-module \(\cA_{1}[-1] = \Delta_1^!\cA[-1]\) has a structure of a chiral algebra, with the Lie algebra map given by the boundary (or gluing) morphism associated to the sequence \[\label{eq:exact-seq-fact} \Delta_*\Delta^!(\cA_{1} \boxtimes \cA_{1}) \to \cA_{1} \boxtimes \cA_{1} \to \jmath_*\jmath^!(\cA_{1} \boxtimes \cA_{1})~.\tag{5}\]

Explicitly, since \(\cA\) is a factorization algebra, we have \(\jmath^!(\cA \boxtimes \cA) \simeq \jmath^!\cA_{2}\). By 2 , we have \(\cA_{1} \iso \Delta^!\cA_{2}\). Therefore, 5 becomes \[\Delta_*\cA_{1} \to \cA_{1} \boxtimes \cA_{1} \to \jmath_*\jmath^!(\cA \boxtimes \cA)\] and the corresponding chiral multiplication map is the boundary morphism, applied to the shift of this sequence: \[\jmath_*\jmath^!(\cA_1[-1] \boxtimes \cA_1[-1]) = \jmath_*\jmath^!(\cA_1 \boxtimes \cA_1)[-2] \to \Delta_*\cA_{1}[-1]~.\]

The equivalence between chiral algebras and factorization algebras over a curve in the absolute case was proven in [3]. This result was generalized to arbitrary smooth schemes in [4], where it was shown to be a special case of Koszul duality between Lie algebras and commutative coalgebras. By adapting the proof of [4] to the relative case, we get:

We will denote this correspondence by \[\begin{align} \cB & \mapsto \cB^{\fact} \\ \cA^{\ch} & \mapsfrom \cA~. \end{align}\]

2.2 Chiral and factorization modules↩︎

Let \(S\) be an algebraic stack locally of finite type. The chiral monoidal structure over \(S\) was defined as a commutative algebra structure on \(\Ran (X/S) \in \PreSt^{\corr}_S\). We therefore have a natural notion of modules over \(\Ran (X/S)\). Such a module is given by an \(S\)-prestack \(\cY / S\), and a commutative action map \[\cY \times_S \Ran (X/S) \from \Act_{\cY} \to \cY\] for some \(S\)-prestack \(\Act_{\cY}\). Let \(\Ran(X/S)\mh\Mod\) be the category of \(\Ran(X/S)\)-modules in \(\PreSt_S^{\corr}\). In the case of augmented modules, the prestack \(\Act_{\cY}\) is uniquely determined:

An important class of modules is that of modules supported at a point, or more generally on any \(Z \to \Ran(X/S)\).

Now recall from Lemma 4 that for \(f: T \to S\), we have a restriction map \[u^{!,\fact}_{f} : \Fact(X/S) \to \Fact(X_T/T)~.\]

By Lemma 5, we also have extensions and restrictions to open sets, namely, for an open subfamily \(f : U/S \to X/S\), we have maps \[f_{\Ran,*}^{\fact} : \Fact(U/S) \leftrightarrows \Fact(X/S) : f_{\Ran}^{!,\fact}~.\]

If \(\cY = \Ran_Z(X/S)\), as in Example 4, we also have a notion of chiral modules:

By adapting the proof of [3] and [4] to the absolute case, we get:

Given a chiral module \(\cN\), we will denote the corresponding factorization module by \[\cN^{\fact} = \Chev^{\ch}(\cN)~.\]

2.3 Chiral Homology↩︎

The existence of \(p_{\Ran,!}\) follows from the fact that in the case where \(X \to S\) is proper, \(\Ran (X/S) \to S\) is pseudo-proper, and in particular \(p_{\Ran,*} = p_{\Ran,!}\) is left adjoint to \(p_{\Ran}^!\). From 1 , we get \[\int_{X/S} \cA \simeq \colim_{I \in \fSet^{\surj,\op}} p_{I,!}\cA_{I}\] where \(p_I : X^{\times_SI}_{S\mh\dR} \to S\) is the projection. Similarly, we have \[\int_{(X/S,Z)} (\cA,\cM) \simeq \colim_{J \subset I \in \fSet^{\subset,\surj,\op}} p_{Z,I,!}\cM^{\fact}_{I}~.\]

