Mirror symmetry and K3 surface zeta functions Ursula Whitcher - PowerPoint PPT Presentation

Mirror symmetry and K3 surface zeta functions Mirror symmetry and K3 surface zeta functions Ursula Whitcher whitchua@uwec.edu University of Wisconsin–Eau Claire October 2015

Mirror symmetry and K3 surface zeta functions Collaborators Tyler Kelly, Charles Doran, Steven Sperber, Ursula Whitcher, John Voight, Adriana Salerno

Mirror symmetry and K3 surface zeta functions Outline Experimental Evidence Classical Mirror Symmetry Arithmetic Mirror Symmetry Alternate Mirrors Berglund-H¨ ubsch-Krawitz Duality Results

Mirror symmetry and K3 surface zeta functions Experimental Evidence Five Interesting Quartics in P 3 Family Equation x 4 0 + x 4 1 + x 4 2 + x 4 F 4 (Fermat/Dwork) 3 x 4 0 + x 4 1 + x 3 2 x 3 + x 3 F 2 L 2 3 x 2 x 4 0 + x 3 1 x 2 + x 3 2 x 3 + x 3 F 1 L 3 (Klein-Mukai) 3 x 1 x 3 0 x 1 + x 3 1 x 0 + x 3 2 x 3 + x 3 L 2 L 2 3 x 2 x 3 0 x 1 + x 3 1 x 2 + x 3 2 x 3 + x 3 L 4 3 x 0

Mirror symmetry and K3 surface zeta functions Experimental Evidence Five Interesting Quartics in P 3 Family Equation x 4 0 + x 4 1 + x 4 2 + x 4 F 4 (Fermat/Dwork) 3 x 4 0 + x 4 1 + x 3 2 x 3 + x 3 F 2 L 2 3 x 2 x 4 0 + x 3 1 x 2 + x 3 2 x 3 + x 3 F 1 L 3 (Klein-Mukai) 3 x 1 x 3 0 x 1 + x 3 1 x 0 + x 3 2 x 3 + x 3 L 2 L 2 3 x 2 x 3 0 x 1 + x 3 1 x 2 + x 3 2 x 3 + x 3 L 4 3 x 0 Warnings ◮ These quartics are not isomorphic.

Mirror symmetry and K3 surface zeta functions Experimental Evidence Five Interesting Quartics in P 3 Family Equation x 4 0 + x 4 1 + x 4 2 + x 4 F 4 (Fermat/Dwork) 3 x 4 0 + x 4 1 + x 3 2 x 3 + x 3 F 2 L 2 3 x 2 x 4 0 + x 3 1 x 2 + x 3 2 x 3 + x 3 F 1 L 3 (Klein-Mukai) 3 x 1 x 3 0 x 1 + x 3 1 x 0 + x 3 2 x 3 + x 3 L 2 L 2 3 x 2 x 3 0 x 1 + x 3 1 x 2 + x 3 2 x 3 + x 3 L 4 3 x 0 Warnings ◮ These quartics are not isomorphic. ◮ These quartics are not Fourier-Mukai partners. ◮ These quartics are not derived equivalent.

Mirror symmetry and K3 surface zeta functions Experimental Evidence Counting Points Prime F 4 F 2 L 2 F 1 L 3 L 2 L 2 L 4 5 0 20 30 80 40 7 64 50 64 64 78 11 144 122 144 144 254 13 128 180 206 336 232 17 600 328 294 600 328 19 400 362 400 400 438 23 576 530 576 576 622 29 768 884 1116 1232 1000 31 1024 962 1024 1024 1334 37 1152 1300 1374 1744 1448

Mirror symmetry and K3 surface zeta functions Experimental Evidence Counting Points Prime F 4 F 2 L 2 F 1 L 3 L 2 L 2 L 4 5 0 20 30 80 40 7 64 50 64 64 78 11 144 122 144 144 254 13 128 180 206 336 232 17 600 328 294 600 328 19 400 362 400 400 438 23 576 530 576 576 622 29 768 884 1116 1232 1000 31 1024 962 1024 1024 1334 37 1152 1300 1374 1744 1448 Equality holds (mod p ) for all p in this table.

Mirror symmetry and K3 surface zeta functions Experimental Evidence Counting Points on Pencils ◮ We can add the deforming monomial − 4 ψ xyzw to each of our quartics to obtain pencils of quartics X ⋄ ,ψ .

