arXiv subjects

Andrei Caldararu

Publications and source records attributed to Andrei Caldararu.

At least 19 recordsLinked to original sources

arXiv2025

Categorical Enumerative Invariants, II: Givental formula

To a pair $(A,s)$ consisting of a smooth, cyclic $A_\infty$-algebra $A$ and a splitting $s$ of the Hodge filtration on its Hochschild homology Costello (2005) associates an invariant which conjecturally generalizes the total descendant Gromov-Witten potential of a symplectic manifold. In this paper we give explicit, computable formulas for Costello's invariants, as Feynman sums over partially directed stable graphs. The formulas use in a crucial way the combinatorial string vertices defined earlier by Costello and the authors. Explicit computations elsewhere confirm in many cases the equality of categorical invariants with known Gromov-Witten, Fan-Jarvis-Ruan-Witten, and Bershadsky-Cecotti-Ooguri-Vafa invariants.

math.SG↗

arXiv2024

Effective Categorical Enumerative Invariants

We introduce enumerative invariants $F_{g,n}$ $(g\geq0$, $n \geq 1)$ associated to a cyclic $A_\infty$ algebra and a splitting of its non-commutative Hodge filtration. These invariants are defined by explicitly computable Feynman sums, and encode the same information as Costello's partition function of the corresponding field theory. Our invariants are stable under Morita equivalence, and therefore can be associated to a Calabi-Yau category with splitting data. This justifies the name categorical enumerative invariants (CEI) that we use for them. CEI conjecturally generalize all known enumerative invariants in symplectic geometry, complex geometry, and singularity theory. They also provide a framework for stating enumerative mirror symmetry predictions in arbitrary genus, whenever homological mirror symmetry holds.

math.AG↗

arXiv2021

Moonshine at Landau-Ginzburg points

We formulate a conjecture predicting unexpected relationships among the coefficients of the elliptic expansions of Klein's modular j-function around j = 0 and j = 1728. Our conjecture is inspired by recent developments in mirror symmetry, in particular by work of Tu computing categorical enumerative invariants of matrix factorization categories and by work of Li-Shen-Zhou computing FJRW invariants of elliptic curves.

math.NT↗

arXiv2021

The cup product in orbifold Hochschild cohomology

We study the multiplicative structure of orbifold Hochschild cohomology in an attempt to generalize the results of Kontsevich and Calaque-Van den Bergh relating the Hochschild and polyvector field cohomology rings of a smooth variety. We introduce the concept of linearized derived scheme, and we argue that when $X$ is a smooth algebraic variety and $G$ is a finite abelian group acting on $X$, the derived fixed locus $\widetilde{X^G}$ admits an HKR linearization. This allows us to define a product on the cohomology of polyvector fields of the orbifold $[X/G]$. We analyze the obstructions to associativity of this product and show that they vanish in certain special cases. We conjecture that in these cases the resulting polyvector field cohomology ring is isomorphic to the Hochschild cohomology of $[X/G]$. Inspired by mirror symmetry we introduce a bigrading on the Hochschild homology of Calabi-Yau orbifolds. We propose a conjectural product which respects this bigrading and simplifies the previously introduced product.

math.AG↗

arXiv2020

Categorical Enumerative Invariants, I: String vertices

We define combinatorial counterparts to the geometric string vertices of Sen-Zwiebach and Costello-Zwiebach, which are certain closed subsets of the moduli spaces of curves. Our combinatorial vertices contain the same information as the geometric ones, are effectively computable, and act on the Hochschild chains of a cyclic $A_\infty$-algebra. This is the first in a series of two papers where we define enumerative invariants associated to a pair consisting of a cyclic $A_\infty$-algebra and a splitting of the Hodge filtration on its cyclic homology. These invariants conjecturally generalize the Gromov-Witten and Fan-Jarvis-Ruan-Witten invariants from symplectic geometry, and the Bershadsky-Cecotti-Ooguri-Vafa invariants from holomorphic geometry.