2.4 Universal factorization algebras↩︎

Let \(X\) be a curve. The local structure of a chiral algebra \(\cB\) with underlying D-module \(\cB_1 \in D\mh\Mod(X)^{\heartsuit}\) around a smooth point \(x \in X\) can be described in terms of vertex operator algebras. A vertex operator algebra (VOA) is a graded vector space \(V\) equipped with a derivation \(T\), two distinguished vectors \(\pmb{1}, \omega \in V\), and a \(D^{\times}\)-family of multiplications \[Y(-,z) : V \to \operatorname{End}(V)[[z^{\pm 1}]]\] satisfying a list of axioms. The vector \(\pmb{1}\) is the vacuum vector, which behaves like an identity element, while \(\omega\) is the conformal vector, which defines an action of the Virasoro algebra, and therefore strong equivariance with respect to formal deformations. Given a VOA, one can construct a global object – a chiral algebra over the smooth locus \(X^{\sm}\), which has \(V\) as its local structure around any smooth point. This was done in [2]. Such a chiral algebra can be defined over any curve in a compatible way. The resulted structure is that of a universal chiral algebra – a chiral algebra defined over any smooth curve and is compatible with étale pullbacks (Cf. [3] and [8]). The construction is done using Gelfand-Kazhdan descent, namely using pullback along the map \[X \to B\Aut_*\hat{\cO}\] obtaind from the principal \(\Aut_*\hat{\cO}\)-bundle \(\operatorname{Coord}_X \to X\) whose fiber at a point is given by the set of formal coordinates around the point. We will define here a similar construction, that gives a derived version of vertex operator algebras, by defining a "chiral product" on the category of \(\Aut\hat{\cO}\)-representations. This approach is borrowed from [22]:

In general, however, the number of disks in each geometric fiber of an \(R\)-multidisk may not be constant.

The assignments \[I \mapsto \MDisk_I^{\str},\MDisk_I, \quad \alpha \mapsto \Delta_{(\alpha)}\] define lax-monoidal functors \[\fSet^{\surj,\op} \to \operatorname{Stacks} \into \PreSt\] with respect to the disjoint union on the source and the Cartesian product on the target, with the lax structure given by \[\sqcup : \prod_{j \in J}\MDisk_{I_j} \to \MDisk_{\sqcup_j I_j}~.\]

Explicitly, we can represent each \(\cF \in \IndCoh^{!,\ren}(\MDisk_{\Ran})\) by a collection \(\cF_{J} \in \IndCoh^{!,\ren}(\MDisk_J)\) for each finite set \(J\), and identifications \(\Delta_{(\alpha)}^!\cF_{i,J} \simeq \cF_{i,J'}\) for each \(\alpha : J' \onto J\). Given \(\cF_i \in \IndCoh^{!,\ren}(\MDisk_{\Ran})\), we have \[\otimes^{\ch}_{i}\cF_{i,J} = \bigoplus_{\beta : J \onto I} \sqcup_{(\beta),?} \boxtimes_{i \in I} \cF_{i,J_i}~.\]

From [4] we get:

We do not know wether this functor restricts to an isomorphism \[\Ch^{\univ} \iso \Fact^{\univ}\] However, for a universal chiral algebra \(\cB\), the pullback of \(\Chev^{\ch}(\cB)\) to any smooth curve will be a factorization algebra, as we describe below.

