O(d, d)-Symmetry and Ernst Formulation of Einstein–Kalb–Ramond Theory

1997 ◽  
Vol 12 (21) ◽  
pp. 1573-1582 ◽  
Author(s):  
Alfredo Herrera-Aguilar ◽  
Oleg Kechkin
Keyword(s):  
Matrix Representation ◽  
Two Dimensions ◽  
Target Space ◽  
Subsequent Reduction ◽  
R Matrix ◽  
The Real ◽  
Ernst Potential

A Ernst-like matrix representation of (3+d)-dimensional Einstein–Kalb–Ramond theory is developed. The analogy with the Einstein and Einstein–Maxwell–Dilaton–Axion theories is discussed. The subsequent reduction to two dimensions is considered. It is shown that, in this case, the theory allows two different Ernst-like d×d-matrix formulations: the real nondualized target space, and the Hermitian dualized nontarget space one. The O(d, d)-symmetry is written in an SL (2,R) matrix-valued form in both cases. The Kramer–Neugebauer transformation, which algebraically maps the nondualized Ernst potential onto the dualized one, is presented.

CRITICAL BEHAVIOR IN TWO-DIMENSIONAL QUANTUM GRAVITY AND EQUATIONS OF MOTION OF THE STRING

1990 ◽  
Vol 05 (11) ◽  
pp. 799-813 ◽  
Author(s):  
SUMIT R. DAS ◽  
AVINASH DHAR ◽  
SPENTA R. WADIA
Keyword(s):  
Equations Of Motion ◽  
Two Dimensions ◽  
Target Space ◽  
Two Dimensional ◽  
Minimal Models ◽  
Conformally Invariant ◽  
Motion Time ◽  
Dynamical Degree ◽  
Renormalization Group Equations

We show how consistent quantization determines the renormalization of couplings in a quantum field theory coupled to gravity in two dimensions. The special status of couplings corresponding to conformally invariant matter is discussed. In string theory, where the dynamical degree of freedom of the two-dimensional metric plays the role of time in target space, these renormalization group equations are themselves the classical equations of motion. Time independent solutions, like classical vacuua, correspond to the situation in which matter is conformally invariant. Time dependent solutions, like tunnelling configurations between vacuua, correspond to special trajectories in theory space. We discuss an example of such a trajectory in the space containing the c < 1 minimal models. We also discuss the connection between this work and the recent attempts to construct non-perturbative string theories based on matrix models.

scholarly journals Random Spanning Forests and Hyperbolic Symmetry

2020 ◽  
Author(s):  
Roland Bauerschmidt ◽  
Nicholas Crawford ◽  
Tyler Helmuth ◽  
Andrew Swan
Keyword(s):  
Sigma Model ◽  
Two Dimensions ◽  
Target Space ◽  
Random Cluster Model ◽  
Random Cluster ◽  
Reinforced Random Walks ◽  
Spanning Forests ◽  
Bounded Size ◽  
Exact Relationship ◽  
Special Case

AbstractWe study (unrooted) random forests on a graph where the probability of a forest is multiplicatively weighted by a parameter $$\beta >0$$ β > 0 per edge. This is called the arboreal gas model, and the special case when $$\beta =1$$ β = 1 is the uniform forest model. The arboreal gas can equivalently be defined to be Bernoulli bond percolation with parameter $$p=\beta /(1+\beta )$$ p = β / ( 1 + β ) conditioned to be acyclic, or as the limit $$q\rightarrow 0$$ q → 0 with $$p=\beta q$$ p = β q of the random cluster model. It is known that on the complete graph $$K_{N}$$ K N with $$\beta =\alpha /N$$ β = α / N there is a phase transition similar to that of the Erdős–Rényi random graph: a giant tree percolates for $$\alpha > 1$$ α > 1 and all trees have bounded size for $$\alpha <1$$ α < 1 . In contrast to this, by exploiting an exact relationship between the arboreal gas and a supersymmetric sigma model with hyperbolic target space, we show that the forest constraint is significant in two dimensions: trees do not percolate on $${\mathbb {Z}}^2$$ Z 2 for any finite $$\beta >0$$ β > 0 . This result is a consequence of a Mermin–Wagner theorem associated to the hyperbolic symmetry of the sigma model. Our proof makes use of two main ingredients: techniques previously developed for hyperbolic sigma models related to linearly reinforced random walks and a version of the principle of dimensional reduction.