math.AT↗

arXiv2018

Categorical primitive forms and Gromov-Witten invariants of $A_n$ singularities

We introduce a categorical analogue of Saito's notion of primitive forms. Let $W$ denote the potential $\frac{1}{n+1} x^{n+1}$. For the category $MF(W)$ of matrix factorizations of $W$ we prove that there exists a unique, up to non-zero constant, categorical primitive form. The corresponding genus zero categorical Gromov-Witten invariants of $MF(W)$ are shown to match with the invariants defined through unfolding of singularities of $W$.

math.AG↗

arXiv2018

Intersections of two Grassmannians in $\mathbf{P}^9$

We study the intersection of two copies of $\mathrm{Gr}(2,5)$ embedded in $\mathbf{P}^9$, and the intersection of the two projectively dual Grassmannians in the dual projective space. These intersections are deformation equivalent, derived equivalent Calabi-Yau threefolds. We prove that generically they are not birational. As a consequence, we obtain a counterexample to the birational Torelli problem for Calabi-Yau threefolds. We also show that these threefolds give a new pair of varieties whose classes in the Grothendieck ring of varieties are not equal, but whose difference is annihilated by a power of the class of the affine line. Our proof of non-birationality involves a detailed study of the moduli stack of Calabi-Yau threefolds of the above type, which may be of independent interest.

math.AG↗

arXiv2017

Computing a categorical Gromov-Witten invariant

We compute the $g=1, n=1$ B-model Gromov-Witten invariant of an elliptic curve E directly from the derived category D(E). More precisely, we carry out the computation of the categorical Gromov-Witten invariant defined by Costello using as target a cyclic $A_\infty$ model of D(E) described by Polishchuk. This is the first non-trivial computation of a positive genus categorical Gromov-Witten invariant, and the result agrees with the prediction of mirror symmetry: it matches the classical (non-categorical) Gromov-Witten invariants of a symplectic 2-torus computed by Dijkgraaf.

math.AG↗

arXiv2016

Derived intersections and the Hodge theorem

The algebraic Hodge theorem was proved in a beautiful 1987 paper by Deligne and Illusie, using positive characteristic methods. We argue that the central algebraic object of their proof can be understood geometrically as a line bundle on a derived scheme. In this interpretation, the Deligne-Illusie result can be seen as a proof that this line bundle is trivial under certain assumptions. We give a criterion for the triviality of this line bundle in a more general context. The proof uses techniques from derived algebraic geometry, specifically arguments which show the formality of certain derived intersections. Applying our criterion we recover Deligne and Illusie's original result. We also apply these techniques to the result of Barannikov-Kontsevich, Sabbah, and Ogus-Vologodsky concerning the formality of the twisted de Rham complex.

math.AG↗

arXiv2014

Formality of derived intersections and the orbifold HKR isomorphism

We study when the derived intersection of two smooth subvarieties of a smooth variety is formal. As a consequence we obtain a derived base change theorem for non-transversal intersections. We also obtain applications to the study of the derived fixed locus of a finite group action and argue that for a global quotient orbifold the exponential map is an isomorphism between the Lie algebra of the free loop space and the loop space itself. This allows us to give new proofs of the HKR decomposition of orbifold Hochschild (co)homology into twisted sectors.

math.AG↗

arXiv2014

On the Lie algebroid of a derived self-intersection

Let $i:X\hookrightarrow Y$ be a closed embedding of smooth algebraic varieties. Denote by $N$ the normal bundle of $X$ in $Y$. We describe the construction of two Lie-type structures on the shifted bundle $N[-1]$ which encode the information of the formal neighborhood of $X$ inside $Y$. We also present applications of classical Lie theoretic constructions (universal enveloping algebra, Chevalley-Eilenberg complex) to the understanding of the geometry of embeddings.

