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The Quantum Geometry of Polyhedral Surfaces: Variations on Strings and All That

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Quantum Triangulations

Part of the book series: Lecture Notes in Physics ((LNP,volume 942))

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Abstract

This chapter is an introduction to the basic ideas of 2-dimensional quantum field theory and non-critical strings. This is classic material which nevertheless proves useful for illustrating the interplay between quantum field theory, the moduli space of Riemann surfaces, and the properties of polyhedral surfaces which are the leitmotiv of this LNP. At the root of this interplay lies 2D quantum gravity. It is well known that such a theory allows for two complementary descriptions: on the one hand we have a conformal field theory (CFT) living on a 2D world-sheet, a description that emphasizes the geometrical aspects of the Riemann surface associated with the world-sheet; while on the other, the theory can be formulated as a statistical critical field theory over the space of polyhedral surfaces (dynamical triangulations). We show that many properties of such 2D quantum gravity models are related to a geometrical mechanism which allows one to describe a polyhedral surface with N 0 vertices as a Riemann surface with N 0 punctures dressed with a field whose charges describe discretized curvatures (related to the deficit angles of the triangulation). Such a picture calls into play the (compactified) moduli space of genus g Riemann surfaces with N 0 punctures \(\mathfrak {M}_{g;N_0}\), and allows one to prove that the partition function of 2D quantum gravity is directly related to computation of the Weil–Petersson volume of \(\mathfrak {M}_{g;N_0}\). By exploiting the large N 0 asymptotics of such Weil–Petersson volumes, characterized by Manin and Zograf, it is then easy to relate the anomalous scaling properties of pure 2D quantum gravity, the KPZ exponent, to the Weil–Petersson volume of \(\mathfrak {M}_{g;N_0}\). We also show that polyhedral surfaces provide a natural kinematical framework within which we can discuss open/closed string duality. A basic problem in such a setting is to provide an explanation of how open/closed duality is generated dynamically, and in particular how a closed surface is related to a corresponding open surface, with gauge-decorated boundaries, in such a way that the quantization of this correspondence leads to an open/closed duality. In particular, we show that from a closed polyhedral surface we naturally get an open hyperbolic surface with geodesic boundaries. This gives a geometrical mechanism describing the transition between closed and open surfaces. Such a correspondence is promoted to the corresponding moduli spaces: \(\mathfrak {M}_{g;N_0}\times \mathbb {R}_{+}^{N}\), the moduli spaces of N 0-pointed closed Riemann surfaces of genus g whose marked points are decorated with the given set of conical angles, and \(\mathfrak {M}_{g;N_0}(L)\times \mathbb {R}_{+}^{N_0}\), the moduli spaces of open Riemann surfaces of genus g with N 0 geodesic boundaries decorated by the corresponding lengths. Such a correspondence provides a nice kinematical setup for establishing an open/closed string duality, by exploiting the results by M. Mirzakhani on the relation between intersection theory over \(\mathfrak {M}(g;N_0)\) and the geometry of hyperbolic surfaces with geodesic boundaries. The results in this chapter connect directly with many deep issues in 3D geometry, ultimately relating to the volume conjecture in hyperbolic geometry and to the role of knot invariants.

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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 [57].

  2. 2.

    For the elementary notion of string theory we shall exploit we refer freely to the excellent presentation in [40, 57], and [38].

  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 ≥ 2, 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, ≥ 2, respectively.

  5. 5.

    We are restricting for simplicity to the case of surface genus g ≥ 2. In the case g = 0,  1, (5.13) acquires finite–dimensional determinants related to the presence of conformal Killing vectors. A careful analysis of the whole subject is presented in [15, 16].

  6. 6.

    \(S_{L}(\widehat {\gamma },u)\) is actually the difference of the Liouville actions S L (φ) and S L (φ + u) respectively associated with the conformal metrics \(\widehat {\gamma }=e^{\varphi }|dz|{ }^2\) and γ = e φ+u|dz|2. The explicit characterization of S L (φ) is very delicate since e φ 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 [64].

  7. 7.

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

  8. 8.

    However, (5.21) is invariant under the combined action of the above translation and of the conformal rescaling \(\widehat {\gamma }\mapsto \widehat {\gamma }\,e^{2\,w(x)}\).

  9. 9.

    This shows that \(S^{(ren)}_{L}({\gamma },u)\) characterizes a conformal field theory with central charge c u  =  1 +  6 Q 2, [38, 40, 53].

  10. 10.

    We remind the reader that formally the classical limit corresponds to n↘ −∞, see the comments to the expression of the partition function (5.20).

  11. 11.

    When ϕ n+1↗∞ and U(ϕ) is real, (n ≤ 1), the term \(\exp \,U(\phi )\) dominates the action \({\mathcal {S}}_{\widehat {\gamma }}[\phi ; f(\phi ), U(\phi )]\) which then becomes large and positive, suppressing, in the path integral over Map(M, V n+1), the configurations for which ϕ n+1↗∞.

  12. 12.

    The extra dimension is actually time–like if n > 26.

  13. 13.

    The effective action (4.167) is written in the so called string frame. By a conformal transformation it is possible to move to the Einstein frame where (4.167) takes a manifest Einstein–Hilbert structure; see e.g. [40, 53] for details. The picture becomes more complex with the presence of the tachyonic coupling U(ϕ), 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.

  14. 14.

    This remark on diffusive motion is nicely stressed and discussed by W. G. Faris in [27].

  15. 15.

    One can well argue that such a sampling process characterizes V n+1 in a neighborhood of the given point.

  16. 16.

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

  17. 17.

    A formulation of Quantum Liouville theory which takes into care the subtleties of the definition of the Liouville action is discussed in [64].

  18. 18.

    As compared with the Matrix theory describing flat 11–dimensional M–theory in the discrete light–cone quantization–see [40] for a review and relevant references.

  19. 19.

    An excellent review is provided by [18].

  20. 20.

    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 [52].

  21. 21.

    For a nice and clear presentation see [38].

  22. 22.

    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) \(POL_{g,N_0}(A)\) has hampered the use of Regge calculus for regularizing 2D quantum gravity.

  23. 23.

    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 \(e^{\nu _{0}}=(\frac {4^{4}}{3^{3}})\). Also note that the parameter c g does not play any relevant role in 2D quantum gravity.

  24. 24.

    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 P. Menotti and P.P. Peirano, (see [45] and references therein).

  25. 25.

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

  26. 26.

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

  27. 27.

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

  28. 28.

    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, 13, 30, 32–36].

  29. 29.

    Different geometrical aspects of the role of the hyperbolic point of view in open/closed string duality have been discussed also by R. Kaufmann and R. Penner [39].

  30. 30.

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

  31. 31.

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

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  1. Dipartimento di Fisica, Università degli Studi di Pavia, Pavia, Italy

    Mauro Carfora

  2. Dipartimento di Matematica, Università degli Studi di Pavia, Pavia, Italy

    Annalisa Marzuoli

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  1. Mauro Carfora
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Carfora, M., Marzuoli, A. (2017). The Quantum Geometry of Polyhedral Surfaces: Variations on Strings and All That. In: Quantum Triangulations. Lecture Notes in Physics, vol 942. Springer, Cham. https://doi.org/10.1007/978-3-319-67937-2_5

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