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Genus-one string amplitudes from CFT Agnese Bissi Uppsala University

Genus-one string amplitudes from CFT · 2019. 8. 19. · 2 Based on • 1809.10670 • 1706.02388 • 1705.02318 • 1612.03891 with (subsets of) Aharony, Alday and Perlmutter PROGRAM:

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Page 1: Genus-one string amplitudes from CFT · 2019. 8. 19. · 2 Based on • 1809.10670 • 1706.02388 • 1705.02318 • 1612.03891 with (subsets of) Aharony, Alday and Perlmutter PROGRAM:

Genus-one string amplitudes from CFTAgnese Bissi

Uppsala University

Page 2: Genus-one string amplitudes from CFT · 2019. 8. 19. · 2 Based on • 1809.10670 • 1706.02388 • 1705.02318 • 1612.03891 with (subsets of) Aharony, Alday and Perlmutter PROGRAM:

�2

Based on

• 1809.10670 • 1706.02388 • 1705.02318 • 1612.03891

with (subsets of) Aharony, Alday and Perlmutter

PROGRAM: study scattering amplitudes in AdS spaces, using CFT techniques applied to dual correlators.

Page 3: Genus-one string amplitudes from CFT · 2019. 8. 19. · 2 Based on • 1809.10670 • 1706.02388 • 1705.02318 • 1612.03891 with (subsets of) Aharony, Alday and Perlmutter PROGRAM:

Motivation• Scattering amplitudes might uncover hidden structures of a theory, not apparent from its

lagrangian formulation

• Using traditional techniques, scattering amplitudes in curved space are hard to study, already at genus zero

• Understand how much of the full structure of string amplitudes can be recovered/ predicted from the dual CFT, using only symmetries

• Precision test of AdS/CFT correspondence, away from the planar limit and from protected observables.

�3

Page 4: Genus-one string amplitudes from CFT · 2019. 8. 19. · 2 Based on • 1809.10670 • 1706.02388 • 1705.02318 • 1612.03891 with (subsets of) Aharony, Alday and Perlmutter PROGRAM:

This talk• I will discuss a setup to compute/constrain genus-one four graviton amplitude in

string theory on in a low energy expansion.

�4

AdS5 ⇥ S5<latexit sha1_base64="H3qaer3qwmOM5qffwm6G9eFucTA=">AAAB+XicdVDLSsNAFJ34rPUVdelmsAiuQtI2tN1V3bis1D6gjWEymbRDJw9mJoUS+iduXCji1j9x5984aSuo6IELh3Pu5d57vIRRIU3zQ1tb39jc2i7sFHf39g8O9aPjrohTjkkHxyzmfQ8JwmhEOpJKRvoJJyj0GOl5k+vc700JFzSO7uQsIU6IRhENKEZSSa6uX/pt14ZDSUMiYPvedvWSaVTKdrVhQkWqttmo56SWS9AyzAVKYIWWq78P/RinIYkkZkiIgWUm0skQlxQzMi8OU0EShCdoRAaKRkgtcrLF5XN4rhQfBjFXFUm4UL9PZCgUYhZ6qjNEcix+e7n4lzdIZVB3MholqSQRXi4KUgZlDPMYoE85wZLNFEGYU3UrxGPEEZYqrKIK4etT+D/plg2rYpRvq6Xm1SqOAjgFZ+ACWKAGmuAGtEAHYDAFD+AJPGuZ9qi9aK/L1jVtNXMCfkB7+wROR5LR</latexit>

Agenus-oneST (↵0, s, t, u) = Aone-loop

sugra (s, t, u) + ↵03A(1)(s, t, u) + . . .<latexit sha1_base64="+XdETl+KuUwBauaE4vDqhDf+lgw=">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</latexit>

Page 5: Genus-one string amplitudes from CFT · 2019. 8. 19. · 2 Based on • 1809.10670 • 1706.02388 • 1705.02318 • 1612.03891 with (subsets of) Aharony, Alday and Perlmutter PROGRAM:

Setup

�5

type IIB superstring theory

on

four dimensional SYM

with SU(N) gauge groupAdS/CFT correspondence

gs<latexit sha1_base64="PEQntZyx9cV4AFmhA3yO5C1elNM=">AAAB6nicdVBNS8NAEJ3Ur1q/qh69LBbBU0jaaNtb0YvHirYW2lA22026dPPB7kYooT/BiwdFvPqLvPlv3LQVVPTBwOO9GWbmeQlnUlnWh1FYWV1b3yhulra2d3b3yvsHXRmngtAOiXkseh6WlLOIdhRTnPYSQXHocXrnTS5z/+6eCsni6FZNE+qGOIiYzwhWWroJhnJYrlim03Rsu4Es87zZsGrNnFTrzpmNbNOaowJLtIfl98EoJmlII0U4lrJvW4lyMywUI5zOSoNU0gSTCQ5oX9MIh1S62fzUGTrRygj5sdAVKTRXv09kOJRyGnq6M8RqLH97ufiX10+V33AzFiWpohFZLPJTjlSM8r/RiAlKFJ9qgolg+lZExlhgonQ6JR3C16fof9KtmnbNrF47ldbFMo4iHMExnIINdWjBFbShAwQCeIAneDa48Wi8GK+L1oKxnDmEHzDePgHVLo4s</latexit>

↵0 =R2

p�

<latexit sha1_base64="95TJynGEKTK3N1Yj7RuVpkb2nCU=">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</latexit>

N =�

g2YM<latexit sha1_base64="ZtZ9IIddBkQdF/XaNNOneMdjrZs=">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</latexit>

genus expansion

higher derivative expansion

Take both N and large: supergravity approximation�<latexit sha1_base64="1LBU0VTslpUHJMef/j1ijEgZpuY=">AAAB7nicdVDLSsNAFL2pr1pfVZduBovgKiRttO2u6MZlBfuANpTJZNIOnTyYmQgl9CPcuFDErd/jzr9x0lZQ0QMDh3POZe49XsKZVJb1YRTW1jc2t4rbpZ3dvf2D8uFRV8apILRDYh6Lvocl5SyiHcUUp/1EUBx6nPa86XXu9+6pkCyO7tQsoW6IxxELGMFKS70h11Efj8oVy3Sajm03kGVeNhtWrZmTat25sJFtWgtUYIX2qPw+9GOShjRShGMpB7aVKDfDQjHC6bw0TCVNMJniMR1oGuGQSjdbrDtHZ1rxURAL/SKFFur3iQyHUs5CTydDrCbyt5eLf3mDVAUNN2NRkioakeVHQcqRilF+O/KZoETxmSaYCKZ3RWSCBSZKN1TSJXxdiv4n3app18zqrVNpXa3qKMIJnMI52FCHFtxAGzpAYAoP8ATPRmI8Gi/G6zJaMFYzx/ADxtsnvCGP2A==</latexit>

scattering amplitudes of gravitons correlators of Tµ⌫<latexit sha1_base64="Qritgbov9juFkUdncpEfFfx73ik=">AAAB8nicdVDLSsNAFJ34rPVVdelmsAiuQtIHbXdFNy4r9AVJKJPppB06mYR5CCX0M9y4UMStX+POv3HSVlDRAxcO59zLvfeEKaNSOc6HtbG5tb2zW9gr7h8cHh2XTk77MtECkx5OWCKGIZKEUU56iipGhqkgKA4ZGYSzm9wf3BMhacK7ap6SIEYTTiOKkTKS1x1lfqyhz/ViVCo7drVSr7UcaEit7rSaOWnkEnRtZ4kyWKMzKr374wTrmHCFGZLSc51UBRkSimJGFkVfS5IiPEMT4hnKUUxkkC1PXsBLo4xhlAhTXMGl+n0iQ7GU8zg0nTFSU/nby8W/PE+rqBlklKdaEY5XiyLNoEpg/j8cU0GwYnNDEBbU3ArxFAmElUmpaEL4+hT+T/oV263albtauX29jqMAzsEFuAIuaIA2uAUd0AMYJOABPIFnS1mP1ov1umrdsNYzZ+AHrLdPq4aRhg==</latexit>

N = 4<latexit sha1_base64="xMF4y3zRC9iA3L+r/6ivOtEGFmI=">AAAB9XicdVDLSgMxFL1TX7W+qi7dBIvgaphpp7RdCEU3rqSCfUA7lkyatqGZB0lGKUP/w40LRdz6L+78GzNtBRU9EDiccy/35HgRZ1JZ1oeRWVldW9/Ibua2tnd29/L7By0ZxoLQJgl5KDoelpSzgDYVU5x2IkGx73Ha9iYXqd++o0KyMLhR04i6Ph4FbMgIVlq67flYjQnmydXszMn18wXLLBXLTs1Cmjhlq1ZNSSWVkG1acxRgiUY//94bhCT2aaAIx1J2bStSboKFYoTTWa4XSxphMsEj2tU0wD6VbjJPPUMnWhmgYSj0CxSaq983EuxLOfU9PZmmlL+9VPzL68ZqWHUTFkSxogFZHBrGHKkQpRWgAROUKD7VBBPBdFZExlhgonRRaQlfP0X/k1bRtEtm8dop1M+XdWThCI7hFGyoQB0uoQFNICDgAZ7g2bg3Ho0X43UxmjGWO4fwA8bbJzNFklM=</latexit>

