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Questions tagged [cohomology]

Cohomology is a general term for a sequence of abelian groups associated with topological spaces. For example, Lie algebra cohomology is a theory used for the study of the topology of Lie groups and homogeneous spaces. Homology also has applications in quantum field theory, for example the BRST quantization, and other areas of mathematical physics.

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Integrating over non-trivial fiber bundles - Chern-Simons Theory

I have been reading Tong's notes on QHE and Gauge Theories, specifically the part about quantizing the Abelian U(1) Chern-Simons level at finite temperature in the presence of a monopole (These discussions begin on pages 150 and 391 in the documents…
Stratiev
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How is group cohomology in SPT's related to the 't Hooft anomaly on the boundary?

I understand that group cohomology description for symmetry protected topological phases (SPT) comes from discrete nonlinear sigma models. A tutorial on this can be found in the excellent lectures by Prof Xiao-Gang Wen:…
pathintegral
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How is cohomology theory used in quantum field theory?

Quantum field theory uses a large amount of mathematics and I was wondering about some applications of cohomology theory in QFT, I understand it has applications in string theory but I was wondering about possible applications in more traditional…
Unmotivated L-function
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Non-locality and Twistor functions

Is there a nice intuitive way to visualize the concept of non-locality associated to twistor functions? And how is it related to the type of non-locality we encounter in Quantum Mechanics?
KP99
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Why 't Hooft anomaly can be described by some characteristic class?

In some recent papers, such as Zohar PhysRevB.97.054418, Zohar arXiv:1705.04786, Metlitski PhysRevB.98.085140, the authors state that the anomaly inflow term/ topological action can be expressed in cocycles, and these terms have the form like $\int…
ZJX
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Topological Descent Equation

Assume that we have a cohomological field theory, with an odd symmetry generated by an odd operator $Q$ and an exact energy momentum tensor $T_{\mu\nu}=[Q,G_{\mu\nu}]$. Then by integrating over an spatial slice we define $G_\mu=\int d^3\vec{x}…
Ivan Burbano
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Is the plane-wave solution to the Maxwell equation an instanton?

The top answer in this post Why are there no instantons in the gauge group $U(1)$? stated that, the reason why there is no U(1) instanton in R^4 is the second cohomology group of R^4 is trivial. I can understand the logic of this answer(and I think…
Gu Xia
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A reference for the fact that the second cohomology of the full Poincare algebra is zero

S. Weinberg in his book "The quantum theory of fields" vol. I says in page 86 that the full Poincare algebra is not semi-simple but its central charges can be eliminated (as he showed in the book). For every finite dimensional semi-simple Lie…
Mahtab
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Cohomology of the Koszul-Tate complex for an irreducible symmetry vanishes in degree $-2$

There must be something really obvious that I am missing here but any help is appreciated. Suppose I have a theory with some action $S$ on some fields $\phi$ such that any function vanishing on-shell has the form $\frac{\partial…
Ivan Burbano
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Hilbert Space in Categorical Gauged TQFT

I am trying to understand how gauge theory interacts with the categorical formulation of TQFT. I will formulate my doubts in two different questions. I have understood gauging a TQFT in different ways: as cohomology elements $H^1(M,G)$, as charged…
Badillo
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What does cohomology of $Q_B$ mean in BRST quantization in Polchinski?

While proving no-ghost theorem ($4.4$ Polchinski) the term cohomology of $Q_B$ is used quite a lot of time. From what I understand this has to be a set since "cohomology of $Q_B$" is isomorphic to Hilbert space of OCQ or light cone quantization.…
aitfel
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Four dimensional massless spectra of type IIA/B compactified on $\mathcal{M}_{4} \times {\rm CY}_3$

I am following “String Theory and M-Theory” by Becker, Becker, and Schwarz and I am currently studying chapter 9. I have a question - or better yet a point of confusion - regarding the derivation of the massless four-dimensional spectra when…
user172341
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De Rham current associated with knot in abelian CS theory on a generic manifold

I'm studying TQFT and I'm stucked on this part of the paper of my teacher: My teacher didn't explain a lot about it and I've never followed an advanced course on differential geometry or algebraic topology, so I'm spending days in studying…
polology
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Anomalies, 2-cocycles and (D+1)-cocycles

I'm learning about anomalies and I'm a bit confused about their relationships to 2-cocycles and 3-cocycles (in the group cohomology $H^{\bullet}(G, U(1))$). The below might only apply to 't Hooft anomalies. I read in some places (e.g. this answer)…
quixot
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Symmetry group of Haldane chain?

After having seen some videos and read some articles, I am having some confusion about the symmetry group G of spin 1 Haldane chain. Being composed of spin 1 sites, it seems natural to consider SO(3) as the symmetry group of the bulk of the Haldane…
baba26
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