Abstract
The notion of a phantom category is a recently coined term, referring to a nontrivial triangulated or dg-category with vanishing Grothendieck group and/or Hochschild homology. In this note, we refer to categories with vanishing Grothendieck group as K-phantoms and categories with vanishing Hochschild homology and HH-phantoms. When both of these invariants vanish, we follow [13], and refer to these categories simply as phantoms.
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1 Introduction
The notion of a phantom category is a recently coined term, referring to a nontrivial triangulated or dg-category with vanishing Grothendieck group and/or Hochschild homology. In this note, we refer to categories with vanishing Grothendieck group as K-phantoms and categories with vanishing Hochschild homology and HH-phantoms. When both of these invariants vanish, we follow [13], and refer to these categories simply as phantoms.
The existence of phantom Fukaya–Seidel categories was conjectured in [10]. This conjecture was based on a combination of the seminal works of Donaldson, Kotschick, and Okonek and van de Ven [11, 18, 20], who distinguished smooth structures on Barlow surfaces and Del Pezzo surfaces of degree one using the moduli space of instanton bundles. It was also inspired by the study of the behavior of D-branes under phase transition following Witten and Aspinwall [4, 26].
By the homological mirror symmetry conjecture, it stands to reason that phantom categories also appear as triangulated categories coming from algebraic geometry. Indeed, the first example of such a category was revealed in [9] where the authors construct an admissible subcategory of the derived category of coherent sheaves on a Godeaux surface with vanishing Hochschild homology (an HH-phantom). A different construction of an HH-phantom was provided in [3], and strengthened in [13] which provides the first example of a phantom admissible subcategory of a bounded derived category of coherent sheaves on a smooth projective variety. In fact, in this example it is shown that both the Hochschild homology and all of the algebraic K-groups of the admissible subcategory vanish. A different example of a geometric phantom is constructed in [8]. In that case the phantom is an admissible subcategory of the derived category of a determinantal Barlow surface.
In this note we show that geometric phantom categories are not accidental and can naturally appear in an infinite sequence of components of the moduli of dg categories. We provide a basic but non-trivial set of examples of K-phantom categories: matrix factorizations for odd-dimensional A 2m -singularities. An important property of our example is that its Orlov spectrum is not a consecutive sequence of integers, a phenomenon connected to birational geometry in [7]. We discuss how this example fits with results from string theory appearing in [2, 4] and outline a description of geometric transformations which might produce phantoms which we plan to explore in future work.
The content of the paper is arranged as follows. In Sect. 3 we discuss the properties of the category \(\mathrm{MF}(k[[x,y]],x^{n+1} + y^{2})\), and show that it has vanishing K0 group. In Sect. 4, we demonstrate that the Orlov Spectrum of this example is not a consecutive sequence of integers for n > 7. In Sect. 5 we discuss the geometric significance of phantom categories in a conjectural framework.
2 Adding Quadratic Terms to the Potential
Suppose \((R,\mathfrak{m})\) is a regular local ring and \(f \in \mathfrak{m}\). Following Eisenbud [12] we consider matrix factorizations
where P 0 and P 1 are free R-modules of finite rank. Matrix factorizations form a differential \(\mathbb{Z}/(2)\)-graded category MF(R, f), c.f. Orlov [21], and the homotopy category H(MF(R, f)) is triangulated. We will denote a quadruple as in (1) by \((P_{0},P_{1},p_{0},p_{1})\).
The aim of this section is to discuss the relation between the categories
We begin by describing an adjunction
with F left adjoint to G. The functor F assigns to an object \((P_{0},P_{1},p_{0},p_{1})\) of \(\mathcal{D}\) the object
of \(\mathcal{C}\), to a morphism \((f_{0},f_{1})\) in degree 0 the morphism
and to a morphism \((f_{0},f_{1})\) in degree 1 the morphism
The functor G maps an object \((P_{0},P_{1},p_{0},p_{1})\) of \(\mathcal{C}\) to the object
of \(\mathcal{D}\), where \(\bar{p}_{i}\) denote the induced maps, and to a morphism \((f_{0},f_{1})\), of either degree, the induced morphism \((\bar{f}_{0},\bar{f}_{1})\). On an object \((P_{0},P_{1},p_{0},p_{1})\) of \(\mathcal{C}\) the counit \(\varepsilon\) is defined by
where
On an object \((P_{0},P_{1},p_{0},p_{1})\) of \(\mathcal{D}\) the unit η is defined by
We note that
and thus for objects X, Y of \(\mathcal{D}\)
which shows that the full subcategory of \(\mathcal{C}\) generated by objects in the image of F is the category of orbits under the action of \(\mathbb{Z}/(2)\) on \(\mathcal{D}\) via the shift functor, c.f. [15].