AdS5 ⇥ S5<latexit sha1_base64="ZmlO4WWE1AaLDmPBXGBTBjD5WXg=">AAAB+nicdVDLTsJAFJ36RHwVXbqZSExcNS3QADvUjUsM8kigNtPpABOmj8xMNaTyKW5caIxbv8Sdf+MUMFGjJ7nJyTn35t57vJhRIU3zQ1tZXVvf2Mxt5bd3dvf29cJBR0QJx6SNIxbxnocEYTQkbUklI72YExR4jHS9yUXmd28JFzQKr+U0Jk6ARiEdUoykkly9cOa3XBsOJA2IgK0bO+/qRdMol+xK3YSKVGyzXstINZOgZZhzFMESTVd/H/gRTgISSsyQEH3LjKWTIi4pZmSWHySCxAhP0Ij0FQ2R2uSk89Nn8EQpPhxGXFUo4Vz9PpGiQIhp4KnOAMmx+O1l4l9eP5HDmpPSME4kCfFi0TBhUEYwywH6lBMs2VQRhDlVt0I8RhxhqdLKQvj6FP5POiXDKhulq0qxcb6MIweOwDE4BRaogga4BE3QBhjcgQfwBJ61e+1Re9FeF60r2nLmEPyA9vYJh8CS5Q==</latexit>

g2s ⇠ 1

N2<latexit sha1_base64="2Da9EtFpoNR1gTzQMH2oTSAt0uk=">AAACAXicdVDLSsNAFJ3UV62vqBvBzWARXIUkjbbdFd24kgr2AX0xmU7aoTNJmJkIJdSNv+LGhSJu/Qt3/o3Th6CiBy4czrmXe+/xY0alsu0PI7O0vLK6ll3PbWxube+Yu3t1GSUCkxqOWCSaPpKE0ZDUFFWMNGNBEPcZafiji6nfuCVC0ii8UeOYdDgahDSgGCkt9cyDQU92XdiWlMN2IBBOnUl61XUnPTNvW17Zc5wStK2zcskulKfELXqnDnQse4Y8WKDaM9/b/QgnnIQKMyRly7Fj1UmRUBQzMsm1E0lihEdoQFqahogT2UlnH0zgsVb6MIiErlDBmfp9IkVcyjH3dSdHaih/e1PxL6+VqKDUSWkYJ4qEeL4oSBhUEZzGAftUEKzYWBOEBdW3QjxEOgelQ8vpEL4+hf+Tums5Bcu99vKV80UcWXAIjsAJcEARVMAlqIIawOAOPIAn8GzcG4/Gi/E6b80Yi5l98APG2yf3mpab</latexit>

Page 6: Genus-one string amplitudes from CFT · 2019. 8. 19. · 2 Based on • 1809.10670 • 1706.02388 • 1705.02318 • 1612.03891 with (subsets of) Aharony, Alday and Perlmutter PROGRAM:

Spectrum at strong coupling• Protected (1/2 BPS) single trace operators

• Unprotected single trace operators

• Double trace operators

�6

Op ⇠ Tr(�i1 . . .�ip)<latexit sha1_base64="9f1sNgUv98MBiwnuRyZhrUKn5K8=">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</latexit>

Op1⇤n@`Op2

<latexit sha1_base64="B3yRcvJ+Fzp8+VO12ngoaoOLmBI=">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</latexit>

� = p<latexit sha1_base64="NwdDJK3JbXndpzZmtetPXezzzQc=">AAAB8HicdVDLSgNBEOz1GeMr6tHLYBA8Lbt5kOQgBPXgMYJ5SLKE2clsMmR2dpmZFULIV3jxoIhXP8ebf+NsEkFFCxqKqm66u/yYM6Ud58NaWV1b39jMbGW3d3b39nMHhy0VJZLQJol4JDs+VpQzQZuaaU47saQ49Dlt++PL1G/fU6lYJG71JKZeiIeCBYxgbaS73hXlGp/HqJ/LO3axUC7VHGRIqezUqimppBJybWeOPCzR6Ofee4OIJCEVmnCsVNd1Yu1NsdSMcDrL9hJFY0zGeEi7hgocUuVN5wfP0KlRBiiIpCmh0Vz9PjHFoVKT0DedIdYj9dtLxb+8bqKDqjdlIk40FWSxKEg40hFKv0cDJinRfGIIJpKZWxEZYYmJNhllTQhfn6L/Satgu0W7cFPK1y+WcWTgGE7gDFyoQB2uoQFNIBDCAzzBsyWtR+vFel20rljLmSP4AevtE4WGkD0=</latexit>

� ⇠ �1/4<latexit sha1_base64="VjpWlkLpINnVZi8Hee0XadQpn1Q=">AAACA3icdVDLSsNAFJ3UV62vqDvdDBbBVUzalLa7oi5cVrAPaGKZTKbt0MmDmYlQQsGNv+LGhSJu/Ql3/o2TtoKKHhg4nHMPd+7xYkaFNM0PLbe0vLK6ll8vbGxube/ou3ttESUckxaOWMS7HhKE0ZC0JJWMdGNOUOAx0vHG55nfuSVc0Ci8lpOYuAEahnRAMZJK6usHzgVhEkFH0AA6TAV9dJNap/YU9vWiaZRLFbtuQkXsilmvZaSaSdAyzBmKYIFmX393/AgnAQklZkiInmXG0k0RlxQzMi04iSAxwmM0JD1FQxQQ4aazG6bwWCk+HERcvVDCmfo9kaJAiEngqckAyZH47WXiX14vkYOam9IwTiQJ8XzRIGFQRjArBPqUEyzZRBGEOVV/hXiEOMJS1VZQJXxdCv8n7ZJhlY3SlV1snC3qyINDcAROgAWqoAEuQRO0AAZ34AE8gWftXnvUXrTX+WhOW2T2wQ9ob5/BoZb3</latexit>

� = p1 + p2 + 2n+ `+�(1)(�)

N2+

�(2)(�)

N4+ . . .

<latexit sha1_base64="KE8Z8dK0vgk8qGxlH4wDeCPDIB4=">AAACTXicdVFNSxtBGJ5NbdX0w7QevQwNhUgg7K4R9VAQ7aEnsdCokI3Lu7OzcXBmdpl5txCW/YO9FHrrv+jFQ0WKszFCG9oXBh6ej/l4JimksOj7P7zWk5Wnz1bX1tvPX7x8tdF5/ebM5qVhfMRymZuLBCyXQvMRCpT8ojAcVCL5eXJ93OjnX7ixItefcVbwiYKpFplggI6KO2n0gUuE97SIg34Rh30aUt2PuJS0H2UGWBVNQSm4rHrBdt2LpNs6he26OrkM62VHuOQYOkeao407XX+wE+4OD3zqwHDXP9hvwF5D0WDgz6dLFnMad767HCsV18gkWDsO/AInFRgUTPK6HZWWF8CuYcrHDmpQ3E6qeRs1feeYlGa5cUsjnbN/JipQ1s5U4pwK8Mouaw35L21cYrY/qYQuSuSaPRyUlZJiTptqaSoMZyhnDgAzwt2VsitwDaH7gLYr4fGl9P/gLBwEO4Pw07B7eLSoY41skbekRwKyRw7JR3JKRoSRr+Qn+UVuvW/ejXfn/X6wtrxFZpP8Na3VezNKslg=</latexit>

p = 2<latexit sha1_base64="wHSkd55NBdNkpL4fRKNZ6Pp+zy4=">AAAB63icdVBNS8NAEN3Ur1q/qh69LBbBU0jSaNuDUPTisYKthTaUzXbTLt3dhN2NUEL/ghcPinj1D3nz37hpK6jog4HHezPMzAsTRpV2nA+rsLK6tr5R3Cxtbe/s7pX3DzoqTiUmbRyzWHZDpAijgrQ11Yx0E0kQDxm5CydXuX93T6SisbjV04QEHI0EjShGOpeSCw8OyhXH9hu+69ahY5836k61kROv5p+50LWdOSpgidag/N4fxjjlRGjMkFI910l0kCGpKWZkVuqniiQIT9CI9AwViBMVZPNbZ/DEKEMYxdKU0HCufp/IEFdqykPTyZEeq99eLv7l9VId1YOMiiTVRODFoihlUMcwfxwOqSRYs6khCEtqboV4jCTC2sRTMiF8fQr/Jx3Pdqu2d+NXmpfLOIrgCByDU+CCGmiCa9ACbYDBGDyAJ/BscevRerFeF60FazlzCH7AevsEov2N/A==</latexit>

p > 2<latexit sha1_base64="HHNMEuWEJzxOu5G445bwggML7hc=">AAAB63icdVBNS8NAEN3Ur1q/qh69LBbBU0jSaNuLFL14rGBroQ1ls920S3c3YXcjlNC/4MWDIl79Q978N27aCir6YODx3gwz88KEUaUd58MqrKyurW8UN0tb2zu7e+X9g46KU4lJG8cslt0QKcKoIG1NNSPdRBLEQ0buwslV7t/dE6loLG71NCEBRyNBI4qRzqXkwoODcsWx/YbvunXo2OeNulNt5MSr+WcudG1njgpYojUov/eHMU45ERozpFTPdRIdZEhqihmZlfqpIgnCEzQiPUMF4kQF2fzWGTwxyhBGsTQlNJyr3ycyxJWa8tB0cqTH6reXi395vVRH9SCjIkk1EXixKEoZ1DHMH4dDKgnWbGoIwpKaWyEeI4mwNvGUTAhfn8L/Scez3art3fiV5uUyjiI4AsfgFLigBprgGrRAG2AwBg/gCTxb3Hq0XqzXRWvBWs4cgh+w3j4BpION/Q==</latexit>

STRESS TENSOR

KALUZA KLEIN MODES

Page 7: Genus-one string amplitudes from CFT · 2019. 8. 19. · 2 Based on • 1809.10670 • 1706.02388 • 1705.02318 • 1612.03891 with (subsets of) Aharony, Alday and Perlmutter PROGRAM:

Spectrum at strong coupling• Protected (1/2 BPS) single trace operators

• Unprotected single trace operators

• Double trace operators

�6

Op ⇠ Tr(�i1 . . .�ip)<latexit sha1_base64="9f1sNgUv98MBiwnuRyZhrUKn5K8=">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</latexit>

Op1⇤n@`Op2

<latexit sha1_base64="B3yRcvJ+Fzp8+VO12ngoaoOLmBI=">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</latexit>

� = p<latexit sha1_base64="NwdDJK3JbXndpzZmtetPXezzzQc=">AAAB8HicdVDLSgNBEOz1GeMr6tHLYBA8Lbt5kOQgBPXgMYJ5SLKE2clsMmR2dpmZFULIV3jxoIhXP8ebf+NsEkFFCxqKqm66u/yYM6Ud58NaWV1b39jMbGW3d3b39nMHhy0VJZLQJol4JDs+VpQzQZuaaU47saQ49Dlt++PL1G/fU6lYJG71JKZeiIeCBYxgbaS73hXlGp/HqJ/LO3axUC7VHGRIqezUqimppBJybWeOPCzR6Ofee4OIJCEVmnCsVNd1Yu1NsdSMcDrL9hJFY0zGeEi7hgocUuVN5wfP0KlRBiiIpCmh0Vz9PjHFoVKT0DedIdYj9dtLxb+8bqKDqjdlIk40FWSxKEg40hFKv0cDJinRfGIIJpKZWxEZYYmJNhllTQhfn6L/Satgu0W7cFPK1y+WcWTgGE7gDFyoQB2uoQFNIBDCAzzBsyWtR+vFel20rljLmSP4AevtE4WGkD0=</latexit>

