<<

Conformal Bootstrap: Old and New

Apratim Kaviraj ENS, Paris PhD: CHEP, Indian Institute of Science I thank….

• My PhD Supervisor: Prof. Aninda Sinha

• My Collaborators, and members of CHEP

• My Family Outline of the talk

• Importance of Conformal Theories • Basics of CFT • Conformal Bootstrap • Polyakov/Mellin Bootstrap RG flow in

AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEeqx6MVjC/YD2lA220m7drMJuxuhhP4CLx4U8epP8ua/cdvmoK0vLDy8M8POvEEiuDau++0UNja3tneKu6W9/YPDo/LxSVvHqWLYYrGIVTegGgWX2DLcCOwmCmkUCOwEk7t5vfOESvNYPphpgn5ER5KHnFFjreZoUK64VXchsg5eDhXI1RiUv/rDmKURSsME1brnuYnxM6oMZwJnpX6qMaFsQkfYsyhphNrPFovOyIV1hiSMlX3SkIX7eyKjkdbTKLCdETVjvVqbm//VeqkJb/yMyyQ1KNnyozAVxMRkfjUZcoXMiKkFyhS3uxI2pooyY7Mp2RC81ZPXoX1V9Sw3ryv12zyOIpzBOVyCBzWowz00oAUMEJ7hFd6cR+fFeXc+lq0FJ585hT9yPn8AzHeM6w== g

mAAAB6HicbZBNSwMxEIZn/az1q+rRS7AInsquCHosevHYgv2AdinZdNrGJtklyQpl6S/w4kERr/4kb/4b03YP2vpC4OGdGTLzRongxvr+t7e2vrG5tV3YKe7u7R8clo6OmyZONcMGi0Ws2xE1KLjChuVWYDvRSGUksBWN72b11hNqw2P1YCcJhpIOFR9wRq2z6rJXKvsVfy6yCkEOZchV65W+uv2YpRKVZYIa0wn8xIYZ1ZYzgdNiNzWYUDamQ+w4VFSiCbP5olNy7pw+GcTaPWXJ3P09kVFpzERGrlNSOzLLtZn5X62T2sFNmHGVpBYVW3w0SAWxMZldTfpcI7Ni4oAyzd2uhI2opsy6bIouhGD55FVoXlYCx/WrcvU2j6MAp3AGFxDANVThHmrQAAYIz/AKb96j9+K9ex+L1jUvnzmBP/I+fwDVj4zx

Fixed point ( function vanishes)

AAAB7HicbZBNS8NAEIYn9avWr6pHL8EieCqJCHosevFYwdRCG8pmO2mXbjZhdyKU0t/gxYMiXv1B3vw3btsctPWFhYd3ZtiZN8qkMOR5305pbX1jc6u8XdnZ3ds/qB4etUyaa44BT2Wq2xEzKIXCgARJbGcaWRJJfIxGt7P64xNqI1L1QOMMw4QNlIgFZ2StoBshsV615tW9udxV8AuoQaFmr/rV7ac8T1ARl8yYju9lFE6YJsElTivd3GDG+IgNsGNRsQRNOJkvO3XPrNN341Tbp8idu78nJiwxZpxEtjNhNDTLtZn5X62TU3wdToTKckLFFx/FuXQpdWeXu32hkZMcW2BcC7ury4dMM042n4oNwV8+eRVaF3Xf8v1lrXFTxFGGEziFc/DhChpwB00IgIOAZ3iFN0c5L86787FoLTnFzDH8kfP5A8PYjqQ=

• A Conformal Field Theory (CFT) is a QFT with enhanced symmetries

µ µ µ

• Translation (P) xAAACAHicbZDLSsNAFIZP6q3WW9SFCzeDRRCEkoigy6IblxXsBZpYJtNJO3QyCTMTsYRufBU3LhRx62O4822ctFlo64FhPv5zDjP/HyScKe0431ZpaXllda28XtnY3NresXf3WipOJaFNEvNYdgKsKGeCNjXTnHYSSXEUcNoORtd5v/1ApWKxuNPjhPoRHggWMoK1kXr2weO9F6XI0zGa0SnC+d2zq07NmRZaBLeAKhTV6NlfXj8maUSFJhwr1XWdRPsZlpoRTicVL1U0wWSEB7RrUOCIKj+bGpigY6P0URhLc4RGU/X3RoYjpcZRYCYjrIdqvpeL//W6qQ4v/YyJJNVUkNlDYcqRsZungfpMUqL52AAmkpm/IjLEEhNtMquYENx5y4vQOqu5hm/Pq/WrIo4yHMIRnIALF1CHG2hAEwhM4Ble4c16sl6sd+tjNlqyip19+FPW5w9pA5Wg x + a ! Lorentz (M) xµ ⇤µx⌫ • AAACCXicbVC7TsMwFHXKq5RXgJHFokJiqhKEBGMFCwNDkehDakLkOE5r1XEiPxBV1JWFX2FhACFW/oCNv8FpM0DLlWwdn3OPru8JM0alcpxvq7K0vLK6Vl2vbWxube/Yu3sdmWqBSRunLBW9EEnCKCdtRRUjvUwQlISMdMPRZaF374mQNOW3apwRP0EDTmOKkTJUYMOHOy/R0FMp9K6NLULFO/C4LhRzB3bdaTjTgovALUEdlNUK7C8vSrFOCFeYISn7rpMpP0dCUczIpOZpSTKER2hA+gZylBDp59NNJvDIMBGMU2EOV3DK/nbkKJFynISmM0FqKOe1gvxP62sVn/s55ZlWhOPZoFgzaPYuYoERFQQrNjYAYUHNXyEeIoGwMuHVTAju/MqLoHPScA2+Oa03L8o4quAAHIJj4IIz0ARXoAXaAINH8AxewZv1ZL1Y79bHrLVilZ598Keszx+riZmq ! ⌫ µ µ • Dilatation (Scaling) (D) xAAACAXicbVDLSsNAFL2pr1pfUTeCm8EiuCqJCLosunFZwT6giWUymbRDZ5IwMxFLqBt/xY0LRdz6F+78G6dtFtp6YOBwzj3cuSdIOVPacb6t0tLyyupaeb2ysbm1vWPv7rVUkklCmyThiewEWFHOYtrUTHPaSSXFIuC0HQyvJn77nkrFkvhWj1LqC9yPWcQI1kbq2QcPd57IkKcT5HETCzGaKT276tScKdAicQtShQKNnv3lhQnJBI014Vipruuk2s+x1IxwOq54maIpJkPcp11DYyyo8vPpBWN0bJQQRYk0L9Zoqv5O5FgoNRKBmRRYD9S8NxH/87qZji78nMVppmlMZouijCNz76QOFDJJieYjQzCRzPwVkQGWmGhTWsWU4M6fvEhapzXX8Juzav2yqKMMh3AEJ+DCOdThGhrQBAKP8Ayv8GY9WS/Wu/UxGy1ZRWYf/sD6/AF6+ZY/ x ! µ µ x • Special Conformal ransformations (K) x 2 AAACBXicbVC7TsMwFL3hWcorwAiDRYXEVCUVEowVLIxFog+pCZXjOq1VO4lsB1FFWVj4FRYGEGLlH9j4G9y0A7RcyfLxOffq+pwg4Uxpx/m2lpZXVtfWSxvlza3tnV17b7+l4lQS2iQxj2UnwIpyFtGmZprTTiIpFgGn7WB0NdHb91QqFke3epxQX+BBxEJGsDZUzz56uPNEijwdIy+UmGTFOzdXLUc9u+JUnaLQInBnoAKzavTsL68fk1TQSBOOleq6TqL9DEvNCKd52UsVTTAZ4QHtGhhhQZWfFS5ydGKYPgpjaU6kUcH+nsiwUGosAtMpsB6qeW1C/qd1Ux1e+BmLklTTiEwXhSlHxvMkEtRnkhLNxwZgIpn5KyJDbMLQJriyCcGdt7wIWrWqa/DNWaV+OYujBIdwDKfgwjnU4Roa0AQCj/AMr/BmPVkv1rv1MW1dsmYzB/CnrM8fX+aYeg== ! x

[D, Kµ]= iKµ AAAB+nicbZDLSsNAFIYnXmu9pbp0M1gEF1oSEXQjFHUhuKlgL5CGMJlO2qEzkzAzUUrso7hxoYhbn8Sdb+M0zUJbfxj4+M85nDN/mDCqtON8WwuLS8srq6W18vrG5ta2XdlpqTiVmDRxzGLZCZEijArS1FQz0kkkQTxkpB0Oryb19gORisbiXo8S4nPUFzSiGGljBXbFuz66Dbo89S+OaQ6BXXVqTi44D24BVVCoEdhf3V6MU06Exgwp5blOov0MSU0xI+NyN1UkQXiI+sQzKBAnys/y08fwwDg9GMXSPKFh7v6eyBBXasRD08mRHqjZ2sT8r+alOjr3MyqSVBOBp4uilEEdw0kOsEclwZqNDCAsqbkV4gGSCGuTVtmE4M5+eR5aJzXX8N1ptX5ZxFECe2AfHAIXnIE6uAEN0AQYPIJn8ArerCfrxXq3PqatC1Yxswv+yPr8AeuVkyA= [D, Pµ]=iPµ

AAAB+XicbZDLSsNAFIZPvNZ6i7p0M1gEF1ISEXQjFHXhsoK9QBrCZDpph04mYWZSKKFv4saFIm59E3e+jdM0C209MMzH/5/DnPnDlDOlHefbWlldW9/YrGxVt3d29/btg8O2SjJJaIskPJHdECvKmaAtzTSn3VRSHIecdsLR3czvjKlULBFPepJSP8YDwSJGsDZSYNve/Xkz6MWZf8OKO7BrTt0pCi2DW0INymoG9levn5AspkITjpXyXCfVfo6lZoTTabWXKZpiMsID6hkUOKbKz4vNp+jUKH0UJdIcoVGh/p7IcazUJA5NZ4z1UC16M/E/z8t0dO3nTKSZpoLMH4oyjnSCZjGgPpOUaD4xgIlkZldEhlhiok1YVROCu/jlZWhf1F3Dj5e1xm0ZRwWO4QTOwIUraMADNKEFBMbwDK/wZuXWi/VufcxbV6xy5gj+lPX5A41dkvM=

[Kµ,P⌫ ]=2i(⌘µ⌫ D Mµ⌫ ) AAACFHicbZDLSsNAFIYn9VbrLerSzWARKmpJiqAboagLQYQK9gJJCJPptB06mYSZiVBCH8KNr+LGhSJuXbjzbZy2QbT1h4GP/5zDmfMHMaNSWdaXkZubX1hcyi8XVlbX1jfMza2GjBKBSR1HLBKtAEnCKCd1RRUjrVgQFAaMNIP+xajevCdC0ojfqUFMvBB1Oe1QjJS2fPPAufbdMDms+S5PvLMKLblEIT/VnjaGl0c3P7zvm0WrbI0FZ8HOoAgy1Xzz021HOAkJV5ghKR3bipWXIqEoZmRYcBNJYoT7qEscjRyFRHrp+Kgh3NNOG3YioR9XcOz+nkhRKOUgDHRniFRPTtdG5n81J1GdUy+lPE4U4XiyqJMwqCI4Sgi2qSBYsYEGhAXVf4W4hwTCSudY0CHY0yfPQqNStjXfHher51kcebADdkEJ2OAEVMEVqIE6wOABPIEX8Go8Gs/Gm/E+ac0Z2cw2+CPj4xsU4p44sha1_base64="ck8pdC+ekZH4nUmSP+ZG7r8lEyk=">AAAB2XicbZDNSgMxFIXv1L86Vq1rN8EiuCozbnQpuHFZwbZCO5RM5k4bmskMyR2hDH0BF25EfC93vo3pz0JbDwQ+zknIvSculLQUBN9ebWd3b/+gfugfNfzjk9Nmo2fz0gjsilzl5jnmFpXU2CVJCp8LgzyLFfbj6f0i77+gsTLXTzQrMMr4WMtUCk7O6oyaraAdLMW2IVxDC9YaNb+GSS7KDDUJxa0dhEFBUcUNSaFw7g9LiwUXUz7GgUPNM7RRtRxzzi6dk7A0N+5oYkv394uKZ9bOstjdzDhN7Ga2MP/LBiWlt1EldVESarH6KC0Vo5wtdmaJNChIzRxwYaSblYkJN1yQa8Z3HYSbG29D77odOn4MoA7ncAFXEMIN3MEDdKALAhJ4hXdv4r15H6uuat66tDP4I+/zBzjGijg=sha1_base64="kiUXlwQ/DkE7FTbBb3+1/hRzF0g=">AAACCXicbZDLSsNAGIX/1FutVaNbN8EiVNSSdKMbQdCFIEIFe4EkhMl00g6dTMLMRCihD+HGV3HjQhFfwJ1v4/SCaOuBgY9zZvjnP2HKqFS2/WUUlpZXVteK66WN8ubWtrlTbskkE5g0ccIS0QmRJIxy0lRUMdJJBUFxyEg7HFyO8/YDEZIm/F4NU+LHqMdpRDFS2grMI/cm8OLsuBF4PPPP67TqEYWCXHvaGF2d3P7wYWBW7Jo9kbUIzgwqMFMjMD+9boKzmHCFGZLSdexU+TkSimJGRiUvkyRFeIB6xNXIUUykn0+WGlkH2ulaUSL04cqauL9f5CiWchiH+maMVF/OZ2Pzv8zNVHTm55SnmSIcTwdFGbNUYo0bsrpUEKzYUAPCguq/WriPBMJK91jSJTjzKy9Cq15zNN/ZUIQ92IcqOHAKF3ANDWgChkd4hld4M56MF+N9WlfBmPW2C39kfHwDSoGcuQ==sha1_base64="1uhNWWRAa0TLZRYyPUb0wpqGm1Y=">AAACFHicbZDLSsNAFIYn9VbrLerSTbAIFS8k3ehGKOpCEKGCvUASwmQ6aYfOTMLMRCihD+HGV3HjQhG3Ltz5Nk7bINr6w8DHf87hzPnDhBKpbPvLKMzNLywuFZdLK6tr6xvm5lZTxqlAuIFiGot2CCWmhOOGIoridiIwZCHFrbB/Maq37rGQJOZ3apBgn8EuJxFBUGkrMA/c68Bj6WE98Hjqn1VJxcMKBpn2tDG8PLr54f3ALNvH9ljWLDg5lEGuemB+ep0YpQxzhSiU0nXsRPkZFIogioclL5U4gagPu9jVyCHD0s/GRw2tPe10rCgW+nFljd3fExlkUg5YqDsZVD05XRuZ/9XcVEWnfkZ4kirM0WRRlFJLxdYoIatDBEaKDjRAJIj+q4V6UECkdI4lHYIzffIsNKvHjuZbu1w7z+Mogh2wCyrAASegBq5AHTQAAg/gCbyAV+PReDbejPdJa8HIZ7bBHxkf3xOinjQ= [Kµ,M⌫⇢]=2i(⌘µ⌫ K⇢ ⌘⇢⌫ Pµ) AAACKHicbZDLSsNAFIYn9VbrrerSTbAIFbQkRdCNWHQjiFDBXqAJYTKdtkNnJmEuQgl9HDe+ihsRRbr1SZy0WWjrgYGf7z+HM+cPY0qkcpyJlVtaXlldy68XNja3tneKu3tNGWmBcANFNBLtEEpMCccNRRTF7VhgyEKKW+HwJvVbT1hIEvFHNYqxz2Cfkx5BUBkUFK86d4HH9Ml9kHhce2IQjf3LKil7WEGDmDZ0bFqMcZoxI1NYT+eOg2LJqTjTsheFm4kSyKoeFN+9boQ0w1whCqXsuE6s/AQKRRDF44KnJY4hGsI+7hjJIcPST6aHju0jQ7p2LxLmcWVP6e+JBDIpRyw0nQyqgZz3Uvif19Gqd+EnhMdaYY5mi3qa2iqy09TsLhEYKToyAiJBzF9tNIACImWyLZgQ3PmTF0WzWnGNfjgr1a6zOPLgAByCMnDBOaiBW1AHDYDAM3gFH+DTerHerC9rMmvNWdnMPvhT1vcPI3qnQg== Conformal Field Theory

• A Conformal Field Theory (CFT) is a QFT with enhanced symmetries

• In Theory, 2d CFT-s describe the formulation

• AdS/CFT correspondence - Vast importance in modern literature Conformal Field Theory

AdS/CFT correspondence

(classical theory) (quantum theory)

Low energy QFT in Leading (large N) part in CFTAAAB7nicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRYLIjHCv2CNpTNZtou3WzC7kYooT/CiwdFvPp7vPlv3LY5aOsLCw/vzLAzb5AIro3rfjuFjc2t7Z3ibmlv/+DwqHx80tZxqhi2WCxi1Q2oRsEltgw3AruJQhoFAjvBpD6vd55QaR7Lppkm6Ed0JPmQM2qs1anfNwdZOBuUK27VXYisg5dDBXI1BuWvfhizNEJpmKBa9zw3MX5GleFM4KzUTzUmlE3oCHsWJY1Q+9li3Rm5sE5IhrGyTxqycH9PZDTSehoFtjOiZqxXa3Pzv1ovNcNbP+MySQ1KtvxomApiYjK/nYRcITNiaoEyxe2uhI2poszYhEo2BG/15HVoX1U9y4/XldpdHkcRzuAcLsGDG6jBAzSgBQwm8Ayv8OYkzovz7nwsWwtOPnMKf+R8/gADM49Y d AdSAAAB8HicbZDLSgMxFIbP1Futt6pLN8EiCEKZkYIuq25cVrQXaYeSyWTa0CQzJBmhDH0KNy4UcevjuPNtTNtZaOsPgY//nEPO+YOEM21c99sprKyurW8UN0tb2zu7e+X9g5aOU0Vok8Q8Vp0Aa8qZpE3DDKedRFEsAk7bwehmWm8/UaVZLB/MOKG+wAPJIkawsdbjVXjfz8Izb9IvV9yqOxNaBi+HCuRq9MtfvTAmqaDSEI617npuYvwMK8MIp5NSL9U0wWSEB7RrUWJBtZ/NFp6gE+uEKIqVfdKgmft7IsNC67EIbKfAZqgXa1Pzv1o3NdGlnzGZpIZKMv8oSjkyMZpej0KmKDF8bAETxeyuiAyxwsTYjEo2BG/x5GVonVc9y3e1Sv06j6MIR3AMp+DBBdThFhrQBAICnuEV3hzlvDjvzse8teDkM4fwR87nDwb5j+M= d+1 ⇔

