Moduli spaces of cold holographic matter Jo ao N. Laia in - - PowerPoint PPT Presentation

Introduction Technique Conclusions

Moduli spaces of cold holographic matter

Jo˜ ao N. Laia

in collaboration with Martin Ammon, Kristan Jensen, Keun-Young Kim and Andy O’Bannon based on arXiv:1208.3197

October 2, 2012

Jo˜ ao N. Laia Moduli spaces of cold holographic matter

Introduction Technique Conclusions

Outline

1 Introduction

Motivation Moduli spaces Results

2 Technique

D3/D7 D3/D5

3 Conclusions

Summary Future directions Speculation

Jo˜ ao N. Laia Moduli spaces of cold holographic matter

Introduction Technique Conclusions Motivation Moduli spaces Results

Motivation

Holography Duality between String Theory in d+1 dimensions and Quantum Field Theory in d-dimensions Why are we using holography? Different way of thinking about quantum systems: new understanding; Strongly coupled quantum systems are hard to describe with standard methods. Two different approaches bottom-up; top-down.

Jo˜ ao N. Laia Moduli spaces of cold holographic matter

Introduction Technique Conclusions Motivation Moduli spaces Results

Deconstructing the title

Cold holographic matter Cold: zero temperature Holographic matter: compressible states from holography Compressible states charge density smooth with chemical potential turn on gauge field component At Examples in nature superfluids solids fermi liquids

Jo˜ ao N. Laia Moduli spaces of cold holographic matter

Introduction Technique Conclusions Motivation Moduli spaces Results

Moduli spaces

We study moduli spaces of cold holographic matter What is a moduli space? Geometric space in which points are objects of a certain kind Instanton moduli space each point of the manifold is a different solution to the self dual equations Moduli space of vacua (in a field theory) each point is a possible ground state String theory connects them

Jo˜ ao N. Laia Moduli spaces of cold holographic matter

Introduction Technique Conclusions Motivation Moduli spaces Results

Moduli space of vacua

Consider N = 4, SU(Nc) gauge theory with N = 2 matter N = 4 vector multiplet (contains Φ1, Φ2, Φ3 − → Nc × Nc matrices) Nf ≪ Nc fundamental hypermultiplets of N = 2 (contain ˜ Qi, Qi − → Nc-legged vectors, i = 1, . . . , Nf ) Superpotential is W = ˜ QiΦ3Qi + Tr(ǫIJKΦIΦJΦK) Minimize it! Coulomb branch: Qi = 0 = ˜ Qi, Φ1, Φ2, Φ3 mutually commuting Higgs branch: Qi, ˜ Qi nonzero, Φ3 = 0

Jo˜ ao N. Laia Moduli spaces of cold holographic matter

Introduction Technique Conclusions Motivation Moduli spaces Results

Moduli space of vacua - brane perspective

Gauge theory content Φ1, Φ2, Φ3 from 3-3 strings ˜ Qi, Qi from 3-7 strings Coulomb branch Higgs branch

Jo˜ ao N. Laia Moduli spaces of cold holographic matter

Introduction Technique Conclusions Motivation Moduli spaces Results

Probe branes

Probe approximation (Nc ≫ Nf ) can be summarized as

[Erdmenger, Evans, Kirsch, Threlfall ’07]

In probe approximation S = SD7 + T7

• P[C4] ∧ F ∧ F

Instanton on D7 − → sources

• P[C4] as if it was a D3

Higgs branch ≡ instanton moduli space

Jo˜ ao N. Laia Moduli spaces of cold holographic matter

Introduction Technique Conclusions Motivation Moduli spaces Results

What we have done

We have studied the Higgs branch in 4ND probe systems, with Nf = 1 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 D3 X X X X D7 X X X X X X X X x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 D3 X X X X D5 X X X X X X x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 D3 X X X X D3′ X X X X

Jo˜ ao N. Laia Moduli spaces of cold holographic matter

Introduction Technique Conclusions Motivation Moduli spaces Results

Probe branes

We’ll always assume Nc ≫ Nf (probe approximation); Nf = 1 D3 leave stack in small numbers: Nc − 1 ∼ Nc (background unchanged) non zero baryon density, zero temperature: cold holographic matter

Jo˜ ao N. Laia Moduli spaces of cold holographic matter

Introduction Technique Conclusions Motivation Moduli spaces Results

Findings

It is known that gauge theory dual to D3/D7 has no moduli space of vacua (when Nf = 1) gauge theories dual to D3/D5 and D3/D3 have a moduli space We will see that, with non zero baryon density moduli space emerges in dual to D3/D7 moduli space still exists in duals to D3/D5 [Chang, Karch ’12] and D3/D3

Jo˜ ao N. Laia Moduli spaces of cold holographic matter

Introduction Technique Conclusions Motivation Moduli spaces Results

Outline

1 Introduction

Motivation Moduli spaces Results

2 Technique

D3/D7 D3/D5

3 Conclusions

Summary Future directions Speculation

Jo˜ ao N. Laia Moduli spaces of cold holographic matter

Introduction Technique Conclusions D3/D7 D3/D5

The D3 background

Consider a stack of Nc D3s x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 D3 X X X X Low energy physics is N = 4 SYM with SU(Nc) gauge group. Describe this brane using holography Nc → ∞, gYM → 0, λ = g2