� ⇠ �1/4<latexit sha1_base64="VjpWlkLpINnVZi8Hee0XadQpn1Q=">AAACA3icdVDLSsNAFJ3UV62vqDvdDBbBVUzalLa7oi5cVrAPaGKZTKbt0MmDmYlQQsGNv+LGhSJu/Ql3/o2TtoKKHhg4nHMPd+7xYkaFNM0PLbe0vLK6ll8vbGxube/ou3ttESUckxaOWMS7HhKE0ZC0JJWMdGNOUOAx0vHG55nfuSVc0Ci8lpOYuAEahnRAMZJK6usHzgVhEkFH0AA6TAV9dJNap/YU9vWiaZRLFbtuQkXsilmvZaSaSdAyzBmKYIFmX393/AgnAQklZkiInmXG0k0RlxQzMi04iSAxwmM0JD1FQxQQ4aazG6bwWCk+HERcvVDCmfo9kaJAiEngqckAyZH47WXiX14vkYOam9IwTiQJ8XzRIGFQRjArBPqUEyzZRBGEOVV/hXiEOMJS1VZQJXxdCv8n7ZJhlY3SlV1snC3qyINDcAROgAWqoAEuQRO0AAZ34AE8gWftXnvUXrTX+WhOW2T2wQ9ob5/BoZb3</latexit>

� = p1 + p2 + 2n+ `+�(1)(�)

N2+

�(2)(�)

N4+ . . .

<latexit sha1_base64="KE8Z8dK0vgk8qGxlH4wDeCPDIB4=">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</latexit>

p = 2<latexit sha1_base64="wHSkd55NBdNkpL4fRKNZ6Pp+zy4=">AAAB63icdVBNS8NAEN3Ur1q/qh69LBbBU0jSaNuDUPTisYKthTaUzXbTLt3dhN2NUEL/ghcPinj1D3nz37hpK6jog4HHezPMzAsTRpV2nA+rsLK6tr5R3Cxtbe/s7pX3DzoqTiUmbRyzWHZDpAijgrQ11Yx0E0kQDxm5CydXuX93T6SisbjV04QEHI0EjShGOpeSCw8OyhXH9hu+69ahY5836k61kROv5p+50LWdOSpgidag/N4fxjjlRGjMkFI910l0kCGpKWZkVuqniiQIT9CI9AwViBMVZPNbZ/DEKEMYxdKU0HCufp/IEFdqykPTyZEeq99eLv7l9VId1YOMiiTVRODFoihlUMcwfxwOqSRYs6khCEtqboV4jCTC2sRTMiF8fQr/Jx3Pdqu2d+NXmpfLOIrgCByDU+CCGmiCa9ACbYDBGDyAJ/BscevRerFeF60FazlzCH7AevsEov2N/A==</latexit>

p > 2<latexit sha1_base64="HHNMEuWEJzxOu5G445bwggML7hc=">AAAB63icdVBNS8NAEN3Ur1q/qh69LBbBU0jSaNuLFL14rGBroQ1ls920S3c3YXcjlNC/4MWDIl79Q978N27aCir6YODx3gwz88KEUaUd58MqrKyurW8UN0tb2zu7e+X9g46KU4lJG8cslt0QKcKoIG1NNSPdRBLEQ0buwslV7t/dE6loLG71NCEBRyNBI4qRzqXkwoODcsWx/YbvunXo2OeNulNt5MSr+WcudG1njgpYojUov/eHMU45ERozpFTPdRIdZEhqihmZlfqpIgnCEzQiPUMF4kQF2fzWGTwxyhBGsTQlNJyr3ycyxJWa8tB0cqTH6reXi395vVRH9SCjIkk1EXixKEoZ1DHMH4dDKgnWbGoIwpKaWyEeI4mwNvGUTAhfn8L/Scez3art3fiV5uUyjiI4AsfgFLigBprgGrRAG2AwBg/gCTxb3Hq0XqzXRWvBWs4cgh+w3j4BpION/Q==</latexit>

STRESS TENSOR

KALUZA KLEIN MODES

SUPERGRAVITY APPROXIMATION

Page 8: Genus-one string amplitudes from CFT · 2019. 8. 19. · 2 Based on • 1809.10670 • 1706.02388 • 1705.02318 • 1612.03891 with (subsets of) Aharony, Alday and Perlmutter PROGRAM:

Spectrum at strong coupling• Protected (1/2 BPS) single trace operators

• Unprotected single trace operators

• Double trace operators

�6

Op ⇠ Tr(�i1 . . .�ip)<latexit sha1_base64="9f1sNgUv98MBiwnuRyZhrUKn5K8=">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</latexit>

Op1⇤n@`Op2

<latexit sha1_base64="B3yRcvJ+Fzp8+VO12ngoaoOLmBI=">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</latexit>

� = p<latexit sha1_base64="NwdDJK3JbXndpzZmtetPXezzzQc=">AAAB8HicdVDLSgNBEOz1GeMr6tHLYBA8Lbt5kOQgBPXgMYJ5SLKE2clsMmR2dpmZFULIV3jxoIhXP8ebf+NsEkFFCxqKqm66u/yYM6Ud58NaWV1b39jMbGW3d3b39nMHhy0VJZLQJol4JDs+VpQzQZuaaU47saQ49Dlt++PL1G/fU6lYJG71JKZeiIeCBYxgbaS73hXlGp/HqJ/LO3axUC7VHGRIqezUqimppBJybWeOPCzR6Ofee4OIJCEVmnCsVNd1Yu1NsdSMcDrL9hJFY0zGeEi7hgocUuVN5wfP0KlRBiiIpCmh0Vz9PjHFoVKT0DedIdYj9dtLxb+8bqKDqjdlIk40FWSxKEg40hFKv0cDJinRfGIIJpKZWxEZYYmJNhllTQhfn6L/Satgu0W7cFPK1y+WcWTgGE7gDFyoQB2uoQFNIBDCAzzBsyWtR+vFel20rljLmSP4AevtE4WGkD0=</latexit>

� ⇠ �1/4<latexit sha1_base64="VjpWlkLpINnVZi8Hee0XadQpn1Q=">AAACA3icdVDLSsNAFJ3UV62vqDvdDBbBVUzalLa7oi5cVrAPaGKZTKbt0MmDmYlQQsGNv+LGhSJu/Ql3/o2TtoKKHhg4nHMPd+7xYkaFNM0PLbe0vLK6ll8vbGxube/ou3ttESUckxaOWMS7HhKE0ZC0JJWMdGNOUOAx0vHG55nfuSVc0Ci8lpOYuAEahnRAMZJK6usHzgVhEkFH0AA6TAV9dJNap/YU9vWiaZRLFbtuQkXsilmvZaSaSdAyzBmKYIFmX393/AgnAQklZkiInmXG0k0RlxQzMi04iSAxwmM0JD1FQxQQ4aazG6bwWCk+HERcvVDCmfo9kaJAiEngqckAyZH47WXiX14vkYOam9IwTiQJ8XzRIGFQRjArBPqUEyzZRBGEOVV/hXiEOMJS1VZQJXxdCv8n7ZJhlY3SlV1snC3qyINDcAROgAWqoAEuQRO0AAZ34AE8gWftXnvUXrTX+WhOW2T2wQ9ob5/BoZb3</latexit>

� = p1 + p2 + 2n+ `+�(1)(�)

N2+

�(2)(�)

N4+ . . .

<latexit sha1_base64="KE8Z8dK0vgk8qGxlH4wDeCPDIB4=">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</latexit>

p = 2<latexit sha1_base64="wHSkd55NBdNkpL4fRKNZ6Pp+zy4=">AAAB63icdVBNS8NAEN3Ur1q/qh69LBbBU0jSaNuDUPTisYKthTaUzXbTLt3dhN2NUEL/ghcPinj1D3nz37hpK6jog4HHezPMzAsTRpV2nA+rsLK6tr5R3Cxtbe/s7pX3DzoqTiUmbRyzWHZDpAijgrQ11Yx0E0kQDxm5CydXuX93T6SisbjV04QEHI0EjShGOpeSCw8OyhXH9hu+69ahY5836k61kROv5p+50LWdOSpgidag/N4fxjjlRGjMkFI910l0kCGpKWZkVuqniiQIT9CI9AwViBMVZPNbZ/DEKEMYxdKU0HCufp/IEFdqykPTyZEeq99eLv7l9VId1YOMiiTVRODFoihlUMcwfxwOqSRYs6khCEtqboV4jCTC2sRTMiF8fQr/Jx3Pdqu2d+NXmpfLOIrgCByDU+CCGmiCa9ACbYDBGDyAJ/BscevRerFeF60FazlzCH7AevsEov2N/A==</latexit>

p > 2<latexit sha1_base64="HHNMEuWEJzxOu5G445bwggML7hc=">AAAB63icdVBNS8NAEN3Ur1q/qh69LBbBU0jSaNuLFL14rGBroQ1ls920S3c3YXcjlNC/4MWDIl79Q978N27aCir6YODx3gwz88KEUaUd58MqrKyurW8UN0tb2zu7e+X9g46KU4lJG8cslt0QKcKoIG1NNSPdRBLEQ0buwslV7t/dE6loLG71NCEBRyNBI4qRzqXkwoODcsWx/YbvunXo2OeNulNt5MSr+WcudG1njgpYojUov/eHMU45ERozpFTPdRIdZEhqihmZlfqpIgnCEzQiPUMF4kQF2fzWGTwxyhBGsTQlNJyr3ycyxJWa8tB0cqTH6reXi395vVRH9SCjIkk1EXixKEoZ1DHMH4dDKgnWbGoIwpKaWyEeI4mwNvGUTAhfn8L/Scez3art3fiV5uUyjiI4AsfgFLigBprgGrRAG2AwBg/gCTxb3Hq0XqzXRWvBWs4cgh+w3j4BpION/Q==</latexit>

