Machine Learning for
Calabi-Yau metrics
FABIAN RUEHLE (CERN & UNIVERSITY OF OXFORD)
Deep Learning and Physics 201902 November 2019
Based on work in progress withLara Anderson, James Gray, Sven Krippendorf, Nikhil Raghuram

Outline‣ String compactifications on Calabi-Yau manifolds• Introduction• The necessity of knowing the metric‣ Calculation of Calabi-Yau metrics• Numerical computation• Machine learning approaches❖ Phrase as optimization problem and solve with GD❖ Use supervised ML (regression) to interpolate metrics❖ Learn metric directly using boosting‣ Conclusion

1. String Compactifications

String compactifications‣ String theory requires 10 spacetime dimensions‣ We only observe 4 spacetime dimensions‣ Assume 6 dimensions are small and compact andthus avoid detection‣ Mathematical framework of string theory puts strong

constraints on the properties of the compactification
spaceString compactifications
Mathematical framework of string theory puts strong
constraints on the properties of the compact space[Candelas, Horowitz, Strominger and Witten `85]

‣ The space must be complex

structure globally) Can define complex<latexit sha1_base64="FUIxtivFMN5Jpm6QFJ9RPJxb79c=">AAAB8nicbVBNS8NAEN3Ur1q/qh69LBbBU0ms+HErevFYxdpCGspmu2mXbnbD7kQpoT/DiwdFvPprvPlvTNIgan0w8Hhvhpl5fiS4Adv+tEoLi0vLK+XVytr6xuZWdXvnzqhYU9amSijd9YlhgkvWBg6CdSPNSOgL1vHHl5nfuWfacCVvYRIxLyRDyQNOCaSS27vhwxEQrdVDv1qz63YOPE+cgtRQgVa/+tEbKBqHTAIVxBjXsSPwEqKBU8GmlV5sWETomAyZm1JJQma8JD95ig9SZYADpdOSgHP150RCQmMmoZ92hgRG5q+Xif95bgzBmZdwGcXAJJ0tCmKBQeHsfzzgmlEQk5QQqnl6K6YjogmFNKVKHsJ5hpPvl+fJ3VHdadQb18e15kURRxntoX10iBx0iproCrVQG1Gk0CN6Ri8WWE/Wq/U2ay1Zxcwu+gXr/Qup45Gk</latexit>

U1

U2<latexit sha1_base64="ug4OtSrnhm6CKoWRZ8trjXlNJvM=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0ms+HErevFY0bSFNpTNdtMu3WzC7kYooT/BiwdFvPqLvPlv3KRB1Ppg4PHeDDPz/JgzpW370yotLa+srpXXKxubW9s71d29tooSSahLIh7Jro8V5UxQVzPNaTeWFIc+px1/cp35nQcqFYvEvZ7G1AvxSLCAEayNdOcOnEG1ZtftHGiROAWpQYHWoPrRH0YkCanQhGOleo4day/FUjPC6azSTxSNMZngEe0ZKnBIlZfmp87QkVGGKIikKaFRrv6cSHGo1DT0TWeI9Vj99TLxP6+X6ODCS5mIE00FmS8KEo50hLK/0ZBJSjSfGoKJZOZWRMZYYqJNOpU8hMsMZ98vL5L2Sd1p1Bu3p7XmVRFHGQ7gEI7BgXNowg20wAUCI3iEZ3ixuPVkvVpv89aSVczswy9Y71/vF420</latexit><latexit sha1_base64="A8qAMOAx2nS1XVSKOZ5XTI55D3U=">AAAB6nicbVBNS8NAEJ34WetX1aOXxSJ4KkkrftyKXjxWNG2hDWWz3bRLN5uwuxFK6E/w4kERr/4ib/4bN2kQtT4YeLw3w8w8P+ZMadv+tJaWV1bX1ksb5c2t7Z3dyt5+W0WJJNQlEY9k18eKciaoq5nmtBtLikOf044/uc78zgOVikXiXk9j6oV4JFjACNZGunMH9UGlatfsHGiROAWpQoHWoPLRH0YkCanQhGOleo4day/FUjPC6azcTxSNMZngEe0ZKnBIlZfmp87QsVGGKIikKaFRrv6cSHGo1DT0TWeI9Vj99TLxP6+X6ODCS5mIE00FmS8KEo50hLK/0ZBJSjSfGoKJZOZWRMZYYqJNOuU8hMsMZ98vL5J2veY0ao3b02rzqoijBIdwBCfgwDk04QZa4AKBETzCM7xY3HqyXq23eeuSVcwcwC9Y71/wm421</latexit>

f12 holomorphic

String compactifications
Mathematical framework of string theory puts strong
constraints on the properties of the compact space[Candelas, Horowitz, Strominger and Witten `85]

‣ The space must be Kähler

gab̄ = @a @ b̄ K

real function called the Kähler potential
i
2
a
b̄
a
b̄
ds = gab̄ dz ⌦ dz̄
J = gab̄ dz ^ dz
2
Kähler form
<latexit sha1_base64="oP4am18ODl7OhCSxKWoPvQu+FGc=">AAACHnicbVDLSgMxFM3UV62vqks3wVJwVabW50IouhHcVLAP6AzDnTRtQzMPkoxQhvkSN/6KGxeKCK70b8y0o6j1QOBwzr039x435Ewq0/wwcnPzC4tL+eXCyura+kZxc6slg0gQ2iQBD0THBUk582lTMcVpJxQUPJfTtju6SP32LRWSBf6NGofU9mDgsz4joLTkFA8HTgyWCyJ2k+TMCkEoBtwBK9Bd6dD4S0ucOK3DboKvnGLJrJgT4FlSzUgJZWg4xTerF5DIo74iHKTsVs1Q2XE6mXCaFKxI0hDICAa0q6kPHpV2PDkvwWWt9HA/EPr5Ck/Unx0xeFKOPb1Y2QM1lH+9VPzP60aqf2LHzA8jRX0y/agfcawCnGaFe0xQovhYEyCC6V0xGYIAonSihUkIpymOvk+eJa39SrVWqV0flOrnWRx5tIN20R6qomNUR5eogZqIoDv0gJ7Qs3FvPBovxuu0NGdkPdvoF4z3T5CopBo=</latexit><latexit sha1_base64="RyIkbnV3c19loLeSU88gqTme3qk=">AAACFnicbVDJSgNBEO1xjXGLevTSGAJeDJNEXA5C0IvHCGaBzCTU9HSSJj0L3T1CMuQrvPgrXjwo4lW8+Td2JoOo8UHB470qquo5IWdSmeansbC4tLyymlnLrm9sbm3ndnYbMogEoXUS8EC0HJCUM5/WFVOctkJBwXM4bTrDq6nfvKNCssC/VaOQ2h70fdZjBJSWurkjV3bKF/1uDJYDAjsT7I47YAWKeVRiNxHHnTg1u7m8WTQT4HlSSkkepah1cx+WG5DIo74iHKRsl8xQ2TEIxQink6wVSRoCGUKftjX1QW+14+StCS5oxcW9QOjyFU7UnxMxeFKOPH1zwQM1kH+9qfif145U78yOmR9GivpktqgXcawCPM0Iu0xQovhIEyCC6VsxGYAAonSS2SSE8ylOvl+eJ41ysVQpVm6O89XLNI4M2kcH6BCV0CmqomtUQ3VE0D16RM/oxXgwnoxX423WumCkM3voF4z3L7Wrnzw=</latexit><latexit sha1_base64="ZIu8S1jhCJcXGGTvZVTdDiKgJjU=">AAACHHicbZBLSwMxFIUzvq2vqks3wSK4KlMrPhBBdCOuFKwKnVruZO7U0MyDJKPUYfwfbvwrblwo4saF4L8xUwfxdSDwcc4NyT1uLLjStv1uDQwODY+Mjo2XJianpmfKs3MnKkokwwaLRCTPXFAoeIgNzbXAs1giBK7AU7e7l+enlygVj8Jj3YuxFUAn5D5noI3VLtcPth1fAkt5lq5kN7TTTsFxQVI3c7aod30OzhV6HcwxLYJ2uWJX7b7oX6gVUCGFDtvlV8eLWBJgqJkApZo1O9atFKTmTGBWchKFMbAudLBpMIQAVSvtL5fRJeN41I+kOaGmfff7jRQCpXqBayYD0Bfqd5ab/2XNRPsbrZSHcaIxZJ8P+YmgOqJ5U9TjEpkWPQPAJDd/pewCTFfa9Fnql7CZa+1r5b9wslKt1av1o9XKzm5RxxhZIItkmdTIOtkh++SQNAgjt+SePJIn6856sJ6tl8/RAau4M09+yHr7AKh4od0=</latexit>

String compactifications
Mathematical framework of string theory puts strong
constraints on the properties of the compact space[Candelas, Horowitz, Strominger and Witten `85]

‣ The space must be Kähler

gab̄ = @a @ b̄ K

real function called the Kähler potential
i
2
a
b̄
a
b̄
ds = gab̄ dz ⌦ dz̄
J = gab̄ dz ^ dz
2
Kähler form
<latexit sha1_base64="oP4am18ODl7OhCSxKWoPvQu+FGc=">AAACHnicbVDLSgMxFM3UV62vqks3wVJwVabW50IouhHcVLAP6AzDnTRtQzMPkoxQhvkSN/6KGxeKCK70b8y0o6j1QOBwzr039x435Ewq0/wwcnPzC4tL+eXCyura+kZxc6slg0gQ2iQBD0THBUk582lTMcVpJxQUPJfTtju6SP32LRWSBf6NGofU9mDgsz4joLTkFA8HTgyWCyJ2k+TMCkEoBtwBK9Bd6dD4S0ucOK3DboKvnGLJrJgT4FlSzUgJZWg4xTerF5DIo74iHKTsVs1Q2XE6mXCaFKxI0hDICAa0q6kPHpV2PDkvwWWt9HA/EPr5Ck/Unx0xeFKOPb1Y2QM1lH+9VPzP60aqf2LHzA8jRX0y/agfcawCnGaFe0xQovhYEyCC6V0xGYIAonSihUkIpymOvk+eJa39SrVWqV0flOrnWRx5tIN20R6qomNUR5eogZqIoDv0gJ7Qs3FvPBovxuu0NGdkPdvoF4z3T5CopBo=</latexit><latexit sha1_base64="RyIkbnV3c19loLeSU88gqTme3qk=">AAACFnicbVDJSgNBEO1xjXGLevTSGAJeDJNEXA5C0IvHCGaBzCTU9HSSJj0L3T1CMuQrvPgrXjwo4lW8+Td2JoOo8UHB470qquo5IWdSmeansbC4tLyymlnLrm9sbm3ndnYbMogEoXUS8EC0HJCUM5/WFVOctkJBwXM4bTrDq6nfvKNCssC/VaOQ2h70fdZjBJSWurkjV3bKF/1uDJYDAjsT7I47YAWKeVRiNxHHnTg1u7m8WTQT4HlSSkkepah1cx+WG5DIo74iHKRsl8xQ2TEIxQink6wVSRoCGUKftjX1QW+14+StCS5oxcW9QOjyFU7UnxMxeFKOPH1zwQM1kH+9qfif145U78yOmR9GivpktqgXcawCPM0Iu0xQovhIEyCC6VsxGYAAonSS2SSE8ylOvl+eJ41ysVQpVm6O89XLNI4M2kcH6BCV0CmqomtUQ3VE0D16RM/oxXgwnoxX423WumCkM3voF4z3L7Wrnzw=</latexit><latexit sha1_base64="ZIu8S1jhCJcXGGTvZVTdDiKgJjU=">AAACHHicbZBLSwMxFIUzvq2vqks3wSK4KlMrPhBBdCOuFKwKnVruZO7U0MyDJKPUYfwfbvwrblwo4saF4L8xUwfxdSDwcc4NyT1uLLjStv1uDQwODY+Mjo2XJianpmfKs3MnKkokwwaLRCTPXFAoeIgNzbXAs1giBK7AU7e7l+enlygVj8Jj3YuxFUAn5D5noI3VLtcPth1fAkt5lq5kN7TTTsFxQVI3c7aod30OzhV6HcwxLYJ2uWJX7b7oX6gVUCGFDtvlV8eLWBJgqJkApZo1O9atFKTmTGBWchKFMbAudLBpMIQAVSvtL5fRJeN41I+kOaGmfff7jRQCpXqBayYD0Bfqd5ab/2XNRPsbrZSHcaIxZJ8P+YmgOqJ5U9TjEpkWPQPAJDd/pewCTFfa9Fnql7CZa+1r5b9wslKt1av1o9XKzm5RxxhZIItkmdTIOtkh++SQNAgjt+SePJIn6856sJ6tl8/RAau4M09+yHr7AKh4od0=</latexit>‣ Note: Symmetry of Kähler transformations

K !K +f +f<latexit sha1_base64="8KSJwA6jLyr+qjeJpy7H/FuSyb8=">AAACCXicbVDNS8MwHE3n15xfVY9egkMQBqN14sdt6EXYZYL7gLWMNEu3sLQpSaqMsqsX/xUvHhTx6n/gzf/GtCuizgeBx3u/l+T3vIhRqSzr0ygsLC4trxRXS2vrG5tb5vZOW/JYYNLCnHHR9ZAkjIakpahipBsJggKPkY43vkz9zi0RkvLwRk0i4gZoGFKfYqS01DdhwxF0OFJICH4HG7ACfVhxuI6kNyb+tG+WraqVAc4TOydlkKPZNz+cAcdxQEKFGZKyZ1uRchMkFMWMTEtOLEmE8BgNSU/TEAVEukm2yRQeaGUAfS70CRXM1J+JBAVSTgJPTwZIjeRfLxX/83qx8s/chIZRrEiIZw/5MYOKw7QWOKCCYMUmmiAsqP4rxCMkEFa6vFJWwnmKk++V50n7qGrXqrXr43L9Iq+jCPbAPjgENjgFdXAFmqAFMLgHj+AZvBgPxpPxarzNRgtGntkFv2C8fwF8+Zm3</latexit>)<latexit sha1_base64="FUIxtivFMN5Jpm6QFJ9RPJxb79c=">AAAB8nicbVBNS8NAEN3Ur1q/qh69LBbBU0ms+HErevFYxdpCGspmu2mXbnbD7kQpoT/DiwdFvPprvPlvTNIgan0w8Hhvhpl5fiS4Adv+tEoLi0vLK+XVytr6xuZWdXvnzqhYU9amSijd9YlhgkvWBg6CdSPNSOgL1vHHl5nfuWfacCVvYRIxLyRDyQNOCaSS27vhwxEQrdVDv1qz63YOPE+cgtRQgVa/+tEbKBqHTAIVxBjXsSPwEqKBU8GmlV5sWETomAyZm1JJQma8JD95ig9SZYADpdOSgHP150RCQmMmoZ92hgRG5q+Xif95bgzBmZdwGcXAJJ0tCmKBQeHsfzzgmlEQk5QQqnl6K6YjogmFNKVKHsJ5hpPvl+fJ3VHdadQb18e15kURRxntoX10iBx0iproCrVQG1Gk0CN6Ri8WWE/Wq/U2ay1Zxcwu+gXr/Qup45Gk</latexit>

g!g<latexit sha1_base64="URV/DoHRTfuLZPWg+F/N5G8tk/o=">AAAB+HicbVDLSsNAFJ3UV62PRl26GSyCq5JY8bErunFZwT6gDWUynaRDJzNhZqLU0C9x40IRt36KO//GSRpErQcuHM65l3vv8WNGlXacT6u0tLyyulZer2xsbm1X7Z3djhKJxKSNBROy5yNFGOWkralmpBdLgiKfka4/ucr87h2Rigp+q6cx8SIUchpQjLSRhnY1hANJw7FGUop7GA7tmlN3csBF4hakBgq0hvbHYCRwEhGuMUNK9V0n1l6KpKaYkVllkCgSIzxBIekbylFElJfmh8/goVFGMBDSFNcwV39OpChSahr5pjNCeqz+epn4n9dPdHDupZTHiSYczxcFCYNawCwFOKKSYM2mhiAsqbkV4jGSCGuTVSUP4SLD6ffLi6RzXHcb9cbNSa15WcRRBvvgABwBF5yBJrgGLdAGGCTgETyDF+vBerJerbd5a8kqZvbAL1jvX5E1kys=</latexit>(anti-) holomorphic function

String compactifications
Mathematical framework of string theory puts strong
constraints on the properties of the compact space[Candelas, Horowitz, Strominger and Witten `85]

‣ The space must be Ricci-flat!

0 = Ricab =<latexit sha1_base64="igUPO4dUO+qUC1xSTLGviITiyp8=">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</latexit>

Ricci form

i@a @ b̄ log det gij̄Kähler metricString compactifications‣ Note: The choice of metric on a manifold non-uniqueString compactifications‣ Note: The choice of metric on a manifold non-unique‣ Question: Does there exist a Ricci-flat Kähler metric?

String compactifications‣ Note: The choice of metric on a manifold non-unique[Image surce: https://commons.wikimedia.org/wiki/File:Eugenio_Calabi.jpeg]‣ Question: Does there exist a Ricci-flat Kähler metric?

I conjecture this to be true if
1c1 (X) =trR = 02⇡<latexit sha1_base64="QHIAi9WKzRlEUUAdurcFjg4a65s=">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</latexit>[Calabi `54]

String compactifications‣ Note: The choice of metric on a manifold non-unique[Image surce: https://commons.wikimedia.org/wiki/File:Eugenio_Calabi.jpeg]‣ Question: Does there exist a Ricci-flat Kähler metric?

I conjecture this to be true if
1c1 (X) =trR = 02⇡<latexit sha1_base64="QHIAi9WKzRlEUUAdurcFjg4a65s=">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</latexit>[Calabi `54]

Indeed, I can prove this.
But I cannot tell you what
the metric will look like.
[Yau `77]

String compactifications‣ Metric will depend on moduli gab̄ = gab̄ (ti ,<latexit sha1_base64="FIxJ6LL7zDWC8H00DaKf4ZaQ/9A=">AAACDHicbVDLSsNAFJ34rPVVdelmsAgVpCRWfCyEohuXFewDmhAm00k7djIJMxOhhH6AG3/FjQtF3PoB7vwbJ2mQaj0wcDjnXO7c40WMSmWaX8bc/MLi0nJhpbi6tr6xWdrabskwFpg0cchC0fGQJIxy0lRUMdKJBEGBx0jbG16lfvueCElDfqtGEXEC1OfUpxgpLbmlct9NkO0hAb3xxRSvKJce2pGk7t2BTplVMwOcJVZOyiBHwy192r0QxwHhCjMkZdcyI+UkSCiKGRkX7ViSCOEh6pOuphwFRDpJdswY7mulB/1Q6McVzNTpiQQFUo4CTycDpAbyr5eK/3ndWPlnTkJ5FCvC8WSRHzOoQpg2A3tUEKzYSBOEBdV/hXiABMJK91fMSjhPcfJz8ixpHVWtWrV2c1yuX+Z1FMAu2AMVYIFTUAfXoAGaAIMH8ARewKvxaDwbb8b7JDpn5DM74BeMj2/xAJsL</latexit>

Kähler moduli tTi(size of manifold)<latexit sha1_base64="xDVBH1r495HJFlyTG9gwy0RGaxc=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69BIvgqaRW/LgVvXisaD+gDWWz3bRLN5uwOxFK6E/w4kERr/4ib/4bN2kQtT4YeLw3w8w8LxJco+N8WoWl5ZXVteJ6aWNza3unvLvX1mGsKGvRUISq6xHNBJeshRwF60aKkcATrONNrlO/88CU5qG8x2nE3ICMJPc5JWikOxzwQbniVJ0M9iKp5aQCOZqD8kd/GNI4YBKpIFr3ak6EbkIUcirYrNSPNYsInZAR6xkqScC0m2SnzuwjowxtP1SmJNqZ+nMiIYHW08AznQHBsf7rpeJ/Xi9G/8JNuIxiZJLOF/mxsDG007/tIVeMopgaQqji5labjokiFE06pSyEyxRn3y8vkvZJtVav1m9PK42rPI4iHMAhHEMNzqEBN9CEFlAYwSM8w4slrCfr1XqbtxasfGYffsF6/wJzQI4L</latexit>

Complex structure moduli P j(shape of manifold)1c1 (X) =trR = 02⇡<latexit sha1_base64="QHIAi9WKzRlEUUAdurcFjg4a65s=">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</latexit>j)<latexit sha1_base64="vIyFg7pi9Fto4iYl3Nm74kRTnp0=">AAAB7XicbVDLSgNBEOyNrxhfUY9eBoPgKeyq+LgFvXiMYB6QLGF2MptMMjuzzMwKIeQfvHhQxKv/482/cXaziBoLGoqqbrq7gpgzbVz30yksLa+srhXXSxubW9s75d29ppaJIrRBJJeqHWBNORO0YZjhtB0riqOA01Ywvkn91gNVmklxbyYx9SM8ECxkBBsrNbuxZr1Rr1xxq24GtEi8nFQgR71X/uj2JUkiKgzhWOuO58bGn2JlGOF0VuommsaYjPGAdiwVOKLan2bXztCRVfoolMqWMChTf05McaT1JApsZ4TNUP/1UvE/r5OY8NKfMhEnhgoyXxQmHBmJ0tdRnylKDJ9Ygoli9lZEhlhhYmxApSyEqxTn3y8vkuZJ1Tutnt6dVWrXeRxFOIBDOAYPLqAGt1CHBhAYwSM8w4sjnSfn1XmbtxacfGYffsF5/wK7P49e</latexit>

