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The Quantum Geometry of Polyhedral Surfaces

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Quantum Triangulations

Part of the book series: Lecture Notes in Physics ((LNP,volume 845))

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Abstract

Among the many significant ideas and developments that connect Mathematics with contemporary Physics one of the most intriguing is the role that Quantum Field Theory (QFT) plays in Geometry and Topology. We can argue back and forth on the relevance of such a role, but the perspective QFT offers is often surprising and far reaching. Examples abound, and a fine selection is provided by the revealing insights offered by Yang–Mills theory into the topology of 4-manifolds, by the relation between Knot Theory and topological QFT, and most recently by the interaction between Strings, Riemann moduli space, and enumerative geometry. Doubtless many of the most striking connections suggested by physicists failed to pass the censorship of the Department of Mathematics, and so do not appear in the above official list. As ill-defined these techniques may be, if we give them some degree of mathematical acceptance then the geometrical perspective they afford is always quite non-trivial and extremely rich. It is within such a framework that we shall examine in this and following chapters some aspects of the relation between an important class of QFTs and polyhedral surfaces. We start with a rather general introduction on geometrical aspects of QFT that will allow us to introduce naturally a notion of Quantum Geometry.

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Notes

  1. 1.

    The concept of tachyon is slightly misleading in non-critical string theory, where low dimensions are sampled and the usual instability associated with tachyons in critical strings is not present. For details see vol. I, Chap. 9 of [61].

  2. 2.

    For the elementary notion of string theory we shall exploit we refer freely to the excellent presentation in [41, 43] and [61].

  3. 3.

    There are clearly too many g’s around, however no confusion should arise since it will be always clear from the context when we are dealing with the genus g of the surface M or with the running metric g of the target manifold \(V^n.\)

  4. 4.

    According to Riemann–Roch theorem \(\dim Ker P_1-\dim Ker P_1^\dagger =6-6g.\) Since \(\dim\;{Ker}\; P_1=6\) for the sphere, \(=2\) for the torus, \(=0\) for surfaces with \(g\geq2,\)it follows that in the case of surfaces \(\dim Ker P_1^\dagger\) is finite-dimensional, and is given by \(=0,2, 6g-6\) when \(g=0,1, {\geq}2,\)respectively.

  5. 5.

    We are restricting for simplicity to the case of surface genus \(g\geq 2.\) In the case \(g=0,1,\) (4.27) acquires finite-dimensional determinants related to the presence of conformal Killing vectors. A careful analysis of the whole subject is presented in [16, 17].

  6. 6.

    \(S_{L}(\hat{\gamma},u)\) is actually the difference of the Liouville actions \(S_L(\varphi )\) and \(S_L(\varphi+u )\) respectively associated with the conformal metrics \(\hat{\gamma}=e^{\varphi }|dz|^2\) and \(\gamma=e^{\varphi +u}|dz|^2.\) The explicit characterization of \(S_L(\varphi )\) is very delicate since \(e^{\varphi }\) is not a function but the \((1,1)\) component of the metric tensor. A thorough analysis of the subject, with the relevant references, is discussed in [69].

  7. 7.

    For notational ease, we have chosen units for the fields \(\phi^k\) such that \({{1}\over {4\pi l_s^2}}=1.\)

  8. 8.

    However, (4.35) is invariant under the combined action of the above translation and of the conformal rescaling \(\hat{\gamma}\mapsto \hat{\gamma} e^{2 w(x)}.\)

  9. 9.

    A fully rigorous derivation of (4.40) is still an open mathematical problem.

  10. 10.

    This shows that \(S^{(ren)}_{L}({\gamma},u)\) characterizes a conformal field theory with central charge \(c_u = 1+ 6 Q^2,\)[41, 43, 57].

  11. 11.

    We remind the reader that formally the classical limit corresponds to \(n\searrow -\infty ,\) see the comments to the expression of the partition function (4.34).

  12. 12.

    When \(\phi^{n\,+\,1}\nearrow \infty\) and \(U(\phi)\) is real, \((n\leq 1),\)the term \({\hbox{exp}}\;U(\phi)\) dominates the action \({{\fancyscript{S}}}_{\hat{\gamma}}[\phi; f(\phi), U(\phi)]\) which then becomes large and positive, suppressing, in the path integral over \(Map(M, V^{n\,+\,1}),\) the configurations for which \(\phi^{n\,+\,1}\nearrow \infty.\)

  13. 13.

    The extra dimension is actually time-like if \(n>26.\)

  14. 14.

