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Jan Ambjorn

Publications and source records attributed to Jan Ambjorn.

At least 37 records · Page 2Linked to original sources
arXiv2015

Multi-point functions of weighted cubic maps

We study the geodesic two- and three-point functions of random weighted cubic maps, which are obtained by assigning random edge lengths to random cubic planar maps. Explicit expressions are obtained by taking limits of recently established bivariate multi-point functions of general planar maps. We give an alternative interpretation of the two-point function in terms of an Eden model exploration process on a random planar triangulation. Finally, the scaling limits of the multi-point functions are studied, showing in particular that the two- and three-point functions of the Brownian map are recovered as the number of faces is taken to infinity.

math-ph↗
arXiv2014

Geodesic distances in Liouville quantum gravity

In order to study the quantum geometry of random surfaces in Liouville gravity, we propose a definition of geodesic distance associated to a Gaussian free field on a regular lattice. This geodesic distance is used to numerically determine the Hausdorff dimension associated to shortest cycles of 2d quantum gravity on the torus coupled to conformal matter fields, showing agreement with a conjectured formula by Y. Watabiki. Finally, the numerical tools are put to test by quantitatively comparing the distribution of lengths of shortest cycles to the corresponding distribution in large random triangulations.

hep-th↗
arXiv2014

The matrix model for hypergeometric Hurwitz numbers

We present the multi-matrix models that are the generating functions for branched covers of the complex projective line ramified over $n$ fixed points $z_i$, $i=1,\dots,n$, (generalized Grotendieck's dessins d'enfants) of fixed genus, degree, and the ramification profiles at two points, $z_1$ and $z_n$. We take a sum over all possible ramifications at other $n-2$ points with the fixed length of the profile at $z_2$ and with the fixed total length of profiles at the remaining $n-3$ points. All these models belong to a class of hypergeometric Hurwitz models thus being tau functions of the Kadomtsev--Petviashvili (KP) hierarchy. In the case described above, we can present the obtained model as a chain of matrices with a (nonstandard) nearest-neighbor interaction of the type $\tr M_iM_{i+1}^{-1}$. We describe the technique for evaluating spectral curves of such models, which opens the possibility of applying the topological recursion for developing $1/N^2$-expansions of these model. These spectral curves turn out to be of an algebraic type.

hep-th↗
arXiv2014

The effective action in 4-dim CDT. The transfer matrix approach

We measure the effective action in all three phases of 4-dimensional Causal Dynamical Triangulations (CDT) using the transfer matrix method. The transfer matrix is parametrized by the total 3-volume of the CDT universe at a given (discrete) time. We present a simple effective model based on the transfer matrix measured in the de Sitter phase. It allows us to reconstruct the results of full CDT in this phase. We argue that the transfer matrix method is valid not only inside the de Sitter phase ('C') but also in the other two phases. A parametrization of the measured transfer matrix / effective action in the 'A' and 'B' phases is proposed and the relation to phase transitions is explained. We discover a potentially new 'bifurcation' phase separating the de Sitter phase ('C') and the 'collapsed' phase ('B').

hep-th↗
arXiv2013

Trees and spatial topology change in CDT

Generalized causal dynamical triangulations (generalized CDT) is a model of two-dimensional quantum gravity in which a limited number of spatial topology changes is allowed to occur. We solve the model at the discretized level using bijections between quadrangulations and trees. In the continuum limit (scaling limit) the amplitudes are shown to agree with known formulas and explicit expressions are obtained for loop propagators and two-point functions. It is shown that from a combinatorial point of view generalized CDT can be viewed as the scaling limit of planar maps with a finite number of faces and we determine the distance function on this ensemble of planar maps. Finally, the relation with planar maps is used to illuminate a mysterious identity of certain continuum cylinder amplitudes.

hep-th↗
arXiv2013

The toroidal Hausdorff dimension of 2d Euclidean quantum gravity

The lengths of shortest non-contractible loops are studied numerically in 2d Euclidean quantum gravity on a torus coupled to conformal field theories with central charge less than one. We find that the distribution of these geodesic lengths displays a scaling in agreement with a Hausdorff dimension given by the formula of Y. Watabiki.

hep-th↗
arXiv2013

2d CDT is 2d Horava-Lifshitz quantum gravity

Causal Dynamical Triangulations (CDT) is a lattice theory where aspects of quantum gravity can be studied. Two-dimensional CDT can be solved analytically and the continuum (quantum) Hamiltonian obtained. In this article we show that this continuum Hamiltonian is the one obtained by quantizing two-dimensional projectable Horava-Lifshitz gravity.

