Calabi-Yau object


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See at cobordism hypothesis – For non-compact cobordisms.



For CC a symmetric monoidal (infinity,2)-category, a Calabi-Yau object in CC is

This is (Lurie 09, def. 4.2.6).


Calabi-Yau algebras


Let S\mathbf{S} be a good symmetric monoidal (∞,1)-category. Write Alg(S)Alg(\mathbf{S}) for the symmetric monoidal (∞,2)-category whose objects are algebra objects in S\mathbf{S} and whose morphisms are bimodule objects.

Then a Calabi-Yau object in Alg(S)Alg(\mathbf{S}) is an algebra object AA equipped with an SO(2)SO(2)-equivariant morphism

tr: S 1A1 tr \colon \int_{S^1} A \to 1

from the Hochschild homology S 1AA AAA\int_{S^1} A \simeq A \otimes_{A \otimes A} A, satisfying the condition that the composite morphism

AA S 0A S 1Atr1 A \otimes A \simeq \int_{S^0} A \to \int_{S^1} A \stackrel{tr}{\to} 1

exhibits AA as its own dual object A A^\vee.

Such an algebra object is called a Calabi-Yau algebra object.

This is (Lurie 09, example 4.2.8).


Relation to extended 2d TQFT (TCFT) and the Cobordism hypothesis

A version of the cobordism hypothesis says that symmetric monoidal (,2)(\infty,2)-functors

Z:Bord 2 nc𝒞 Z : Bord_2^{nc} \to \mathcal{C}

out of a version of the (infinity,2)-category of cobordisms where all 2-cobordisms have at least one outgoing (ingoing) boundary component, are equivalently given by their value on the point, which is a Calabi-Yau object in 𝒞\mathcal{C}.

This is (Lurie 09, theorem 4.2.11).

Here the trace condition translates to the cobordism which is the “disappearance of a circle”.

* \array{ && \longleftarrow \\ & \swarrow && \nwarrow \\ & \searrow && \nearrow \\ && \longrightarrow } \;\;\;\;\;\; \Rightarrow \;\;\;\;\;\; \ast

Its would-be adjoint, the “appearance of a circle” is not included in Bord 2 ncBord_2^{nc}.

This is closely related to the description of 2d TQFT as TCFTs (Lurie 09, theorem 4.2.13).

2d TQFT (“TCFT”)coefficientsalgebra structure on space of quantum states
open topological stringVect k{}_kFrobenius algebra AAfolklore+(Abrams 96)
open topological string with closed string bulk theoryVect k{}_kFrobenius algebra AA with trace map BZ(A)B \to Z(A) and Cardy condition(Lazaroiu 00, Moore-Segal 02)
non-compact open topological stringCh(Vect)Calabi-Yau A-∞ algebra(Kontsevich 95, Costello 04)
non-compact open topological string with various D-branesCh(Vect)Calabi-Yau A-∞ category
non-compact open topological string with various D-branes and with closed string bulk sectorCh(Vect)Calabi-Yau A-∞ category with Hochschild cohomology
local closed topological string2Mod(Vect k{}_k) over field kkseparable symmetric Frobenius algebras(SchommerPries 11)
non-compact local closed topological string2Mod(Ch(Vect))Calabi-Yau A-∞ algebra(Lurie 09, section 4.2)
non-compact local closed topological string2Mod(S)(\mathbf{S}) for a symmetric monoidal (∞,1)-category S\mathbf{S}Calabi-Yau object in S\mathbf{S}(Lurie 09, section 4.2)


Revised on January 3, 2015 17:08:05 by David Corfield (