The necessity of knowing the metric
1. Yukawa couplings‣ Definition of mass of a particle
2
1 µ⌫
1
!
2
L = g @µ @⌫ ¯m ¯L=g µ⌫ @µ @⌫ ¯
2
2
2

1( m)2 ¯2<latexit sha1_base64="kNu3zI4zPxqXQp1li8qGoggWgmU=">AAACA3icbVDLSsNAFJ34rPUVdaebwSK4KokVH7uiG5cV7AOaUCaTSTN0MgkzE6WEght/xY0LRdz6E+78GydpELUeGDiccw937vESRqWyrE9jbn5hcWm5slJdXVvf2DS3tjsyTgUmbRyzWPQ8JAmjnLQVVYz0EkFQ5DHS9UaXud+9JULSmN+ocULcCA05DShGSksDc9dJQuoIOgwVEiK+c5jO+ihXB2bNqlsF4CyxS1IDJVoD88PxY5xGhCvMkJR920qUmyGhKGZkUnVSSRKER2hI+ppyFBHpZsUNE3igFR8GsdCPK1ioPxMZiqQcR56ejJAK5V8vF//z+qkKztyM8iRVhOPpoiBlUMUwLwT6VBCs2FgThAXVf4U4RAJhpWurFiWc5zj5PnmWdI7qdqPeuD6uNS/KOipgD+yDQ2CDU9AEV6AF2gCDe/AInsGL8WA8Ga/G23R0zigzO+AXjPcv+W+Ygg==</latexit><latexit sha1_base64="X/SFnqHwlQnXNyajEtJNicMJt1A=">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</latexit><latexit sha1_base64="Kzrs717URqSZgUVSviflNGNiOj8=">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</latexit>‣ Mass from Higgs mechanism + Yukawa couplingL<latexit sha1_base64="LqfpA1T6Cx2ghHRTYTsf/bYGVT0=">AAACC3icbVDLSsNAFJ3UV62vqEs3Q4vgqiQqPnZFNy5cVLAPaEKYTCftNJNkmJkIIXTvxl9x40IRt/6AO//GJA2i1gP3cjjnXmbucTmjUhnGp1ZZWFxaXqmu1tbWNza39O2droxigUkHRywSfRdJwmhIOooqRvpcEBS4jPRc/zL3e3dESBqFtyrhxA7QKKQexUhlkqPXry0Zc0kUTJyUTvypxcfUoRaX1JkU3Xf0htE0CsB5YpakAUq0Hf3DGkY4DkioMENSDkyDKztFQlHMyLRmxZJwhH00IoOMhigg0k6LW6ZwP1OG0ItEVqGChfpzI0WBlEngZpMBUmP518vF/7xBrLwzO6UhjxUJ8ewhL2ZQRTAPBg6pIFixJCMIC5r9FeIxEgirLL5aEcJ5jpPvk+dJ97BpHjWPbo4brYsyjirYA3VwAExwClrgCrRBB2BwDx7BM3jRHrQn7VV7m41WtHJnF/yC9v4FRZyb+A==</latexit>

yijk ijk<latexit sha1_base64="LHrJRlv6DIA4alQkxsaqWdJJ/BE=">AAACCnicbZDLSsNAFIYnXmu9RV26GS2Cq5JY8bIrunFZwV6gCWUynSRDJ5MwM1FK6NqNr+LGhSJufQJ3vo2TNIhafxj4+M85nDm/lzAqlWV9GnPzC4tLy5WV6ura+samubXdkXEqMGnjmMWi5yFJGOWkrahipJcIgiKPka43uszr3VsiJI35jRonxI1QwKlPMVLaGph7ThJSR9AgVEiI+M5hiAeMTN0CB2bNqluF4CzYJdRAqdbA/HCGMU4jwhVmSMq+bSXKzZBQFDMyqTqpJAnCIxSQvkaOIiLdrDhlAg+0M4R+LPTjChbuz4kMRVKOI093RkiF8m8tN/+r9VPln7kZ5UmqCMfTRX7KoIphngscUkGwYmMNCAuq/wpxiATCSqdXLUI4z3XyffIsdI7qdqPeuD6uNS/KOCpgF+yDQ2CDU9AEV6AF2gCDe/AInsGL8WA8Ga/G27R1zihndsAvGe9fuAubrw==</latexit>!h iyijk h i iL<latexit sha1_base64="XSlTfPwV5X1giZak4XponzDI628=">AAACG3icbVDLSgMxFM34rPVVdekmWARXZdqKj13RjQsXFewDOmXIpLdt2kwmJBmhDO13uPFX3LhQxJXgwr9xOh1ErQcSTs65l5t7PMmZNrb9aS0sLi2vrGbWsusbm1vbuZ3dug5CRaFGAx6opkc0cCagZpjh0JQKiO9xaHjDy6nfuAOlWSBuzUhC2yc9wbqMEhNLbq507ehQajB4MnIjNhiOHU5Ej4Mj+8xljkoeE0dq5g6Se+jm8nbBToDnSTEleZSi6ubenU5AQx+EoZxo3Sra0rQjogyjHMZZJ9QgCR2SHrRiKogPuh0lu43xYax0cDdQ8REGJ+rPjoj4Wo98L670ienrv95U/M9rhaZ71o6YkKEBQWeDuiHHJsDToHCHKaCGj2JCqGLxXzHtE0WoiePMJiGcT3HyvfI8qZcKxXKhfHOcr1ykcWTQPjpAR6iITlEFXaEqqiGK7tEjekYv1oP1ZL1ab7PSBSvt2UO/YH18AbX+ozg=</latexit>jk

Mass term‣ Need to canonically normalize kinetic term

yijk =<latexit sha1_base64="/YNweFQCJnLSj+Ws2LY2d6vcesA=">AAACGHicbVDLSgMxFM3UV62vqks3wSK4qlMrPhZC0Y07K9gHdIYhk962aTOZIckoZehnuPFX3LhQxG13/o3ptIivA7kczrmXm3v8iDOlbfvDyszNLywuZZdzK6tr6xv5za26CmNJoUZDHsqmTxRwJqCmmebQjCSQwOfQ8AeXE79xB1KxUNzqYQRuQLqCdRgl2khe/mDoJaw/GJ07TGiviZ3rALrEuYd2F1pOpJjH0tpP68D18gW7aKfAf0lpRgpohqqXHzvtkMYBCE05UapVsiPtJkRqRjmMck6sICJ0QMw+QwUJQLlJetgI7xmljTuhNE9onKrfJxISKDUMfNMZEN1Tv72J+J/XinXn1E2YiGINgk4XdWKOdYgnKeE2k0A1HxpCqGTmr5j2iCRUmyxzaQhnExx/nfyX1A+LpXKxfHNUqFzM4siiHbSL9lEJnaAKukJVVEMUPaAn9IJerUfr2Xqz3qetGWs2s41+wBp/AiMIoTk=</latexit>

Z

X

⌦^[ i

j

k]

Ni,j̄ =
<latexit sha1_base64="o4hn01gK6naVsLZ+RhLH2XchXAs=">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</latexit>

Z

X

J ^J ^[ i!

j̄ ] =

ij̄

[Douglas `15]

The necessity of knowing the metric
2. Massive string spectrum and backreaction‣ Mass from non-harmonic forms upon compactificationL<latexit sha1_base64="NSq9VrgHuDyvJFKnK6SzX4EamXM=">AAACEnicbVA7T8MwGHTKq5RXgJHFokKCpUoo4qEuVWFgYCgSfUhNFDmu21p14sh2EFXU38DCX2FhACFWJjb+DU5aIaCcZOl0d5/t7/yIUaks69PIzc0vLC7llwsrq2vrG+bmVlPyWGDSwJxx0faRJIyGpKGoYqQdCYICn5GWPzxP/dYtEZLy8EaNIuIGqB/SHsVIackzD64cGUeSKOhwnUuvcaIBdSpOjd95iW1djJ0KTCXPLFolKwOcJfaUFMEUdc/8cLocxwEJFWZIyo5tRcpNkFAUMzIuOLEkEcJD1CcdTUMUEOkm2UpjuKeVLuxxoU+oYKb+nEhQIOUo8HUyQGog/3qp+J/XiVXv1E1oGMWKhHjyUC9mUHGY9gO7VBCs2EgThAXVf4V4gATCSrdYyEo4S3H8vfIsaR6W7HKpfH1UrNamdeTBDtgF+8AGJ6AKLkEdNAAG9+ARPIMX48F4Ml6Nt0k0Z0xntsEvGO9f1u2dsQ==</latexit>

compactify⇤10DL<latexit sha1_base64="Kdx8es84pGZ7O5AYMemiwKuQn+o=">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</latexit>(⇤4D +

6D )‣ Find eigenvalues of Laplace operator
6D

a b̄

= gab̄ @ @<latexit sha1_base64="NzL/Ic5umfO+dn7LTB2ngt+C+Os=">AAACRXicbVBLSwMxGMz6tr6qHr0Ei+CpbKv4OAiiHjwqWBWadvk2zbah2QdJVihh/XFevHvzH3jxoIhXza5VfA0EhpnvyyTjJ4Ir7br3zsjo2PjE5NR0aWZ2bn6hvLh0ruJUUtagsYjlpQ+KCR6xhuZasMtEMgh9wS78/mHuX1wxqXgcnelBwlohdCMecAraSl6ZkCMmNHhm6yi7JkmP4z3c9QwQHyT2M5KA1BxEG0hsr8lTzKeWtc3XVLFHAgnUhO16ZuqF5pUrbtUtgP+S2pBU0BAnXvmOdGKahizSVIBSzZqb6JbJ86hgWYmkiiVA+9BlTUsjCJlqmaKFDK9ZpYODWNoTaVyo3zcMhEoNQt9OhqB76reXi/95zVQHOy3DoyTVLKIfQUEqsI5xXinucMmoFgNLgEpu34ppD2wX2hZfKkrYzbH19eW/5LxerW1UN043K/sHwzqm0ApaReuohrbRPjpGJ6iBKLpBD+gJPTu3zqPz4rx+jI44w51l9APO2zt/OrOb</latexit>‣ Effective mass term forL<latexit sha1_base64="YIlwJfyoK0TmCha9MRMK3+irPqo=">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</latexit><latexit sha1_base64="pvYCNsI0u0mQasGs5uhFUruhqb8=">AAAB63icbVDLSsNAFL2pr1pfVZduBovgqiS2+NgV3bisYB/QhjKZTpqhM5MwMxFK6C+4caGIW3/InX9jkgZR64ELh3Pu5d57vIgzbWz70yqtrK6tb5Q3K1vbO7t71f2Drg5jRWiHhDxUfQ9rypmkHcMMp/1IUSw8Tnve9Cbzew9UaRbKezOLqCvwRDKfEWwyaRgFbFSt2XU7B1omTkFqUKA9qn4MxyGJBZWGcKz1wLEj4yZYGUY4nVeGsaYRJlM8oYOUSiyodpP81jk6SZUx8kOVljQoV39OJFhoPRNe2imwCfRfLxP/8wax8S/dhMkoNlSSxSI/5siEKHscjZmixPBZSjBRLL0VkQArTEwaTyUP4SrD+ffLy6R7Vnca9cZds9a6LuIowxEcwyk4cAEtuIU2dIBAAI/wDC+WsJ6sV+tt0VqyiplD+AXr/QssUI52</latexit>

⇤4D

m2
=
2

m2
+
2 [Braun,Brelidze,Douglas,Ovrut `08]

‣ Higher derivative corrections, backreaction of flux, …

The necessity of knowing the metric3. Consistency of internal gauge flux
‣ Can have vacuum expectation value of field strength

Fab 6= 0 in the 6D compact dimensions<latexit sha1_base64="nTHQHctfX3sCiSkiekqFjuQT2ds=">AAAB/nicbVDLSsNAFJ34rPUVFVduBovgqiRWfOyKgrisYB/QhDCZTtqhk0mcmQglBPwVNy4Ucet3uPNvnKRB1Hpg4HDOvdwzx48ZlcqyPo25+YXFpeXKSnV1bX1j09za7sgoEZi0ccQi0fORJIxy0lZUMdKLBUGhz0jXH1/mfveeCEkjfqsmMXFDNOQ0oBgpLXnmrhMiNcKIpVeZlyI/czi5szyzZtWtAnCW2CWpgRItz/xwBhFOQsIVZkjKvm3Fyk2RUBQzklWdRJIY4TEakr6mHIVEumkRP4MHWhnAIBL6cQUL9edGikIpJ6GvJ/Ow8q+Xi/95/UQFZ25KeZwowvH0UJAwqCKYdwEHVBCs2EQThAXVWSEeIYGw0o1VixLOc5x8f3mWdI7qdqPeuDmuNS/KOipgD+yDQ2CDU9AE16AF2gCDFDyCZ/BiPBhPxqvxNh2dM8qdHfALxvsXqViWEg==</latexit>‣ Equations of motion require

Fab = 0 , Fāb̄ = 0 , gab̄ F ab̄ = 0<latexit sha1_base64="DgS3EYgPUP+WtSG3H0NjPD/Rimk=">AAAB+3icbVDLSsNAFJ3UV62vWJduBovgqiQqPhZCURCXFewD2hBuppN26GQSZiZiCfkVNy4UceuPuPNvTNogaj0wcDjnXu6Z40WcKW1Zn0ZpYXFpeaW8Wllb39jcMrerbRXGktAWCXkoux4oypmgLc00p91IUgg8Tjve+Cr3O/dUKhaKOz2JqBPAUDCfEdCZ5JrVfgB6RIAn16mbgJdeWK5Zs+rWFHie2AWpoQJN1/zoD0ISB1RowkGpnm1F2klAakY4TSv9WNEIyBiGtJdRAQFVTjLNnuL9TBlgP5TZExpP1Z8bCQRKTQIvm8yTqr9eLv7n9WLtnzkJE1GsqSCzQ37MsQ5xXgQeMEmJ5pOMAJEsy4rJCCQQndVVmZZwnuPk+8vzpH1Yt4/qR7fHtcZlUUcZ7aI9dIBsdIoa6AY1UQsR9IAe0TN6MVLjyXg13majJaPY2UG/YLx/AQhXlJE=</latexit><latexit sha1_base64="hfARrXZgsYCBrBm5WYXxAjdaFxA=">AAACBXicbVDLSsNAFJ3UV62vqEtdDBbBVUms+FgIRUFcVrAPaEK4mU7aoZMHMxOhhGzc+CtuXCji1n9w59+YpEXUemCGwzn3cu89bsSZVIbxqZXm5hcWl8rLlZXVtfUNfXOrLcNYENoiIQ9F1wVJOQtoSzHFaTcSFHyX0447usz9zh0VkoXBrRpH1PZhEDCPEVCZ5Oi7lg9qSIAnV6mTWC4IDMXvpueGo1eNmlEAzxJzSqpoiqajf1j9kMQ+DRThIGXPNCJlJyAUI5ymFSuWNAIyggHtZTQAn0o7Ka5I8X6m9LEXiuwFChfqz44EfCnHvptV5jvLv14u/uf1YuWd2gkLoljRgEwGeTHHKsR5JLjPBCWKjzMCRLBsV0yGIICoLLhKEcJZjuPvk2dJ+7Bm1mv1m6Nq42IaRxntoD10gEx0ghroGjVRCxF0jx7RM3rRHrQn7VV7m5SWtGnPNvoF7f0L8ziYVw==</latexit><latexit sha1_base64="GeDT/+ZUF0QPjutY5vwM1QlF7kg=">AAACC3icbVDLSsNAFJ3UV62vqEs3Q4vgqiRWfCyEoiAuK9gHNDHcTKft0MmDmYlQQvZu/BU3LhRx6w+4829M0lLUeuDC4Zx7ufceN+RMKsP40goLi0vLK8XV0tr6xuaWvr3TkkEkCG2SgAei44KknPm0qZjitBMKCp7LadsdXWZ++54KyQL/Vo1Dansw8FmfEVCp5OjlgROD5YLAbmJ5oIYEeHyV3M3Ec8PRK0bVyIHniTklFTRFw9E/rV5AIo/6inCQsmsaobJjEIoRTpOSFUkaAhnBgHZT6oNHpR3nvyR4P1V6uB+ItHyFc/XnRAyelGPPTTuzc+VfLxP/87qR6p/aMfPDSFGfTBb1I45VgLNgcI8JShQfpwSIYOmtmAxBAFFpfKU8hLMMx7OX50nrsGrWqrWbo0r9YhpHEe2hMjpAJjpBdXSNGqiJCHpAT+gFvWqP2rP2pr1PWgvadGYX/YL28Q3jmpsT</latexit>‣ Can solve Donaldson-Uhlenbeck-Yau equations forpolystable vector bundles of slope zero, but this isvery hard to check in practice[Donaldson `85; Uhlenbeck, Yau `86]

2. Calabi-Yau metrics

Donaldson’s algorithm‣

Donaldson provided an algorithm to approximate CY
metrics [Donaldson `01; Donaldson `05; Douglas, Karp, Lukic, Reinbacher `06]•

The algorithm is numerical•

It requires fixing a single point in Kähler and complex structure moduli
space (and we do not a priori know which point we should choose)•

Numerical computation extremely slow‣

Can we use ML to learn the CY metric for arbitrarycomplex structure?

‣

Note: last week a paper appeared that uses ML for
regression on det g for fixed moduli [Ashmore, He, Ovrut `19]<latexit sha1_base64="k24yrZBARA3+naHhLwR4/fR3d+o=">AAAB7XicbVDLSsNAFJ3UV62vqks3g0VwVRIrPnZFNy4r2Ae0oUwmk3bsJBNmboQS+g9uXCji1v9x5984SYOo9cCFwzn3cu89Xiy4Btv+tEpLyyura+X1ysbm1vZOdXevo2WiKGtTKaTqeUQzwSPWBg6C9WLFSOgJ1vUm15nffWBKcxndwTRmbkhGEQ84JWCkzsBngEfDas2u2znwInEKUkMFWsPqx8CXNAlZBFQQrfuOHYObEgWcCjarDBLNYkInZMT6hkYkZNpN82tn+MgoPg6kMhUBztWfEykJtZ6GnukMCYz1Xy8T//P6CQQXbsqjOAEW0fmiIBEYJM5exz5XjIKYGkKo4uZWTMdEEQomoEoewmWGs++XF0nnpO406o3b01rzqoijjA7QITpGDjpHTXSDWqiNKLpHj+gZvVjSerJerbd5a8kqZvbRL1jvXz/4jw0=</latexit>‣

We want to learn either Kähler potential K or metric gab̄<latexit sha1_base64="N0Gyr1fZncsdM900Vklly4cNS64=">AAAB6HicbVDLSsNAFL2pr1pfVZduBovgqiRWfOyKbgQ3LdgHtKFMppN27GQSZiZCCf0CNy4UcesnufNvnKRB1HrgwuGce7n3Hi/iTGnb/rQKS8srq2vF9dLG5tb2Tnl3r63CWBLaIiEPZdfDinImaEszzWk3khQHHqcdb3Kd+p0HKhULxZ2eRtQN8EgwnxGsjdS8HZQrdtXOgBaJk5MK5GgMyh/9YUjigApNOFaq59iRdhMsNSOczkr9WNEIkwke0Z6hAgdUuUl26AwdGWWI/FCaEhpl6s+JBAdKTQPPdAZYj9VfLxX/83qx9i/chIko1lSQ+SI/5kiHKP0aDZmkRPOpIZhIZm5FZIwlJtpkU8pCuExx9v3yImmfVJ1atdY8rdSv8jiKcACHcAwOnEMdbqABLSBA4RGe4cW6t56sV+tt3lqw8pl9+AXr/Qu6zY0G</latexit><latexit sha1_base64="JxI/glx90bwjuYS0FlRrEYkdnJQ=">AAAB8nicbVDLSsNAFJ3UV62vqks3g0VwVRIVH7uiG5cV7APSUG6m03boZCbMTIQS+hluXCji1q9x5984SYOo9cCFwzn3cu89YcyZNq776ZSWlldW18rrlY3Nre2d6u5eW8tEEdoikkvVDUFTzgRtGWY47caKQhRy2gknN5nfeaBKMynuzTSmQQQjwYaMgLGSP+qn0AtB4XDWr9bcupsDLxKvIDVUoNmvfvQGkiQRFYZw0Nr33NgEKSjDCKezSi/RNAYygRH1LRUQUR2k+ckzfGSVAR5KZUsYnKs/J1KItJ5Goe2MwIz1Xy8T//P8xAwvg5SJODFUkPmiYcKxkTj7Hw+YosTwqSVAFLO3YjIGBcTYlCp5CFcZzr9fXiTtk7p3Wj+9O6s1ros4yugAHaJj5KEL1EC3qIlaiCCJHtEzenGM8+S8Om/z1pJTzOyjX3DevwAsDZFR</latexit>

