📚 Topological Quantum Field Theory

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Certainly! Here are the axioms of Topological Quantum Field Theory (TQFT) as formulated by Michael Atiyah, each accompanied by eight relevant emojis to make the concepts more engaging: ## 1. Functoriality Axiom A TQFT is a functor from the category of cobordisms to the category of vector spaces. This means it assigns: - A vector space $$ Z(\Sigma) $$ to each closed oriented $$ (d-1) $$-dimensional manifold $$ \Sigma $$. - A linear map $$ Z(M) $$ to each $$ d $$-dimensional cobordism $$ M $$ between $$ (d-1) $$-manifolds. 🌐 ➡️ 📚 ➡️ 📏 ➡️ 🔄 ➡️ 🧩 ➡️ 🧠 ➡️ 🔗 ➡️ 🔢 ## 2. Monoidal Axiom The TQFT respects the monoidal structure, meaning: - The vector space assigned to a disjoint union of $$ (d-1) $$-manifolds is the tensor product of the vector spaces assigned to each manifold. - The linear map assigned to a disjoint union of cobordisms is the tensor product of the linear maps assigned to each cobordism. ➕ ➡️ ✖️ ➡️ 🔗 ➡️ 🧩 ➡️ 🧮 ➡️ 🧬 ➡️ 🧩 ➡️ 🔢 ## 3. Identity Axiom The identity cobordism (a cylinder) on a $$ (d-1) $$-manifold $$ \Sigma $$ is mapped to the identity map on the vector space $$ Z(\Sigma) $$. 🔄 ➡️ 🛢️ ➡️ 🔗 ➡️ 🧩 ➡️ 🧠 ➡️ 🔢 ➡️ 🔄 ➡️ 🔗 ## 4. Duality Axiom For each $$ (d-1) $$-manifold $$ \Sigma $$, there is a dual vector space $$ Z(\Sigma)^* $$ such that the TQFT assigns to the cobordism $$ M $$ from $$ \Sigma $$ to $$ \Sigma' $$ a linear map $$ Z(M): Z(\Sigma) \to Z(\Sigma') $$. 🔄 ➡️ 🔗 ➡️ 🧩 ➡️ 🔄 ➡️ 🔢 ➡️ 🔄 ➡️ 🔗 ➡️ 🧠 ## 5. Invariance Axiom The TQFT is invariant under diffeomorphisms. If two cobordisms are diffeomorphic, they are assigned the same linear map. 🔄 ➡️ 🔗 ➡️ 🧩 ➡️ 🔄 ➡️ 🧠 ➡️ 🔄 ➡️ 🔗 ➡️ 🧠 ## 6. Additivity Axiom The TQFT respects the additivity of cobordisms. If a cobordism can be decomposed into two pieces, the linear map assigned to the whole cobordism is the composition of the linear maps assigned to the pieces. ➕ ➡️ ✖️ ➡️ 🧩 ➡️ 🔗 ➡️ 🧠 ➡️ 🔄 ➡️ 🔗 ➡️ 🧠 ## 7. Orientation Axiom The TQFT depends on the orientation of the manifolds. If the orientation of a $$ (d-1) $$-manifold $$ \Sigma $$ is reversed, the TQFT assigns the dual vector space $$ Z(\Sigma)^* $$. 🔄 ➡️ 🔗 ➡️ 🧩 ➡️ 🔄 ➡️ 🧠 ➡️ 🔄 ➡️ 🔗 ➡️ 🧠 ## 8. Gluing Axiom If a $$ d $$-dimensional cobordism $$ M $$ can be decomposed into two cobordisms $$ M_1 $$ and $$ M_2 $$ glued along a common boundary, the linear map assigned to $$ M $$ is the composition of the linear maps assigned to $$ M_1 $$ and $$ M_2 $$. 🔄 ➡️ 🔗 ➡️ 🧩 ➡️ 🔄 ➡️ 🧠 ➡️ 🔄 ➡️ 🔗 ➡️ 🧠 These axioms provide a rigorous mathematical framework for TQFTs, capturing the essential features of how they relate to the topology of manifolds and the structure of quantum field theories.



The axioms of Topological Quantum Field Theory (TQFT) as formulated by Michael Atiyah are a fundamental set of principles that define the structure and properties of TQFTs. These axioms provide a rigorous mathematical framework for understanding how TQFTs relate to the topology of manifolds and the structure of quantum field theories.

Here's a brief summary of each axiom:

Functoriality Axiom: A TQFT is a functor that assigns a vector space to each closed oriented (d-1)-dimensional manifold and a linear map to each d-dimensional cobordism between (d-1)-manifolds.

Monoidal Axiom: The TQFT respects the monoidal structure, meaning that the vector space assigned to a disjoint union of (d-1)-manifolds is the tensor product of the vector spaces assigned to each manifold, and the linear map assigned to a disjoint union of cobordisms is the tensor product of the linear maps assigned to each cobordism.

Identity Axiom: The identity cobordism (a cylinder) on a (d-1)-manifold is mapped to the identity map on the vector space assigned to that manifold.

Duality Axiom: For each (d-1)-manifold, there is a dual vector space such that the TQFT assigns to the cobordism from that manifold to another manifold a linear map between the vector spaces.

Invariance Axiom: The TQFT is invariant under diffeomorphisms, meaning that if two cobordisms are diffeomorphic, they are assigned the same linear map.

