Geometric approach for 3d interfaces at strong coupling Based on: 2005.05983 with J.J. Heckman, T. B. Rochais, E. Torres String Pheno Seminar - July 7, 2020 Markus Dierigl
<latexit sha1_base64="1wtfwR5PwcVzZNWfZ2Sq0H1BsWY=">ACEHicbVA9SwNBEN2LXzF+RS1tFoOoIOEuCGonWKhgEdGomAthbzMxS3bvjt05MRwB/4CNf8XGQhFbSzv/jZuPQo0PBh7vzTAzL4ilMOi6X05mbHxicio7nZuZnZtfyC8uXZgo0RwqPJKRvgqYASlCqKBACVexBqYCZdB+6DnX96CNiIKz7ETQ02xm1A0BWdopXp+3T9kSjHqmyQwgNRHuMP07KS7UdqivmLYCoL0urtZzxfcotsHSXekBTIEOV6/tNvRDxRECKXzJiq58ZYS5lGwSV0c35iIGa8zW6gamnIFJha2n+oS9es0qDNSNsKkfbVnxMpU8Z0VGA7eyeav15P/M+rJtjcraUijBOEkA8WNRNJMaK9dGhDaOAoO5YwroW9lfIW04yjzTBnQ/D+vjxKLkpFb7u4d7pd2D+H8SRJStklWwQj+yQfXJEyqRCOHkgT+SFvDqPzrPz5rwPWjPOMJl8gvOxze8BpzY</latexit> Outline • Non-dynamical example: Topological insulator • The fundamental domain of Maxwell theory and its T-invariant subspace • Generalization to other duality groups Γ ⊂ SL(2 , Z ) • A six-dimensional interpretation • Other construction of interfaces in four dimensions • Conclusions and Outlook
<latexit sha1_base64="6onpVS6bq4P7a0AxN+mSK0pra8Q=">AB6HicbVA9SwNBEJ2LXzF+RS1tFoNgFe5EULuIjWCTgPmA5Ah7m7lkzd7esbsnhCNgb2OhiK0/yc5/4+aj0MQHA4/3ZpiZFySCa+O6305uZXVtfSO/Wdja3tndK+4fNHScKoZ1FotYtQKqUXCJdcONwFaikEaBwGYwvJn4zUdUmsfy3owS9CPalzkjBor1a67xZJbdqcgy8SbkxLMUe0Wvzq9mKURSsME1brtuYnxM6oMZwLHhU6qMaFsSPvYtlTSCLWfTQ8dkxOr9EgYK1vSkKn6eyKjkdajKLCdETUDvehNxP+8dmrCSz/jMkNSjZbFKaCmJhMviY9rpAZMbKEMsXtrYQNqKLM2GwKNgRv8eVl0jgre+flq9p5qXL3NIsjD0dwDKfgwQVU4BaqUAcGCM/wCm/Og/PivDsfs9acM4/wEP7A+fwBv3KNWg=</latexit> <latexit sha1_base64="wCIdgcR682YCgE0aL8WB61f6Xxc=">ACIXicbVDLSgMxFM34rPVdekmWAQXUmakYN0V3AhuKtgqNKVk0lsbzGSG5I5YhoJf4sZfceNCke7EnzF9LT1QOBwzslN7gkTJS36/pe3sLi0vLKaW8uvb2xubRd2dhs2To2AuohVbG5DbkFJDXWUqOA2McCjUMFNeH8+8m8ewFgZ62vsJ9CK+J2WXSk4OqldqDsAXLKpKYs848pSyQbUAbaJlwAZQiPmEVx5d2Mgq1C0W/5I9B50kwJUyRa1dGLJOLNINArFrW0GfoKtjBuUQsEgz1ILbvw9v4Omo5pHYFvZeMBPXRKh3Zj45GOlZ/38h4ZG0/Cl0y4tizs95I/M9rptitDKpkxRBi8lD3VRjOmoLtqRBgSqviNcGOn+SkWPGy7QlZp3JQSzK8+TxkpKJfOrsrF6uXTpI4c2ScH5IgE5JRUyQWpkToR5Jm8knfy4b14b96nN5xEF7xphXvkD7zvH3YGo4w=</latexit> <latexit