Example: The quintic
1. Start from complex 4D ambient space P

4<latexit sha1_base64="qpdezEnIBrftgIXHorvsRqFXcOg=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgqiRafOyKblxWsA9oYplMJ+3QySTMTIQS+htuXCji1p9x5984SYOo9cDA4Zx7uWeOH3OmtG1/WqWl5ZXVtfJ6ZWNza3unurvXUVEiCW2TiEey52NFORO0rZnmtBdLikOf064/uc787gOVikXiTk9j6oV4JFjACNZGct0Q67Hvp63ZfWNQrdl1OwdaJE5BalCgNah+uMOIJCEVmnCsVN+xY+2lWGpGOJ1V3ETRGJMJHtG+oQKHVHlpnnmGjowyREEkzRMa5erPjRSHSk1D30xmGdVfLxP/8/qJDi68lIk40VSQ+aEg4UhHKCsADZmkRPOpIZhIZrIiMsYSE21qquQlXGY4+/7yIumc1J3T+ulto9a8KuoowwEcwjE4cA5NuIEWtIFADI/wDC9WYj1Zr9bbfLRkFTv78AvW+xcC85HN</latexit>z0 ≠ 0z0 ≠ 0 z ≠ 01[z0 : z1 : z2 : z3 : z4]≃ [λz0 : λz1 : λz2 : λz3 : λz4]

z1 ≠ 0K FS = − log(ziz̄i)∑iFS¯
g = ∂a∂b̄K
ab̄

Example: The quintic
1. Start from complex 4D ambient space P

4<latexit sha1_base64="qpdezEnIBrftgIXHorvsRqFXcOg=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgqiRafOyKblxWsA9oYplMJ+3QySTMTIQS+htuXCji1p9x5984SYOo9cDA4Zx7uWeOH3OmtG1/WqWl5ZXVtfJ6ZWNza3unurvXUVEiCW2TiEey52NFORO0rZnmtBdLikOf064/uc787gOVikXiTk9j6oV4JFjACNZGct0Q67Hvp63ZfWNQrdl1OwdaJE5BalCgNah+uMOIJCEVmnCsVN+xY+2lWGpGOJ1V3ETRGJMJHtG+oQKHVHlpnnmGjowyREEkzRMa5erPjRSHSk1D30xmGdVfLxP/8/qJDi68lIk40VSQ+aEg4UhHKCsADZmkRPOpIZhIZrIiMsYSE21qquQlXGY4+/7yIumc1J3T+ulto9a8KuoowwEcwjE4cA5NuIEWtIFADI/wDC9WYj1Zr9bbfLRkFTv78AvW+xcC85HN</latexit>

2. Consider complex 3D hyper surface of degree 5!
55
1 z1 + 2 z2 + . . . + 126 z1 z2 z3 z4 z5 = 0<latexit sha1_base64="GtLOVDE+Kd9IWGHhtsglGC6HR3I=">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</latexit>gab̄ = ∂a∂¯ b̄Kiab̄
J = gab̄ dz ^ dz
22ab̄ds = gab̄ dz ⌦ dz
<latexit sha1_base64="5XXkmGMWUsYigAUsne9LyySldFY=">AAACGHicbZDJSgNBEIZ73I1b1KOXxiB4ipMoLgdB9CKeFEwUMjHU9NQkjT0L3T1KHOYxvPgqXjwo4tWbb2NPMojbDw0ff3VRVb8bC660bX9YI6Nj4xOTU9Olmdm5+YXy4lJTRYlk2GCRiOSlCwoFD7GhuRZ4GUuEwBV44V4f5fWLG5SKR+G57sfYDqAbcp8z0MbqlDdO9h1fAkt5ltazbicFxwVJ3Yx6d1fg3KLXxRzTwu6UK3bVHoj+hVoBFVLotFN+d7yIJQGGmglQqlWzY91OQWrOBGYlJ1EYA7uGLrYMhhCgaqeDwzK6ZhyP+pE0L9R04H7vSCFQqh+YbdcC0D31u5ab/9VaifZ32ykP40RjyIaD/ERQHdE8JepxiUyLvgFgkptdKeuByUmbLEuDEPZybX+d/Bea9Wpts7p5tlU5OCzimCIrZJWskxrZIQfkmJySBmHknjySZ/JiPVhP1qv1Nvw6YhU9y+SHrPdP/UCggA==</latexit><latexit sha1_base64="pULOCY2LA9DIZ0j4VyjyjTNxMBQ=">AAACEXicbZDJSgNBEIZ74hbjFvXopTEEcgozibgchKAXjxHMAtmo6elJmvQsdPcIccgrePFVvHhQxKs3b76NPckgavyh4eOvLqrqt0POpDLNTyOztLyyupZdz21sbm3v5Hf3mjKIBKENEvBAtG2QlDOfNhRTnLZDQcGzOW3Z48uk3rqlQrLAv1GTkPY8GPrMZQSUtgb5kiP7lfPhIIauDQLbU+zc9aEbKOZRmXCc+oN8wSybM+FFsFIooFT1Qf6j6wQk8qivCAcpO5YZql4MQjHC6TTXjSQNgYxhSDsafdADe/HsoikuasfBbiD08xWeuT87YvCknHh63aIHaiT/1hLzv1onUu5pL2Z+GCnqk/kgN+JYBTiJBztMUKL4RAMQwfSumIxAAFE6xNwshLNEx98nL0KzUraq5er1UaF2kcaRRQfoEJWQhU5QDV2hOmoggu7RI3pGL8aD8WS8Gm/zrxkj7dlHv2S8fwEeyZ1Z</latexit>

g FS,X = i*(g FS)ab̄
ab̄

¯
FS,X
j
i
FS
g (z1, z2, z3) = 𝒥ag (1,z1, z2, z3, z4(z1, z2, z3))ij¯𝒥ab̄b̄Example: The quintic
1. Start from complex 4D ambient space P

4<latexit sha1_base64="qpdezEnIBrftgIXHorvsRqFXcOg=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgqiRafOyKblxWsA9oYplMJ+3QySTMTIQS+htuXCji1p9x5984SYOo9cDA4Zx7uWeOH3OmtG1/WqWl5ZXVtfJ6ZWNza3unurvXUVEiCW2TiEey52NFORO0rZnmtBdLikOf064/uc787gOVikXiTk9j6oV4JFjACNZGct0Q67Hvp63ZfWNQrdl1OwdaJE5BalCgNah+uMOIJCEVmnCsVN+xY+2lWGpGOJ1V3ETRGJMJHtG+oQKHVHlpnnmGjowyREEkzRMa5erPjRSHSk1D30xmGdVfLxP/8/qJDi68lIk40VSQ+aEg4UhHKCsADZmkRPOpIZhIZrIiMsYSE21qquQlXGY4+/7yIumc1J3T+ulto9a8KuoowwEcwjE4cA5NuIEWtIFADI/wDC9WYj1Zr9bbfLRkFTv78AvW+xcC85HN</latexit>

2. Consider complex 3D hyper surface of degree 5!
55
1 z1 + 2 z2 + . . . + 126 z1 z2 z3 z4 z5 = 0<latexit sha1_base64="GtLOVDE+Kd9IWGHhtsglGC6HR3I=">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</latexit>

3. This hypersurface will have vanishing c1 (X)) can find a CY metric and a hol. (3,0) form Ω<latexit sha1_base64="/mxHv3Ma/itiZCozyeKYrdgwqvU=">AAAB7XicbVDLSsNAFL2pr1pfVZduBotQNyWx4mNXdOOygn1AG8pkOmnHTjJhZiKU0H9w40IRt/6PO//GSRpErQcuHM65l3vv8SLOlLbtT6uwtLyyulZcL21sbm3vlHf32krEktAWEVzIrocV5SykLc00p91IUhx4nHa8yXXqdx6oVEyEd3oaUTfAo5D5jGBtpDYZONXu8aBcsWt2BrRInJxUIEdzUP7oDwWJAxpqwrFSPceOtJtgqRnhdFbqx4pGmEzwiPYMDXFAlZtk187QkVGGyBfSVKhRpv6cSHCg1DTwTGeA9Vj99VLxP68Xa//CTVgYxZqGZL7IjznSAqWvoyGTlGg+NQQTycytiIyxxESbgEpZCJcpzr5fXiTtk5pTr9VvTyuNqzyOIhzAIVTBgXNowA00oQUE7uERnuHFEtaT9Wq9zVsLVj6zD79gvX8BduyOiQ==</latexit><latexit sha1_base64="FUIxtivFMN5Jpm6QFJ9RPJxb79c=">AAAB8nicbVBNS8NAEN3Ur1q/qh69LBbBU0ms+HErevFYxdpCGspmu2mXbnbD7kQpoT/DiwdFvPprvPlvTNIgan0w8Hhvhpl5fiS4Adv+tEoLi0vLK+XVytr6xuZWdXvnzqhYU9amSijd9YlhgkvWBg6CdSPNSOgL1vHHl5nfuWfacCVvYRIxLyRDyQNOCaSS27vhwxEQrdVDv1qz63YOPE+cgtRQgVa/+tEbKBqHTAIVxBjXsSPwEqKBU8GmlV5sWETomAyZm1JJQma8JD95ig9SZYADpdOSgHP150RCQmMmoZ92hgRG5q+Xif95bgzBmZdwGcXAJJ0tCmKBQeHsfzzgmlEQk5QQqnl6K6YjogmFNKVKHsJ5hpPvl+fJ3VHdadQb18e15kURRxntoX10iBx0iproCrVQG1Gk0CN6Ri8WWE/Wq/U2ay1Zxcwu+gXr/Qup45Gk</latexit>

Example: The quintic
1. Start from complex 4D ambient space P

4<latexit sha1_base64="qpdezEnIBrftgIXHorvsRqFXcOg=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgqiRafOyKblxWsA9oYplMJ+3QySTMTIQS+htuXCji1p9x5984SYOo9cDA4Zx7uWeOH3OmtG1/WqWl5ZXVtfJ6ZWNza3unurvXUVEiCW2TiEey52NFORO0rZnmtBdLikOf064/uc787gOVikXiTk9j6oV4JFjACNZGct0Q67Hvp63ZfWNQrdl1OwdaJE5BalCgNah+uMOIJCEVmnCsVN+xY+2lWGpGOJ1V3ETRGJMJHtG+oQKHVHlpnnmGjowyREEkzRMa5erPjRSHSk1D30xmGdVfLxP/8/qJDi68lIk40VSQ+aEg4UhHKCsADZmkRPOpIZhIZrIiMsYSE21qquQlXGY4+/7yIumc1J3T+ulto9a8KuoowwEcwjE4cA5NuIEWtIFADI/wDC9WYj1Zr9bbfLRkFTv78AvW+xcC85HN</latexit>

2. Consider complex 3D hyper surface of degree 5!
55
1 z1 + 2 z2 + . . . + 126 z1 z2 z3 z4 z5 = 0<latexit sha1_base64="GtLOVDE+Kd9IWGHhtsglGC6HR3I=">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</latexit>

3. This hypersurface will have vanishing c1 (X)) can find a CY metric and a hol. (3,0) form Ω<latexit sha1_base64="/mxHv3Ma/itiZCozyeKYrdgwqvU=">AAAB7XicbVDLSsNAFL2pr1pfVZduBotQNyWx4mNXdOOygn1AG8pkOmnHTjJhZiKU0H9w40IRt/6PO//GSRpErQcuHM65l3vv8SLOlLbtT6uwtLyyulZcL21sbm3vlHf32krEktAWEVzIrocV5SykLc00p91IUhx4nHa8yXXqdx6oVEyEd3oaUTfAo5D5jGBtpDYZONXu8aBcsWt2BrRInJxUIEdzUP7oDwWJAxpqwrFSPceOtJtgqRnhdFbqx4pGmEzwiPYMDXFAlZtk187QkVGGyBfSVKhRpv6cSHCg1DTwTGeA9Vj99VLxP68Xa//CTVgYxZqGZL7IjznSAqWvoyGTlGg+NQQTycytiIyxxESbgEpZCJcpzr5fXiTtk5pTr9VvTyuNqzyOIhzAIVTBgXNowA00oQUE7uERnuHFEtaT9Wq9zVsLVj6zD79gvX8BduyOiQ==</latexit><latexit sha1_base64="FUIxtivFMN5Jpm6QFJ9RPJxb79c=">AAAB8nicbVBNS8NAEN3Ur1q/qh69LBbBU0ms+HErevFYxdpCGspmu2mXbnbD7kQpoT/DiwdFvPprvPlvTNIgan0w8Hhvhpl5fiS4Adv+tEoLi0vLK+XVytr6xuZWdXvnzqhYU9amSijd9YlhgkvWBg6CdSPNSOgL1vHHl5nfuWfacCVvYRIxLyRDyQNOCaSS27vhwxEQrdVDv1qz63YOPE+cgtRQgVa/+tEbKBqHTAIVxBjXsSPwEqKBU8GmlV5sWETomAyZm1JJQma8JD95ig9SZYADpdOSgHP150RCQmMmoZ92hgRG5q+Xif95bgzBmZdwGcXAJJ0tCmKBQeHsfzzgmlEQk5QQqnl6K6YjogmFNKVKHsJ5hpPvl+fJ3VHdadQb18e15kURRxntoX10iBx0iproCrVQG1Gk0CN6Ri8WWE/Wq/U2ay1Zxcwu+gXr/Qup45Gk</latexit>

4. Consider simplified case
5
5
5
5
5
p = z1 + z2 + z3 + z4 + z5 + z1 z2 z3 z4 z5 = 0<latexit sha1_base64="p/48P/lBZ+5mzPOJEbvO5qR9fGY=">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</latexit>

Example: The quintic
1. Start from complex 4D ambient space P

4<latexit sha1_base64="qpdezEnIBrftgIXHorvsRqFXcOg=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgqiRafOyKblxWsA9oYplMJ+3QySTMTIQS+htuXCji1p9x5984SYOo9cDA4Zx7uWeOH3OmtG1/WqWl5ZXVtfJ6ZWNza3unurvXUVEiCW2TiEey52NFORO0rZnmtBdLikOf064/uc787gOVikXiTk9j6oV4JFjACNZGct0Q67Hvp63ZfWNQrdl1OwdaJE5BalCgNah+uMOIJCEVmnCsVN+xY+2lWGpGOJ1V3ETRGJMJHtG+oQKHVHlpnnmGjowyREEkzRMa5erPjRSHSk1D30xmGdVfLxP/8/qJDi68lIk40VSQ+aEg4UhHKCsADZmkRPOpIZhIZrIiMsYSE21qquQlXGY4+/7yIumc1J3T+ulto9a8KuoowwEcwjE4cA5NuIEWtIFADI/wDC9WYj1Zr9bbfLRkFTv78AvW+xcC85HN</latexit>

2. Consider complex 3D hyper surface of degree 5!
55
1 z1 + 2 z2 + . . . + 126 z1 z2 z3 z4 z5 = 0<latexit sha1_base64="GtLOVDE+Kd9IWGHhtsglGC6HR3I=">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</latexit>

3. This hypersurface will have vanishing c1 (X)) can find a CY metric and a hol. (3,0) form Ω<latexit sha1_base64="/mxHv3Ma/itiZCozyeKYrdgwqvU=">AAAB7XicbVDLSsNAFL2pr1pfVZduBotQNyWx4mNXdOOygn1AG8pkOmnHTjJhZiKU0H9w40IRt/6PO//GSRpErQcuHM65l3vv8SLOlLbtT6uwtLyyulZcL21sbm3vlHf32krEktAWEVzIrocV5SykLc00p91IUhx4nHa8yXXqdx6oVEyEd3oaUTfAo5D5jGBtpDYZONXu8aBcsWt2BrRInJxUIEdzUP7oDwWJAxpqwrFSPceOtJtgqRnhdFbqx4pGmEzwiPYMDXFAlZtk187QkVGGyBfSVKhRpv6cSHCg1DTwTGeA9Vj99VLxP68Xa//CTVgYxZqGZL7IjznSAqWvoyGTlGg+NQQTycytiIyxxESbgEpZCJcpzr5fXiTtk5pTr9VvTyuNqzyOIhzAIVTBgXNowA00oQUE7uERnuHFEtaT9Wq9zVsLVj6zD79gvX8BduyOiQ==</latexit><latexit sha1_base64="FUIxtivFMN5Jpm6QFJ9RPJxb79c=">AAAB8nicbVBNS8NAEN3Ur1q/qh69LBbBU0ms+HErevFYxdpCGspmu2mXbnbD7kQpoT/DiwdFvPprvPlvTNIgan0w8Hhvhpl5fiS4Adv+tEoLi0vLK+XVytr6xuZWdXvnzqhYU9amSijd9YlhgkvWBg6CdSPNSOgL1vHHl5nfuWfacCVvYRIxLyRDyQNOCaSS27vhwxEQrdVDv1qz63YOPE+cgtRQgVa/+tEbKBqHTAIVxBjXsSPwEqKBU8GmlV5sWETomAyZm1JJQma8JD95ig9SZYADpdOSgHP150RCQmMmoZ92hgRG5q+Xif95bgzBmZdwGcXAJJ0tCmKBQeHsfzzgmlEQk5QQqnl6K6YjogmFNKVKHsJ5hpPvl+fJ3VHdadQb18e15kURRxntoX10iBx0iproCrVQG1Gk0CN6Ri8WWE/Wq/U2ay1Zxcwu+gXr/Qup45Gk</latexit>

4. Consider simplified case
5
5
5
5
5
p = z1 + z2 + z3 + z4 + z5 + z1 z2 z3 z4 z5 = 0<latexit sha1_base64="p/48P/lBZ+5mzPOJEbvO5qR9fGY=">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</latexit>

5. Can construct volume form in patch (z0 ≠ 0):
dz1 ∧ dz2 ∧ dz3, vol(X) =Ω ∧ Ω̄ ∼ J ∧ J ∧ J
Ω=∫X∫X∂z 4 pNumerical algorithm[Braun,Brelidze,Douglas,Ovrut `08]

1. Choose an order k<latexit sha1_base64="LJtdnj+M3emAK5lySPkAzLsYW1A=">AAAB6HicbVDJSgNBEK2JW4xb1KOXxiB4ChMVl1vQi8cEzALJEHo6NUmbnoXuHiEM+QIvHhTx6id582/smQyixgcFj/eqqKrnRoIrbdufVmFpeWV1rbhe2tjc2t4p7+61VRhLhi0WilB2XapQ8ABbmmuB3Ugi9V2BHXdyk/qdB5SKh8Gdnkbo+HQUcI8zqo3UnAzKFbtqZyCLpJaTCuRoDMof/WHIYh8DzQRVqlezI+0kVGrOBM5K/VhhRNmEjrBnaEB9VE6SHTojR0YZEi+UpgJNMvXnREJ9paa+azp9qsfqr5eK/3m9WHuXTsKDKNYYsPkiLxZEhyT9mgy5RKbF1BDKJDe3EjamkjJtsillIVylOP9+eZG0T6q10+pp86xSv87jKMIBHMIx1OAC6nALDWgBA4RHeIYX6956sl6tt3lrwcpn9uEXrPcv602NJg==</latexit>

Numerical algorithm[Braun,Brelidze,Douglas,Ovrut `08]

1. Choose an order k<latexit sha1_base64="LJtdnj+M3emAK5lySPkAzLsYW1A=">AAAB6HicbVDJSgNBEK2JW4xb1KOXxiB4ChMVl1vQi8cEzALJEHo6NUmbnoXuHiEM+QIvHhTx6id582/smQyixgcFj/eqqKrnRoIrbdufVmFpeWV1rbhe2tjc2t4p7+61VRhLhi0WilB2XapQ8ABbmmuB3Ugi9V2BHXdyk/qdB5SKh8Gdnkbo+HQUcI8zqo3UnAzKFbtqZyCLpJaTCuRoDMof/WHIYh8DzQRVqlezI+0kVGrOBM5K/VhhRNmEjrBnaEB9VE6SHTojR0YZEi+UpgJNMvXnREJ9paa+azp9qsfqr5eK/3m9WHuXTsKDKNYYsPkiLxZEhyT9mgy5RKbF1BDKJDe3EjamkjJtsillIVylOP9+eZG0T6q10+pp86xSv87jKMIBHMIx1OAC6nALDWgBA4RHeIYX6956sl6tt3lrwcpn9uEXrPcv602NJg==</latexit>