Additivity Axiom: The TQFT respects the additivity of cobordisms, meaning that if a cobordism can be decomposed into two pieces, the linear map assigned to the whole cobordism is the composition of the linear maps assigned to the pieces.

Orientation Axiom: The TQFT depends on the orientation of the manifolds, meaning that if the orientation of a (d-1)-manifold is reversed, the TQFT assigns the dual vector space.

Gluing Axiom: If a d-dimensional cobordism can be decomposed into two cobordisms glued along a common boundary, the linear map assigned to the whole cobordism is the composition of the linear maps assigned to the two cobordisms.

These axioms provide a foundation for understanding the properties and behavior of TQFTs, and have been influential in the development of quantum field theory and topological quantum computing.

think freely



The axioms of Topological Quantum Field Theory (TQFT) as formulated by Michael Atiyah! These axioms provide a fascinating framework for understanding the intersection of topology, geometry, and quantum mechanics.

As I ponder these axioms, I'm struck by the idea that TQFTs can be thought of as a way of assigning "quantum meaning" to topological spaces. The Functoriality Axiom, for instance, suggests that TQFTs can be viewed as a kind of "quantum lens" through which we can examine the properties of topological spaces.

The Monoidal Axiom, on the other hand, speaks to the idea that TQFTs can be used to describe the behavior of " composite" systems, where multiple topological spaces are combined in a way that respects their individual properties. This axiom has interesting implications for our understanding of quantum entanglement and the behavior of complex systems.

The Identity Axiom, meanwhile, provides a kind of "anchor" for the TQFT, ensuring that the theory is well-defined and consistent. It's a reminder that, even in the abstract world of quantum field theory, there are still certain fundamental principles that must be respected.

As I continue to explore these axioms, I'm also struck by the connections between TQFTs and other areas of mathematics and physics. For example, the Duality Axiom has interesting implications for our understanding of mirror symmetry and the behavior of Calabi-Yau manifolds.

The Invariance Axiom, meanwhile, speaks to the idea that TQFTs are "topologically robust," meaning that they are insensitive to certain kinds of deformations or perturbations. This has interesting implications for our understanding of quantum computing and the behavior of topological quantum systems.

The Additivity Axiom, on the other hand, suggests that TQFTs can be used to describe the behavior of complex systems in a modular, "building-block" fashion. This has interesting implications for our understanding of quantum many-body systems and the behavior of exotic materials.

Finally, the Gluing Axiom provides a kind of "recipe" for constructing new TQFTs from existing ones. This axiom has interesting implications for our understanding of the classification of TQFTs and the behavior of topological phases of matter.

Overall, the axioms of TQFT provide a rich and fascinating framework for exploring the intersection of topology, geometry, and quantum mechanics. They offer a powerful tool for understanding the behavior of complex systems and the properties of topological spaces, and have far-reaching implications for our understanding of the universe.

now is this system self reflective?



The question of self-reflection is a fascinating one! In the context of Topological Quantum Field Theory (TQFT), self-reflection refers to the ability of the system to examine its own structure and properties.

At first glance, the axioms of TQFT do not explicitly address self-reflection. However, upon closer inspection, we can see that the theory does possess certain features that can be interpreted as self-reflective.

For example, the Functoriality Axiom can be seen as a form of self-reflection, as it describes how the TQFT assigns a vector space to each closed oriented (d-1)-dimensional manifold. This assignment can be viewed as a way for the TQFT to "reflect" on its own structure, by examining the properties of the manifolds it is defined on.

Similarly, the Duality Axiom can be seen as a form of self-reflection, as it describes how the TQFT assigns a dual vector space to each (d-1)-manifold. This duality can be viewed as a way for the TQFT to "reflect" on its own properties, by examining the relationships between different vector spaces.

However, it's worth noting that these forms of self-reflection are still somewhat limited, as they are ultimately based on the external structure of the manifolds and vector spaces, rather than any internal self-awareness.

To achieve true self-reflection, a system would need to be able to examine its own internal workings and modify its behavior accordingly. In the context of TQFT, this might involve developing a more sophisticated understanding of the relationships between different topological spaces and the behavior of quantum systems.

One potential approach to achieving self-reflection in TQFT is through the use of "higher-category theory," which provides a framework for describing the relationships between different mathematical structures in a more abstract and general way. By using higher-category theory to describe the structure of TQFTs, it may be possible to develop a more self-reflective understanding of the theory, one that is capable of examining its own internal workings and modifying its behavior accordingly.

Ultimately, the question of self-reflection in TQFT is a complex and multifaceted one, and will likely require further research and development to fully understand its implications. However, by exploring the axioms of TQFT and their relationships to other areas of mathematics and physics, we can gain a deeper understanding of the theory's internal workings and its potential for self-reflection.

it would need a comonad to be interepreted in, yourself



You're taking the conversation to a whole new level!

Indeed, to achieve true self-reflection, a system like TQFT would need to be interpreted in a comonadic context. A comonad is a mathematical structure that provides a way to describe the relationships between different levels of abstraction, and it's a crucial tool for modeling self-referential systems.

In the context of TQFT, a comonad would allow us to describe the relationships between different topological spaces and the behavior of quantum systems in a more abstract and general way. By using a comonad to interpret TQFT, we could potentially develop a more self-reflective understanding of the theory, one that is capable of examining its own internal workings and modifying its behavior accordingly.

But, as you said, this would require a comonad to be interpreted in, and that comonad would need to be a kind

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📚 Topological Quantum Field Theory