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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{motives in physics} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{motivic_cohomology}{}\paragraph*{{Motivic cohomology}}\label{motivic_cohomology} [[!include motivic cohomology - contents]] \hypertarget{physics}{}\paragraph*{{Physics}}\label{physics} [[!include physicscontents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{ViaPerturbativeAlgebraicDeformationQuantization}{Via perturbative algebraic deformation quantization}\dotfill \pageref*{ViaPerturbativeAlgebraicDeformationQuantization} \linebreak \noindent\hyperlink{via_nonperturbative_geometric_quantization}{Via non-perturbative geometric quantization}\dotfill \pageref*{via_nonperturbative_geometric_quantization} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \noindent\hyperlink{ReferencesPerturbativeAlgebraic}{On motivic structures in perturbative quantum field theory}\dotfill \pageref*{ReferencesPerturbativeAlgebraic} \linebreak \noindent\hyperlink{ReferencesNonPerturbativeGeometric}{On motivic structures in non-perturbative local quantum field theory}\dotfill \pageref*{ReferencesNonPerturbativeGeometric} \linebreak \noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} [[motive|Motivic]] structures enters [[quantum physics]] in two dual guises, related, on the one hand, to [[algebraic deformation quantization]] and, on the other hand, to [[geometric quantization]]. \hypertarget{ViaPerturbativeAlgebraicDeformationQuantization}{}\subsubsection*{{Via perturbative algebraic deformation quantization}}\label{ViaPerturbativeAlgebraicDeformationQuantization} In the first case one observes that [[formal deformation quantization]] of $n$-dimensional [[prequantum field theory]] amounts to choosing an inverse [[equivalence]] to the [[formality]] map from [[En-algebras]] to [[Pn-algebras]], this is explained at \emph{\href{deformation+quantization#MotivicGaloisGroup}{Motivic Galois group action on the space of quantizations}}. The [[automorphism infinity-group]] of either side therefore naturally [[infinity-action|acts]] on the space of formal deformation quantization choices obtained this way and one shows (conjectured by [[Maxim Kontsevich]] (\hyperlink{Kontsevich99}{Kontsevich 99}), recently proven by Dolgushev) that the connected component group of this is the [[Grothendieck-Teichmüller group]], a [[quotient]] of the [[motivic Galois group]]. Related to this in some way is [[Alain Connes]]` ``[[cosmic Galois group]]'' [[action|acting]] on the space of [[renormalization|renormalizations]] of [[perturbative quantum field theory]]. According to Kontsevich, this explains the role of motivic [[periods]] in [[correlation functions]] and hence in [[scattering amplitudes]] (see also at \emph{[[amplituhedron]]}) in [[perturbative field theory]]. More concretely, [[scattering amplitudes]] typically are expressions in [[multiple zeta values]] and handling them involves intricate combinatorics. By just re-expressing these in terms of [[motivic multiple zeta values]] (see the rerferences) much of the combinatorics becomes more tractable (in terms of some [[Hopf algebra]]). This is how ``motivic'' structures are used by many practicioners. The actual [[motives]] do not play much of a role in these computations, but one makes use of the combinatorial simplification obtained by re-expressing multiple zeta values by their motivic version. For more see the motive-related references at \emph{[[Feynman diagram]]}, at \emph{[[motivic multiple zeta values]]} and at \emph{[[motivic L-function]]}. \hypertarget{via_nonperturbative_geometric_quantization}{}\subsubsection*{{Via