Blobbed Topological Recursion for Tensor Models

Transcription

1 Quantum Gravity in Paris Institut Henri Poincaré, Paris 2017

2 Outline The case for a tensor model 4 5

3 Matrix models for 2d quantum gravity Integrals of matrices with Feynman graphs = poly-angulations of surfaces. p = 2 p = 4 p = 3 p = 5 Figure: Example of a poly-angulation and its dual Feynman Graph.

4 Observables of the model: {O p = tr(m p ) p N} Represent boundary states of the triangulated surfaces. Product of O p multiple boundary components. Transition amplitudes = numbers of triangulations of surfaces with corresponding boundaries. Higher dimensions? Tensor models. Generalize the techniques of matrix models to tensor models.

5 Schwinger-Dyson Equations One Matrix model: Z = dm exp( NTrV (M)), V (M) = polynomial potential p max p=2 t px p. Change of variables in matrix integral implies k 1 1 Tr(M k1 1 m )Tr(M m ) m=0 + n Tr(M k i ) i=2 n k j Tr(M k j +k 1 1) j=2 n i=2,i j N Tr(M k 1 V (M)) Tr(M k i ) n Tr(M k i ) = 0 i=2

6 Loop Equations Outline In the perturbative (formal) setting: n i=1 Tr(M ki ) c = g 0 Cumulants have a 1/N expansion. N 2 2g n n Tr(M k i ) g c. i=1 From cumulants dene generating functions of observables: (g, n) s.t. 2g 2+n 2, W g n (x 1,..., x n ) = p i 0 i tr(mp i ) g c i x p i +1 i

7 S-D Equations become Loop equations: W g 1 n+1 (x, x, x I ) + 0 h g J I W h 1+ J (x, x J)W g h 1+ I J (x, x I J ) + W g (x, x n 1 2,..., ˆx i,..., x n ) W g (x n 1 2,..., x n ) x i (x x i ) 2 i I V (x)wn g (x, x I ) + Pn g (x; x 2,..., x n ) = 0. P g n (x; x 2,..., x n ) a polynomial in x.

8 Solution of the loop equations Compute W 0 1 and W 0 2. W 0 (x) = (V (x) V (x) 2 4P 0(x)) 1 The form of W 0 tells us about a function (covering) 1 x : Σ C \ i γ i. In the 1-cut case: V (x) 2 4P 0(x) = M(x) (x a)(x b) 1 M(x) a polynomial not vanishing at a, b. a, b = ramication points. Σ is a sphere. x(z) = a+b + a b (z + 1/z). Dene 2 4 ωn g = Wn g (x(z 1 ),..., x(z n ))dx 1... dx n. One shows ω 0 2(z 1, z 2 ) = dz 1dz 2 (z 1 z 2 ) 2.

9 Solutions of the loop equations Solve the loop equations by the following recurrence formula: ω g n (z 1,..., z n ) = p i Res z p i K(z, z 1 ) [ ω g 1 n+1 (z, ι(z), z 2,..., z n ) + 0 h g J I ω h 1+ J (z, z J) ω g h 1+ I J (ι(z), z I J) ]. with x ι = x, ι(z) = 1/z (ι=deck transformation of the cover) and z K(z, z 1 ) = 1 ι(z) (, z ω0 2 1) 2 ω 0(z) (ι(z)) 1 ω0 1

10 Idea of proof Outline Loop Equations Linear Loop equations (LLE) + Quadratic Loop Equations (QLE). LLE: Introduce S, linear operator acts on forms f (z) by Sf (z) = f (z) + f (1/z). S z1 ω g n (z 1,..., z n ) = 0, (g, n) (0, 1), (0, 2). Obtained from loop equations.

11 Idea of proof Outline QLE: Qn g = ω g 1 (z n+1 1, ι(z 1 ), z 2,..., z n ) + ω1+ J h (z 1, z J ) ω g h 1+ I J (ι(z 1), z I J ) 0 h g J I has a double zero when z 1 x 1 (a, b). Shown from loop equations.

12 Idea of proof Outline Construct element of ker S. "Cauchy Formula": ( z ) ωn g (z 1,..., z n ) = Res ω z p 2(, 0 z 1 ) ωn g (z,..., z n ). p {x 1 (a),x 1 (b)} Using Sω 0 2 = dx(z 1)dx(z 2 ) (x(z 1 ) x(z 2 )) 2 and ι(x 1 (a, b)) = x 1 (a, b), one shows indeed S z1 Res z p p {x 1 (a),x 1 (b)} ( z ) ω2(, 0 z 1 ) ωn g (z,..., z n ) = 0

13 Idea of proof Outline Using QLE one shows Res z p p {x 1 (a),x 1 (b)} p ( z ) ω2(, 0 z 1 ) ωn g (z,..., z n ) = Res K(z, z 1) [ ω g 1 (z, ι(z), z n+1 2,..., z n ) z p + 0 h g J I ω h 1+ J (z, z J) ω g h 1+ I J (ι(z), z I J ) ].

14 A combinatorial representation of solution. z out z in z z ι(z out ) Res zout p i K(z out, z in ) ω 0 2(z, z ) Figure: Building blocks of the Topological Recursion Graphs. g = 0 graphs are trees. Adding loops on these trees g.

15 : the graphs. Integral of tensor variable T, T (C N ) d. Feynman graphs = (d + 1)-colored graphs. Encode PL-manifolds Figure: Vertex of a 4-colored graph and its dual tetrahedra.