2. Compute all monomials of degree k<latexit sha1_base64="LJtdnj+M3emAK5lySPkAzLsYW1A=">AAAB6HicbVDJSgNBEK2JW4xb1KOXxiB4ChMVl1vQi8cEzALJEHo6NUmbnoXuHiEM+QIvHhTx6id582/smQyixgcFj/eqqKrnRoIrbdufVmFpeWV1rbhe2tjc2t4p7+61VRhLhi0WilB2XapQ8ABbmmuB3Ugi9V2BHXdyk/qdB5SKh8Gdnkbo+HQUcI8zqo3UnAzKFbtqZyCLpJaTCuRoDMof/WHIYh8DzQRVqlezI+0kVGrOBM5K/VhhRNmEjrBnaEB9VE6SHTojR0YZEi+UpgJNMvXnREJ9paa+azp9qsfqr5eK/3m9WHuXTsKDKNYYsPkiLxZEhyT9mgy5RKbF1BDKJDe3EjamkjJtsillIVylOP9+eZG0T6q10+pp86xSv87jKMIBHMIx1OAC6nALDWgBA4RHeIYX6956sl6tt3lrwcpn9uEXrPcv602NJg==</latexit>

monomials

#

zi

k=1

5

zi2, zizj (i < j)

k=2

15

3 2
zi , zi zj, zizj zk
zi4, zi3zj, zi2zj2, zi2zj zk, zizj zk zl

k=3
k=4

zi5, zi4zj, zi3zj2, zi3zj zk, zi2zj2zk, zi2zj zk zl, zizj zk zl zm

k=5<latexit sha1_base64="p/48P/lBZ+5mzPOJEbvO5qR9fGY=">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</latexit>

5
5
5
5
5
p = z1 + z2 + z3 + z4 + z 5 +35
70
126 - 1z1 z2 z3 z4 z5 = 0

Numerical algorithm[Braun,Brelidze,Douglas,Ovrut `08]

1. Choose an order k<latexit sha1_base64="LJtdnj+M3emAK5lySPkAzLsYW1A=">AAAB6HicbVDJSgNBEK2JW4xb1KOXxiB4ChMVl1vQi8cEzALJEHo6NUmbnoXuHiEM+QIvHhTx6id582/smQyixgcFj/eqqKrnRoIrbdufVmFpeWV1rbhe2tjc2t4p7+61VRhLhi0WilB2XapQ8ABbmmuB3Ugi9V2BHXdyk/qdB5SKh8Gdnkbo+HQUcI8zqo3UnAzKFbtqZyCLpJaTCuRoDMof/WHIYh8DzQRVqlezI+0kVGrOBM5K/VhhRNmEjrBnaEB9VE6SHTojR0YZEi+UpgJNMvXnREJ9paa+azp9qsfqr5eK/3m9WHuXTsKDKNYYsPkiLxZEhyT9mgy5RKbF1BDKJDe3EjamkjJtsillIVylOP9+eZG0T6q10+pp86xSv87jKMIBHMIx1OAC6nALDWgBA4RHeIYX6956sl6tt3lrwcpn9uEXrPcv602NJg==</latexit>

2. Compute all monomials of degree k<latexit sha1_base64="LJtdnj+M3emAK5lySPkAzLsYW1A=">AAAB6HicbVDJSgNBEK2JW4xb1KOXxiB4ChMVl1vQi8cEzALJEHo6NUmbnoXuHiEM+QIvHhTx6id582/smQyixgcFj/eqqKrnRoIrbdufVmFpeWV1rbhe2tjc2t4p7+61VRhLhi0WilB2XapQ8ABbmmuB3Ugi9V2BHXdyk/qdB5SKh8Gdnkbo+HQUcI8zqo3UnAzKFbtqZyCLpJaTCuRoDMof/WHIYh8DzQRVqlezI+0kVGrOBM5K/VhhRNmEjrBnaEB9VE6SHTojR0YZEi+UpgJNMvXnREJ9paa+azp9qsfqr5eK/3m9WHuXTsKDKNYYsPkiLxZEhyT9mgy5RKbF1BDKJDe3EjamkjJtsillIVylOP9+eZG0T6q10+pp86xSv87jKMIBHMIx1OAC6nALDWgBA4RHeIYX6956sl6tt3lrwcpn9uEXrPcv602NJg==</latexit>

3. Sample random points on X
<latexit sha1_base64="MR47EM00aXvJAcec3gKWyzZ0UTs=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lU/LgVvXhswX5AG8pmO2nXbjZhdyOU0l/gxYMiXv1J3vw3JmkQtT4YeLw3w8w8LxJcG9v+tApLyyura8X10sbm1vZOeXevpcNYMWyyUISq41GNgktsGm4EdiKFNPAEtr3xTeq3H1BpHso7M4nQDehQcp8zahKp0emXK3bVzkAWiZOTCuSo98sfvUHI4gClYYJq3XXsyLhTqgxnAmelXqwxomxMh9hNqKQBaneaHTojR4kyIH6okpKGZOrPiSkNtJ4EXtIZUDPSf71U/M/rxsa/dKdcRrFByeaL/FgQE5L0azLgCpkRk4RQpnhyK2EjqigzSTalLISrFOffLy+S1knVOa2eNs4qtes8jiIcwCEcgwMXUINbqEMTGCA8wjO8WPfWk/Vqvc1bC1Y+sw+/YL1/Ac6BjRM=</latexit>

4
Fix
two
randompointsinambientℙ•and find the line through these points

Numerical algorithm[Braun,Brelidze,Douglas,Ovrut `08]

1. Choose an order k<latexit sha1_base64="LJtdnj+M3emAK5lySPkAzLsYW1A=">AAAB6HicbVDJSgNBEK2JW4xb1KOXxiB4ChMVl1vQi8cEzALJEHo6NUmbnoXuHiEM+QIvHhTx6id582/smQyixgcFj/eqqKrnRoIrbdufVmFpeWV1rbhe2tjc2t4p7+61VRhLhi0WilB2XapQ8ABbmmuB3Ugi9V2BHXdyk/qdB5SKh8Gdnkbo+HQUcI8zqo3UnAzKFbtqZyCLpJaTCuRoDMof/WHIYh8DzQRVqlezI+0kVGrOBM5K/VhhRNmEjrBnaEB9VE6SHTojR0YZEi+UpgJNMvXnREJ9paa+azp9qsfqr5eK/3m9WHuXTsKDKNYYsPkiLxZEhyT9mgy5RKbF1BDKJDe3EjamkjJtsillIVylOP9+eZG0T6q10+pp86xSv87jKMIBHMIx1OAC6nALDWgBA4RHeIYX6956sl6tt3lrwcpn9uEXrPcv602NJg==</latexit>

2. Compute all monomials of degree k<latexit sha1_base64="LJtdnj+M3emAK5lySPkAzLsYW1A=">AAAB6HicbVDJSgNBEK2JW4xb1KOXxiB4ChMVl1vQi8cEzALJEHo6NUmbnoXuHiEM+QIvHhTx6id582/smQyixgcFj/eqqKrnRoIrbdufVmFpeWV1rbhe2tjc2t4p7+61VRhLhi0WilB2XapQ8ABbmmuB3Ugi9V2BHXdyk/qdB5SKh8Gdnkbo+HQUcI8zqo3UnAzKFbtqZyCLpJaTCuRoDMof/WHIYh8DzQRVqlezI+0kVGrOBM5K/VhhRNmEjrBnaEB9VE6SHTojR0YZEi+UpgJNMvXnREJ9paa+azp9qsfqr5eK/3m9WHuXTsKDKNYYsPkiLxZEhyT9mgy5RKbF1BDKJDe3EjamkjJtsillIVylOP9+eZG0T6q10+pp86xSv87jKMIBHMIx1OAC6nALDWgBA4RHeIYX6956sl6tt3lrwcpn9uEXrPcv602NJg==</latexit>

3. Sample random points on X
<latexit sha1_base64="MR47EM00aXvJAcec3gKWyzZ0UTs=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lU/LgVvXhswX5AG8pmO2nXbjZhdyOU0l/gxYMiXv1J3vw3JmkQtT4YeLw3w8w8LxJcG9v+tApLyyura8X10sbm1vZOeXevpcNYMWyyUISq41GNgktsGm4EdiKFNPAEtr3xTeq3H1BpHso7M4nQDehQcp8zahKp0emXK3bVzkAWiZOTCuSo98sfvUHI4gClYYJq3XXsyLhTqgxnAmelXqwxomxMh9hNqKQBaneaHTojR4kyIH6okpKGZOrPiSkNtJ4EXtIZUDPSf71U/M/rxsa/dKdcRrFByeaL/FgQE5L0azLgCpkRk4RQpnhyK2EjqigzSTalLISrFOffLy+S1knVOa2eNs4qtes8jiIcwCEcgwMXUINbqEMTGCA8wjO8WPfWk/Vqvc1bC1Y+sw+/YL1/Ac6BjRM=</latexit>

4
Fix
two
randompointsinambientℙ•and find the line through these points

x

x

x

• Find 5 intersection points with the CY

surfaceNumerical algorithm[Braun,Brelidze,Douglas,Ovrut `08]

1. Choose an order k<latexit sha1_base64="LJtdnj+M3emAK5lySPkAzLsYW1A=">AAAB6HicbVDJSgNBEK2JW4xb1KOXxiB4ChMVl1vQi8cEzALJEHo6NUmbnoXuHiEM+QIvHhTx6id582/smQyixgcFj/eqqKrnRoIrbdufVmFpeWV1rbhe2tjc2t4p7+61VRhLhi0WilB2XapQ8ABbmmuB3Ugi9V2BHXdyk/qdB5SKh8Gdnkbo+HQUcI8zqo3UnAzKFbtqZyCLpJaTCuRoDMof/WHIYh8DzQRVqlezI+0kVGrOBM5K/VhhRNmEjrBnaEB9VE6SHTojR0YZEi+UpgJNMvXnREJ9paa+azp9qsfqr5eK/3m9WHuXTsKDKNYYsPkiLxZEhyT9mgy5RKbF1BDKJDe3EjamkjJtsillIVylOP9+eZG0T6q10+pp86xSv87jKMIBHMIx1OAC6nALDWgBA4RHeIYX6956sl6tt3lrwcpn9uEXrPcv602NJg==</latexit>

2. Compute all monomials of degree k<latexit sha1_base64="LJtdnj+M3emAK5lySPkAzLsYW1A=">AAAB6HicbVDJSgNBEK2JW4xb1KOXxiB4ChMVl1vQi8cEzALJEHo6NUmbnoXuHiEM+QIvHhTx6id582/smQyixgcFj/eqqKrnRoIrbdufVmFpeWV1rbhe2tjc2t4p7+61VRhLhi0WilB2XapQ8ABbmmuB3Ugi9V2BHXdyk/qdB5SKh8Gdnkbo+HQUcI8zqo3UnAzKFbtqZyCLpJaTCuRoDMof/WHIYh8DzQRVqlezI+0kVGrOBM5K/VhhRNmEjrBnaEB9VE6SHTojR0YZEi+UpgJNMvXnREJ9paa+azp9qsfqr5eK/3m9WHuXTsKDKNYYsPkiLxZEhyT9mgy5RKbF1BDKJDe3EjamkjJtsillIVylOP9+eZG0T6q10+pp86xSv87jKMIBHMIx1OAC6nALDWgBA4RHeIYX6956sl6tt3lrwcpn9uEXrPcv602NJg==</latexit>

3. Sample random points on X
<latexit sha1_base64="MR47EM00aXvJAcec3gKWyzZ0UTs=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lU/LgVvXhswX5AG8pmO2nXbjZhdyOU0l/gxYMiXv1J3vw3JmkQtT4YeLw3w8w8LxJcG9v+tApLyyura8X10sbm1vZOeXevpcNYMWyyUISq41GNgktsGm4EdiKFNPAEtr3xTeq3H1BpHso7M4nQDehQcp8zahKp0emXK3bVzkAWiZOTCuSo98sfvUHI4gClYYJq3XXsyLhTqgxnAmelXqwxomxMh9hNqKQBaneaHTojR4kyIH6okpKGZOrPiSkNtJ4EXtIZUDPSf71U/M/rxsa/dKdcRrFByeaL/FgQE5L0azLgCpkRk4RQpnhyK2EjqigzSTalLISrFOffLy+S1knVOa2eNs4qtes8jiIcwCEcgwMXUINbqEMTGCA8wjO8WPfWk/Vqvc1bC1Y+sw+/YL1/Ac6BjRM=</latexit>

4
Fix
two
randompointsinambientℙ•and find the line through these points

x

x

x

• Find 5 intersection points with the CY

surface• Random sample in ambient space

Random sample on CY surface≠

Numerical algorithm[Braun,Brelidze,Douglas,Ovrut `08]

1. Choose an order k<latexit sha1_base64="LJtdnj+M3emAK5lySPkAzLsYW1A=">AAAB6HicbVDJSgNBEK2JW4xb1KOXxiB4ChMVl1vQi8cEzALJEHo6NUmbnoXuHiEM+QIvHhTx6id582/smQyixgcFj/eqqKrnRoIrbdufVmFpeWV1rbhe2tjc2t4p7+61VRhLhi0WilB2XapQ8ABbmmuB3Ugi9V2BHXdyk/qdB5SKh8Gdnkbo+HQUcI8zqo3UnAzKFbtqZyCLpJaTCuRoDMof/WHIYh8DzQRVqlezI+0kVGrOBM5K/VhhRNmEjrBnaEB9VE6SHTojR0YZEi+UpgJNMvXnREJ9paa+azp9qsfqr5eK/3m9WHuXTsKDKNYYsPkiLxZEhyT9mgy5RKbF1BDKJDe3EjamkjJtsillIVylOP9+eZG0T6q10+pp86xSv87jKMIBHMIx1OAC6nALDWgBA4RHeIYX6956sl6tt3lrwcpn9uEXrPcv602NJg==</latexit>

2. Compute all monomials of degree k<latexit sha1_base64="LJtdnj+M3emAK5lySPkAzLsYW1A=">AAAB6HicbVDJSgNBEK2JW4xb1KOXxiB4ChMVl1vQi8cEzALJEHo6NUmbnoXuHiEM+QIvHhTx6id582/smQyixgcFj/eqqKrnRoIrbdufVmFpeWV1rbhe2tjc2t4p7+61VRhLhi0WilB2XapQ8ABbmmuB3Ugi9V2BHXdyk/qdB5SKh8Gdnkbo+HQUcI8zqo3UnAzKFbtqZyCLpJaTCuRoDMof/WHIYh8DzQRVqlezI+0kVGrOBM5K/VhhRNmEjrBnaEB9VE6SHTojR0YZEi+UpgJNMvXnREJ9paa+azp9qsfqr5eK/3m9WHuXTsKDKNYYsPkiLxZEhyT9mgy5RKbF1BDKJDe3EjamkjJtsillIVylOP9+eZG0T6q10+pp86xSv87jKMIBHMIx1OAC6nALDWgBA4RHeIYX6956sl6tt3lrwcpn9uEXrPcv602NJg==</latexit>

3. Sample random points on X
<latexit sha1_base64="MR47EM00aXvJAcec3gKWyzZ0UTs=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lU/LgVvXhswX5AG8pmO2nXbjZhdyOU0l/gxYMiXv1J3vw3JmkQtT4YeLw3w8w8LxJcG9v+tApLyyura8X10sbm1vZOeXevpcNYMWyyUISq41GNgktsGm4EdiKFNPAEtr3xTeq3H1BpHso7M4nQDehQcp8zahKp0emXK3bVzkAWiZOTCuSo98sfvUHI4gClYYJq3XXsyLhTqgxnAmelXqwxomxMh9hNqKQBaneaHTojR4kyIH6okpKGZOrPiSkNtJ4EXtIZUDPSf71U/M/rxsa/dKdcRrFByeaL/FgQE5L0azLgCpkRk4RQpnhyK2EjqigzSTalLISrFOffLy+S1knVOa2eNs4qtes8jiIcwCEcgwMXUINbqEMTGCA8wjO8WPfWk/Vqvc1bC1Y+sw+/YL1/Ac6BjRM=</latexit>

4
Fix
two
randompointsinambientℙ•and find the line through these points

x

x

x

• Find 5 intersection points with the CY

surface• Random sample in ambient space

Random sample on CY surface≠• Measure on CY given by pullback of
FS form, dA = 𝔼P∼ℙ4[Q ∩ L] = i*(J FS)3
[Shiffman, Zelditch `99]

Numerical algorithm[Braun,Brelidze,Douglas,Ovrut `08]

1. Choose an order k<latexit sha1_base64="LJtdnj+M3emAK5lySPkAzLsYW1A=">AAAB6HicbVDJSgNBEK2JW4xb1KOXxiB4ChMVl1vQi8cEzALJEHo6NUmbnoXuHiEM+QIvHhTx6id582/smQyixgcFj/eqqKrnRoIrbdufVmFpeWV1rbhe2tjc2t4p7+61VRhLhi0WilB2XapQ8ABbmmuB3Ugi9V2BHXdyk/qdB5SKh8Gdnkbo+HQUcI8zqo3UnAzKFbtqZyCLpJaTCuRoDMof/WHIYh8DzQRVqlezI+0kVGrOBM5K/VhhRNmEjrBnaEB9VE6SHTojR0YZEi+UpgJNMvXnREJ9paa+azp9qsfqr5eK/3m9WHuXTsKDKNYYsPkiLxZEhyT9mgy5RKbF1BDKJDe3EjamkjJtsillIVylOP9+eZG0T6q10+pp86xSv87jKMIBHMIx1OAC6nALDWgBA4RHeIYX6956sl6tt3lrwcpn9uEXrPcv602NJg==</latexit>

2. Compute all monomials of degree k<latexit sha1_base64="LJtdnj+M3emAK5lySPkAzLsYW1A=">AAAB6HicbVDJSgNBEK2JW4xb1KOXxiB4ChMVl1vQi8cEzALJEHo6NUmbnoXuHiEM+QIvHhTx6id582/smQyixgcFj/eqqKrnRoIrbdufVmFpeWV1rbhe2tjc2t4p7+61VRhLhi0WilB2XapQ8ABbmmuB3Ugi9V2BHXdyk/qdB5SKh8Gdnkbo+HQUcI8zqo3UnAzKFbtqZyCLpJaTCuRoDMof/WHIYh8DzQRVqlezI+0kVGrOBM5K/VhhRNmEjrBnaEB9VE6SHTojR0YZEi+UpgJNMvXnREJ9paa+azp9qsfqr5eK/3m9WHuXTsKDKNYYsPkiLxZEhyT9mgy5RKbF1BDKJDe3EjamkjJtsillIVylOP9+eZG0T6q10+pp86xSv87jKMIBHMIx1OAC6nALDWgBA4RHeIYX6956sl6tt3lrwcpn9uEXrPcv602NJg==</latexit>

3. Sample random points on X
<latexit sha1_base64="MR47EM00aXvJAcec3gKWyzZ0UTs=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lU/LgVvXhswX5AG8pmO2nXbjZhdyOU0l/gxYMiXv1J3vw3JmkQtT4YeLw3w8w8LxJcG9v+tApLyyura8X10sbm1vZOeXevpcNYMWyyUISq41GNgktsGm4EdiKFNPAEtr3xTeq3H1BpHso7M4nQDehQcp8zahKp0emXK3bVzkAWiZOTCuSo98sfvUHI4gClYYJq3XXsyLhTqgxnAmelXqwxomxMh9hNqKQBaneaHTojR4kyIH6okpKGZOrPiSkNtJ4EXtIZUDPSf71U/M/rxsa/dKdcRrFByeaL/FgQE5L0azLgCpkRk4RQpnhyK2EjqigzSTalLISrFOffLy+S1knVOa2eNs4qtes8jiIcwCEcgwMXUINbqEMTGCA8wjO8WPfWk/Vqvc1bC1Y+sw+/YL1/Ac6BjRM=</latexit>

4. Make an ansatz for K : K<latexit sha1_base64="N0Gyr1fZncsdM900Vklly4cNS64=">AAAB6HicbVDLSsNAFL2pr1pfVZduBovgqiRWfOyKbgQ3LdgHtKFMppN27GQSZiZCCf0CNy4UcesnufNvnKRB1HrgwuGce7n3Hi/iTGnb/rQKS8srq2vF9dLG5tb2Tnl3r63CWBLaIiEPZdfDinImaEszzWk3khQHHqcdb3Kd+p0HKhULxZ2eRtQN8EgwnxGsjdS8HZQrdtXOgBaJk5MK5GgMyh/9YUjigApNOFaq59iRdhMsNSOczkr9WNEIkwke0Z6hAgdUuUl26AwdGWWI/FCaEhpl6s+JBAdKTQPPdAZYj9VfLxX/83qx9i/chIko1lSQ+SI/5kiHKP0aDZmkRPOpIZhIZm5FZIwlJtpkU8pCuExx9v3yImmfVJ1atdY8rdSv8jiKcACHcAwOnEMdbqABLSBA4RGe4cW6t56sV+tt3lqw8pl9+AXr/Qu6zY0G</latexit><latexit sha1_base64="3fIaVNy8coGpQhq+LSQwXSXrBWE=">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</latexit>

k = 1, h

ab̄

=δ

ab̄

(k)