non-perturbative geometric quantization}}\label{via_nonperturbative_geometric_quantization} On the other hand, in full [[non-perturbative quantum field theory|non-perturbative]] [[geometric quantization]] in its modern cohomological form as \href{http://ncatlab.org/nlab/show/geometric+quantization+by+push-forward}{geometric quantization by push-forward} one finds a ``[[cohesion|cohesive]]'' form of actual [[motivic cohomology]] exhibited by ``[[cohesive]] [[pure motives]]''. In effect, a [[local action functional]] on a [[moduli space]] of [[field (physics)|field]] [[trajectories]] is exhibited by a [[correspondence]] with the [[local action functional]] itself exhibited by a [[twisted bivariant cohomology|twisted bivariant cocycle]] on the correspondence space, and the [[motivic quantization|motivic path integral quantization]] of this corresponds to the induced pull-push index transform. See the \hyperlink{ReferencesNonPerturbativeGeometric}{references below}. \hypertarget{references}{}\subsection*{{References}}\label{references} We list references \begin{enumerate}% \item \hyperlink{ReferencesPerturbativeAlgebraic}{on motives in perturbative algebraic quantization} \item \hyperlink{ReferencesNonPerturbativeGeometric}{on motives in non-perturbative geometric quantization} \end{enumerate} \hypertarget{ReferencesPerturbativeAlgebraic}{}\subsubsection*{{On motivic structures in perturbative quantum field theory}}\label{ReferencesPerturbativeAlgebraic} The action of a [[motivic Galois group]] (``[[cosmic Galois group]]'') on the space of choices in [[deformation quantization]] was first observed/conjectured in \begin{itemize}% \item [[Maxim Kontsevich]], \emph{Operads and Motives in Deformation Quantization}, Lett. Math. Phys.48:35-72,1999 (\href{http://arxiv.org/abs/math/9904055}{arXiv:math/9904055}) \end{itemize} A quick and rough survey of this and vaguely related motivic structures in physics is in the appendix of \begin{itemize}% \item A. Rej, [[Matilde Marcolli]], \emph{Motives, an introductory survey for physicists} (\href{http://www.its.caltech.edu/~matilde/ObiMotivesSurveyFinal.pdf}{pdf}) \end{itemize} A detailed review of the motivic [[cosmic Galois group]] [[action]] on the space of [[renormalization]] procedures is in section 7 of \begin{itemize}% \item [[Alain Connes]], [[Matilde Marcolli]], \emph{[[Noncommutative Geometry, Quantum Fields and Motives]]} \end{itemize} Discussion of motivic structure in [[periods]] in [[scattering amplitudes]] is also the lecture \begin{itemize}% \item [[Spencer Bloch]], \emph{Dilogarithm motives appearing in physics} (\href{http://blip.tv/pifagorov/spencer-bloch-dilogarithm-motives-arising-in-physics-3854827}{video}); S. Bloch, H. Esnault, D. Kreimer, \emph{On motives associated to graph polynomials} \href{http://www.math.uchicago.edu/~bloch/graphpoly050928b.pdf}{pdf}; Spencer Bloch, Pierre Vanhove, \emph{The elliptic dilogarithm for the sunset graph}, IHES preprint P-13-24, \href{http://preprints.ihes.fr/2013/P/P-13-24.pdf}{pdf} \item Conference \emph{Amplitudes and Periods}, Dec 3-7, 2012, IH\'E{}S, $<$http://pagesperso.ihes.fr/{\tt \symbol{126}}vanhove/QFT2012{\tt \symbol{62}} (has online videos of talks) \end{itemize} See also the material linked at \begin{itemize}% \item Conference \emph{Amplitudes and Periods}, Dec 3-7, 2012, IH\'E{}S (\href{http://pagesperso.ihes.fr/~vanhove/QFT2012}{web}) \end{itemize} More recent developments on motivic structures in [[scattering amplitudes]] include for instance \begin{itemize}% \item John Golden, Alexander B. Goncharov, Marcus Spradlin, Cristian Vergu, Anastasia Volovich, \emph{Motivic Amplitudes and