A Witten diagram

CFT boundary

AdS bulk Conformal Field Theory

AdS/CFT correspondence

(classical theory) (quantum theory)

Low energy QFT in Leading (large N) part in CFTAAAB7nicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfRYLIjHCv2CNpTNZtou3WzC7kYooT/CiwdFvPp7vPlv3LY5aOsLCw/vzLAzb5AIro3rfjuFjc2t7Z3ibmlv/+DwqHx80tZxqhi2WCxi1Q2oRsEltgw3AruJQhoFAjvBpD6vd55QaR7Lppkm6Ed0JPmQM2qs1anfNwdZOBuUK27VXYisg5dDBXI1BuWvfhizNEJpmKBa9zw3MX5GleFM4KzUTzUmlE3oCHsWJY1Q+9li3Rm5sE5IhrGyTxqycH9PZDTSehoFtjOiZqxXa3Pzv1ovNcNbP+MySQ1KtvxomApiYjK/nYRcITNiaoEyxe2uhI2poszYhEo2BG/15HVoX1U9y4/XldpdHkcRzuAcLsGDG6jBAzSgBQwm8Ayv8OYkzovz7nwsWwtOPnMKf+R8/gADM49Y d AdSAAAB8HicbZDLSgMxFIbP1Futt6pLN8EiCEKZkYIuq25cVrQXaYeSyWTa0CQzJBmhDH0KNy4UcevjuPNtTNtZaOsPgY//nEPO+YOEM21c99sprKyurW8UN0tb2zu7e+X9g5aOU0Vok8Q8Vp0Aa8qZpE3DDKedRFEsAk7bwehmWm8/UaVZLB/MOKG+wAPJIkawsdbjVXjfz8Izb9IvV9yqOxNaBi+HCuRq9MtfvTAmqaDSEI617npuYvwMK8MIp5NSL9U0wWSEB7RrUWJBtZ/NFp6gE+uEKIqVfdKgmft7IsNC67EIbKfAZqgXa1Pzv1o3NdGlnzGZpIZKMv8oSjkyMZpej0KmKDF8bAETxeyuiAyxwsTYjEo2BG/x5GVonVc9y3e1Sv06j6MIR3AMp+DBBdThFhrQBAICnuEV3hzlvDjvzse8teDkM4fwR87nDwb5j+M= d+1 ⇔

A Witten diagram

CFT boundary

Duality is found in

large N CFTs ( Eg: AAAB9HicbVDLSgMxFL1TX7W+qi7dBIvgqsxIQTdC0Y0rqWAf0A4lk2ba0EwyJplCGfodblwo4taPceffmGlnoa0HAodz7uWenCDmTBvX/XYKa+sbm1vF7dLO7t7+QfnwqKVloghtEsml6gRYU84EbRpmOO3EiuIo4LQdjG8zvz2hSjMpHs00pn6Eh4KFjGBjJb8XYTMimKf3s+tav1xxq+4caJV4OalAjka//NUbSJJEVBjCsdZdz42Nn2JlGOF0VuolmsaYjPGQdi0VOKLaT+ehZ+jMKgMUSmWfMGiu/t5IcaT1NArsZBZSL3uZ+J/XTUx45adMxImhgiwOhQlHRqKsATRgihLDp5ZgopjNisgIK0yM7alkS/CWv7xKWhdVz/KHWqV+k9dRhBM4hXPw4BLqcAcNaAKBJ3iGV3hzJs6L8+58LEYLTr5zDH/gfP4AgouR6Q== =4 SYM ) N AdS bulk Conformal Field Theory

• A Conformal Field Theory (CFT) is a QFT with enhanced symmetries

• In , CFT-s describe the worldsheet formulation

• AdS/CFT correspondence - Vast importance in modern literature

• CFTs arise in the study of quantum critical points Conformal Field Theory

In Critical phenomena

• Many systems show conformal symmetry (CFT) at critical points of continuous phase transitions.

• Universality: At critical points, systems with same symmetries are equivalent (in the IR) irrespective of Lagrangian. Conformal Field Theory

In Critical phenomena

• Many systems show conformal symmetry (CFT) at critical points of continuous phase transitions.

• Universality: At critical points, systems with same symmetries are equivalent (in the IR) irrespective of Lagrangian.

4 Model in 3d Water critical point 3d Ising critical point AAAB7XicbZDLSgMxFIbP1Futt6pLN8EiuCozUtBl0Y3LCvYC7VgyaaaNzSQhyQhl6Du4caGIW9/HnW9j2s5CW38IfPznHHLOHynOjPX9b6+wtr6xuVXcLu3s7u0flA+PWkammtAmkVzqToQN5UzQpmWW047SFCcRp+1ofDOrt5+oNkyKeztRNEzwULCYEWyd1eqpEXuo9csVv+rPhVYhyKECuRr98ldvIEmaUGEJx8Z0A1/ZMMPaMsLptNRLDVWYjPGQdh0KnFATZvNtp+jMOQMUS+2esGju/p7IcGLMJIlcZ4LtyCzXZuZ/tW5q46swY0Kllgqy+ChOObISzU5HA6YpsXziABPN3K6IjLDGxLqASi6EYPnkVWhdVAPHd7VK/TqPowgncArnEMAl1OEWGtAEAo/wDK/w5knvxXv3PhatBS+fOYY/8j5/AD5PjuU= at fixed point

Critical point

⟺ ⟺

Same IR Physics ➟ Same critical exponents Conformal Field Theory

In Critical phenomena

• RG approach in critical phenomena uses this principle.

• Critical exponents are related to correlation functions.

4 Model in 3d Water critical point 3d Ising critical point AAAB7XicbZDLSgMxFIbP1Futt6pLN8EiuCozUtBl0Y3LCvYC7VgyaaaNzSQhyQhl6Du4caGIW9/HnW9j2s5CW38IfPznHHLOHynOjPX9b6+wtr6xuVXcLu3s7u0flA+PWkammtAmkVzqToQN5UzQpmWW047SFCcRp+1ofDOrt5+oNkyKeztRNEzwULCYEWyd1eqpEXuo9csVv+rPhVYhyKECuRr98ldvIEmaUGEJx8Z0A1/ZMMPaMsLptNRLDVWYjPGQdh0KnFATZvNtp+jMOQMUS+2esGju/p7IcGLMJIlcZ4LtyCzXZuZ/tW5q46swY0Kllgqy+ChOObISzU5HA6YpsXziABPN3K6IjLDGxLqASi6EYPnkVWhdVAPHd7VK/TqPowgncArnEMAl1OEWGtAEAo/wDK/w5knvxXv3PhatBS+fOYY/8j5/AD5PjuU= at fixed point

Critical point

⟺ ⟺

Same IR Physics ➟ Same critical exponents Conformal Field Theory

In Critical phenomena

• RG approach in critical phenomena uses this principle.

• Critical exponents are related to correlation functions.

2 ⌘ d Example: (x)(y) x y AAACIXicbVDLSsNAFJ34rPVVdelmsAi6qCQi6FJ047KCVaGJZTK5aQdnJmFmIoa0v+LGX3HjQpHuxJ9xGrPwdWC4h3Pu5c49YcqZNq777kxNz8zOzdcW6otLyyurjbX1S51kikKHJjxR1yHRwJmEjmGGw3WqgIiQw1V4ezrxr+5AaZbIC5OnEAjSlyxmlBgr9RpHPieyzwH76YDt3O+WJd/1VaViXzNhCx7et/LhTbHf8sGQVjTqNZrunlsC/yVeRZqoQrvXGPtRQjMB0lBOtO56bmqCgijDKIdR3c80pITekj50LZVEgA6K8sIR3rZKhONE2ScNLtXvEwURWucitJ2CmIH+7U3E/7xuZuKjoGAyzQxI+rUozjg2CZ7EhSOmgBqeW0KoYvavmA6IItTYUOs2BO/3yX/J5f6eZ/n5QfP4pIqjhjbRFtpBHjpEx+gMtVEHUfSAntALenUenWfnzRl/tU451cwG+gHn4xNT8KJb : order parameter

h i⇠| | AAAB63icbZDNSgMxFIVv6l+tf1WXboJFcFVmRNBl0Y3LCrYW2qFk0kwnNMkMSUYoQ1/BjQtF3PpC7nwbM+0stPVA4OPce8m9J0wFN9bzvlFlbX1jc6u6XdvZ3ds/qB8edU2Saco6NBGJ7oXEMMEV61huBeulmhEZCvYYTm6L+uMT04Yn6sFOUxZIMlY84pTYwhqkMR/WG17Tmwuvgl9CA0q1h/WvwSihmWTKUkGM6fteaoOcaMupYLPaIDMsJXRCxqzvUBHJTJDPd53hM+eMcJRo95TFc/f3RE6kMVMZuk5JbGyWa4X5X62f2eg6yLlKM8sUXXwUZQLbBBeH4xHXjFoxdUCo5m5XTGOiCbUunpoLwV8+eRW6F03f8f1lo3VTxlGFEziFc/DhClpwB23oAIUYnuEV3pBEL+gdfSxaK6icOYY/Qp8/E4qOPw== Conformal Field Theory

In Critical phenomena

• RG approach in critical phenomena uses this principle.

• Critical exponents are related to correlation functions.

2 ⌘ d Example: (x)(y) x y AAACIXicbVDLSsNAFJ34rPVVdelmsAi6qCQi6FJ047KCVaGJZTK5aQdnJmFmIoa0v+LGX3HjQpHuxJ9xGrPwdWC4h3Pu5c49YcqZNq777kxNz8zOzdcW6otLyyurjbX1S51kikKHJjxR1yHRwJmEjmGGw3WqgIiQw1V4ezrxr+5AaZbIC5OnEAjSlyxmlBgr9RpHPieyzwH76YDt3O+WJd/1VaViXzNhCx7et/LhTbHf8sGQVjTqNZrunlsC/yVeRZqoQrvXGPtRQjMB0lBOtO56bmqCgijDKIdR3c80pITekj50LZVEgA6K8sIR3rZKhONE2ScNLtXvEwURWucitJ2CmIH+7U3E/7xuZuKjoGAyzQxI+rUozjg2CZ7EhSOmgBqeW0KoYvavmA6IItTYUOs2BO/3yX/J5f6eZ/n5QfP4pIqjhjbRFtpBHjpEx+gMtVEHUfSAntALenUenWfnzRl/tU451cwG+gHn4xNT8KJb : order parameter

h i⇠| | AAAB63icbZDNSgMxFIVv6l+tf1WXboJFcFVmRNBl0Y3LCrYW2qFk0kwnNMkMSUYoQ1/BjQtF3PpC7nwbM+0stPVA4OPce8m9J0wFN9bzvlFlbX1jc6u6XdvZ3ds/qB8edU2Saco6NBGJ7oXEMMEV61huBeulmhEZCvYYTm6L+uMT04Yn6sFOUxZIMlY84pTYwhqkMR/WG17Tmwuvgl9CA0q1h/WvwSihmWTKUkGM6fteaoOcaMupYLPaIDMsJXRCxqzvUBHJTJDPd53hM+eMcJRo95TFc/f3RE6kMVMZuk5JbGyWa4X5X62f2eg6yLlKM8sUXXwUZQLbBBeH4xHXjFoxdUCo5m5XTGOiCbUunpoLwV8+eRW6F03f8f1lo3VTxlGFEziFc/DhClpwB23oAIUYnuEV3pBEL+gdfSxaK6icOYY/Qp8/E4qOPw==

Shortcut: Use 4 To get this

AAAB9XicbVDLSgMxFL1TX7W+qi7dBIvgqsyIoMuiG5cV7AM605LJZNrQJDMkGaUM/Q83LhRx67+4829M21lo64HA4ZxzuTcnTDnTxnW/ndLa+sbmVnm7srO7t39QPTxq6yRThLZIwhPVDbGmnEnaMsxw2k0VxSLktBOOb2d+55EqzRL5YCYpDQQeShYzgo2V+j630QgjPx2x/uWgWnPr7hxolXgFqUGB5qD65UcJyQSVhnCsdc9zUxPkWBlGOJ1W/EzTFJMxHtKepRILqoN8fvUUnVklQnGi7JMGzdXfEzkWWk9EaJMCm5Fe9mbif14vM/F1kDOZZoZKslgUZxyZBM0qQBFTlBg+sQQTxeytiIywwsTYoiq2BG/5y6ukfVH3LL+/rDVuijrKcAKncA4eXEED7qAJLSCg4Ble4c15cl6cd+djES05xcwx/IHz+QPZWJIS theory for this for Ising/Water CFT Conformal Field Theory

In Critical phenomena

2

AAAB7nicbVDLSsNAFL2pr1pfVZduBovgqiQi1GXRjcsK9gFtKDeTSTt0MgkzE6GEfoQbF4q49Xvc+TdO2yy09cDA4ZxzmXtPkAqujet+O6WNza3tnfJuZW//4PCoenzS0UmmKGvTRCSqF6BmgkvWNtwI1ksVwzgQrBtM7uZ+94kpzRP5aKYp82McSR5xisZK3YGw0RCH1Zpbdxcg68QrSA0KtIbVr0GY0Cxm0lCBWvc9NzV+jspwKtisMsg0S5FOcMT6lkqMmfbzxbozcmGVkESJsk8aslB/T+QYaz2NA5uM0Yz1qjcX//P6mYlu/JzLNDNM0uVHUSaIScj8dhJyxagRU0uQKm53JXSMCqmxDVVsCd7qyeukc1X3LH+4rjVvizrKcAbncAkeNKAJ99CCNlCYwDO8wpuTOi/Ou/OxjJacYuYU/sD5/AE8x499 + + AAAB8HicbVDLSgMxFL1TX7W+qi7dBIvgqswUQZdFNy4r2Ie0Y8lkMm1okhmSjFCGfoUbF4q49XPc+Tdm2llo64HA4Zxzyb0nSDjTxnW/ndLa+sbmVnm7srO7t39QPTzq6DhVhLZJzGPVC7CmnEnaNsxw2ksUxSLgtBtMbnK/+0SVZrG8N9OE+gKPJIsYwcZKDwNuoyF+bAyrNbfuzoFWiVeQGhRoDatfgzAmqaDSEI617ntuYvwMK8MIp7PKINU0wWSCR7RvqcSCaj+bLzxDZ1YJURQr+6RBc/X3RIaF1lMR2KTAZqyXvVz8z+unJrryMyaT1FBJFh9FKUcmRvn1KGSKEsOnlmCimN0VkTFWmBjbUcWW4C2fvEo6jbpn+d1FrXld1FGGEziFc/DgEppwCy1oAwEBz/AKb45yXpx352MRLTnFzDH8gfP5A2bQkCE= + ….

2 ⌘ d Example: (x)(y) x y AAACIXicbVDLSsNAFJ34rPVVdelmsAi6qCQi6FJ047KCVaGJZTK5aQdnJmFmIoa0v+LGX3HjQpHuxJ9xGrPwdWC4h3Pu5c49YcqZNq777kxNz8zOzdcW6otLyyurjbX1S51kikKHJjxR1yHRwJmEjmGGw3WqgIiQw1V4ezrxr+5AaZbIC5OnEAjSlyxmlBgr9RpHPieyzwH76YDt3O+WJd/1VaViXzNhCx7et/LhTbHf8sGQVjTqNZrunlsC/yVeRZqoQrvXGPtRQjMB0lBOtO56bmqCgijDKIdR3c80pITekj50LZVEgA6K8sIR3rZKhONE2ScNLtXvEwURWucitJ2CmIH+7U3E/7xuZuKjoGAyzQxI+rUozjg2CZ7EhSOmgBqeW0KoYvavmA6IItTYUOs2BO/3yX/J5f6eZ/n5QfP4pIqjhjbRFtpBHjpEx+gMtVEHUfSAntALenUenWfnzRl/tU451cwG+gHn4xNT8KJb : order parameter

h i⇠| | AAAB63icbZDNSgMxFIVv6l+tf1WXboJFcFVmRNBl0Y3LCrYW2qFk0kwnNMkMSUYoQ1/BjQtF3PpC7nwbM+0stPVA4OPce8m9J0wFN9bzvlFlbX1jc6u6XdvZ3ds/qB8edU2Saco6NBGJ7oXEMMEV61huBeulmhEZCvYYTm6L+uMT04Yn6sFOUxZIMlY84pTYwhqkMR/WG17Tmwuvgl9CA0q1h/WvwSihmWTKUkGM6fteaoOcaMupYLPaIDMsJXRCxqzvUBHJTJDPd53hM+eMcJRo95TFc/f3RE6kMVMZuk5JbGyWa4X5X62f2eg6yLlKM8sUXXwUZQLbBBeH4xHXjFoxdUCo5m5XTGOiCbUunpoLwV8+eRW6F03f8f1lo3VTxlGFEziFc/DhClpwB23oAIUYnuEV3pBEL+gdfSxaK6icOYY/Qp8/E4qOPw==

Shortcut: Use 4 To get this

AAAB9XicbVDLSgMxFL1TX7W+qi7dBIvgqsyIoMuiG5cV7AM605LJZNrQJDMkGaUM/Q83LhRx67+4829M21lo64HA4ZxzuTcnTDnTxnW/ndLa+sbmVnm7srO7t39QPTxq6yRThLZIwhPVDbGmnEnaMsxw2k0VxSLktBOOb2d+55EqzRL5YCYpDQQeShYzgo2V+j630QgjPx2x/uWgWnPr7hxolXgFqUGB5qD65UcJyQSVhnCsdc9zUxPkWBlGOJ1W/EzTFJMxHtKepRILqoN8fvUUnVklQnGi7JMGzdXfEzkWWk9EaJMCm5Fe9mbif14vM/F1kDOZZoZKslgUZxyZBM0qQBFTlBg+sQQTxeytiIywwsTYoiq2BG/5y6ukfVH3LL+/rDVuijrKcAKncA4eXEED7qAJLSCg4Ble4c15cl6cd+djES05xcwx/IHz+QPZWJIS theory for this for Ising/Water CFT Conformal Field Theory

In Critical phenomena

2

AAAB7nicbVDLSsNAFL2pr1pfVZduBovgqiQi1GXRjcsK9gFtKDeTSTt0MgkzE6GEfoQbF4q49Xvc+TdO2yy09cDA4ZxzmXtPkAqujet+O6WNza3tnfJuZW//4PCoenzS0UmmKGvTRCSqF6BmgkvWNtwI1ksVwzgQrBtM7uZ+94kpzRP5aKYp82McSR5xisZK3YGw0RCH1Zpbdxcg68QrSA0KtIbVr0GY0Cxm0lCBWvc9NzV+jspwKtisMsg0S5FOcMT6lkqMmfbzxbozcmGVkESJsk8aslB/T+QYaz2NA5uM0Yz1qjcX//P6mYlu/JzLNDNM0uVHUSaIScj8dhJyxagRU0uQKm53JXSMCqmxDVVsCd7qyeukc1X3LH+4rjVvizrKcAbncAkeNKAJ99CCNlCYwDO8wpuTOi/Ou/OxjJacYuYU/sD5/AE8x499 + + AAAB8HicbVDLSgMxFL1TX7W+qi7dBIvgqswUQZdFNy4r2Ie0Y8lkMm1okhmSjFCGfoUbF4q49XPc+Tdm2llo64HA4Zxzyb0nSDjTxnW/ndLa+sbmVnm7srO7t39QPTzq6DhVhLZJzGPVC7CmnEnaNsxw2ksUxSLgtBtMbnK/+0SVZrG8N9OE+gKPJIsYwcZKDwNuoyF+bAyrNbfuzoFWiVeQGhRoDatfgzAmqaDSEI617ntuYvwMK8MIp7PKINU0wWSCR7RvqcSCaj+bLzxDZ1YJURQr+6RBc/X3RIaF1lMR2KTAZqyXvVz8z+unJrryMyaT1FBJFh9FKUcmRvn1KGSKEsOnlmCimN0VkTFWmBjbUcWW4C2fvEo6jbpn+d1FrXld1FGGEziFc/DgEppwCy1oAwEBz/AKb45yXpx352MRLTnFzDH8gfP5A2bQkCE= + ….