1
(k) ab̄ (k)=log sa h sb̄
k⇡: Fubini Study metricNumerical algorithm[Braun,Brelidze,Douglas,Ovrut `08]

1. Choose an order k<latexit sha1_base64="LJtdnj+M3emAK5lySPkAzLsYW1A=">AAAB6HicbVDJSgNBEK2JW4xb1KOXxiB4ChMVl1vQi8cEzALJEHo6NUmbnoXuHiEM+QIvHhTx6id582/smQyixgcFj/eqqKrnRoIrbdufVmFpeWV1rbhe2tjc2t4p7+61VRhLhi0WilB2XapQ8ABbmmuB3Ugi9V2BHXdyk/qdB5SKh8Gdnkbo+HQUcI8zqo3UnAzKFbtqZyCLpJaTCuRoDMof/WHIYh8DzQRVqlezI+0kVGrOBM5K/VhhRNmEjrBnaEB9VE6SHTojR0YZEi+UpgJNMvXnREJ9paa+azp9qsfqr5eK/3m9WHuXTsKDKNYYsPkiLxZEhyT9mgy5RKbF1BDKJDe3EjamkjJtsillIVylOP9+eZG0T6q10+pp86xSv87jKMIBHMIx1OAC6nALDWgBA4RHeIYX6956sl6tt3lrwcpn9uEXrPcv602NJg==</latexit>

2. Compute all monomials of degree k<latexit sha1_base64="LJtdnj+M3emAK5lySPkAzLsYW1A=">AAAB6HicbVDJSgNBEK2JW4xb1KOXxiB4ChMVl1vQi8cEzALJEHo6NUmbnoXuHiEM+QIvHhTx6id582/smQyixgcFj/eqqKrnRoIrbdufVmFpeWV1rbhe2tjc2t4p7+61VRhLhi0WilB2XapQ8ABbmmuB3Ugi9V2BHXdyk/qdB5SKh8Gdnkbo+HQUcI8zqo3UnAzKFbtqZyCLpJaTCuRoDMof/WHIYh8DzQRVqlezI+0kVGrOBM5K/VhhRNmEjrBnaEB9VE6SHTojR0YZEi+UpgJNMvXnREJ9paa+azp9qsfqr5eK/3m9WHuXTsKDKNYYsPkiLxZEhyT9mgy5RKbF1BDKJDe3EjamkjJtsillIVylOP9+eZG0T6q10+pp86xSv87jKMIBHMIx1OAC6nALDWgBA4RHeIYX6956sl6tt3lrwcpn9uEXrPcv602NJg==</latexit>

3. Sample random points on X
<latexit sha1_base64="MR47EM00aXvJAcec3gKWyzZ0UTs=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lU/LgVvXhswX5AG8pmO2nXbjZhdyOU0l/gxYMiXv1J3vw3JmkQtT4YeLw3w8w8LxJcG9v+tApLyyura8X10sbm1vZOeXevpcNYMWyyUISq41GNgktsGm4EdiKFNPAEtr3xTeq3H1BpHso7M4nQDehQcp8zahKp0emXK3bVzkAWiZOTCuSo98sfvUHI4gClYYJq3XXsyLhTqgxnAmelXqwxomxMh9hNqKQBaneaHTojR4kyIH6okpKGZOrPiSkNtJ4EXtIZUDPSf71U/M/rxsa/dKdcRrFByeaL/FgQE5L0azLgCpkRk4RQpnhyK2EjqigzSTalLISrFOffLy+S1knVOa2eNs4qtes8jiIcwCEcgwMXUINbqEMTGCA8wjO8WPfWk/Vqvc1bC1Y+sw+/YL1/Ac6BjRM=</latexit>

4. Make an ansatz for K : K(k)<latexit sha1_base64="N0Gyr1fZncsdM900Vklly4cNS64=">AAAB6HicbVDLSsNAFL2pr1pfVZduBovgqiRWfOyKbgQ3LdgHtKFMppN27GQSZiZCCf0CNy4UcesnufNvnKRB1HrgwuGce7n3Hi/iTGnb/rQKS8srq2vF9dLG5tb2Tnl3r63CWBLaIiEPZdfDinImaEszzWk3khQHHqcdb3Kd+p0HKhULxZ2eRtQN8EgwnxGsjdS8HZQrdtXOgBaJk5MK5GgMyh/9YUjigApNOFaq59iRdhMsNSOczkr9WNEIkwke0Z6hAgdUuUl26AwdGWWI/FCaEhpl6s+JBAdKTQPPdAZYj9VfLxX/83qx9i/chIko1lSQ+SI/5kiHKP0aDZmkRPOpIZhIZm5FZIwlJtpkU8pCuExx9v3yImmfVJ1atdY8rdSv8jiKcACHcAwOnEMdbqABLSBA4RGe4cW6t56sV+tt3lqw8pl9+AXr/Qu6zY0G</latexit><latexit sha1_base64="3fIaVNy8coGpQhq+LSQwXSXrBWE=">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</latexit>

5. Compute normalization:Z<latexit sha1_base64="VG8j61XLu78ge44tiOfq+ZuhQAk=">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</latexit>

1(k) ab̄ (k)=log sa h sb̄
k⇡
Z

J3 /
X

X

⌦ ^ ⌦ = constNumerical algorithm[Braun,Brelidze,Douglas,Ovrut `08]

1. Choose an order k<latexit sha1_base64="LJtdnj+M3emAK5lySPkAzLsYW1A=">AAAB6HicbVDJSgNBEK2JW4xb1KOXxiB4ChMVl1vQi8cEzALJEHo6NUmbnoXuHiEM+QIvHhTx6id582/smQyixgcFj/eqqKrnRoIrbdufVmFpeWV1rbhe2tjc2t4p7+61VRhLhi0WilB2XapQ8ABbmmuB3Ugi9V2BHXdyk/qdB5SKh8Gdnkbo+HQUcI8zqo3UnAzKFbtqZyCLpJaTCuRoDMof/WHIYh8DzQRVqlezI+0kVGrOBM5K/VhhRNmEjrBnaEB9VE6SHTojR0YZEi+UpgJNMvXnREJ9paa+azp9qsfqr5eK/3m9WHuXTsKDKNYYsPkiLxZEhyT9mgy5RKbF1BDKJDe3EjamkjJtsillIVylOP9+eZG0T6q10+pp86xSv87jKMIBHMIx1OAC6nALDWgBA4RHeIYX6956sl6tt3lrwcpn9uEXrPcv602NJg==</latexit>

2. Compute all monomials of degree k<latexit sha1_base64="LJtdnj+M3emAK5lySPkAzLsYW1A=">AAAB6HicbVDJSgNBEK2JW4xb1KOXxiB4ChMVl1vQi8cEzALJEHo6NUmbnoXuHiEM+QIvHhTx6id582/smQyixgcFj/eqqKrnRoIrbdufVmFpeWV1rbhe2tjc2t4p7+61VRhLhi0WilB2XapQ8ABbmmuB3Ugi9V2BHXdyk/qdB5SKh8Gdnkbo+HQUcI8zqo3UnAzKFbtqZyCLpJaTCuRoDMof/WHIYh8DzQRVqlezI+0kVGrOBM5K/VhhRNmEjrBnaEB9VE6SHTojR0YZEi+UpgJNMvXnREJ9paa+azp9qsfqr5eK/3m9WHuXTsKDKNYYsPkiLxZEhyT9mgy5RKbF1BDKJDe3EjamkjJtsillIVylOP9+eZG0T6q10+pp86xSv87jKMIBHMIx1OAC6nALDWgBA4RHeIYX6956sl6tt3lrwcpn9uEXrPcv602NJg==</latexit>

3. Sample random points on X
<latexit sha1_base64="MR47EM00aXvJAcec3gKWyzZ0UTs=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lU/LgVvXhswX5AG8pmO2nXbjZhdyOU0l/gxYMiXv1J3vw3JmkQtT4YeLw3w8w8LxJcG9v+tApLyyura8X10sbm1vZOeXevpcNYMWyyUISq41GNgktsGm4EdiKFNPAEtr3xTeq3H1BpHso7M4nQDehQcp8zahKp0emXK3bVzkAWiZOTCuSo98sfvUHI4gClYYJq3XXsyLhTqgxnAmelXqwxomxMh9hNqKQBaneaHTojR4kyIH6okpKGZOrPiSkNtJ4EXtIZUDPSf71U/M/rxsa/dKdcRrFByeaL/FgQE5L0azLgCpkRk4RQpnhyK2EjqigzSTalLISrFOffLy+S1knVOa2eNs4qtes8jiIcwCEcgwMXUINbqEMTGCA8wjO8WPfWk/Vqvc1bC1Y+sw+/YL1/Ac6BjRM=</latexit>

4. Make an ansatz for K : K(k)<latexit sha1_base64="N0Gyr1fZncsdM900Vklly4cNS64=">AAAB6HicbVDLSsNAFL2pr1pfVZduBovgqiRWfOyKbgQ3LdgHtKFMppN27GQSZiZCCf0CNy4UcesnufNvnKRB1HrgwuGce7n3Hi/iTGnb/rQKS8srq2vF9dLG5tb2Tnl3r63CWBLaIiEPZdfDinImaEszzWk3khQHHqcdb3Kd+p0HKhULxZ2eRtQN8EgwnxGsjdS8HZQrdtXOgBaJk5MK5GgMyh/9YUjigApNOFaq59iRdhMsNSOczkr9WNEIkwke0Z6hAgdUuUl26AwdGWWI/FCaEhpl6s+JBAdKTQPPdAZYj9VfLxX/83qx9i/chIko1lSQ+SI/5kiHKP0aDZmkRPOpIZhIZm5FZIwlJtpkU8pCuExx9v3yImmfVJ1atdY8rdSv8jiKcACHcAwOnEMdbqABLSBA4RGe4cW6t56sV+tt3lqw8pl9+AXr/Qu6zY0G</latexit><latexit sha1_base64="3fIaVNy8coGpQhq+LSQwXSXrBWE=">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</latexit>

5. Compute normalization:
Z

6. Replace X<latexit sha1_base64="bBEDyP8NFJMFooXEfHNkesRHpyw=">AAACFXicbVDJSgNBEO2JW4xb1KOXxiB4kDAx4nILevEYwSyQCUNPp5M06ekeumvUMMxPePFXvHhQxKvgzb+xsyBqfFDweK+KqnpBJLgB1/10MnPzC4tL2eXcyura+kZ+c6tuVKwpq1EllG4GxDDBJasBB8GakWYkDARrBIOLkd+4YdpwJa9hGLF2SHqSdzklYCU/f+BxCX4Te5r3+kC0VrfYM3HoJx6wO0giZX2DlcTNNPXzBbfojoFnSWlKCmiKqp//8DqKxiGTQAUxplVyI2gnRAOngqU5LzYsInRAeqxlqSQhM+1k/FWK96zSwV2lbUnAY/XnREJCY4ZhYDtDAn3z1xuJ/3mtGLqn7YTLKAYm6WRRNxYYFB5FhDtcMwpiaAmhmttbMe0TTSjYIHPjEM5GOP5+eZbUD4ulcrF8dVSonE/jyKIdtIv2UQmdoAq6RFVUQxTdo0f0jF6cB+fJeXXeJq0ZZzqzjX7Bef8C6yKgEA==</latexit>!

X

points on X

Z<latexit sha1_base64="VG8j61XLu78ge44tiOfq+ZuhQAk=">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</latexit>

1(k) ab̄ (k)=log sa h sb̄
k⇡
Z

J3 /
X

X

3

3

⌦ ^ ⌦ = const

and d z d z̄ ! w<latexit sha1_base64="mhliw99avaJJ546wA5q3QgPGBzo=">AAACB3icbVDLSsNAFJ3UV62vqEtBBovgqiRWfOCm6MZlBfuAJpbJZNIOnTyYmVja0J0bf8WNC0Xc+gvu/BsnaRC1HriXwzn3MnOPEzEqpGF8aoW5+YXFpeJyaWV1bX1D39xqijDmmDRwyELedpAgjAakIalkpB1xgnyHkZYzuEz91h3hgobBjRxFxPZRL6AexUgqqavvurfVsXWuuuUgnown0OK015eI83AIh129bFSMDHCWmDkpgxz1rv5huSGOfRJIzJAQHdOIpJ0gLilmZFKyYkEihAeoRzqKBsgnwk6yOyZwXyku9EKuKpAwU39uJMgXYuQ7atJHsi/+eqn4n9eJpXdqJzSIYkkCPH3IixmUIUxDgS7lBEs2UgRhTtVfIe4jjrBU0ZWyEM5SHH+fPEuahxWzWqleH5VrF3kcRbAD9sABMMEJqIErUAcNgME9eATP4EV70J60V+1tOlrQ8p1t8Ava+xfIoZlo</latexit>

Numerical algorithm[Braun,Brelidze,Douglas,Ovrut `08]

1. Choose an order k<latexit sha1_base64="LJtdnj+M3emAK5lySPkAzLsYW1A=">AAAB6HicbVDJSgNBEK2JW4xb1KOXxiB4ChMVl1vQi8cEzALJEHo6NUmbnoXuHiEM+QIvHhTx6id582/smQyixgcFj/eqqKrnRoIrbdufVmFpeWV1rbhe2tjc2t4p7+61VRhLhi0WilB2XapQ8ABbmmuB3Ugi9V2BHXdyk/qdB5SKh8Gdnkbo+HQUcI8zqo3UnAzKFbtqZyCLpJaTCuRoDMof/WHIYh8DzQRVqlezI+0kVGrOBM5K/VhhRNmEjrBnaEB9VE6SHTojR0YZEi+UpgJNMvXnREJ9paa+azp9qsfqr5eK/3m9WHuXTsKDKNYYsPkiLxZEhyT9mgy5RKbF1BDKJDe3EjamkjJtsillIVylOP9+eZG0T6q10+pp86xSv87jKMIBHMIx1OAC6nALDWgBA4RHeIYX6956sl6tt3lrwcpn9uEXrPcv602NJg==</latexit>

2. Compute all monomials of degree k<latexit sha1_base64="LJtdnj+M3emAK5lySPkAzLsYW1A=">AAAB6HicbVDJSgNBEK2JW4xb1KOXxiB4ChMVl1vQi8cEzALJEHo6NUmbnoXuHiEM+QIvHhTx6id582/smQyixgcFj/eqqKrnRoIrbdufVmFpeWV1rbhe2tjc2t4p7+61VRhLhi0WilB2XapQ8ABbmmuB3Ugi9V2BHXdyk/qdB5SKh8Gdnkbo+HQUcI8zqo3UnAzKFbtqZyCLpJaTCuRoDMof/WHIYh8DzQRVqlezI+0kVGrOBM5K/VhhRNmEjrBnaEB9VE6SHTojR0YZEi+UpgJNMvXnREJ9paa+azp9qsfqr5eK/3m9WHuXTsKDKNYYsPkiLxZEhyT9mgy5RKbF1BDKJDe3EjamkjJtsillIVylOP9+eZG0T6q10+pp86xSv87jKMIBHMIx1OAC6nALDWgBA4RHeIYX6956sl6tt3lrwcpn9uEXrPcv602NJg==</latexit>

3. Sample random points on X
<latexit sha1_base64="MR47EM00aXvJAcec3gKWyzZ0UTs=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lU/LgVvXhswX5AG8pmO2nXbjZhdyOU0l/gxYMiXv1J3vw3JmkQtT4YeLw3w8w8LxJcG9v+tApLyyura8X10sbm1vZOeXevpcNYMWyyUISq41GNgktsGm4EdiKFNPAEtr3xTeq3H1BpHso7M4nQDehQcp8zahKp0emXK3bVzkAWiZOTCuSo98sfvUHI4gClYYJq3XXsyLhTqgxnAmelXqwxomxMh9hNqKQBaneaHTojR4kyIH6okpKGZOrPiSkNtJ4EXtIZUDPSf71U/M/rxsa/dKdcRrFByeaL/FgQE5L0azLgCpkRk4RQpnhyK2EjqigzSTalLISrFOffLy+S1knVOa2eNs4qtes8jiIcwCEcgwMXUINbqEMTGCA8wjO8WPfWk/Vqvc1bC1Y+sw+/YL1/Ac6BjRM=</latexit>

4. Make an ansatz for K : K(k)<latexit sha1_base64="N0Gyr1fZncsdM900Vklly4cNS64=">AAAB6HicbVDLSsNAFL2pr1pfVZduBovgqiRWfOyKbgQ3LdgHtKFMppN27GQSZiZCCf0CNy4UcesnufNvnKRB1HrgwuGce7n3Hi/iTGnb/rQKS8srq2vF9dLG5tb2Tnl3r63CWBLaIiEPZdfDinImaEszzWk3khQHHqcdb3Kd+p0HKhULxZ2eRtQN8EgwnxGsjdS8HZQrdtXOgBaJk5MK5GgMyh/9YUjigApNOFaq59iRdhMsNSOczkr9WNEIkwke0Z6hAgdUuUl26AwdGWWI/FCaEhpl6s+JBAdKTQPPdAZYj9VfLxX/83qx9i/chIko1lSQ+SI/5kiHKP0aDZmkRPOpIZhIZm5FZIwlJtpkU8pCuExx9v3yImmfVJ1atdY8rdSv8jiKcACHcAwOnEMdbqABLSBA4RGe4cW6t56sV+tt3lqw8pl9+AXr/Qu6zY0G</latexit><latexit sha1_base64="3fIaVNy8coGpQhq+LSQwXSXrBWE=">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</latexit>

5. Compute normalization:
Z

6. Replace X!

X

points on X

Z<latexit sha1_base64="VG8j61XLu78ge44tiOfq+ZuhQAk=">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</latexit>

1(k) ab̄ (k)=log sa h sb̄
k⇡
Z

J3 /
X

X

3

⌦ ^ ⌦ = const

3and d z d z̄ ! w<latexit sha1_base64="mhliw99avaJJ546wA5q3QgPGBzo=">AAACB3icbVDLSsNAFJ3UV62vqEtBBovgqiRWfOCm6MZlBfuAJpbJZNIOnTyYmVja0J0bf8WNC0Xc+gvu/BsnaRC1HriXwzn3MnOPEzEqpGF8aoW5+YXFpeJyaWV1bX1D39xqijDmmDRwyELedpAgjAakIalkpB1xgnyHkZYzuEz91h3hgobBjRxFxPZRL6AexUgqqavvurfVsXWuuuUgnown0OK015eI83AIh129bFSMDHCWmDkpgxz1rv5huSGOfRJIzJAQHdOIpJ0gLilmZFKyYkEihAeoRzqKBsgnwk6yOyZwXyku9EKuKpAwU39uJMgXYuQ7atJHsi/+eqn4n9eJpXdqJzSIYkkCPH3IixmUIUxDgS7lBEs2UgRhTtVfIe4jjrBU0ZWyEM5SHH+fPEuahxWzWqleH5VrF3kcRbAD9sABMMEJqIErUAcNgME9eATP4EV70J60V+1tOlrQ8p1t8Ava+xfIoZlo</latexit>NXNsa (pi )sb (p̄i )wi
7. Compute T (h)ab̄ = vol(X)
s(p
)
·
h
·
s(p̄
)
i
i
i=1
<latexit sha1_base64="bBEDyP8NFJMFooXEfHNkesRHpyw=">AAACFXicbVDJSgNBEO2JW4xb1KOXxiB4kDAx4nILevEYwSyQCUNPp5M06ekeumvUMMxPePFXvHhQxKvgzb+xsyBqfFDweK+KqnpBJLgB1/10MnPzC4tL2eXcyura+kZ+c6tuVKwpq1EllG4GxDDBJasBB8GakWYkDARrBIOLkd+4YdpwJa9hGLF2SHqSdzklYCU/f+BxCX4Te5r3+kC0VrfYM3HoJx6wO0giZX2DlcTNNPXzBbfojoFnSWlKCmiKqp//8DqKxiGTQAUxplVyI2gnRAOngqU5LzYsInRAeqxlqSQhM+1k/FWK96zSwV2lbUnAY/XnREJCY4ZhYDtDAn3z1xuJ/3mtGLqn7YTLKAYm6WRRNxYYFB5FhDtcMwpiaAmhmttbMe0TTSjYIHPjEM5GOP5+eZbUD4ulcrF8dVSonE/jyKIdtIv2UQmdoAq6RFVUQxTdo0f0jF6cB+fJeXXeJq0ZZzqzjX7Bef8C6yKgEA==</latexit><latexit sha1_base64="23R9j3T68lRzyho4s0y4VlHoBFE=">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</latexit>

Numerical algorithm[Braun,Brelidze,Douglas,Ovrut `08]