STRESS TENSOR

KALUZA KLEIN MODES

SUPERGRAVITY APPROXIMATION

LOOP CORRECTIONS TO SUGRA

Page 9: Genus-one string amplitudes from CFT · 2019. 8. 19. · 2 Based on • 1809.10670 • 1706.02388 • 1705.02318 • 1612.03891 with (subsets of) Aharony, Alday and Perlmutter PROGRAM:

Spectrum at strong coupling• Protected (1/2 BPS) single trace operators

• Unprotected single trace operators

• Double trace operators

�6

Op ⇠ Tr(�i1 . . .�ip)<latexit sha1_base64="9f1sNgUv98MBiwnuRyZhrUKn5K8=">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</latexit>

Op1⇤n@`Op2

<latexit sha1_base64="B3yRcvJ+Fzp8+VO12ngoaoOLmBI=">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</latexit>

� = p<latexit sha1_base64="NwdDJK3JbXndpzZmtetPXezzzQc=">AAAB8HicdVDLSgNBEOz1GeMr6tHLYBA8Lbt5kOQgBPXgMYJ5SLKE2clsMmR2dpmZFULIV3jxoIhXP8ebf+NsEkFFCxqKqm66u/yYM6Ud58NaWV1b39jMbGW3d3b39nMHhy0VJZLQJol4JDs+VpQzQZuaaU47saQ49Dlt++PL1G/fU6lYJG71JKZeiIeCBYxgbaS73hXlGp/HqJ/LO3axUC7VHGRIqezUqimppBJybWeOPCzR6Ofee4OIJCEVmnCsVNd1Yu1NsdSMcDrL9hJFY0zGeEi7hgocUuVN5wfP0KlRBiiIpCmh0Vz9PjHFoVKT0DedIdYj9dtLxb+8bqKDqjdlIk40FWSxKEg40hFKv0cDJinRfGIIJpKZWxEZYYmJNhllTQhfn6L/Satgu0W7cFPK1y+WcWTgGE7gDFyoQB2uoQFNIBDCAzzBsyWtR+vFel20rljLmSP4AevtE4WGkD0=</latexit>

� ⇠ �1/4<latexit sha1_base64="VjpWlkLpINnVZi8Hee0XadQpn1Q=">AAACA3icdVDLSsNAFJ3UV62vqDvdDBbBVUzalLa7oi5cVrAPaGKZTKbt0MmDmYlQQsGNv+LGhSJu/Ql3/o2TtoKKHhg4nHMPd+7xYkaFNM0PLbe0vLK6ll8vbGxube/ou3ttESUckxaOWMS7HhKE0ZC0JJWMdGNOUOAx0vHG55nfuSVc0Ci8lpOYuAEahnRAMZJK6usHzgVhEkFH0AA6TAV9dJNap/YU9vWiaZRLFbtuQkXsilmvZaSaSdAyzBmKYIFmX393/AgnAQklZkiInmXG0k0RlxQzMi04iSAxwmM0JD1FQxQQ4aazG6bwWCk+HERcvVDCmfo9kaJAiEngqckAyZH47WXiX14vkYOam9IwTiQJ8XzRIGFQRjArBPqUEyzZRBGEOVV/hXiEOMJS1VZQJXxdCv8n7ZJhlY3SlV1snC3qyINDcAROgAWqoAEuQRO0AAZ34AE8gWftXnvUXrTX+WhOW2T2wQ9ob5/BoZb3</latexit>

� = p1 + p2 + 2n+ `+�(1)(�)

N2+

�(2)(�)

N4+ . . .

<latexit sha1_base64="KE8Z8dK0vgk8qGxlH4wDeCPDIB4=">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</latexit>

p = 2<latexit sha1_base64="wHSkd55NBdNkpL4fRKNZ6Pp+zy4=">AAAB63icdVBNS8NAEN3Ur1q/qh69LBbBU0jSaNuDUPTisYKthTaUzXbTLt3dhN2NUEL/ghcPinj1D3nz37hpK6jog4HHezPMzAsTRpV2nA+rsLK6tr5R3Cxtbe/s7pX3DzoqTiUmbRyzWHZDpAijgrQ11Yx0E0kQDxm5CydXuX93T6SisbjV04QEHI0EjShGOpeSCw8OyhXH9hu+69ahY5836k61kROv5p+50LWdOSpgidag/N4fxjjlRGjMkFI910l0kCGpKWZkVuqniiQIT9CI9AwViBMVZPNbZ/DEKEMYxdKU0HCufp/IEFdqykPTyZEeq99eLv7l9VId1YOMiiTVRODFoihlUMcwfxwOqSRYs6khCEtqboV4jCTC2sRTMiF8fQr/Jx3Pdqu2d+NXmpfLOIrgCByDU+CCGmiCa9ACbYDBGDyAJ/BscevRerFeF60FazlzCH7AevsEov2N/A==</latexit>

p > 2<latexit sha1_base64="HHNMEuWEJzxOu5G445bwggML7hc=">AAAB63icdVBNS8NAEN3Ur1q/qh69LBbBU0jSaNuLFL14rGBroQ1ls920S3c3YXcjlNC/4MWDIl79Q978N27aCir6YODx3gwz88KEUaUd58MqrKyurW8UN0tb2zu7e+X9g46KU4lJG8cslt0QKcKoIG1NNSPdRBLEQ0buwslV7t/dE6loLG71NCEBRyNBI4qRzqXkwoODcsWx/YbvunXo2OeNulNt5MSr+WcudG1njgpYojUov/eHMU45ERozpFTPdRIdZEhqihmZlfqpIgnCEzQiPUMF4kQF2fzWGTwxyhBGsTQlNJyr3ycyxJWa8tB0cqTH6reXi395vVRH9SCjIkk1EXixKEoZ1DHMH4dDKgnWbGoIwpKaWyEeI4mwNvGUTAhfn8L/Scez3art3fiV5uUyjiI4AsfgFLigBprgGrRAG2AwBg/gCTxb3Hq0XqzXRWvBWs4cgh+w3j4BpION/Q==</latexit>

STRESS TENSOR

KALUZA KLEIN MODES

SUPERGRAVITY APPROXIMATION

LOOP CORRECTIONS TO SUGRA

STRINGY CORRECTIONS TO SUGRA

Page 10: Genus-one string amplitudes from CFT · 2019. 8. 19. · 2 Based on • 1809.10670 • 1706.02388 • 1705.02318 • 1612.03891 with (subsets of) Aharony, Alday and Perlmutter PROGRAM:

Expansions

�7

hO2(x1)O2(x2)O2(x3)O2(x4)i =G(u, v)

x412x

434

<latexit sha1_base64="AvpYvHd9YlNJSdEmAVL+GHncO5M=">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</latexit>

Page 11: Genus-one string amplitudes from CFT · 2019. 8. 19. · 2 Based on • 1809.10670 • 1706.02388 • 1705.02318 • 1612.03891 with (subsets of) Aharony, Alday and Perlmutter PROGRAM:

Expansions

�7

hO2(x1)O2(x2)O2(x3)O2(x4)i =G(u, v)

x412x

434

<latexit sha1_base64="AvpYvHd9YlNJSdEmAVL+GHncO5M=">AAACcXicdZFdS8MwFIbT+j2/5seNiBIcwoYy2q6iXgiiF3qnglNhnSXN0hlM05Kk4ii99/d555/wxj9gOqeo0wMhL885Jzl5EySMSmVZL4Y5Mjo2PjE5VZqemZ2bLy8sXsk4FZg0ccxicRMgSRjlpKmoYuQmEQRFASPXwf1xkb9+IELSmF+qXkLaEepyGlKMlEZ++Ql6DPEuIxB6EVJ3GLHsLPcdWH307doQcoZRYxi5NU/0Dz3wQoFw9lVwklfT7Ydanj36me3kt67eG67e85Jfrlj1hrPj7ltQC3fH2t8rxG6BoF23+lEBgzj3y89eJ8ZpRLjCDEnZsq1EtTMkFMWM5CUvlSRB+B51SUtLjiIi21nfsRxuatKBYSz04gr26feODEVS9qJAVxazy9+5Av6Va6Uq3GtnlCepIhx/XBSmDKoYFvbDDhUEK9bTAmFB9awQ3yFtktKfVJjw+VL4v7hy6naj7ly4lcOjgR2TYBVsgCqwwS44BKfgHDQBBq/GsrFmrBtv5ooJzY2PUtMY9CyBH2FuvQPhYLsW</latexit>

G(u, v) = G(0)(u, v) +1

N2G(1)(u, v,�) +

1

N4G(2)(u, v,�) + . . .

<latexit sha1_base64="Jkn77FE/mziGa+u852CAQxe4mbo=">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</latexit>

Large N expansion

Genus expansion

Page 12: Genus-one string amplitudes from CFT · 2019. 8. 19. · 2 Based on • 1809.10670 • 1706.02388 • 1705.02318 • 1612.03891 with (subsets of) Aharony, Alday and Perlmutter PROGRAM:

Expansions

�7

hO2(x1)O2(x2)O2(x3)O2(x4)i =G(u, v)

x412x

434

<latexit sha1_base64="AvpYvHd9YlNJSdEmAVL+GHncO5M=">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</latexit>

G(u, v) = G(0)(u, v) +1

N2G(1)(u, v,�) +

1

N4G(2)(u, v,�) + . . .