16⇡2

In dAAAB83icbZDLSgMxFIbP1Futt6pLN4NFcGOZkYJuhKIblxXsBTpDyWQybWgmCUlGKENfw40LRdz6Mu58G9N2Ftr6Q+DjP+dwTv5IMqqN5307pbX1jc2t8nZlZ3dv/6B6eNTRIlOYtLFgQvUipAmjnLQNNYz0pCIojRjpRuO7Wb37RJSmgj+aiSRhioacJhQjY60gvmlcBERqygQfVGte3ZvLXQW/gBoUag2qX0EscJYSbjBDWvd9T5owR8pQzMi0EmSaSITHaEj6FjlKiQ7z+c1T98w6sZsIZR837tz9PZGjVOtJGtnOFJmRXq7NzP9q/cwk12FOucwM4XixKMmYa4Q7C8CNqSLYsIkFhBW1t7p4hBTCxsZUsSH4y19ehc5l3bf80Kg1b4s4ynACp3AOPlxBE+6hBW3AIOEZXuHNyZwX5935WLSWnGLmGP7I+fwBcbeRSg== =4 ✏ dimension = ✏ (for CFT / fixed point)

AAACCnicbVC7TsMwFHV4lvIqMLIYKiSmKikIWJAqWBiLRB9SEyrHcVqrjmPZDlIVZWbhV1gYQIiVL2Djb3DaDNByJEtH55yr63t8wajStv1tLSwuLa+sltbK6xubW9uVnd22ihOJSQvHLJZdHynCKCctTTUjXSEJinxGOv7oOvc7D0QqGvM7PRbEi9CA05BipI3Urxy4zIQDdOmGEuHUOXMFva9n6UnmEqEoyzNVu2ZPAOeJU5AqKNDsV77cIMZJRLjGDCnVc2yhvRRJTTEjWdlNFBEIj9CA9AzlKCLKSyenZPDIKAEMY2ke13Ci/p5IUaTUOPJNMkJ6qGa9XPzP6yU6vPBSykWiCcfTRWHCoI5h3gsMqCRYs7EhCEtq/grxEJlOtGmvbEpwZk+eJ+16zTH89rTauCrqKIF9cAiOgQPOQQPcgCZoAQwewTN4BW/Wk/VivVsf0+iCVczsgT+wPn8AJiuajA== 3 Conformal Field Theory

In Critical phenomena

2

AAAB7nicbVDLSsNAFL2pr1pfVZduBovgqiQi1GXRjcsK9gFtKDeTSTt0MgkzE6GEfoQbF4q49Xvc+TdO2yy09cDA4ZxzmXtPkAqujet+O6WNza3tnfJuZW//4PCoenzS0UmmKGvTRCSqF6BmgkvWNtwI1ksVwzgQrBtM7uZ+94kpzRP5aKYp82McSR5xisZK3YGw0RCH1Zpbdxcg68QrSA0KtIbVr0GY0Cxm0lCBWvc9NzV+jspwKtisMsg0S5FOcMT6lkqMmfbzxbozcmGVkESJsk8aslB/T+QYaz2NA5uM0Yz1qjcX//P6mYlu/JzLNDNM0uVHUSaIScj8dhJyxagRU0uQKm53JXSMCqmxDVVsCd7qyeukc1X3LH+4rjVvizrKcAbncAkeNKAJ99CCNlCYwDO8wpuTOi/Ou/OxjJacYuYU/sD5/AE8x499 + + AAAB8HicbVDLSgMxFL1TX7W+qi7dBIvgqswUQZdFNy4r2Ie0Y8lkMm1okhmSjFCGfoUbF4q49XPc+Tdm2llo64HA4Zxzyb0nSDjTxnW/ndLa+sbmVnm7srO7t39QPTzq6DhVhLZJzGPVC7CmnEnaNsxw2ksUxSLgtBtMbnK/+0SVZrG8N9OE+gKPJIsYwcZKDwNuoyF+bAyrNbfuzoFWiVeQGhRoDatfgzAmqaDSEI617ntuYvwMK8MIp7PKINU0wWSCR7RvqcSCaj+bLzxDZ1YJURQr+6RBc/X3RIaF1lMR2KTAZqyXvVz8z+unJrryMyaT1FBJFh9FKUcmRvn1KGSKEsOnlmCimN0VkTFWmBjbUcWW4C2fvEo6jbpn+d1FrXld1FGGEziFc/DgEppwCy1oAwEBz/AKb45yXpx352MRLTnFzDH8gfP5A2bQkCE= + ….

16⇡2

In dAAAB83icbZDLSgMxFIbP1Futt6pLN4NFcGOZkYJuhKIblxXsBTpDyWQybWgmCUlGKENfw40LRdz6Mu58G9N2Ftr6Q+DjP+dwTv5IMqqN5307pbX1jc2t8nZlZ3dv/6B6eNTRIlOYtLFgQvUipAmjnLQNNYz0pCIojRjpRuO7Wb37RJSmgj+aiSRhioacJhQjY60gvmlcBERqygQfVGte3ZvLXQW/gBoUag2qX0EscJYSbjBDWvd9T5owR8pQzMi0EmSaSITHaEj6FjlKiQ7z+c1T98w6sZsIZR837tz9PZGjVOtJGtnOFJmRXq7NzP9q/cwk12FOucwM4XixKMmYa4Q7C8CNqSLYsIkFhBW1t7p4hBTCxsZUsSH4y19ehc5l3bf80Kg1b4s4ynACp3AOPlxBE+6hBW3AIOEZXuHNyZwX5935WLSWnGLmGP7I+fwBcbeRSg== =4 ✏ dimension = ✏ (for CFT / fixed point)

AAACCnicbVC7TsMwFHV4lvIqMLIYKiSmKikIWJAqWBiLRB9SEyrHcVqrjmPZDlIVZWbhV1gYQIiVL2Djb3DaDNByJEtH55yr63t8wajStv1tLSwuLa+sltbK6xubW9uVnd22ihOJSQvHLJZdHynCKCctTTUjXSEJinxGOv7oOvc7D0QqGvM7PRbEi9CA05BipI3Urxy4zIQDdOmGEuHUOXMFva9n6UnmEqEoyzNVu2ZPAOeJU5AqKNDsV77cIMZJRLjGDCnVc2yhvRRJTTEjWdlNFBEIj9CA9AzlKCLKSyenZPDIKAEMY2ke13Ci/p5IUaTUOPJNMkJ6qGa9XPzP6yU6vPBSykWiCcfTRWHCoI5h3gsMqCRYs7EhCEtq/grxEJlOtGmvbEpwZk+eJ+16zTH89rTauCrqKIF9cAiOgQPOQQPcgCZoAQwewTN4BW/Wk/VivVsf0+iCVczsgT+wPn8AJiuajA== 3

✏2 109✏3

⇒ ⌘ = + + Put ✏AAAB83icbZDLSgMxFIbP1Futt6pLN8EiuCozIuhGKLpxWcFeoDOUTJppQzNJSDJCGfoablwo4taXcefbmLaz0NYfAh//OYdz8seKM2N9/9srra1vbG6Vtys7u3v7B9XDo7aRmSa0RSSXuhtjQzkTtGWZ5bSrNMVpzGknHt/N6p0nqg2T4tFOFI1SPBQsYQRbZ4UhVYZxKdANCvrVml/350KrEBRQg0LNfvUrHEiSpVRYwrExvcBXNsqxtoxwOq2EmaEKkzEe0p5DgVNqonx+8xSdOWeAEqndExbN3d8TOU6NmaSx60yxHZnl2sz8r9bLbHId5UyozFJBFouSjCMr0SwANGCaEssnDjDRzN2KyAhrTKyLqeJCCJa/vArti3rg+OGy1rgt4ijDCZzCOQRwBQ24hya0gICCZ3iFNy/zXrx372PRWvKKmWP4I+/zB/bAkPY= =1 for 3d !! AAACJnicbZDLSgMxFIYzXmu9VV26CRZBEGSmragLoejGpYJVoVNLJj2joZlkSM4IZejTuPFV3LhQRNz5KKYXxNuBwM//ncPJ+aNUCou+/+5NTE5Nz8wW5orzC4tLy6WV1QurM8OhwbXU5ipiFqRQ0ECBEq5SAyyJJFxG3eMBv7wDY4VW59hLoZWwGyViwRk6q106DAHZYRgbxvMQUiukVteVfr5b62+P3MA/oF+k6sh+teIY72i07VLZ3/GHRf+KYCzKZFyn7dJz2NE8S0Ahl8zaZuCn2MqZQcEl9IthZiFlvMtuoOmkYgnYVj48s083ndOhsTbuKaRD9/tEzhJre0nkOhOGt/Y3G5j/sWaG8X4rFyrNEBQfLYozSVHTQWa0IwxwlD0nGDfC/ZXyW+bCQZds0YUQ/D75r7io7AROn9XK9aNxHAWyTjbIFgnIHqmTE3JKGoSTe/JInsmL9+A9ea/e26h1whvPrJEf5X18ArJ4pUE= 54 5832 ··· Conformal Field Theory

In Critical phenomena

2

AAAB7nicbVDLSsNAFL2pr1pfVZduBovgqiQi1GXRjcsK9gFtKDeTSTt0MgkzE6GEfoQbF4q49Xvc+TdO2yy09cDA4ZxzmXtPkAqujet+O6WNza3tnfJuZW//4PCoenzS0UmmKGvTRCSqF6BmgkvWNtwI1ksVwzgQrBtM7uZ+94kpzRP5aKYp82McSR5xisZK3YGw0RCH1Zpbdxcg68QrSA0KtIbVr0GY0Cxm0lCBWvc9NzV+jspwKtisMsg0S5FOcMT6lkqMmfbzxbozcmGVkESJsk8aslB/T+QYaz2NA5uM0Yz1qjcX//P6mYlu/JzLNDNM0uVHUSaIScj8dhJyxagRU0uQKm53JXSMCqmxDVVsCd7qyeukc1X3LH+4rjVvizrKcAbncAkeNKAJ99CCNlCYwDO8wpuTOi/Ou/OxjJacYuYU/sD5/AE8x499 + + AAAB8HicbVDLSgMxFL1TX7W+qi7dBIvgqswUQZdFNy4r2Ie0Y8lkMm1okhmSjFCGfoUbF4q49XPc+Tdm2llo64HA4Zxzyb0nSDjTxnW/ndLa+sbmVnm7srO7t39QPTzq6DhVhLZJzGPVC7CmnEnaNsxw2ksUxSLgtBtMbnK/+0SVZrG8N9OE+gKPJIsYwcZKDwNuoyF+bAyrNbfuzoFWiVeQGhRoDatfgzAmqaDSEI617ntuYvwMK8MIp7PKINU0wWSCR7RvqcSCaj+bLzxDZ1YJURQr+6RBc/X3RIaF1lMR2KTAZqyXvVz8z+unJrryMyaT1FBJFh9FKUcmRvn1KGSKEsOnlmCimN0VkTFWmBjbUcWW4C2fvEo6jbpn+d1FrXld1FGGEziFc/DgEppwCy1oAwEBz/AKb45yXpx352MRLTnFzDH8gfP5A2bQkCE= + ….

16⇡2

In dAAAB83icbZDLSgMxFIbP1Futt6pLN4NFcGOZkYJuhKIblxXsBTpDyWQybWgmCUlGKENfw40LRdz6Mu58G9N2Ftr6Q+DjP+dwTv5IMqqN5307pbX1jc2t8nZlZ3dv/6B6eNTRIlOYtLFgQvUipAmjnLQNNYz0pCIojRjpRuO7Wb37RJSmgj+aiSRhioacJhQjY60gvmlcBERqygQfVGte3ZvLXQW/gBoUag2qX0EscJYSbjBDWvd9T5owR8pQzMi0EmSaSITHaEj6FjlKiQ7z+c1T98w6sZsIZR837tz9PZGjVOtJGtnOFJmRXq7NzP9q/cwk12FOucwM4XixKMmYa4Q7C8CNqSLYsIkFhBW1t7p4hBTCxsZUsSH4y19ehc5l3bf80Kg1b4s4ynACp3AOPlxBE+6hBW3AIOEZXuHNyZwX5935WLSWnGLmGP7I+fwBcbeRSg== =4 ✏ dimension = ✏ (for CFT / fixed point)

AAACCnicbVC7TsMwFHV4lvIqMLIYKiSmKikIWJAqWBiLRB9SEyrHcVqrjmPZDlIVZWbhV1gYQIiVL2Djb3DaDNByJEtH55yr63t8wajStv1tLSwuLa+sltbK6xubW9uVnd22ihOJSQvHLJZdHynCKCctTTUjXSEJinxGOv7oOvc7D0QqGvM7PRbEi9CA05BipI3Urxy4zIQDdOmGEuHUOXMFva9n6UnmEqEoyzNVu2ZPAOeJU5AqKNDsV77cIMZJRLjGDCnVc2yhvRRJTTEjWdlNFBEIj9CA9AzlKCLKSyenZPDIKAEMY2ke13Ci/p5IUaTUOPJNMkJ6qGa9XPzP6yU6vPBSykWiCcfTRWHCoI5h3gsMqCRYs7EhCEtq/grxEJlOtGmvbEpwZk+eJ+16zTH89rTauCrqKIF9cAiOgQPOQQPcgCZoAQwewTN4BW/Wk/VivVsf0+iCVczsgT+wPn8AJiuajA== 3

✏2 109✏3

⇒ ⌘ = + + Put ✏AAAB83icbZDLSgMxFIbP1Futt6pLN8EiuCozIuhGKLpxWcFeoDOUTJppQzNJSDJCGfoablwo4taXcefbmLaz0NYfAh//OYdz8seKM2N9/9srra1vbG6Vtys7u3v7B9XDo7aRmSa0RSSXuhtjQzkTtGWZ5bSrNMVpzGknHt/N6p0nqg2T4tFOFI1SPBQsYQRbZ4UhVYZxKdANCvrVml/350KrEBRQg0LNfvUrHEiSpVRYwrExvcBXNsqxtoxwOq2EmaEKkzEe0p5DgVNqonx+8xSdOWeAEqndExbN3d8TOU6NmaSx60yxHZnl2sz8r9bLbHId5UyozFJBFouSjCMr0SwANGCaEssnDjDRzN2KyAhrTKyLqeJCCJa/vArti3rg+OGy1rgt4ijDCZzCOQRwBQ24hya0gICCZ3iFNy/zXrx372PRWvKKmWP4I+/zB/bAkPY= =1 for 3d !! AAACJnicbZDLSgMxFIYzXmu9VV26CRZBEGSmragLoejGpYJVoVNLJj2joZlkSM4IZejTuPFV3LhQRNz5KKYXxNuBwM//ncPJ+aNUCou+/+5NTE5Nz8wW5orzC4tLy6WV1QurM8OhwbXU5ipiFqRQ0ECBEq5SAyyJJFxG3eMBv7wDY4VW59hLoZWwGyViwRk6q106DAHZYRgbxvMQUiukVteVfr5b62+P3MA/oF+k6sh+teIY72i07VLZ3/GHRf+KYCzKZFyn7dJz2NE8S0Ahl8zaZuCn2MqZQcEl9IthZiFlvMtuoOmkYgnYVj48s083ndOhsTbuKaRD9/tEzhJre0nkOhOGt/Y3G5j/sWaG8X4rFyrNEBQfLYozSVHTQWa0IwxwlD0nGDfC/ZXyW+bCQZds0YUQ/D75r7io7AROn9XK9aNxHAWyTjbIFgnIHqmTE3JKGoSTe/JInsmL9+A9ea/e26h1whvPrJEf5X18ArJ4pUE= 54 5832 ···

⇒ (Compare with lattice: ~ 0.036 )

AAAB+HicbZDNSgMxFIXv1L9afzrq0k2wCK7KjAp1WXTjsoKthc5QMmmmDU0yQ5IR6tAnceNCEbc+ijvfxrSdhbZeCPk4515yc6KUM20879spra1vbG6Vtys7u3v7VffgsKOTTBHaJglPVDfCmnImadsww2k3VRSLiNOHaHwz8x8eqdIskfdmktJQ4KFkMSPYWKnvVgNqMAo0E8irexeNvluz97zQKvgF1KCoVt/9CgYJyQSVhnCsdc/3UhPmWBlGOJ1WgkzTFJMxHtKeRYkF1WE+X3yKTq0yQHGi7JEGzdXfEzkWWk9EZDsFNiO97M3E/7xeZuKrMGcyzQyVZPFQnHFkEjRLAQ2YosTwiQVMFLO7IjLCChNjs6rYEPzlL69C57zuW767rDWvizjKcAwncAY+NKAJt9CCNhDI4Ble4c15cl6cd+dj0Vpyipkj+FPO5w+P05G0 0.037 ⇠ Conformal Field Theory

What is the most natural way to study CFT -s ?

