RAQIS-2016 NIKITA NEKRASOV Geneva, August 23, 2016 RAQIS2016 - - PowerPoint PPT Presentation

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RAQIS-2016

NIKITA NEKRASOV Geneva, August 23, 2016

RAQIS’2016 Geneva August 23 Nikita Nekrasov

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Plan

• Quantum integrable systems from gauge theories
• Gauge theories from branes
• Y, Q and ˜

Q-observables

• A1-type models
• Gauge origami model
• Observables from gauge origami
• Conclusions and outlook

♦

Bethe/gauge correspondence

Gauge theories with N = (2, 2) d = 2 super-Poincare invariance ⇔ Quantum integrable systems ♦

Bethe/gauge correspondence

NN, S.Shatashvili circa 2007

Supersymmetric vacua (in finite volume) of gauge theory ⇔ Stationary states of the QIS ♦

Bethe/gauge correspondence

Qa|vacα = 0, ¯ Q ˙

a|vacα = 0,

• Hgauge|vacα = 0

|vacα ⇔ Ψα Oi|vacα = ǫi|vacα ⇔

• HiΨα = ǫiΨα

♦

Bethe/gauge correspondence

Twisted chiral ring ⇔ quantum integrals of motion Oi ∼

• k

hi,k (2πi)kk!Trσk ⇔

• Hi

dOi = {Q, Ri }, [Oi , Oj ] = 0 Hvac is separated by the spectrum of {Oi }

♦

Bethe/gauge correspondence

Effective twisted superpotential W(σ) ⇔ The YY-functional ♦

Bethe/gauge correspondence

Equations for vacua from minimization of the effective potential ∂ W(σ) ∂σi = 2πini , i = 1, . . . , r ♦

Bethe/gauge correspondence

Equations for vacua from minimization of the effective potential ∂ W(σ) ∂σi = 2πini , i = 1, . . . , r ⇔ Bethe equations of the QIS ♦

Bethe/gauge correspondence

Twisted chiral ring = deformation of the theory

• Wtree −

→ Wtree +

• k

TkOk ∂ Weff(σ; T ) ∂σi = 2πini , i = 1, . . . , r ǫk = ∂ Weff(σ; T ) ∂Tk ⇔ Bethe eigenvalues of the QIS ♦

Bethe/gauge correspondence

From Yang-Mills to Yang-Baxter and back ♦

Bethe/gauge correspondence

U(N) gauge theory: FI term r and θ-angle t = θ + i r 2π , q = exp 2πit L fundamental hypers (Qf , ˜ Qf ): L twisted masses: µf , f = 1, . . . , L

• L. Alvarez-Gaume and D. Freedman ’1983
• S. J. Gates, C. M. Hull and M. Rocek ’1984

twisted mass u corresponding to (Qf , ˜ Qf ) → (eiαQf , eiα ˜ Qf ) = ⇒ Vacua on the Coulomb branch

L

• f =1

σi − µf + u σi − µf − u = q

• j=i

σi − σj + 2u σi − σj − 2u (1) ♦

Bethe/gauge correspondence

Length L spin chain, N magnon sector λi = iσi 2u rapidity of magnons νf = iµf 2u inhomogeneities q twist parameter Sf +L = q− σ3

2 Sf q σ3 2

Bethe equations:

L

• f =1

λi − νf + i

2

λi − νf − i

2

= q

• j=i

λi − λj + i λi − λj − i ♦

Bethe/gauge correspondence

Length L spin chain, N magnon sector Bethe equations:

L

• f =1

λi − νf + i

2

λi − νf − i

2

= q

• j=i

λi − λj + i λi − λj − i NB the magic:

N

• i=1

dlog (Equationi) ∧ dλi = 0 Hence there is a potential: the Yang-Yang (YY) functional

• C. N. Yang and C. P. Yang’ 1969

♦

Bethe/gauge correspondence

Coulomb moduli ↔ rapidities of magnons Twisted masses of fundamentals ↔ inhomogeneities Instanton amplitude ↔ twist parameters ♦

