MatrixCosmologyAnIntroduction

单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,Matrix Cosmology,An Introduction,Miao Li,University of Science and Technology,Institute of Theoretical Physics,Chinese Academy of Science,1st Asian Winter School, Phoenix Park,1,Contents:,A toy model,Matrix description,A class of generalizations,More generalizations,Quantum computations,2,Motivations:,String theory faces the following challenges posed,by cosmology:,Formulate string theory in a time-dependent,background in general.,2. Explain the origin of the universe, in particular, the,nature of the big bang singularity.,3. Understand the nature of dark energy.,3,None of the above problems is easy.,1. A toy model,Recently, in paper,hep-th/0506180,Craps, Sethi and Verlinde consider the “simple”,background:,4,This background is not as simple as it appears, since,the Einstein metric,has a null singularity at . The spacetime,Looks like a cone:,lightcone time,5,CSV shows that perturbative string description breaks,down near the null singularity. In fact, the scattering,amplitudes diverge at any finite order.,I suspect that string S-matrix,does not exist,.,Nevertheless, CSV shows that a variation of matrix,Theory can be a good effective description.,6,In the 11 dimensional perspective, the metric is,locates in a finite distance away in terms of,the affine parameter if we follow a null geodesic.,If , then,These quantities blow up at .,7,More comments on the singularity later.,String vertex operator,With a constant dilaton, a vertex operator assumes,The form,with the on-shell condition:,8,With , we need to attach a factor,to the vertex operator,The on-shell condition for k is the same as,before.,The vertex operator blows up at,9,Scattering amplitudes,Blows up whenever 2g-2+n0.,Thus, the string perturbative S-matrix is ill-defined.,10,2. Matrix description,CSV propose to use a matrix model to describe the,Physics in this background, since,The background preserves half of all SUSY,if,There is a decoupling argument a la Seiberg and,Sen.,11, The proposal,In the IIA matrix string model, for a sector with a,Fixed longitudinal momentum,Where , the matrix action,12,The Yang-Mills coupling constant is related to the,string couping constant through,Now with,We simply have,Thus,13,So near the “big bang” singularity, the SYM is a free,nonabelian theory. On the other hand, near,the theory tends to a CFT with an orbifold target,space.,Strings are in the twisted sectors.,14,More details:,For , in the gauge,For,Residual gauge symmetry is permutations of the,eigen-values of the matrices,15,3. A class of generalizations,In,hep-th/0506260,I showed that the CSV model,is a special case of a,large class of models.,In terms of the 11 dimensional M theory picture, the,metric assumes the form,where there are 9 transverse coordinates, grouped,into 9-d and d .,16,This metric in general breaks half of supersymmetry.,Next we specify to the special case when both f and,g are linear function of :,If d=9 and one takes the minus sign in the above, we,get a flat background.,The null singularity still locates at .,17,Again, perturbative string description breaks down,near the singularity. To see this, compacitfy one,spatial direction, say , to obtain a string theory.,Start with the light-cone world-sheet action,We use the light-cone gauge in which , we,see that there are two effective string tensions:,18,As long as d is not 1, there is in general no plane,wave vertex operator, unless we restrict to the special,situation when the vertex operator is independent of,. For instance, consider a massless scalar satisfying,The momentum component contains a imaginary,Part thus the vertex operator contains a factor,diverging near the singularity.,19,Since each vertex operator is weighted by the string,coupling constant, one may say that the effective,string coupling constant diverges. In fact, the,effective Newton constant also diverges:,We conjecture that in this class of string background,there is no S-matrix at all.,However, one may use D0-branes to describe the,theory, since the Seiberg decoupling argument,applies.,20,We shall not present that argument here, instead,We simply display the matrix action. It contains,the bosonic part and fermionic part,This action is quite rich. Lets discuss the general,conclusions one can draw without doing any,calculation.,21,Case 1.,The kinetic term of is always simple, but the,kinetic term of vanishes at the singularity, this,implies that these coordinates fluctuate wildly. Also,coefficient of all other terms vanish, so all matrices,are fully nonabelian.,As , the coefficients of interaction terms blow,up, so all bosonic matrices are forced to be,Commuting.,22,Case 2.,At the big bang, are independent of time, and,are nonabelian moduli if d4. There is no constraint,on other commutators of bosonic matrices.,As , if d4, all matrices have to be commuting.,For d4, are nonabelian.,23,4. More generalizations,Bin Chen in hep-th/0508191 considers the following,More general background,where,This class of backgrounds all preserve half of SUSY,24,Bin Chens background has to satisfy only one diff.,equation. However, it is not clear whether one can,write down a matrix model.,Das and Michelson in hep-th/0508068 study a,background appears to be a special case of Bin,Chen:,25,Das and Michelson claim that one can write down,a matrix model for this background.,It is interesting that these authors noted that, a,String which appears to be weakly coupled at later,times is actually a fuzzy cylinder at early times.,Das, Michelson, Narayan and Trivedi in,hep-th/0602107 constructed a model in IIB string,Theory which is a deformation of , this,work overlaps with the work of Chu and Ho.,26,Ishino and Ohta in hep-th/0603215 study the,matrix string description of the following,background:,Again, all functions are functions of only u. They,are subject to a single equation,27,Finally, Chu and Ho in hep-th/0602054 consider,the following class of time-dependent deformation,of the AdS solution:,Where,Also subject to a single equation.,28,Chu and Ho propose that string theory in this,background is dual to a generalized super Yang-,Mills theory in 3+1 dimension with both time-,dependent metric and time-dependent coupling.,29,5. Quantum computations,To check whether these matrix descriptions are really,correct, we need to compute at least the interaction,between two D0-branes. This is done in,hep-th/0507185,by myself and my student Wei Song.,There, we use the shock wave to represent the,background generated by a D0-brane which carries,a net stress tensor .,30,In fact, the most general ansatz is,for multiple D0-branes localized in the transverse space , but smeared in the transverse space . The background metric of the shock wave is,with,31,The probe action of a D0-brane in such a background,is,with,We see that in the big bang, the second term in the,square root blows up, thus the perturbative expansion,in terms of small v and large r breaks down.,32,The breaking-down of this expansion implies the,breaking-down the loop perturbation in the matrix,calculation. This is not surprising, since for instance,some nonabelian degrees of freedom become light,at the big bang as the term,in the CSV model shows.,33,Therefore, it is of no surprising that some,Computations done so far have not correctly,reproduce the previous result.,In hep-th/0512335, Wei Song and myself used,Matrix model to compute interaction between two,D0-branes, we find a null static potential, however,there is a complex term, signaling an,instability.,34,Craps, Rajaraman and Sethi in hep-th/0601062,also computed the interaction at the one loop level,and found a different result.,They found a static potential decays at later times.,Why these results are different? Possible answers:,1. Results depend sensitively on the method of,calculation: initial conditions can be subtle.,2. D0-branes and associated potential are not good,observables.,35,Conclusion:,Time-dependent backgrounds are beasts hard to,tame in string theory.,36,