Mirror symmetry of Calabi-Yau four-folds with non-trivial cohomology of odd degrees Sebastian Greiner Master’s Thesis supervised by Thomas Grimm Max-Planck-Institut f¨ ur Physik IMPRS Workshop June 2015 Sebastian Greiner (MPP Munich) Mirror symmetry of CY 4 IMPRS Workshop June 2015 1 / 18
Introduction In general, to apply concepts of string theory to the real world, we need to compactify our string theory on a manifold M to four space-time dimensions. Sebastian Greiner (MPP Munich) Mirror symmetry of CY 4 IMPRS Workshop June 2015 2 / 18
Introduction In general, to apply concepts of string theory to the real world, we need to compactify our string theory on a manifold M to four space-time dimensions. The physics (field content, gauge groups,...) of the resulting effective supergravity theory are determined by the geometry of M . Therefore, we need to understand its structure. Sebastian Greiner (MPP Munich) Mirror symmetry of CY 4 IMPRS Workshop June 2015 2 / 18
Introduction In general, to apply concepts of string theory to the real world, we need to compactify our string theory on a manifold M to four space-time dimensions. The physics (field content, gauge groups,...) of the resulting effective supergravity theory are determined by the geometry of M . Therefore, we need to understand its structure. Of special interest to phenomenology is F-Theory, since it allows insights into strongly-coupled behavior of string-theory. This (in some sense) twelve-dimensional string theory needs to be compactified on a so called (elliptically fibered) Calabi-Yau four-fold. ( CY 4 ) Sebastian Greiner (MPP Munich) Mirror symmetry of CY 4 IMPRS Workshop June 2015 2 / 18
Outline 1 Dimensional Reduction 2 Moduli 3 Mirror Symmetry of the Torus 4 Mirror Symmetry on CY 4 Sebastian Greiner (MPP Munich) Mirror symmetry of CY 4 IMPRS Workshop June 2015 3 / 18
Dimensional Reduction D -dim. gravity theory coupled to matter M D − n M n Sebastian Greiner (MPP Munich) Mirror symmetry of CY 4 IMPRS Workshop June 2015 4 / 18
Dimensional Reduction D -dim. gravity theory coupled to matter ⇒ n -dim. effective theory M D − n M n Sebastian Greiner (MPP Munich) Mirror symmetry of CY 4 IMPRS Workshop June 2015 4 / 18
Dimensional Reduction D -dim. gravity theory coupled to matter ⇒ n -dim. effective theory M D − n At every point of the curved spacetime M n there is a very small deformable internal manifold M D − n whose eigenmodes around a stable ground state correspond to fields on M n . M n 1 mass ∼ size ( M D − n ) Sebastian Greiner (MPP Munich) Mirror symmetry of CY 4 IMPRS Workshop June 2015 4 / 18
Dimensional Reduction Since the internal manifold M D − n is very small, we can neglect the massive modes for the effective theory. M D − n M n Sebastian Greiner (MPP Munich) Mirror symmetry of CY 4 IMPRS Workshop June 2015 5 / 18
Dimensional Reduction Since the internal manifold M D − n is very small, we can neglect the massive modes for the effective theory. M D − n ⇒ Consider only massless eigenmodes! M n Sebastian Greiner (MPP Munich) Mirror symmetry of CY 4 IMPRS Workshop June 2015 5 / 18
Dimensional Reduction Since the internal manifold M D − n is very small, we can neglect the massive modes for the effective theory. M D − n ⇒ Consider only massless eigenmodes! The massless oscillations around a stable ground state of M D − n preserve the geometric structure. M n Sebastian Greiner (MPP Munich) Mirror symmetry of CY 4 IMPRS Workshop June 2015 5 / 18
Dimensional Reduction Since the internal manifold M D − n is very small, we can neglect the massive modes for the effective theory. M D − n ⇒ Consider only massless eigenmodes! The massless oscillations around a stable ground state of M D − n preserve the geometric structure. M n Parameters of a geometrical object preserving its structure are called moduli . Sebastian Greiner (MPP Munich) Mirror symmetry of CY 4 IMPRS Workshop June 2015 5 / 18
Dimensional Reduction Since the internal manifold M D − n is very small, we can neglect the massive modes for the effective theory. M D − n ⇒ Consider only massless eigenmodes! The massless oscillations around a stable ground state of M D − n preserve the geometric structure. M n Parameters of a geometrical object preserving its structure are called moduli . ⇒ Study moduli! Sebastian Greiner (MPP Munich) Mirror symmetry of CY 4 IMPRS Workshop June 2015 5 / 18
