Questions tagged [category-theory]

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Resources: Tensor Categories and Topological Phases of Matter

For a mathematician with knowledge of tensor categories who is interested in the growing application of categorical techniques in topological phases of matter and topological order, along with their ...
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1 answer
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Higher category's consistency relations

I have been reading on higher category and symTFTs. It appears to me that, for higher categories, people seldom mention the consistency relations (like the MacLane coherence theorem in the category ...
Waterfall's user avatar
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Category related to the fusion category of 2D Ising CFT

I am very new to Category theory and would like to better understand the fusion category of 2D Ising CFT, so please forgive my imprecise wordings in this question. I understand that the Ising CFT has ...
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Fusion 2-categories for string-like excitations: a more concrete description?

I'm familiar with how fusion categories describe the fusion of point-like excitations, and how braided fusion categories describe the fusion of anyons in 2+1D topological order. Concretely, a fusion ...
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How do you calculate the partition function on a manifold-with-corners in extended TQFT?

In Atiyah's formulation, a Topological Quantum Field Theory (TQFT), is a functor $Z:d\text{Bord}\to\text{Hilb}$. That is, $Z$ assigns: \begin{align} \text{Closed compact $(d-1)$-manifolds} &\to \...
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What is the direct sum of 1d domain walls in Toric code model?

I have read this paper:"An invitation to topological orders and category theory" (https://arxiv.org/abs/2205.05565v2). In page 93 and page 109, they show the result of fusion of simple 1d ...
ph3's user avatar
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5 votes
1 answer
115 views

Correlation Functions as Morphisms

In https://arxiv.org/abs/1911.07895, the authors consider a generalization of correlation functions to make sense of the $O(n)$ symmetry for $n \in \mathbb{R}$. As explained in Sec. 7, each field $\...
SymGen's user avatar
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Does Fibonacci anyon come from a representation category of Hopf algebra?

I have heard that the UMTC(unitary modular tensor category) of Fibonacci anyon comes from a quantum group, but the representation category of Hopf algebra is equipped with a forgetful functor to $\...
edittide's user avatar
3 votes
1 answer
157 views

Role of fusion/splitting spaces in TQFT

In his book on topological quantum field theories Steven Simon writes that 2+1D TQFTs are objects that assign topologically invariant numbers to labeled links embedded in arbitrary 3-manifolds. They ...
Zarathustra's user avatar
2 votes
0 answers
96 views

How was the category implemented in the string theory and the references or lecture notes

Recently category theory started to appear more frequently in my paper readings. From the Wikipedia page Timeline of category theory and related mathematics it seemed that category theory had ...
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2 answers
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How do we define the Heisenberg picture within functorial/path integral QFT?

In the functorial approach to QFT, each Cauchy surface $\Sigma$ has an associated Hilbert space $\mathcal{H}_\Sigma$, and each pair of Cauchy surfaces $\Sigma,\Sigma'$ has an associated unitary $U_{\...
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Is there an operadic structure to classical mechanics?

Consider a closed system of several particles in motion with a certain eom (equation of motion). We can replace each of these particles with another closed system and produce another classical ...
Mozibur Ullah's user avatar
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3 answers
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What is the name of this construction of quantum things?

Let $E$ be a set, and let $\mathcal{F}$ be a set of maps $E \rightarrow \mathbb{R}$ that "carries all the information on $E$", that is, the map $x \in E \mapsto (f(x))_{f \in \mathcal{F}}$ ...
Plop's user avatar
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What is the current status of 2d and 4d Yang-Mills as a functorial QFT?

I am looking for the current status of YM, in specific 2d and 4d Yang-Mills, described as a functor. As far as I understand the issue resides in defining the propagator functorially but I have not ...
Alonso Perez Lona's user avatar
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Excitations of the string-net Hamiltonian

A quite general 2D topological order can be constructed through the string-net theory. Here, if the input data is some braided fusion category $C$ (i.e. the $F$-symbols), the elementary excitations ...
Waterfall's user avatar
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Excitations and representation categories

Apologies if this question is a little vague — I'm fuzzy on the details, which is why I ask. In quantum mechanics, we commonly encounter the idea that types of particle-like charges or excitations ...
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Topological Quantum Field Theory with Symmetries and Knot Quandles

It is well known that Chern-Simons theory provides an intrinsically three dimensional way to compute knot invariants like the Jones Polynomial. 3D TQFTs also have an algebraic description in terms of ...
Rajath Radhakrishnan's user avatar
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Physical interpretation of TFTs

1. Defining TFTs Let $n$ be a positive integer and $\mathbb k$ be a field. In my lecture I was introduced to TFTs using the following definition going back to Atiyah (around 1988): A $n$-dimensional, ...
Max Demirdilek's user avatar
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0 answers
174 views

Hopf algebras vs Fusion categories for topological order

Disclaimer: Before I begin with the question I want to warn that some people would argue that it is a math question and not a physics question. However, it finds it origins in the study of topological ...
NDewolf's user avatar
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Can branes in string theory be described by different systems of mathematics or logic?