Parallel Visual Coding in Three Dimensions

Perception ◽  
1994 ◽  
Vol 23 (4) ◽  
pp. 453-470 ◽  
Author(s):  
Glyn W Humphreys ◽  
Nicole Keulers ◽  
Nick Donnelly
Keyword(s):  
Three Dimensional ◽  
Spatial Relations ◽  
Image Features ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Linear Perspective ◽  
Size Difference ◽  
The Real ◽  
Local Background ◽  
Real Size

Evidence from visual-search experiments is discussed that indicates that there is spatially parallel encoding based on three-dimensional (3-D) spatial relations between complex image features. In one paradigm, subjects had to detect an odd part of cube-like figures, formed by grouping of corner junctions. Performance with cube-like figures was unaffected by the number of corner junctions present, though performance was affected when the corners did not configure into a cube. It is suggested from the data that junctions can be grouped to form 3-D shapes in a spatially parallel manner. Further, performance with cube-like figures was more robust to noncollinearity between junctions than was performance when junctions grouped to form two-dimensional planes. In the second paradigm, subjects searched for targets defined by their size. Performance was affected by a size illusion, induced by linear-perspective cues from local background neighbourhoods. Search was made more efficient when the size illusion was consistent with the real size difference between targets and nontargets, and it was made less efficient when the size illusion was inconsistent with the real size difference. This last result occurred even though search was little affected by the display size in a control condition. We suggest that early, parallel visual processes are influenced by 3-D spatial relations between visual elements, that grouping based on 3-D spatial relations is relatively robust to noncollinearity between junctions, and that, at least in some circumstances, 3-D relations dominate those coded in two-dimensions.

The logic of absolute value inequalities

1964 ◽  
Vol 57 (2) ◽  
pp. 73-74
Author(s):  
Marvin L. Chachere
Keyword(s):  
Real Line ◽  
Two Dimensions ◽  
Elementary Algebra ◽  
Absolute Value ◽  
The Real ◽  
Definition Of ◽  
Absolute Value Inequalities

Teachers motivated by the need to express the distance of any point on the real line from zero find the definition of absolute value useful and interesting. It is used in elementary algebra to define addition and multiplication of “directed” numbers and to describe the graphs of broken lines in one and two dimensions. Recent emphasis on inequalities and the inclusion in newer textbooks of absolute value inequalities raise several questions that cannot be handled in a perfunctory manner.

scholarly journals STRING THEORETICAL INTERPRETATION FOR FINITE N YANG–MILLS THEORY IN TWO DIMENSIONS

2005 ◽  
Vol 20 (01) ◽  
pp. 29-41 ◽  
Author(s):  
TOSHIHIRO MATSUO ◽  
SO MATSUURA
Keyword(s):  
Asymptotic Expansion ◽  
Closed String ◽  
Two Dimensions ◽  
Perturbative Expansion ◽  
Target Space ◽  
Theoretical Interpretation ◽  
Two Dimensional ◽  
Yang Mills ◽  
Large N ◽  
Residual Part

We discuss the equivalence between a string theory and the two-dimensional Yang–Mills theory with SU (N) gauge group for finite N. We find a sector which can be interpreted as a sum of covering maps from closed string worldsheets to the target space, whose covering number is less than N. This gives an asymptotic expansion of 1/N whose large N limit becomes the chiral sector defined by Gross and Taylor. We also discuss that the residual part of the partition function provides the nonperturbative corrections to the perturbative expansion.

scholarly journals Sp(4, R)/GL(2, R) Matrix Structure of Geodesic Solutions for Einstein–Maxwell–Dilaton–Axion Theory

1997 ◽  
Vol 12 (24) ◽  
pp. 4357-4368 ◽  
Author(s):  
Oleg Kechkin ◽  
Maria Yurova
Keyword(s):  
Mass Parameter ◽  
Matrix Operator ◽  
Target Space ◽  
Matrix Structure ◽  
Naked Singularities ◽  
R Matrix ◽  
The Family ◽  
Geodesic Lines ◽  
Isotropic Geodesic

The Sp (4, R)/ GL (2, R) matrix operator defining the family of isotropic geodesic lines in the target space of the stationary D = 4 Einstein–Maxwell–dilaton–axion theory is constructed. This operator is used to derive a class of solutions describing a system of point centers with nontrivial values of mass, parameter NUT, as well as electric, magnetic, dilaton and axion charges. It is shown that this class contains the Majumdar–Papapetrou-like solutions and also the solutions for massless naked singularities.

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