sha1_base64="Kr4epEKbNWoHvKdAs7kyl3An5cw=">ACnicbZBLSwMxFIUzPmt9jbp0Ey2CqzJTCupOEIrgRsHaQqeWTHqnDc08SO4oZSi4c+NfceNCEbf+Anf+G9PHQlsPBA7n3JDcz0+k0Og439bc/MLi0nJuJb+6tr6xaW9t3+g4VRyqPJaxqvtMgxQRVFGghHqigIW+hJrfOxv2tTtQWsTRNfYTaIasE4lAcIYmatl7XqAYzsArJBdky9RNyWBrRCvXtod4BWnbBKToj0VnjTkyBTHTZsr+8dszTECLkmndcJ0EmxlTKLiEQd5LNSM91gHGsZGLATdzEarDOiBSdo0iJU5EdJR+vtGxkKt+6FvJkOGXT3dDcP/ukaKwXEzE1GSIkR8/FCQSoxHXKhbaGAo+wbw7gS5q+Ud5lhg4Ze3kBwp1eNTelolsunlyVC6cXD2McObJL9skhckROSXn5JUCSeP5Jm8kjfryXqx3q2P8eicNUG4Q/7I+vwBIN2aVQ=</latexit> Topological insulator [Kane, Mele ’05,’06], [Bernevig, Hughes, Zhang ’06], [Moore, Balents, ’06], … System with unbroken global U(1) and time-reversal symmetry. Couple U(1) to background gauge field with local term A θ 8 π 2 F ∧ F Breaks time-reversal invariance unless θ ∈ { 0 , π } mod 2 π trivial phase topological insulator phase
<latexit sha1_base64="FUCe/0udYATUgB6qdpD2vPS0ai8=">ACA3icbZBLSwMxFIUzPmt9jbrTbAIrsqMFKy7iCmwr2AZ2hZDJ32tDMgySjlGgG/+KGxeKuPVPuPfmD4W2nogcDjnhuR+XsKZVJb1bSwtr6yurRc2iptb2zu75t5+U8apoNCgMY9F2yMSOIugoZji0E4EkNDj0PIGV+O+9QBCsji6V8ME3JD0IhYwSpSOuahEwhCMzvPqthJWI4vsfMIfg/wdcsWVrIrxo7JkpoZnqXfPL8WOahApyomUHdtKlJsRoRjlkBedVEJC6ID0oKNtREKQbjbZIcnOvFxEAt9IoUn6e8bGQmlHIaengyJ6sv5bhz+13VSFVTdjEVJqiCi04eClGMV4zEQ7DMBVPGhNoQKpv+KaZ9oKEpjK2oI9vzKi6Z5VrYr5Yu7Sql2O5riKAjdIxOkY3OUQ3doDpqIpG6Bm9ojfjyXgx3o2P6eiSMUN4gP7I+PwBfEuXOQ=</latexit> <latexit