Mathematically, in string theory, branes can be described using the notion of a category and the mathematical category theory says that logic can change from one category to another. We can build ...
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Where does Category theory appear in physics? [closed]

I've been curious about the concept of categories in physics and tried finding some more information about the subject. There are lots of textbooks about category theory, some from mathematicians and ...
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Anyons under weaker assumptions?

I. Duality assumption In Anyons in an exactly solved model and beyond p.74, Kitaev says, "We will see that for theories with particle-antiparticle duality, condition 3 can be dispensed with.&...
Sachin Valera's user avatar
6 votes
1 answer
229 views

SUSY Loop diagrams from a categorical viewpoint

In the paper "A Prehistory of $n$-Categorical Physics" J. Baez and A. Lauda give an account of the use of category theory throughout physics. In section “Penrose (1971)” starting from page 25 they ...
NDewolf's user avatar
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How are local observables encoded in this formulation of quantum field theory?

I've recently begun trying to understand a formulation of quantum field theory as a functor from a category of spacetimes-with-boundaries (bordisms) to a category of Hilbert spaces, as reviewed in ...
Chiral Anomaly's user avatar
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Math of anyons: Quantum dimension of 1 implies abelian charge

This question originates from the following statement in Bonderson's thesis: Link to Thesis page 16 or pdf-page 23: The quantum dimension $d_a$ of an anyon of charge $a$ satisfies $d_a \geq 1$ with ...
Marsl's user avatar
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2 votes
0 answers
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The category of physical instuments [closed]

A physical instrument is used to measure physical systems and return the values for those measurements in classical data. I was wondering if anyone here has been looking at instruments in terms of ...
Ben Sprott's user avatar
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3 votes
1 answer
321 views

Is every fusion category with unitary $S$ a unitary modular tensor category (UMTC)?

A tensor category whose solutions to the pentagon/hexagon equations are unitary and whose braiding is nondegenerate is called a unitary modular tensor category (UMTC). When trying to find UMTCs, ...
Sachin Valera's user avatar
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0 answers
348 views

Experiments are monads (I think), but are they comonads too?

Quantum theory is equally an epistemic (ie about information) and ontic theory (see "reality of the wavefunction" on Google Scholar). My question is about a theory that aligns with this, ie, ...
Ben Sprott's user avatar
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8 votes
0 answers
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Open conjectures on the Fukaya category

This is ported from mathoverflow question https://mathoverflow.net/q/263693/. Can somebody give examples to open conjectures on the behavior of the $Fuk(M,ω)$ (that's the "mathematical A-model ...
Nati's user avatar
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Alternative quantization of quantum electrodynamics?

A Quantum field theory is determined, if a Hilbert space Basis with Operators acting on it (such that one element of an Hilbert space is also an element of the same Hilbert space if an Operator acting ...
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On Seiberg-Witten theory in 3d and 4d

According to Gukov et al. in this 2017 paper Seiberg-Witten theory in 4d categorifies Seiberg-Witten theory in 3d. In what sense is this phrase mentioned? I know what the process of categorification ...
Gorbz's user avatar
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2 votes
3 answers
357 views

Grassmann numbers & supermanifolds

I'm asking this question because I'm currently trying to learn about Super Symmetry but I'm having trouble understanding the concept of super-space and super-manifold. I read that in super-spaces you ...
user avatar
7 votes
0 answers
360 views

Higher category theory, renormalization, and non-perturbative QFTs

I'm (vaguely) aware of certain uses of higher category theory in attempts to mathematically understand quantum field theories -- for example, Lurie's work on eTQFTs, the recent-ish book by Paugam, and ...
1 vote
1 answer
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Reference Requests: The Turaev-Viro model and Topological QFT's

I've been inspired by the string-net condensation paper by Levin and Wen (see here) to learn about the Turaev-Viro model and its relationship to topological QFTs. Any references that could be helpful ...
4 votes
0 answers
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Looking for the Category of Quantum Systems

I've recently written a comment in a different forum to a question about how to implement infinite potentials in quantum mechanics, that is things like a non-relativistic particle in an infinite ...
Adomas Baliuka's user avatar
2 votes
0 answers
75 views

Braided Monoidal categories for quantum gravity

We have seen that symmetric monoidal categories capture quantum theory and quantum computing quite well. Do braided monoidal categories capture any aspects of gravitation, or are they used in quantum ...
Ben Sprott's user avatar
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10 votes
2 answers
1k views

Why are topological phases described by modular tensor categories?