3 Moduli spaces of nodal curves↩︎

3.1 Stable weighted pointed curves↩︎

Given \(g,n \in \ZZ_{\geq 0}\), the moduli stack \(\cM_{g,n}\) of \(n\)-pointed, smooth, projective, genus \(g\) curves, admits a well known compactification, given by the Deligne-Mumford stack \(\bcM_{g,n}\): For a \(k\)-algebra \(R\), its \(R\)-points are given by flat and proper families \(X \to \Spec R\), together with \(n\) sections \(p_1,\dots,p_n : \Spec R \to X\), such that \(X\) has at worst nodal singularities, namely points whose formal neighborhood is isomorphic to \(k[[x,y]]/(xy)\), and such that the configuration \((X,p_1,\dots,p_n)\) is stable. A pointed curve is stable if for each irreducible component \(X_0\) of genus \(g_0\), the number of nodes and marked points is greater than \(2-2g_0\). In particular, \(\bcM_{0,n}\) is empty for \(n<3\), and \(\bcM_{1,n}\) is empty for \(n=0\).

However, there exist other compactifications of \(\cM_{g,n}\). A family of such compactifications, which we will use in this paper, is given by Hassett’s moduli space of weighted pointed curves [26]:

Over geometric points, this means the following: The total weight at any given point is \(\leq 1\), where a node has weight \(1\), and the total weight of an irreducible component of genus \(h\) is greater than \(2 - 2h\). The stacks \(\bcM_{g,\pmb{w}}\) have the usual stable reduction maps, as well as stable reduction maps \(\bcM_{g,\pmb{w}} \to \bcM_{g,\pmb{w}'}\) whenever \(w_i' \leq w_i\) for all \(i\). It is proven in [26] that these spaces are smooth and proper Deligne-Mumford stacks.

We will mostly be interested in a special case of this construction, where we only have two classes of marked points: Those with weight one, which behaves like marked points on the usual compactification \(\bcM_{g,n}\), and those with weight \(\epsilon > 0\), which may collide with each other (but not with nodes or weight one points), thus behave like Ran points.

A special feature of the stacks \(\bcM_{g,n,I}\) is that, when forgetting the weight \(\epsilon\) points, the result is a semistable curve:

We will mostly be interested in sheaves and spaces over \(\bcM_{g,n,I}\) which have a flat connection with respect to changing the (weight \(\epsilon\)) marked points, but not with respect to changing the geometry of the curve (we regard a weight one marked point as a puncture, so changing it would mean changing the geometry). Thus, we define:

For a surjective map of finite sets \(\alpha : I \to J\), let \[\Delta_{(\alpha)} : (\bcM_{g,n,J})_{\dR/} \to (\bcM_{g,n,I})_{\dR/}\] be the map \[(X,x_1,\dots,x_n,\{y_j^{\red}\}_{j \in J}) \mapsto (X,x_1,\dots,x_n,\{y_{\alpha(i)}^{\red}\}_{i \in I})~.\]

By definition, \[\IndCoh(\bcM_{g,n,\Ran}) = \lim_{I \in \fSet^{\surj}} \IndCoh((\bcM_{g,n,I})_{\dR/})~,\] where the limit is taken with respect to \(!\)-pullbacks.

Since the maps \(\Delta_{(\alpha)}\) are closed embeddings, \(\Delta_{(\alpha)}^!\) admits a left adjoint \(\Delta_{(\alpha),!} \simeq \Delta_{(\alpha),*}\), and we get:

Assume now \(2g+n-2>0\). Then we have a projection map \[\bcM_{g,n,\Ran} \to \bcM_{g,n}\] given by forgetting the weight-\(\epsilon\) points, followed by stable reduction. Furthermore, this projection can be lifted to the Ran space of the universal curve:

We can identify an object in the fiber of \(\bcX_{g,n} \to \bcM_{g,n}\) as a marked point of weight \(0\). In particular, we have a stable reduction map from points of weight \(\epsilon\): \[\label{eq:Pi95gn} \Pi_{g,n} : \bcM_{g,n,\Ran} \to \Ran_{g,n}~.\tag{7}\]

Finally, we also have a semistable version of \(\bcM_{g,n,\Ran}\):

