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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*{2-spectral triple} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{noncommutative_geometry}{}\paragraph*{{Noncommutative geometry}}\label{noncommutative_geometry} [[!include noncommutative geometry - contents]] \hypertarget{functorial_quantum_field_theory}{}\paragraph*{{Functorial quantum field theory}}\label{functorial_quantum_field_theory} [[!include functorial quantum field theory - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{Idea}{Idea}\dotfill \pageref*{Idea} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{flop_transition}{Flop transition}\dotfill \pageref*{flop_transition} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak \hypertarget{Idea}{}\subsection*{{Idea}}\label{Idea} An ordinary [[spectral triple]] is, as discussed there, the abstract algebraic data characterizing [[supersymmetric quantum mechanics]] on a [[worldline]] and thereby spectrally encoding an effective (possibly [[non-commutative geometry|non-commutative]], hence ``[[non-geometric background|non-geometric]]'') [[target space]] [[geometry]]. Ordinary [[Riemannian geometry]] with [[spin structure]] is the special case of this where the [[Hilbert space]] in the spectral triple is that of [[square integrable function|square integrable]] [[sections]] of the [[spinor bundle]] and the operator $D$ acting on that is the standard [[Dirac operator]], hence the ``supercharge'' of the worldline supersymmetry of the [[spinning particle]]. In generalization of this, a ``2-spectral triple'' should be the analogous algebraic data that encodes the [[worldsheet]] theory of a [[superstring]] propagating on a [[target space]] geometry which is a generalization of [[Riemannian geometry]] with ([[twisted string structure]]) [[string structure]]. Of course such data is just that of a [[2d superconformal field theory]], realized locally for instance by a [[vertex operator algebra]] or by a [[conformal net]] of [[local net of observables|local observables]]. But for emphasis it may be useful to speak of such data as constituting a ``2-spectral triple'', for emphasizing more the important and intricate relation to the concept of [[spectral triples]], which in much of the literature seems to be unduly ignored. \begin{tabular}{l|l|l|l} [[quantum system]]&[[supercharge]]&formalization&algebra\\ \hline quantum [[spinning particle]]&[[Dirac operator]]&[[spectral triple]]&[[operator algebra]]\\ quantum [[spinning string]]&[[Dirac-Ramond operator]]&[[2d SCFT]]&[[vertex operator algebra]]\\ \end{tabular} That the 0-mode sector of a [[2d SCFT]] -- hence the quantum point [[particle]] limit of a quantum [[superstring]] dynamics -- yields a [[spectral triple]] was maybe first highlighted in (\hyperlink{FroehlichGawedzki93}{Fr\"o{}hlich-Gawdzki 93}) by way of a series of concrete examples, such as the [[WZW model]]. Here the role of the [[Dirac operator]] of the spectral triple is played by the [[Dirac-Ramond operator]] of the [[superstring]], hence the operator whose [[index]] (in the [[large volume limit]]) is the [[Witten genus]]. That hence the superstring quantum theory should be regarded as a kind of higher spectral triple was maybe first suggested in (\hyperlink{Chamseddine97}{Chamseddine 97}), together with arguments that the associated [[spectral action]] indeed reproduces the [[action functional]] of the string's [[target space]] [[effective quantum field theory|effective]] [[supergravity]] theory. An exposition of this perspective is in (\hyperlink{FroehlichGrandjeanRecknagel97}{Fr\"o{}hlich-Grandjean-Recknagel 97, section 7.2}). Later it was shown more formally (\hyperlink{RoggenkampWendland03}{Roggenkamp-Wendland 03}), reviewed in (\hyperlink{RoggenkampWendland08}{Roggenkamp-Wendland 08}), that there is a precise algebraically formalization of taking the ``point particle limit'' of a quantum string, by sending its [[vertex operator algebra]] to a spectral triple obtained by suitably retaining only [[worldsheet]] 0-modes. In (\hyperlink{Soibelman11}{Soibelman 11}) this was used as a means to systematically study the large volume limit of [[effective quantum field theory|effective]] string [[spacetimes]] (and hence aspects of the [[landscape of string theory vacua]]) by studying the spectral geometries (i.e. the Connes-style noncommutative geometries) of the spectral triples arising from the string's point particle limit this way. Now, since there is