1. Choose an order k<latexit sha1_base64="LJtdnj+M3emAK5lySPkAzLsYW1A=">AAAB6HicbVDJSgNBEK2JW4xb1KOXxiB4ChMVl1vQi8cEzALJEHo6NUmbnoXuHiEM+QIvHhTx6id582/smQyixgcFj/eqqKrnRoIrbdufVmFpeWV1rbhe2tjc2t4p7+61VRhLhi0WilB2XapQ8ABbmmuB3Ugi9V2BHXdyk/qdB5SKh8Gdnkbo+HQUcI8zqo3UnAzKFbtqZyCLpJaTCuRoDMof/WHIYh8DzQRVqlezI+0kVGrOBM5K/VhhRNmEjrBnaEB9VE6SHTojR0YZEi+UpgJNMvXnREJ9paa+azp9qsfqr5eK/3m9WHuXTsKDKNYYsPkiLxZEhyT9mgy5RKbF1BDKJDe3EjamkjJtsillIVylOP9+eZG0T6q10+pp86xSv87jKMIBHMIx1OAC6nALDWgBA4RHeIYX6956sl6tt3lrwcpn9uEXrPcv602NJg==</latexit>

2. Compute all monomials of degree k<latexit sha1_base64="LJtdnj+M3emAK5lySPkAzLsYW1A=">AAAB6HicbVDJSgNBEK2JW4xb1KOXxiB4ChMVl1vQi8cEzALJEHo6NUmbnoXuHiEM+QIvHhTx6id582/smQyixgcFj/eqqKrnRoIrbdufVmFpeWV1rbhe2tjc2t4p7+61VRhLhi0WilB2XapQ8ABbmmuB3Ugi9V2BHXdyk/qdB5SKh8Gdnkbo+HQUcI8zqo3UnAzKFbtqZyCLpJaTCuRoDMof/WHIYh8DzQRVqlezI+0kVGrOBM5K/VhhRNmEjrBnaEB9VE6SHTojR0YZEi+UpgJNMvXnREJ9paa+azp9qsfqr5eK/3m9WHuXTsKDKNYYsPkiLxZEhyT9mgy5RKbF1BDKJDe3EjamkjJtsillIVylOP9+eZG0T6q10+pp86xSv87jKMIBHMIx1OAC6nALDWgBA4RHeIYX6956sl6tt3lrwcpn9uEXrPcv602NJg==</latexit>

3. Sample random points on X
<latexit sha1_base64="MR47EM00aXvJAcec3gKWyzZ0UTs=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lU/LgVvXhswX5AG8pmO2nXbjZhdyOU0l/gxYMiXv1J3vw3JmkQtT4YeLw3w8w8LxJcG9v+tApLyyura8X10sbm1vZOeXevpcNYMWyyUISq41GNgktsGm4EdiKFNPAEtr3xTeq3H1BpHso7M4nQDehQcp8zahKp0emXK3bVzkAWiZOTCuSo98sfvUHI4gClYYJq3XXsyLhTqgxnAmelXqwxomxMh9hNqKQBaneaHTojR4kyIH6okpKGZOrPiSkNtJ4EXtIZUDPSf71U/M/rxsa/dKdcRrFByeaL/FgQE5L0azLgCpkRk4RQpnhyK2EjqigzSTalLISrFOffLy+S1knVOa2eNs4qtes8jiIcwCEcgwMXUINbqEMTGCA8wjO8WPfWk/Vqvc1bC1Y+sw+/YL1/Ac6BjRM=</latexit>

4. Make an ansatz for K : K(k)<latexit sha1_base64="N0Gyr1fZncsdM900Vklly4cNS64=">AAAB6HicbVDLSsNAFL2pr1pfVZduBovgqiRWfOyKbgQ3LdgHtKFMppN27GQSZiZCCf0CNy4UcesnufNvnKRB1HrgwuGce7n3Hi/iTGnb/rQKS8srq2vF9dLG5tb2Tnl3r63CWBLaIiEPZdfDinImaEszzWk3khQHHqcdb3Kd+p0HKhULxZ2eRtQN8EgwnxGsjdS8HZQrdtXOgBaJk5MK5GgMyh/9YUjigApNOFaq59iRdhMsNSOczkr9WNEIkwke0Z6hAgdUuUl26AwdGWWI/FCaEhpl6s+JBAdKTQPPdAZYj9VfLxX/83qx9i/chIko1lSQ+SI/5kiHKP0aDZmkRPOpIZhIZm5FZIwlJtpkU8pCuExx9v3yImmfVJ1atdY8rdSv8jiKcACHcAwOnEMdbqABLSBA4RGe4cW6t56sV+tt3lqw8pl9+AXr/Qu6zY0G</latexit><latexit sha1_base64="3fIaVNy8coGpQhq+LSQwXSXrBWE=">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</latexit>

5. Compute normalization:
Z

6. Replace X!

X

points on X

Z<latexit sha1_base64="VG8j61XLu78ge44tiOfq+ZuhQAk=">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</latexit>

1(k) ab̄ (k)=log sa h sb̄
k⇡
Z

J3 /
X

X

3

⌦ ^ ⌦ = const

3and d z d z̄ ! w<latexit sha1_base64="mhliw99avaJJ546wA5q3QgPGBzo=">AAACB3icbVDLSsNAFJ3UV62vqEtBBovgqiRWfOCm6MZlBfuAJpbJZNIOnTyYmVja0J0bf8WNC0Xc+gvu/BsnaRC1HriXwzn3MnOPEzEqpGF8aoW5+YXFpeJyaWV1bX1D39xqijDmmDRwyELedpAgjAakIalkpB1xgnyHkZYzuEz91h3hgobBjRxFxPZRL6AexUgqqavvurfVsXWuuuUgnown0OK015eI83AIh129bFSMDHCWmDkpgxz1rv5huSGOfRJIzJAQHdOIpJ0gLilmZFKyYkEihAeoRzqKBsgnwk6yOyZwXyku9EKuKpAwU39uJMgXYuQ7atJHsi/+eqn4n9eJpXdqJzSIYkkCPH3IixmUIUxDgS7lBEs2UgRhTtVfIe4jjrBU0ZWyEM5SHH+fPEuahxWzWqleH5VrF3kcRbAD9sABMMEJqIErUAcNgME9eATP4EV70J60V+1tOlrQ8p1t8Ava+xfIoZlo</latexit>NXNsa (pi )sb (p̄i )wi
7. Compute T (h)ab̄ = vol(X)
s(p
)
·
h
·
s(p̄
)
i
i
i=1
<latexit sha1_base64="bBEDyP8NFJMFooXEfHNkesRHpyw=">AAACFXicbVDJSgNBEO2JW4xb1KOXxiB4kDAx4nILevEYwSyQCUNPp5M06ekeumvUMMxPePFXvHhQxKvgzb+xsyBqfFDweK+KqnpBJLgB1/10MnPzC4tL2eXcyura+kZ+c6tuVKwpq1EllG4GxDDBJasBB8GakWYkDARrBIOLkd+4YdpwJa9hGLF2SHqSdzklYCU/f+BxCX4Te5r3+kC0VrfYM3HoJx6wO0giZX2DlcTNNPXzBbfojoFnSWlKCmiKqp//8DqKxiGTQAUxplVyI2gnRAOngqU5LzYsInRAeqxlqSQhM+1k/FWK96zSwV2lbUnAY/XnREJCY4ZhYDtDAn3z1xuJ/3mtGLqn7YTLKAYm6WRRNxYYFB5FhDtcMwpiaAmhmttbMe0TTSjYIHPjEM5GOP5+eZbUD4ulcrF8dVSonE/jyKIdtIv2UQmdoAq6RFVUQxTdo0f0jF6cB+fJeXXeJq0ZZzqzjX7Bef8C6yKgEA==</latexit><latexit sha1_base64="23R9j3T68lRzyho4s0y4VlHoBFE=">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</latexit>

1
8. Set T (h)ab̄ = hab̄ and repeat until convergence<latexit sha1_base64="J5a04VvWO75M7lyo+wajBm1CvqQ=">AAACCHicbVDLSsNAFJ3UV62vqEsXDhahLiyJFR8LoejGZYW+oI1hMp02QycPZiZCCVm68VfcuFDErZ/gzr9xkgbfBy4czrmXe+9xQkaFNIx3rTAzOze/UFwsLS2vrK7p6xttEUQckxYOWMC7DhKEUZ+0JJWMdENOkOcw0nHGF6nfuSFc0MBvyklILA+NfDqkGEkl2fp2s+Lu2THqO4hDJzlzr+N9M/kSbL1sVI0M8C8xc1IGORq2/tYfBDjyiC8xQ0L0TCOUVoy4pJiRpNSPBAkRHqMR6SnqI48IK84eSeCuUgZwGHBVvoSZ+n0iRp4QE89RnR6SrvjtpeJ/Xi+SwxMrpn4YSeLj6aJhxKAMYJoKHFBOsGQTRRDmVN0KsYs4wlJlV8pCOE1x9PnyX9I+qJq1au3qsFw/z+Mogi2wAyrABMegDi5BA7QABrfgHjyCJ+1Oe9CetZdpa0HLZzbBD2ivH1/YmRQ=</latexit>

Numerical algorithm - Error measure
1. Use final hab̄ to compute K

(k)

1
=
log sa(k)h ab̄s (k)b̄kπ2.Go to a patch (wlog z0 = 1) and pull back ambient
space expression for gab̄ = ∂a∂b̄K to CY by solving p
for z4, z4 = z4(z1, z2, z3) (wlog) and compute Jab̄
3. Measure deviation from Ricci-flat metricN

J 3(pi)⟨Ω ∧ Ω̄⟩
σ=
1−
w
i
3
∑
⟨J
⟩
N⟨Ω ∧ Ω̄⟩ i=1
Ω(pi) ∧ Ω̄(pi)
1

3
J
(pi)
1
3
⟨Ω ∧ Ω̄⟩ =Ω(pi) ∧ Ω̄(pi) , ⟨J ⟩ =wi∑∑ Ω(p ) ∧ Ω̄(p )
N
i
i
i
i

Numerical algorithm
16000
14000
12000
10000
8000
6000
4000
2000
0

Number of points needed# points# params in h

Number of parameters

20000018000016000014000012000010000080000
60000

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0kk

Grows like (k!)2

Grows like (k!)2<latexit sha1_base64="4bxnGdlrzNXBxl++NSc6T4h6IUc=">AAAB7XicbVDLSsNAFL3xWeur6tLNaBHqpqSt+NgV3bisYB/QxjKZTtqxk0yYmQgl9B/cuFDErf/jzr9xkgZR64ELh3Pu5d573JAzpW3701pYXFpeWc2t5dc3Nre2Czu7LSUiSWiTCC5kx8WKchbQpmaa004oKfZdTtvu+Crx2w9UKiaCWz0JqePjYcA8RrA2Uqs0Pji+q/YLRbtsp0DzpJKRImRo9AsfvYEgkU8DTThWqluxQ+3EWGpGOJ3me5GiISZjPKRdQwPsU+XE6bVTdGSUAfKENBVolKo/J2LsKzXxXdPpYz1Sf71E/M/rRto7d2IWhJGmAZkt8iKOtEDJ62jAJCWaTwzBRDJzKyIjLDHRJqB8GsJFgtPvl+dJq1qu1Mq1m5Ni/TKLIwf7cAglqMAZ1OEaGtAEAvfwCM/wYgnryXq13matC1Y2swe/YL1/AS6Rjlo=</latexit><latexit sha1_base64="4bxnGdlrzNXBxl++NSc6T4h6IUc=">AAAB7XicbVDLSsNAFL3xWeur6tLNaBHqpqSt+NgV3bisYB/QxjKZTtqxk0yYmQgl9B/cuFDErf/jzr9xkgZR64ELh3Pu5d573JAzpW3701pYXFpeWc2t5dc3Nre2Czu7LSUiSWiTCC5kx8WKchbQpmaa004oKfZdTtvu+Crx2w9UKiaCWz0JqePjYcA8RrA2Uqs0Pji+q/YLRbtsp0DzpJKRImRo9AsfvYEgkU8DTThWqluxQ+3EWGpGOJ3me5GiISZjPKRdQwPsU+XE6bVTdGSUAfKENBVolKo/J2LsKzXxXdPpYz1Sf71E/M/rRto7d2IWhJGmAZkt8iKOtEDJ62jAJCWaTwzBRDJzKyIjLDHRJqB8GsJFgtPvl+dJq1qu1Mq1m5Ni/TKLIwf7cAglqMAZ1OEaGtAEAvfwCM/wYgnryXq13matC1Y2swe/YL1/AS6Rjlo=</latexit>

Error

1000

0.35

800

0.30

600

0.25

æ

t[s]

Computation time

400

0.20

200

0.15
0.10

0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0kk

Grows exponentially

Decreases slowly

Numerical algorithm
16000
14000
12000
10000
8000
6000
4000
2000
0

Number of points needed# points# params in h

Number of parameters

20000018000016000014000012000010000080000
60000

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0kk

Bottleneck:
Numerical integration
Grows
like (k!)2
Grows like (k!)2<latexit sha1_base64="4bxnGdlrzNXBxl++NSc6T4h6IUc=">AAAB7XicbVDLSsNAFL3xWeur6tLNaBHqpqSt+NgV3bisYB/QxjKZTtqxk0yYmQgl9B/cuFDErf/jzr9xkgZR64ELh3Pu5d573JAzpW3701pYXFpeWc2t5dc3Nre2Czu7LSUiSWiTCC5kx8WKchbQpmaa004oKfZdTtvu+Crx2w9UKiaCWz0JqePjYcA8RrA2Uqs0Pji+q/YLRbtsp0DzpJKRImRo9AsfvYEgkU8DTThWqluxQ+3EWGpGOJ3me5GiISZjPKRdQwPsU+XE6bVTdGSUAfKENBVolKo/J2LsKzXxXdPpYz1Sf71E/M/rRto7d2IWhJGmAZkt8iKOtEDJ62jAJCWaTwzBRDJzKyIjLDHRJqB8GsJFgtPvl+dJq1qu1Mq1m5Ni/TKLIwf7cAglqMAZ1OEaGtAEAvfwCM/wYgnryXq13matC1Y2swe/YL1/AS6Rjlo=</latexit>in computation of hComputation time

Error

0.35

800

0.30

600

0.25

æ

t[s]

1000

<latexit sha1_base64="4bxnGdlrzNXBxl++NSc6T4h6IUc=">AAAB7XicbVDLSsNAFL3xWeur6tLNaBHqpqSt+NgV3bisYB/QxjKZTtqxk0yYmQgl9B/cuFDErf/jzr9xkgZR64ELh3Pu5d573JAzpW3701pYXFpeWc2t5dc3Nre2Czu7LSUiSWiTCC5kx8WKchbQpmaa004oKfZdTtvu+Crx2w9UKiaCWz0JqePjYcA8RrA2Uqs0Pji+q/YLRbtsp0DzpJKRImRo9AsfvYEgkU8DTThWqluxQ+3EWGpGOJ3me5GiISZjPKRdQwPsU+XE6bVTdGSUAfKENBVolKo/J2LsKzXxXdPpYz1Sf71E/M/rRto7d2IWhJGmAZkt8iKOtEDJ62jAJCWaTwzBRDJzKyIjLDHRJqB8GsJFgtPvl+dJq1qu1Mq1m5Ni/TKLIwf7cAglqMAZ1OEaGtAEAvfwCM/wYgnryXq13matC1Y2swe/YL1/AS6Rjlo=</latexit>

400

0.20

200

0.15
0.10

0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0kk

Grows exponentially

Decreases slowly

Idea 1: Phrase as
optimization Problem

Idea 1: Optimize with GD‣ We can phrase finding the CY metric as an

optimization problem: Find h such that σ = 0

Idea 1: Optimize with GD‣ We can phrase finding the CY metric as an

optimization problem: Find h such that σ = 0‣ Perform (stochastic) gradient descent with loss

function σ

w = h, b = 0<latexit sha1_base64="mLxlxYr4EoZaL5DzIbERp0SLlKU=">AAAB73icbVDLSsNAFL2pr1pfVZduBovgQkpqxceiUHTjsoJ9QBvKZDpph04mcWailNCfcONCEbf+jjv/xkkaRK0HLhzOuZd773FDzpS27U8rt7C4tLySXy2srW9sbhW3d1oqiCShTRLwQHZcrChngjY105x2Qkmx73LadsdXid++p1KxQNzqSUgdHw8F8xjB2kidh9roCLk1u18s2WU7BZonlYyUIEOjX/zoDQIS+VRowrFS3YodaifGUjPC6bTQixQNMRnjIe0aKrBPlROn907RgVEGyAukKaFRqv6ciLGv1MR3TaeP9Uj99RLxP68bae/ciZkII00FmS3yIo50gJLn0YBJSjSfGIKJZOZWREZYYqJNRIU0hIsEp98vz5PWcblSLVdvTkr1yyyOPOzBPhxCBc6gDtfQgCYQ4PAIz/Bi3VlP1qv1NmvNWdnMLvyC9f4Fr+ePOA==</latexit>

s1 (p)<latexit sha1_base64="yRoPyFWpXFVZ6y4vWxa46cZdg0M=">AAAB7XicbVDLSsNAFL2pr1pfVZdugkWom5JY8bErunFZwT6gDWUynbRjJzNhZiKU0H9w40IRt/6PO//GSRpErQcuHM65l3vv8SNGlXacT6uwtLyyulZcL21sbm3vlHf32krEEpMWFkzIro8UYZSTlqaakW4kCQp9Rjr+5Dr1Ow9EKir4nZ5GxAvRiNOAYqSN1FYDtxodD8oVp+ZksBeJm5MK5GgOyh/9ocBxSLjGDCnVc51IewmSmmJGZqV+rEiE8ASNSM9QjkKivCS7dmYfGWVoB0Ka4trO1J8TCQqVmoa+6QyRHqu/Xir+5/ViHVx4CeVRrAnH80VBzGwt7PR1e0glwZpNDUFYUnOrjcdIIqxNQKUshMsUZ98vL5L2Sc2t1+q3p5XGVR5HEQ7gEKrgwjk04Aaa0AIM9/AIz/BiCevJerXe5q0FK5/Zh1+w3r8As/SOsQ==</latexit>s2 (p)<latexit sha1_base64="6kHgBQnrUeagVMhlOM+5l77Os8s=">AAACGHicbVBLSwMxGMz6rPW16tFLsAgepO723VvRi8cK9gHtUrJp2oZmN0uSFcqyP8OLf8WLB0W89ua/Md0uotaBwDAzX/Jl3IBRqSzr01hb39jc2s7sZHf39g8OzaPjtuShwKSFOeOi6yJJGPVJS1HFSDcQBHkuIx13erPwOw9ESMr9ezULiOOhsU9HFCOlpYF5FfWTS3pi7DqRlbcSXK6QuM+GXMl4YOasfL1eq9glmJBqsbwklXIB2mk6B1I0B+a8P+Q49IivMENS9mwrUE6EhKKYkTjbDyUJEJ6iMelp6iOPSCdKdorhuVaGcMSFPr6CifpzIkKelDPP1UkPqYn86y3E/7xeqEY1J6J+ECri4+VDo5BBxeGiJTikgmDFZpogLKjeFeIJEggr3WU2LUGj8v3lVdIu5O1ivnhXyjWu0zoy4BScgQtggypogFvQBC2AwSN4Bq/gzXgyXox342MZXTPSmRPwC8b8CyE+nN8=</latexit>...