• Symmetries • Lagrangian independent Conformal Bootstrap

CFT Basics

• Every state is created by an operator O = O vac |AAACAnicbZDLSgMxFIYz9VbrbdSVuAkWwVWZEUE3QtGNu1awF2iHkknPtKGZzJBkCmVa3Pgqblwo4tancOfbmLaz0NYfAl/+cw7J+f2YM6Ud59vKrayurW/kNwtb2zu7e/b+QV1FiaRQoxGPZNMnCjgTUNNMc2jGEkjoc2j4g9tpvTEEqVgkHvQoBi8kPcECRok2Vsc+GlfakogeB3yNK3g8JDS7d+yiU3JmwsvgZlBEmaod+6vdjWgSgtCUE6VarhNrLyVSM8phUmgnCmJCB6QHLYOChKC8dLbCBJ8ap4uDSJojNJ65vydSEio1Cn3TGRLdV4u1qflfrZXo4MpLmYgTDYLOHwoSjnWEp3ngLpNANR8ZIFQy81dM+0QSqk1qBROCu7jyMtTPS67h+4ti+SaLI4+O0Qk6Qy66RGV0h6qohih6RM/oFb1ZT9aL9W59zFtzVjZziP7I+vwBVDqWvA== i | i • Every operator O is labelled by two quantum numbers

Spin `AAAB63icbZBNSwMxEIZn61etX1WPXoJF8FR2RdBj0YvHCrYW2qVk09k2NMkuSVYoS/+CFw+KePUPefPfmLZ70NYXAg/vzJCZN0oFN9b3v73S2vrG5lZ5u7Kzu7d/UD08apsk0wxbLBGJ7kTUoOAKW5ZbgZ1UI5WRwMdofDurPz6hNjxRD3aSYijpUPGYM2pnVg+F6Fdrft2fi6xCUEANCjX71a/eIGGZRGWZoMZ0Az+1YU615UzgtNLLDKaUjekQuw4VlWjCfL7rlJw5Z0DiRLunLJm7vydyKo2ZyMh1SmpHZrk2M/+rdTMbX4c5V2lmUbHFR3EmiE3I7HAy4BqZFRMHlGnudiVsRDVl1sVTcSEEyyevQvuiHji+v6w1boo4ynACp3AOAVxBA+6gCS1gMIJneIU3T3ov3rv3sWgtecXMMfyR9/kDDWiOOw==

Scaling dimension (Eigenvalue of D) AAAB7XicbZBNSwMxEIZn/az1q+rRS7AInsquCHos6sFjBfsB7VKyabaNzSZLMiuU0v/gxYMiXv0/3vw3pu0etPWFwMM7M2TmjVIpLPr+t7eyura+sVnYKm7v7O7tlw4OG1ZnhvE601KbVkQtl0LxOgqUvJUaTpNI8mY0vJnWm0/cWKHVA45SHia0r0QsGEVnNTq3XCLtlsp+xZ+JLEOQQxly1bqlr05PsyzhCpmk1rYDP8VwTA0KJvmk2MksTykb0j5vO1Q04TYcz7adkFPn9EisjXsKycz9PTGmibWjJHKdCcWBXaxNzf9q7Qzjq3AsVJohV2z+UZxJgppMTyc9YThDOXJAmRFuV8IG1FCGLqCiCyFYPHkZGueVwPH9Rbl6ncdRgGM4gTMI4BKqcAc1qAODR3iGV3jztPfivXsf89YVL585gj/yPn8AYI6O/A== Conformal Bootstrap

CFT Basics

• Every state is created by an operator O = O vac |AAACAnicbZDLSgMxFIYz9VbrbdSVuAkWwVWZEUE3QtGNu1awF2iHkknPtKGZzJBkCmVa3Pgqblwo4tancOfbmLaz0NYfAl/+cw7J+f2YM6Ud59vKrayurW/kNwtb2zu7e/b+QV1FiaRQoxGPZNMnCjgTUNNMc2jGEkjoc2j4g9tpvTEEqVgkHvQoBi8kPcECRok2Vsc+GlfakogeB3yNK3g8JDS7d+yiU3JmwsvgZlBEmaod+6vdjWgSgtCUE6VarhNrLyVSM8phUmgnCmJCB6QHLYOChKC8dLbCBJ8ap4uDSJojNJ65vydSEio1Cn3TGRLdV4u1qflfrZXo4MpLmYgTDYLOHwoSjnWEp3ngLpNANR8ZIFQy81dM+0QSqk1qBROCu7jyMtTPS67h+4ti+SaLI4+O0Qk6Qy66RGV0h6qohih6RM/oFb1ZT9aL9W59zFtzVjZziP7I+vwBVDqWvA== i | i • Every operator O is labelled by two quantum numbers

Spin `AAAB63icbZBNSwMxEIZn61etX1WPXoJF8FR2RdBj0YvHCrYW2qVk09k2NMkuSVYoS/+CFw+KePUPefPfmLZ70NYXAg/vzJCZN0oFN9b3v73S2vrG5lZ5u7Kzu7d/UD08apsk0wxbLBGJ7kTUoOAKW5ZbgZ1UI5WRwMdofDurPz6hNjxRD3aSYijpUPGYM2pnVg+F6Fdrft2fi6xCUEANCjX71a/eIGGZRGWZoMZ0Az+1YU615UzgtNLLDKaUjekQuw4VlWjCfL7rlJw5Z0DiRLunLJm7vydyKo2ZyMh1SmpHZrk2M/+rdTMbX4c5V2lmUbHFR3EmiE3I7HAy4BqZFRMHlGnudiVsRDVl1sVTcSEEyyevQvuiHji+v6w1boo4ynACp3AOAVxBA+6gCS1gMIJneIU3T3ov3rv3sWgtecXMMfyR9/kDDWiOOw==

Scaling dimension (Eigenvalue of D) AAAB7XicbZBNSwMxEIZn/az1q+rRS7AInsquCHos6sFjBfsB7VKyabaNzSZLMiuU0v/gxYMiXv0/3vw3pu0etPWFwMM7M2TmjVIpLPr+t7eyura+sVnYKm7v7O7tlw4OG1ZnhvE601KbVkQtl0LxOgqUvJUaTpNI8mY0vJnWm0/cWKHVA45SHia0r0QsGEVnNTq3XCLtlsp+xZ+JLEOQQxly1bqlr05PsyzhCpmk1rYDP8VwTA0KJvmk2MksTykb0j5vO1Q04TYcz7adkFPn9EisjXsKycz9PTGmibWjJHKdCcWBXaxNzf9q7Qzjq3AsVJohV2z+UZxJgppMTyc9YThDOXJAmRFuV8IG1FCGLqCiCyFYPHkZGueVwPH9Rbl6ncdRgGM4gTMI4BKqcAc1qAODR3iGV3jztPfivXsf89YVL585gj/yPn8AYI6O/A==

• All 2-point and 3-point functions have precise forms

ij Oi(x)Oj(y) = (x y)2i 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 i Cijk Oi(x)Oj(y)Ok(z) = (x y)k i j (x z)j i k (y z)i j k

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 i Conformal Bootstrap

CFT Basics

Operator Product Expansion (OPE) : For any two operators,

O (x) O (y) C f(x y)O i ⇥ j ⇠ ijk k k= ,` 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 X{ } In any n-point function

O O = O h··· i ⇥ j ···i h··· k ···i k= ,` 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 X{ } • All 2-point and 3-point functions have precise forms

ij Oi(x)Oj(y) = (x y)2i 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 i Cijk Oi(x)Oj(y)Ok(z) = (x y)k i j (x z)j i k (y z)i j k

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 i Conformal Bootstrap

CFT Basics

Operator Product Expansion (OPE) : For any two operators,

O (x) O (y) C f(x y)O i ⇥ j ⇠ ijk k k= ,` 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 X{ } In any n-point function

O O = O h··· i ⇥ j ···i h··· k ···i k= ,` 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 X{ }

Solving a CFT ⇔ Finding all possible `, ,C {AAACAHicbZDLSsNAFIYn9VbrLerChZtgEVyUkoigy2JduKxgL9CEMJmetGMnkzAzEUrIxldx40IRtz6GO9/GaZuFtv4w8PGfczhz/iBhVCrb/jZKK6tr6xvlzcrW9s7unrl/0JFxKgi0Scxi0QuwBEY5tBVVDHqJABwFDLrBuDmtdx9BSBrzezVJwIvwkNOQEqy05ZtHbuYCYzX3BpjCtaaf0Ydx7ua+WbXr9kzWMjgFVFGhlm9+uYOYpBFwRRiWsu/YifIyLBQlDPKKm0pIMBnjIfQ1chyB9LLZAbl1qp2BFcZCP66smft7IsORlJMo0J0RViO5WJua/9X6qQqvvIzyJFXAyXxRmDJLxdY0DWtABRDFJhowEVT/1SIjLDBROrOKDsFZPHkZOud1R/PdRbVxXcRRRsfoBJ0hB12iBrpFLdRGBOXoGb2iN+PJeDHejY95a8koZg7RHxmfP3aZllA= ijk} Conformal Bootstrap

Operator Product Expansion (OPE) : For any two operators,

O (x) O (y) C f(x y)O i ⇥ j ⇠ ijk k k= ,` 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 X{ }

Consider the 4-point function

O(x )O(x )O(x )O(x ) = C gg(u, v(u,) v) AAAB/XicbZDLSsNAFIYn9VbrLV52boJFqFBKIoIui7pwWcFeoAlhMj1ph04mYWZSqKH4Km5cKOLW93Dn2zhts9DWHwY+/nMO58wfJIxKZdvfRmFldW19o7hZ2tre2d0z9w9aMk4FgSaJWSw6AZbAKIemoopBJxGAo4BBOxjeTOvtEQhJY/6gxgl4Ee5zGlKClbZ886jvZ+4tMIWrLjA2qaTV0Zlvlu2aPZO1DE4OZZSr4Ztfbi8maQRcEYal7Dp2orwMC0UJg0nJTSUkmAxxH7oaOY5Aetns+ol1qp2eFcZCP66smft7IsORlOMo0J0RVgO5WJua/9W6qQqvvIzyJFXAyXxRmDJLxdY0CqtHBRDFxhowEVTfapEBFpgoHVhJh+AsfnkZWuc1R/P9Rbl+ncdRRMfoBFWQgy5RHd2hBmoigh7RM3pFb8aT8WK8Gx/z1oKRzxyiPzI+fwCVsJSk ,` h 1 2 3 4 i ,` AAACPnicbVBNSwMxFMz6WetX1aOXYBFaENnVgl6EYj14s4LbCt2yZNPXGsxmlyRbLEt/mRd/gzePXjwo4tWj6bYHWx3IYzLzHsmbIOZMadt+sebmFxaXlnMr+dW19Y3NwtZ2Q0WJpODSiEfyNiAKOBPgaqY53MYSSBhwaAb3tZHf7INULBI3ehBDOyQ9wbqMEm0kv+B6nIgeB3xVevCd8qgeZfU4q5WyJ8c+PsOeSkI/9S6Aa3LgAefD2vQVe7hXSg76Zb9QtA/tDPgvcSakiCao+4VnrxPRJAShKSdKtRw71u2USM0oh2HeSxTEhN6THrQMFSQE1U6z9Yd43ygd3I2kOULjTP09kZJQqUEYmM6Q6Ds1643E/7xWorun7ZSJONEg6PihbsKxjvAoS9xhEqjmA0MIlcz8FdM7IgnVJvG8CcGZXfkvaRwdOoZfV4rV80kcObSL9lAJOegEVdElqiMXUfSIXtE7+rCerDfr0/oat85Zk5kdNAXr+wciRawn X,` 2 2 2 2 2 x12x34 x14x23 C,` = C u = v = AAACAnicbZDLSsNAFIYn9VbrLepK3ASL4EJKUgTdCMW6cFnBXqCJYTI9bcdOJmFmIpQQ3Pgqblwo4tancOfbOK1ZaOsPA9/85xxmzh/EjEpl219GYWFxaXmluFpaW9/Y3DK3d1oySgSBJolYJDoBlsAoh6aiikEnFoDDgEE7GNUn9fY9CEkjfqPGMXghHnDapwQrbfnmXt1P3UtgCh+7wFh2ru/0bpTdVn2zbFfsqax5cHIoo1wN3/x0exFJQuCKMCxl17Fj5aVYKEoYZCU3kRBjMsID6GrkOATppdMVMutQOz2rHwl9uLKm7u+JFIdSjsNAd4ZYDeVsbWL+V+smqn/mpZTHiQJOfh7qJ8xSkTXJw+pRAUSxsQZMBNV/tcgQC0yUTq2kQ3BmV56HVrXiaL4+Kdcu8jiKaB8doCPkoFNUQ1eogZqIoAf0hF7Qq/FoPBtvxvtPa8HIZ3bRHxkf36R3lu8= ijk x2 x2 x2 x2 AAACFHicbZBNS8NAEIY3ftb6FfXoZbEIglCStKAXoejFYwX7AW0Mm+2mXbrZhN2NWEJ+hBf/ihcPinj14M1/4zbNQVsHlnl5ZobZef2YUaks69tYWl5ZXVsvbZQ3t7Z3ds29/baMEoFJC0csEl0fScIoJy1FFSPdWBAU+ox0/PHVtN65J0LSiN+qSUzcEA05DShGSiPPPE0u+oFAOH3wUtvJ7hyda3Wds5zUZsTJiWdWrKqVB1wUdiEqoIimZ371BxFOQsIVZkjKnm3Fyk2RUBQzkpX7iSQxwmM0JD0tOQqJdNP8qAweazKAQST04wrm9PdEikIpJ6GvO0OkRnK+NoX/1XqJCs7dlPI4UYTj2aIgYVBFcOoQHFBBsGITLRAWVP8V4hHSHintY1mbYM+fvCjaTtXW+qZeaVwWdpTAITgCJ8AGZ6ABrkETtAAGj+AZvII348l4Md6Nj1nrklHMHIA/YXz+AM3ingM= 13 24 AAACFHicbVDLSsNAFJ3UV62vqEs3g0UQhJK0Bd0IRTcuK9gHtLFMppN26GQSZibFEvIRbvwVNy4UcevCnX/jJM1CWw8M93DOvdy5xw0Zlcqyvo3Cyura+kZxs7S1vbO7Z+4ftGUQCUxaOGCB6LpIEkY5aSmqGOmGgiDfZaTjTq5TvzMlQtKA36lZSBwfjTj1KEZKSwPzbHrZ9wTC8cMgtuvJfVXXak3XJFNquZI6ycAsWxUrA1wmdk7KIEdzYH71hwGOfMIVZkjKnm2FyomRUBQzkpT6kSQhwhM0Ij1NOfKJdOLsqASeaGUIvUDoxxXM1N8TMfKlnPmu7vSRGstFLxX/83qR8i6cmPIwUoTj+SIvYlAFME0IDqkgWLGZJggLqv8K8RjpjJTOsaRDsBdPXibtasXW/LZeblzlcRTBETgGp8AG56ABbkATtAAGj+AZvII348l4Md6Nj3lrwchnDsEfGJ8/z5+eBA== 13 24

( x = x x ) AAACAHicbZDLSgMxFIbP1Futt1EXLtwEi1AXlhkRdCMU3bisYC/QDkMmzbRpMxeSjLQMs/FV3LhQxK2P4c63MW1nodUDIR//fw7J+b2YM6ks68soLC2vrK4V10sbm1vbO+buXlNGiSC0QSIeibaHJeUspA3FFKftWFAceJy2vNHN1G89UCFZFN6rSUydAPdD5jOClZZc86CCumjspmyYXembnY7doVZOXLNsVa1Zob9g51CGvOqu+dntRSQJaKgIx1J2bCtWToqFYoTTrNRNJI0xGeE+7WgMcUClk84WyNCxVnrIj4Q+oUIz9edEigMpJ4GnOwOsBnLRm4r/eZ1E+ZdOysI4UTQk84f8hCMVoWkaqMcEJYpPNGAimP4rIgMsMFE6s5IOwV5c+S80z6q25rvzcu06j6MIh3AEFbDhAmpwC3VoAIEMnuAFXo1H49l4M97nrQUjn9mHX2V8fAMV05TH ij i j Conformal Bootstrap

Conformal block (a known function)