We specialize to the case of a power series ring \(R = k[[x_{0},\ldots,x_{m}]]\) over an algebraically closed field k of characteristic zero. Under these assumptions, Knörrer [17] shows that for an indecomposable object \(X \in H(\mathcal{D})\) with \(X\not\cong X[1]\) the object F(X) is indecomposable as well, while for an indecomposable X with \(X\mathop{\cong}X[1]\) we have that \(F(X)\mathop{\cong}Y \oplus Y [1]\) for some indecomposable \(Y \in H(\mathcal{C})\). The same property also holds for the functor G. Moreover, an object \(Y \in H(\mathcal{C})\) is in the essential image of F if and only if \(Y \mathop{\cong}Y [1]\).
3 The A 2m Singularity
Let k be an algebraically closed field of characteristic zero and fix an even integer n = 2m. Consider the category of matrix factorizations
associated with the A n curve singularity.
For 0 ≤ i ≤ n + 1 define matrix factorizations
where R = k[[x, y]] and
By [25] these are all the indecomposable objects and
in the triangulated homotopy category \(H(\mathcal{C})\). Moreover, for 0 < i < n + 1 there are triangles
as follows from [25].
We claim that
Indeed, writing \([X] \in K_{0}(\mathcal{C})\) for the K-theory class of an object \(X \in \mathcal{C}\), we see from (17) and (18) that [W 0] = 0, \([W_{i}] = i[W_{1}]\) for 0 < i < n + 1, so [W 1] is a generator. But \((n - 1)[W_{1}] = 0\), and since \(W_{i}[1]\mathop{\cong}W_{i}\) we also have 2[W 1] = 0, so [W 1] = 0, since n was assumed to be even.
We note that, by Knörrer periodicity (see [17]), \(\mathcal{C}\) is equivalent to any of the categories
with m odd. The relation to the corresponding category for even m is a special instance of the discussion in the previous section with
Indecomposable objects of \(H(\mathcal{D})\) are given by
for 1 ≤ i ≤ n, see for example [21]. We have
so in this case the essential image of F is all of \(\mathcal{C}\), hence \(\mathcal{C}\) is the orbit category of \(\mathcal{D}\) under the action of \(\mathbb{Z}/(2)\) via the shift functor.
Remark 1.
Let \(\mathcal{A} = \mathit{MF}_{\mathit{gr}}(k[[X]],x^{n+1})\) denote the dg-category of graded matrix factorizations of the A n -singularity. The group \(\mathbb{Z}\) acts on \(\mathcal{A}\) via grading shift, and the corresponding orbit dg-category, in the sense of [15], is equivalent to \(\mathit{MF}(k[[X]],x^{n+1})\) by a general result from [16]. The category \(\mathcal{A}\) itself coincides with the bounded derived category of finite-dimensional representations of an A n quiver, as was show in [22].
4 Comparison of Orlov Spectra
Let us recall the following definitions. For a more complete treatment see, [7, 23]. Let \(\mathcal{T}\) be a triangulated category. For a full subcategory, \(\mathcal{I}\), of \(\mathcal{T}\) we denote by \(\langle \mathcal{I}\rangle\) the full subcategory of \(\mathcal{T}\) whose objects are isomorphic to summands of finite coproducts of shifts of objects in \(\mathcal{I}\). In other words, \(\langle \mathcal{I}\rangle\) is the smallest full subcategory containing \(\mathcal{I}\) which is closed under isomorphisms, shifting, and taking finite coproducts and summands. For two full subcategories, \(\mathcal{I}_{1}\) and \(\mathcal{I}_{2}\), we denote by \(\mathcal{I}_{1} {\ast}\mathcal{I}_{2}\) the full subcategory of objects, T, such that there is a distinguished triangle,
with \(I_{i} \in \mathcal{I}_{i}\). Set
and, for n ≥ 1, inductively define,
Similarly we define
For an object, \(X \in \mathcal{T}\), we notationally identify X with the full subcategory consisting of E in writing, \(\langle X\rangle _{n}\). The reader is warned that, in some of the previous literature, \(\langle \mathcal{I}\rangle _{0}:= 0\) and \(\langle \mathcal{I}\rangle _{1}:=\langle \mathcal{I}\rangle\). We follow the notation in [5, 7]. With our convention, the index equals the number of cones allowed.