In other words, we have \[\fM_{g,n,\Ran}^{\sst} = \Ran_{\emptyset}(\mathring{\fX}_{g,n}^{\sst}/\fM_{g,n}^{\sst})~,\] where \(\fX_{g,n}^{\sst} \to \fM_{g,n}^{\sst}\) is the universal curve, and \(\mathring{\fX}_{g,n}^{\sst} = \fX_{g,n}^{\sst,\sm} \backslash \bigcup_{i=1}^n \sigma_i^*\fM_{g,n}^{\sst}\) is the smooth locus of the corresponding punctured curve. Here \(\sigma_i : \fM_{g,n}^{\sst} \to \fX_{g,n}^{\sst}\) are the canonical sections. We then have a natural inclusion \[\label{eq:Phi95gn} \Phi_{g,n} : \bcM_{g,n,\Ran} \to \fM_{g,n,\Ran}^{\sst}\tag{8}\]

and a left inverse over the non-empty locus, given by stable reduction \[\label{eq:Psi95gn} \Psi_{g,n} : \fM_{g,n,\Ran}^{\sst} \backslash \fM_{0,2,\emptyset}^{\sst} \to \bcM_{g,n,\Ran}~.\tag{9}\]

3.2 Moduli of semistable modifications for \((g,n) = (0,2)\)↩︎

In this section we review some results about the moduli of semistable modifications in the case of two-pointed genus zero curves. The basic object of study will be the moduli stack \(\fM_{0,2}^{\sst}\), parametrizing chains of genus zero curves (“rational chains”), with the two endpoints marked. A general semistable modification of an \(n\)-pointed genus \(g\) curve, either at a marked point or at a node, is given by inserting a rational chain at that point, hence the importance of this special case.

A basic object in the description of this stack is the quotient stack \(\AA^1/\GG_m\), parametrizing (virtual) Cartier divisors: A map \(S \to \AA^1/\GG_m\) from a scheme \(S\) is equivalent to a choice of a Cartier divisor \(D \subset S\), given by pulling back the universal Cartier divisor \(B\GG_m = \{0\}/\GG_m \into \AA^1/\GG_m\): \[D = S \times_{\AA^1/\GG_m} B\GG_m~.\]

Over \(\AA^1\), we have the family \[\overline{\WW}[1] = \left\{([x_0:x_1],[y_0:y_1],t) \vcentcolon x_0y_0 = tx_1y_1\right\} \subset \PP^1 \times \PP^1 \times \AA^1 \xrightarrow{t} \AA^1\] (or: the closure of \(\{xy=t\} \subset \AA^3\) inside \(\PP^1 \times \PP^1 \times \AA^1\)). We have two sections \[0,\infty : \AA^1 \to \overline{\WW}[1]\] given by \[0,\infty : t \mapsto ([1:0],[0:1],t),([0:1],[1:0],t)\]respectively. This is a family of semistable, two pointed, genus zero curves, hence defines a map \[\tilde{\tau}_1 : \AA^1 \to \fM_{0,2}^{\sst}~.\]

The curve \(\overline{\WW}[1] \to \AA^1\) admits an action by \(\GG^2_m\), given by \[(\lambda_0,\lambda_1) \cdot ([x_0:x_1],[y_0:y_1],t) = ([\lambda_0^{-1}x_0:x_1],[\lambda_1y_0:y_1],\lambda_0^{-1}t\lambda_1)\] and so we get a map \[\tau_1 : \fA[1] \mathrel{\vcenter{:}}= \GG_m \backslash \AA^1 / \GG_m \to \fM_{0,2}^{\sst}~.\] As we will see, this map is an open embedding of the substack of curves with at most two irreducible components. More generally, Li [28] defined families of semistable two-pointed genus zero curves \[\overline{\WW}[\ell] \to \AA^{\ell}\] for each \({\ell}>0\), having at most \(\ell\) nodes (or: at most \(\ell+1\) irreducible components). It has the property that the fiber over a point \((t_1,\dots,t_{\ell})\) with \(i\) zeros has exactly \(i\) nodes, and it can be equipped with a \(\GG_m^{\ell+1}\) action.

