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Calabi-Yau object

Context

Functorial quantum field theory

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

Contents

Idea

See at cobordism hypothesis – For non-compact cobordisms.

Definition

Definition

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

This is (Lurie 09, def. 4.2.6).

Examples

Calabi-Yau algebras

Example

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).

Properties

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’s 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)

References

Revised on November 18, 2014 08:58:01 by Urs Schreiber (217.155.201.6)