sha1_base64="OwWs2KkMn/ipLUupJ5HldxshJs=">AB6nicbVDLSgNBEOyNrxhfUY9eBoPgKexqQL0FvAheIpoHJEuYncwmQ2Znl5leMSwBf8CLB0W8+kXe/Bsnj4NGCxqKqm6u4JECoOu+XklpZXVtfy64WNza3tneLuXsPEqWa8zmIZ61ZADZdC8ToKlLyVaE6jQPJmMLyc+M17ro2I1R2OEu5HtK9EKBhFK90+dE+7xZJbdqcgf4k3JyWYo9YtfnZ6MUsjrpBJakzbcxP0M6pRMnHhU5qeELZkPZ521JFI278bHrqmBxZpUfCWNtSKbqz4mMRsaMosB2RhQHZtGbiP957RTDcz8TKkmRKzZbFKaSYEwmf5Oe0JyhHFlCmRb2VsIGVFOGNp2CDcFbfPkvaZyUvUr54qZSql4/zuLIwEcwjF4cAZVuIa1IFBH57gBV4d6Tw7b87rDXnzCPch19wPr4BO02ONw=</latexit> <latexit sha1_base64="NgeB9xmEe8slVR2w6uYodpFLvnI=">AB8XicbVDLSgNBEJyNrxhfUY9eBoPgKexKQD0IAS+ClwjmgckSZie9yZDZ2WmVwhLwI/w4kERr/6N/GyeOgiQUNRVU3V1BIoVB1/12ciura+sb+c3C1vbO7l5x/6Bh4lRzqPNYxroVMANSKijQAmtRAOLAgnNYHg98ZuPoI2I1T2OEvAj1lciFJyhlR46OABk9Iq63WLJLbtT0GXizUmJzFHrFr86vZinESjkhnT9twE/YxpFzCuNBJDSMD1kf2pYqFoHxs+nFY3pilR4NY21LIZ2qvycyFhkzigLbGTEcmEVvIv7ntVML/xMqCRFUHy2KEwlxZhO3qc9oYGjHFnCuBb2VsoHTDONqSCDcFbfHmZNM7KXqV8eVcpVW+fZnHkyRE5JqfEI+ekSm5IjdQJ4o8k1fy5hjnxXl3PmatOWce4SH5A+fzB3VkJI=</latexit> <latexit sha1_base64="rbksVT0LT/12jrXwQ0465Abhjmc=">AB83icbZBLSwMxFIUz9VXrq+rSTbAIrsqMCOpCKLgR3FSwD+gMJZPeaUMzMyG5I5Sh4K9w40IRt/4Zd/4b08dCWw8EPs65ITcnVFIYdN1vp7Cyura+UdwsbW3v7O6V9w+aJs0hwZPZarbITMgRQINFCihrTSwOJTQCoc3k7z1CNqINHnAkYIgZv1ERIztJbv4wCQ0WvqK9EtV9yqOxVdBm8OFTJXvVv+8nspz2JIkEtmTMdzFQY50yi4hHJzwoxoesDx2LCYvBPl05zE9sU6PRqm2J0E6dX/fyFlszCgO7WTMcGAWs4n5X9bJMLoMcpGoDCHhs4eiTFJM6aQA2hMaOMqRBca1sLtSPmCacbQ1lWwJ3uKXl6F5VvXOq1f35Xa3dOsjiI5IsfklHjkgtTILamTBuFEkWfySt6czHlx3p2P2WjBmVd4SP7I+fwBUgqRqw=</latexit> Constructing an interface Induces a Chern-Simons term: θ = π θ = 0 1 8 π A ∧ F x 3 breaks time-reversal invariance (half-integer level) To restore it, one needs to add fields on the interface - charged 3d Dirac fermion - topological field theory (gapped phases) e.g. [Seiberg, Witten ‘16]
<latexit sha1_base64="MLTnbLpj0im+l0p8OhPpnUoUK+E=">ACGnicbZBLSwMxFIUz9VXra9Slm2ARBKHMlIK6EApuBDcV7AM6tWTSTBuaeZDcEcow4L9w419x40IRd+LGf2OmnYW2HgczrkhuZ8bCa7Asr6NwtLyupacb20sbm1vWPu7rVUGEvKmjQUoey4RDHBA9YEDoJ1IsmI7wrWdseXWd+Z1LxMLiFScR6PhkG3OUgI76pu0AifEFdjxJaFLDTsQxT5PhXTXFJ3nqwIgBSZNq1qZ9s2xVrKnworFzU0a5Gn3z0xmENPZAFQpbq2FUEvIRI4FSwtObFiEaFjMmRdbQPiM9VLpqul+EgnA+yFUp8A8DT9fSMhvlIT39WTPoGRmu+y8L+uG4N31kt4EMXAjp7yIsFhBnPCAS0ZBTLQhVHL9V0xHROMATbOkIdjzKy+aVrVi1yrnN7Vy/fphqOIDtAhOkY2OkV1dIUaqIkoekTP6BW9GU/Gi/FufMxGC0aOcB/9kfH1A7fxoIo=</latexit> <latexit