Mirror symmetry and K3 surface zeta functions Experimental Evidence Counting Points on Pencils ◮ We can add the deforming monomial − 4 ψ xyzw to each of our quartics to obtain pencils of quartics X ⋄ ,ψ . ◮ We can count the number of points on X ⋄ ,ψ over F p for 0 ≤ ψ < p .

Mirror symmetry and K3 surface zeta functions Experimental Evidence Counting Points on Pencils ◮ We can add the deforming monomial − 4 ψ xyzw to each of our quartics to obtain pencils of quartics X ⋄ ,ψ . ◮ We can count the number of points on X ⋄ ,ψ over F p for 0 ≤ ψ < p . ◮ For each ψ , the point counts on X ⋄ ,ψ agree (mod p ).

Mirror symmetry and K3 surface zeta functions Experimental Evidence What about the zeta functions? The zeta function for any smooth quartic will have the form 1 Z ( X / F p , T ) = (1 − T )(1 − pT )(1 − p 2 T ) P ( T ) .

Mirror symmetry and K3 surface zeta functions Experimental Evidence What about the zeta functions? The zeta function for any smooth quartic will have the form 1 Z ( X / F p , T ) = (1 − T )(1 − pT )(1 − p 2 T ) P ( T ) . Let’s use Edgar Costa’s code to look at P ( T ) for X ⋄ ,ψ when p = 41.

Mirror symmetry and K3 surface zeta functions Experimental Evidence F 4 ψ F 4 (1 − 41 T ) 19 (1 − 18 T + 41 2 T 2 ) 0 1, 9, 32, 40 not smooth (1 − 41 T ) 3 (1 + 41 T ) 16 (1 − 50 T + 41 2 T 2 ) 2, 18, 23, 39 (1 − 41 T ) 19 (1 + 78 T + 41 2 T 2 ) 3, 14, 27, 38 (1 − 41 T ) 13 (1 + 41 T ) 6 (1 + 66 T + 41 2 T 2 ) 4, 5, 36, 37 (1 − 41 T ) 3 (1 + 41 T ) 16 (1 − 50 T + 41 2 T 2 ) 6, 13, 28, 35 (1 − 41 T ) 3 (1 + 41 T ) 16 (1 + 46 T + 41 2 T 2 ) 7, 19, 22, 34 (1 − 41 T ) 13 (1 + 41 T ) 6 (1 − 62 T + 41 2 T 2 ) 8, 10, 31, 33 (1 − 41 T ) 5 (1 + 41 T ) 16 11, 17, 24, 30 (1 − 41 T ) 3 (1 + 41 T ) 16 (1 − 50 T + 41 2 T 2 ) 12, 15, 26, 29 (1 − 41 T ) 3 (1 + 41 T ) 16 (1 − 50 T + 41 2 T 2 ) 16, 20, 21, 25

Mirror symmetry and K3 surface zeta functions Experimental Evidence F 2 L 2 ψ F 2 L 2 (1 − 41 T ) 11 (1 + 41 T ) 8 (1 − 18 T + 41 2 T 2 ) 0 1, 9, 32, 40 not smooth (1 − 41 T ) 11 (1 + 41 T ) 8 (1 − 50 T + 41 2 T 2 ) 2, 18, 23, 39 (1 − 41 T ) 19 (1 + 78 T + 41 2 T 2 ) 3, 14, 27, 38 (1 − 41 T ) 9 (1 + 41 T ) 2 (1 + 41 2 T 2 ) 4 (1 + 66 T + 41 2 T 2 ) 4, 5, 36, 37 (1 − 41 T ) 11 (1 + 41 T ) 8 (1 − 50 T + 41 2 T 2 ) 6, 13, 28, 35 (1 − 41 T ) 11 (1 + 41 T ) 8 (1 + 46 T + 41 2 T 2 ) 7, 19, 22, 34 (1 − 41 T ) 9 (1 + 41 T ) 2 (1 + 41 2 T 2 ) 4 (1 − 62 T + 41 2 T 2 ) 8, 10, 31, 33 (1 − 41 T ) 13 (1 + 41 T ) 8 11, 17, 24, 30 (1 − 41 T ) 11 (1 + 41 T ) 8 (1 − 50 T + 41 2 T 2 ) 12, 15, 26, 29 (1 − 41 T ) 11 (1 + 41 T ) 8 (1 − 50 T + 41 2 T 2 ) 16, 20, 21, 25