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<latexit sha1_base64="uvQCdp13UEFUIDqSumYLUENZk5k=">AAAB7XicbVDLSsNAFL3xWeur6tLNYBHqpiSt+NgV3bisYB/QhjKZTtqxk5kwMxFK6D+4caGIW//HnX9jkgZR64ELh3Pu5d57vJAzbWz701paXlldWy9sFDe3tnd2S3v7bS0jRWiLSC5V18OaciZoyzDDaTdUFAcepx1vcp36nQeqNJPizkxD6gZ4JJjPCDaJ1NaDWiU8GZTKdtXOgBaJk5My5GgOSh/9oSRRQIUhHGvdc+zQuDFWhhFOZ8V+pGmIyQSPaC+hAgdUu3F27QwdJ8oQ+VIlJQzK1J8TMQ60ngZe0hlgM9Z/vVT8z+tFxr9wYybCyFBB5ov8iCMjUfo6GjJFieHThGCiWHIrImOsMDFJQMUshMsUZ98vL5J2rerUq/Xb03LjKo+jAIdwBBVw4BwacANNaAGBe3iEZ3ixpPVkvVpv89YlK585gF+w3r8AtXuOsg==</latexit>sM (p)<latexit sha1_base64="d9i35T2RoltL5kakGtC9RoRa3KA=">AAAB7XicbVDLSsNAFL2pr1pfVZdugkWom5JY8bErunEjVLAPaEOZTCft2MlMmJkIJfQf3LhQxK3/486/cZIG8XXgwuGce7n3Hj9iVGnH+bAKC4tLyyvF1dLa+sbmVnl7p61ELDFpYcGE7PpIEUY5aWmqGelGkqDQZ6TjTy5Tv3NPpKKC3+ppRLwQjTgNKEbaSG01uK5Gh4Nyxak5Gey/xM1JBXI0B+X3/lDgOCRcY4aU6rlOpL0ESU0xI7NSP1YkQniCRqRnKEchUV6SXTuzD4wytAMhTXFtZ+r3iQSFSk1D33SGSI/Vby8V//N6sQ7OvITyKNaE4/miIGa2Fnb6uj2kkmDNpoYgLKm51cZjJBHWJqBSFsJ5ipOvl/+S9lHNrdfqN8eVxkUeRxH2YB+q4MIpNOAKmtACDHfwAE/wbAnr0XqxXuetBSuf2YUfsN4+Ad64js0=</latexit>

dot<latexit sha1_base64="BDFsIL5aw0LB7cvxbXwgnzitS+A=">AAAB+nicbVDLSgMxFM3UV62vqS7dBItQN2XGio9d0Y3LCvYB7TBk0rQNzSRDklHK2E9x40IRt36JO//GzHQQtR4IHM65h3tzgohRpR3n0yosLa+srhXXSxubW9s7dnm3rUQsMWlhwYTsBkgRRjlpaaoZ6UaSoDBgpBNMrlK/c0ekooLf6mlEvBCNOB1SjLSRfLvcF8ZO04ny3Wp0NPPtilNzMsBF4uakAnI0ffujPxA4DgnXmCGleq4TaS9BUlPMyKzUjxWJEJ6gEekZylFIlJdkp8/goVEGcCikeVzDTP2ZSFCo1DQMzGSI9Fj99VLxP68X6+G5l1AexZpwPF80jBnUAqY9wAGVBGs2NQRhSc2tEI+RRFibtkpZCRcpTr+/vEjaxzW3XqvfnFQal3kdRbAPDkAVuOAMNMA1aIIWwOAePIJn8GI9WE/Wq/U2Hy1YeWYP/IL1/gUcGJQI</latexit>s2 (p)<latexit sha1_base64="6kHgBQnrUeagVMhlOM+5l77Os8s=">AAACGHicbVBLSwMxGMz6rPW16tFLsAgepO723VvRi8cK9gHtUrJp2oZmN0uSFcqyP8OLf8WLB0W89ua/Md0uotaBwDAzX/Jl3IBRqSzr01hb39jc2s7sZHf39g8OzaPjtuShwKSFOeOi6yJJGPVJS1HFSDcQBHkuIx13erPwOw9ESMr9ezULiOOhsU9HFCOlpYF5FfWTS3pi7DqRlbcSXK6QuM+GXMl4YOasfL1eq9glmJBqsbwklXIB2mk6B1I0B+a8P+Q49IivMENS9mwrUE6EhKKYkTjbDyUJEJ6iMelp6iOPSCdKdorhuVaGcMSFPr6CifpzIkKelDPP1UkPqYn86y3E/7xeqEY1J6J+ECri4+VDo5BBxeGiJTikgmDFZpogLKjeFeIJEggr3WU2LUGj8v3lVdIu5O1ivnhXyjWu0zoy4BScgQtggypogFvQBC2AwSN4Bq/gzXgyXox342MZXTPSmRPwC8b8CyE+nN8=</latexit>...

<latexit sha1_base64="kRmTAhzFz+wUB0lrer5P6WRftjM=">AAAB+nicbVDLSgMxFM34rPU11aWbYBHqpkxb8bErunFZwT6gHYZMmmlDM8mQZJQy9lPcuFDErV/izr8xMx1ErQcCh3Pu4d4cP2JUacf5tJaWV1bX1gsbxc2t7Z1du7TXUSKWmLSxYEL2fKQIo5y0NdWM9CJJUOgz0vUnV6nfvSNSUcFv9TQibohGnAYUI20kzy4NhLHTdKK8eiU6nnl22ak6GeAiqeWkDHK0PPtjMBQ4DgnXmCGl+jUn0m6CpKaYkVlxECsSITxBI9I3lKOQKDfJTp/BI6MMYSCkeVzDTP2ZSFCo1DT0zWSI9Fj99VLxP68f6+DcTSiPYk04ni8KYga1gGkPcEglwZpNDUFYUnMrxGMkEdamrWJWwkWK0+8vL5JOvVprVBs3J+XmZV5HARyAQ1ABNXAGmuAatEAbYHAPHsEzeLEerCfr1Xqbjy5ZeWYf/IL1/gUdoJQJ</latexit>

s1 (p)<latexit sha1_base64="CT42Enq7LYEs/E2wjD/i+f6Cbbo=">AAAB+nicbVDLSgMxFM34rPU11aWbYBHqpkyt+NgV3bgRKtgHtMOQSTNtaCYZkoxSxn6KGxeKuPVL3Pk3ZqaDqPVA4HDOPdyb40eMKu04n9bC4tLyymphrbi+sbm1bZd22krEEpMWFkzIro8UYZSTlqaakW4kCQp9Rjr++DL1O3dEKir4rZ5ExA3RkNOAYqSN5NmlvjB2mk6Ud12JDqeeXXaqTgY4T2o5KYMcTc/+6A8EjkPCNWZIqV7NibSbIKkpZmRa7MeKRAiP0ZD0DOUoJMpNstOn8MAoAxgIaR7XMFN/JhIUKjUJfTMZIj1Sf71U/M/rxTo4cxPKo1gTjmeLgphBLWDaAxxQSbBmE0MQltTcCvEISYS1aauYlXCe4uT7y/OkfVSt1av1m+Ny4yKvowD2wD6ogBo4BQ1wBZqgBTC4B4/gGbxYD9aT9Wq9zUYXrDyzC37Bev8CRviUJA==</latexit>sM (p)

=s·h·s<latexit sha1_base64="ZPj5fMrO+dHhrqp+1bjs1tGiPLA=">AAACCnicbVDLSgMxFM34rPU16tJNtAiuytSKj4VQdOOygn1AZyiZTKYNzSRDkhHK0LUbf8WNC0Xc+gXu/Bsz00HUeiDkcO49N7nHjxlV2nE+rbn5hcWl5dJKeXVtfWPT3tpuK5FITFpYMCG7PlKEUU5ammpGurEkKPIZ6fijq6zeuSNSUcFv9TgmXoQGnIYUI22kvr3njlAcowvl4kBoOJxerjCebGSqJn274lSdHHCW1ApSAQWaffvDDQROIsI1ZkipXs2JtZciqSlmZFJ2E0VihEdoQHqGchQR5aX5KhN4YJQAhkKawzXM1Z+OFEVKjSPfdEZID9XfWib+V+slOjzzUsrjRBOOpw+FCYNawCwXGFBJsGZjQxCW1PwV4iGSCGuTXjkP4TzDyffKs6R9VK3Vq/Wb40rjsoijBHbBPjgENXAKGuAaNEELYHAPHsEzeLEerCfr1Xqbts5ZhWcH/IL1/gXWV5sk</latexit>Idea 1: Optimize with GD‣ Parameter update hab̄ ! hab̄<latexit sha1_base64="ERh3sIfwNoUmRvC5pshMtRHARp0=">AAACOHicbVDLSgMxFM3UV62vqks3wVJwY5la8bErunFnBfuATil30kwbmnmQZIQyzGe58TPciRsXirj1C8xMh1KtBwLnnptzk3vsgDOpTPPFyC0tr6yu5dcLG5tb2zvF3b2W9ENBaJP43BcdGyTlzKNNxRSnnUBQcG1O2/b4Oum3H6iQzPfu1SSgPReGHnMYAaWlfvF21I/AskFgO7aUj+dKfIwt4MEIsOUIIJEVgFAMuCXZ0IV4Vs974n6xZFbMFHiRVDNSQhka/eKzNfBJ6FJPEQ5SdqtmoHpRMppwGhesUNIAyBiGtKupBy6VvShdPMZlrQyw4wt9PIVTdd4RgSvlxNW7lF1QI/m3l4j/9bqhci56EfOCUFGPTB9yQo51QkmKeMAEJYpPNAEimP4rJiPQKSmddSEN4TLB2WzlRdI6qVRrldrdaal+lcWRRwfoEB2hKjpHdXSDGqiJCHpEr+gdfRhPxpvxaXxNr+aMzLOPfsH4/gH31a3m</latexit>

@
↵
@hab̄

Idea 1: Optimize with GD‣ Parameter update hab̄ ! hab̄<latexit sha1_base64="ERh3sIfwNoUmRvC5pshMtRHARp0=">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</latexit>

@
↵
@hab̄

Remember…
1. Use final hab̄ to compute K

(k)

1
=
log sa(k)h ab̄ s (k)b̄kπ2.Go to a patch (wlog z0 = 1) and pull back ambient
space expression for gab̄ = ∂a∂b̄ K to CY by solving p
for z4, z4 = z4(z1, z2, z3) (wlog) and compute Jab̄
3. Measure deviation from Ricci-flat metricN

J 3(pi)⟨Ω ∧ Ω̄⟩
σ=
1−
wi
3
∑
⟨J ⟩
N⟨Ω ∧ Ω̄⟩ i=1
Ω(pi) ∧ Ω̄(pi)
1

3
J
(pi)
1
3
⟨Ω ∧ Ω̄⟩ =Ω(pi) ∧ Ω̄(pi) , ⟨J ⟩ =
wi
∑
∑
N i
Ω(pi) ∧ Ω̄(pi)i

Idea 1: Optimize with GD‣ Parameter update hab̄ ! hab̄@‣ To computeneed to@hab̄<latexit sha1_base64="ERh3sIfwNoUmRvC5pshMtRHARp0=">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</latexit>

@
↵
@hab̄

<latexit sha1_base64="Ja6WVuzev1RY1wnJsj+xhYVhmjc=">AAACFHicbVDLSgMxFM3UV62vUZdugkUQhDK14mNXdOOygn1Ap5Q7aaYNzWSGJCOUYT7Cjb/ixoUibl2482/MtIOo9UDg5Jx7b3KPF3GmtON8WoWFxaXlleJqaW19Y3PL3t5pqTCWhDZJyEPZ8UBRzgRtaqY57USSQuBx2vbGV5nfvqNSsVDc6klEewEMBfMZAW2kvn3k+hJI4kYgNQPuKjYMIP2+41E/AdcDib007dtlp+JMgedJNSdllKPRtz/cQUjigApNOCjVrTqR7iXZaMJpWnJjRSMgYxjSrqECAqp6yXSpFB8YZYD9UJojNJ6qPzsSCJSaBJ6pDECP1F8vE//zurH2z3sJE1GsqSCzh/yYYx3iLCE8YJISzSeGAJHM/BWTEZiUtMmxNA3hIsPp98rzpHVcqdYqtZuTcv0yj6OI9tA+OkRVdIbq6Bo1UBMRdI8e0TN6sR6sJ+vVepuVFqy8Zxf9gvX+BWVPn8o=</latexit>

1. Compute<latexit sha1_base64="87Nv9W8QkKz6btPK6tqPwzauMVU=">AAACIXicbVDLSgMxFM3UV62vqks3wSK4KlMrWhdCUReCmwr2AZ1S7qSZNjQzE5KMUIb5FTf+ihsXinQn/oyZthS1HrhwOOfe5N7jCs6Utu1PK7O0vLK6ll3PbWxube/kd/caKowkoXUS8lC2XFCUs4DWNdOctoSk4LucNt3hdeo3H6lULAwe9EjQjg/9gHmMgDZSN19xbijXgO8uHU8CiR0BUjPgDg/7zhCEgGSu4UE3BscFid0k6eYLdtGeAC+S0owU0Ay1bn7s9EIS+TTQhINS7ZItdCdOnyacJjknUlQAGUKftg0NwKeqE08uTPCRUXrYC6WpQOOJ+nMiBl+pke+aTh/0QP31UvE/rx1pr9KJWSAiTQMy/ciLONYhTuPCPSYp0XxkCBDJzK6YDMAkpU2ouUkIFynO5icvksZJsVQulu9PC9WrWRxZdIAO0TEqoXNURbeohuqIoCf0gt7Qu/VsvVof1njamrFmM/voF6yvbwgkpNQ=</latexit>

2. Compute<latexit sha1_base64="9i6TT8LugbXw/aLX2bDCTSSF2HA=">AAACIHicbVDLSgMxFM3UV62vqks3wSK4KlMrVhdCUReCmwr2AZ1huJOmbWjmQZIRyjCf4sZfceNCEd3p15hpR1HrgcDhnPvIPW7ImVSm+W7k5uYXFpfyy4WV1bX1jeLmVksGkSC0SQIeiI4LknLm06ZiitNOKCh4Lqdtd3Se+u1bKiQL/Bs1DqntwcBnfUZAackp1qwLyhXgwakVglAMuANWoDvSgfGXljix5YLAbpKVXznFklk2J8CzpJKREsrQcIpvVi8gkUd9RThI2a2YobLjdD7hNClYkaQhkBEMaFdTHzwq7XhyYIL3tNLD/UDo5ys8UX92xOBJOfZcXemBGsq/Xir+53Uj1T+2Y+aHkaI+mS7qRxyrAKdp4R4TlCg+1gSIYPqvmAxBAFE608IkhJMUR98nz5LWQblSLVevD0v1syyOPNpBu2gfVVAN1dElaqAmIugOPaAn9GzcG4/Gi/E6Lc0ZWc82+gXj4xP1jKQ3</latexit>@ log K=@hab̄
g = @a @ b̄ K

3. Go to a patch and pull back<latexit sha1_base64="fkcbjG5wrxhYz3NotzgEqq5rCwM=">AAAB73icbVDLSgNBEOyNrxhfUY9eBoPgKewa8XEL6sFjBGMCyRJmJ73JkNmHM7NCWPITXjwo4tXf8ebfONksosaChqKqm+4uLxZcadv+tAoLi0vLK8XV0tr6xuZWeXvnTkWJZNhkkYhk26MKBQ+xqbkW2I4l0sAT2PJGl1O/9YBS8Si81eMY3YAOQu5zRrWR2t0rFJqSQa9csat2BjJPnJxUIEejV/7o9iOWBBhqJqhSHceOtZtSqTkTOCl1E4UxZSM6wI6hIQ1QuWl274QcGKVP/EiaCjXJ1J8TKQ2UGgee6QyoHqq/3lT8z+sk2j9zUx7GicaQzRb5iSA6ItPnSZ9LZFqMDaFMcnMrYUMqKdMmolIWwvkUJ98vz5O7o6pTq9Zujiv1izyOIuzBPhyCA6dQh2toQBMYCHiEZ3ix7q0n69V6m7UWrHxmF37Bev8ClImPzg==</latexit>

4. Computedet(g) = det( g)<latexit sha1_base64="pOCMGc6ULh8V5p7R3JxrroyRzxw=">AAACBXicbVDJSgNBEO1xjXGLetRDYxCSS5gYcTkIQT14jGAWyAyhp1OZNOlZ6K4RQsjFi7/ixYMiXv0Hb/6Nk8kganxQ1OO9KrrrOaEUGk3z05ibX1hcWs6sZFfX1jc2c1vbDR1EikOdBzJQLYdpkMKHOgqU0AoVMM+R0HQGlxO/eQdKi8C/xWEItsdcX/QEZxhLndyedQUSmdUFLLjF86RPJeoWO7m8WTIT0FlSTkmepKh1ch9WN+CRBz5yybRul80Q7RFTKLiEcdaKNISMD5gL7Zj6zANtj5IrxvQgVrq0F6i4fKSJ+nNjxDyth54TT3oM+/qvNxH/89oR9k7tkfDDCMHn04d6kaQY0EkktCsUcJTDmDCuRPxXyvtMMY5xcNkkhLMJjr9PniWNw1K5UqrcHOWrF2kcGbJL9kmBlMkJqZJrUiN1wsk9eSTP5MV4MJ6MV+NtOjpnpDs75BeM9y+mopeF</latexit>

5. Compute<latexit sha1_base64="BnNlpRcY3x7RexdAkWnvrllFclw=">AAAB9HicbVDLSsNAFL2pr1pfVZduBovgqiRWfOyKunBZwT6gCWUynbRDZ5I4MymU0O9w40IRt36MO//GaRpErQcuHM65l3vv8WPOlLbtT6uwtLyyulZcL21sbm3vlHf3WipKJKFNEvFIdnysKGchbWqmOe3EkmLhc9r2R9czvz2mUrEovNeTmHoCD0IWMIK1kTz3hnKNkavYQOBeuWJX7QxokTg5qUCORq/84fYjkggaasKxUl3HjrWXYqkZ4XRachNFY0xGeEC7hoZYUOWl2dFTdGSUPgoiaSrUKFN/TqRYKDURvukUWA/VX28m/ud1Ex1ceCkL40TTkMwXBQlHOkKzBFCfSUo0nxiCiWTmVkSGWGKiTU6lLITLGc6+X14krZOqU6vW7k4r9as8jiIcwCEcgwPnUIdbaEATCDzAIzzDizW2nqxX623eWrDymX34Bev9C3C3kgY=</latexit>gIdea 1: Optimize with GD‣ Can build in symmetries•

 is real ) h is hermitian<latexit sha1_base64="tPTXRfVwoWvqYKm4sBCAnGCgKPk=">AAAB7XicbVDLSgNBEJyNrxhfUY9eFoPgKWyM+LgFvXiMYB6QLKF3MpuMmZ0ZZmaFsOQfvHhQxKv/482/cXaziBoLGoqqbrq7AsmoNp736RSWlldW14rrpY3Nre2d8u5eW4tYYdLCggnVDUATRjlpGWoY6UpFIAoY6QST69TvPBClqeB3ZiqJH8GI05BiMFZq9ycgJQzKFa/qZXAXSS0nFZSjOSh/9IcCxxHhBjPQulfzpPETUIZiRmalfqyJBDyBEelZyiEi2k+ya2fukVWGbiiULW7cTP05kUCk9TQKbGcEZqz/eqn4n9eLTXjhJ5TL2BCO54vCmLlGuOnr7pAqgg2bWgJYUXuri8egABsbUCkL4TLF2ffLi6R9Uq3Vq/Xb00rjKo+jiA7QITpGNXSOGugGNVELYXSPHtEzenGE8+S8Om/z1oKTz+yjX3DevwCu9I9W</latexit><latexit sha1_base64="FUIxtivFMN5Jpm6QFJ9RPJxb79c=">AAAB8nicbVBNS8NAEN3Ur1q/qh69LBbBU0ms+HErevFYxdpCGspmu2mXbnbD7kQpoT/DiwdFvPprvPlvTNIgan0w8Hhvhpl5fiS4Adv+tEoLi0vLK+XVytr6xuZWdXvnzqhYU9amSijd9YlhgkvWBg6CdSPNSOgL1vHHl5nfuWfacCVvYRIxLyRDyQNOCaSS27vhwxEQrdVDv1qz63YOPE+cgtRQgVa/+tEbKBqHTAIVxBjXsSPwEqKBU8GmlV5sWETomAyZm1JJQma8JD95ig9SZYADpdOSgHP150RCQmMmoZ92hgRG5q+Xif95bgzBmZdwGcXAJJ0tCmKBQeHsfzzgmlEQk5QQqnl6K6YjogmFNKVKHsJ5hpPvl+fJ3VHdadQb18e15kURRxntoX10iBx0iproCrVQG1Gk0CN6Ri8WWE/Wq/U2ay1Zxcwu+gXr/Qup45Gk</latexit><latexit sha1_base64="TVgB+ZbllFTWp7MpQSAN1Adyj4U=">AAACE3icbVBLSwMxGMzWV62vVY9egkUQkWXbSru9Fb14rGAfsF1KNs22odkHSVYoy/4HL/4VLx4U8erFm//GdLuIWgcCw8x8yZdxI0aFNM1PrbCyura+UdwsbW3v7O7p+wddEcYckw4OWcj7LhKE0YB0JJWM9CNOkO8y0nOnV3O/d0e4oGFwK2cRcXw0DqhHMZJKGupnySC7xOZj10lMw8xwvkTSSTrUy6bRbDYssw4z0qw1FsSqWrCSB8sgR3uofwxGIY59EkjMkBB2xYykkyAuKWYkLQ1iQSKEp2hMbEUD5BPhJNk6KTxRygh6IVcnkDBTf04kyBdi5rsq6SM5EX+9ufifZ8fSs5yEBlEsSYAXD3kxgzKE84LgiHKCJZspgjCnaleIJ4gjLFWNpbwEhfr3l5dJt2pUakbt5qLcuszrKIIjcAxOQQU0QAtcgzboAAzuwSN4Bi/ag/akvWpvi2hBy2cOwS9o718GJ5qc</latexit>•

