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String Theory and the Ads/CFT Correspondence

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String Theory and the Ads/CFT Correspondence String Theory and the AdS/CFT correspondence Matthias Gaberdiel ETH Zürich PSI 26 September 2019 Generalities What string theory is (and what it is not)? It is a promising approach towards a quantum theory incorporating high energy physics and general relativity. However, it is not a `complete’ theory: there are many conceptual problems (in particular its off-shell formulation) that are not well understood. Generalities What string theory is (and what it is not)? There are also (mainly technical) problems in understanding string theory for time-dependent and cosmological configurations, in particular for de Sitter backgrounds. On the other hand, string theory seems to provide, for example, a convincing explanation of black hole entropy in terms of microscopic states (at least for certain black holes). Generalities What string theory is (and what it is not)? String theory makes very specific and testable predictions… However, as for any theory of quantum gravity, these predictions are only `sharp’ near the Planck scale, and direct experimental verification seems currently out of reach. Generalities As a consequence, supporting evidence for string theory has appeared more indirectly, e.g. ‣ AdS/CFT correspondence has led to testable insights into conventional QFTs ‣ Triggered new developments in mathematics, e.g. mirror symmetry. ‣ Inspiration for model building, e.g. higher dimensions, supersymmetry, etc. ‣ Given rise to new techniques for perturbative and non-perturbative analysis of (susy) QFTs, e.g. Seiberg-Witten, etc. Generalities As a consequence, supporting evidence for string theory has appeared more indirectly, e.g. ‣ AdS/CFT correspondence has led to testable insights into conventional QFTs ‣ Triggered new developments in mathematics, e.g. mirror symmetry. ‣ Inspiration for model building, e.g. higher dimensions, supersymmetry, etc. ‣ Given rise to new techniques for perturbative and non-perturbative analysis of (susy) QFTs, e.g. Seiberg-Witten, etc. String basics The starting point of string theory is rather unconventional (and certainly not what you would begin with if you were to try and quantise gravity). [In fact, the history of the subject is rather convoluted, and it only become clear at some later stage that string theory actually includes gravity.] String basics The basic idea is that the fundamental objects, in terms of which the theory is being formulated, are one-dimensional strings. These strings can either be or closed open String basics They propagate (initially) in a given fixed background. The consistency conditions (conformal beta equations) require that this background satisfies a modification of the Einstein equations Thus string theory incorporates general relativity. String basics Furthermore, the excitation spectrum of the string contains the `graviton‘ that describes fluctuations of the background. Thus string theory may in fact be background independent. String basics The other vibrational excitations of the string contain the quanta that correspond to the familiar elementary particles. Thus string theory incorporates (some extension of) the standard model of particle physics. String basics instead of This smooth description suggests that string theory has better UV properties than conventional quantum field theories. [On the other hand, string field theory is not that well developed. In particular, one does not yet understand how to derive the `Feynman rules’ of string field theory from first principles.] String basics So far I have described the standard string folklore…. But how does one actually describe string theory explicitly? Quantitative strings In order to describe strings quantiatively, think of them in terms of a 2-dimensional field theory that is defined on the world-sheet of the string. Xµ(s, t) <latexit sha1_base64="OAiS0CfTNr0/I2AKIEy9kn5gusQ=">AAAB8XicbVBNSwMxEJ2tX7V+VT16CRahgpTdKuix6MVjBfuB7VqyabYNTbJLkhXK0n/hxYMiXv033vw3pu0etPXBwOO9GWbmBTFn2rjut5NbWV1b38hvFra2d3b3ivsHTR0litAGiXik2gHWlDNJG4YZTtuxolgEnLaC0c3Ubz1RpVkk7804pr7AA8lCRrCx0kP7sSuSsj4zp71iya24M6Bl4mWkBBnqveJXtx+RRFBpCMdadzw3Nn6KlWGE00mhm2gaYzLCA9qxVGJBtZ/OLp6gE6v0URgpW9Kgmfp7IsVC67EIbKfAZqgXvan4n9dJTHjlp0zGiaGSzBeFCUcmQtP3UZ8pSgwfW4KJYvZWRIZYYWJsSAUbgrf48jJpViveeaV6d1GqXWdx5OEIjqEMHlxCDW6hDg0gIOEZXuHN0c6L8+58zFtzTjZzCH/gfP4An9WQOg==</latexit> Sigma