Definition 1.
Let X be an object \(\mathcal{T}\). If there is an \(n\) with \(\langle X\rangle _{n} = \mathcal{T}\), we set
Otherwise, we set . We call the generation time of X.
Definition 2.
Let \(X\) be an object of a triangulated category, \(\mathcal{T}\). The Orlov spectrum of \(\mathcal{T}\), denoted \(\text{OSpec}\mathcal{T}\), is the set,
Let \(F: \mathcal{T} \rightarrow \mathcal{R}\) be an exact functor between triangulated categories. If every object in \(\mathcal{R}\) is isomorphic to a direct summand of an object in the image of F, we say that F is dense.
Lemma 1.
If \(F: \mathcal{T} \rightarrow \mathcal{R}\) is dense and \(X\) is a strong generator, then,
Proof.
If X is a generator of \(\mathcal{T}\) with minimal generation time \(t\), then \(\mathcal{T} =\langle X\rangle _{t}\). Now as F is an exact functor,
Since every object of \(\mathcal{R}\) is a summand of an object \(F(\mathcal{T} )\), we see that \(\mathcal{R} =\langle F(X)\rangle _{t}\) and the formula follows.
Proposition 1.
Let R be a complete regular ring. For any f ∈ R one has:
Proof.
We demonstrate that,
Notice that F is dense as, by Proposition 2.6 of [17], for any object, \(A \in \text{MF}(R[[y]],f + y^{2})\), A ⊕ A[1] is in the image of F. Meanwhile, G is dense by (11). Therefore,
where the first equality follows from the (11) and the two inequalities are applications of Lemma 1. Therefore the map,
provides an inclusion of Orlov spectra.
By the symmetry of the situation which follows from Knörrer Periodicity (Theorem 3.1 of [17]), we have the other inclusion as well.
Remark 2.
Proposition 1 can also be obtained as a consequence of viewing \(\text{MF}(R,f)\) as a \(\mathbb{Z}_{2}\)-orbit category of \(\text{MF}(R[[y]],f + y^{2})\) and applying Proposition 9.8 of [6].
Corollary 1.
The category, \(\text{MF}(k[[x,y]],x^{2m+1})\) has the following properties:
-
1.
\(\text{MF}(k[[x,y]],x^{2m+1})\) is a K-phantom i.e. \(\text{K}_{0}(\text{MF}(k[[x,y]],x^{2m+1}) = 0\)
-
2.
\(\text{OSpec}(\text{MF}(k[[x,y]],x^{2m+1}) = \left \{0,1,\ldots,\left \lceil \frac{m} {s} \right \rceil - 1,\ldots,\left \lceil \frac{m} {2} \right \rceil - 1,m - 1\right \}\)
Proof.
1) was explained in the paragraph following (19). 2) Follows from Proposition 1 and Theorem 4.14 of [7]
5 Conjectures
In this section, we propose that the following conjectual procedures could lead to the creation of phantom categories:
-
1.
Rational blow downs and smoothings of surfaces.
-
2.
Degenerations and smoothings of threefolds.
These ideas were inspired by [9], which provided the first example of an HH-phantom as a subcategory of the derived category of coherent sheaves on the classical Godeaux surface. This led the third author to immediately conjecture that the derived category of coherent sheaves on a Barlow surface would contain a phantom subcategory in the stronger sense. This intuition was brought to fruition in [8] for generic determinantal Barlow surfaces and was based on the following rationale.