Mirror symmetry and K3 surface zeta functions Experimental Evidence L 2 L 2 ψ L 2 L 2 (1 − 41 T ) 19 (1 − 18 T + 41 2 T 2 ) 0 1, 9, 32, 40 not smooth (1 − 41 T ) 11 (1 + 41 T ) 8 (1 − 50 T + 41 2 T 2 ) 2, 18, 23, 39 (1 − 41 T ) 15 (1 + 41 T ) 4 (1 + 78 T + 41 2 T 2 ) 3, 14, 27, 38 (1 − 41 T ) 13 (1 + 41 T ) 6 (1 + 66 T + 41 2 T 2 ) 4, 5, 36, 37 (1 − 41 T ) 11 (1 + 41 T ) 8 (1 − 50 T + 41 2 T 2 ) 6, 13, 28, 35 (1 − 41 T ) 15 (1 + 41 T ) 4 (1 + 46 T + 41 2 T 2 ) 7, 19, 22, 34 (1 − 41 T ) 13 (1 + 41 T ) 6 (1 − 62 T + 41 2 T 2 ) 8, 10, 31, 33 (1 − 41 T ) 17 (1 + 41 T ) 4 11, 17, 24, 30 (1 − 41 T ) 11 (1 + 41 T ) 8 (1 − 50 T + 41 2 T 2 ) 12, 15, 26, 29 (1 − 41 T ) 11 (1 + 41 T ) 8 (1 − 50 T + 41 2 T 2 ) 16, 20, 21, 25

Mirror symmetry and K3 surface zeta functions Experimental Evidence Experimental evidence foreshadows . . . ◮ We see interesting shared quadratic factors in the zeta functions (also for other values of ⋄ ).

Mirror symmetry and K3 surface zeta functions Experimental Evidence Experimental evidence foreshadows . . . ◮ We see interesting shared quadratic factors in the zeta functions (also for other values of ⋄ ). ◮ In fact, these are part of shared cubic factors.

Mirror symmetry and K3 surface zeta functions Experimental Evidence Experimental evidence foreshadows . . . ◮ We see interesting shared quadratic factors in the zeta functions (also for other values of ⋄ ). ◮ In fact, these are part of shared cubic factors. ◮ We can also make predictions about the structure of the other factors.

Mirror symmetry and K3 surface zeta functions Classical Mirror Symmetry Why? The arithmetic patterns we observe are a consequence of mirror symmetry.

Mirror symmetry and K3 surface zeta functions Classical Mirror Symmetry Are Strings the Answer? String Theory proposes that “fundamental” particles are strings.

Mirror symmetry and K3 surface zeta functions Classical Mirror Symmetry Extra Dimensions For string theory to work as a consistent theory of quantum mechanics, it must allow the strings to vibrate in extra, compact dimensions.

Mirror symmetry and K3 surface zeta functions Classical Mirror Symmetry Building a Model Locally, space-time should look like M 3 , 1 × V . ◮ M 3 , 1 is four-dimensional space-time ◮ V is a d -dimensional complex manifold ◮ Physicists require d = 3 (6 real dimensions) ◮ V is a Calabi-Yau manifold

Mirror symmetry and K3 surface zeta functions Classical Mirror Symmetry Calabi-Yau Manifolds We can define an n -dimensional Calabi-Yau manifold as a simply connected, smooth . . . ◮ Variety with trivial canonical bundle ◮ Ricci-flat K¨ ahler-Einstein manifold ◮ Complex manifold with a unique (up to scaling) holomorphic n -form

Mirror symmetry and K3 surface zeta functions Classical Mirror Symmetry Calabi-Yau Manifolds We can define an n -dimensional Calabi-Yau manifold as a simply connected, smooth . . . ◮ Variety with trivial canonical bundle ◮ Ricci-flat K¨ ahler-Einstein manifold ◮ Complex manifold with a unique (up to scaling) holomorphic n -form Calabi-Yau 2-folds are also known as K3 surfaces.

Mirror symmetry and K3 surface zeta functions Classical Mirror Symmetry A-Model or B-Model? The worldsheet of a string is a Riemann surface. We consider maps from our string worldsheet to our space-time. Choosing Variables ◮ z = a + ib , w = c + id ◮ z = a + ib , w = c − id