16 : the graphs. 3 Every PL pseudo-manifolds have colored graphs representation.

17 Observables of the models are invariants under the natural U(N) d action on tensors. They are indexed by d-colored graphs. {B(T, T ) B {d colored graphs}} Tensor partition function: Z = dtd T e Nd 1 T T +N d 1 V (T, T ), with V (T, T ) = B N 2 (d 2)! ϖ(b) t B B(T, T ).

18 : Topological recursion? n i=0 B i(t, T ) have 1/N expansion. Schwinger-Dyson equation exists. Describe the combinatorics of triangulation in dimensions greater than 2. But no natural set of W g n generating functions...

19 : Topological recursion? Solution: Study simple tensor model! Consider the model with V (T, T ) = d Q m,i (T, T ) = (T î i=1 T ) i (T î λ i 2 Q m,i(t, T ) where T ). d d Figure: Q m,1 invariant

20 . After some work one shows the simplest tensor model reformulates, Z[{α c }, N] = d dm c e N 2 f,hn d c=1 d c=1 tr(m2 c ) e tr log 2 [ 1 d N d 2 2 d c=1 αcmc ]. with M c = 1 (c 1) M c 1 (d c). plenty of matrices M c. We focus here on d = 6 as this implies the following slides.

21 Loop equations Plenty of matrices: W g n W g k, k N6 General loop equations: let us write a part of it... We h 1+q(x, x q )W g h e 1+r (x, x r ) + W g 1 2e (x, x, x 1+k k) g h 0 q+r=k q,r,k N d=6 = Some multi-linear operator on the W h q s.t. 2h 2 + q 2g 2 + k This multi-linear operator does basically two operations: 1 construct combinations of derivatives of W g k e 1. 2 Taylor expand the generating function at in some variables and select one coecient of this Taylor expansion.

22 Blobbed Topological Recursion How do we deal with these equations? We introduce one copy of C by colors and consider variables x c C c. Doing so we can construct Wn g such that if k 1 variables live in C 1, k 2 in C 2,... We have Wn g = W g, k = (k k 1, k 2,..., k d ), k c = n. c The planar resolvent W 0 1 has one cut on each copy of C.

23 Blobbed Topological Recursion Loop equations still involved but look more familiar W g 1 n+1 (x, x, x I ) + + i I 0 h g J I W h 1+ J (x, x J)W g h 1+ I J (x, x I J ) W g (x, x n 1 2,..., ˆx i,..., x n ) W g (x n 1 2,..., x n ) = x i (x x i ) 2 Multi-linear operator on W g n with 2g 2 + n 2g 2 + n.

24 Blobbed Topological Recursion for tensors How to solve? Still exist Linear Loop Equations and Quadratic Loop Equations. LLE: S z1 ω g n (z 1,..., z n ) + O z1 ω g n (z 1,..., z n ) = l g n (z 1,..., z n ), (1) (g, n) (0, 1), (0, 2). Important point: ln g (z 1,..., z n ) only depends on ω g n with 2g 2 + n < 2g 2 + n and is explicitly determined (just quite heavy to write). QLE: Same as before...

25 Blobbed Topological Recursion for tensors Approach(due to Borot and Shadrin): project ω g n on ker(s + O). Call this projection Pω g n. One has Pω g n = Polar part. Hω g n = Holomorphic part. ω g n = Pω g n + Hω g n (2)

26 Polar Part Outline Pω g n = p {x 1 i (a i ),x 1 i since this time we have Then using QLE Res z p (b i )} i=1...d ( z ) ω2(, 0 z 1 ) ωn g (z,..., z n ), (S + O)ω 0 2 = dx(z 1)dx(z 2 ) (z 1 z 2 ) 2. (3) Pω g n = p Res z p K(z, z 1) [ ω g 1 n+1 (z, ι(z), z 2,..., z n ) + 0 h g J I ω h 1+ J (z, z J) ω g h 1+ I J (ι(z), z I J ) ].

27 Holomorphic Part Outline Hωn g = 1 z 2iπ Γ= ω2(, 0 z 1 )ln g (z; z 2,..., z n ), c Γc which is a fully closed expression since it only depends on ω g n 2g 2 + n < 2g 2 + n. with There is a graphical expansion of the full formula. Will not describe it.

28 Expression of observables from W n Question: How to extract expressions of the form m B i (T, T ) i=1 from W g n? Answer: Consider the following invariant: what is its mean value?

29 Expression of observables from W n Perform Wick pairings:

30 For each Wick paired graph, To each cycle f p (i) of color 0i, associate a cycle variable x p (i). To each line l of color 0 associate a "propagator": 1 N 2 d 2 1 c α cx (c), p l where x (c) p l means that the cycle f p (c) to which x p (c) is attached pass through l. Important: The propagator term has to be understood as a formal series in negative powers of N. multiply all propagator weights together and multiply by W n ({cycles variables}). For each Wick paired graph: get a weight. Integrate the cycle variables of these weight around the corresponding cuts. Cycle variable of color c is integrated against a cut Γ c. Then sum all the integrals corresponding to each Wick paired graph.

31 Outline Two possible tracks: Generalize to more/all tensor models? Possible hope "The ABCD of topological recursion" (Andersen,Borot,Chekhov,Orantin). 8 days old paper. However their work applies to quadratic dierential operators, while tensor models, a priori, need d-linear dierential operators. Group Field Theory?

32 Thank you! Enjoy the conference!