Bethe/gauge correspondence

Bethe/vacuum equations:

L

• f =1

λi − νf + i

2

λi − νf − i

2

= q

• j=i

λi − λj + i λi − λj − i ⇔

L

• f =1

σi − µf + u σi − µf − u = q

• j=i

σi − σj + 2u σi − σj − 2u (2) NB the analyticity: all the parameters are naturally complex in the gauge theory setting The first hint a complex phase space is in play ♦

Bethe/gauge correspondence

Spin 1

2 – very quantum system, cannot see the phase space

♦

Bethe/gauge correspondence

What about quantum systems with classical limits? ♦

Bethe/gauge correspondence

What about quantum systems with classical limits? Gauge theory parameter? ↔ ♦

Quantum mechanics from 4d gauge theory

It turns out one should look at four dimensional theories e.g. N = 2 super-Yang-Mills theory in four dimensions ♦

Quantum mechanics from 4d gauge theory

It turns out one should look at four dimensional theories e.g. N = 2 super-Yang-Mills theory in four dimensions But view them as two dimensional theories ♦

Quantum mechanics from 4d gauge theory

It turns out one should look at four dimensional theories e.g. N = 2 super-Yang-Mills theory in four dimensions But view them as two dimensional theories = ⇒ SO(2) R-symmetry ♦

Quantum mechanics from 4d gauge theory

It turns out one should look at four dimensional theories e.g. N = 2 super-Yang-Mills theory in four dimensions But view them as two dimensional theories = ⇒ SO(2) R-symmetry

similar to SO(3) acting on cohomology of Kahler manifolds and SO(5) acting on cohomology of hyper-Kahler

♦

Quantum mechanics from 4d gauge theory

Four dimensional N = 2 theory Viewed as two dimensional theory with SO(2) R-symmetry Turn on the twisted mass for this symmetry = ⇒

NN, S.Shatashvili, 2009

♦

Quantum mechanics from 4d gauge theory

Four dimensional N = 2 theory Viewed as two dimensional theory with SO(2) R-symmetry Turn on the twisted mass for this symmetry = ⇒ Compactify the 1 + 1 dimensional spacetime on R × S1 (finite volume) ♦

Quantum mechanics from 4d gauge theory

Four dimensional N = 2 theory without With finite size effects R1,2 × S1 at low energy = 3d sigma model with hyperk¨ ahler target space The phase space of Seiberg-Witten integrable system ♦

Quantum mechanics from 4d gauge theory

Four dimensional N = 2 theory Compactified onto D × S1 × R1 (cigar × circle × time axis) With Ω-deformation along the cigar D ♦

Quantum mechanics from 4d gauge theory

Four dimensional N = 2 theory Compactified onto D × S1 × R1 (cigar × circle × time axis) With Ω-deformation along the cigar D ♦

Quantum mechanics from 4d gauge theory

Four dimensional N = 2 theory Compactified onto D × S1 × R1 (cigar × circle × time axis) With Ω-deformation along the cigar D At low energy ♦

Quantum mechanics from 4d gauge theory

Four dimensional N = 2 theory Compactified onto D × S1 × R1 (cigar × circle × time axis) × S1 × R1 at low energy ↓ × R1 Becomes 2d sigma model on R+ × R1 ♦

Quantum mechanics from 4d gauge theory

Four dimensional N = 2 theory Compactified onto D × S1 × R1 (cigar × circle × time axis) × S1 × R1 at low energy ↓ × R1 Becomes 2d sigma model on R+ × R1 = ⇒ deformation quantization

NN, E.Witten’2009 Using A.Kapustin,D.Orlov’s branes’2003 introduced by F. Bayen, L. Boutet de Monvel, M. Flato,

• C. Fronsdal, A. Lichnerowicz et D. Sternheimer’78,

existence of formal def.quant. shown by M. Kontsevich’99 using a sigma model further explored by A. Cattaneo and G. Felder’99

♦

♦♦♦

Despite the construction above The dictionary is largely unknown Study the examples