math.AG↗

arXiv2013

Curved A-infinity algebras and Landau-Ginzburg models

We study the Hochschild homology and cohomology of curved A-infinity algebras that arise in the study of Landau-Ginzburg (LG) models in physics. We show that the ordinary Hochschild homology and cohomology of these algebras vanish. To correct this we introduce modified versions of these theories, Borel-Moore Hochschild homology and compactly supported Hochschild cohomology. For LG models the new invariants yield the answer predicted by physics, shifts of the Jacobian ring. We also study the relationship between graded LG models and the geometry of hypersurfaces. We prove that Orlov's derived equivalence descends from an equivalence at the differential graded level, so in particular the CY/LG correspondence is a dg equivalence. This leads us to study the equivariant Hochschild homology of orbifold LG models. The results we get can be seen as noncommutative analogues of the Lefschetz hyperplane and Griffiths transversality theorems.

math.KT↗

arXiv2011

PBW for an inclusion of Lie algebras

Let h \subset g be an inclusion of Lie algebras with quotient h-module n. There is a natural degree filtration on the h-module U(g)/U(g)h whose associated graded h-module is isomorphic to S(n). We give a necessary and sufficient condition for the existence of a splitting of this filtration. In turn such a splitting yields an isomorphism between the h-modules U(g)/U(g)h and S(n). For the diagonal embedding h \subset h \oplus h the condition is automatically satisfied and we recover the classical Poincae-Birkhoff-Witt theorem. The main theorem and its proof are direct translations of results in algebraic geometry, obtained using an ad hoc dictionary. This suggests the existence of a unified framework allowing the simultaneous study of Lie algebras and of algebraic varieties, and a closely related work in this direction is on the way.

math.QA↗

arXiv2011

When is the self-intersection of a subvariety a fibration?

We provide a necessary and sufficient condition for the derived self-intersection of a smooth subscheme inside a smooth scheme to be a fibration over the subscheme. As a consequence we deduce a generalized HKR isomorphism. We also investigate the relationship of our result to path spaces in homotopy theory, Buchweitz-Flenner formality in algebraic geometry, and draw parallels with similar results in Lie theory and symplectic geometry.

math.AG↗

arXiv2007

The Mukai pairing, I: a categorical approach

We study the Hochschild homology of smooth spaces, emphasizing the importance of a pairing which generalizes Mukai's pairing on the cohomology of K3 surfaces. We show that integral transforms between derived categories of spaces induce, functorially, linear maps on homology. Adjoint functors induce adjoint linear maps with respect to the Mukai pairing. We define a Chern character with values in Hochschild homology, and we discuss analogues of the Hirzebruch-Riemann-Roch theorem and the Cardy Condition from physics. This is done in the context of a 2-category which has spaces as its objects and integral kernels as its 1-morphisms.

math.AG↗

arXiv2006

The Pfaffian-Grassmannian derived equivalence

We argue that there exists a derived equivalence between Calabi-Yau threefolds obtained by taking dual hyperplane sections (of the appropriate codimension) of the Grassmannian G(2, 7) and the Pfaffian Pf(7). The existence of such an equivalence has been conjectured by physicists for almost ten years, as the two families of Calabi-Yau threefolds are believed to have the same mirror. It is the first example of a derived equivalence between Calabi-Yau threefolds which are provably non-birational.

math.AG↗

arXiv2006

Non-birational Calabi-Yau threefolds that are derived equivalent

We argue that the existence of genus one fibrations with multisections of high degree on certain Calabi-Yau threefolds implies the existence of pairs of such varieties that are not birational, but are derived equivalent. It also (likely) implies the existence of non-birational counterexamples to the Torelli problem for Calabi-Yau threefolds.

math.AG↗

arXiv2006

Fiberwise stable bundles on elliptic threefolds with relative Picard number one

We show that fiberwise stable vector bundles are preserved by relative Fourier-Mukai transforms between elliptic threefolds with relative Picard number one. Using these bundles we define new invariants of elliptic fibrations, and we relate the invariants of a space with those of a relative moduli space of stable sheaves on it. As a byproduct, we calculate the intersection form of a certain new example of an elliptic Calabi-Yau threefold.

math.AG↗