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Zz Branes from a Worldsheet Perspective∗ Vol. 38 (2007) ACTA PHYSICA POLONICA B No 13 ZZ BRANES FROM A WORLDSHEET PERSPECTIVE ∗ Jan Ambjørn a,b, Jens Anders Gesser a aThe Niels Bohr Institute, Copenhagen University Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark bInstitute for Theoretical Physics, Utrecht University Leuvenlaan 4, NL-3584 CE Utrecht, The Netherlands (Received September 20, 2007) We show how non-compact space-time (ZZ branes) emerges as a limit of compact space-time (FZZT branes) for specific ratios between the square of the boundary cosmological constant and the bulk cosmological constant in the (2, 2m 1) minimal model coupled to two-dimensional Euclidean quantum gravity.− Furthermore, we show that the principal (r, s) ZZ brane can be viewed as the basic (1,1) ZZ boundary state tensored with a (r, s) Cardy boundary state for a general (p, q) minimal model coupled to two- dimensional quantum gravity. In this sense there exists only one ZZ bound- ary state, the basic (1,1) boundary state. PACS numbers: 11.25.Pm, 11.25.Hf, 04.60.Nc, 04.60.–m 1. Introduction Two-dimensional Euclidean quantum gravity serves as a good laboratory for the study of potential theories of quantum gravity in higher dimensions. Although it contains no dynamical gravitons and does not face the problem of being non-renormalizable, it can address many of the other conceptional questions which confronts a quantum field theory of gravity. How does one define the concept of distance in a theory where one is instructed to integrate over all geometries, how does one define the concept of correlation functions when one couples matter to gravity and the resulting theory is supposed to be diffeomorphism invariant? These are just two of many questions which can be addressed successfully and which are as difficult to answer in two dimensions as in higher dimensions. In addition two-dimensional quantum ∗ Presented at the Conference on Random Matrix Theory, “From Fundamental Physics to Applications”, Kraków, Poland, May 2–6, 2007. (3993) 3994 J. Ambjørn, J.A. Gesser gravity coupled to a minimal conformal field theory is nothing but a so-called non-critical string theory and serves as a good laboratory for the study of non-perturbative effects in string theory. The quantization of 2d gravity was first carried out for compact two- dimensional geometries using matrix models, combinatorial methods and methods from conformal field theory. Later on Zamolodchikov, Zamolod- chikov and Fateev and also Teschner (FZZT) used Liouville quantum field theory to quantize the disk geometry [2], thereby reproducing results already obtained from matrix models. Then the Zamolodchikovs (ZZ) turned their attention to a previously unadressed question crucial to quantum gravity, namely how to quantize non-compact 2d Euclidean geometries [1]. They asked if the quantization of the disk geometry could be generalized to the Lobachevskiy plane, also known as Euclidean AdS2 or the pseudosphere. The pseudosphere is a non-compact space with no genuine boundary. How- ever, one has to impose suitable boundary conditions at infinity in order to obtain a conformal field theory. The crucial difference compared to the quantization of the compact disk is that one can invoke the assumption of factorization when discussing the correlator of two operators. One assumes that (x) (y) (x) (y) (1) hO1 O2 i → hO1 ihO2 i when the geodesic distance on the pseudosphere between x and y goes to infinity. This additional requirement results in a number of self-consistent boundary conditions at infinity compatible with the conformal invariance of quantum Liouville theory. It is the purpose of this article to address the question of quantizing non-compact 2D Euclidean geometries using a different approach than the Zamolodchikovs. However, our results will relate to the random geometries obtained by the Zamolodchikovs. In modern string terminology boundary conditions are almost synony- mous to “branes” and in this spirit the conventional partition function for the disk and the partition function for the pseudosphere were reinterpre- tated in non-critical string theory as FZZT and ZZ branes, respectively. In this context it was first noticed that there is an intriguing relationship be- tween the FZZT and the ZZ branes [3–6], as well as between the analytical continuation of the disk amplitude to the complex plane and the space of conformal invariant boundary conditions one can impose. It is the objective of this article to analyze these relations from a world- sheet perspective. ZZ Branes from a Worldsheet