♦

♦♦♦

Today’s goal Translate the notion of Q-operator to gauge theory

♦

♦♦♦

Today’s goal Translate the notion of Q-operators to gauge theory

♦

♦♦♦

Large supply of examples D-brane realizations of supersymmetric gauge theories

♦

♦♦♦

D-brane realization: open strings

become gauge bosons and their superpartners at low energies ♦

♦♦♦

D-brane realization: scalars in the gauge multiplet

describe fluctuations in the transverse directions ♦

♦♦♦

D-brane realization: instantons as additional branes

additional open strings = ⇒ ADHM data ♦

♦♦♦

SUPERSYMMETRIC GAUGE THEORY

in FOUR DIMENSIONS + equivariant localisation = ⇒

statistical mechanical model

♦♦♦

Seed data for a theory on one copy of C2

Vector space N ≈ Cn Coulomb parameters: a = (a1, . . . , an) ∈ Cn Ω-deformation parameteres: (ε′, ε′′|˜ ε′, ˜ ε′′) ∈ C4 ε′ + ε′′ + ˜ ε′ + ˜ ε′′ = 0 Fugacity q ∈ C×, |q| < 1

♦♦♦

Seed data for a theory on one copy of C2

The Coulomb parameters are the growth points: a1, . . . , an ∈ C

♦♦♦

Random variables

The growth points sprout descendents: aα + ε′(i − 1) + ε′′(j − 1) ∈ C, i, j ≥ 1, α = 1, . . . , n

♦♦♦

Random variables

The growth points sprout descendents: aα + ε′(i − 1) + ε′′(j − 1) ∈ C, i, j ≥ 1, α = 1, . . . , n

♦♦♦

Random variables

The growth points sprout descendents: aα + ε′(i − 1) + ε′′(j − 1) ∈ C, i, j ≥ 1, α = 1, . . . , n

♦♦♦

Random variables

The growth points sprout descendents: aα + ε′(i − 1) + ε′′(j − 1) ∈ C, i, j ≥ 1, α = 1, . . . , n

♦♦♦

Random variables

The growth points sprout descendents: aα + ε′(i − 1) + ε′′(j − 1) ∈ C, i, j ≥ 1, α = 1, . . . , n

♦♦♦ Random variables: n partitions, Λ =

• λ(1), . . . , λ(n)

λ(α) = (λ(α)

1

≥ λ(α)

2

≥ . . . λ(α)

ℓ(λ(α)) > 0)

|λ(α)| =

ℓ(λ(α))

• i=1

λ(α)

i

α = 1, . . . , n

♦♦♦ Random variables: Young diagrams λ(1), . . . , λ(n)

♦♦♦ Random variables: |λ| colored points zα,i,j In the growth picture zα,i,j = aα + ε′(i − 1) + ε′′(j − 1) 1 ≤ i ≤ ℓλ(α), 1 ≤ j ≤ λ(α)

i

blue α = 1, . . . , n

♦♦♦ Random variables: n partitions Λ =

• λ(1), . . . , λ(n)

The role of the fugacity: extra weight q|Λ| =

n

• α=1

ℓλ(α)

• i=1

qλ(α)

i

♦♦♦

Boltzmann weights

Energy of charges µΛ =

• c∈∂Λ,˜

c∈∂Λ

e−ch(c)ch(˜

c)·(uε′,ε′′(zc−z˜

c)−uε′,ε′′(zc−z˜ c+˜

ε′))

with the charges ch(c) = ±1 as in the pictures

♦♦♦

Boltzmann weights

Energy of charges µΛ =

• c∈∂Λ,˜

c∈∂Λ

e−ch(c)ch(˜

c)·(uε′,ε′′(zc−z˜

c)−uε′,ε′′(zc−z˜ c+˜

ε′))

with the charges ch(c) = ±1 as in the pictures ch(c) = +1 for new growth points ch(c) = −1 for decay points