Consider the 4-point function

O(x )O(x )O(x )O(x ) = C gg(u, v(u,) v) AAAB/XicbZDLSsNAFIYn9VbrLV52boJFqFBKIoIui7pwWcFeoAlhMj1ph04mYWZSqKH4Km5cKOLW93Dn2zhts9DWHwY+/nMO58wfJIxKZdvfRmFldW19o7hZ2tre2d0z9w9aMk4FgSaJWSw6AZbAKIemoopBJxGAo4BBOxjeTOvtEQhJY/6gxgl4Ee5zGlKClbZ886jvZ+4tMIWrLjA2qaTV0Zlvlu2aPZO1DE4OZZSr4Ztfbi8maQRcEYal7Dp2orwMC0UJg0nJTSUkmAxxH7oaOY5Aetns+ol1qp2eFcZCP66smft7IsORlOMo0J0RVgO5WJua/9W6qQqvvIzyJFXAyXxRmDJLxdY0CqtHBRDFxhowEVTfapEBFpgoHVhJh+AsfnkZWuc1R/P9Rbl+ncdRRMfoBFWQgy5RHd2hBmoigh7RM3pFb8aT8WK8Gx/z1oKRzxyiPzI+fwCVsJSk ,` h 1 2 3 4 i ,` 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 X,` 2 2 2 2 2 x12x34 x14x23 C,` = C u = v = AAACAnicbZDLSsNAFIYn9VbrLepK3ASL4EJKUgTdCMW6cFnBXqCJYTI9bcdOJmFmIpQQ3Pgqblwo4tancOfbOK1ZaOsPA9/85xxmzh/EjEpl219GYWFxaXmluFpaW9/Y3DK3d1oySgSBJolYJDoBlsAoh6aiikEnFoDDgEE7GNUn9fY9CEkjfqPGMXghHnDapwQrbfnmXt1P3UtgCh+7wFh2ru/0bpTdVn2zbFfsqax5cHIoo1wN3/x0exFJQuCKMCxl17Fj5aVYKEoYZCU3kRBjMsID6GrkOATppdMVMutQOz2rHwl9uLKm7u+JFIdSjsNAd4ZYDeVsbWL+V+smqn/mpZTHiQJOfh7qJ8xSkTXJw+pRAUSxsQZMBNV/tcgQC0yUTq2kQ3BmV56HVrXiaL4+Kdcu8jiKaB8doCPkoFNUQ1eogZqIoAf0hF7Qq/FoPBtvxvtPa8HIZ3bRHxkf36R3lu8= ijk x2 x2 x2 x2 AAACFHicbZBNS8NAEIY3ftb6FfXoZbEIglCStKAXoejFYwX7AW0Mm+2mXbrZhN2NWEJ+hBf/ihcPinj14M1/4zbNQVsHlnl5ZobZef2YUaks69tYWl5ZXVsvbZQ3t7Z3ds29/baMEoFJC0csEl0fScIoJy1FFSPdWBAU+ox0/PHVtN65J0LSiN+qSUzcEA05DShGSiPPPE0u+oFAOH3wUtvJ7hyda3Wds5zUZsTJiWdWrKqVB1wUdiEqoIimZ371BxFOQsIVZkjKnm3Fyk2RUBQzkpX7iSQxwmM0JD0tOQqJdNP8qAweazKAQST04wrm9PdEikIpJ6GvO0OkRnK+NoX/1XqJCs7dlPI4UYTj2aIgYVBFcOoQHFBBsGITLRAWVP8V4hHSHintY1mbYM+fvCjaTtXW+qZeaVwWdpTAITgCJ8AGZ6ABrkETtAAGj+AZvII348l4Md6Nj1nrklHMHIA/YXz+AM3ingM= 13 24 AAACFHicbVDLSsNAFJ3UV62vqEs3g0UQhJK0Bd0IRTcuK9gHtLFMppN26GQSZibFEvIRbvwVNy4UcevCnX/jJM1CWw8M93DOvdy5xw0Zlcqyvo3Cyura+kZxs7S1vbO7Z+4ftGUQCUxaOGCB6LpIEkY5aSmqGOmGgiDfZaTjTq5TvzMlQtKA36lZSBwfjTj1KEZKSwPzbHrZ9wTC8cMgtuvJfVXXak3XJFNquZI6ycAsWxUrA1wmdk7KIEdzYH71hwGOfMIVZkjKnm2FyomRUBQzkpT6kSQhwhM0Ij1NOfKJdOLsqASeaGUIvUDoxxXM1N8TMfKlnPmu7vSRGstFLxX/83qR8i6cmPIwUoTj+SIvYlAFME0IDqkgWLGZJggLqv8K8RjpjJTOsaRDsBdPXibtasXW/LZeblzlcRTBETgGp8AG56ABbkATtAAGj+AZvII348l4Md6Nj3lrwchnDsEfGJ8/z5+eBA== 13 24 OPE coefficients ( x = x x ) AAACAHicbZDLSgMxFIbP1Futt1EXLtwEi1AXlhkRdCMU3bisYC/QDkMmzbRpMxeSjLQMs/FV3LhQxK2P4c63MW1nodUDIR//fw7J+b2YM6ks68soLC2vrK4V10sbm1vbO+buXlNGiSC0QSIeibaHJeUspA3FFKftWFAceJy2vNHN1G89UCFZFN6rSUydAPdD5jOClZZc86CCumjspmyYXembnY7doVZOXLNsVa1Zob9g51CGvOqu+dntRSQJaKgIx1J2bCtWToqFYoTTrNRNJI0xGeE+7WgMcUClk84WyNCxVnrIj4Q+oUIz9edEigMpJ4GnOwOsBnLRm4r/eZ1E+ZdOysI4UTQk84f8hCMVoWkaqMcEJYpPNGAimP4rIgMsMFE6s5IOwV5c+S80z6q25rvzcu06j6MIh3AEFbDhAmpwC3VoAIEMnuAFXo1H49l4M97nrQUjn9mHX2V8fAMV05TH ij i j Conformal Bootstrap

g (u, v) O(x1)O(x2)O(x3)O(x4) = CCC,`,,``ggAAAB/XicbZDLSsNAFIYn9VbrLV52boJFqFBKIoIui7pwWcFeoAlhMj1ph04mYWZSqKH4Km5cKOLW93Dn2zhts9DWHwY+/nMO58wfJIxKZdvfRmFldW19o7hZ2tre2d0z9w9aMk4FgSaJWSw6AZbAKIemoopBJxGAo4BBOxjeTOvtEQhJY/6gxgl4Ee5zGlKClbZ886jvZ+4tMIWrLjA2qaTV0Zlvlu2aPZO1DE4OZZSr4Ztfbi8maQRcEYal7Dp2orwMC0UJg0nJTSUkmAxxH7oaOY5Aetns+ol1qp2eFcZCP66smft7IsORlOMo0J0RVgO5WJua/9W6qQqvvIzyJFXAyXxRmDJLxdY0CqtHBRDFxhowEVTfapEBFpgoHVhJh+AsfnkZWuc1R/P9Rbl+ncdRRMfoBFWQgy5RHd2hBmoigh7RM3pFb8aT8WK8Gx/z1oKRzxyiPzI+fwCVsJSk g((u,v,,`, v` u())v, u) h i

AAACFXicbZDLSgMxFIYz9VbrbdSlm2ARKpQyI4Iui3XhsoK9QGcYMumZNjRzIckUytCXcOOruHGhiFvBnW9j2s5CW38IfPnPOSTn9xPOpLKsb6Owtr6xuVXcLu3s7u0fmIdHbRmngkKLxjwWXZ9I4CyClmKKQzcRQEKfQ8cfNWb1zhiEZHH0oCYJuCEZRCxglChteWbVkWnoZc4tcEWqDnA+xY2lu4PxoDKupueeWbZq1lx4FewcyihX0zO/nH5M0xAiRTmRsmdbiXIzIhSjHKYlJ5WQEDoiA+hpjEgI0s3mW03xmXb6OIiFPpHCc/f3REZCKSehrztDooZyuTYz/6v1UhVcuxmLklRBRBcPBSnHKsaziHCfCaCKTzQQKpj+K6ZDIghVOsiSDsFeXnkV2hc1W/P9Zbl+k8dRRCfoFFWQja5QHd2hJmohih7RM3pFb8aT8WK8Gx+L1oKRzxyjPzI+fwCZJZ3G ,`

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 ,` XAAACIXicbZDLSgMxFIYz9VbrbdSlm2ARKpQyI4JdFuvCZQV7gc4wZNLTNjRzIckUytBXceOruHGhSHfiy5i2s7CtPwS+/OcckvP7MWdSWda3kdva3tndy+8XDg6Pjk/M07OWjBJBoUkjHomOTyRwFkJTMcWhEwsggc+h7Y/q83p7DEKyKHxWkxjcgAxC1meUKG15ZtWRSeClzgNwRcoOcD6tr16xgwerTmlcTq49s2hVrIXwJtgZFFGmhmfOnF5EkwBCRTmRsmtbsXJTIhSjHKYFJ5EQEzoiA+hqDEkA0k0XG07xlXZ6uB8JfUKFF+7fiZQEUk4CX3cGRA3lem1u/lfrJqpfdVMWxomCkC4f6iccqwjP48I9JoAqPtFAqGD6r5gOiSBU6VALOgR7feVNaN1UbM1Pt8XafRZHHl2gS1RCNrpDNfSIGqiJKHpBb+gDfRqvxrvxZcyWrTkjmzlHKzJ+fgEHMKNg X,` t-channel

Should be equal

s-channel O(x )O(x )O(x )O(x ) = C gg(u, v(u,) v) AAAB/XicbZDLSsNAFIYn9VbrLV52boJFqFBKIoIui7pwWcFeoAlhMj1ph04mYWZSqKH4Km5cKOLW93Dn2zhts9DWHwY+/nMO58wfJIxKZdvfRmFldW19o7hZ2tre2d0z9w9aMk4FgSaJWSw6AZbAKIemoopBJxGAo4BBOxjeTOvtEQhJY/6gxgl4Ee5zGlKClbZ886jvZ+4tMIWrLjA2qaTV0Zlvlu2aPZO1DE4OZZSr4Ztfbi8maQRcEYal7Dp2orwMC0UJg0nJTSUkmAxxH7oaOY5Aetns+ol1qp2eFcZCP66smft7IsORlOMo0J0RVgO5WJua/9W6qQqvvIzyJFXAyXxRmDJLxdY0CqtHBRDFxhowEVTfapEBFpgoHVhJh+AsfnkZWuc1R/P9Rbl+ncdRRMfoBFWQgy5RHd2hBmoigh7RM3pFb8aT8WK8Gx/z1oKRzxyiPzI+fwCVsJSk ,` h 1 2 3 4 i ,` 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 X,` 2 2 2 2 2 x12x34 x14x23 C,` = C u = v = AAACAnicbZDLSsNAFIYn9VbrLepK3ASL4EJKUgTdCMW6cFnBXqCJYTI9bcdOJmFmIpQQ3Pgqblwo4tancOfbOK1ZaOsPA9/85xxmzh/EjEpl219GYWFxaXmluFpaW9/Y3DK3d1oySgSBJolYJDoBlsAoh6aiikEnFoDDgEE7GNUn9fY9CEkjfqPGMXghHnDapwQrbfnmXt1P3UtgCh+7wFh2ru/0bpTdVn2zbFfsqax5cHIoo1wN3/x0exFJQuCKMCxl17Fj5aVYKEoYZCU3kRBjMsID6GrkOATppdMVMutQOz2rHwl9uLKm7u+JFIdSjsNAd4ZYDeVsbWL+V+smqn/mpZTHiQJOfh7qJ8xSkTXJw+pRAUSxsQZMBNV/tcgQC0yUTq2kQ3BmV56HVrXiaL4+Kdcu8jiKaB8doCPkoFNUQ1eogZqIoAf0hF7Qq/FoPBtvxvtPa8HIZ3bRHxkf36R3lu8= ijk x2 x2 x2 x2 AAACFHicbZBNS8NAEIY3ftb6FfXoZbEIglCStKAXoejFYwX7AW0Mm+2mXbrZhN2NWEJ+hBf/ihcPinj14M1/4zbNQVsHlnl5ZobZef2YUaks69tYWl5ZXVsvbZQ3t7Z3ds29/baMEoFJC0csEl0fScIoJy1FFSPdWBAU+ox0/PHVtN65J0LSiN+qSUzcEA05DShGSiPPPE0u+oFAOH3wUtvJ7hyda3Wds5zUZsTJiWdWrKqVB1wUdiEqoIimZ371BxFOQsIVZkjKnm3Fyk2RUBQzkpX7iSQxwmM0JD0tOQqJdNP8qAweazKAQST04wrm9PdEikIpJ6GvO0OkRnK+NoX/1XqJCs7dlPI4UYTj2aIgYVBFcOoQHFBBsGITLRAWVP8V4hHSHintY1mbYM+fvCjaTtXW+qZeaVwWdpTAITgCJ8AGZ6ABrkETtAAGj+AZvII348l4Md6Nj1nrklHMHIA/YXz+AM3ingM= 13 24 AAACFHicbVDLSsNAFJ3UV62vqEs3g0UQhJK0Bd0IRTcuK9gHtLFMppN26GQSZibFEvIRbvwVNy4UcevCnX/jJM1CWw8M93DOvdy5xw0Zlcqyvo3Cyura+kZxs7S1vbO7Z+4ftGUQCUxaOGCB6LpIEkY5aSmqGOmGgiDfZaTjTq5TvzMlQtKA36lZSBwfjTj1KEZKSwPzbHrZ9wTC8cMgtuvJfVXXak3XJFNquZI6ycAsWxUrA1wmdk7KIEdzYH71hwGOfMIVZkjKnm2FyomRUBQzkpT6kSQhwhM0Ij1NOfKJdOLsqASeaGUIvUDoxxXM1N8TMfKlnPmu7vSRGstFLxX/83qR8i6cmPIwUoTj+SIvYlAFME0IDqkgWLGZJggLqv8K8RjpjJTOsaRDsBdPXibtasXW/LZeblzlcRTBETgGp8AG56ABbkATtAAGj+AZvII348l4Md6Nj3lrwchnDsEfGJ8/z5+eBA== 13 24

( x = x x ) AAACAHicbZDLSgMxFIbP1Futt1EXLtwEi1AXlhkRdCMU3bisYC/QDkMmzbRpMxeSjLQMs/FV3LhQxK2P4c63MW1nodUDIR//fw7J+b2YM6ks68soLC2vrK4V10sbm1vbO+buXlNGiSC0QSIeibaHJeUspA3FFKftWFAceJy2vNHN1G89UCFZFN6rSUydAPdD5jOClZZc86CCumjspmyYXembnY7doVZOXLNsVa1Zob9g51CGvOqu+dntRSQJaKgIx1J2bCtWToqFYoTTrNRNJI0xGeE+7WgMcUClk84WyNCxVnrIj4Q+oUIz9edEigMpJ4GnOwOsBnLRm4r/eZ1E+ZdOysI4UTQk84f8hCMVoWkaqMcEJYpPNGAimP4rIgMsMFE6s5IOwV5c+S80z6q25rvzcu06j6MIh3AEFbDhAmpwC3VoAIEMnuAFXo1H49l4M97nrQUjn9mHX2V8fAMV05TH ij i j Conformal Bootstrap

C,` g,`(u, v)= C,` g,`(v, u)

XAAACYHicnVFLS0JBGJ177WGmqbWrzZAECiL3RlCbQLJFS4N8gFdk7vipg3MfzMOQi3+yXYs2/ZLGxyK1VR98cOac7/DNnPFjzqRynE/LTh0cHh2nTzKn2dxZvlA8b8tICwotGvFIdH0igbMQWoopDt1YAAl8Dh1/2ljqnRkIyaLwTc1j6AdkHLIRo0QZalB496QOBon3DFyRqgecLxrbR+zh8TZT1tVZxdCPpv9jn1V1ZVAoOTVnVXgfuBtQQptqDgof3jCiOoBQUU6k7LlOrPoJEYpRDouMpyXEhE7JGHoGhiQA2U9WAS3wjWGGeBQJ06HCK/a3IyGBlPPAN5MBURO5qy3Jv7SeVqOHfsLCWCsI6XrRSHOsIrxMGw+ZAKr43ABCBTN3xXRCBKHK/EnGhODuPnkftG9rrsGvd6X60yaONLpC16iMXHSP6ugFNVELUfRlpayslbO+7bSdt4vrUdvaeC7QVtmXP10ztNc= ,` X,`

[Ratazzi, Rychkov, Tonni, Vichi (2008)]

[el-Showk, Paulos, Poland, Rychkov, Simmons-Duffin (2012)]

2 2 2 2 2 x12x34 x14x23 C,` = C u = v = AAACAnicbZDLSsNAFIYn9VbrLepK3ASL4EJKUgTdCMW6cFnBXqCJYTI9bcdOJmFmIpQQ3Pgqblwo4tancOfbOK1ZaOsPA9/85xxmzh/EjEpl219GYWFxaXmluFpaW9/Y3DK3d1oySgSBJolYJDoBlsAoh6aiikEnFoDDgEE7GNUn9fY9CEkjfqPGMXghHnDapwQrbfnmXt1P3UtgCh+7wFh2ru/0bpTdVn2zbFfsqax5cHIoo1wN3/x0exFJQuCKMCxl17Fj5aVYKEoYZCU3kRBjMsID6GrkOATppdMVMutQOz2rHwl9uLKm7u+JFIdSjsNAd4ZYDeVsbWL+V+smqn/mpZTHiQJOfh7qJ8xSkTXJw+pRAUSxsQZMBNV/tcgQC0yUTq2kQ3BmV56HVrXiaL4+Kdcu8jiKaB8doCPkoFNUQ1eogZqIoAf0hF7Qq/FoPBtvxvtPa8HIZ3bRHxkf36R3lu8= ijk x2 x2 x2 x2 AAACFHicbZBNS8NAEIY3ftb6FfXoZbEIglCStKAXoejFYwX7AW0Mm+2mXbrZhN2NWEJ+hBf/ihcPinj14M1/4zbNQVsHlnl5ZobZef2YUaks69tYWl5ZXVsvbZQ3t7Z3ds29/baMEoFJC0csEl0fScIoJy1FFSPdWBAU+ox0/PHVtN65J0LSiN+qSUzcEA05DShGSiPPPE0u+oFAOH3wUtvJ7hyda3Wds5zUZsTJiWdWrKqVB1wUdiEqoIimZ371BxFOQsIVZkjKnm3Fyk2RUBQzkpX7iSQxwmM0JD0tOQqJdNP8qAweazKAQST04wrm9PdEikIpJ6GvO0OkRnK+NoX/1XqJCs7dlPI4UYTj2aIgYVBFcOoQHFBBsGITLRAWVP8V4hHSHintY1mbYM+fvCjaTtXW+qZeaVwWdpTAITgCJ8AGZ6ABrkETtAAGj+AZvII348l4Md6Nj1nrklHMHIA/YXz+AM3ingM= 13 24 AAACFHicbVDLSsNAFJ3UV62vqEs3g0UQhJK0Bd0IRTcuK9gHtLFMppN26GQSZibFEvIRbvwVNy4UcevCnX/jJM1CWw8M93DOvdy5xw0Zlcqyvo3Cyura+kZxs7S1vbO7Z+4ftGUQCUxaOGCB6LpIEkY5aSmqGOmGgiDfZaTjTq5TvzMlQtKA36lZSBwfjTj1KEZKSwPzbHrZ9wTC8cMgtuvJfVXXak3XJFNquZI6ycAsWxUrA1wmdk7KIEdzYH71hwGOfMIVZkjKnm2FyomRUBQzkpT6kSQhwhM0Ij1NOfKJdOLsqASeaGUIvUDoxxXM1N8TMfKlnPmu7vSRGstFLxX/83qR8i6cmPIwUoTj+SIvYlAFME0IDqkgWLGZJggLqv8K8RjpjJTOsaRDsBdPXibtasXW/LZeblzlcRTBETgGp8AG56ABbkATtAAGj+AZvII348l4Md6Nj3lrwchnDsEfGJ8/z5+eBA== 13 24