Perspective 3995 2. From compact to non-compact geometry The disk and cylinder amplitudes for generic values of the coupling con- stants in minimal string theory were first calculated using matrix model techniques. In order to compare with continuum calculations performed in the context of Liouville theory, it is necessary to work in the so-called con- formal background [7]. In the following we will, for simplicity, concentrate on the disk and the cylinder amplitudes in the (2, 2m 1) minimal conformal field theories coupled to 2d quantum gravity. In the− conformal background the disk amplitude is given by: w (x) = ( 1)mPˆ (x, √µ) x + √µ = ( 1)m (√µ)(2m−1)/2 P (t)√t + 1 , µ − m − m q (2) where t = x/√µ and where [4, 7] 2 2−2m Pm(t) (t + 1) = 2 (T2m−1(t) + 1) . (3) Tp(t) being the first kind of Chebyshev polynomial of degree p. In Eq. (2) x denotes the boundary cosmological coupling constant and µ the bulk cosmo- logical coupling constant, the theory viewed as 2d quantum gravity coupled to the (2, 2m 1) minimal CFT. The zeros of the polynomial Pm(t) are all located on the− real axis between 1 and 1 and more explicitly we can write: − m−1 P (t) = (t t ) , m − n nY=1 2nπ t = cos , 1 n m 1 . (4) n − 2m 1 ≤ ≤ − − The zeros of Pm(t) can be associated with the m 1 principal ZZ branes in the notation of [4]. In order to understand this, i.e.−in order to understand why the special values tn (and only these values) of the boundary cosmolog- ical constant are related to non-compact worldsheet geometries, it is useful to invoke the so-called loop–loop propagator Gµ(x,y; d) [12–16]. It describes the amplitude of an “exit” loop with boundary cosmological constant y to be separated a distance d from an “entrance” loop with boundary cosmological constant x (the entrance loop conventionally assumed to have one marked point). Gµ(x,y; d) satisfies the following equation: ∂ ∂ G (x,y; d)= w (x)G (x,y; d) , (5) ∂d µ −∂x µ µ 3996 J. Ambjørn, J.A. Gesser with the following solution: x ′ wµ(¯x(d)) 1 dx Gµ(x,y; d)= , d = ′ , (6) wµ(x) x¯(d)+ y Z wµ(x ) x¯(d) where x¯(d) is called the running boundary coupling constant. For the (2, 2m 1) minimal model coupled to 2d gravity (6) reads: − ¯ m−1 ¯ ′ 1 1 1+ t(d) n=1 (t(d) tn) Gµ(t,t ; d) − , (7) ∝ √µ t¯(d)+ t′ p √1+ t Qm−1 (t t ) n=1 − n Q where we use the notation of (2), i.e. t = x/√µ, t′ = y/√µ and t¯(d) = x¯(d)/√µ. For m = 2, i.e. pure gravity d measures the geodesic distance. For m > 2 this is not true. Rather, it is a distance measured in terms of matter excitations. This is explicit by construction in some models of quantum gravity with matter, for instance the Ising model and the c = 2 model formulated as an O( 2) model [18, 19]. However, we can still use−d as a measure of distance and− we will do so in the following. When d it follows from (6) that the running boundary coupling constant t¯(d) converges→∞ to one of the zeros of the polynomial Pm(t), i.e. 2kπ t¯(d) tk , tk = cos . (8) −−−→d→∞ − 2m 1 − The cylinder amplitude (7) vanishes for generic values of t′ in the limit d . However, as shown in [8] we have a unique situation when we choose ′ →∞ ′ t = tk since in this case the term 1/(t¯(d)+ t ) in (7) becomes singular for d − . After some algebra we obtain the following expression: →∞ m−1 ′ 1 1 n 2nπ Gµ(t,t = tk, d ) ( 1) sin − →∞ ∝ √µ √1+ t − 2m 1 Xn=1 − 1 1 . (9) × √1+ t + √1+ t − √1+ t √1+ t n − n ′ Notice, Gµ(t,t = tk, d ) is independent of which zero tk the running boundary coupling− constant→∞ approaches in the limit d , apart from an overall constant of proportionality. → ∞ Formula (9) describes an AdS-like non-compact space with cosmological constant µ and with one compact boundary with boundary cosmological constant x as explained in [8] in the case of pure gravity. In the last section we will comment on the fact that we have to set t′ = t in order to generate − k ZZ Branes from a Worldsheet Perspective 3997 an AdS-like non-compact space in the limit d and that tk serves as an attractive fixed point for the running boundary→∞ coupling constant. Now, we will explain how the cylinder amplitude (9) is related to the conventional FZZT–ZZ cylinder amplitude in the Liouville approach to quantum gravity. 3. The cylinder amplitudes Like the disk amplitude (2), the cylinder amplitude in the (2, 2m 1) minimal CFT coupled to 2d quantum gravity was first calculated using− the one-matrix model. Quite remarkable it was found to be universal, i.e. the same in all the (2, 2m 1) minimal models coupled to quantum gravity [7,17]: − 2 Zµ(t1,t2)= log √t1 +1+ √t2 + 1 √µ a , (10) − where a is a (lattice) cut-off.
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