The potential uε′,ε′′ (z) = d ds

• s=0

1 Γ(s) ∞ dt t ts e−tz (1 − etε′)(1 − etε′′) solves uε′,ε′′ (z) + uε′,ε′′ z − ε′ − ε′′ − uε′,ε′′ z − ε′ − uε′,ε′′ z − ε′′ = log (z) (3) ♦

♦♦♦

Boltzmann weights

Energy of charges µΛ =

• c∈∂Λ,˜

c∈∂Λ

e−ch(c)ch(˜

c)·(uε′,ε′′(zc−z˜

c)−uε′,ε′′(zc−z˜ c+˜

ε′))

• A0-type model

♦♦♦

Boltzmann weights: generalization

Graph γ with the set of vertices Vertγ and the set of edges Edgeγ n → (ni)i∈Vertγ , Λ =

• λ(α)α=1...n

→ (Λi)i∈Vertγ =

• λ(i,α)α=1,...,ni

i∈Vertγ

Assignment of masses me ∈ C to the edges e ∈ Edgeγ µλ =

• i∈Vertγ
• c∈∂λi,˜

c∈∂λi

e−ch(c)ch(˜

c)uε′,ε′′(zc−z˜

c)

× ×

• e∈Edgeγ
• c∈∂λs(e),˜

c∈∂λt(e)

ech(c)ch(˜

c)uε′,ε′′(zc−z˜

c+me)

This is a γ-quiver model

♦♦♦ When γ is of the ADE type µλ =

• i∈Vertγ
• c∈∂λi,˜

c∈∂λi

e−ch(c)ch(˜

c)uε′,ε′′(zc−z˜

c) ×

×

• e∈Edgeγ
• c∈∂λs(e),˜

c∈∂λt(e)

ech(c)ch(˜

c)uε′,ε′′(zc−z˜

c+me)

(4) The γADE-quiver model can be obtained from the A0 model by “orbifolding”

♦♦♦ When γ is of the ADE type µλ =

• i∈Vertγ
• c∈∂λi,˜

c∈∂λi

e−ch(c)ch(˜

c)uε′,ε′′(zc−z˜

c) ×

×

• e∈Edgeγ
• c∈∂λs(e),˜

c∈∂λt(e)

ech(c)ch(˜

c)uε′,ε′′(zc−z˜

c+me)

(5) by a subgroup Γγ of SU(2)

♦♦♦

Proliferation of fugacities

q − → (qi)i∈Vertγ q|λ| − →

• i∈Vertγ

q|λi|

i

♦

♦♦♦

New models by taking limits

e.g. qi → 0 for some i ∈ Vertγ and/or ai,α → ∞ for some (i, α) ♦

♦♦♦

New models by taking limits

e.g. qi → 0 for some i ∈ Vertγ and/or ai,α → ∞ for some (i, α) For example, A1 model ♦

♦♦♦ The A1 model Random variable, again Λ =

• λ(1), . . . , λ(n)

♦

♦♦♦

The A1 model

Random variable, again Λ =

• λ(1), . . . , λ(n)

The measure is different µΛ =

• c∈∂Λ,˜

c∈∂Λ

e−ch(c)ch(˜

c)·uε′,ε′′(zc−z˜

c) ×

• c∈Λ

P(zc) , Mass polynomial P(x) =

2n

• f =1

(x − mf ) mf ∈ C ♦

♦♦♦

The A1 model

Random variable, again Λ =

• λ(1), . . . , λ(n)

The measure is different µΛ =

• c∈∂Λ,˜

c∈∂Λ

e−ch(c)ch(˜

c)·uε′,ε′′(zc−z˜

c) ×

• c∈Λ

P(zc) , Mass polynomial P(x) =

2n

• f =1

(x − mf ) = P+(x + ε1 + ε2)P−(x) mf ∈ C P±(x) =

n

• f =1

(x − m±

f )