( x = x x ) AAACAHicbZDLSgMxFIbP1Futt1EXLtwEi1AXlhkRdCMU3bisYC/QDkMmzbRpMxeSjLQMs/FV3LhQxK2P4c63MW1nodUDIR//fw7J+b2YM6ks68soLC2vrK4V10sbm1vbO+buXlNGiSC0QSIeibaHJeUspA3FFKftWFAceJy2vNHN1G89UCFZFN6rSUydAPdD5jOClZZc86CCumjspmyYXembnY7doVZOXLNsVa1Zob9g51CGvOqu+dntRSQJaKgIx1J2bCtWToqFYoTTrNRNJI0xGeE+7WgMcUClk84WyNCxVnrIj4Q+oUIz9edEigMpJ4GnOwOsBnLRm4r/eZ1E+ZdOysI4UTQk84f8hCMVoWkaqMcEJYpPNGAimP4rIgMsMFE6s5IOwV5c+S80z6q25rvzcu06j6MIh3AEFbDhAmpwC3VoAIEMnuAFXo1H49l4M97nrQUjn9mHX2V8fAMV05TH ij i j Conformal Bootstrap

C,` g,`(u, v)= C,` g,`(v, u)

XAAACYHicnVFLS0JBGJ177WGmqbWrzZAECiL3RlCbQLJFS4N8gFdk7vipg3MfzMOQi3+yXYs2/ZLGxyK1VR98cOac7/DNnPFjzqRynE/LTh0cHh2nTzKn2dxZvlA8b8tICwotGvFIdH0igbMQWoopDt1YAAl8Dh1/2ljqnRkIyaLwTc1j6AdkHLIRo0QZalB496QOBon3DFyRqgecLxrbR+zh8TZT1tVZxdCPpv9jn1V1ZVAoOTVnVXgfuBtQQptqDgof3jCiOoBQUU6k7LlOrPoJEYpRDouMpyXEhE7JGHoGhiQA2U9WAS3wjWGGeBQJ06HCK/a3IyGBlPPAN5MBURO5qy3Jv7SeVqOHfsLCWCsI6XrRSHOsIrxMGw+ZAKr43ABCBTN3xXRCBKHK/EnGhODuPnkftG9rrsGvd6X60yaONLpC16iMXHSP6ugFNVELUfRlpayslbO+7bSdt4vrUdvaeC7QVtmXP10ztNc= ,` X,`

Non-perturbative, lagrangian free equation !! Conformal Bootstrap

C,` g,`(u, v)= C,` g,`(v, u)

XAAACYHicnVFLS0JBGJ177WGmqbWrzZAECiL3RlCbQLJFS4N8gFdk7vipg3MfzMOQi3+yXYs2/ZLGxyK1VR98cOac7/DNnPFjzqRynE/LTh0cHh2nTzKn2dxZvlA8b8tICwotGvFIdH0igbMQWoopDt1YAAl8Dh1/2ljqnRkIyaLwTc1j6AdkHLIRo0QZalB496QOBon3DFyRqgecLxrbR+zh8TZT1tVZxdCPpv9jn1V1ZVAoOTVnVXgfuBtQQptqDgof3jCiOoBQUU6k7LlOrPoJEYpRDouMpyXEhE7JGHoGhiQA2U9WAS3wjWGGeBQJ06HCK/a3IyGBlPPAN5MBURO5qy3Jv7SeVqOHfsLCWCsI6XrRSHOsIrxMGw+ZAKr43ABCBTN3xXRCBKHK/EnGhODuPnkftG9rrsGvd6X60yaONLpC16iMXHSP6ugFNVELUfRlpayslbO+7bSdt4vrUdvaeC7QVtmXP10ztNc= ,` X,`

Non-perturbative, lagrangian free equation !!

How to Solve Numerically? • Assume symmetries and a dimension

Throw in all possible , `,C • {AAACCHicbZDLSsNAFIYn9VbrLerShcEiuCglEUGXxbpwWcFeoAlhMj1ph04mYWYilJClG1/FjQtF3PoI7nwbp20W2vrDwDf/OYeZ8wcJo1LZ9rdRWlldW98ob1a2tnd298z9g46MU0GgTWIWi16AJTDKoa2oYtBLBOAoYNANxs1pvfsAQtKY36tJAl6Eh5yGlGClLd88djP3BpjCNRcYqzX939fczX2zatftmaxlcAqookIt3/xyBzFJI+CKMCxl37ET5WVYKEoY5BU3lZBgMsZD6GvkOALpZbNFcutUOwMrjIU+XFkz9/dEhiMpJ1GgOyOsRnKxNjX/q/VTFV55GeVJqoCT+UNhyiwVW9NUrAEVQBSbaMBEUP1Xi4ywwETp7Co6BGdx5WXonNcdzXcX1cZ1EUcZHaETdIYcdIka6Ba1UBsR9Iie0St6M56MF+Pd+Ji3loxi5hD9kfH5AytgmW0= ,`} • Check for consistency , `,C

Obtain bounds for consistent AAACCHicbZDLSsNAFIYn9VbrLerShcEiuCglEUGXxbpwWcFeoAlhMj1ph04mYWYilJClG1/FjQtF3PoI7nwbp20W2vrDwDf/OYeZ8wcJo1LZ9rdRWlldW98ob1a2tnd298z9g46MU0GgTWIWi16AJTDKoa2oYtBLBOAoYNANxs1pvfsAQtKY36tJAl6Eh5yGlGClLd88djP3BpjCNRcYqzX939fczX2zatftmaxlcAqookIt3/xyBzFJI+CKMCxl37ET5WVYKEoY5BU3lZBgMsZD6GvkOALpZbNFcutUOwMrjIU+XFkz9/dEhiMpJ1GgOyOsRnKxNjX/q/VTFV55GeVJqoCT+UNhyiwVW9NUrAEVQBSbaMBEUP1Xi4ywwETp7Co6BGdx5WXonNcdzXcX1cZ1EUcZHaETdIYcdIka6Ba1UBsR9Iie0St6M56MF+Pd+Ji3loxi5hD9kfH5AytgmW0= ,` • { } Conformal Bootstrap

C,` g,`(u, v)= C,` g,`(v, u)

XAAACYHicnVFLS0JBGJ177WGmqbWrzZAECiL3RlCbQLJFS4N8gFdk7vipg3MfzMOQi3+yXYs2/ZLGxyK1VR98cOac7/DNnPFjzqRynE/LTh0cHh2nTzKn2dxZvlA8b8tICwotGvFIdH0igbMQWoopDt1YAAl8Dh1/2ljqnRkIyaLwTc1j6AdkHLIRo0QZalB496QOBon3DFyRqgecLxrbR+zh8TZT1tVZxdCPpv9jn1V1ZVAoOTVnVXgfuBtQQptqDgof3jCiOoBQUU6k7LlOrPoJEYpRDouMpyXEhE7JGHoGhiQA2U9WAS3wjWGGeBQJ06HCK/a3IyGBlPPAN5MBURO5qy3Jv7SeVqOHfsLCWCsI6XrRSHOsIrxMGw+ZAKr43ABCBTN3xXRCBKHK/EnGhODuPnkftG9rrsGvd6X60yaONLpC16iMXHSP6ugFNVELUfRlpayslbO+7bSdt4vrUdvaeC7QVtmXP10ztNc= ,` X,` Sample Result: (3d Ising) [Rychkov, Poland, Vichi (2018)] [Poland, Simmons-Duffin (2016)]

AAAB+nicbZDLSgMxFIYzXmu9TXXpJlgEV2VGBF0WdeGygr1AOwyZ9EwbmkmGJKOUsY/ixoUibn0Sd76NaTsLbf0h8PGfczgnf5Rypo3nfTsrq2vrG5ulrfL2zu7evls5aGmZKQpNKrlUnYho4ExA0zDDoZMqIEnEoR2Nrqf19gMozaS4N+MUgoQMBIsZJcZaoVvp3QA3JMx7kGrGpZiEbtWreTPhZfALqKJCjdD96vUlzRIQhnKiddf3UhPkRBlGOUzKvUxDSuiIDKBrUZAEdJDPTp/gE+v0cSyVfcLgmft7IieJ1uMksp0JMUO9WJua/9W6mYkvg5yJNDMg6HxRnHFsJJ7mgPtMATV8bIFQxeytmA6JItTYtMo2BH/xy8vQOqv5lu/Oq/WrIo4SOkLH6BT56ALV0S1qoCai6BE9o1f05jw5L8678zFvXXGKmUP0R87nD7WllEg= ✏ Lowest dimension in spectrum

Dimension of O

AAAB+HicbZDLSgMxFIYz9VbrpaMu3QSL4KrMiKDLoi5cVrAX6AxDJj1tQ5PMkGSEOvRJ3LhQxK2P4s63MW1noa0/BD7+cw7n5I9TzrTxvG+ntLa+sblV3q7s7O7tV92Dw7ZOMkWhRROeqG5MNHAmoWWY4dBNFRARc+jE45tZvfMISrNEPphJCqEgQ8kGjBJjrcitBrfADYnyQLOhINPIrXl1by68Cn4BNVSoGblfQT+hmQBpKCda93wvNWFOlGGUw7QSZBpSQsdkCD2LkgjQYT4/fIpPrdPHg0TZJw2eu78nciK0nojYdgpiRnq5NjP/q/UyM7gKcybTzICki0WDjGOT4FkKuM8UUMMnFghVzN6K6YgoQo3NqmJD8Je/vArt87pv+f6i1rgu4iijY3SCzpCPLlED3aEmaiGKMvSMXtGb8+S8OO/Ox6K15BQzR+iPnM8f/aiTSw== Conformal Bootstrap

C,` g,`(u, v)= C,` g,`(v, u)

XAAACYHicnVFLS0JBGJ177WGmqbWrzZAECiL3RlCbQLJFS4N8gFdk7vipg3MfzMOQi3+yXYs2/ZLGxyK1VR98cOac7/DNnPFjzqRynE/LTh0cHh2nTzKn2dxZvlA8b8tICwotGvFIdH0igbMQWoopDt1YAAl8Dh1/2ljqnRkIyaLwTc1j6AdkHLIRo0QZalB496QOBon3DFyRqgecLxrbR+zh8TZT1tVZxdCPpv9jn1V1ZVAoOTVnVXgfuBtQQptqDgof3jCiOoBQUU6k7LlOrPoJEYpRDouMpyXEhE7JGHoGhiQA2U9WAS3wjWGGeBQJ06HCK/a3IyGBlPPAN5MBURO5qy3Jv7SeVqOHfsLCWCsI6XrRSHOsIrxMGw+ZAKr43ABCBTN3xXRCBKHK/EnGhODuPnkftG9rrsGvd6X60yaONLpC16iMXHSP6ugFNVELUfRlpayslbO+7bSdt4vrUdvaeC7QVtmXP10ztNc= ,` X,`

Can we use this philosophy to study CFT-s analytically?

Different setup ? ➔ Can we start with crossing symmetry and demand OPE structure

[Polyakov (1979)] Polyakov / Mellin Bootstrap

Start with this:

O(x )O(x )O(x )O(x ) = F (x ,x ,x ,x )+F (x x )+F (x x ) hAAACY3icdVFdS9xAFJ3EVu1u1ah9k8LQpeCiLMmu0L4IYqH4VgvuKmyWZTJ7kx2cTMLMje0a/JO++eZL/0cn2Ty0fly4l8M55zIzZ6JcCoO+/+C4K2/erq6tv2u1329sbnnbOyOTFZrDkGcy01cRMyCFgiEKlHCVa2BpJOEyuv5W6Zc3oI3I1AUucpikLFEiFpyhpabebSiZSiTQH/u/p0G3mv16Dup51KWhXhpCemz7e2U7tCbbg8PKcFBzfRpKiFGLZI5M6+wXrZdfFwfdqdfxe35d9DkIGtAhTZ1PvftwlvEiBYVcMmPGgZ/jpGQaBZdw1woLAznj1yyBsYWKpWAmZZ3RHf1smRmNM21bIa3ZfzdKlhqzSCPrTBnOzVOtIl/SxgXGXyelUHmBoPjyoLiQFDNaBU5nQgNHubCAcS3sXSmfM8042m9p2RCCp09+Dkb9XmDxz6POyWkTxzrZI5/IPgnIF3JCzsg5GRJOHp1VZ8vxnD9u291xPyytrtPs7JL/yv34F4E0sOE= 1 2 3 4 i 1 2 3 4 2 $ 4 2 $ 3

??

O(x )O(x )O(x )O(x ) = C gg(u, v(u,) v)

1 2 3 4 ,` AAAB/XicbZDLSsNAFIYn9VbrLV52boJFqFBKIoIui7pwWcFeoAlhMj1ph04mYWZSqKH4Km5cKOLW93Dn2zhts9DWHwY+/nMO58wfJIxKZdvfRmFldW19o7hZ2tre2d0z9w9aMk4FgSaJWSw6AZbAKIemoopBJxGAo4BBOxjeTOvtEQhJY/6gxgl4Ee5zGlKClbZ886jvZ+4tMIWrLjA2qaTV0Zlvlu2aPZO1DE4OZZSr4Ztfbi8maQRcEYal7Dp2orwMC0UJg0nJTSUkmAxxH7oaOY5Aetns+ol1qp2eFcZCP66smft7IsORlOMo0J0RVgO5WJua/9W6qQqvvIzyJFXAyXxRmDJLxdY0CqtHBRDFxhowEVTfapEBFpgoHVhJh+AsfnkZWuc1R/P9Rbl+ncdRRMfoBFWQgy5RHd2hBmoigh7RM3pFb8aT8WK8Gx/z1oKRzxyiPzI+fwCVsJSk ,` h i 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 X,` (Note: very specific structure)

Different setup ? ➔ Can we start with crossing symmetry and demand OPE structure Polyakov / Mellin Bootstrap

Start with this:

O(x )O(x )O(x )O(x ) = F (x ,x ,x ,x )+F (x x )+F (x x ) 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 1 2 3 4 i 1 2 3 4 2 $ 4 2 $ 3

??

O(x )O(x )O(x )O(x ) = C gg(u, v(u,) v)

1 2 3 4 ,` AAAB/XicbZDLSsNAFIYn9VbrLV52boJFqFBKIoIui7pwWcFeoAlhMj1ph04mYWZSqKH4Km5cKOLW93Dn2zhts9DWHwY+/nMO58wfJIxKZdvfRmFldW19o7hZ2tre2d0z9w9aMk4FgSaJWSw6AZbAKIemoopBJxGAo4BBOxjeTOvtEQhJY/6gxgl4Ee5zGlKClbZ886jvZ+4tMIWrLjA2qaTV0Zlvlu2aPZO1DE4OZZSr4Ztfbi8maQRcEYal7Dp2orwMC0UJg0nJTSUkmAxxH7oaOY5Aetns+ol1qp2eFcZCP66smft7IsORlOMo0J0RVgO5WJua/9W6qQqvvIzyJFXAyXxRmDJLxdY0CqtHBRDFxhowEVTfapEBFpgoHVhJh+AsfnkZWuc1R/P9Rbl+ncdRRMfoBFWQgy5RHd2hBmoigh7RM3pFb8aT8WK8Gx/z1oKRzxyiPzI+fwCVsJSk ,` h i 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 X,` (Note: very specific structure)

Possible if F -s are Witten diagrams W (u, v)

AAAB/XicbZDLSsNAFIYn9VbrLV52boJFqFBKIoIui7pwWcFeoAlhMj1ph04mYWZSqKH4Km5cKOLW93Dn2zhts9DWHwY+/nMO58wfJIxKZdvfRmFldW19o7hZ2tre2d0z9w9aMk4FgSaJWSw6AZbAKIemoopBJxGAo4BBOxjeTOvtEQhJY/6gxgl4Ee5zGlKClbZ886jtZ+4tMIWrLjA2qaTV0Zlvlu2aPZO1DE4OZZSr4Ztfbi8maQRcEYal7Dp2orwMC0UJg0nJTSUkmAxxH7oaOY5Aetns+ol1qp2eFcZCP66smft7IsORlOMo0J0RVgO5WJua/9W6qQqvvIzyJFXAyXxRmDJLxdY0CqtHBRDFxhowEVTfapEBFpgoHVhJh+AsfnkZWuc1R/P9Rbl+ncdRRMfoBFWQgy5RHd2hBmoigh7RM3pFb8aT8WK8Gx/z1oKRzxyiPzI+fwB8QJSU ,` Polyakov / Mellin Bootstrap

[, AK, Kallol Sen, Aninda Sinha (2016)]

O(x )O(x )O(x )O(x ) = F (x ,x ,x ,xs )+F (x xt )+F (x xu ) AAACY3icdVFdS9xAFJ3EVu1u1ah9k8LQpeCiLMmu0L4IYqH4VgvuKmyWZTJ7kx2cTMLMje0a/JO++eZL/0cn2Ty0fly4l8M55zIzZ6JcCoO+/+C4K2/erq6tv2u1329sbnnbOyOTFZrDkGcy01cRMyCFgiEKlHCVa2BpJOEyuv5W6Zc3oI3I1AUucpikLFEiFpyhpabebSiZSiTQH/u/p0G3mv16Dup51KWhXhpCemz7e2U7tCbbg8PKcFBzfRpKiFGLZI5M6+wXrZdfFwfdqdfxe35d9DkIGtAhTZ1PvftwlvEiBYVcMmPGgZ/jpGQaBZdw1woLAznj1yyBsYWKpWAmZZ3RHf1smRmNM21bIa3ZfzdKlhqzSCPrTBnOzVOtIl/SxgXGXyelUHmBoPjyoLiQFDNaBU5nQgNHubCAcS3sXSmfM8042m9p2RCCp09+Dkb9XmDxz6POyWkTxzrZI5/IPgnIF3JCzsg5GRJOHp1VZ8vxnD9u291xPyytrtPs7JL/yv34F4E0sOE= 1 2 3 4 = 1C 2 W3 4 (u, v)+2 W 4 (u, v)+2 W 3 (u, v) h i ,` ,` $ ,` $ ,` 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 X,` h i 1 4 1 4 1 4

3 2 3 3 2 2 Polyakov / Mellin Bootstrap

[Rajesh Gopakumar, AK, Kallol Sen, Aninda Sinha (2016)]

O(x )O(x )O(x )O(x ) = F (x ,x ,x ,xs )+F (x xt )+F (x xu ) 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 1 2 3 4 = 1C 2 W3 4 (u, v)+2 W 4 (u, v)+2 W 3 (u, v) h i ,` ,` $ ,` $ ,` AAACZHicdZFLSwMxFIUz47u+RsWVIMEiKEqZEUE3grQuXCpYK3TGkklv29DMg+RGKEP/pDuXbvwdpo+FtXoh8HHOvXmcxLkUGn3/w3EXFpeWV1bXSusbm1vb3s7us86M4lDnmczUS8w0SJFCHQVKeMkVsCSW0Ij7tZHfeAOlRZY+4SCHKGHdVHQEZ2illlfc0FCbpFWEdyCRnYcg5ZCGtDarhFXRbdLGrPha6OGJOX87PZsz8D/DTIzRflHLK/sVf1x0HoIplMm0Hlree9jOuEkgRS6Z1s3AzzEqmELBJQxLodGQM95nXWhaTFkCOirGIQ3psVXatJMpu1KkY/XnRMESrQdJbDsThj392xuJf3lNg53rqBBpbhBSPjmoYyTFjI4Sp22hgKMcWGBcCXtXyntMMY72X0o2hOD3k+fh+aISWH68LN9Wp3GskgNyRE5IQK7ILbknD6ROOPl0VhzP2XG+3A13z92ftLrOdGaPzJR7+A2Z7rb8 X,` h i 1 4 1 4 1 4

3 2 3 3 2 2 Why it works?