Cluster Coordinates} (\href{http://arxiv.org/abs/1305.1617}{arXiv:1305.1617}) \end{itemize} \hypertarget{ReferencesNonPerturbativeGeometric}{}\subsubsection*{{On motivic structures in non-perturbative local quantum field theory}}\label{ReferencesNonPerturbativeGeometric} The formulation of [[quantization]] of [[local prequantum field theory]] as a passage to [[cohesive]] generalized [[pure motives]], hence ``[[motivic quantization]]'' is formulated as such in the last section of \begin{itemize}% \item [[Urs Schreiber]], \emph{[[schreiber:differential cohomology in a cohesive topos]]} (\href{http://arxiv.org/abs/1310.7930}{arXiv:1310.7930}) \end{itemize} based on \begin{itemize}% \item [[Joost Nuiten]], \emph{[[schreiber:master thesis Nuiten|Cohomological quantization of local prequantum boundary field theory]]} \end{itemize} The following is a list of ``precursors'' of aspects of the idea [[motivic quantization]] as laid out there $\,$ The formulation of [[geometric quantization]] as an [[index]] map in [[K-theory]] is attributed to [[Raoul Bott]]. The proposal that the natural [[domain]] for geometric quantization are [[Lagrangian correspondences]] is due to \begin{itemize}% \item [[Lars Hörmander]], \emph{Fourier Integral Operators I.}, Acta Math. \textbf{127} (1971) \end{itemize} \begin{itemize}% \item [[Alan Weinstein]], \emph{Symplectic manifolds and their lagrangian submanifolds}, Advances in Math. \textbf{6} (1971) \end{itemize} In [[semiclassical quantization]] the precursor to this are works of [[Maslov]] in 1960s (which are quite related to Hoermander's and Sato's work). With the recognition of [[supersymmetric quantum mechanics]] in the 1980s, [[index theory]] (hence [[push-forward in generalized cohomology]] to the point) was understood to be about [[partition functions]] of systems of [[supersymmetric quantum mechanics]] in \begin{itemize}% \item [[Luis Alvarez-Gaumé]], \emph{Supersymmetry and the Atiyah-Singer index theorem}, Comm. Math. Phys. \textbf{90}:2 (1983) 161-173, \href{http://projecteuclid.org/euclid.cmp/1103940278}{euclid} \item [[Ezra Getzler]], \emph{Pseudodifferential operators on supermanifolds and the Atiyah-Singer index theorem}, Comm. Math. Phys. 92 (1983), 163-178. (\href{http://math.northwestern.edu/~getzler/Papers/1103940796.pdf}{pdf}) \end{itemize} \begin{itemize}% \item [[Ezra Getzler]], \emph{A short proof of the Atiyah-Singer index theorem}, Topology 25 (1986), 111-117 (\href{http://math.northwestern.edu/~getzler/Papers/local.pdf}{pdf}) \end{itemize} In higher analogy to this but much more subtly, the [[partition function]] of the [[heterotic string]], hence the [[Witten genus]], was understood to be the [[push-forward in generalized cohomology|push-forward]] to the point in [[tmf]]: \begin{itemize}% \item [[Matthew Ando]], [[Mike Hopkins]], [[Charles Rezk]], \emph{Multiplicative orientations of KO-theory and the spectrum of topological modular forms}, 2010 (\href{http://www.math.uiuc.edu/~mando/papers/koandtmf.pdf}{pdf}) \end{itemize} The general perspective of the [[path integral as a pull-push transform]] was originally laid out, somewhat implicitly, in \begin{itemize}% \item [[Daniel Freed]], \emph{Quantum groups from path integrals}, \href{http://xxx.lanl.gov/abs/q-alg/9501025}{arXiv:q-alg/9501025}; \emph{Higher algebraic structures and quantization}, \href{http://arxiv.org/abs/hep-th/9212115}{arXiv:hep-th/9212115} \end{itemize} and then fully explicitly in \begin{itemize}% \item [[Daniel Freed]], \emph{Twisted K-theory and the Verlinde ring}, Andrejewski lecture (\href{http://www.ma.utexas.edu/users/dafr/Andrejewski%20Lectures.html}{pdf slides}) \end{itemize} Discussion