For a surjective order preserving map \(\alpha : [\ell] \onto [\ell']\), we have maps \[\fA[\ell'] \to \fA[\ell]\] given by \[(t_1,\dots,t_{\ell'}) \mapsto (t_{[\alpha(0)+1,\alpha(1)]},\dots,t_{[\alpha(\ell-1)+1,\alpha(\ell)]})~.\]

The following is a combination of [29]:

We will also need the following variants:

The following is essentially [29]:

In particular, we get projection maps \[\fM_{0,2+1}^{\sst}, \fM_{0,1+2}^{\sst} \to \AA^1/\GG_m\] and \[\fM_{0,2+2}^{\sst} \to \AA^1\] which maps the smooth locus to \(\GG_m\) and the singular locus to \(\{0\}\).

3.3 Moduli of semistable modifications for general \((g,n)\)↩︎

In this section, we describe the structure of semistable modifications at a point (either smooth or nodal) in terms of the stack \(\fM_{0,2}^{\sst}\) and its variants above. Informally, a semistable modification of a curve \(X\) at a smooth point \(x\) is given by attaching a (possibly empty) rational tail at \(x\). Since this construction is local, it can be identified with attaching a rational tail to \(\PP^1\) at \(\infty\). This data is equivalent to a choice of an object \(X \in \fM_{0,2}^{\sst}(k)\), together with identification of the component of \(0\) with \(\PP^1\). Similarly, a semistable modification at a node is given by inserting a rational bridge, which is equivalent to the insertion of a rational bridge at \(\infty \sim 0' \in \PP^1 \cup_{\infty \sim 0'} \PP^1\). The latter is equivalent, up to identifying the components of \(0,\infty\) with \(\PP^1\), with an object of \(\fM_{0,2}^{\sst}\). The precise statement was described in [29] (under the names “expanded pairs” and “expanded degenerations”). We will follow their description, and explain the corresponding structure on Ran spaces.

4 Chiral algebras on semistable modifications↩︎

4.1 Chiral action on stable curves↩︎

In this section we discuss the chiral monoidal structure, defined in Section 2, in the context of families of stable and semistable curves. First note that, since \(\fM_{g,n,\Ran} \to \fM_{g,n}\) is a family of smooth curves over the stack \(\fM_{g,n}\), we have a chiral product \[\fM_{g,n,\Ran} \times_{\fM_{g,n}^{\sst}} \fM_{g,n,\Ran} \xleftarrow{\jmath} (\fM_{g,n,\Ran} \times_{\fM_{g,n}^{\sst}} \fM_{g,n,\Ran})_{\disj} \xrightarrow{\sqcup} \fM_{g,n,\Ran}^{\sst}~.\]

Similarly, \(\mathring{\Ran}_{g,n} \to \bcM_{g,n}\) (see Definition 28) is a family of smooth curves over the stack \(\bcM_{g,n}\), and so we have a chiral multiplication \[\mathring{\Ran}_{g,n} \times_{\bcM_{g,n}} \mathring{\Ran}_{g,n} \xleftarrow{\jmath} (\mathring{\Ran}_{g,n} \times_{\bcM_{g,n}} \mathring{\Ran}_{g,n})_{\disj} \xrightarrow{\sqcup} \mathring{\Ran}_{g,n}~.\]

Given a universal factorization algebra \(\cA\), our goal in this section is to construct a factorization module \(\Vac(\cA)_{g,n}^0\) defined over nodal curves \(\Ran_{g,n}\), with respect to the action of \(\cA_{g,n}^0\), defined over the smooth, punctured locus \(\mathring{\Ran}_{g,n}\).