Moduli - Circle The only adjustable parameter of a circle R preserving its structure is its radius R . Sebastian Greiner (MPP Munich) Mirror symmetry of CY 4 IMPRS Workshop June 2015 6 / 18
Moduli - Circle The only adjustable parameter of a circle R preserving its structure is its radius R . Therefore, only the deformations δ R are δ R possible. Sebastian Greiner (MPP Munich) Mirror symmetry of CY 4 IMPRS Workshop June 2015 6 / 18
Moduli - Torus In order to obtain supersymmetry, we need to consider Calabi-Yau spaces as internal manifolds. Sebastian Greiner (MPP Munich) Mirror symmetry of CY 4 IMPRS Workshop June 2015 7 / 18
Moduli - Torus In order to obtain supersymmetry, we need to consider Calabi-Yau spaces as internal manifolds. The simpelst example of a Calabi-Yau manifold is a torus T 2 . It is given by a product of two circles: Sebastian Greiner (MPP Munich) Mirror symmetry of CY 4 IMPRS Workshop June 2015 7 / 18
Moduli - Torus In order to obtain supersymmetry, we need to consider Calabi-Yau spaces as internal manifolds. The simpelst example of a Calabi-Yau manifold is a torus T 2 . It is given by a product of two circles: a b R a R b × a b Therefore, we haven now two moduli: R a , R b . Sebastian Greiner (MPP Munich) Mirror symmetry of CY 4 IMPRS Workshop June 2015 7 / 18
Moduli - Torus In order to obtain supersymmetry, we need to consider Calabi-Yau spaces as internal manifolds. The simpelst example of a Calabi-Yau manifold is a torus T 2 . It is given by a product of two circles: a b R a R b × a b Therefore, we haven now two moduli: R a , R b . ⇒ Two independent eigenmodes to excite! Sebastian Greiner (MPP Munich) Mirror symmetry of CY 4 IMPRS Workshop June 2015 7 / 18
Moduli - Torus We choose a special basis for the moduli: a b Sebastian Greiner (MPP Munich) Mirror symmetry of CY 4 IMPRS Workshop June 2015 8 / 18
Moduli - Torus We choose a special basis for the moduli: Volume: V = R a · R b a b Sebastian Greiner (MPP Munich) Mirror symmetry of CY 4 IMPRS Workshop June 2015 8 / 18
Moduli - Torus We choose a special basis for the moduli: Volume: V = R a · R b a and R = R a b Ratio: R b Sebastian Greiner (MPP Munich) Mirror symmetry of CY 4 IMPRS Workshop June 2015 8 / 18
Moduli - Torus We have two basic eigenmodes: R = R a R b constant ratio: = const. R b R a (equal phases) Sebastian Greiner (MPP Munich) Mirror symmetry of CY 4 IMPRS Workshop June 2015 9 / 18
Moduli - Torus We have two basic eigenmodes: R = R a R b constant ratio: = const. R b R a (equal phases) R b constant volume: V = R a · R b = const. R a (opposite phases) Sebastian Greiner (MPP Munich) Mirror symmetry of CY 4 IMPRS Workshop June 2015 9 / 18
Mirror Symmetry T 2 defined by Consider now the torus ˆ R b = 1 R b = R a ˆ ˆ ⇒ ˆ V = ˆ R a ˆ , R a = R a . = R ! R b R b Sebastian Greiner (MPP Munich) Mirror symmetry of CY 4 IMPRS Workshop June 2015 10 / 18
Mirror Symmetry T 2 defined by Consider now the torus ˆ R b = 1 R b = R a ˆ ˆ ⇒ ˆ V = ˆ R a ˆ , R a = R a . = R ! R b R b Therefore, we have the correspondence of eigenmodes: ˆ R b R b ˆ R a R a ˆ R =const. V =const. Sebastian Greiner (MPP Munich) Mirror symmetry of CY 4 IMPRS Workshop June 2015 10 / 18
Mirror Symmetry ˆ R b R b ˆ R a R a ˆ R =const. V =const. Due to this correspondence, on both geometries (stable ground states) we have the same content of massless eigenmodes. Sebastian Greiner (MPP Munich) Mirror symmetry of CY 4 IMPRS Workshop June 2015 11 / 18
Mirror Symmetry ˆ R b R b ˆ R a R a ˆ R =const. V =const. Due to this correspondence, on both geometries (stable ground states) we have the same content of massless eigenmodes. ⇒ We have the same physics on both configurations! Sebastian Greiner (MPP Munich) Mirror symmetry of CY 4 IMPRS Workshop June 2015 11 / 18
Mirror Symmetry T 2 is the mirror of T 2 . This symmetry is called mirror symmetry and ˆ R �→ ˆ The map V is called mirror map . Sebastian Greiner (MPP Munich) Mirror symmetry of CY 4 IMPRS Workshop June 2015 12 / 18
Mirror Symmetry T 2 is the mirror of T 2 . This symmetry is called mirror symmetry and ˆ R �→ ˆ The map V is called mirror map . Mirror symmetry has the advantage that 1 δ ˆ δ R b ≫ 1 , ⇒ R b ∼ ≪ 1 , δ R b i.e. non-perturbative effects map to perturbative effects and are therefore acessible for calculations. Sebastian Greiner (MPP Munich) Mirror symmetry of CY 4 IMPRS Workshop June 2015 12 / 18
Recommend
More recommend