along these lines of a pull-push quantization over [[KU]] of a 2-dimensional [[Chern-Simons theory]]-like [[gauge theory]] is in \begin{itemize}% \item [[Daniel Freed]], [[Michael Hopkins]], [[Constantin Teleman]], \emph{Consistent Orientation of Moduli Spaces} (\href{http://arxiv.org/abs/0711.1909}{arXiv:0711.1909}), chapter XIX, pages 395-419 in: Oscar Garcia-Prada, Jean Pierre Bourguignon, Simon Salamon (eds.) \emph{The Many Facets of Geometry: A Tribute to Nigel Hitchin}, Oxford University Press 2010 (\href{http://www.oxfordscholarship.com/view/10.1093/acprof:oso/9780199534920.001.0001/acprof-9780199534920}{doi:10.1093/acprof:oso/9780199534920.001.0001}) \end{itemize} More in detail a functorial quantization of suitable correspondences of smooth manifolds to [[KK-theory]] by pull-push has been given in (\hyperlink{ConnesSkandalis84}{Connes-Skandalis 84}) and the generalization of that to [[equivariant K-theory]] (hence to [[groupoid K-theory]] of [[action groupoids]]) is in \begin{itemize}% \item [[Heath Emerson]], [[Ralf Meyer]], \emph{Bivariant K-theory via correspondences}, Adv. Math. 225 (2010), 2883-2919 (\href{http://arxiv.org/abs/0812.4949}{arXiv:0812.4949}) \end{itemize} The point of view that the [[path integral as a pull-push transform|pull-push quantization]] of [[Gromov-Witten theory]] should be thought of as a theory of [[Chow motives]] of [[Deligne-Mumford stacks]] is expressed in \begin{itemize}% \item [[Kai Behrend]], [[Yuri Manin]], \emph{Stacks of Stable Maps and Gromov-Witten Invariants} (\href{http://arxiv.org/abs/alg-geom/9506023}{arXiv:alg-geom/9506023}) \end{itemize} \begin{itemize}% \item [[Bertrand Toën]], \emph{On motives for Deligne-Mumford stacks}, International Mathematics Research Notices 2000, 17 (2000) 909-928 (\href{http://arxiv.org/abs/math/0006160}{arXiv:math/0006160}, \href{http://hal.archives-ouvertes.fr/hal-00773027}{web}, \href{http://hal.archives-ouvertes.fr/docs/00/77/30/27/PDF/motdm.pdf}{pdf}) \end{itemize} The proposal that the natural [[codomain]] for geometric quantization is [[KK-theory]] is due to \begin{itemize}% \item [[Klaas Landsman]], \emph{Functorial quantization and the Guillemin-Sternberg conjecture} Proc. Bialowieza 2002 (\href{http://arxiv.org/abs/math-ph/0307059}{arXiv:math-ph/0307059}) \item [[Klaas Landsman]], \emph{Functoriality of quantization: a KK-theoretic approach}, talk at ECOAS, ECOAS, Dartmouth College, October 2010 (\href{http://www.academia.edu/1992202/Functoriality_of_quantization_a_KK-theoretic_approach}{web}) \end{itemize} The point of view that pull-push along [[correspondences]] equipped with [[operator K-theory]] [[cycles]] in [[KK-theory]] is a [[K-theory]]-analog of [[motives]] was amplified in \begin{itemize}% \item [[Alain Connes]], [[Georges Skandalis]], \emph{The longitudinal index theorem for foliations}. Publ. Res. Inst. Math. Sci. 20, no. 6, 1139--1183 (1984) (\href{http://www.alainconnes.org/docs/longitudinal.pdf}{pdf}) \end{itemize} \begin{itemize}% \item [[Alain Connes]], Caterina Consani, [[Matilde Marcolli]], \emph{Noncommutative geometry and motives: the thermodynamics of endomotives} (\href{http://arxiv.org/abs/math/0512138}{arXiv:math/0512138}) \end{itemize} The proof that the [[universal property]] that characterizes [[noncommutative motives]] is the analog in [[noncommutative algebraic geometry]] of the [[universal property]] that characterizes [[KK-theory]] in [[noncommutative topology]] is due to \begin{itemize}% \item [[Gonçalo Tabuada]], \emph{K-theory via universal invariants}, Duke Math. J. 145 (2008), no.1, 121--206. \end{itemize} \begin{itemize}% \item [[Denis-Charles Cisinski]], [[Gonçalo