After blowing-up, a determinantal Barlow surface degenerates to a two sheeted covering of \(\mathbb{P}^{1} \times \mathbb{P}^{1}\) with a singular ramification curve. These degenerations and blow-ups replace a cohomology class from h 2, 0 with one in h 1, 1. The new h 1, 1 class is spanned by a single exceptional object. In the mirror this procedure creates a deep singularity and the phantom subcategory, \(\mathcal{A}\subseteq \text{D}^{\text{b}}(\text{coh}X)\), can be viewed through Homological Mirror Symmetry, as the Fukaya–Seidel category of this singularity.
In general, from the perspective of [14], the Hochschild homology of the Fukaya–Seidel category of a hypersurface singularity is computed as the hypercohomology of a sheaf of vanishing cycles on the singular locus associated to the potential. Therefore, the fact that the Fukaya–Seidel category contains an HH-phantom is equivalent to the existence of a connected component of the singular locus whose associated sheaf of vanishing cycles has trivial cohomology. Hence, we can look for symplectic procedures which create such singular fibers.
Indeed, the required geometric procedures in the case of the Barlow surface, appear naturally and robustly in physics where they are known as conifold and extremal exoflop transitions [4, 26]. Pushing the envelope, these procedures suggest a general framework for producing phantom categories.
Let us consider an example appearing in [24]: a conic bundle over \(\mathbb{P}^{2}\) with a discriminant curve of degree 12. There is a degeneration consisting of a sequence of exoflops [4], which reduces the intermediate Jacobian of this conic bundle to zero. This is the analog of the transition which on the Barlow surface modifies an h 2, 0 class into an h 1, 1 class. While the intermediate Jacobian is eliminated, the singular fiber in the mirror has degenerated but not disappeared. However, all the cohomology has Hodge degree (p, p), hence the presence of an exceptional collection forces the existence of an HH-phantom.
Conjecture 1.
Let X be the threefold obtained from the conic bundle after degeneration and smoothing described above. There is a semi-orthogonal decomposition,
where \(\mathcal{A}\) is a phantom category and E i are exceptional objects.
Semi-orthogonal components have also been described as a categorical analog of the Griffiths–Clemens component by Kuznetsov, see for example [19]. The conic bundle above is not rational by [24]. Therefore, the example above indicates that phantom categories, at least in some cases, can be a finer invariant than the classical Griffith–Clemens component. This suggests a method to understanding birational geometry for three dimensional conic bundles if the following conjecture holds:
Conjecture 2.
For a generic three dimensional conic bundle over a rational surface, there exists a deformation of the complex structure and resolution of singularities, X, such that \(\text{D}^{\text{b}}(\text{coh}X)\) contains an admissible phantom subcategory.
This conjecture is based on ideas from [1, 2]. Indeed, [1] provides a mirror symmetry construction for conic bundles which can be degenerated as in [2] so that the mirror still contains a deep singular fiber but whose Hodge cycles are all of type (p, p). In the example above the phantom subcategory corresponds to a component of the singular locus of this mirror given by an elliptic curve whose associated sheaf of vanishing cycles F has trivial hypercohomology.
Conversely, the following conjecture follows from the Jordan–Hölder property for semi-orthogonal decompositions conjectured by Kuznetsov see for example, [19].
Conjecture 3.
For any rational threefold, X, Db(cohX) does not contain an admissible phantom subcategory.
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Acknowledgements
We would like to whole-heartedly thank C. Böhning, H. Graf von Bothmer, P. Sosna, D. Orlov, S. Galkin, A. Kuznetsov, D. Auroux, C. Vafa, E. Sharpe, and S. Gukov, for inspiring these ideas through many extremely useful discussions. The first and second authors were funded by FWF Grant P20778, and an ERC Grant (GEMIS). The third author was funded by NSF Grant DMS0600800, NSF FRG Grant DMS-0652633, FWF Grant P20778, and an ERC Grant (GEMIS).
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Favero, D., Haiden, F., Katzarkov, L. (2014). An Orbit Construction of Phantoms, Orlov Spectra, and Knörrer Periodicity. In: Castano-Bernard, R., Catanese, F., Kontsevich, M., Pantev, T., Soibelman, Y., Zharkov, I. (eds) Homological Mirror Symmetry and Tropical Geometry. Lecture Notes of the Unione Matematica Italiana, vol 15. Springer, Cham. https://doi.org/10.1007/978-3-319-06514-4_2
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