nLab Calabi-Yau category

1. Idea

The notion of Calabi-Yau category is a horizontal categorification of that of Frobenius algebra – a Frobenius algebroid. Their name derives from the fact that the definition of Calabi-Yau categories have been originally studied as an abstract version of the derived category of coherent sheaves on a Calabi-Yau manifold.

2. Definition

1-categorical

A Calabi-Yau category is a Vect-enriched category $C$ equipped for each object $c \in C$ with a trace-like map

$$Tr_C : C(c,c) \to k$$

to the ground field, such that for all objects $d \in C$ the induced pairing

$$\langle -,-\rangle_{c,d} : C(c,d) \otimes C(d,c) \to k$$

given by

$$\langle f,g \rangle = Tr(g \circ f)$$

is symmetric and non-degenerate.

[Question: this 1-categorical definition seems to allow for different Frobenius structures on the endomorphism algebras of isomorphic objects. Would it be better to define it as a dinatural transformation from the Hom-functor to the constant functor with value the ground field ?]

A Calabi-Yau category with a single object is the same (or rather: is equivalently) the pointed monoid delooping of a Frobenius algebra.

$(\infty,1)$-categorical

A Calabi-Yau $A_\infty$-category of dimension $d \in \mathbb{N}$ is an A-∞ category $C$ equipped with, for each pair $a,b$ of objects, a morphism of chain complexes

$$\langle -,-\rangle_{a,b} : C(a,b) \otimes C(b,a) \to k[d]$$

such that

1. this is non-degenerate and is symmetric in that

   $$\langle - , - \rangle_{a,b} = \langle - , - \rangle_{b,a} \circ \sigma_{a,b}$$

   for $\sigma_{a,b} : C(a,b)\otimes C(b,a) \to C(b,a) \otimes C(a,b)$ the symmetry isomorphism of the symmetric monoidal category of chain complexes;

2. this is cyclically invariant in that for all elements $(\alpha_i)$ is the respective hom-complexes we have

   $$\langle m_{n-1}(\alpha_0 \otimes \cdots \otimes \alpha_{n-2}), \alpha_{n-1} \rangle = (-1)^{(n+1)+ |\alpha_0| \sum_{i = 1}^{n-1}|\alpha_i|} \langle m_{n-1}(\alpha_1 \otimes \cdots \otimes \alpha_{n-2}), \alpha_0 \rangle \,.$$

3. Examples

From Calabi-Yau varieties

• Let $X$ be a smooth projective Calabi-Yau variety of dimension $d$. Write $D^b(X)$ for the bounded derived category of that of coherent sheaves on $X$.

  Then $D^b(X)$ is a CY $A_\infty$-category in a naive way:

  • the non-binary composition maps are all trivial;

  • the pairing is given by Serre duality (one needs also a choice of trivialization of the canonical bundle of $X$)

This is however not the morally correct CY $A_\infty$-structure associated with a Calabi-Yau. A correct choice is, for example, the Dolbeault dg-enhancement of the derived category

(Costello 04, 7.2, Costello 05, 2.2)

From symplectic manifolds

The Fukaya category associated with a symplectic manifold $X$. But see this MO discussion for more.

From string topology

string topology: for $X$ a compact simply connected oriented manifold, its cohomology $H^{\bullet}(X)$ is naturally a Calabi-Yau $A_\infty$-category with a single object. The $A_\infty$ structure comes from the homological perturbation lemma. One could also use the dg algebra of cochains $C^\bullet(X)$.

4. Properties

Classification of 2d TQFT

Calabi-Yau $A_\infty$-categories classify TCFTs. This remarkable result is what actually one should expect. Indeed, TCFTs originally arose as an abstract version of the CFTs constructed from sigma-models whose targets are Calabi-Yau spaces.

2d TQFT (“TCFT”)	coefficients	algebra structure on space of quantum states
open topological string	Vect${}_k$	Frobenius algebra $A$	folklore+(Abrams 96)
open topological string with closed string bulk theory	Vect${}_k$	Frobenius algebra $A$ with trace map $B \to Z(A)$ and Cardy condition	(Lazaroiu 00, Moore-Segal 02)
non-compact open topological string	Ch(Vect)	Calabi-Yau A-∞ algebra	(Kontsevich 95, Costello 04)
non-compact open topological string with various D-branes	Ch(Vect)	Calabi-Yau A-∞ category	“
non-compact open topological string with various D-branes and with closed string bulk sector	Ch(Vect)	Calabi-Yau A-∞ category with Hochschild cohomology	“
local closed topological string	2Mod(Vect${}_k$) over field $k$	separable symmetric Frobenius algebras	(SchommerPries 11)
non-compact local closed topological string	2Mod(Ch(Vect))	Calabi-Yau A-∞ algebra	(Lurie 09, section 4.2)
non-compact local closed topological string	2Mod$(\mathbf{S})$ for a symmetric monoidal (∞,1)-category $\mathbf{S}$	Calabi-Yau object in $\mathbf{S}$	(Lurie 09, section 4.2)

5. Related concepts

• Calabi-Yau object, Calabi-Yau algebra, pre-Calabi-Yau algebra

6. References

• Kevin Costello, Topological conformal field theories and Calabi-Yau categories (arXiv:math/0412149)

• Kevin Costello, The Gromov-Witten potential associated to a TCFT (arXiv:0509264)

• Maxim Kontsevich, Yan Soibelman. Notes on A-infinity algebras, A-infinity categories and non-commutative geometry, arXiv:math/0606241

• Lee Cho, Notes on Kontsevich-Soibelman’s theorem about cyclic A-infinity algebras (arXiv:1002.3653)

• Jacob Lurie, section 4.2 of On the Classification of Topological Field Theories (arXiv:0905.0465)

A relative version is defined for functors instead of categories,

• Christopher Brav , Tobias Dyckerhoff, Relative Calabi–Yau structures, Compositio Math. 155 (2019) 372–412 doi; Relative Calabi–Yau structures II: shifted Lagrangians in the moduli of objects, Sel. Math. New Ser. 27, 63 (2021). doi
• Christopher Brav, Nick Rozenblyum, The cyclic Deligne conjecture and Calabi-Yau structures, arXiv:2305.10323
• Bernhard Keller, Yu Wang, An introduction to relative Calabi-Yau structures, arXiv:2111.10771
• Christopher Kuo, Wenyuan Li, Relative Calabi-Yau structure on microlocalization, arXiv:2408.04085
• Ludmil Katzarkov, Pranav Pandit, Theodore Spaide, Calabi-Yau structures, spherical functors, and shifted symplectic structures, Adv. Math. 392 (2021) 108037 doi