Tabuada]], \emph{Non connective K-theory via universal invariants}. Compositio Mathematica 147 (2011), 1281--1320 (\href{http://arxiv.org/abs/0903.3717}{arXiv:0903.3717}) \end{itemize} \begin{itemize}% \item [[Andrew Blumberg]], [[David Gepner]], [[Gonçalo Tabuada]], \emph{A universal characterization of higher algebraic K-theory}, Geometry and Topology (\href{http://arxiv.org/abs/1001.2282}{arXiv:1001.2282}) \end{itemize} The description of [[string topology]] operations as an [[HQFT]] defined by pull-push transforms in [[ordinary homology]]/[[ordinary cohomology]] was originally realized in \begin{itemize}% \item [[Ralph Cohen]], [[Veronique Godin]], \emph{[[A Polarized View of String Topology]]} (\href{http://arxiv.org/abs/math/0303003}{arXiv:math/0303003}) \item Hirotaka Tamanoi, \emph{Loop coproducts in string topology and triviality of higher genus TQFT operations} (2007) (\href{http://arxiv.org/abs/0706.1276}{arXiv:0706.1276}) \end{itemize} A detailed discussion and generalization to [[open strings]] and an open-closed [[HQFT]] in the presence of a single space-filling [[brane]] is in \begin{itemize}% \item [[Veronique Godin]], \emph{Higher string topology operations} (2007)(\href{http://arxiv.org/abs/0711.4859}{arXiv:0711.4859}) \end{itemize} and for arbitrary [[branes]] in \begin{itemize}% \item [[Alexander Kupers]], \emph{String topology operations} MS thesis (2011) (\href{http://math.stanford.edu/~kupers/thesis7thjune2011.pdf}{pdf}) \end{itemize} That [[D-brane charge]] and [[T-duality]] is naturally understood in terms of pull-push/[[indices]] along [[correspondences]] in [[noncommutative topology]]/[[KK-theory]] was amplified in \begin{itemize}% \item [[Jacek Brodzki]], [[Varghese Mathai]], [[Jonathan Rosenberg]], [[Richard Szabo]], \emph{Noncommutative correspondences, duality and D-branes in bivariant K-theory}, Adv. Theor. Math. Phys.13:497-552,2009 (\href{http://arxiv.org/abs/0708.2648}{arXiv:0708.2648}) \end{itemize} The general analogy between such [[KK-theory]] cocycles and [[pure motives]] is noted explicitly in \begin{itemize}% \item [[Alain Connes]], Caterina Consani, [[Matilde Marcolli]], \emph{Noncommutative geometry and motives: the thermodynamics of endomotives} (\href{http://arxiv.org/abs/math/0512138}{arXiv:math/0512138}) \end{itemize} \begin{itemize}% \item [[Alain Connes]], [[Matilde Marcolli]], \emph{[[Noncommutative Geometry, Quantum Fields and Motives]]} \end{itemize} This analogy is given a precise form in \begin{itemize}% \item [[Snigdhayan Mahanta]], \emph{Higher nonunital Quillen $K'$-theory, KK-dualities, and applications to topological T-duality}, (\href{http://wwwmath.uni-muenster.de/u/snigdhayan.mahanta/papers/KQ.pdf}{pdf}, \href{http://wwwmath.uni-muenster.de/u/snigdhayan.mahanta/papers/KKTD.pdf}{talk notes}) \end{itemize} where it is shown that there is a universal functor $KK \longrightarrow NCC_{dg}$ from [[KK-theory]] to the category of [[noncommutative motives]], which is the category of [[dg-categories]] and dg-[[profunctors]] up to homotopy between them. This is given by sending a [[C\emph{-algebra]] to the [[dg-category]] of [[perfect complexes]] of (the unitalization of) its underlying [[associative algebra]].