Random tensors, a functional integral point of view R azvan Gur - PowerPoint PPT Presentation

Random tensors, a “functional integral” point of view R˘ azvan Gur˘ au IHP, 2016

Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition Introduction Tensor Models The quartic model The continuum limit is a phase transition 2

Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition Why random discretized spaces? 3

Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition Why random discretized spaces? Try to make sense of: � � D g (metrics) D X matter e − S topologies � √ gR − κ V � √ g + κ m S m . S ∼ κ R 3

Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition Why random discretized spaces? Try to make sense of: � � D g (metrics) D X matter e − S topologies � √ gR − κ V � √ g + κ m S m . S ∼ κ R Idea: Replace the sum over topologies and metrics with a sum over random discretizations � � � D g (metrics) → . topologies random discretizations 3

Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition Why random discretized spaces? Try to make sense of: � � D g (metrics) D X matter e − S topologies � √ gR − κ V � √ g + κ m S m . S ∼ κ R Idea: Replace the sum over topologies and metrics with a sum over random discretizations � � � D g (metrics) → . topologies random discretizations But what measure should one use over the random discretizations? 3

Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition Matrix and tensor models 4

Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition Matrix and tensor models Matrix and tensor models are generating functions for random discrete geometries 4

Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition Matrix and tensor models Matrix and tensor models are generating functions for random discrete geometries Field theories with no kinetic term (probability measures) invariant under conjugation by the unitary group U ( N ) ◮ “path integrals” for matrix (tensor) “fields” ◮ Feynman graphs dual to discretized D -dimensional spaces ◮ the weight of a discretization is fixed by the Feynman rules: canonical measures over random discretizations. 4

Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition Matrix and tensor models Matrix and tensor models are generating functions for random discrete geometries Field theories with no kinetic term (probability measures) invariant under conjugation by the unitary group U ( N ) ◮ “path integrals” for matrix (tensor) “fields” ◮ Feynman graphs dual to discretized D -dimensional spaces ◮ the weight of a discretization is fixed by the Feynman rules: canonical measures over random discretizations. ◮ bonus: the perturbative expansion can be reorganized in powers of 1 / N . ◮ the perturbative series diverges, but the series at fixed order in 1 / N converges. 4

Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition From Matrix to Tensor Models Invariant action for a complex, generic Invariant action for a matrix M ab tensor T a 1 ... a D ribbon graphs ↔ surfaces colored graphs ↔ D dimensional spaces 1 1 2 2 3 0 0 g ( G ) ≥ 0 genus ω ( G ) ≥ 0 degree 2 1 / N expansion A ( G ) = N D − ( D − 1)! ω ( G ) 1 / N expansion A ( G ) = N 2 − 2 g ( G ) leading order: g ( G ) = 0, spheres. leading order: ω ( G ) = 0, spheres. 5

Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition Introduction Tensor Models The quartic model The continuum limit is a phase transition 6

Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition Tensor invariants as Edge Colored Graphs 7

Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition Tensor invariants as Edge Colored Graphs Building blocks: tensors with no symmetry transforming as � ¯ � U (1) b 1 a 1 . . . U ( D ) ¯ U (1) p 1 q 1 . . . ¯ U ( D ) p D q D ¯ T ′ T ′ b 1 ... b D = b D a D T a 1 ... a D , p 1 ... p D = T q 1 ... q D 7

Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition Tensor invariants as Edge Colored Graphs Building blocks: tensors with no symmetry transforming as � ¯ � U (1) b 1 a 1 . . . U ( D ) ¯ U (1) p 1 q 1 . . . ¯ U ( D ) p D q D ¯ T ′ T ′ b 1 ... b D = b D a D T a 1 ... a D , p 1 ... p D = T q 1 ... q D T 1 1 Invariants: colored graphs 2 2 2 2 3 3 3 T 3 1 1 1 1 D 2 2 � � � � � Tr B ( T , ¯ ¯ T ) = T a 1 T q 1 δ a c 1 w q c v ... a D v ... q D 1 w ¯ v ¯ v ¯ 3 2 2 2 2 3 v v ¯ c =1 e c =( w , ¯ w ) 3 3 3 1 1 1 1 3 2 2 ◮ White (black) vertices for T ( ¯ T ). ◮ Edges for δ a c q c colored by c , the position of the index. 7

Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition Invariant Actions for Tensor Models 8

Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition Invariant Actions for Tensor Models The most general single trace tensor model D T ) = 1 � � � S ( T , ¯ T a 1 ... a D ¯ t B Tr B ( ¯ δ a c q c + T q 1 ... q D T , T ) λ c =1 B � TdT ] e − N D − 1 S ( T , ¯ [ d ¯ T ) Z TM ( t B ) = 8

Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition Invariant Actions for Tensor Models The most general single trace tensor model D T ) = 1 � � � S ( T , ¯ T a 1 ... a D ¯ t B Tr B ( ¯ δ a c q c + T q 1 ... q D T , T ) λ c =1 B � TdT ] e − N D − 1 S ( T , ¯ [ d ¯ T ) Z TM ( t B ) = Feynman graphs: “vertices” B . T 1 1 2 2 2 2 3 3 3 T � e − N D − 1 � c =1 δ ac qc � � T a 1 ... aD ¯ 3 1 � D 1 1 1 1 T q 1 ... qD λ 2 2 ¯ T , T 1 Tr B 1 ( ¯ T , T )Tr B 2 ( ¯ T , T ) . . . 1 3 2 2 2 2 3 3 3 3 1 1 1 1 3 2 2 8

Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition Invariant Actions for Tensor Models The most general single trace tensor model D T ) = 1 � � � S ( T , ¯ T a 1 ... a D ¯ t B Tr B ( ¯ δ a c q c + T q 1 ... q D T , T ) λ c =1 B � TdT ] e − N D − 1 S ( T , ¯ [ d ¯ T ) Z TM ( t B ) = Feynman graphs: “vertices” B . T 1 1 2 2 2 2 3 3 3 � T e − N D − 1 � c =1 δ ac qc � � T a 1 ... aD ¯ 3 1 � D T q 1 ... qD 1 1 1 1 λ 2 2 ¯ T , T ��� 1 � T a 1 a 2 a 3 ¯ δ ... T p 1 p 2 p 3 . . . 1 3 2 2 2 2 3 3 3 3 1 1 1 1 3 2 2 8

Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition Invariant Actions for Tensor Models The most general single trace tensor model D T ) = 1 � � � S ( T , ¯ T a 1 ... a D ¯ t B Tr B ( ¯ δ a c q c + T q 1 ... q D T , T ) λ c =1 B � TdT ] e − N D − 1 S ( T , ¯ [ d ¯ T ) Z TM ( t B ) = Feynman graphs: “vertices” B . Gaussian integral: Wick contractions of T and ¯ T (“propagators”) → dashed edges to which we assign the fictitious color 0. T 1 1 0 2 � 2 2 e − N D − 1 � c =1 δ ac qc � � T a 1 ... aD ¯ 2 3 1 � D 3 3 T q 1 ... qD T λ 3 1 1 1 1 ¯ T , T 2 ��� 2 � T a 1 a 2 a 3 ¯ δ ... . . . T p 1 p 2 p 3 1 1 � �� � 3 2 2 2 2 λ 3 ∼ ND − 1 δ a 1 p 1 δ a 2 p 2 δ a 3 p 3 3 3 3 1 1 1 1 3 2 2 8