♦

♦♦♦

The A1 model is a limit of the A2-model

q±1 → 0 a±1,α → m±

α

Masses from frozen Coulomb moduli α = 1, . . . n q0 → q q|Λ|µΛ =

• c∈∂Λ,˜

c∈∂Λ

e−ch(c)ch(˜

c)·uε′,ε′′(zc−z˜

c) ×

• c∈Λ

(qP(zc)) , ♦

♦♦♦

OBSERVABLES

Yi(x) = xni exp −

∞

• k=1

1 kxk TrΦk

i

↑ four dimensional definition ♦

♦♦♦

Y-operators the statistical model

Yi(x)[λ] =

ni

• α=1
• ∈∂+λ(i,α)(x − zc)
• ∈∂−λ(i,α)(x − zc)

zc = aα + ε′(i − 1) + ε′′(j − 1), c = (α, ), = (i, j) are the + charges, are the − charges #{ ∈ ∂+λ} − #{ ∈ ∂−λ} = 1 ♦

♦♦♦

Q-operators in the statistical model

Q(1)

i

(x)[λ] =

ni

• α=1

(−ε′)

x−ai,α ε′

Γ

• − x−ai,α

ε′

• c∈Λ(i)

x − zc − ε′′ x − zc Q(2)

i

(x)[λ] =

ni

• α=1

(−ε′′)

x−ai,α ε′′

Γ

• − x−ai,α

ε′′

• c∈Λ(i)

x − zc − ε′ x − zc ♦

♦♦♦

Q versus Y

Q(1)

i

(x)[λ] =

ni

• α=1
• −ε′ x−ai,α

ε′

Γ

• −

x−ai,α ε′

• c∈Λ(i)

x − zc − ε′′ x − zc Q(2)

i

(x)[λ] =

ni

• α=1
• −ε′′ x−ai,α

ε′′

Γ

• −

x−ai,α ε′′

• c∈Λ(i)

x − zc − ε′ x − zc

Q(1)

i

(x) Q(1)

i

(x − ε′) = Q(2)

i

(x) Q(2)

i

(x − ε′′) = Yi(x) ♦

♦♦♦

Nonperturbative Dyson-Schwinger equation

NN’15

in the A1 case

• Y(x + ε′ + ε′′) + q P(x)

Y(x)

• has no singularities in x

♦

♦♦♦

Nonperturbative Dyson-Schwinger equation

in the A1 case

• Y(x + ε′ + ε′′) + q P(x)

Y(x)

• = T(x)

is a degree n polynomial in x ♦

♦♦♦

Back to quantum integrable system

Take a limit ε′ → 0, ε′′ = - finite

NS = NN-Shatashvili limit’09

Limit shape phenomenon

NN, A. Okounkov’03 NN, V. Pestun, S. Shatashvili’14

Y(x)−1 = Y (x)−1, Y(x) = Y (x) Y (x + ) + q P(x) Y (x) = T(x) is a degree n polynomial in x ♦

♦♦♦

Back to quantum integrable system

Take a limit ε′ → 0, ε′′ = - finite Limit shape phenomenon = ⇒ linear difference equation on Q(x) Q(2)(x)−1 = Q(x)−1, Q(2)(x) = Q(x) Q(x + ) + q P(x)Q(x − ) = T(x)Q(x) The celebrated T − Q equation

• R. Baxter
• L. Faddeev, L. Takhtadzhan
• E. Sklyanin

♦

♦♦♦

Back to quantum integrable system

Take a limit ε′ → 0, ε′′ = - finite The celebrated T − Q equation Q(x + ) + q P(x)Q(x − ) = T(x)Q(x) Has two solutions over quasiconstants ♦

♦♦♦

Back to quantum integrable system

Take a limit ε′ → 0, ε′′ = - finite The celebrated T − Q equation Q(x + ) + q P(x)Q(x − ) = T(x)Q(x)

• Q(x + ) + q P(x)

Q(x − ) = T(x) Q(x) Has two solutions over quasiconstants Quantum (discrete) Wronskian W (x) = Q(x + /2) Q(x − /2) − Q(x − /2) Q(x + /2) = quasi-constant up to normalization ♦

♦♦♦

Back to quantum integrable system

Take a limit ε′ → 0, ε′′ = - finite The celebrated T − Q equation Q(x + ) + q P(x)Q(x − ) = T(x)Q(x)

• Q(x + ) + q P(x)

Q(x − ) = T(x) Q(x) Both solutions are used in the functional Bethe ansatz

• E. Sklyanin

Where is Q(x) in gauge theory?