} Linearization of correspondences of [[discrete groupoid|geometrically discrete groupoids]] was considered in \begin{itemize}% \item [[John Baez]], [[Jim Dolan]], \emph{Groupoidification}, November 2009 (\href{http://math.ucr.edu/home/baez/groupoidification/}{web}) \end{itemize} and applied to a pull-push quantization of [[Dijkgraaf-Witten theory]] in \begin{itemize}% \item [[Jeffrey Morton]], \emph{2-Vector Spaces and Groupoids} (\href{http://arxiv.org/abs/0810.2361}{arXiv:0810.2361}) \item [[Jeffrey Morton]], \emph{Cohomological Twisting of 2-Linearization and Extended TQFT} (\href{http://arxiv.org/abs/1003.5603v4}{arXiv:1003.5603v4}) \end{itemize} following previous work by [[Daniel Freed]] and [[Frank Quinn]]. An unpublished predecessor note on quantization of correspondences of moduli stacks of fields is \begin{itemize}% \item [[Urs Schreiber]], \emph{[[schreiber:Nonabelian cocycles and their quantum symmetries]]} (2008) \end{itemize} Quantization of [[correspondences]] of [[perfect ∞-stacks]] by pull-push of [[stable (∞,1)-categories]] of [[quasicoherent sheaves]] is discussed in \begin{itemize}% \item [[David Ben-Zvi]], [[John Francis]], [[David Nadler]], \emph{[[geometric function theory|Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry]]}, J. Amer. Math. Soc. \textbf{23} (2010), no. 4, 909-966, (\href{http://arxiv.org/abs/0805.0157}{arXiv:0805.0157}) \end{itemize} Linearization of [[higher correspondences]] of [[discrete ∞-groupoids]] as the quantizaton of [[∞-Dijkgraaf-Witten theories]] is indicated in section 3 and 8 of \begin{itemize}% \item [[Daniel Freed]], [[Mike Hopkins]], [[Jacob Lurie]], [[Constantin Teleman]], \emph{[[Topological Quantum Field Theories from Compact Lie Groups]], in P. Kotiuga (ed.),}A Celebration of the Mathematical Legacy of Raoul Bott, AMS, (2010) (\href{http://arxiv.org/abs/0905.0731}{arXiv:0905.0731}) \end{itemize} and in \begin{itemize}% \item [[Jacob Lurie]], \emph{[[Ambidexterity in K(n)-Local Stable Homotopy Theory]]}, talk at Notre Dame Graduate Summer School on Topology and Field Theories and Harvard lecture 2012 (\href{http://www.math.harvard.edu/~lurie/papers/Ambidexterity.pdf}{pdf}) \end{itemize} A clear picture of [[fiber integration]] in [[twisted cohomology]] is developed in \begin{itemize}% \item [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], \emph{Twists of K-theory and TMF}, in Robert S. Doran, Greg Friedman, [[Jonathan Rosenberg]], \emph{Superstrings, Geometry, Topology, and $C^*$-algebras}, Proceedings of Symposia in Pure Mathematics \href{http://www.ams.org/bookstore-getitem/item=PSPUM-81}{vol 81}, American Mathematical Society (\href{http://arxiv.org/abs/1002.3004}{arXiv:1002.3004}) \end{itemize} A proposal to axiomatize [[perturbative field theory|perturbative]] [[prequantum field theory]] by functors from [[cobordisms]] to a [[symplectic category]] of [[symplectic manifolds]] and [[Lagrangian correspondences]] is in \begin{itemize}% \item [[Alberto Cattaneo]], [[Pavel Mnev]], [[Nicolai Reshetikhin]], \emph{Classical and quantum Lagrangian field theories with boundary} (\href{http://arxiv.org/abs/1207.0239}{arXiv:1207.0239}, \href{http://users.uoa.gr/~iandroul/GAP%202012%20WEBPAGE/GAP2012_Cattaneo.pdf}{pdf slides}) \end{itemize} \begin{itemize}% \item [[Damien Calaque]], \emph{Lagrangian structures on mapping stacks and semi-classical TFTs} (\href{http://arxiv.org/abs/1306.3235}{arXiv:1306.3235}) \end{itemize} \hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries} \begin{itemize}% \item [[geometry of physics]] \item [[fiber bundles in physics]] \item [[higher category theory and physics]] \item [[string theory FAQ]] \item [[twisted smooth cohomology in string theory]] \item [[Hilbert's sixth problem]] \item [[model theory and physics]] \item [[L-infinity algebras in physics]] \item [[motivation for sheaves, cohomology and higher stacks]] \item [[applications of (higher) category theory]] \item [[motivation for higher differential geometry]] \item [[motivation for cohesion]] \end{itemize} [[!redirects motives and physics]] [[!redirects motivic cohomology in physics]] \end{document}