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10 jui 2019 · Magnificent Four with Colors and Beyond Eleven Dimensions Nikita Nekrasov They may teach us about eleven dimensional super-gravity

Magnicent Four with Colors

NIKITA NEKRASOV

Moscow, June 10, 2019

Magnicent Four with Colors, and Beyond Eleven Dimensions

Nikita Nekrasov

Simons Center for Geometry and Physics

ITMPh

Mosco w

June 10, 2019

A popular approach to quantum gravity

is to approximate the space-time geometry by some discrete structure

A popular approach to quantum gravity

is to approximate the space-time geometry by some discrete structure Then develop tools for summing over these discrete structures

A popular approach to quantum gravity

is to approximate the space-time geometry by some discrete structure Then develop tools for summing over these discrete structures

Tuning the parameters so as to get, in some limit

Smooth geometries

To some extent

two dimensional quantum gravity is successfully solved in this fashion using matrix models log Z

NNdM eNtrV(M)X

fatgraphs$triangulatedRiemannsurfaces

To some extent

two dimensional quantum gravity is successfully solved in this fashion using matrix models log Z

NNdM eNtrV(M)1X

g=0N 22gX
genusg Riemannsurfaces

Going up in dimension

proves dicult

Going up in dimension

proves dicult { the obvious generalization of a matrix model is the so-called tensor theory M ij!ijk There is no analogue of genus expansion for general three-manifolds

Going up in dimension

proves dicult { the obvious generalization of a matrix model is the so-called tensor theory M ij!ijk There is no analogue of genus expansion for general three-manifolds However an interesting largeNscaling has been recently found

In the context of the SYK model,g7!Gurau index

In this talk

I will discuss two models of random three dimensional geometries

In this talk

I will discuss two models of random three dimensional geometries I do not claim they quantize three dimensional Einstein gravity

In this talk

I will discuss two models of random three dimensional geometries They may teach us about eleven dimensional super-gravity

In this talk

I will discuss two models of random three dimensional geometries They may teach us about eleven dimensional super-gravity

M-theory, and beyond

Geometries from partitions

One way to generate ad-dimensional random geometry

Is from some local growth model ind+ 1-dimensions

Geometries from partitions

For example, start with the simplest \mathematical" problem

Geometries from partitions

For example, start with the simplest problem: counting natural numbers

1;2;3;:::

Geometries from partitions

For example, start with the simplest problem: accounting

1;2;3;:::}}}}

Geometries from partitions

Accounting for objects without structure

1;2;3;:::}}}}

Geometries from partitions

Now add the simplest structure: partitions of integers (1); (2);(1;1); (3);(2;1);(1;1;1);:::

Geometries from partitions

Now add the simplest structure: partitions of integers (1); (2);(1;1); (3);(2;1);(1;1;1);:::}}}}

Geometries from partitions

The structure: partitions of integers as bound states (1); (2);(1;1); (3);(2;1);(1;1;1);:::}}}}

Partitions as growth model

Gardening and bricks}}}}

Partitions as growth model

Gardening and bricks}}}}

Partitions as growth model

Gardening and bricks}}}}

Partitions as growth model

Gardening and bricks}}}}

Partitions as growth model

Gardening and bricks}}}}

Partitions as growth model

Partition (1) made of brick}}}}

Partitions as growth model

Gardening and bricks}}}}

Partitions as growth model

Gardening and bricks}}}}

Partitions as growth model

Gardening and bricks}}}}

Partitions as a growth model

Gardening and bricks}}}}

Partitions as a growth model

Partition (2) made of bricks}}}}

Partitions as a growth model

Gardening and bricks}}}}

Partitions as a growth model

Gardening and bricks}}}}

Partitions as a growth model

Gardening and bricks}}}}

Partitions as a growth model

Gardening and bricks}}}}

Partitions as a growth model

Partition (2;1) made of bricks}}}}

Partitions as a growth model

Partition (3;1) made of bricks}}}}

Partitions as a growth model

The probability of a given partition, e.g. (3;1), is determined by the equalityof the chances of jumps from one partition

e.g. from (3;1) to another, e.g. (3;2) or (4;1), or (3;1;1)

Possibilities of growth: Young graph}}}}

Partitions as a growth model

Thus the probabilitypof a given partition

is proportional to the # of ways it can be built out of the nothing times the # of ways it can be reduced to nothing

Partitions as a growth model

Thus the probabilitypof a given partition

is proportional to the # of ways it can be built out of the nothing times the # of ways it can be reduced to nothing: quantum bricks

Plancherel measure: symmetry factors

One can calculate this to be equal to

p =dim()jj! 2 2jje2 =e2 Y

2hooklength of!

For example,p3;1=11

2224212=164

Supersymmetric gauge theory

Remarkably,pis the simplest example

of an instanton measure p = (sDetA)12 i.e. the one-loop (exact) contribution of an instantonA=A inN= 2 supersymmetric gauge theory

Supersymmetric gauge theory and random partitions

ConsiderN= 2 supersymmetric gauge theory in four dimensions

The elds of a vector multiplet are

A m,m= 1;2;3;4;i,= 1;2 andi= 1;2;; with the supersymmetry transformations, schematically

A+; ;

(;)(F++DA;F+DA) + [;]

Supersymmetric gauge theory and random partitions

Supersymmetric partition function of the theory can be computed exactly by localizing on the-invariant eld congurations, i.e.F+ A= 0 Z=X k 2NkZ M kinstanton measure of some eective measure, including the regularization factors

Supersymmetric gauge theory and random partitions

The integral over the moduli space can be further simplied by by deforming the supersymmetry using the rotational symmetry ofR4 Z=X k 2NkX ;jj=kp The deformed path integral is computed by exact saddle point analysis withenumerating the saddle points

Supersymmetric gauge theory and random partitions

Generic rotation ofR4:grot= exp0

B@0"10 0

"10 0 0

0 0 0"2

0 0"201

C CA Z=X k 2NkX ;jj=kp ("1;"2) The deformed path integral is computed by exact saddle point