sha1_base64="Wt/6s+dvYR3lbDqdwPc5KcRLu9I=">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</latexit> <latexit sha1_base64="iVjr5ALdBCquXhM9lh1kEfpvhpU=">ACMXicbVBNSwMxFMz6bf2qevQSLIKHWnZFUG+CF8FLBdsK3VLepmkbmt2syVulLAV/kRf/iXjpQRGv/gmzbQWtDgSGmUle3gSxFAZd+jMzM7NLywuLedWVtfWN/KbW1WjEs14hSmp9E0AhksR8QoKlPwm1hzCQPJa0DvP/Nod10ao6Br7MW+E0IlEWzBAKzXzFz52OQL1teh0EbRW9/SAfovFIvVvE2hZAZLpjLvZmPTzBw08wW35I5A/xJvQgpkgnIz/+y3FEtCHiGTYEzdc2NspKBRMkHOT8xPAbWgw6vWxpByE0jHW08oHtWadG20vZESEfqzxsphMb0w8AmQ8CumfYy8T+vnmD7pJGKE6QR2w8qJ1Iiopm9dGW0Jyh7FsCTAv7V8q6oIGhLTlnS/CmV/5Lqocl76h0enVUOLt8GNexRHbILtknHjkmZ+SClEmFMPJIXsgreXOenKHz7nyMozPOpMJt8gvO5xfIdKp3</latexit> <latexit sha1_base64="NyKtr3IYrXcbdGHp/cU5Cld62Zo=">ACrHicjVFNa9wEJWdtE23X9v02MuQpaWlYbFLoO0t0Eugh6SQTbaszDKWx7sisuxIctLFGPrf8g967+J9qO02eSQAaHvHma0Zu0UtK6KPoThBubDx4+2nrcefL02fMX3ZfbJ7asjaCBKFVphilaUlLTwEmnaFgZwiJVdJqefZ3zpxdkrCz1sZtVlBQ40TKXAp1PjbtX3GEN3MjJ1KEx5SXw3KBoEBbEB0jbRvzFWQt8dxf4eY0Z8JQmUjdVgc7Iny2cjwk491cBnHT2j7jx+JoI4S2kc5nwIFsX3rPDuNuL+tEi4DaIV6DHVnE07v7mWSnqgrQTCq0dxVHlkgaNk0JR2+G1pQrFGU5o5KHGgmzSLMxu4Y3PZJCXxh/tYJH9X9FgYe2sSH2ln29q17l58i5uVLv8c9JIXdWOtFg2ymsFroT5iCThoRTMw9QGOlnBTFvyzn9vxJsTrX74NTj72473+l+97vf1v5Z2bLHXbIe9YzH7xPbZATtiAyaC98FhMAx+hP3wOByFybI0DFYWvmI3IsyvAblA0xI=</latexit> What if U(1) is dynamical? θ 1 L = 2 g 2 F ∧ ∗ F + 8 π 2 F ∧ F Has SL(2,Z) duality [Montonen, Olive ’77], [Witten ’95] ✓ q e ◆ ✓ q e ◆ ✓ a ◆ τ → a τ + b b c τ + d , → q m c d q m τ = 4 π i g 2 + θ with complexified coupling: 2 π Time-reversal acts as: θ → − θ , τ → − τ anti-holomorphic involution
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