    The effective action (4.78) is written in the so called string frame. By a conformal transformation it is possible to move to the Einstein frame where (4.78) takes a manifest Einstein–Hilbert structure; see e.g., [43, 57] for details. The picture becomes more complex with the presence of the tachyonic coupling \(U(\phi),\)and in general the implementation of conformal invariance just at leading order is not believed to be sufficient for a reliable effective field theory description.

  15. 15.

    This remark on diffusive motion is nicely stressed and discussed by Faris in [28].

  16. 16.

    One can well argue that such a sampling process characterizes \(V^{n\,+\,1}\) in a neighborhood of the given point.

  17. 17.

    Our definition of \(\Gamma_{\it KPZ}\) is concocted in such a way to explicitly keep track of the genus g. Another standard definition of the string susceptibility exponent \(\Gamma_{\it string}\) is via the asymptotic scaling \(Z_{g}[A] \sim A^{(\Gamma_{\it string}-2) (1-g)-1}.\) The two critical exponents are related by \(\Gamma_{\it string}=(\Gamma_{\it KPZ}-2g)/(1-g).\)

  18. 18.

    A mathematically rigorous formulation of Quantum Liouville theory which takes into care the subtleties of the definition of the Liouville action is discussed in [69].

  19. 19.

    As compared with the Matrix theory describing flat 11-dimensional M-theory in the discrete light-cone quantization—see [43] for a review and relevant references.

  20. 20.

    An excellent review is provided by [19].

  21. 21.

    The light-cone cellular decomposition of the N-pointed Teichmüller space arises from the structure theory of abelian differential of the third kind. As in the case of quadratic differentials associated to ribbon graphs, also here we get a graph structure yielding for a cellular decomposition which descends to Riemann moduli space and exhibits certain computational advantages with respect to the ribbon graph cellularization [56].

  22. 22.

    For a nice and clear presentation see [41].

  23. 23.

    The situation is apparently similar to what happens in standard two-dimensional Regge calculus. There, however, a poor understanding of the correct measure to use over (a badly selected part of) \({\it POL}_{g,N_0}(A)\) has hampered the use of Regge calculus for regularizing 2D quantum gravity.

  24. 24.

    Here we deal with generalized triangulations, barycentrically dual to trivalent graphs; in the case of regular triangulations in place of \(108\sqrt{3}\) we would get \(e^{ v _{0}}=({{4^{4}}\over {3^{3}}}).\) Also note that the parameter \(c_g\) does not play any relevant role in 2D quantum gravity.

  25. 25.

    In this respect, the situation is here quite simpler than that described in the delicate and prescient analysis of the measure issue in Regge calculus addressed in a series of paper by Menotti and Peirano, (see [49] and references therein).

  26. 26.

    With respect to the statement of this result in Appendix B we have slightly specialized the notation.

  27. 27.

    The ordering is important for characterizing the diffeomorphism group relevant to the problem: \({\fancyscript{D}}iff(M,N_0)\) if we are injecting \((T_{(1)}, M)\) into \((T_{(2)}, M)\) so as to consider the neighborhoods of the vertices \(\{q_h\}\in (T_{(2)}, M)\) as (conformally) smooth as seen by \((T_{(1)}, M),\) whereas \({\fancyscript{D}}iff(M,\hat{N}_0)\) is the appropriate group when we inject \((T_{(2)}, M)\) into \((T_{(1)}, M).\)

  28. 28.

    As the name suggests, this is basically a free energy.

  29. 29.

    This is a vast subject with thousands of relevant papers. A nice selection, among those emphasizing the connection with combinatorial aspects, is provided by [1, 2, 15, 32, 34–38].

  30. 30.

    Different geometrical aspects of the role of the hyperbolic point of view in open/closed string duality have been discussed also by Kaufmann and Penner [42].

  31. 31.

    Recall that these boundary components correspond, under hyperbolic completion, to the ideal vertices \(\sigma^0_{hyp}(k) := v ^0(k).\)

  32. 32.

    See [4] for a deep analysis and the relevant references.

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  1. Annalisa Marzuoli

    Present address: Dipto. di Matematica “F. Casorati”, Università degli Studi di Pavia, Via Ferrata 1, 27100, Pavia, Italy

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  1. Dipto. Fisica Nucleare e Teorica, Istituto Nazionale di Fisica Nucleare e Teorica, Sez. di Pavia, Università degli Studi di Pavia, Via Bassi 6, 27100, Pavia, Italy

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Carfora, M., Marzuoli, A. (2012). The Quantum Geometry of Polyhedral Surfaces. In: Quantum Triangulations. Lecture Notes in Physics, vol 845. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-24440-7_4

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