Singularities and in the University

+Singularities and Gauge Theory Phases Mboyo Esole Harvard University With Shu-Heng Shao Shing-Tung Yau String 2014, Princeton arXiv:1402.6331 + Other teams n  Kumar, Park, Taylor n  Grimm, Hayashi n  Krause, Mayrhofer, Weigand n  Hayashi, Lawrie, Morisson , Schafer-Nameki + Other teams n  Kumar, Park, Taylor n  Grimm, Hayashi n  Krause, Mayrhofer, Weigand n  Hayashi, Lawrie, Morisson , Schafer-Nameki Love to my other collaborators: Paolo Aluffi Patrick Jefferson Michele Del Zotto, Jonathan Heckman, Cumrun Vafa + At the menu: n  Geometry n  Gauge theories n  Representation theory n  Singularities n  Resolutions of singularities It also comes as a combo: …an elliptic fibration + Tale of two worlds Gauge theories Elliptic fibrations Gauge algebra Codimension one singularities Representations Codimension two singularities Yukawa Codimension three singularities Coulomb phases Crepant resolutions Walls Partial resolutions Phase transitions Flops + 5D supersymmetric gauge theories with supercharges Matter content: Gravity multiplet Vector multiplets Hypermultiplets Tensor multplets + 5D supersymmetric gauge theories with supercharges Geometry: Vector multiplets Hypermultiplets Very Special Geometry Quaternionic-K¨ ahler Hyperk¨ ahler + Coulomb branch of 5D gauge theory n  Vector multiplets-> Weyl chamber n  Massless Hypers at singularities -> sub-chamber structure F( ) = m0 Tr( 2 ccl )+ Tr( @ X )+ |( , ↵)|3 12 ↵:roots X w:weights |( , w)|3 A + Incidence geometry of a representation R n  g is a Lie algebra with Cartan sub-algebra h n  Roots of g define Weyl chambers n  R : a representation of g n  Weights of R refine the Weyl chambers by adding a subchamber structure + Elliptic fibrations n  Elliptic curves are some of the oldest but yet most prominent objects across mathematics n  n  Number theory Algebraic geometry n  Cryptography Geometric design n  Physics n  + The case of SU(3) Tree of resolutions E0 (x, y, e0 �e1 ) E1 (y e 2) � , e1 (s, e �e 2) T+ flop T− + The case of SU(3) Fiber structure SU (3) P3 = 1 1 I3 I4 a1 = 1 a1 = 1 IV P3 = 1 I∗0 codim codim codim + The case of SU(3) µ1 O C + C− W w2 µ2 E0 (x, y, e0 �e1 ) E1 (y e 2) � , e1 (s, e �e 2) T+ flop T− + The case of SU(3) µ1 O C + C− W w2 E0 (x, y, e0 �e1 ) E1 (y µ2 A perfect match! e 2) � , e1 (s, e �e 2) T+ flop T− + The case of SU(4) p0 µ2 p+ Incidence geometry W0 + C+ C+ O W+ µ3 `+ p C+ W ` L C µ3 ` + The case of SU(4) p0 W0 + C+ C+ p p+ W W + C C+ `+ ` ` + The case of SU(4) p0 Incidence geometry W0 + C+ C+ p p+ W W+ C C+ `+ C + ` ` W+ + C+ W0 C+ W C W+ L W0 W Tree of resolutions for SU(4) Resolutions e1 � e2 ) T+ ) �e e (r, (x, e �e 3) (y , + E0 (x, y, e0 �e1 ) E1 (x, y, e1 �e2 ) B �e 3) e (y , (s, e �e 3) e (s, ) �e T− ) �e e (x, (r, e �e 3) T−+ T++ B+ flop B− T+− T−− flop + The case of SU(4) T+ E0 E1 B T− T−+ T++ T+− T−− flop flop + Tree of resolutions for SU(4) Fiber structure SU (4) P4 = 1 1 1 a1 = I4 1 I∗1 a1 = 1 codim I5 P4 = a22,1 ∗ I0 codim − 4a4,2 = 1 1 I�+ codim + The case of SU(4) A perfect match! p0 µ2 p+ T+ W0 + C+ C+ O W+ E0 µ3 `+ E1 B p C+ W L T− C ` p0 µ3 ` W0 + C+ C C+ W W+ C C+ `+ ` + C+ W0 C+ W C T++ flop ` flop T+− T−− W+ L W0 p p+ + W+ T−+ W + The case of SU(4) p0 µ2 p+ T+ W0 + C+ C+ O W+ E0 µ3 `+ E1 B p C+ W L T− C ` p0 µ3 ` W0 + C+ C C+ W W+ C C+ `+ ` + C+ W0 C+ W C T++ flop ` flop T+− T−− W+ L W0 p p+ + W+ T−+ W + The case of SU(4) C + W + + C+ W C+ W C W+ L W0 W + The case of SU(5) B1,3 B1,2 B1,3 B2,3 B3,2 B1,3 B3,2 B2,3 B3,1 B2,1 B3,1 B3,1 + THANK YOU! ... Geometry n  Gauge theories n  Representation theory n  Singularities n  Resolutions of singularities It also comes as a combo: …an elliptic fibration + Tale of two worlds Gauge theories Elliptic... i @ 1i 2 IV ? i j j + Collision of singularities (Miranda models) J =1 : In + Im ⇤ I2n + Im ! ! In+ m ⇤ In+ m j @ @ 2j 2j 2j I2n+1 + J =0 J =1 : : ⇤ Im ! ⇤+ In+ m+1 j 2j II + IV ! II + I0⇤ ! IV... z + Singular fibers n  Kodaira n  Weierstrass model n  Néron models n  Tate’s algorithm n  Miranda models (collisions of singularities) n  Szydlo (generalization of Miranda’s model and Tate’s