Charges and Electromagnetic Radiation as Topological Excitations

<div>Stereographic view with parallel eyes at a fibre neighbourhood. (a) Fibre <svg height="8.97279pt" id="M244" style="vertical-align:-0.04980946pt" version="1.1" viewbox="-0.0498162 -8.92298 12.0914 8.97279" width="12.0914pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M896 662C863 643 826 624 783 624C682 624 539 680 429 680C328 680 253 659 202 616C163 583 138 537 138 484C138 399 208 349 283 349C396 349 466 417 492 554L472 555C436 442 385 379 286 379C234 379 179 410 179 491C179 578 257 634 382 634C475 634 622 574 731 574C796 574 862 610 907 647L896 662ZM657 568C577 499 520 440 477 341L463 310C368 277 286 233 247 186L260 171C320 211 380 237 437 256C349 77 281 31 184 31C124 31 83 59 78 94H80C87 83 101 81 109 81C137 81 153 104 153 131S130 180 103 180C68 180 43 154 43 111C43 34 123 0 192 0C335 0 469 75 546 282C599 291 640 293 661 294C653 267 650 244 650 224V206L709 226C706 235 706 245 706 252C706 271 706 289 714 312C758 333 786 368 786 387C786 403 775 409 763 409C744 409 712 391 685 352C644 351 604 345 564 337C595 433 622 506 671 554L657 568Z" id="g198-7"></path></g></svg> on a torus in <svg height="11.6412pt" id="M245" style="vertical-align:-0.04979992pt" version="1.1" viewbox="-0.0498162 -11.5914 14.7935 11.6412" width="14.7935pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M712 0L497 285C582 312 628 378 628 466C628 609 518 662 390 662H70V0H236V271H310L505 0H712ZM491 604C551 577 584 532 584 464C584 401 550 345 491 321V604ZM447 320C428 317 411 315 392 315H236V618H397C412 618 432 613 447 610V320ZM626 44H528L363 271C392 271 422 270 451 275L626 44ZM192 44H114V618H192V44Z" id="g197-29"></path></g><g transform="matrix(.0091,0,0,-0.0091,9.711,-5.741)"><path d="M290 377C321 398 342 415 358 430C378 450 389 473 389 502C389 578 329 635 238 635H237C184 635 137 610 109 578L64 515L88 493C112 529 154 573 208 573S303 542 303 482C303 409 233 370 141 341L149 308C165 313 190 319 215 319C272 319 341 283 341 193C342 98 292 43 222 43C163 43 122 72 96 94C88 101 79 100 70 94C61 87 47 73 46 60C44 47 48 37 62 23C76 10 118 -12 165 -12C238 -12 430 62 430 223C430 297 379 359 290 375V377Z" id="g50-52"></path></g></svg>. Dreibeins of red tangential vectors and blue and cyan tangential vectors are drawn at three positions. (b) Gauß map <svg height="12.1958pt" id="M246" style="vertical-align:-3.181479pt" version="1.1" viewbox="-0.0498162 -9.01432 19.3918 12.1958" width="19.3918pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M449 645C392 682 336 687 293 687C136 687 37 594 37 499C37 397 122 342 207 329C176 273 159 217 159 158C159 48 247 -15 356 -15C417 -15 478 3 530 39S626 128 626 187C626 256 580 290 508 290C402 290 311 187 311 65L333 63C344 171 423 247 491 247C528 247 562 225 562 173C562 97 470 15 360 15C284 15 243 61 243 125C243 176 255 240 312 324C552 324 781 473 781 594C781 642 748 669 685 669C547 669 336 527 229 361C151 369 78 413 78 501C78 604 200 657 301 657C336 657 381 657 437 626L449 645ZM331 355C443 513 593 632 690 632C723 632 740 619 740 592C740 534 671 464 576 416C501 377 411 354 332 354L331 355Z" id="g198-4"></path></g><g transform="matrix(.0091,0,0,-0.0091,10.361,3.132)"><path d="M896 662C863 643 826 624 783 624C682 624 539 680 429 680C328 680 253 659 202 616C163 583 138 537 138 484C138 399 208 349 283 349C396 349 466 417 492 554L472 555C436 442 385 379 286 379C234 379 179 410 179 491C179 578 257 634 382 634C475 634 622 574 731 574C796 574 862 610 907 647L896 662ZM657 568C577 499 520 440 477 341L463 310C368 277 286 233 247 186L260 171C320 211 380 237 437 256C349 77 281 31 184 31C124 31 83 59 78 94H80C87 83 101 81 109 81C137 81 153 104 153 131S130 180 103 180C68 180 43 154 43 111C43 34 123 0 192 0C335 0 469 75 546 282C599 291 640 293 661 294C653 267 650 244 650 224V206L709 226C706 235 706 245 706 252C706 271 706 289 714 312C758 333 786 368 786 387C786 403 775 409 763 409C744 409 712 391 685 