<latexit sha1_base64="Jkn77FE/mziGa+u852CAQxe4mbo=">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</latexit>

Large N expansion

Genus expansion

G(1,1)(u, v) +G(1,2)(u, v)

�32

+ . . .<latexit sha1_base64="5CRznirKhJA9cQiCNqYQoLmj3tw=">AAACPnicdVDLSgMxFM34tr6qLt0Ei9BSKTPTluqu6EKXFWwVOrVk0owGMw+SjFBCvsyN3+DOpRsXirh1aaatoEUvBA7nwc09fsKokLb9ZM3Mzs0vLC4t51ZW19Y38ptbHRGnHJM2jlnML30kCKMRaUsqGblMOEGhz8iFf3uc6Rd3hAsaR+dymJBeiK4jGlCMpKH6+bYXInmDEVMn+koVnX2npIvp/l2p7AUcYTUtuxNZK4+ZLQN0pcbGqlau1rrsDWIp+vmCXam69dqhDQ2o1e3Dgww0Mgo6FXs0BTCZVj//aHI4DUkkMUNCdB07kT2FuKSYEZ3zUkEShG/RNekaGKGQiJ4ana/hnmEGMIi5eZGEI/ZnQqFQiGHoG2d2jZjWMvIvrZvK4KCnaJSkkkR4vChIGZQxzLqEA8oJlmxoAMKcmr9CfINMG9I0njMlfF8K/wcdt+JUK+5ZrdA8mtSxBHbALigCBzRAE5yCFmgDDO7BM3gFb9aD9WK9Wx9j64w1yWyDX2N9fgESNq8s</latexit>

G(2,1)(u, v) +G(2,2)(u, v)

�32

+ . . .<latexit sha1_base64="6hJwA6TBpP/mL6/dHNRk9Avmm+0=">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</latexit>

↵0<latexit sha1_base64="5r0JDGPdpb9luIHGa7o42aLv+t0=">AAAB7nicdVDLSsNAFL3xWeur6tLNYBFdhaQP2u6KblxWsA9oQ5lMJ+3QySTMTIQS+hFuXCji1u9x5984aSuo6IELh3Pu5d57/JgzpR3nw1pb39jc2s7t5Hf39g8OC0fHHRUlktA2iXgkez5WlDNB25ppTnuxpDj0Oe360+vM795TqVgk7vQspl6Ix4IFjGBtpO4A83iCL4aFomOXS9VKw0GGVKpOo56RWiYh13YWKMIKrWHhfTCKSBJSoQnHSvVdJ9ZeiqVmhNN5fpAoGmMyxWPaN1TgkCovXZw7R+dGGaEgkqaERgv1+0SKQ6VmoW86Q6wn6reXiX95/UQHdS9lIk40FWS5KEg40hHKfkcjJinRfGYIJpKZWxGZYImJNgnlTQhfn6L/Sadku2W7dFspNq9WceTgFM7gElyoQRNuoAVtIDCFB3iCZyu2Hq0X63XZumatZk7gB6y3T2WYj58=</latexit>

expansion

Large expansion�<latexit sha1_base64="BmymfYePB7xX1o94+nJC4jRwhb8=">AAAB7nicdVDLSgMxFL1TX7W+qi7dBIvgapi+aLsrunFZwT6gHUomk2lDM5khyQhl6Ee4caGIW7/HnX9jpq2gogcCh3POJfceL+ZMacf5sHIbm1vbO/ndwt7+weFR8fikp6JEEtolEY/kwMOKciZoVzPN6SCWFIcep31vdp35/XsqFYvEnZ7H1A3xRLCAEayN1B9xE/XxuFhy7GqlXms5yJBa3Wk1M9LIJFS2nSVKsEZnXHwf+RFJQio04VipYdmJtZtiqRnhdFEYJYrGmMzwhA4NFTikyk2X6y7QhVF8FETSPKHRUv0+keJQqXnomWSI9VT99jLxL2+Y6KDppkzEiaaCrD4KEo50hLLbkc8kJZrPDcFEMrMrIlMsMdGmoYIp4etS9D/pVexy1a7c1krtq3UdeTiDc7iEMjSgDTfQgS4QmMEDPMGzFVuP1ov1uormrPXMKfyA9fYJtGGP0w==</latexit>

Page 13: Genus-one string amplitudes from CFT · 2019. 8. 19. · 2 Based on • 1809.10670 • 1706.02388 • 1705.02318 • 1612.03891 with (subsets of) Aharony, Alday and Perlmutter PROGRAM:

OPE decomposition

�8

G(u, v) = 1 +X

�,`

a�,`u��`

2 g�,`(u, v)

<latexit sha1_base64="qRKq/qv+pYZGIpNsz5tkU+sYdZs=">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</latexit>

conformal blocks 3 pt functionsidentity

OPE decomposition

Page 14: Genus-one string amplitudes from CFT · 2019. 8. 19. · 2 Based on • 1809.10670 • 1706.02388 • 1705.02318 • 1612.03891 with (subsets of) Aharony, Alday and Perlmutter PROGRAM:

OPE decomposition

�8

G(u, v) = 1 +X

�,`

a�,`u��`

2 g�,`(u, v)

<latexit sha1_base64="qRKq/qv+pYZGIpNsz5tkU+sYdZs=">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</latexit>

conformal blocks 3 pt functionsidentity

O�,`<latexit sha1_base64="pkA4Ja5xowK7y57nprtgTXN/dzU=">AAACAnicdVDLSgNBEJyNrxhfUU/iZTAIHmTZTaJJbkE9eDOCeUA2hNlJJxky+2BmVghL8OKvePGgiFe/wpt/42wSQUULGoqqbrq73JAzqSzrw0gtLC4tr6RXM2vrG5tb2e2dhgwiQaFOAx6IlkskcOZDXTHFoRUKIJ7LoemOzhO/eQtCssC/UeMQOh4Z+KzPKFFa6mb3HI+oISU8vpp0Y+cCuCLHDnA+6WZzllmsFG27jC3ztFK2CpWE5EvFExvbpjVFDs1R62bfnV5AIw98RTmRsm1boerERChGOUwyTiQhJHREBtDW1CceyE48fWGCD7XSw/1A6PIVnqrfJ2LiSTn2XN2ZHCx/e4n4l9eOVL/ciZkfRgp8OlvUjzhWAU7ywD0mgCo+1oRQwfStmA6JIFTp1DI6hK9P8f+kkTftgpm/LuaqZ/M40mgfHaAjZKMSqqJLVEN1RNEdekBP6Nm4Nx6NF+N11poy5jO76AeMt08OhJff</latexit>

In this regime are : •protected single traces •unprotected double trace

OPE decomposition

Page 15: Genus-one string amplitudes from CFT · 2019. 8. 19. · 2 Based on • 1809.10670 • 1706.02388 • 1705.02318 • 1612.03891 with (subsets of) Aharony, Alday and Perlmutter PROGRAM:

OPE decomposition

�8

G(u, v) = 1 +X

�,`

a�,`u��`

2 g�,`(u, v)

<latexit sha1_base64="qRKq/qv+pYZGIpNsz5tkU+sYdZs=">AAACTHicdZBLaxsxEMe1buukzstpj72ImkJCHbPrB4kPBZMW2mMKdRLwOmZWnrWFtQ/0CJhlP2AvPeSWT5FLDy0lEO3GLW1IBwT/+c2MpPkHqeBKu+61U3ny9Fl1bf15bWNza3unvvviVCVGMhyyRCTyPACFgsc41FwLPE8lQhQIPAsW74v62SVKxZP4i16mOI5gFvOQM9AWTerMj0DPGYjsY75nmpf777y3vjLRJPM/oNDQ9FGIHP6ktMzNReaHEtiKHpQwa+f57EFjeeWk3nBbnXav23epFd2e2z8qxGGBqNdyy2iQVZxM6lf+NGEmwlgzAUqNPDfV4wyk5kxgXvONwhTYAmY4sjKGCNU4K83I6RtLpjRMpD2xpiX9eyKDSKllFNjOYnX1sFbAx2ojo8Ojccbj1GiM2f1DoRFUJ7Rwlk65RKbF0gpgktu/UjYH65K2/tesCb83pf8Xp+2W12m1P3cbg+OVHevkFXlN9ohHDsmAfCInZEgY+UpuyA/y0/nmfHd+Obf3rRVnNfOS/BOV6h0SLLSO</latexit>

conformal blocks 3 pt functionsidentity

O�,`<latexit sha1_base64="pkA4Ja5xowK7y57nprtgTXN/dzU=">AAACAnicdVDLSgNBEJyNrxhfUU/iZTAIHmTZTaJJbkE9eDOCeUA2hNlJJxky+2BmVghL8OKvePGgiFe/wpt/42wSQUULGoqqbrq73JAzqSzrw0gtLC4tr6RXM2vrG5tb2e2dhgwiQaFOAx6IlkskcOZDXTHFoRUKIJ7LoemOzhO/eQtCssC/UeMQOh4Z+KzPKFFa6mb3HI+oISU8vpp0Y+cCuCLHDnA+6WZzllmsFG27jC3ztFK2CpWE5EvFExvbpjVFDs1R62bfnV5AIw98RTmRsm1boerERChGOUwyTiQhJHREBtDW1CceyE48fWGCD7XSw/1A6PIVnqrfJ2LiSTn2XN2ZHCx/e4n4l9eOVL/ciZkfRgp8OlvUjzhWAU7ywD0mgCo+1oRQwfStmA6JIFTp1DI6hK9P8f+kkTftgpm/LuaqZ/M40mgfHaAjZKMSqqJLVEN1RNEdekBP6Nm4Nx6NF+N11poy5jO76AeMt08OhJff</latexit>

In this regime are : •protected single traces •unprotected double trace

OPE decomposition

Crossing symmetry G(u, v) =⇣uv

⌘2G(v, u)

<latexit sha1_base64="2iLCVFiP1KrOM+LAbMr/ZZTaIXE=">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</latexit>

Page 16: Genus-one string amplitudes from CFT · 2019. 8. 19. · 2 Based on • 1809.10670 • 1706.02388 • 1705.02318 • 1612.03891 with (subsets of) Aharony, Alday and Perlmutter PROGRAM:

Analytic bootstrap• As the correlator develops singularities.