model The propagation of the string is described by specifying to which point in target space (space-time) every point on the world-sheet is mapped to. This map is controlled by the sigma-model action T mn µ ⌫ S = ds dt ph h @mX @nX Gµ⌫ <latexit sha1_base64="+tMxUSG6Ck9QSYmpwkorQlQhavA=">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</latexit> 2 Z world-sheet space-time metric metric Sigma model The sigma model is classically invariant under reparametrisations and Weyl rescalings of the metric. Can use this symmetry to go to `conformal gauge’: take world-sheet metric to be proportional to usual 2d Minkowski metric. Then the residual symmetry is the conformal (Weyl) symmetry — the resulting 2d field theory is therefore a conformal field theory. Conformal transformations The conformal transformations preserve the metric up to rescalings. They therefore preserve angles, but not necessarily lengths. For a conformally invariant theory it therefore does not matter whether one looks like this or like Virasoro algebra The conformal symmetry in 2 dimensions is thus very powerful! In fact, the algebra of infinitesimal conformal transformations is infinite dimensional: c 2 [Lm,Ln]=(m n)Lm+n + m(m 1)δm, n <latexit sha1_base64="PqKSENkPE6PFoMMcHpV/aIRrRcA=">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</latexit> − 12 − − [Virasoro algebra] c: central charge Conformal symmetry Because of this large symmetry, 2d conformal field theories can be solved essentially based on symmetry considerations alone. [In fact, 2d conformal field theories have a very rich mathematical structure: they define vertex operator algebras, and have had a significant impact on many areas of modern mathematics, such as group theory, number theory, algebra, etc.] Critical dimension From the point of view of string theory, the conformal symmetry is a gauge symmetry. Just as in electrodynamics, this gauge symmetry can then remove the negative norm states (ghosts) that the covariant theory initially has. In the present context, this requires that c<latexit sha1_base64="9RzkchyuETaDbG/59nxiGdCR74c=">AAAB9HicbVDLSgNBEOz1GeMr6tHLYBA8LbtR1GPQi8cI5gHZJcxOepMhs4/MzAZCyHd48aCIVz/Gm3/jJNmDJhY0FFXddHcFqeBKO863tba+sbm1Xdgp7u7tHxyWjo4bKskkwzpLRCJbAVUoeIx1zbXAViqRRoHAZjC4n/nNEUrFk/hJj1P0I9qLecgZ1UbymSdwSCrXxCM26ZTKju3MQVaJm5My5Kh1Sl9eN2FZhLFmgirVdp1U+xMqNWcCp0UvU5hSNqA9bBsa0wiVP5kfPSXnRumSMJGmYk3m6u+JCY2UGkeB6Yyo7qtlbyb+57UzHd76Ex6nmcaYLRaFmSA6IbMESJdLZFqMDaFMcnMrYX0qKdMmp6IJwV1+eZU0KrZ7aVcer8rVuzyOApzCGVyACzdQhQeoQR0YDOEZXuHNGlkv1rv1sWhds/KZE/gD6/MHQJGQcw==</latexit> 26 . [Goddard, Thorn ’72] Critical dimension If the target space is flat then c=dimension. Thus D=26 is the critical dimension of (bosonic) string theory. In order to describe 4d physics, the idea is then that 26<latexit sha1_base64="WO3jf6KuSRx584Pa9GRUgd1xuLI=">AAACJHicbVDLSsNAFJ3Ud31VXbq5WARBKUksKohQdOOygq1CE8JkMrVDJ5N0ZiKU0I9x46+4ceEDF278FqcPwdeBC2fOuZe594QpZ0rb9rtVmJqemZ2bXyguLi2vrJbW1psqySShDZLwRF6HWFHOBG1opjm9TiXFccjpVdg9G/pXt1QqlohL3U+pH+MbwdqMYG2koHTsHsAJAAlyT8ZQjQaw+/VgQg/Agz3wer0MRz/kE3BdCEplu2KPAH+JMyFlNEE9KL14UUKymApNOFaq5dip9nMsNSOcDopepmiKSRff0JahAsdU+fnoyAFsGyWCdiJNCQ0j9ftEjmOl+nFoOmOsO+q3NxT/81qZbh/5ORNppqkg44/aGQedwDAxiJikRPO+IZhIZnYF0sESE21yLZoQnN8n/yVNt+LsV9yLarl2OoljHm2iLbSDHHSIaugc1VEDEXSHHtATerburUfr1XobtxasycwG+gHr4xP8B6Fb</latexit> = c4d + cint ,cint = 22 i.e. 22 dimensions are compactified on some internal manifold — extra dimensions. Compactification The `internal’ theory may again be defined in terms of a non-linear sigma model, but in general it may be any conformal field theory with c=22. It is therefore not really appropriate to talk about an `internal manifold’ — the relevant conformal field theory may or may not have a geometric interpretation. Fermions The analysis for the (world-sheet) fermionic string is similar: in this case the relevant gauge symmetry is the N=1 superconformal symmetry that contains in addition to the Virasoro generator a superfield G c 2 [Lm,Ln]=(m n)Lm+n + m(m 1)δm, n − 12 − − m [L ,G ]= r G m r 2 − m+r c 2 1 Gr,Gs =2Lr+2 + r δr, s . {<latexit sha1_base64="hcUoRtFCLDZRWCrXBUXJAQo0Z/E=">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</latexit>
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