Explicitly, an \(R\)-point of \(\Ran_{g,n}^{\sst}\) is given by an \(n\)-pointed semistable curve \((X,\overline{p})\), and a finite and reduced subset \(\{x_i^{\red}\}_{i \in I} \subset X^{\operatorname{st}}(R^{\red})\) of its stable reduction. Denote by \(\operatorname{St} : X \to X^{\operatorname{st}}\) the projection. Then a point as above lies within \(\mathring{\Ran}_{g,n}^{\sst}\) if the image of \(\{x_i^{\red}\}_{i \in I}\) is contained in \(X^{\operatorname{st,sm}} \backslash \{\operatorname{St}(p_1),\dots,\operatorname{St}(p_n)\}\).

Therefore, the chiral monoidal structure on the family \(\mathring{\Ran}_{g,n}^{\sst}\) is obtained by base-change from the chiral multiplication on \(\mathring{\Ran}_{g,n}\). By Lemma 7, \(\Ran_{g,n}^{\sst} \in \mathring{\Ran}_{g,n}^{\sst}\mh\Mod^{\et}\) is a factorization module space over it.

Define a map \[\Upsilon_{g,n} : \mathring{\Ran}_{g,n}^{\sst} \to \fM_{g,n,\Ran}^{\sst}\] by \[(X,p_1,\dots,p_n,\{x_i^{\red}\}_{i \in I}) \mapsto \left(X,p_1,\dots,p_n,\left\{\operatorname{St}^{-1}(x_i^{\red})\right\}_{i\in I}\right)~.\] Note that this is well-defined, since \(\operatorname{St}\) is an isomorphism away from the exceptional divisor. From Lemma 5 and Lemma 8, with \(S = \fM_{g,n}^{\sst}\), \(\Ran(X/S) = \fM^{\sst}_{g,n,\Ran}\) and \(\Ran(U/S) = \mathring{\Ran}_{g,n}^{\sst}\), we get:

From Theorem 5, we get for each universal factorization algebra \(\cA\) a family of factorization algebras over these spaces.

Now from the map \(\Pi_{g,n}\) of 7 and the map \(\bcM_{g,n,\Ran} \to \fM_{g,n}^{\sst}\), we get a map \[\tilde{\Pi}_{g,n} : \bcM_{g,n,\Ran} \to \Ran_{g,n}^{\sst}\] Explicitly, for an \(R\)-point \((X,x_1,\dots,x_n,\{y_i^{\red}\}_{i \in I}) \in \bcM_{g,n,\Ran}(R)\), let \[\operatorname{St} : (X,x_1,\dots,x_n) \to (X^{\operatorname{st}},x'_1,\dots,x'_n)\] be the stable reduction map. Then \[\tilde{\Pi}_{g,n} : \left(X,x_1,\dots,x_n,\left\{y_i^{\red}\right\}_{i \in I}\right) \mapsto \left(X,x_1,\dots,x_n,\left\{\operatorname{St}(y_i^{\red})\right\}_{i \in I}\right)~.\]

Combining Lemma 18 with Lemma 6, we get:

Finally, using Lemma 4 and Lemma 7, with \(\Psi_{g,n} : T = \fM^{\sst}_{g,n} \to \bcM_{g,n} = S\) (see 9 ) and \(\Ran(X/S) = \mathring{\Ran}_{g,n}\), we get:

The module \(\Vac(\cA)^0_{g,n}\) will be our extension of a universal factorization algebra to nodal curves.

Note that by base-change, we have \[\Vac(\cA)_{(X,x_1,\dots,x_n)}^0 = \Pi_{(X,x_1,\dots,x_n),!}\Vac(\cA)_{(X,x_1,\dots,x_n)}^{\epsilon}\] where \(\Pi_{(X,x_1,\dots,x_n)}\) is as in Example 11.