• The whole OPE data/correlator can be reconstructed from such singularities

�9

Sing[G(u, v)] ! {an,`, �n,`} ! G(u, v)<latexit sha1_base64="7SYWpA2+ozixwqM9NP9WElIbu5k=">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</latexit>

v ! 0<latexit sha1_base64="Z6rjkh8WABgj4vGO/4/th8AniBw=">AAAB7XicdVDLSgMxFM3UV62vqks3wSK4GtIXbXdFNy4r2Ae0Q8mkmTY2kwxJplCG/oMbF4q49X/c+Tdm2goqeiBwOOdecs/xI860QejDyWxsbm3vZHdze/sHh0f545OOlrEitE0kl6rnY005E7RtmOG0FymKQ5/Trj+9Tv3ujCrNpLgz84h6IR4LFjCCjZU6s4GREA3zBeSWS9VKA0FLKlXUqKeklkqw6KIlCmCN1jD/PhhJEodUGMKx1v0iioyXYGUY4XSRG8SaRphM8Zj2LRU4pNpLltcu4IVVRjCQyj5h4FL9vpHgUOt56NvJEJuJ/u2l4l9ePzZB3UuYiGJDBVl9FMQc2ohpdDhiihLD55Zgopi9FZIJVpgYW1DOlvCVFP5POiW3WHZLt5VC82pdRxacgXNwCYqgBprgBrRAGxBwDx7AE3h2pPPovDivq9GMs945BT/gvH0CdmWPEQ==</latexit>

Komargodski, Zhiboedov

• The structure of the singularities comes from crossing symmetry and the OPE content.

Caron-Huot Alday

Fitzpatrick, Kaplan,Poland, Simmons-Duffin

Page 17: Genus-one string amplitudes from CFT · 2019. 8. 19. · 2 Based on • 1809.10670 • 1706.02388 • 1705.02318 • 1612.03891 with (subsets of) Aharony, Alday and Perlmutter PROGRAM:

Divergences, leading order

�10

X

�,`

a�,`u��`

2 g�,`(u, v) =⇣uv

⌘2 �1 + a2,0vg2,0(v, u) + a4,0v

2g4,0(v, u) + . . .�

<latexit sha1_base64="vdcREU9YLpCBxu2TkJYktaJ15t8=">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</latexit>

identity 1/2 BPS op p=2 double traces

For v ! 0<latexit sha1_base64="9nXOT2NnvAalxeSiQaanE5TUUCM=">AAAB7nicdVDLSgMxFL3js9ZX1aWbYBFcDTPtaNtd0Y3LCvYB7VAyadqGZjJDkimUoR/hxoUibv0ed/6NmbaCih4IHM65l9xzgpgzpR3nw1pb39jc2s7t5Hf39g8OC0fHLRUlktAmiXgkOwFWlDNBm5ppTjuxpDgMOG0Hk5vMb0+pVCwS93oWUz/EI8GGjGBtpPYU9XSEnH6h6NhezXPdKnLsq1rVKdcyUqp4ly5ybWeBIqzQ6Bfee4OIJCEVmnCsVNd1Yu2nWGpGOJ3ne4miMSYTPKJdQwUOqfLTxblzdG6UARpG0jyh0UL9vpHiUKlZGJjJEOux+u1l4l9eN9HDqp8yESeaCrL8aJhwZCJm2dGASUo0nxmCiWTmVkTGWGKiTUN5U8JXUvQ/aZVst2yX7rxi/XpVRw5O4QwuwIUK1OEWGtAEAhN4gCd4tmLr0XqxXpeja9Zq5wR+wHr7BNSpj0A=</latexit>

• Low twist operators are associated to power-law divergences

• Double trace operators (infinitely many) with large spin

Page 18: Genus-one string amplitudes from CFT · 2019. 8. 19. · 2 Based on • 1809.10670 • 1706.02388 • 1705.02318 • 1612.03891 with (subsets of) Aharony, Alday and Perlmutter PROGRAM:

• Matching the divergences allows fixing the OPE data, up to finite spin ambiguities.

• These ambiguities correspond to crossing symmetric solutions without divergences. For such solutions for

• They correspond to quartic vertices in AdS

• In addition to power law singularities , there are logarithmic singularities ( )

�11

Divergences

⇠ log2 v<latexit sha1_base64="rvLr6gtdXSgK0GSOBMC6SxhSsiQ=">AAAB9HicdVBNS8NAEN3Ur1q/qh69LBbBU0jSaNtb0YvHCtYWmlg22027dLOJu5tCCf0dXjwo4tUf481/46atoKIPBh7vzTAzL0gYlcqyPozCyura+kZxs7S1vbO7V94/uJVxKjBp45jFohsgSRjlpK2oYqSbCIKigJFOML7M/c6ECEljfqOmCfEjNOQ0pBgpLfmepBH0WDy8c+CkX65YpttwbbsOLfO8UbeqjZw4NffMhrZpzVEBS7T65XdvEOM0IlxhhqTs2Vai/AwJRTEjs5KXSpIgPEZD0tOUo4hIP5sfPYMnWhnAMBa6uIJz9ftEhiIpp1GgOyOkRvK3l4t/eb1UhXU/ozxJFeF4sShMGVQxzBOAAyoIVmyqCcKC6lshHiGBsNI5lXQIX5/C/8mtY9pV07l2K82LZRxFcASOwSmwQQ00wRVogTbA4B48gCfwbEyMR+PFeF20FozlzCH4AePtE2e1keA=</latexit>

• Equivalently, the whole correlator can be reconstructed from the double discontinuity (dDisc). The same type of ambiguities arise also in this approach.

�n,`, an,` 6= 0<latexit sha1_base64="tXzJMWpEnRHr05gyK1oJ5nuz6as=">AAACC3icdVA9SwNBEN3zM8avqKXNYhAs5LhLokm6oI1lBKNCLoS9zSRZsrt37u4J4Uhv41+xsVDE1j9g579xEyN+oA8GHu/NMDMvjDnTxvPenJnZufmFxcxSdnlldW09t7F5rqNEUWjQiEfqMiQaOJPQMMxwuIwVEBFyuAgHx2P/4hqUZpE8M8MYWoL0JOsySoyV2rmdoEeEIO1U7gfA+Wgff3EcSLjCXjuX99xSteT7Fey5h9WKV6yOSaFcOvCx73oT5NEU9XbuNehENBEgDeVE66bvxaaVEmUY5TDKBomGmNAB6UHTUkkE6FY6+WWEd63Swd1I2ZIGT9TvEykRWg9FaDsFMX392xuLf3nNxHQrrZTJODEg6ceibsKxifA4GNxhCqjhQ0sIVczeimmfKEKNjS9rQ/j8FP9PzguuX3QLp6V87WgaRwZtox20h3xURjV0guqogSi6QXfoAT06t8698+Q8f7TOONOZLfQDzss7Ormaig==</latexit>

` = 0, 2, . . . L<latexit sha1_base64="ec3qzsftd978nuIPjn00QEWV3Jk=">AAAB+XicdVDLSsNAFJ3UV62vqEs3g0VwUUKSRtsuhKIbFy4qWFtoQ5lMJ+3QySTMTAol9E/cuFDErX/izr9x+hBU9MDA4Zx7uHdOkDAqlW1/GLmV1bX1jfxmYWt7Z3fP3D+4l3EqMGnimMWiHSBJGOWkqahipJ0IgqKAkVYwupr5rTERksb8Tk0S4kdowGlIMVJa6plmlzB2YZfcUrcfKwlvembRtrya5zhVaFvntapdrs2IW/HOHOhY9hxFsESjZ77rJE4jwhVmSMqOYyfKz5BQFDMyLXRTSRKER2hAOppyFBHpZ/PLp/BEK30YxkI/ruBc/Z7IUCTlJAr0ZITUUP72ZuJfXidVYdXPKE9SRTheLApTBlUMZzXAPhUEKzbRBGFB9a0QD5FAWOmyCrqEr5/C/8m9azlly731ivXLZR15cASOwSlwQAXUwTVogCbAYAwewBN4NjLj0XgxXhejOWOZOQQ/YLx9Aigjkrg=</latexit>

Caron-Huot

Heemskerk, Penedones, Polchinski, Sully

Page 19: Genus-one string amplitudes from CFT · 2019. 8. 19. · 2 Based on • 1809.10670 • 1706.02388 • 1705.02318 • 1612.03891 with (subsets of) Aharony, Alday and Perlmutter PROGRAM:

Strategy

�12

• Expand the OPE data for large N

� = 4 + 2n+ `+�(1)n,`(�)

N2+

�(2)n,`(�)

N4+ . . .

<latexit sha1_base64="l6IApGCgr6yV00WuoH0VpiVfnB0=">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</latexit>

an,` = a(0)n,` +a(1)n,`(�)

N2+

a(2)n,`(�)

N4+ . . .

<latexit sha1_base64="D79bF3PwdCszY1+XGUeyHvlaU2Q=">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</latexit>

•Use crossing symmetry to fix the OPE data by requiring the proper singularity structure.

•Up to order there are power law singularities and we can compute and N�2<latexit sha1_base64="qsCNEjRzRFXJ8rWtS31kcoZea4M=">AAAB7XicdVDLSgMxFM3UV62vqks3wSK4cZjpg7a7ohtXUsE+oB1LJs20sZlkSDJCGfoPblwo4tb/ceffmGkrqOiBC4dz7uXee/yIUaUd58PKrKyurW9kN3Nb2zu7e/n9g7YSscSkhQUTsusjRRjlpKWpZqQbSYJCn5GOP7lI/c49kYoKfqOnEfFCNOI0oBhpI7WvbpOz4myQLzh2qVgp1x1oSLni1GspqaYSdG1njgJYojnIv/eHAsch4RozpFTPdSLtJUhqihmZ5fqxIhHCEzQiPUM5Conykvm1M3hilCEMhDTFNZyr3ycSFCo1DX3TGSI9Vr+9VPzL68U6qHkJ5VGsCceLRUHMoBYwfR0OqSRYs6khCEtqboV4jCTC2gSUMyF8fQr/J+2i7Zbs4nW50DhfxpEFR+AYnAIXVEEDXIImaAEM7sADeALPlrAerRfrddGasZYzh+AHrLdPckCPDw==</latexit>

a(0)n,`<latexit sha1_base64="5gPq5h+lPpLxF0rs5fX7qe95Ctw=">AAAB+XicdVDLSsNAFJ34rPUVdelmsAgVJKQv2u6KblxWsA9oY5hMJ+3QySTMTAol5E/cuFDErX/izr9x0lZQ0QMXDufcy733eBGjUtn2h7G2vrG5tZ3bye/u7R8cmkfHXRnGApMODlko+h6ShFFOOooqRvqRICjwGOl50+vM782IkDTkd2oeESdAY059ipHSkmuayE345ZAwlt4nRfsidc2CbVXKtWrThppUa3azkZF6JsGSZS9QACu0XfN9OApxHBCuMENSDkp2pJwECUUxI2l+GEsSITxFYzLQlKOASCdZXJ7Cc62MoB8KXVzBhfp9IkGBlPPA050BUhP528vEv7xBrPyGk1AexYpwvFzkxwyqEGYxwBEVBCs21wRhQfWtEE+QQFjpsPI6hK9P4f+kW7ZKFat8Wy20rlZx5MApOANFUAJ10AI3oA06AIMZeABP4NlIjEfjxXhdtq4Zq5kT8APG2ycyCZNj</latexit>