Scaling hab̄ ! c hab̄ )  ! c  ) log  ! c0 log <latexit sha1_base64="FUIxtivFMN5Jpm6QFJ9RPJxb79c=">AAAB8nicbVBNS8NAEN3Ur1q/qh69LBbBU0ms+HErevFYxdpCGspmu2mXbnbD7kQpoT/DiwdFvPprvPlvTNIgan0w8Hhvhpl5fiS4Adv+tEoLi0vLK+XVytr6xuZWdXvnzqhYU9amSijd9YlhgkvWBg6CdSPNSOgL1vHHl5nfuWfacCVvYRIxLyRDyQNOCaSS27vhwxEQrdVDv1qz63YOPE+cgtRQgVa/+tEbKBqHTAIVxBjXsSPwEqKBU8GmlV5sWETomAyZm1JJQma8JD95ig9SZYADpdOSgHP150RCQmMmoZ92hgRG5q+Xif95bgzBmZdwGcXAJJ0tCmKBQeHsfzzgmlEQk5QQqnl6K6YjogmFNKVKHsJ5hpPvl+fJ3VHdadQb18e15kURRxntoX10iBx0iproCrVQG1Gk0CN6Ri8WWE/Wq/U2ay1Zxcwu+gXr/Qup45Gk</latexit><latexit sha1_base64="v8DG3IqeJghMuM2vI6RxumwiD7g=">AAACBHicbVDLSsNAFJ3UV62vqMtuBovgqiRWfOCm6MZlBfuAJpSb6bQdOnkwM1FK6MKNv+LGhSJu/Qh3/o2TNIhaD1w4nHMv997jRZxJZVmfRmFhcWl5pbhaWlvf2Nwyt3daMowFoU0S8lB0PJCUs4A2FVOcdiJBwfc4bXvjy9Rv31IhWRjcqElEXR+GARswAkpLPbPsjCGKwBFsOFIgRHiHiXM+E3tmxapaGfA8sXNSQTkaPfPD6Yck9mmgCAcpu7YVKTcBoRjhdFpyYkkjIGMY0q6mAfhUukn2xBTva6WPB6HQFSicqT8nEvClnPie7vRBjeRfLxX/87qxGpy6CQuiWNGAzBYNYo5ViNNEcJ8JShSfaAJEMH0rJiMQQJTOrZSFcJbi+PvledI6rNq1au36qFK/yOMoojLaQwfIRieojq5QAzURQffoET2jF+PBeDJejbdZa8HIZ3bRLxjvXxSSmIE=</latexit><latexit sha1_base64="FUIxtivFMN5Jpm6QFJ9RPJxb79c=">AAAB8nicbVBNS8NAEN3Ur1q/qh69LBbBU0ms+HErevFYxdpCGspmu2mXbnbD7kQpoT/DiwdFvPprvPlvTNIgan0w8Hhvhpl5fiS4Adv+tEoLi0vLK+XVytr6xuZWdXvnzqhYU9amSijd9YlhgkvWBg6CdSPNSOgL1vHHl5nfuWfacCVvYRIxLyRDyQNOCaSS27vhwxEQrdVDv1qz63YOPE+cgtRQgVa/+tEbKBqHTAIVxBjXsSPwEqKBU8GmlV5sWETomAyZm1JJQma8JD95ig9SZYADpdOSgHP150RCQmMmoZ92hgRG5q+Xif95bgzBmZdwGcXAJJ0tCmKBQeHsfzzgmlEQk5QQqnl6K6YjogmFNKVKHsJ5hpPvl+fJ3VHdadQb18e15kURRxntoX10iBx0iproCrVQG1Gk0CN6Ri8WWE/Wq/U2ay1Zxcwu+gXr/Qup45Gk</latexit><latexit sha1_base64="sZmvE6hzcV7GpR0qSGKKOOPZB+0=">AAACEXicbVDLSsNAFJ3UV62vqEs3g0XoqiQqPnBTdOOygn1AU8rNdNIMnTyYmSgl5Bfc+CtuXCji1p07/8ZJW6RaD1w4nHMv997jxpxJZVlfRmFhcWl5pbhaWlvf2Nwyt3eaMkoEoQ0S8Ui0XZCUs5A2FFOctmNBIXA5bbnDq9xv3VEhWRTeqlFMuwEMQuYxAkpLPbPi91JwXBCpm2WOYANfgRDRPSbOBZ71embZqlpj4HliT0kZTVHvmZ9OPyJJQENFOEjZsa1YdVMQihFOs5KTSBoDGcKAdjQNIaCym44/yvCBVvrYi4SuUOGxOjuRQiDlKHB1ZwDKl3+9XPzP6yTKO+umLIwTRUMyWeQlHKsI5/HgPhOUKD7SBIhg+lZMfBBAlA6xNA7hPMfJz8vzpHlYtY+qRzfH5drlNI4i2kP7qIJsdIpq6BrVUQMR9ICe0At6NR6NZ+PNeJ+0FozpzC76BePjG9MJnmc=</latexit><latexit sha1_base64="sMpRsVWl7NmxmGpDiFbP0rhzdvw=">AAACC3icbVDLSsNAFJ34rPUVdelmaBFdlcSKj13RjcsK9gFNKTfTaTt0kgkzE6WE7t34K25cKOLWH3Dn3zhpg6964MLhnHu59x4/4kxpx/mw5uYXFpeWcyv51bX1jU17a7uuRCwJrRHBhWz6oChnIa1ppjltRpJC4HPa8IcXqd+4oVIxEV7rUUTbAfRD1mMEtJE6dsHjou8NIYrAk6w/0CCluMVk/1vv2EWn5EyAZ4mbkSLKUO3Y715XkDigoSYclGq5TqTbCUjNCKfjvBcrGgEZQp+2DA0hoKqdTH4Z4z2jdHFPSFOhxhP150QCgVKjwDedAeiB+uul4n9eK9a903bCwijWNCTTRb2YYy1wGgzuMkmJ5iNDgEhmbsVkABKINvHlJyGcpTj+enmW1A9LbrlUvjoqVs6zOHJoFxXQAXLRCaqgS1RFNUTQHXpAT+jZurcerRfrddo6Z2UzO+gXrLdPqfmbkw==</latexit>

corresponds to trivial Kähler transformation K → K + const
Can be accounted for by using weight normalization‣ Works, but σ(h) is too complicated to backpropagate

efficiently‣ Just doing numerical integration + fix point iteration isfasterIdea 2: Perform
integration and use
regression for ψ

Idea 2: Regression NN‣ Use supervised ML (regression)•Compute h using Donaldson’s algorithm for several complex structures ψ•

Use NN to learn interpolating h‣ How bad is the error without regression?
<latexit sha1_base64="4DENYJfOEPzF8MR8kv7f3bu8SlI=">AAAB83icbVDLSsNAFJ3UV62vqks3g0VwVRIrPnZFXbisYB/QhHIznbRDZ5IwMxFK6G+4caGIW3/GnX/jJA2i1gMXDufcy733+DFnStv2p1VaWl5ZXSuvVzY2t7Z3qrt7HRUlktA2iXgkez4oyllI25ppTnuxpCB8Trv+5Drzuw9UKhaF93oaU0/AKGQBI6CN5Lo3lGtwFRsJGFRrdt3OgReJU5AaKtAaVD/cYUQSQUNNOCjVd+xYeylIzQins4qbKBoDmcCI9g0NQVDlpfnNM3xklCEOImkq1DhXf06kIJSaCt90CtBj9dfLxP+8fqKDCy9lYZxoGpL5oiDhWEc4CwAPmaRE86khQCQzt2IyBglEm5gqeQiXGc6+X14knZO606g37k5rzasijjI6QIfoGDnoHDXRLWqhNiIoRo/oGb1YifVkvVpv89aSVczso1+w3r8AGDeR3A==</latexit>

Comments• Symmetry:<latexit sha1_base64="QmLE+z/k4Y4gth3pegqHHNJG1ds=">AAACIXicbVDLSgMxFM3UV62vqks3wSK4aZmxonVXdOPCRQX7gE4ZMmmmDc08SO4otfRX3Pgrblwo0p34M2amRar1QODknHNJ7nEjwRWY5qeRWVpeWV3Lruc2Nre2d/K7ew0VxpKyOg1FKFsuUUzwgNWBg2CtSDLiu4I13cFV4jfvmVQ8DO5gGLGOT3oB9zgloCUnX7EjxW3Je30gUoYPuJgI2L5hHsypj858qKivTr5glswUeJFYM1JAM9Sc/MTuhjT2WQBUEKXalhlBZ0QkcCrYOGfHikWEDkiPtTUNiM9UZ5RuOMZHWuliL5T6BIBTdX5iRHylhr6rkz6BvvrrJeJ/XjsGr9IZ8SCKgQV0+pAXCwwhTurCXS4ZBTHUhFDJ9V8x7RNJKOhSc2kJFwnOflZeJI2TklUulW9PC9XLWR1ZdIAO0TGy0DmqomtUQ3VE0RN6QW/o3Xg2Xo0PYzKNZozZzD76BePrG3+ipRo=</latexit>!

• Absolute error, zi !

zi

gets worse
for higher k ⇒ large relative
error<latexit sha1_base64="FTZh40z7Ro3/X+wahJF7Sa+2YGA=">AAAB63icbVDLSsNAFL2pr1pfVZduBovgqiQqPnZFNy4r2Ae0oUymk3bozCTMTIQS+gtuXCji1h9y5984SYOo9cCFwzn3cu89QcyZNq776ZSWlldW18rrlY3Nre2d6u5eW0eJIrRFIh6pboA15UzSlmGG026sKBYBp51gcpP5nQeqNIvkvZnG1Bd4JFnICDaZ1I81G1Rrbt3NgRaJV5AaFGgOqh/9YUQSQaUhHGvd89zY+ClWhhFOZ5V+ommMyQSPaM9SiQXVfprfOkNHVhmiMFK2pEG5+nMixULrqQhsp8BmrP96mfif10tMeOmnTMaJoZLMF4UJRyZC2eNoyBQlhk8twUQxeysiY6wwMTaeSh7CVYbz75cXSfuk7p3WT+/Oao3rIo4yHMAhHIMHF9CAW2hCCwiM4RGe4cURzpPz6rzNW0tOMbMPv+C8fwE9B46B</latexit>

k=2

k=4

<latexit sha1_base64="4DENYJfOEPzF8MR8kv7f3bu8SlI=">AAAB83icbVDLSsNAFJ3UV62vqks3g0VwVRIrPnZFXbisYB/QhHIznbRDZ5IwMxFK6G+4caGIW3/GnX/jJA2i1gMXDufcy733+DFnStv2p1VaWl5ZXSuvVzY2t7Z3qrt7HRUlktA2iXgkez4oyllI25ppTnuxpCB8Trv+5Drzuw9UKhaF93oaU0/AKGQBI6CN5Lo3lGtwFRsJGFRrdt3OgReJU5AaKtAaVD/cYUQSQUNNOCjVd+xYeylIzQins4qbKBoDmcCI9g0NQVDlpfnNM3xklCEOImkq1DhXf06kIJSaCt90CtBj9dfLxP+8fqKDCy9lYZxoGpL5oiDhWEc4CwAPmaRE86khQCQzt2IyBglEm5gqeQiXGc6+X14knZO606g37k5rzasijjI6QIfoGDnoHDXRLWqhNiIoRo/oGb1YifVkvVpv89aSVczso1+w3r8AGDeR3A==</latexit>Idea 2: Regression NN‣

k = 2 : h is a 15 × 15 complex matrix ⇒ 450 real dofs
(note that half are related since h is hermitian)‣ We do not build in this symmetry but “hermitianize”

1
the output h → (h + h †) to average out fluctuations
2‣ Engineer features: h depends (implicitly) on complex

structure (since the points P inserted in s and s̄ do)‣ More explicitly, it depends on ψ, | ψ | , ψ 1/5 and theirconjugates‣ Provide the real and imaginary parts as their inputs

Idea 2: Regression NN
Re(Ψ)

Comments

Im(Ψ)• Train for 100 epochs

Re(Ψ2 )

• Dropout layer

regularization at lastlayer

Im(Ψ2 )
Re(Ψ1/5 )Im(Ψ1/5 )•650, Tanh
200, ReLU

450500, ReLU

L1 lossIdea 2: Regression NN<latexit sha1_base64="4DENYJfOEPzF8MR8kv7f3bu8SlI=">AAAB83icbVDLSsNAFJ3UV62vqks3g0VwVRIrPnZFXbisYB/QhHIznbRDZ5IwMxFK6G+4caGIW3/GnX/jJA2i1gMXDufcy733+DFnStv2p1VaWl5ZXSuvVzY2t7Z3qrt7HRUlktA2iXgkez4oyllI25ppTnuxpCB8Trv+5Drzuw9UKhaF93oaU0/AKGQBI6CN5Lo3lGtwFRsJGFRrdt3OgReJU5AaKtAaVD/cYUQSQUNNOCjVd+xYeylIzQins4qbKBoDmcCI9g0NQVDlpfnNM3xklCEOImkq1DhXf06kIJSaCt90CtBj9dfLxP+8fqKDCy9lYZxoGpL5oiDhWEc4CwAPmaRE86khQCQzt2IyBglEm5gqeQiXGc6+X14knZO606g37k5rzasijjI6QIfoGDnoHDXRLWqhNiIoRo/oGb1YifVkvVpv89aSVczso1+w3r8AGDeR3A==</latexit><latexit sha1_base64="4DENYJfOEPzF8MR8kv7f3bu8SlI=">AAAB83icbVDLSsNAFJ3UV62vqks3g0VwVRIrPnZFXbisYB/QhHIznbRDZ5IwMxFK6G+4caGIW3/GnX/jJA2i1gMXDufcy733+DFnStv2p1VaWl5ZXSuvVzY2t7Z3qrt7HRUlktA2iXgkez4oyllI25ppTnuxpCB8Trv+5Drzuw9UKhaF93oaU0/AKGQBI6CN5Lo3lGtwFRsJGFRrdt3OgReJU5AaKtAaVD/cYUQSQUNNOCjVd+xYeylIzQins4qbKBoDmcCI9g0NQVDlpfnNM3xklCEOImkq1DhXf06kIJSaCt90CtBj9dfLxP+8fqKDCy9lYZxoGpL5oiDhWEc4CwAPmaRE86khQCQzt2IyBglEm5gqeQiXGc6+X14knZO606g37k5rzasijjI6QIfoGDnoHDXRLWqhNiIoRo/oGb1YifVkvVpv89aSVczso1+w3r8AGDeR3A==</latexit><latexit sha1_base64="FTZh40z7Ro3/X+wahJF7Sa+2YGA=">AAAB63icbVDLSsNAFL2pr1pfVZduBovgqiQqPnZFNy4r2Ae0oUymk3bozCTMTIQS+gtuXCji1h9y5984SYOo9cCFwzn3cu89QcyZNq776ZSWlldW18rrlY3Nre2d6u5eW0eJIrRFIh6pboA15UzSlmGG026sKBYBp51gcpP5nQeqNIvkvZnG1Bd4JFnICDaZ1I81G1Rrbt3NgRaJV5AaFGgOqh/9YUQSQaUhHGvd89zY+ClWhhFOZ5V+ommMyQSPaM9SiQXVfprfOkNHVhmiMFK2pEG5+nMixULrqQhsp8BmrP96mfif10tMeOmnTMaJoZLMF4UJRyZC2eNoyBQlhk8twUQxeysiY6wwMTaeSh7CVYbz75cXSfuk7p3WT+/Oao3rIo4yHMAhHIMHF9CAW2hCCwiM4RGe4cURzpPz6rzNW0tOMbMPv+C8fwE9B46B</latexit>‣ Very small training set and time needed‣ Learns functional dependence of metric on CS‣ Reflects the symmetry ψ → − ψ in the problem (yellow region)‣ Can extrapolate somewhat (red region, 1 order of magnitude)‣ But size of k (and hence output layer grows with (k!)2<latexit sha1_base64="FTZh40z7Ro3/X+wahJF7Sa+2YGA=">AAAB63icbVDLSsNAFL2pr1pfVZduBovgqiQqPnZFNy4r2Ae0oUymk3bozCTMTIQS+gtuXCji1h9y5984SYOo9cCFwzn3cu89QcyZNq776ZSWlldW18rrlY3Nre2d6u5eW0eJIrRFIh6pboA15UzSlmGG026sKBYBp51gcpP5nQeqNIvkvZnG1Bd4JFnICDaZ1I81G1Rrbt3NgRaJV5AaFGgOqh/9YUQSQaUhHGvd89zY+ClWhhFOZ5V+ommMyQSPaM9SiQXVfprfOkNHVhmiMFK2pEG5+nMixULrqQhsp8BmrP96mfif10tMeOmnTMaJoZLMF4UJRyZC2eNoyBQlhk8twUQxeysiY6wwMTaeSh7CVYbz75cXSfuk7p3WT+/Oao3rIo4yHMAhHIMHF9CAW2hCCwiM4RGe4cURzpPz6rzNW0tOMbMPv+C8fwE9B46B</latexit>Idea 2: Regression NN
0.03

h1,1(ψ)1.5

h1,2(ψ)

0.02
0.01

1.0

20

40

60

80

100

80

100

-0.01
0.5

-0.02
-0.03
20

40

60

80

100

0.03

h2,2(ψ)

2.5

2.0

-0.04

h2,1(ψ)

0.02
0.01

1.5

1.0

20
-0.01

0.5

-0.02
20

40

60

80

100

40

60

Idea 2: Regression NNh(ψ) =

Idea 3: Learn g directly
by boosting Fubini Study
metricIdea 3: Learn g directly from FS‣ Note: gab̄ is always a hermitian 3 × 3 matrix‣ Two approaches••

Learn gab̄ from scratch
Use Fubini Study as starting point and find CT metric asFS=g+ δga,b̄ (works much better)correction (boosting): ga,CYb̄
a,b̄‣ Good behavior at overlaps built in the previous appraoch, need tomake sure that works here

Can find flat
metric here

Can find flat
metric here

U1
<latexit sha1_base64="ug4OtSrnhm6CKoWRZ8trjXlNJvM=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0ms+HErevFY0bSFNpTNdtMu3WzC7kYooT/BiwdFvPqLvPlv3KRB1Ppg4PHeDDPz/JgzpW370yotLa+srpXXKxubW9s71d29tooSSahLIh7Jro8V5UxQVzPNaTeWFIc+px1/cp35nQcqFYvEvZ7G1AvxSLCAEayNdOcOnEG1ZtftHGiROAWpQYHWoPrRH0YkCanQhGOleo4day/FUjPC6azSTxSNMZngEe0ZKnBIlZfmp87QkVGGKIikKaFRrv6cSHGo1DT0TWeI9Vj99TLxP6+X6ODCS5mIE00FmS8KEo50hLK/0ZBJSjSfGoKJZOZWRMZYYqJNOpU8hMsMZ98vL5L2Sd1p1Bu3p7XmVRFHGQ7gEI7BgXNowg20wAUCI3iEZ3ixuPVkvVpv89aSVczswy9Y71/vF420</latexit>

U2<latexit sha1_base64="A8qAMOAx2nS1XVSKOZ5XTI55D3U=">AAAB6nicbVBNS8NAEJ34WetX1aOXxSJ4KkkrftyKXjxWNG2hDWWz3bRLN5uwuxFK6E/w4kERr/4ib/4bN2kQtT4YeLw3w8w8P+ZMadv+tJaWV1bX1ksb5c2t7Z3dyt5+W0WJJNQlEY9k18eKciaoq5nmtBtLikOf044/uc78zgOVikXiXk9j6oV4JFjACNZGunMH9UGlatfsHGiROAWpQoHWoPLRH0YkCanQhGOleo4day/FUjPC6azcTxSNMZngEe0ZKnBIlZfmp87QsVGGKIikKaFRrv6cSHGo1DT0TWeI9Vj99TLxP6+X6ODCS5mIE00FmS8KEo50hLK/0ZBJSjSfGoKJZOZWRMZYYqJNOuU8hMsMZ98vL5J2veY0ao3b02rzqoijBIdwBCfgwDk04QZa4AKBETzCM7xY3HqyXq23eeuSVcwcwC9Y71/wm421</latexit>

Does it glue to a global property here?

Idea 3: Learn g directly from FS‣ Define two losses

1. Ricci flat: L1 =2. Gluing:‣ Input:

L2 =J3
Ω ∧ Ω̄
∑overlapT− 1 (with J computed from ga,FSb̄ + δga,b̄ )ab(1) c̄d¯
gbc̄ T − g (2)¯ad[z1, z2, z3, z4, ψ] (in patch z0 ≠ 0)

‣ Output: δgab̄

Idea 3: Learn g directly from FS
L1: Ricci flatness condition
Re(Ψ)

Re(δg11 )Im(Ψ)

Re(δg12 )Re(z1 )Im(δg12 )Induced Fubini-Study
NN metricIm(z1 )
..
.

..
.

..
.
Re(z4 )..
.

Re(δg33 )Im(z4 )

10

9100, ReLU

100, ReLU

100, ReLU

Comments• Train w/ 50’000 points for 1000 epochs•

L2 weight normalization

L2: Error in overlap condition
Induced Fubini-Study
NN metric

Conclusions‣ Metric important to compute• Yukawa couplings and hence Standard Model masses• Massive string spectrum• Bundle stability‣ Numerical algorithm constrained at fixed point in moduli space(unclear a priori which value to take)‣ Use ML• To replace FP iteration (slow)• As a regressor to interpolate between different points in moduli
space• To directly learn the metric (subtleties on overlaps)

Thank you