♦

♦♦♦

Nonperturbative Dyson-Schwinger equation

in the A1 case

• Y(x + ε′ + ε′′) + q P(x)

Y(x)

• = T(x)

has no singularities in x

can be shown directly (residue matching)

♦

♦♦♦

Nonperturbative Dyson-Schwinger equation

in the A1 case

• Y(x + ε′ + ε′′) + q P(x)

Y(x)

• = T(x)

has no singularities in x

can be shown conceptually

♦

♦♦♦

Nonperturbative Dyson-Schwinger equation

in the A1 case X1,x = Y(x + ε′ + ε′′) + q P(x) Y(x)

A1 fundamental qq-character

♦

♦♦♦

THE qq-CHARACTERS Xw,ν = PARTITION FUNCTION

OF A POINT-LIKE DEFECT Dw,ν ♦

♦♦♦

THE qq-CHARACTERS Xw,ν = PARTITION FUNCTION

OF A POINT-LIKE DEFECT Dw,ν which can be engineered using intersecting branes ♦

♦♦♦

Brane-world scenarios

♦

♦♦♦

Local model: ∪a<b C2

ab ⊂ C4

For example, when 1 ≤ a, b ≤ 3 ♦

♦♦♦

Local IIB string model:

D5’s and D5’s spanning

• ∪a<b C2

ab

• × R1,1 ⊂ R1,9

When 1 ≤ a, b ≤ 4 = ⇒ (0, 2) susy in R1,1 ♦

♦♦♦

Chan-Paton spaces Nab for the stack of branes spanning C2

ab

with complex coordinates za, zb ♦

♦♦♦

Useful pictures

n12 = dimN12 branes along C2

12

♦

♦♦♦

Useful pictures

n12 = dimN12 branes along C2

12 and n23 = dimN23 branes alond C2 23

♦

♦♦♦ n12 = dimN12 branes along C2

12,

n23 = dimN23 branes along C2

23,

n13 = dimN13 branes along C2

13,

n24 = dimN23 branes along C2

24,

n14 = dimN14 branes along C2

14,

n34 = dimN34 branes along C2

34

♦♦♦ = ♦

♦♦♦

Integrate out the degrees of freedom

• n all but one of the stacks

To produce observables on the remaining stack of branes ♦

♦♦♦

How one integrates out the degrees of freedom? Using unbroken supersymmetry and localisation

♦

♦♦♦

Degrees of freedom associated with k instantons Chan-Paton spaces

K = Ck=#instanton charge, Nab = Cnab ♦

♦♦♦

Degrees of freedom associated with instantons:

Rectangular complex Iab, Jab matrices, 1 ≤ a < b ≤ 4 ♦

♦♦♦

Degrees of freedom associated with instantons

Square complex matrices Ba, a = 1, . . . , 4 ♦

♦♦♦

Local ADHM data

♦

♦♦♦

Partition function of gauge origami

♦

♦♦♦

Partition Z-function of gauge origami

Equivariant integral over the space of solutions

• f generalized ADHM equations

♦

♦♦♦

Generalized ADHM equations

µab + εabcdµ†

cd = 0

µab = [Ba, Bb] + IabJab ♦

♦♦♦

Generalized ADHM equations I.

µab + εabcdµ†

cd = 0

µab = [Ba, Bb] + IabJab

• a

[Ba, B†

a] +

• a<b

IabI †

ab − J† abJab = ζ · 1K

7 hermitian k × k matrix equations ♦

♦♦♦

Generalized ADHM equations I.

µab + εabcdµ†

cd = 0,

for all 1 ≤ a < b ≤ 4, where µab = [Ba, Bb] + IabJab and µ ≡

• a

[Ba, B†

a] +

• a<b

IabI †

ab − J† abJab = ζ · 1K

7 hermitian k × k matrix equations Divide by the U(k) action ♦

♦♦♦

Generalized ADHM equations II.