• Basis of Witten diagrams are well behaved at large dimensions • Witten diagrams encode the conformal blocks Polyakov / Mellin Bootstrap

O(x )O(x )O(x )O(x ) = F (x ,x ,x ,xs )+F (x xt )+F (x xu ) 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 1 2 3 4 = 1C 2 W3 4 (u, v)+2 W 4 (u, v)+2 W 3 (u, v) h i ,` ,` $ ,` $ ,` 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 X,` h i

= C,` g,`(u, v)

AAACKXicbZBLS0JBFMfn9jR73WrZZkgCA5F7I6hNINqipUE+wCsydzzq4NwH8xDk4tdp01dpU1BU275Io95Fan8Y+PM753Dm/P2YM6kc58taW9/Y3NrO7GR39/YPDu2j47qMtKBQoxGPRNMnEjgLoaaY4tCMBZDA59Dwh5VpvTECIVkUPqpxDO2A9EPWY5Qogzp26daTOugk3h1wRQoecD7BXpn1W7iyTHF/keR1YXTRsXNO0ZkJrxo3NTmUqtqx37xuRHUAoaKcSNlynVi1EyIUoxwmWU9LiAkdkj60jA1JALKdzC6d4HNDurgXCfNChWf070RCAinHgW86A6IGcrk2hf/VWlr1btoJC2OtIKTzRT3NsYrwNDbcZQKo4mNjCBXM/BXTARGEKhNu1oTgLp+8auqXRdf4h6tcqZzGkUGn6AzlkYuuUQndoyqqIYqe0At6Rx/Ws/VqfVrf89Y1K505QQuyfn4BOSel9g== X,` h

+ qn(, `)O +2n+`(u, v)+ q˜n(, `)g20 O +2n+`(u, v)

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 n n X X i

: Dimension of O AAAB73icbZBNS8NAEIY39avWr6pHL4tF8FQSEfRY1IM3K9gPaEPZbCft0s0m7k6EEvonvHhQxKt/x5v/xm2bg7a+sPDwzgw78waJFAZd99sprKyurW8UN0tb2zu7e+X9g6aJU82hwWMZ63bADEihoIECJbQTDSwKJLSC0fW03noCbUSsHnCcgB+xgRKh4Ayt1e7egETWu+uVK27VnYkug5dDheSq98pf3X7M0wgUcsmM6Xhugn7GNAouYVLqpgYSxkdsAB2LikVg/Gy274SeWKdPw1jbp5DO3N8TGYuMGUeB7YwYDs1ibWr+V+ukGF76mVBJiqD4/KMwlRRjOj2e9oUGjnJsgXEt7K6UD5lmHG1EJRuCt3jyMjTPqp7l+/NK7SqPo0iOyDE5JR65IDVyS+qkQTiR5Jm8kjfn0Xlx3p2PeWvByWcOyR85nz+3Ho++ O Polyakov / Mellin Bootstrap

O(x )O(x )O(x )O(x ) = F (x ,x ,x ,xs )+F (x xt )+F (x xu ) 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 1 2 3 4 = 1C 2 W3 4 (u, v)+2 W 4 (u, v)+2 W 3 (u, v) h i ,` ,` $ ,` $ ,` 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 X,` h i

OPE

= C,` g,`(u, v)

AAACKXicbZBLS0JBFMfn9jR73WrZZkgCA5F7I6hNINqipUE+wCsydzzq4NwH8xDk4tdp01dpU1BU275Io95Fan8Y+PM753Dm/P2YM6kc58taW9/Y3NrO7GR39/YPDu2j47qMtKBQoxGPRNMnEjgLoaaY4tCMBZDA59Dwh5VpvTECIVkUPqpxDO2A9EPWY5Qogzp26daTOugk3h1wRQoecD7BXpn1W7iyTHF/keR1YXTRsXNO0ZkJrxo3NTmUqtqx37xuRHUAoaKcSNlynVi1EyIUoxwmWU9LiAkdkj60jA1JALKdzC6d4HNDurgXCfNChWf070RCAinHgW86A6IGcrk2hf/VWlr1btoJC2OtIKTzRT3NsYrwNDbcZQKo4mNjCBXM/BXTARGEKhNu1oTgLp+8auqXRdf4h6tcqZzGkUGn6AzlkYuuUQndoyqqIYqe0At6Rx/Ws/VqfVrf89Y1K505QQuyfn4BOSel9g== X,` h

+ qn(, `)g2O +2n+`(u, v)+ q˜n(, `)g20 O +2n+`(u, v)

{ 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 n n X X i

• Extra terms • Must Vanish for a true CFT : Dimension of O AAAB73icbZBNS8NAEIY39avWr6pHL4tF8FQSEfRY1IM3K9gPaEPZbCft0s0m7k6EEvonvHhQxKt/x5v/xm2bg7a+sPDwzgw78waJFAZd99sprKyurW8UN0tb2zu7e+X9g6aJU82hwWMZ63bADEihoIECJbQTDSwKJLSC0fW03noCbUSsHnCcgB+xgRKh4Ayt1e7egETWu+uVK27VnYkug5dDheSq98pf3X7M0wgUcsmM6Xhugn7GNAouYVLqpgYSxkdsAB2LikVg/Gy274SeWKdPw1jbp5DO3N8TGYuMGUeB7YwYDs1ibWr+V+ukGF76mVBJiqD4/KMwlRRjOj2e9oUGjnJsgXEt7K6UD5lmHG1EJRuCt3jyMjTPqp7l+/NK7SqPo0iOyDE5JR65IDVyS+qkQTiR5Jm8kjfn0Xlx3p2PeWvByWcOyR85nz+3Ho++ O Polyakov / Mellin Bootstrap

O(x )O(x )O(x )O(x ) = F (x ,x ,x ,xs )+F (x xt )+F (x xu ) 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 1 2 3 4 = 1C 2 W3 4 (u, v)+2 W 4 (u, v)+2 W 3 (u, v) h i ,` ,` $ ,` $ ,` AAACZHicdZFLSwMxFIUz47u+RsWVIMEiKEqZEUE3grQuXCpYK3TGkklv29DMg+RGKEP/pDuXbvwdpo+FtXoh8HHOvXmcxLkUGn3/w3EXFpeWV1bXSusbm1vb3s7us86M4lDnmczUS8w0SJFCHQVKeMkVsCSW0Ij7tZHfeAOlRZY+4SCHKGHdVHQEZ2illlfc0FCbpFWEdyCRnYcg5ZCGtDarhFXRbdLGrPha6OGJOX87PZsz8D/DTIzRflHLK/sVf1x0HoIplMm0Hlree9jOuEkgRS6Z1s3AzzEqmELBJQxLodGQM95nXWhaTFkCOirGIQ3psVXatJMpu1KkY/XnRMESrQdJbDsThj392xuJf3lNg53rqBBpbhBSPjmoYyTFjI4Sp22hgKMcWGBcCXtXyntMMY72X0o2hOD3k+fh+aISWH68LN9Wp3GskgNyRE5IQK7ILbknD6ROOPl0VhzP2XG+3A13z92ftLrOdGaPzJR7+A2Z7rb8 X,` h i = C g (u, v) ,` ,` + qn(, `)g2O +2n+`(u, v)+ q˜n(, `)g20 O +2n+`(u, v)

AAACKXicbZBLS0JBFMfn9jR73WrZZkgCA5F7I6hNINqipUE+wCsydzzq4NwH8xDk4tdp01dpU1BU275Io95Fan8Y+PM753Dm/P2YM6kc58taW9/Y3NrO7GR39/YPDu2j47qMtKBQoxGPRNMnEjgLoaaY4tCMBZDA59Dwh5VpvTECIVkUPqpxDO2A9EPWY5Qogzp26daTOugk3h1wRQoecD7BXpn1W7iyTHF/keR1YXTRsXNO0ZkJrxo3NTmUqtqx37xuRHUAoaKcSNlynVi1EyIUoxwmWU9LiAkdkj60jA1JALKdzC6d4HNDurgXCfNChWf070RCAinHgW86A6IGcrk2hf/VWlr1btoJC2OtIKTzRT3NsYrwNDbcZQKo4mNjCBXM/BXTARGEKhNu1oTgLp+8auqXRdf4h6tcqZzGkUGn6AzlkYuuUQndoyqqIYqe0At6Rx/Ws/VqfVrf89Y1K505QQuyfn4BOSel9g== ,` AAACbHicfVFdSxtBFJ1d61fUGrUPLSIMBm0kEnaDoI9i+9A3LTQqZJdldnKzGZyd3czcFcKyT/2HffMn9MXf0MnHQ43igYHDuffcO3MmzqUw6HlPjrv0YXlldW29trG59XG7vrN7a7JCc+jyTGb6PmYGpFDQRYES7nMNLI0l3MUP3yb1u0fQRmTqF45zCFOWKDEQnKGVovrvgLZoQANTpFGpqlGkmsF3kMhOA5DyhCZR2ZkJ0XWro1oTtWoWp48n9KUzQCH7UI6qxQk0+frOjCuRhFG94bW9Kehr4s9Jg8xxE9X/BP2MFyko5JIZ0/O9HMOSaRRcQlULCgM54w8sgZ6liqVgwnIaVkWPrNKng0zbo5BO1f8dJUuNGaex7UwZDs1ibSK+VesVOLgIS6HyAkHx2aJBISlmdJI87QsNHOXYEsa1sHelfMg042j/p2ZD8Bef/Jrcdtq+5T/PGpdX8zjWyD45JE3ik3NySX6QG9IlnPx1tp3Pzhfn2f3k7rsHs1bXmXv2yAu4x/8ASAC3JA== n n X h X X i

qn(, `)=0 q˜n(, `)=0

XAAACUHicbVFNSwMxEJ2t3/Wr6tFLsAgKIrsi6EUQ9eBRwarQLUs2ndZgNrtNZoWy7E/04s3f4cWDommt4EcnhHm8N48kL3GmpCXff/YqE5NT0zOzc9X5hcWl5drK6rVNcyOwIVKVmtuYW1RSY4MkKbzNDPIkVngT358O9JsHNFam+or6GbYS3tWyIwUnR0W1bmjzJCrCM1TEd0JUqmQh60WFLrd+kNvsiPlOGK1xnpCkamPRK8d6o1rd3/WHxf6DYATqMKqLqPYUtlORJ6hJKG5tM/AzahXckBQKy2qYW8y4uOddbDqoeYK2VQwDKdmmY9qskxq3NbEh+9NR8MTafhK7yYTTnf2rDchxWjOnzmGrkDrLCbX4OqiTK0YpG6TL2tKgINV3gAsj3V2ZuOOGC3J/UHUhBH+f/B9c7+0GDl/u149PRnHMwjpswBYEcADHcA4X0AABj/ACb/DuPXmv3kfF+xr97rAGv6pS/QQf2LIF ,` X,`

• Polyakov Bootstrap equations • Completely algebraic (no x dependence) Polyakov / Mellin Bootstrap

Example 4

theory in AAAB83icbZDLSgMxFIbP1Futt6pLN4NFcGOZkYJuhKIblxXsBTpDyWQybWgmCUlGKENfw40LRdz6Mu58G9N2Ftr6Q+DjP+dwTv5IMqqN5307pbX1jc2t8nZlZ3dv/6B6eNTRIlOYtLFgQvUipAmjnLQNNYz0pCIojRjpRuO7Wb37RJSmgj+aiSRhioacJhQjY60gvmlcBERqygQfVGte3ZvLXQW/gBoUag2qX0EscJYSbjBDWvd9T5owR8pQzMi0EmSaSITHaEj6FjlKiQ7z+c1T98w6sZsIZR837tz9PZGjVOtJGtnOFJmRXq7NzP9q/cwk12FOucwM4XixKMmYa4Q7C8CNqSLYsIkFhBW1t7p4hBTCxsZUsSH4y19ehc5l3bf80Kg1b4s4ynACp3AOPlxBE+6hBW3AIOEZXuHNyZwX5935WLSWnGLmGP7I+fwBcbeRSg== (Wilson Fisher CFT)

AAAB+HicbVDLSsNAFL2pr1ofjbp0M1gEVyWRQl0W3bisYB/QxDKZTNqhk0mYmQg19EvcuFDErZ/izr9x2mahrQcGDuecy71zgpQzpR3n2yptbG5t75R3K3v7B4dV++i4q5JMEtohCU9kP8CKciZoRzPNaT+VFMcBp71gcjP3e49UKpaIez1NqR/jkWARI1gbaWhXPW7CIUZeOmYPDTS0a07dWQCtE7cgNSjQHtpfXpiQLKZCE46VGrhOqv0cS80Ip7OKlymaYjLBIzowVOCYKj9fHD5D50YJUZRI84RGC/X3RI5jpaZxYJIx1mO16s3F/7xBpqMrP2cizTQVZLkoyjjSCZq3gEImKdF8aggmkplbERljiYk2XVVMCe7ql9dJ97LuGn7XqLWuizrKcApncAEuNKEFt9CGDhDI4Ble4c16sl6sd+tjGS1ZxcwJ/IH1+QOrppJt d =4 ✏

qn(, `)=0 q˜n(, `)=0

XAAACUHicbVFNSwMxEJ2t3/Wr6tFLsAgKIrsi6EUQ9eBRwarQLUs2ndZgNrtNZoWy7E/04s3f4cWDommt4EcnhHm8N48kL3GmpCXff/YqE5NT0zOzc9X5hcWl5drK6rVNcyOwIVKVmtuYW1RSY4MkKbzNDPIkVngT358O9JsHNFam+or6GbYS3tWyIwUnR0W1bmjzJCrCM1TEd0JUqmQh60WFLrd+kNvsiPlOGK1xnpCkamPRK8d6o1rd3/WHxf6DYATqMKqLqPYUtlORJ6hJKG5tM/AzahXckBQKy2qYW8y4uOddbDqoeYK2VQwDKdmmY9qskxq3NbEh+9NR8MTafhK7yYTTnf2rDchxWjOnzmGrkDrLCbX4OqiTK0YpG6TL2tKgINV3gAsj3V2ZuOOGC3J/UHUhBH+f/B9c7+0GDl/u149PRnHMwjpswBYEcADHcA4X0AABj/ACb/DuPXmv3kfF+xr97rAGv6pS/QQf2LIF ,` X,`

• Polyakov Bootstrap equations • Completely algebraic (no x dependence) Polyakov / Mellin Bootstrap

Example 4

theory in AAAB83icbZDLSgMxFIbP1Futt6pLN4NFcGOZkYJuhKIblxXsBTpDyWQybWgmCUlGKENfw40LRdz6Mu58G9N2Ftr6Q+DjP+dwTv5IMqqN5307pbX1jc2t8nZlZ3dv/6B6eNTRIlOYtLFgQvUipAmjnLQNNYz0pCIojRjpRuO7Wb37RJSmgj+aiSRhioacJhQjY60gvmlcBERqygQfVGte3ZvLXQW/gBoUag2qX0EscJYSbjBDWvd9T5owR8pQzMi0EmSaSITHaEj6FjlKiQ7z+c1T98w6sZsIZR837tz9PZGjVOtJGtnOFJmRXq7NzP9q/cwk12FOucwM4XixKMmYa4Q7C8CNqSLYsIkFhBW1t7p4hBTCxsZUsSH4y19ehc5l3bf80Kg1b4s4ynACp3AOPlxBE+6hBW3AIOEZXuHNyZwX5935WLSWnGLmGP7I+fwBcbeRSg== (Wilson Fisher CFT)

AAAB+HicbVDLSsNAFL2pr1ofjbp0M1gEVyWRQl0W3bisYB/QxDKZTNqhk0mYmQg19EvcuFDErZ/izr9x2mahrQcGDuecy71zgpQzpR3n2yptbG5t75R3K3v7B4dV++i4q5JMEtohCU9kP8CKciZoRzPNaT+VFMcBp71gcjP3e49UKpaIez1NqR/jkWARI1gbaWhXPW7CIUZeOmYPDTS0a07dWQCtE7cgNSjQHtpfXpiQLKZCE46VGrhOqv0cS80Ip7OKlymaYjLBIzowVOCYKj9fHD5D50YJUZRI84RGC/X3RI5jpaZxYJIx1mO16s3F/7xBpqMrP2cizTQVZLkoyjjSCZq3gEImKdF8aggmkplbERljiYk2XVVMCe7ql9dJ97LuGn7XqLWuizrKcApncAEuNKEFt9CGDhDI4Ble4c16sl6sd+tjGS1ZxcwJ/IH1+QOrppJt d =4 ✏ Input: • Conserved stress tensor T AAAB/nicbZDLSsNAFIYn9VbrrSqu3AwWwY0lkYJuhKIuXFboDZpQJtOTduhkEmYmQgkFX8WNC0Xc+hzufBsnbRba+sPAx3/O4Zz5/ZgzpW372yqsrK6tbxQ3S1vbO7t75f2DtooSSaFFIx7Jrk8UcCagpZnm0I0lkNDn0PHHt1m98whSsUg09SQGLyRDwQJGiTZWv3zk3gHXpN/E17h27kKsGM/8il21Z8LL4ORQQbka/fKXO4hoEoLQlBOleo4day8lUjPKYVpyEwUxoWMyhJ5BQUJQXjo7f4pPjTPAQSTNExrP3N8TKQmVmoS+6QyJHqnFWmb+V+slOrjyUibiRIOg80VBwrGOcJYFHjAJVPOJAUIlM7diOiKSUG0SK5kQnMUvL0P7ouoYfqhV6jd5HEV0jE7QGXLQJaqje9RALURRip7RK3qznqwX6936mLcWrHzmEP2R9fkDsf6Uqg== =4 ✏ T 2