352C644 351 604 345 564 337C595 433 622 506 671 554L657 568Z" id="g198-7"></path></g></svg> of <svg height="8.97279pt" id="M247" style="vertical-align:-0.04980946pt" version="1.1" viewbox="-0.0498162 -8.92298 12.0914 8.97279" width="12.0914pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M896 662C863 643 826 624 783 624C682 624 539 680 429 680C328 680 253 659 202 616C163 583 138 537 138 484C138 399 208 349 283 349C396 349 466 417 492 554L472 555C436 442 385 379 286 379C234 379 179 410 179 491C179 578 257 634 382 634C475 634 622 574 731 574C796 574 862 610 907 647L896 662ZM657 568C577 499 520 440 477 341L463 310C368 277 286 233 247 186L260 171C320 211 380 237 437 256C349 77 281 31 184 31C124 31 83 59 78 94H80C87 83 101 81 109 81C137 81 153 104 153 131S130 180 103 180C68 180 43 154 43 111C43 34 123 0 192 0C335 0 469 75 546 282C599 291 640 293 661 294C653 267 650 244 650 224V206L709 226C706 235 706 245 706 252C706 271 706 289 714 312C758 333 786 368 786 387C786 403 775 409 763 409C744 409 712 391 685 352C644 351 604 345 564 337C595 433 622 506 671 554L657 568Z" id="g198-7"></path></g></svg> with red radial vector <svg height="18.3692pt" id="M248" style="vertical-align:-3.834pt" version="1.1" viewbox="-0.0498162 -14.5352 33.4933 18.3692" width="33.4933pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,-5.911)"><path d="M567 230V280H69V230H567Z" id="g117-149"></path></g><g transform="matrix(.013,0,0,-0.013,-1.901,-5.911)"><path d="M901 255C861 290 790 361 716 442L694 427L796 280H69V230H796L694 83L716 68C790 149 861 220 901 255Z" id="g117-148"></path></g><g transform="matrix(.013,0,0,-0.013,2.588,0)"><path d="M391 364C391 409 353 448 295 448C249 448 198 426 152 393C65 331 23 225 23 139C23 14 96 -12 146 -12C198 -12 280 9 367 101L351 124C300 78 242 48 194 48C129 48 109 107 109 162V191C208 213 391 266 391 364ZM313 350C313 305 268 261 113 223C132 334 187 381 217 398C227 404 244 405 261 405C290 405 313 385 313 350Z" id="g113-102"></path></g><g transform="matrix(.0091,0,0,-0.0091,10.578,-8.841)"><path d="M310 541L304 571C290 586 211 619 185 610L80 76L131 52L310 541Z" id="g50-31"></path></g><g transform="matrix(.0091,0,0,-0.0091,10.578,3.784)"><path d="M896 662C863 643 826 624 783 624C682 624 539 680 429 680C328 680 253 659 202 616C163 583 138 537 138 484C138 399 208 349 283 349C396 349 466 417 492 554L472 555C436 442 385 379 286 379C234 379 179 410 179 491C179 578 257 634 382 634C475 634 622 574 731 574C796 574 862 610 907 647L896 662ZM657 568C577 499 520 440 477 341L463 310C368 277 286 233 247 186L260 171C320 211 380 237 437 256C349 77 281 31 184 31C124 31 83 59 78 94H80C87 83 101 81 109 81C137 81 153 104 153 131S130 180 103 180C68 180 43 154 43 111C43 34 123 0 192 0C335 0 469 75 546 282C599 291 640 293 661 294C653 267 650 244 650 224V206L709 226C706 235 706 245 706 252C706 271 706 289 714 312C758 333 786 368 786 387C786 403 775 409 763 409C744 409 712 391 685 352C644 351 604 345 564 337C595 433 622 506 671 554L657 568Z" id="g198-7"></path></g><g transform="matrix(.013,0,0,-0.013,19.471,0)"><path d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z" id="g113-41"></path></g><g transform="matrix(.013,0,0,-0.013,23.969,0)"><path d="M352 391C352 416 319 448 267 448C236 448 173 423 147 400C107 364 96 332 96 304C96 248 143 210 193 181C241 153 258 124 258 100C258 72 232 38 184 38C151 38 107 66 81 108C77 114 64 116 55 111C34 99 23 84 23 65C23 29 81 -12 134 -12C220 -12 325 61 325 141C325 184 297 215 234 256C194 282 161 309 161 346C161 380 188 401 217 401C255 401 279 380 301 353C308 344 313 341 325 347C341 355 352 371 352 391Z" id="g113-116"></path></g><g transform="matrix(.013,0,0,-0.013,28.844,0)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z" id="g113-42"></path></g></svg> and blue and cyan tangential vectors at the corresponding three positions.</div>

Advances in High Energy Physics

Charges and Electromagnetic Radiation as Topological Excitations

Figure 5