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See Appendix [ref] for more details about the quantization of the coefficient matrix $K$.","intent_p":0.84,"contests_p":null}]},{"kind":"method_or_tool","label":"Uses its method or tool","papers":4,"sentences":4,"quotes":[{"arxiv_id":"2112.02092","title":"Symmetry TFTs from String Theory","url":"/paper/2112.02092","section":"Integral Quantization of $G_4$-flux","sentence":"In order to diagnose whether $G_4$ has integral or half-integral periods on a four-cycle $_4$, we can compute the integral of the fourth Stiefel-Whitney class of the tangent bundle $T _11$ on $_4$ [ref] , ∫__4 G_4 = 1/2 ∫__4 w_4(T_11) 1 , where the pullback to $_4$ is implicit.","intent_p":0.81,"contests_p":null},{"arxiv_id":"1908.09916","title":"Consistency of M-Theory on nonorientable manifolds","url":"/paper/1908.09916","section":"The cubic form on spin 12-manifolds","sentence":"We follow Witten's argument in [ref] .","intent_p":0.95,"contests_p":null},{"arxiv_id":"0705.3484","title":"An approach to anomalies in M-theory via KSpin","url":"/paper/0705.3484","section":"The Degree eight `dual' field $Θ$","sentence":"This would come from assuming both factors in the shifted quantization condition [ref] to be in integral cohomology, an so the mod $q$ reduction is in mod $q$ cohomology.] 1/2 [ 1/2P_3^1 G_4 + G_4 ∪G_4 ] = 1/2 [ 1/2P_3^1 ( ρ_3(a) - ρ_3(λ/2) ) + ( ρ_3(a) - ρ_3(λ/2) ) ∪( ρ_3(a) - ρ_3(λ/2) ) ] = ρ_3(a) ∪ρ_3(a) - ρ_3(λ/2) ∪ρ_3(λ/2) + ρ_3(a) ∪ρ_3(a) + ρ_3(λ/2) ∪ρ_3(λ/2) - ρ_3(a) ∪ρ_3(λ) = [ 2 ρ_3(a) ∪ρ_3(a) - ρ_3(a) ∪ρ_3(λ)) ] = Θ( ρ_3(a) ), the DFM class with the degree eight term set to zero.","intent_p":0.84,"contests_p":null},{"arxiv_id":"1503.02068","title":"Three-Family Particle Physics Models from Global F-theory Compactifications","url":"/paper/1503.02068","section":"4D Anomaly Cancelation and Family Structure","sentence":"In this parametrization, we check, for every allowed value for $(n_7,n_9)$, whether there are integral values of $b$ for which the number $n_D3$ of D3-branes 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for $G_4^form∧G_4^form$, previously obtained using $E_8$-gauge theory in [ref] .","intent_p":0.98,"contests_p":null},{"arxiv_id":"1005.1700","title":"Geometry of Spin and Spin^c structures in the M-theory partition function","url":"/paper/1005.1700","section":"Effect of the $Spin^c$ condition on the $C$-field","sentence":"Hence the whole expression is integral and we are essentially back to the result of [ref] .","intent_p":0.85,"contests_p":null},{"arxiv_id":"1108.1759","title":"Extended Charged Events and Chern-Simons Couplings","url":"/paper/1108.1759","section":"Concluding remarks","sentence":"For the case $D=11$, $p=3$, our result for the quantization of the Chern-Simons coupling was obtained previously in [ref] in the context of M-theory (i.e. through the use of a combination of results from 11-dimensional supergravity and string theory).","intent_p":0.93,"contests_p":null},{"arxiv_id":"2510.19935","title":"Anomaly-induced vanishing of brane partition 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to cancel membrane anomalies, as in [ref] .","intent_p":0.96,"contests_p":null},{"arxiv_id":"2112.02092","title":"Symmetry TFTs from String Theory","url":"/paper/2112.02092","section":"Differential Cohomological Formulation of M-theory","sentence":"The fact that the particular combination $I_12$ is nonetheless well-defined stems from the analysis of [ref] , which demonstrates that the total topological action $e^i S_CS$ is well-defined up to a sign, which cancels a potential sign problem in the definition of the Rarita-Schwinger determinant.","intent_p":0.98,"contests_p":null}]},{"kind":"background","label":"Cites it as background","papers":39,"sentences":43,"quotes":[{"arxiv_id":"1203.4542","title":"On Flux Quantization in F-Theory II: Unitary and Symplectic Gauge Groups","url":"/paper/1203.4542","section":"Introduction","sentence":"In this paper we will address the the quantization rule for the M-theory $G_4$ flux [ref] in its relation to the Freed-Witten quantization condition for gauge fluxes on 7-brane configurations [ref] .","intent_p":0.87,"contests_p":null},{"arxiv_id":"1011.6388","title":"On Flux Quantization in F-Theory","url":"/paper/1011.6388","section":"Introduction","sentence":"In his seminal work [ref] , Witten derived the general rule for flux quantization in M-theory: G_4-λ/2 ∈ H^4(M_11,ℤ) , where $λ$ is one half the first Pontryagin class of the eleven-dimensional spacetime.","intent_p":0.84,"contests_p":null},{"arxiv_id":"1508.00162","title":"Fluxes in F-theory Compactifications on Genus-One Fibrations","url":"/paper/1508.00162","section":"Introduction","sentence":"Finally, in section [ref], we address two subtle and related issues, the quantization condition [ref] for gauge fluxes and the cancellation of discrete gauge anomalies.","intent_p":0.82,"contests_p":null},{"arxiv_id":"0705.3484","title":"An approach to anomalies in M-theory via KSpin","url":"/paper/0705.3484","section":"The Quadratic Refinement and Spin K-theory","sentence":"The 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