4.2 The join monoidal structure↩︎

In this section, we will see that for \((g,n) = (0,2)\), the (pre)stacks of semistable modifications (namely \(\fM_{0,2}^{\sst},\fM_{0,2,\Ran},\bcM_{0,2,\Ran}\)) admit an additional structure of an associative algebra, given by concatenation of rational chains, and a universal factorization algebra produces an equivariant sheaf with respect to this algebra structure. From that we will get an associative algebra structure on global sections, which, as we prove later, is the value at a node of the extension \(\Vac(\cA)^0_{g,n}\) of Corollary 6.

This defines a non-unital associative algebra structure on \(\fM_{0,2}^{\sst},\fM_{0,2,\Ran}^{\sst}\). Furthermore, the projection commutes with the multiplication maps, and we have:

The main result of this section is:

Now note that the join product of two stable configurations is again stable, and in particular, the join product restricts to an associative multiplication on \(\bcM_{0,2,\Ran}\):

For general \((g,n)\), the join product acts on each marked point:

This defines \(n\) commuting right actions, and \(n\) commuting left actions. Note, however, that the right and left actions do not commute with each other. The left action can be obtained from the right action by pre-composition with the involution on \(\fM_{0,2}^{\sst}\) obtained by switching \(x_0\) and \(x_\infty\). Similarly to the case of \((g,n) = (0,2)\), we have:

5 Chiral homology for nodal curves↩︎

5.1 Associative algebra structure↩︎

In this section, we define chiral homology for nodal and punctured curves, and prove a gluing result relating the chiral homology of a nodal curve with that of its normalization.

Note that \(p_!^0 \simeq p_*^0\) is well-defined since \(p^0\) is pseudo-proper. Explicitly, \(\int_{g,n}\cA\) is given by integrating the vacuum module over all stable configurations in all semistable modifications: \[\int_{g,n}\cA \simeq p^{\epsilon}_!\Vac(\cA)_{g,n}^{\epsilon}\] and \[\int_{X \backslash \{x_1,\dots,x_n\}} \cA \simeq p_{(X,x_1,\dots,x_n),!}^{\epsilon}\Vac(\cA)_{(X,x_1,\dots,x_n)}^{\epsilon}~,\] where \(\Vac(\cA)_{(X,x_1,\dots,x_n)}^{\epsilon}\) is as in Definition 40, \(p^{\epsilon} : \bcM_{g,n,\Ran} \to \bcM_{g,n}\) the projection, and \(p^{\epsilon}_{X,x_1,\dots,x_n} : \bcM_{(X,x_1,\dots,x_n),\Ran} \to \Spec R\) its base-change along \(f\).

The restriction of \(\Vac(\cA)_{(X,x_1,\dots,x_n)}^{0}\) to \(\mathring{X} \mathrel{\vcenter{:}}= X^{\sm} \backslash \{x_1,\dots,x_n\}\) is just the usual vacuum module, while its fiber over \(x_i\) has a structure of a chiral module \(\fZ_{\cA}^+\) supported at a point, which we describe below. In particular, if \(X\) is smooth, we can realize \(\int_{X \backslash {x_1,\dots,x_n}} \cA\) as factorization homology of \(\cA\) over the proper curve \(X\) with coefficients in chiral module \(\fZ_{\cA}^+\) at each of the point \(x_1,\dots,x_n\). Its value over a node has a structure of a chiral bimodule supported at a point, isomorphic to the bimodule \(\fZ_{\cA}^{\nd}\) which we describe below.

Recall from Definition 41 the monoidal category \(\IndCoh^{\star}(\bcM_{0,2,\Ran})\).

5.2 Gluing formula↩︎

In this section we prove the main results of this paper, namely the expression of chiral homology for nodal curves in terms of chiral homology with coefficients for their normalizations. We start wth the case of a nodal curve given by gluing two pointed curves along marked points.

We have a similar result for self-gluing:

Alternatively, one can deduce the gluing formulas from a gluing formula for the chiral modules \(\fZ_{\cA}^{\pm}\), namely we have the following result, which is proven similarly to Theorem 7.

References↩︎