�(1)n,`

<latexit sha1_base64="tDr736w65FvmBT7FlGV0Jc1uMf4=">AAAB/nicdVDLSgMxFM3UV62vqrhyEyxCBSkzfdB2V3TjsoJ9QGcsmTRtQ5PMkGSEMhT8FTcuFHHrd7jzb8y0FVT0wIXDOfdy7z1+yKjStv1hpVZW19Y30puZre2d3b3s/kFbBZHEpIUDFsiujxRhVJCWppqRbigJ4j4jHX9ymfidOyIVDcSNnobE42gk6JBipI3Uzx65I8Q5uo3zztmsH4tzlzA262dzdqFUrJTrNjSkXLHrtYRUEwk6BXuOHFii2c++u4MAR5wIjRlSqufYofZiJDXFjMwybqRIiPAEjUjPUIE4UV48P38GT40ygMNAmhIaztXvEzHiSk25bzo50mP120vEv7xepIc1L6YijDQReLFoGDGoA5hkAQdUEqzZ1BCEJTW3QjxGEmFtEsuYEL4+hf+TdrHglArF63KucbGMIw2OwQnIAwdUQQNcgSZoAQxi8ACewLN1bz1aL9brojVlLWcOwQ9Yb58RjJWU</latexit>

Page 20: Genus-one string amplitudes from CFT · 2019. 8. 19. · 2 Based on • 1809.10670 • 1706.02388 • 1705.02318 • 1612.03891 with (subsets of) Aharony, Alday and Perlmutter PROGRAM:

Sugra+strings

�13

Fixed by singularities

Truncated solutions

�(1)(�) = �(1)sugra +

�(1)truncated

�3/2+

�(1)truncated

�5/2+ . . .

<latexit sha1_base64="nENPzTG6I018GqIxFSt7j1TDVOU=">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</latexit>

Page 21: Genus-one string amplitudes from CFT · 2019. 8. 19. · 2 Based on • 1809.10670 • 1706.02388 • 1705.02318 • 1612.03891 with (subsets of) Aharony, Alday and Perlmutter PROGRAM:

How to go to one loop

�14

•At order there is a term of this form:

⇣uv

⌘2 X

n,`

a(0)n,`v2+n+

�(1)n,`

N2 +�(2)n,`

N4 gn,`(v, u) ⇠log2 v

N4

X

n,`

a(0)n,`(�(1)n,`)

2v2+ngn,`(v, u)

<latexit sha1_base64="OVwkgjEJut2yVdqwJ2m+TOHWwg4=">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</latexit>

N�4<latexit sha1_base64="Qj2nBUNIght1Jn3VN8H2GU4sKsk=">AAAB7XicdVDLSgMxFM3UV62vqks3wSK4cZi2U9ruim5cSQX7gHYsmTTTxmaSIckIZeg/uHGhiFv/x51/Y6atoKIHLhzOuZd77/EjRpV2nA8rs7K6tr6R3cxtbe/s7uX3D9pKxBKTFhZMyK6PFGGUk5ammpFuJAkKfUY6/uQi9Tv3RCoq+I2eRsQL0YjTgGKkjdS+uk3O3NkgX3Dscqni1h1oiFtx6rWUVFMJFm1njgJYojnIv/eHAsch4RozpFSv6ETaS5DUFDMyy/VjRSKEJ2hEeoZyFBLlJfNrZ/DEKEMYCGmKazhXv08kKFRqGvqmM0R6rH57qfiX14t1UPMSyqNYE44Xi4KYQS1g+jocUkmwZlNDEJbU3ArxGEmEtQkoZ0L4+hT+T9olu1i2S9duoXG+jCMLjsAxOAVFUAUNcAmaoAUwuAMP4Ak8W8J6tF6s10VrxlrOHIIfsN4+AXVKjxE=</latexit>

•The coefficient in front of the singularity is given by a(0)n,`(�(1)n,`)

2<latexit sha1_base64="YVtq0nd45dUrIpAxXTbiVqkyShA=">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</latexit>

•From this information it is possible to compute and the full correlator. �(2)n,`

<latexit sha1_base64="995OsvlP+XsUL0wPv3a/gHWDAcE=">AAAB/nicdVDLSgMxFM3UV62vqrhyEyxCBSkzfdB2V3TjsoJ9QKeWTJppQ5PMkGSEMhT8FTcuFHHrd7jzb8y0FVT0wIXDOfdy7z1eyKjStv1hpVZW19Y30puZre2d3b3s/kFbBZHEpIUDFsiuhxRhVJCWppqRbigJ4h4jHW9ymfidOyIVDcSNnoakz9FIUJ9ipI00yB65I8Q5uo3zxbPZIBbnLmFsNsjm7EKpWCnXbWhIuWLXawmpJhJ0CvYcObBEc5B9d4cBjjgRGjOkVM+xQ92PkdQUMzLLuJEiIcITNCI9QwXiRPXj+fkzeGqUIfQDaUpoOFe/T8SIKzXlnunkSI/Vby8R//J6kfZr/ZiKMNJE4MUiP2JQBzDJAg6pJFizqSEIS2puhXiMJMLaJJYxIXx9Cv8n7WLBKRWK1+Vc42IZRxocgxOQBw6ogga4Ak3QAhjE4AE8gWfr3nq0XqzXRWvKWs4cgh+w3j4BExuVlQ==</latexit>

Page 22: Genus-one string amplitudes from CFT · 2019. 8. 19. · 2 Based on • 1809.10670 • 1706.02388 • 1705.02318 • 1612.03891 with (subsets of) Aharony, Alday and Perlmutter PROGRAM:

�15

⇣�(1)n,`

⌘2=

⇣�(1)sugra + �(1)

truncated

⌘2= �(1)

sugra ⇥ �(1)sugra + �(1)

sugra ⇥ �(1)truncated + �(1)

truncated ⇥ �(1)truncated

<latexit sha1_base64="MaE47IlGTTw1GsED6Np3uEoVPFs=">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</latexit>

1

�32

<latexit sha1_base64="p8db+/mg2dYuvdz/+Aey0JKXHEI=">AAACCHicdVDLSsNAFJ3UV62vqEsXDhbBVUj6oO2u6MZlBfuAJpbJZNIOnTyYmQglZOnGX3HjQhG3foI7/8ZJW0FFDwwczrmHO/e4MaNCmuaHVlhZXVvfKG6WtrZ3dvf0/YOeiBKOSRdHLOIDFwnCaEi6kkpGBjEnKHAZ6bvTi9zv3xIuaBRey1lMnACNQ+pTjKSSRvqx7XOEUytLbaZSHrpJF0o1SytZlo30smlUK/Vay4SK1Opmq5mTRi5ByzDnKIMlOiP93fYinAQklJghIYaWGUsnRVxSzEhWshNBYoSnaEyGioYoIMJJ54dk8FQpHvQjrl4o4Vz9nkhRIMQscNVkgORE/PZy8S9vmEi/6aQ0jBNJQrxY5CcMygjmrUCPcoIlmymCMKfqrxBPkOpBqu5KqoSvS+H/pFcxrKpRuaqV2+fLOorgCJyAM2CBBmiDS9ABXYDBHXgAT+BZu9cetRftdTFa0JaZQ/AD2tsnGV+asA==</latexit>

1

�3<latexit sha1_base64="WELEWC8SykYTj+CAD/VKNdmmhvo=">AAAB/HicdVDLSsNAFJ34rPUV7dLNYBFchaQP2u6KblxWsA9oYplMJu3QyYOZiRBC/BU3LhRx64e482+ctBVU9MDA4Zx7uHeOGzMqpGl+aGvrG5tb26Wd8u7e/sGhfnQ8EFHCMenjiEV85CJBGA1JX1LJyCjmBAUuI0N3fln4wzvCBY3CG5nGxAnQNKQ+xUgqaaJXbJ8jnFl5ZjOV8tBtPZ/oVdOo15qNjgkVaTTNTrsgrUKClmEuUAUr9Cb6u+1FOAlIKDFDQowtM5ZOhrikmJG8bCeCxAjP0ZSMFQ1RQISTLY7P4ZlSPOhHXL1QwoX6PZGhQIg0cNVkgORM/PYK8S9vnEi/7WQ0jBNJQrxc5CcMyggWTUCPcoIlSxVBmFN1K8QzpNqQqq+yKuHrp/B/MqgZVt2oXTeq3YtVHSVwAk7BObBAC3TBFeiBPsAgBQ/gCTxr99qj9qK9LkfXtFWmAn5Ae/sENemVJg==</latexit>

Alday, Caron-Huot

Aprile, Drummond, Heslop, Paul

Page 23: Genus-one string amplitudes from CFT · 2019. 8. 19. · 2 Based on • 1809.10670 • 1706.02388 • 1705.02318 • 1612.03891 with (subsets of) Aharony, Alday and Perlmutter PROGRAM:

�16

G(u, v) = G(0)(u, v) +1

N2

✓G(1)(u, v)sugra +

G(1)(u, v)string

�32

+ . . .

◆+

1

N4

G(2)sugra(u, v) +

G(2)string(u, v)

�32

+ . . .

!

<latexit sha1_base64="etMUlvj/eT9JRM1rZ+jqcfXG7OM=">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</latexit>

Final results

Page 24: Genus-one string amplitudes from CFT · 2019. 8. 19. · 2 Based on • 1809.10670 • 1706.02388 • 1705.02318 • 1612.03891 with (subsets of) Aharony, Alday and Perlmutter PROGRAM:

�16

G(u, v) = G(0)(u, v) +1

N2

✓G(1)(u, v)sugra +

G(1)(u, v)string

�32

+ . . .