hep-th↗
arXiv2012

Semi-classical Dynamical Triangulations

For non-critical string theory the partition function reduces to an integral over moduli space after integrating over matter fields. The moduli integrand is known analytically for genus one surfaces. The formalism of dynamical triangulations provides us with a regularization of non-critical string theory and we show that even for very small triangulations it reproduces very well the continuum integrand when the central charge $c$ of the matter fields is large negative, thus providing a striking example of how the quantum fluctuations of geometry disappear when $c \to -\infty$.

hep-th↗
arXiv2012

The transfer matrix in four-dimensional CDT

The Causal Dynamical Triangulation model of quantum gravity (CDT) has a transfer matrix, relating spatial geometries at adjacent (discrete lattice) times. The transfer matrix uniquely determines the theory. We show that the measurements of the scale factor of the (CDT) universe are well described by an effective transfer matrix where the matrix elements are labeled only by the scale factor. Using computer simulations we determine the effective transfer matrix elements and show how they relate to an effective minisuperspace action at all scales.

hep-th↗
arXiv2012

New multicritical matrix models and multicritical 2d CDT

We define multicritical CDT models of 2d quantum gravity and show that they are a special case of multicritical generalized CDT models obtained from the new scaling limit, the so-called "classical" scaling limit, of matrix models. The multicritical behavior agrees with the multicritical behavior of the so-called branched polymers.

hep-th↗
arXiv2012

Remarks on Holographic Wilson Loops and the Schwinger Effect

We extend Douglas' solution of the problem of finding minimal surfaces to anti-de Sitter space. The case of a circle as a boundary contour is elaborated. We discuss applications to ${\cal N}=4$ super Yang-Mills: a circular Wilson loop and the Schwinger process, where we calculate the $1/\sqrtλ$ correction to the critical value of constant electric field.

hep-th↗
arXiv2011

CDT as a scaling limit of matrix models

It is shown that generalized CDT, the two-dimensional theory of quantum gravity, constructed as a scaling limit from so-called causal dynamical triangulations, can be obtained from a cubic matrix model. It involves taking a new scaling limit of matrix models, which is more natural from a classical point of view.

hep-th↗
arXiv2009

Quantum gravity as sum over spacetimes

A major unsolved problem in theoretical physics is to reconcile the classical theory of general relativity with quantum mechanics. These lectures will deal with an attempt to describe quantum gravity as a path integral over geometries known as "Causal Dynamical Triangulations" (CDT).

gr-qc↗
arXiv2007

ZZ branes from a worldsheet perspective

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 boundary state, the basic (1,1) boundary state.

hep-th↗
arXiv2007

World-sheet dynamics of ZZ branes

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 quantum gravity.

hep-th↗
arXiv2007

Wilson Loops in 2D Noncommutative Euclidean Gauge Theory: 2. 1/θExpansion

We analyze the $1/θ$ and 1/N expansions of the Wilson loop averages $ _{U_θ(N)}$ in the two-dimensional noncommutative $U_θ(N)$ gauge theory with the parameter of noncommutativity $θ$. For a generic rectangular contour $C$, a concise integral representation is derived (non-perturbatively both in the coupling constant $g^{2}$ and in $θ$) for the next-to-leading term of the $1/θ$ expansion. In turn, in the limit when $θ$ is much larger than the area $A(C)$ of the surface bounded by $C$, the large $θ$ asymptote of this representation is argued to yield the next-to-leading term of the $1/θ$ series. For both of the expansions, the next-to-leading contribution exhibits only a power-like decay for areas $A(C)>>σ^{-1}$ (but $A(C)<<θ$) much larger than the inverse of the string tension $σ$ defining the range of the exponential decay of the leading term. Consequently, for large $θ$, it hinders a direct stringy interpretation of the subleading terms of the 1/N expansion in the spirit of Gross-Taylor proposal for the $θ=0$ commutative D=2 gauge theory.

hep-th↗
arXiv2006

Wrapping interactions and a new source of corrections to the spin-chain/string duality

Assuming that the world-sheet sigma-model in the AdS/CFT correspondence is an integrable {\em quantum} field theory, we deduce that there might be new corrections to the spin-chain/string Bethe ansatz paradigm. These come from virtual particles propagating around the circumference of the cylinder and render Bethe ansatz quantization conditions only approximate. We determine the nature of these corrections both at weak and at strong coupling in the near BMN limit, and find that the first corrections behave qualitatively as wrapping interactions at weak coupling.

hep-th↗