BaIbc + εabcdB†

dJ† bc = 0

for all a, b, c, d, s.t. εabcd = 0 2k

a<b nab complex equations

♦

♦♦♦

Generalized ADHM equations III.

JabIcd − I †

abJ† cd = 0

for all a, b, c, d, s.t. εabcd = 0 JabBp−1

b

Ibc = 0 for all 1 ≤ a < b < c ≤ 4, and p ≥ 1 ♦

♦♦♦

THE GAUGE PARTITION FUNCTION

by equivariant localisation becomes

THE STATMECH PARTITION FUNCTION

• f the ensemble of 6 sets
• f random colored partitions Λ(A), A = ab,

1 ≤ a < b ≤ 4 Λ(A) =

• λ(A,α)

1≤α≤nA

, , . . . , ♦

♦♦♦

THE GAUGE PARTITION FUNCTION

by equivariant localisation becomes

THE STATMECH PARTITION FUNCTION

• f the ensemble of 6 sets
• f random colored partitions Λ(A), A = ab,

1 ≤ a < b ≤ 4 Λ(A) =

• λ(A,α)

1≤α≤nA

6 types of growth point clusters a(A,α) ♦

♦♦♦

THE GAUGE PARTITION FUNCTION

by equivariant localisation becomes

THE STATMECH PARTITION FUNCTION

6 types of growth clusters a(A,α) ♦

♦♦♦

GAUGE ORIGAMI BOLTZMANN WEIGHTS Z [

a, ε, q] =

(Λ)

• A

q|Λ(A)| µ(Λ) ( a, ε) A = 12, 13, 14, 23, 24, 34 Λ(A) =

• λ(A,α)nA

α=1

♦

♦♦♦

GAUGE ORIGAMI BOLTZMANN WEIGHTS

µ(Λ) ( a, ε) =

• A=ab
• c∈∂Λ(A),˜

c∈∂Λ(A)

ech(c)ch(˜

c)(uεa,εb(zc−z˜

c+εd)−uεa,εb(zc−z˜ c))

×

• A=ab,B,A∩B={a}
• c∈∂Λ(A),˜

c∈∂Λ(B)

Γ zc − z˜

c − εa − εb

εa ch(c)ch(˜

c)

×

• A=ab,B,A∩B=∅
• c∈∂Λ(A),˜

c∈∂Λ(B)

(zc − z˜

c − εa − εb)ch(c)ch(˜ c)

(6)

d / ∈ A

Λ(A) =

• λ(A,α)nA

α=1

♦

♦♦♦

THE MAIN CLAIM

with a suitably defined perturbative factors

the partition Z-function

• f gauge origami

is an entire function of all

a<b nab Coulomb parameters

♦

♦♦♦

APPLICATIONS THE NONPERTURBATIVE DS EQUATIONS

♦

♦♦♦

APPLICATIONS BPS/CFT CORRESPONDENCE

NN, 2002-2004

Correlators of chiral observables in four dimensional supersymmetric theories are holomorphic blocks (form-factors)

• f some conformal field theory

(or a massive integrable deformation thereof) in two dimensions

♦

♦

Z-FUNCTIONS

OF A-TYPE QUIVER THEORIES, WITH OR WITHOUT DEFECTS

OBEY THE BPZ/KZ-TYPE EQUATIONS

• f chiral algebras for 2d CFT and SCFTs

♦

♦♦♦♦

MORE APPLICATIONS

♦♦♦♦

♦♦♦♦

ORBIFOLD with respect to a subgroup

H ⊂ U(1)3 ⊂ SU(4)

Can also use non-abelian subgroups ˜ H ⊂ SU(2) × U(1) × SU(2) ⊂ SU(4), not in this lecture, though