information lost in passing from a stringy ``2-spectral triple'' (a [[2d SCFT]]) to its underlying point particle [[spectral triple]], not all spectral triples are to be expected to have a lift to a 2-spectral triple (possibly corresponding to a [[UV-completion]] of the corresponding target space [[effective field theories]]). In view of this, it is noteworthy that the spectral triple of the [[Connes-Lott-Chamseddine model]] shares a few key properties with the 2d SCFTs considered in [[string phenomenology]]: The [[Connes-Lott-Chamseddine model]] is an encoding in a spectral triple of the [[standard model of particle physics]] coupled to [[gravity]] realized as a kind of spectral [[Kaluza-Klein compactification]] on a non-commutative fiber space down to ordinary 4d [[Minkowski spacetime]] (or possibly its [[Wick rotation|Wick rotated]] Euclidean version). In order for this to work out, it turns out that the compactified non-commutative fiber space needs to have [[KO-dimension]] equal to $6$. (Here the fiber space is classically just a (``non-commutative'') point, but it appears as the singular collapsing limit of a space of finite dimension. This actual dimension is the [[KO-dimension]].) Hence the claim of the [[Connes-Lott-Chamseddine model]] is that if the [[standard model of particle physics]] is encoded as a singular limit of a [[Kaluza-Klein compactification]] modeled via a [[spectral triple]] then the [[dimensions]] of the [[KK-compactification]] are \begin{displaymath} 4 + 6 \;\;\; (mod\;8) \end{displaymath} with 4-dimensional base space and 6-dimensional fiber space, to a total of a 10-dimensional [[spacetime]] at high energy (after uncompactification of the fiber). This, of course, is precisely the dimensionality of the target spacetime of [[perturbative string theory vacua]] for the critical [[superstring]]. This point was highlighted in \hyperlink{Connes06}{Connes 06, p. 8}: \begin{quote}% When one looks at the table (7.2) of Appendix 7 giving the [[KO-dimension]] of the finite space $[$ i.e. the [[noncommutative geometry|noncommutative]] [[KK-compactification]]-[[fiber]] $F$ $]$ one then finds that its [[KO-dimension]] is now equal to 6 [[modulo]] 8 (!). As a result we see that the [[KO-dimension]] of the [[Cartesian product|product space]] $M \times F$ $[$ i.e. of 4d [[spacetime]] $M$ with the [[noncommutative geometry|noncommutative]] [[KK-compactification]]-[[fiber]] $F$$]$ is in fact equal to $10 \sim 2$ [[modulo]] 8. Of course the above 10 is very reminiscent of string theory, in which the finite space $F$ might bea good candidate for an ``[[effective field theory|effective]]'' [[KK-compactification|compactification]] at least for low energies. But 10 is also 2 [[modulo]] 8 which might be related to the observations of \hyperlink{LauscherReuter06}{Lauscher-Reuter 06} about [[gravity]]. \end{quote} Algebraically, this arises from the fact that the [[BRST complex]] for the [[superstring]] [[worldsheet]] theory is consistent (has BRST differential squaring to 0) precisely if the corresponding [[2d SCFT]] has conformal [[central charge]] 15, and each spacetime dimension contributes $1 \tfrac{1}{2}$ to this central charge (a contribution of 1 from each bosonic direction, and another $\tfrac{1}{2}$ for the corresponding fermionic contribution). \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \hypertarget{flop_transition}{}\subsubsection*{{Flop transition}}\label{flop_transition} There is at least evidence that there is a continuous path in the space of 2-spectral triples that starts and ends at a point describing the ordinary geometry of a complex 3-dimensional [[Calabi-Yau space]] but passes in between through a 2-spectral triple/2d SCFT (a [[Gepner model]]) which is not the $\sigma$-model of an ordinary geometry, hence which describes ``noncommutative 2-geometry'' (to borrow that terminology from the situation of ordinary spectral triples). This is called the [[flop transition]] (alluding to the fact that the geometries at the start and end of this path have different [[topology]]). This was further expanded on and used for the mathematical study of the [[large volume limit]] of [[string theory]] [[vacua]] in (\hyperlink{Soibelman11}{Soibelman 11}). \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[spectral triple]] \item [[spectral action]] \item [[D-brane geometry]] \item [[automorphism of a 2-spectral triple]] \item [[swampland]] \end{itemize} \hypertarget{References}{}\subsection*{{References}}\label{References} An