◆+

1

N4

G(2)sugra(u, v) +

G(2)string(u, v)

�32

+ . . .

!

<latexit sha1_base64="etMUlvj/eT9JRM1rZ+jqcfXG7OM=">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</latexit>

Completely fixed by the singularities arising from protected operators in the OPE

Final results

Page 25: Genus-one string amplitudes from CFT · 2019. 8. 19. · 2 Based on • 1809.10670 • 1706.02388 • 1705.02318 • 1612.03891 with (subsets of) Aharony, Alday and Perlmutter PROGRAM:

Mellin amplitudes• String amplitudes on are given by the Mellin transform M(s,t,u)

�17

AdS5 ⇥ S5<latexit sha1_base64="H3qaer3qwmOM5qffwm6G9eFucTA=">AAAB+XicdVDLSsNAFJ34rPUVdelmsAiuQtI2tN1V3bis1D6gjWEymbRDJw9mJoUS+iduXCji1j9x5984aSuo6IELh3Pu5d57vIRRIU3zQ1tb39jc2i7sFHf39g8O9aPjrohTjkkHxyzmfQ8JwmhEOpJKRvoJJyj0GOl5k+vc700JFzSO7uQsIU6IRhENKEZSSa6uX/pt14ZDSUMiYPvedvWSaVTKdrVhQkWqttmo56SWS9AyzAVKYIWWq78P/RinIYkkZkiIgWUm0skQlxQzMi8OU0EShCdoRAaKRkgtcrLF5XN4rhQfBjFXFUm4UL9PZCgUYhZ6qjNEcix+e7n4lzdIZVB3MholqSQRXi4KUgZlDPMYoE85wZLNFEGYU3UrxGPEEZYqrKIK4etT+D/plg2rYpRvq6Xm1SqOAjgFZ+ACWKAGmuAGtEAHYDAFD+AJPGuZ9qi9aK/L1jVtNXMCfkB7+wROR5LR</latexit>

G(u0, v) =

Z i1

�i1ds dt u0svtM(s, t, u)�2(2� s)�2(2� t)�2(2� u)

<latexit sha1_base64="K1PjvZKgIVmi/Dt73s63oNiy08w=">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</latexit>

with s+t+u=2.

• At genus zero

M (0)(s, t, u) =1

(s� 1)(t� 1)(u� 1)+ (↵0)3a+ (↵0)5

�b�s2 + t2 + u2

�+ b1

�+ . . .

<latexit sha1_base64="QBQGG82ShKX1Dq4Lo2srdsiXGSY=">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</latexit>

polynomials in s,t,u

unknown

SUGRA R4<latexit sha1_base64="43ZqLIzK/f+NQuzqE5Xwd+kHhuQ=">AAAB9HicdVDLTgIxFL2DL8QX6tJNIzFxNZmBUWBHdOMSjTwSGEmnFGjoPGw7JGTCd7hxoTFu/Rh3/o0dwESNnqTJyTn35p4eL+JMKsv6MDIrq2vrG9nN3Nb2zu5efv+gKcNYENogIQ9F28OSchbQhmKK03YkKPY9Tlve+DL1WxMqJAuDWzWNqOvjYcAGjGClJbfrYzUimCc3szunly9YplN1bLuCLPO8WrFK1ZQUy86ZjWzTmqMAS9R7+fduPySxTwNFOJayY1uRchMsFCOcznLdWNIIkzEe0o6mAfapdJN56Bk60UofDUKhX6DQXP2+kWBfyqnv6ck0pPztpeJfXidWg4qbsCCKFQ3I4tAg5kiFKG0A9ZmgRPGpJpgIprMiMsICE6V7yukSvn6K/ifNommXzOK1U6hdLOvIwhEcwynYUIYaXEEdGkDgHh7gCZ6NifFovBivi9GMsdw5hB8w3j4BOjWSaQ==</latexit>

@4R4<latexit sha1_base64="hA0HimGXsaqHX6Lxh/SDSNxvC10=">AAACAHicdVDLSsNAFJ34rPUVdeHCzWARXIWkjbbdFd24rGIf0KRlMp20QycPZiZCCdn4K25cKOLWz3Dn3zhpK6jogYHDOffeufd4MaNCmuaHtrS8srq2Xtgobm5t7+zqe/ttESUckxaOWMS7HhKE0ZC0JJWMdGNOUOAx0vEml7nfuSNc0Ci8ldOYuAEahdSnGEklDfRDJ0ZcUsT6thMgOcaIpTdZ3x7oJdOw67Zl1aBpnNdrZqWek3LVPrOgZZgzlMACzYH+7gwjnAQklJghIXqWGUs3zYdjRrKikwgSIzxBI9JTNEQBEW46OyCDJ0oZQj/i6oUSztTvHSkKhJgGnqrMlxS/vVz8y+sl0q+5KQ3jRJIQzz/yEwZlBPM04JBygiWbKoIwp2pXiMeIIyxVZkUVwtel8H/SLhtWxShf26XGxSKOAjgCx+AUWKAKGuAKNEELYJCBB/AEnrV77VF70V7npUvaoucA/ID29glB55bZ</latexit>

Mack Penedones

Rastelli,Zhou

see also Drummond, Nandan, Paul, Rigatos

Page 26: Genus-one string amplitudes from CFT · 2019. 8. 19. · 2 Based on • 1809.10670 • 1706.02388 • 1705.02318 • 1612.03891 with (subsets of) Aharony, Alday and Perlmutter PROGRAM:

Flat space limit

�18

• Take the flat space limit s, t, u ! 1, ↵0 ! 0 with s↵0, t↵0, u↵0 fixed<latexit sha1_base64="jdhTLj+pKG6lNCAW1syOVKJDafA=">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</latexit>

M (0)(s, t, u) ! M̃ (0)(s, t, u)<latexit sha1_base64="Kaw6UECAia/Ws9Zj7wUxNSNqTNA=">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</latexit>

SHAPIRO-VIRASORO AMPLITUDE

M (0)(s, t, u) =1

(s� 1)(t� 1)(u� 1)+ (↵0)3a+ (↵0)5

�b�s2 + t2 + u2

�+ b1

�+ . . .

<latexit sha1_base64="QBQGG82ShKX1Dq4Lo2srdsiXGSY=">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</latexit>

• The matching with the Shapiro Virasoro amplitude fixes some coefficients:

curvature corrections

• The same can be done for the genus one amplitude

M̃ (1)(s, t, u)<latexit sha1_base64="R9zMhLTdQ83cueF8Alg0k1qy/R4=">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</latexit>

• The genus one matching puts constraints on the curvature corrections b1, . . .<latexit sha1_base64="JGIgChli9FO5OwPkXIihDD4SMDs=">AAAB8XicdVDLSsNAFJ3UV62vqktFBovgQkLSB213RTcuW7APbEOZTCft0MkkzEyEErr0D9y4UMStP9DvcOc3+BNOWgUVPXDhcM493IcbMiqVZb0ZqaXlldW19HpmY3Nreye7u9eSQSQwaeKABaLjIkkY5aSpqGKkEwqCfJeRtju+SPz2DRGSBvxKTULi+GjIqUcxUlq6dvv2GewNAiX72ZxlFvKlYtWCmhRLVrWSkHIiQdu05sjVDmeN99ujWb2ffdU5HPmEK8yQlF3bCpUTI6EoZmSa6UWShAiP0ZB0NeXIJ9KJ5xtP4YlWBtALhC6u4Fz9noiRL+XEd3Wnj9RI/vYS8S+vGymv4sSUh5EiHC8GeRGDKoDJ+XBABcGKTTRBWFC9K8QjJBBW+kkZ/YSvS+H/pJU37YKZb9i52jlYIA0OwDE4BTYogxq4BHXQBBhwcAcewKMhjXvjyXhetKaMz8w++AHj5QP6kZQw</latexit>

Non-analytic part of type IIB genus one string ampl in flat space

see also Binder,Chester,Pufu, Wang

Green,Russo,Vanhove M (1)(s, t, u) !<latexit sha1_base64="118VIQjw2dV0Fn5xx0aTRc4AXnY=">AAAB+3icdVBNS0JBFJ1nX2ZfL1u2GZJAQeQ9P1B3Ups2gUFqoCbzxlEH530wc18kD/9KmxZFtO2PtOvfNE8NKurAhcM593LvPU4guALL+jASa+sbm1vJ7dTO7t7+gXmYbis/lJS1qC98eeMQxQT3WAs4CHYTSEZcR7COMz2P/c4dk4r73jXMAtZ3ydjjI04JaGlgpi9vo6ydm2dVHvJhDvfAH5gZq1AqVsp1C2tSrlj1WkyqsYTtgrVABq3QHJjvvaFPQ5d5QAVRqmtbAfQjIoFTweapXqhYQOiUjFlXU4+4TPWjxe1zfKqVIR75UpcHeKF+n4iIq9TMdXSnS2Cifnux+JfXDWFU60fcC0JgHl0uGoUCg4/jIPCQS0ZBzDQhVHJ9K6YTIgkFHVdKh/D1Kf6ftIsFu1QoXpUzjbNVHEl0jE5QFtmoihroAjVRC1F0jx7QE3o25saj8WK8LlsTxmrmCP2A8fYJJmWTPg==</latexit>

Page 27: Genus-one string amplitudes from CFT · 2019. 8. 19. · 2 Based on • 1809.10670 • 1706.02388 • 1705.02318 • 1612.03891 with (subsets of) Aharony, Alday and Perlmutter PROGRAM:

Conclusions

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• I discussed how to construct genus one string amplitudes on curved space, studying the dual correlator.

• The matching with the flat space limit agrees with known results and give predictions to curvature corrections.

• Vertex operator

• Higher genus

• Heavy states

• Fixing ambiguities (numerical bootstrap?)

Page 28: Genus-one string amplitudes from CFT · 2019. 8. 19. · 2 Based on • 1809.10670 • 1706.02388 • 1705.02318 • 1612.03891 with (subsets of) Aharony, Alday and Perlmutter PROGRAM:

�20