Unknowns of the form AAACJHicbZDLSgMxFIYz9VbrrerSTbAIgiAzRVAQoagLlxVsFTq1ZNJTDc0kQ3JGKKUP48ZXcePCCy7c+Cym7RS8HQj5+P9zSM4fJVJY9P0PLzc1PTM7l58vLCwuLa8UV9fqVqeGQ41rqc1VxCxIoaCGAiVcJQZYHEm4jLonQ//yDowVWl1gL4FmzG6U6AjO0Emt4mF4ChLZ0fhq+TsZBCEkVkit6EQpT5Tr8k7I2xptq1jyd/1R0b8QZFAiWVVbxdewrXkag0IumbWNwE+w2WcGBZcwKISphYTxLruBhkPFYrDN/mjJAd1ySpt2tHFHIR2p3yf6LLa2F0euM2Z4a397Q/E/r5Fi56DZFypJERQfP9RJJUVNh4nRtjDAUfYcMG6E+yvlt8wwji7Xggsh+L3yX6iXdwPH53ulynEWR55skE2yTQKyTyrkjFRJjXByTx7JM3nxHrwn7817H7fmvGxmnfwo7/ML2DWkVA== = 0 + 1✏ + 2✏ + • ··· C = C + C ✏ + C ✏2 + AAACH3icbVDLSgMxFM34rPVVdekmWARBkZki6kYo1oVLBVuFTh0y6a2GZpIhuSOUoX/ixl9x40IRceffmD4EXwcC555zLzf3xKkUFn3/w5uYnJqemS3MFecXFpeWSyurDaszw6HOtdTmKmYWpFBQR4ESrlIDLIklXMbd2sC/vANjhVYX2EuhlbAbJTqCM3RSVNqvRXl4AhLZTghS9o9qkb9di4IQUiukVtQVla/iukK3Q97WaKNS2d/1h6B/STAmZTLGWVR6D9uaZwko5JJZ2wz8FFs5Myi4hH4xzCykjHfZDTQdVSwB28qH9/XpplPatKONewrpUP0+kbPE2l4Su86E4a397Q3E/7xmhp3DVi5UmiEoPlrUySRFTQdh0bYwwFH2HGHcCPdXym+ZYRxdpEUXQvD75L+kUdkNHD/fK1ePx3EUyDrZIFskIAekSk7JGakTTu7JI3kmL96D9+S9em+j1glvPLNGfsD7+ATSLqGY ,` 0 1 2 ···

qn(, `)=0 q˜n(, `)=0

XAAACUHicbVFNSwMxEJ2t3/Wr6tFLsAgKIrsi6EUQ9eBRwarQLUs2ndZgNrtNZoWy7E/04s3f4cWDommt4EcnhHm8N48kL3GmpCXff/YqE5NT0zOzc9X5hcWl5drK6rVNcyOwIVKVmtuYW1RSY4MkKbzNDPIkVngT358O9JsHNFam+or6GbYS3tWyIwUnR0W1bmjzJCrCM1TEd0JUqmQh60WFLrd+kNvsiPlOGK1xnpCkamPRK8d6o1rd3/WHxf6DYATqMKqLqPYUtlORJ6hJKG5tM/AzahXckBQKy2qYW8y4uOddbDqoeYK2VQwDKdmmY9qskxq3NbEh+9NR8MTafhK7yYTTnf2rDchxWjOnzmGrkDrLCbX4OqiTK0YpG6TL2tKgINV3gAsj3V2ZuOOGC3J/UHUhBH+f/B9c7+0GDl/u149PRnHMwjpswBYEcADHcA4X0AABj/ACb/DuPXmv3kfF+xr97rAGv6pS/QQf2LIF ,` X,`

• Polyakov Bootstrap equations • Completely algebraic (no x dependence) Polyakov / Mellin Bootstrap

Example 4

theory in AAAB83icbZDLSgMxFIbP1Futt6pLN4NFcGOZkYJuhKIblxXsBTpDyWQybWgmCUlGKENfw40LRdz6Mu58G9N2Ftr6Q+DjP+dwTv5IMqqN5307pbX1jc2t8nZlZ3dv/6B6eNTRIlOYtLFgQvUipAmjnLQNNYz0pCIojRjpRuO7Wb37RJSmgj+aiSRhioacJhQjY60gvmlcBERqygQfVGte3ZvLXQW/gBoUag2qX0EscJYSbjBDWvd9T5owR8pQzMi0EmSaSITHaEj6FjlKiQ7z+c1T98w6sZsIZR837tz9PZGjVOtJGtnOFJmRXq7NzP9q/cwk12FOucwM4XixKMmYa4Q7C8CNqSLYsIkFhBW1t7p4hBTCxsZUsSH4y19ehc5l3bf80Kg1b4s4ynACp3AOPlxBE+6hBW3AIOEZXuHNyZwX5935WLSWnGLmGP7I+fwBcbeRSg== (Wilson Fisher CFT)

AAAB+HicbVDLSsNAFL2pr1ofjbp0M1gEVyWRQl0W3bisYB/QxDKZTNqhk0mYmQg19EvcuFDErZ/izr9x2mahrQcGDuecy71zgpQzpR3n2yptbG5t75R3K3v7B4dV++i4q5JMEtohCU9kP8CKciZoRzPNaT+VFMcBp71gcjP3e49UKpaIez1NqR/jkWARI1gbaWhXPW7CIUZeOmYPDTS0a07dWQCtE7cgNSjQHtpfXpiQLKZCE46VGrhOqv0cS80Ip7OKlymaYjLBIzowVOCYKj9fHD5D50YJUZRI84RGC/X3RI5jpaZxYJIx1mO16s3F/7xBpqMrP2cizTQVZLkoyjjSCZq3gEImKdF8aggmkplbERljiYk2XVVMCe7ql9dJ97LuGn7XqLWuizrKcApncAEuNKEFt9CGDhDI4Ble4c16sl6sd+tjGS1ZxcwJ/IH1+QOrppJt d =4 ✏ Input: Results:

• Conserved stress tensor T Just solve order by order AAAB/nicbZDLSsNAFIYn9VbrrSqu3AwWwY0lkYJuhKIuXFboDZpQJtOTduhkEmYmQgkFX8WNC0Xc+hzufBsnbRba+sPAx3/O4Zz5/ZgzpW372yqsrK6tbxQ3S1vbO7t75f2DtooSSaFFIx7Jrk8UcCagpZnm0I0lkNDn0PHHt1m98whSsUg09SQGLyRDwQJGiTZWv3zk3gHXpN/E17h27kKsGM/8il21Z8LL4ORQQbka/fKXO4hoEoLQlBOleo4day8lUjPKYVpyEwUxoWMyhJ5BQUJQXjo7f4pPjTPAQSTNExrP3N8TKQmVmoS+6QyJHqnFWmb+V+slOrjyUibiRIOg80VBwrGOcJYFHjAJVPOJAUIlM7diOiKSUG0SK5kQnMUvL0P7ouoYfqhV6jd5HEV0jE7QGXLQJaqje9RALURRip7RK3qznqwX6936mLcWrHzmEP2R9fkDsf6Uqg== T =4 ✏ 2

Unknowns of the form AAACJHicbZDLSgMxFIYz9VbrrerSTbAIgiAzRVAQoagLlxVsFTq1ZNJTDc0kQ3JGKKUP48ZXcePCCy7c+Cym7RS8HQj5+P9zSM4fJVJY9P0PLzc1PTM7l58vLCwuLa8UV9fqVqeGQ41rqc1VxCxIoaCGAiVcJQZYHEm4jLonQ//yDowVWl1gL4FmzG6U6AjO0Emt4mF4ChLZ0fhq+TsZBCEkVkit6EQpT5Tr8k7I2xptq1jyd/1R0b8QZFAiWVVbxdewrXkag0IumbWNwE+w2WcGBZcwKISphYTxLruBhkPFYrDN/mjJAd1ySpt2tHFHIR2p3yf6LLa2F0euM2Z4a397Q/E/r5Fi56DZFypJERQfP9RJJUVNh4nRtjDAUfYcMG6E+yvlt8wwji7Xggsh+L3yX6iXdwPH53ulynEWR55skE2yTQKyTyrkjFRJjXByTx7JM3nxHrwn7817H7fmvGxmnfwo7/ML2DWkVA== = 0 + 1✏ + 2✏ + • ··· C = C + C ✏ + C ✏2 +

AAACH3icbVDLSgMxFM34rPVVdekmWARBkZki6kYo1oVLBVuFTh0y6a2GZpIhuSOUoX/ixl9x40IRceffmD4EXwcC555zLzf3xKkUFn3/w5uYnJqemS3MFecXFpeWSyurDaszw6HOtdTmKmYWpFBQR4ESrlIDLIklXMbd2sC/vANjhVYX2EuhlbAbJTqCM3RSVNqvRXl4AhLZTghS9o9qkb9di4IQUiukVtQVla/iukK3Q97WaKNS2d/1h6B/STAmZTLGWVR6D9uaZwko5JJZ2wz8FFs5Myi4hH4xzCykjHfZDTQdVSwB28qH9/XpplPatKONewrpUP0+kbPE2l4Su86E4a397Q3E/7xmhp3DVi5UmiEoPlrUySRFTQdh0bYwwFH2HGHcCPdXym+ZYRxdpEUXQvD75L+kUdkNHD/fK1ePx3EUyDrZIFskIAekSk7JGakTTu7JI3kmL96D9+S9em+j1glvPLNGfsD7+ATSLqGY Perturbative results ,` 0 1 2 ··· of Feynman diagrams

qn(, `)=0 q˜n(, `)=0

XAAACUHicbVFNSwMxEJ2t3/Wr6tFLsAgKIrsi6EUQ9eBRwarQLUs2ndZgNrtNZoWy7E/04s3f4cWDommt4EcnhHm8N48kL3GmpCXff/YqE5NT0zOzc9X5hcWl5drK6rVNcyOwIVKVmtuYW1RSY4MkKbzNDPIkVngT358O9JsHNFam+or6GbYS3tWyIwUnR0W1bmjzJCrCM1TEd0JUqmQh60WFLrd+kNvsiPlOGK1xnpCkamPRK8d6o1rd3/WHxf6DYATqMKqLqPYUtlORJ6hJKG5tM/AzahXckBQKy2qYW8y4uOddbDqoeYK2VQwDKdmmY9qskxq3NbEh+9NR8MTafhK7yYTTnf2rDchxWjOnzmGrkDrLCbX4OqiTK0YpG6TL2tKgINV3gAsj3V2ZuOOGC3J/UHUhBH+f/B9c7+0GDl/u149PRnHMwjpswBYEcADHcA4X0AABj/ACb/DuPXmv3kfF+xr97rAGv6pS/QQf2LIF ,` X,`

• Polyakov Bootstrap equations • Completely algebraic (no x dependence) Polyakov / Mellin Bootstrap

Sampling of results

C & AAAB7XicbZBNSwMxEIZn/az1q+rRS7AInsquCHos6sFjBfsB7VKyabaNzSZLMiuU0v/gxYMiXv0/3vw3pu0etPWFwMM7M2TmjVIpLPr+t7eyura+sVnYKm7v7O7tlw4OG1ZnhvE601KbVkQtl0LxOgqUvJUaTpNI8mY0vJnWm0/cWKHVA45SHia0r0QsGEVnNTq3XCLtlsp+xZ+JLEOQQxly1bqlr05PsyzhCpmk1rYDP8VwTA0KJvmk2MksTykb0j5vO1Q04TYcz7adkFPn9EisjXsKycz9PTGmibWjJHKdCcWBXaxNzf9q7Qzjq3AsVJohV2z+UZxJgppMTyc9YThDOXJAmRFuV8IG1FCGLqCiCyFYPHkZGueVwPH9Rbl6ncdRgGM4gTMI4BKqcAc1qAODR3iGV3jztPfivXsf89YVL585gj/yPn8AYI6O/A== for AAAB+HicbZDLSsNAFIYn9VbrpVGXbgaL4EJKIoIui3XhsoK9QBPCZHrSDp1MwsxEqKFP4saFIm59FHe+jdM2C239YeDjP+dwzvxhypnSjvNtldbWNza3ytuVnd29/ap9cNhRSSYptGnCE9kLiQLOBLQ10xx6qQQShxy64bg5q3cfQSqWiAc9ScGPyVCwiFGijRXY1WaQe7fANTn3gPNpYNecujMXXgW3gBoq1ArsL2+Q0CwGoSknSvVdJ9V+TqRmlMO04mUKUkLHZAh9g4LEoPx8fvgUnxpngKNEmic0nru/J3ISKzWJQ9MZEz1Sy7WZ+V+tn+no2s+ZSDMNgi4WRRnHOsGzFPCASaCaTwwQKpm5FdMRkYRqk1XFhOAuf3kVOhd11/D9Za1xU8RRRsfoBJ0hF12hBrpDLdRGFGXoGb2iN+vJerHerY9Fa8kqZo7QH1mfP2HykuY= ,`

(Low dim. Scalar) 2 Wilson Fisher CFT,

AAAB7nicbZBNSwMxEIZn/az1q+rRS7AInspuEfRY9OKxgv2Adi3ZNG1Ds9mQzApl6Y/w4kERr/4eb/4b03YP2vpC4OGdGTLzRloKi77/7a2tb2xubRd2irt7+weHpaPjpk1Sw3iDJTIx7YhaLoXiDRQoeVsbTuNI8lY0vp3VW0/cWJGoB5xoHsZ0qMRAMIrOanX1SDxWSa9U9iv+XGQVghzKkKveK311+wlLY66QSWptJ/A1hhk1KJjk02I3tVxTNqZD3nGoaMxtmM3XnZJz5/TJIDHuKSRz9/dERmNrJ3HkOmOKI7tcm5n/1TopDq7DTCidIlds8dEglQQTMrud9IXhDOXEAWVGuF0JG1FDGbqEii6EYPnkVWhWK4Hj+8ty7SaPowCncAYXEMAV1OAO6tAABmN4hld487T34r17H4vWNS+fOYE/8j5/AJLijw0= } O(N) vector models - new results (Spin �) @µ @µ AAACHnicbVBLSwMxGMzWV62vVY9egkXwVHZF0WPRi8cK9gHdZclm0zY0m4QkK5Slv8SLf8WLB0UET/pvzLZ70NaBwDDzTZJvYsmoNp737VRWVtfWN6qbta3tnd09d/+go0WmMGljwYTqxUgTRjlpG2oY6UlFUBoz0o3HN4XffSBKU8HvzUSSMEVDTgcUI2OlyL0I5IjCQCJlKGJRHqRZ5E9hgBNh9IIeEMasZQORW/ca3gxwmfglqYMSrcj9DBKBs5RwgxnSuu970oR5cTtmZFoLMk0kwmM0JH1LOUqJDvPZelN4YpUEDoSyhxs4U38ncpRqPUljO5kiM9KLXiH+5/UzM7gKc8plZgjH84cGGYNGwKIrmFBFsGETSxBW1P4V4hFSCBvbaM2W4C+uvEw6Zw3f8rvzevO6rKMKjsAxOAU+uARNcAtaoA0weATP4BW8OU/Oi/PufMxHK06ZOQR/4Hz9AEynozQ= 1 ` ··· [Gopakumar, AK, Sen, Sinha (2016)] [Parijat Dey, AK, Aninda Sinha (2016)] (High dim. Scalar) 4

AAAB7XicbZDLSgMxFIbP1Futt6pLN8EiuCozUtBl0Y3LCvYC7VgyaaaNzSQhyQhl6Du4caGIW9/HnW9j2s5CW38IfPznHHLOHynOjPX9b6+wtr6xuVXcLu3s7u0flA+PWkammtAmkVzqToQN5UzQpmWW047SFCcRp+1ofDOrt5+oNkyKeztRNEzwULCYEWyd1eqpEXuo9csVv+rPhVYhyKECuRr98ldvIEmaUGEJx8Z0A1/ZMMPaMsLptNRLDVWYjPGQdh0KnFATZvNtp+jMOQMUS+2esGju/p7IcGLMJIlcZ4LtyCzXZuZ/tW5q46swY0Kllgqy+ChOObISzU5HA6YpsXziABPN3K6IjLDGxLqASi6EYPnkVWhdVAPHd7VK/TqPowgncArnEMAl1OEWGtAEAo/wDK/w5knvxXv3PhatBS+fOYY/8j5/AD5PjuU= Wilson Fisher CFT - new results 2 2 } (Spin 2) @ @ AAACFXicbVBNS8MwGE79nPOr6tFLcQgeZLRD0OPQi8cJ7gPWWtIs3cKSNCSpMEr/hBf/ihcPingVvPlvTLce3OYDgSfP875v8j6RoERp1/2xVlbX1jc2K1vV7Z3dvX374LCjklQi3EYJTWQvggpTwnFbE01xT0gMWURxNxrfFH73EUtFEn6vJwIHDA45iQmC2kihfe6LEXlo+AJKTSANM5+loZfP3xv5rCq0a27dncJZJl5JaqBEK7S//UGCUoa5RhQq1fdcoYOsmI0ozqt+qrCAaAyHuG8ohwyrIJtulTunRhk4cSLN4dqZqn87MsiUmrDIVDKoR2rRK8T/vH6q46sgI1ykGnM0eyhOqaMTp4jIGRCJkaYTQyCSxPzVQSMoIdImyKoJwVtceZl0GnXP8LuLWvO6jKMCjsEJOAMeuARNcAtaoA0QeAIv4A28W8/Wq/Vhfc5KV6yy5wjMwfr6BXLOn5o= µ1 µ2 [Rajesh Gopakumar, Aninda Sinha (2018)] Polyakov / Mellin Bootstrap Highlights

• Algebraic - can be Automated easily • More efficient than Feynman diagram techniques • Numerics possible and more powerful • Connection to AdS/CFT and holography

even for weakly coupled CFTs (like Ising) Polyakov / Mellin Bootstrap Recent developments

• Direct link to usual bootstrap for 1d [Dalimil Mazac, Miguel Paulos (2018)]

• Connection to Holographic bootstrap [Rajesh Gopakumar, Aninda Sinha (2018)]

• Setup for boundary CFTs [Apratim Kaviraj, Miguel Paulos (2018)]

Thanks !!