Quantum Mirror Map for Del Pezzo Geometries Yuji Sugimoto (Univ. - PowerPoint PPT Presentation

Quantum Mirror Map for Del Pezzo Geometries Yuji Sugimoto (Univ. of Science and Technology of China) with Sanefumi Moriyama Tomohiro Furukawa (Osaka City Univ.) arXiv:190?.?????

Susy. Gauge Theory encoded in algebraic curve Mirror Map redefining the variables well studied not well studied A-period B-period Free energy relating to BPS indices <latexit sha1_base64="v525ikpCJY0awz2BguTKZ9/oYU=">ACg3ichVHLSsNAFD3GV62vqhtBkGBRlEKY1IiCD42LujKlgJSRx1MC+StGCLO1f+gAtXCiLqUv/AjT/gwk8QlwpuXHibBkRFvSFz5y586ZGcOzRBAy9tgNTY1t7Qm2pLtHZ1d3ame3tXALfkmL5iu5frh5wSzi8EIrQ4uez3XbsPiasTdfW18rcz8QrMS7nt809Z3HLEtTD0kSksNFvNCmx2tjBUDYcumpsoVOUM5K1cyiqJoqTRTWBTyT6DGI048m7qAkVswYWJEmxwOAgJW9AR0LcBFQwecZuoEucTEtE6xwGSpC1RFacKndg9GndothGzDs1rPYNIbdIuFv0+KWUMswd2yV7YPbtmT+z917VqEfNyz5lo67lntZ91L/89q/Kphxi91P1p+cQ25iMvAry7kVM7RmXV+uHL8sTy0NV0fYGXsm/6fskd3RCZzyq3m+yJdOkKQHUL9f90+wmlXUcSW7mEvPzMVPkcAhjBK9z2BGSwgjwLte4gr3OBWapYyUlbK1UulhljThy8hTX8AMJ6SWA=</latexit> Π A ( z ) ∼ c 1 z + c 2 z + ...

Susy. Gauge Theory Mirror Map redefining the variables not well studied <latexit sha1_base64="v525ikpCJY0awz2BguTKZ9/oYU=">ACg3ichVHLSsNAFD3GV62vqhtBkGBRlEKY1IiCD42LujKlgJSRx1MC+StGCLO1f+gAtXCiLqUv/AjT/gwk8QlwpuXHibBkRFvSFz5y586ZGcOzRBAy9tgNTY1t7Qm2pLtHZ1d3ame3tXALfkmL5iu5frh5wSzi8EIrQ4uez3XbsPiasTdfW18rcz8QrMS7nt809Z3HLEtTD0kSksNFvNCmx2tjBUDYcumpsoVOUM5K1cyiqJoqTRTWBTyT6DGI048m7qAkVswYWJEmxwOAgJW9AR0LcBFQwecZuoEucTEtE6xwGSpC1RFacKndg9GndothGzDs1rPYNIbdIuFv0+KWUMswd2yV7YPbtmT+z917VqEfNyz5lo67lntZ91L/89q/Kphxi91P1p+cQ25iMvAry7kVM7RmXV+uHL8sTy0NV0fYGXsm/6fskd3RCZzyq3m+yJdOkKQHUL9f90+wmlXUcSW7mEvPzMVPkcAhjBK9z2BGSwgjwLte4gr3OBWapYyUlbK1UulhljThy8hTX8AMJ6SWA=</latexit> We need this to obtain physical quntities and reach important results. encoded in algebraic curve A-period Π A ( z ) ∼ c 1 z + c 2 z + ...

ABJM theory (U(N) k x U(N) -k ) <latexit sha1_base64="AJRuSImHKoCnLtYIHOG/XNRLmvY=">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</latexit> <latexit sha1_base64="9xBWkbLZxOgSEDpiWJNTaNKejU=">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</latexit> <latexit sha1_base64="I0eCl6BtDUOanoMPdONcNquLP8s=">ACcXichVFNSwJBGH7cvsw+tLoUXSQpikJGC4pOVpeOaqmBluxuUw6u8vuKpT0B/oDBZ0UIqKf0aU/0MGfEB0NunTodV2IkuodZuaZ97nWdmFMTtsNYyf19Q8MDvmHAyOjY+PB0MRk1jaqlsozqEZ1oEi21wTOs84wtH4gWlxuaJoPKeUdzr7uRq3bGHo+86ZyQ8r8qkuToQqO0QdFZKiuLV4vlIoKbK1VAxFWJS5Ee4FMQ9E4EXSCN2hgGMYUFBRw6HMIaZNjU8oiBwSTuEHXiLELC3e4QIC0VcrilCETW6bxlFZ5j9Vp3alpu2qVTtGoW6QMY549s3vWZk/sgb2wj19r1d0aHS9nNCtdLTeLwcvpvfd/VRWaHZS+VH96dnCDderIO+my3RuoXb1tfOr9t5mer6+wJrslfw3WIs90g302pt6m+LpGwToA2I/n7sXZOPR2Go0nlqLJLa9r/BjFnNYpPdeRwK7SCJD51q4RgNX1uakcLSXDdV8nmaKXwLafkTVw2Oqw=</latexit> <latexit sha1_base64="xOCzbF3lZnTWlcaxC8VOtgDKRQ8=">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</latexit> (see also Moriyama, Kubo, Furukawa’s slides&Poster) [Aharony-Bergman-Jafferis-Maldacena 2008] Quantum Mirror Map <latexit sha1_base64="+y9bkQKOnHuXTbvdKZzeuXW7oL8=">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</latexit> encoded in local P 1 x P 1 with quantization ⇣ Q − 1 / 2 ⌘ ⇣ P − 1 / 2 ⌘ Q b b P = q b P b b Q 1 / 2 + b b P 1 / 2 + b b H = Q Π A ( z ) Π A ( z, ~ ) q = e i ~ � �

U(N) k x U(N) -k x U(N) k x U(N) -k ABJM theory (U(N) k x U(N) -k ) <latexit sha1_base64="9xBWkbLZxOgSEDpiWJNTaNKejU=">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</latexit> [Aharony-Bergman-Jafferis-Maldacena 2008] <latexit sha1_base64="AJRuSImHKoCnLtYIHOG/XNRLmvY=">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</latexit> (1,1,1,1) model encoded in local P 1 x P 1 with quantization (see also Moriyama, Kubo, Furukawa’s slides&Poster) [Honda, Moriyama, 2014] <latexit sha1_base64="/h01glb193CpEtEnr7M8lsf9ZLU=">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</latexit> generalization ⇣ Q − 1 / 2 ⌘ ⇣ P − 1 / 2 ⌘ Q b b P = q b P b b Q 1 / 2 + b b P 1 / 2 + b b H = Q h⇣ Q 1 / 2 + Q − 1 / 2 ⌘ ⇣ P 1 / 2 + P − 1 / 2 ⌘i 2 b H =