early observation that the 0-mode sector of a [[2d SCFT]] is a [[spectral triple]], demonstrated in a series of concrete examples, is \begin{itemize}% \item [[Jürg Fröhlich]], [[Krzysztof Gawędzki]], \emph{Conformal Field Theory and Geometry of Strings}, extended lecture notes for lecture given at the Mathematical Quantum Theory Conference, Vancouver, Canada, August 4-8 (\href{http://arxiv.org/abs/hep-th/9310187}{arXiv:hep-th/9310187}) \end{itemize} The suggestion to understand, conversely, the [[string theory|string]]`s [[worldvolume]] [[2d SCFT]] as a higher spectral triple is due to \begin{itemize}% \item [[Ali Chamseddine]], \emph{An Effective Superstring Spectral Action}, Phys.Rev. D56 (1997) 3555-3567 (\href{http://arxiv.org/abs/hep-th/9705153}{arXiv:hep-th/9705153}), \end{itemize} which claims to show that the corresponding [[spectral action]] reproduces the correct effective background action known in [[string theory]]. A more expository account of this perspective is in \begin{itemize}% \item [[Jürg Fröhlich]], Oliver Grandjean, [[Andreas Recknagel]], section 7 of \emph{Supersymmetric quantum theory, non-commutative geometry, and gravitation} Lecture Notes Les Houches (1995) (\href{http://arxiv.org/abs/hep-th/9706132}{arXiv:hep-th/9706132}). \end{itemize} A more formal derivation of how ordinary [[spectral triples]] arise as point particle limits of [[vertex operator algebra]]s for [[2d SCFTs]] then appears in \begin{itemize}% \item [[Daniel Roggenkamp]], [[Katrin Wendland]], \emph{Limits and Degenerations of Unitary Conformal Field Theories}, Commun.Math.Phys. 251 (2004) 589-643 (\href{http://arxiv.org/abs/hep-th/0308143}{arXiv:hep-th/0308143}) \end{itemize} summarized in \begin{itemize}% \item [[Daniel Roggenkamp]], [[Katrin Wendland]], \emph{Decoding the geometry of conformal field theories}, Proceedings of the 7th International Workshop ``Lie Theory and Its Applications in Physics'', Varna, Bulgaria (\href{http://arxiv.org/abs/0803.0657}{arXiv:0803.0657}) \end{itemize} A brief indication of some ideas of [[Yan Soibelman]] and [[Maxim Kontsevich]] on this matter is at \begin{itemize}% \item [[Urs Schreiber]], \href{http://golem.ph.utexas.edu/category/2007/06/had_the_pleasure_of_talking.html}{\emph{Spectral triples and graph field theory}} \end{itemize} Further development of this and application to the study of the [[large volume limit]] of [[superstring]] [[vacua]] is in \begin{itemize}% \item [[Yan Soibelman]], \emph{Collapsing CFTs, spaces with non-negative Ricci curvature and nc-geometry} (\href{http://www.math.ksu.edu/~soibel/nc-riem-3.pdf}{pdf}), in [[Hisham Sati]], [[Urs Schreiber]] (eds.), \emph{[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]}, Proceedings of Symposia in Pure Mathematics, AMS (2001) \end{itemize} based on \begin{itemize}% \item [[Maxim Kontsevich]], [[Yan Soibelman]], section 2 of \emph{Homological mirror symmetry and torus fibrations}, Proceedings of KIAS Annual International Conference on Symplectic Geometry and Mirror Symmetry (\href{http://arxiv.org/abs/math/0011041}{arXiv:math/0011041}, \href{http://inspirehep.net/record/536540/}{spire}) \end{itemize} (discussing aspects of [[homological mirror symmetry]]). Analogous detailed discussion based not on the [[vertex operator algebra]] description of local [[CFT]] but on the [[AQFT]] description via [[conformal nets]] is in \begin{itemize}% \item [[Sebastiano Carpi]], Robin Hillier, [[Yasuyuki Kawahigashi]], [[Roberto Longo]], \emph{Spectral triples and the super-Virasoro algebra}, Commun.Math.Phys.295:71-97 (2010) (\href{http://arxiv.org/abs/0811.4128}{arXiv:0811.4128}) \end{itemize} where [[2d SCFTs]] are related essentially to [[local nets]] of [[spectral triples]]. Exposition of these results is in \begin{itemize}% \item [[Urs Schreiber]], \emph{\href{https://www.physicsforums.com/insights/spectral-standard-model-string-compactifications/}{Spectral Standard Model and String Compactifications}}, PhysicsForums--Insights (2016) \end{itemize} See also the references at [[geometric model for elliptic cohomology]]. That the [[Connes-Lott models]] could be [[effective field theory]]-limits of [[perturbative string theory vacua]] is also mentioned in \begin{itemize}% \item [[Alain Connes]], p. 8 of \emph{Noncommutative Geometry and the standard model with neutrino mixing}, JHEP0611:081,2006 (\href{http://arxiv.org/abs/hep-th/0608226}{arXiv:hep-th/0608226}) \end{itemize} [[!redirects 2-spectral triples]] \end{document}