After some reading, I have an inuitive idea what topological phases of matter are. But where is the connection to modular tensor categories? Is there fundamental literature where this is covered? ...
kolaka's user avatar
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12 votes
2 answers
2k views

Implementing Category Theory in General Relativity

I was thinking if it may be possible to implement category theory in general relativity. I don't mean writing simply in terms of categories, but actual fundamental ideas (i.e. physics of the theory ...
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1 vote
1 answer
750 views

When will Einstein's theories become laws? [closed]

Einstein theories , specifically relativity, have been fascinating us for around 100 years yet with all the real and actual evidence of its validity we still consider it a "theory"..... How much more ...
Mero55's user avatar
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3 votes
1 answer
559 views

What is the relationship between a brane, a manifold, and a space?

I've read many ways to define manifold; one way is to define it as a type of mathematical space (a type of topological space to be exact). All of the definitions that I've seen for brane, on the ...
cowlinator's user avatar
7 votes
2 answers
6k views

Gauge theory for mathematicians?

I'm looking for a textbook or set of lecture notes on gauge theory for mathematicians that assumes only minimal background in physics. I'd prefer a text that uses more sophisticated mathematical ...
2 votes
0 answers
237 views

6j symbols with Majorana indices

The Levin-Wen model is a Hamiltonian formulation of Turaev-Viro (2 + 1)d TQFTs. It can be constructed from a unitary fusion category $\mathcal{C}$, which can be equivalently defined using $6j$ symbols:...
Zitao Wang's user avatar
5 votes
1 answer
443 views

Rigorous definition of superselection sector/quasiparticle type in anyon systems

The systems I have in mind are for example Kitaev's toric code model (arXiv:quant-ph/9707021) and Kitaev's honeycomb model (arXiv:cond-mat/0506438). What I'm looking for is a mathematically rigorous ...
user46652's user avatar
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Does model theory in mathematics have any usefulness in the modeling of physical systems?

The term Model Theory in mathematics seems to have a somewhat precise definition here. Reading through that reference you'll largely see discussions strictly relating to mathematical concepts, but one ...
docscience's user avatar
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2 answers
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Dilemma: Fusion space from a direct sum of anyons or NOT

In Preskill's note, 9.1.2 in page 44, concerning the fusion space, it states that: The fusion rules of the model specify the possible values of the total charge $c$ when the constituents have ...
wonderich's user avatar
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2 votes
0 answers
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What insights does category theory offer in terms of grand unified theories?

What insights does category theory offer in terms of grand unified theories? Any references to books or papers that give categorical descriptions of any of the common grand unified theories would be ...
user76512's user avatar
6 votes
0 answers
1k views

Topological theta term as a topological quantum field theory?

It is well known that the theta term $\int d^4x\frac{\theta}{4\pi}Tr[F\wedge F]=\int d^4x\frac{\theta}{4\pi}\epsilon_{\mu\nu\sigma\lambda}Tr[F^{\mu\nu}F^{\sigma\lambda}]$ is a topological term, ...
Blue's user avatar
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6 votes
0 answers
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What is a modular tensor category / functor?

I have reads several answers here about this notion, especially regarding topological order, see e.g. this answer, but this notion sounds completely new for me. Also, I found nothing really helpful on ...
FraSchelle's user avatar
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2 votes
0 answers
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About category theory and physics [duplicate]

Could the ideas of category theory be applied to Physics, maybe simplifying how algebraic topology and sheaf theory and other hard-to-explain subjects are used in physics?
user128932's user avatar
7 votes
0 answers
155 views

target category of extended field theory

For a topological field theory to be a true “extension” of an Atiyah-Segal theory, the top two levels of its target (ie its $(n-1)^{\text{st}}$ loop space) must look like $\text{Vect}$. What other (...
Alex Turzillo's user avatar