Physics and the Monster Quantum Gravity Modular Forms Faber

Physics and the Monster Quantum Gravity, Modular Forms & Faber Polynomials Alex Maloney, McGill Concordia University, 3 October 2007 A. Maloney & E. Witten, to appear E. Witten, arxiv:0706.3359 An Apology: I’m a physicist, not a mathematician. My goal is not to describe new mathematics, but to provide a new physical interpretation of math you already know: Elementary results in the theory of modular functions have deep implications for certain theories of quantum gravity. Elementary facts about the behavior of quantum gravity lead to interesting (and possibly new) observations about certain modular functions. The connection with the Monster group & Moonshine is not entirely clear... The Plan for Today: • Overview: Quantum Gravity and Modular Forms • The Partition Functions of Three Dimensional Gravity • Fourier Coefficients of Modular Forms =⇒ Black Hole Physics • Physics of Phase Transitions =⇒ Zeroes of Modular Functions Overview The Physics: Gravity (i.e. general relativity) is a theory of dynamical metrics. It is notoriously difficult to quantize. But in D = 3, the problem simplifies: Locally, the metric g is fixed by the equations of motion. We are left with the global degrees of freedom of constant curvature metrics. Three types of theories: R > 0, R = 0, R < 0. We will focus on the third case. The theory is parameterized by Curvature radius Coupling constant G (the cosmological constant). (Newton’s constant). 16G . which can be combined into a dimensionless constant k = Quantum Gravity with R < 0 To define our theory we specify a class of metrics. We do this by fixing boundary conditions at asymptotic infinity: The prototypical Riemannian R < 0 metric is the Poincare ball H 3 , whose asymptotic boundary is S 2 . The isometry group SL(2, C) of H 3 acts on this S 2 by conformal transformations. The corresponding Lorentzian R < 0 metric is Anti-de Sitter space AdS3 , with asymptotic boundary R 1 × S 1 . Likewise, the isometry group of AdS3 acts on this boundary by conformal transformations. Our goal is to integrate over all manifolds with the same conformal structure at infinity, and compute a partition function Z= Dg e −S(g ) = Dg e −kVol(g ) The Partition Function We will focus on the case where the boundary is T 2 . The partition function Z is a function of the conformal structure of the boundary, i.e. of the modular parameter τ describing the conformal structure of T 2 . So Z = Z (τ ) is a modular function. We can compute the “saddle-point” approximation to Z (τ ): Z (τ ) = Dg e −kVol(g ) ∼ e −kVol(M3 ) ∼ e −2πikτ + . . . where M3 = H 3 /Z is a genus one handle-body endowed with a constant negative curvature metric. AdS/CFT We can do more than just compute the volume of the saddle point. By studying the class of metrics with fixed asymptotic boundary conditions it follows that Z (τ ) is the partition function of a conformal field theory: Z (τ ) = Tr H e −2πiτ L0 where the Hilbert space H is a representation of the Virasoro algebra c [Lm , Ln ] = (m − n)Lm+n + (m3−m )δm+n 12 with c = 24k. This dual conformal field theory lives on the two dimensional conformal boundary of the geometry, and provides an equivalent description of the three dimensional quantum gravity. Brown & Henneaux, Maldacena Modular Partition Functions Partition Function I To summarize, we seek a Z (τ ) that: is SL(2, Z) invariant reduces to Z ∼ q −k as Im(τ ) → ∞ is consistent with the Virasoro algebra is holomorphic in τ (this is a strong assumption) This partition function tells us the spectrum of the theory: Z (τ ) = ∆≥−k F (∆)q ∆ , q = e 2πiτ where F (∆) is the number of states with L0 = ∆. Physically, ∆ is the energy of a state, and τ is the temperature of the system. Partition Functions II Such Z (τ ) exist only under certain conditions: k is an integer. The ∆ are integers. From a physics point of view, these elementary facts have profound implications: the allowed values of the curvature radius are quantized. the allowed masses of states in the theory are quantized. In particular, the allowed masses of black holes are quantized. Witten Case: k = 1 Here, the partition function is unique: Z (τ ) = J(τ ) = q −1 + 196884q + . . . The conformal field theory with partition function J(τ ) is known. It is the famous “Monster CFT” of Frenkel, Lepowski & Meurman, with central charge c = 24. The states of the CFT form representations of the Monster group M. For k > 1 it is not known whether analogous CFTs exist, but one can still construct the associated partition functions. The conjecture that such CFTs exist has survived several consistency checks. Witten, Gaiotto & Yin, . . . Case: k > 1 Start with the action of the Hecke operator: d−1 Tn J = d|n b=0 J nτ + bd d2 = q −n + O(q) As a partition function, Tk J itself is not consistent with the Virasoro algebra. But k Z (τ ) = n=0 an Tn J is consistent, for certain (non-unique) choices of an . The coefficients F (∆) are dimensions of representations of M. So one might suspect that M is secretly the symmetry group of three dimensional quantum gravity. Black Holes Black Holes I The coefficients of the modular function Z (τ ) grow exponentially at large ∆ √ F (∆) ∼ e 4π k∆ + . . . What are all these states? AdS3 gravity contains black holes. The black hole entropy predicted by Bekenstein and Hawking SBH = √ A = 4π k∆ 4G agrees precisely with this asymptotic growth. For example, the entropy of the k = 1, ∆ = 1 states is 4π ∼ log (196883) The subleading terms in the asymptotic growth can be matched with known corrections to the black hole entropy formula. Black Holes II The Modular function Z (τ ) = ∆≥−k F (∆)q ∆ is determined completely by its polar part: F (∆), −k ≤ ∆ ≤ 0 This means that we can choose our an so that the first k Laurent coefficients F (∆) of Z (τ ) are small, but the F (∆) become (exponentially) large once ∆ > 0. Physically, this corresponds to the fact that there is a “gap” in the spectrum of black holes in AdS3 . Empty AdS3 has energy ∆ = −k Black Holes all have energy ∆ > 0 Phase Transitions Poincare Series I In fact, for a given boundary T 2 , there are many different constant curvature manifolds M3 with this T 2 as a boundary: A B One for each choice of cycle cA + dB which is contractible in M3 . Under (1, 0) → (c, d), Vol(M3 ) transforms under SL(2, Z). The full saddle point approximation is a Poincare series: Z (τ ) = Dg e −Vol(g ) ∼ c,d exp −2πik aτ + b cτ + d Poincare Series II This sum over Z\SL(2, Z) diverges, but can be regularized in a way consistent with SL(2, Z) invariance: Z (τ ) = E (0, τ ) E (s, τ ) = c,d Im(τ ) cτ + d 2s exp −2πik aτ + b cτ + d The answer is just Tk J, the “saddle-point” partition function. One can include the corrections to this saddle point answer to reproduce the full partition function k Z (τ ) = n=0 an Tn J Complex Geometries Although I’ve focused on the holomorphic part Z (τ ), the full partition function of quantum gravity is real. It is the square |Z (τ )|2 . We reproduced the partition function by summing over Z\SL(2, Z) ¯ τ separately for both holomorphic Z (τ ) and anti-holomorphic Z (¯) sectors. Physically, this means that we are summing over complex geometries, which is not usually done in general relativity. The sum over real geometries can be done, but the answer does not have a sensible physical interpretation – it does not reproduce the black hole entropy. Phase Transitions The different saddles M3 have different physical interpretations. T X When X is contractible, the geometry describes a thermal gas of particles, with temperature Im(τ ). When T is contractible, the geometry describes a black hole with temperature Im(τ ). When a combination of T and X is contractible, the geometry describes a rotating black hole with temperature Im(τ ) and angular potential Re(τ ). Phase Diagram For different values of τ , different saddles M3 dominate the partition function. The tessellation of H2 by the action of Z\SL(2, Z) is the phase diagram of quantum gravity in three dimensions. Hawking & Page, Maldacena & Strominger Phase Transitions As τ moves from one fundamental region to another we expect a phase transition. There is a beautiful theory of phase transition in two dimensions developed by Lee & Yang and Fischer. In the classical (k → ∞) limit, the partition function Z (τ ) should become non-analytic along the phase boundaries. This non-analyticity arises because at finite k, Z (τ ) has k zeroes on the phase boundary. These zeroes become dense on this curve in the k → ∞ limit. We can prove that these facts are true for the modular functions under consideration, at least for physically reasonable choices of an . Phase Transitions II For example, it can be proven that all zeroes of Tn J lie along phase boundaries in the τ plane, and that these zeroes become dense in the n → ∞. 0.2 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Asai, Kaneko & Ninomiya 0.6 0.8 1.2 0.4 1.4 1 Conclusion We can learn a lot about quantum gravity in three dimensions by thinking about modular forms: Modular functions J(τ ), Tn J(τ ), . . . =⇒ spectrum of quantum gravity Large ∆ Laurent coefficients =⇒ Black Hole entropy Zeroes of Modular functions =⇒ Phase Transitions Dual CFTs with Monster Symmetry? Hopefully, this new physical interpretation can give insight into mathematics as well.