♦♦♦♦

♦♦♦♦

ORBIFOLD with respect to a subgroup

H ⊂ U(1)3 ⊂ SU(4) Chan-Paton spaces NA − →

• ω

NA,ω ⊗ Rω Fugacities q − → (qω) ω’s ↔ Rω = irreps of H ♦♦♦♦

♦♦♦♦

For example, do the Z3-orbifold

(z1, z2, z3, z4) →

• z1, z2, e

2πim 3 z3, e− 2πim 3 z4

• ,

m = 0, 1, 2 ♦♦♦♦

♦♦♦♦

For example, do the Z3-orbifold

(z1, z2, z3, z4) →

• z1, z2, e

2πim 3 z3, e− 2πim 3 z4

• ,

m = 0, 1, 2 Three fugacities q−1, q0, q+ ♦♦♦♦

♦♦♦♦

For example, do the Z3-orbifold

(z1, z2, z3, z4) →

• z1, z2, e

2πim 3 z3, e− 2πim 3 z4

• ,

m = 0, 1, 2 Three fugacities q−1, q0, q+ ♦♦♦♦

♦♦♦♦

The Z3-orbifold, in the limit q±1 → 0

(z1, z2, z3, z4) →

• z1, z2, e

2πim 3 z3, e− 2πim 3 z4

• ,

m = 0, 1, 2 ♦♦♦♦

♦♦♦♦

The qq-character X1,x

N±1

12 = M±e±ε3 ,

N0

12 = N ,

N0

34 = ex

♦♦♦♦

♦♦♦♦

The qq-character X1,x

N±

12 = M±e±ε3,

N0

12 = N,

N0

34 = ex

The two terms in Y(x + ε1 + ε2) + q P(x)

Y(x) come from

k34 = 0 and k34 = 1, respectively ♦♦♦♦

♦♦♦♦

The ˜ Q(x)-operator ˜ Q(2)

1,x

N±1

12 = M±e±ε3,

N0

12 = N,

N0

23 = ex

Surface operator

♦♦♦♦

♦♦♦♦

The ˜ Q(x)-operator ˜ Q(2)

1,x

Explicitly ˜ Q(2)

1,x = ∞

• n=0

qn Q(2)

1,x−ε2

Q(2)

1,x+nε2+ε1Q(2) 1,x+(n−1)ε2 n

• j=1
• 1 + ε1

jε2

• P(x + (j − 1)ε2)

♦♦♦♦

♦♦♦♦

The Q(x)-operator Q(2)

1,x

N±1

12 = M±e±ε3,

N0

12 = N,

N+

23 = ex+ε3

Another surface operator

♦♦♦♦

♦♦♦♦

The QIS limit ε1 → 0, ε2 →

˜ Q(2)

1,x = ˜

Q(x) , Q(2)

1,x = Q(x)

˜ Q(x) =

∞

• n=0

qn Q(x − ) Q(x + n)Q(x + (n − 1))

n

• j=1

P(x + (j − 1)) ♦♦♦♦

♦♦♦♦

The QIS limit ε1 → 0, ε2 →

˜ Q(x) =

∞

• n=0

qn Q(x − ) Q(x + n)Q(x + (n − 1))

n

• j=1

P(x + (j − 1)) So that ˜ Q(x)Q(x) = 1 + qP(x)Q(x − ) ˜ Q(x + ) ♦♦♦♦

♦♦♦♦

The QIS limit ε1 → 0, ε2 →

˜ Q(x) =

∞

• n=0

qn Q(x − ) Q(x + n)Q(x + (n − 1))

n

• j=1

P(x + (j − 1)) So that ˜ Q(x)Q(x) − qP(x)Q(x − ) ˜ Q(x + ) = 1 Quantum Wronskian (up to some normalization) ♦♦♦♦

♦♦♦♦

CONCLUSIONS AND OUTLOOK

The Bethe/gauge dictionary is a powerful tool, which can be used to expand both domains of knowledge: low-energy dynamics of gauge theories, spectra and eigenstates of quantum integrable systems More to follow.... ♦♦♦♦

♦♦♦♦

THANK YOU

♦♦♦♦