YMNc fixed

λ → ∞ The background is (R4 = 4πgsNcα′2) ds2 = Z −1/2ηµνdxµdxν + Z 1/2 dr2 + r2ds2

S5

• ,

Z(r) ≡ R4/r4, F5 = 4 R (volAdS5 + volS5) , F5 = dC4

Jo˜ ao N. Laia Moduli spaces of cold holographic matter

Introduction Technique Conclusions D3/D7 D3/D5

D7 probe

We probe this system with a D7 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 D3 X X X X D7 X X X X X X X X The low energy description is N = 4 SYM with gauge group SU(Nc), together with N = 2 fundamental matter. Dual is D7 in the D3 background S7 = −T7

• d8ξ
• det(−P[G]ab + Fab) + 1

2T7

• P[C4] ∧ F ∧ F.

Ansatz A(ξ) = A0(z)dx0 + Ai(z)dzi, (z1, z2, z3, z4) = (x4, x5, x6, x7)

Jo˜ ao N. Laia Moduli spaces of cold holographic matter

Introduction Technique Conclusions D3/D7 D3/D5

D7 probe

Recall: what we will show is the existence of a moduli space On shell action the same for different configurations No dependence on the field theory coordinates integrate them Write the action as an action in 4d (z1, z2, z3, z4) s7 = −T7

• d4z
• det(gij + Z −1/2fij) − 1

8Z −1˜ ǫijklfijfkl

• ,

gij is an effective metric.

[Chen, Hashimoto, Matsuura ’09]

gij ≡ δij − ∂iA0∂jA0, fij ≡ ∂iAj − ∂jAi.

Jo˜ ao N. Laia Moduli spaces of cold holographic matter

Introduction Technique Conclusions D3/D7 D3/D5

The most important formula of the presentation

For general 4 × 4 symmetric G and antisymmetric F

• det (Gij + Fij) ≥
• det Gij + 1

8

• ˜

ǫijklFijFkl

• [Gibbons, Hashimoto ’00]

Inequality saturated for self dual F with respect to metric G Fij = ±1 2ǫijklFkl, ǫijkl ≡ ˜ ǫijkl/

• det Gij.

uses the fact that G and F are 4 × 4 matrices: will be useful for 4ND systems useless for non 4ND systems. No moduli spaces there. Next step Use this to show the existence of degenerate ground states: moduli space!

Jo˜ ao N. Laia Moduli spaces of cold holographic matter

Introduction Technique Conclusions D3/D7 D3/D5

Back to our setting

Our action is reduced to s7 ≤ −T7

• d4z
• det gij + 1

8Z −1 |˜ ǫijklfijfkl| − ˜ ǫijklfijfkl

• .

bound saturated for self dual f wrt effective metric g self dual f extremizes action − → solves equation of motion self dual f maximizes action − → minimizes Helmholtz free energy s7 = −T7

• d4z
• det gij,

Recall: gij ≡ δij − ∂iA0∂jA0 Interpretation

• F ∧ F is a number −

→ D3 brane charge in the D7, sources

• P[C4]

Jo˜ ao N. Laia Moduli spaces of cold holographic matter

Introduction Technique Conclusions D3/D7 D3/D5

Recipe

fij out of the action is very convenient Makes it very easy to get solutions To get explicit solutions solve electrostatic problem. Get A0(z) A0(z) defines the effective metric get Ai(z) from self dual equations In particular, vacuum energy independent of fij (as it should...) But, don’t forget The moduli space exists only if we find normalizable solutions

Jo˜ ao N. Laia Moduli spaces of cold holographic matter

Introduction Technique Conclusions D3/D7 D3/D5

Example

The purpose of the example Show you why there is no moduli space at zero density Show you what happens when we turn on the density Possible A0(z) that describes a compressible state is A′

0(ρ) =

1

• 1 + ρ6/ρ6

, ρ6

0 related to density

ρ is the radial coordinate (boundary at ρ → ∞) solves equations of motion Effective metric is then gij dzidzj = ρ6 ρ6 + ρ6 dρ2 + ρ2ds2

S3 .

Jo˜ ao N. Laia Moduli spaces of cold holographic matter

Introduction Technique Conclusions D3/D7 D3/D5

Effective metric gij dzidzj = ρ6 ρ6 + ρ6 dρ2 + ρ2ds2

S3 .

Metric is conformally flat gij dzidzj = Ω(¯ ρ)2 d ¯ ρ2 + ¯ ρ2ds2

S3

• ,

Ω(¯ ρ) =

• 1 − ρ6

0/(4¯

ρ6) 1/3 Range of new coordinate is ¯ ρ ∈ [2−1/3ρ0, ∞) Solve self dual equation in this background fij = 1 2ǫijklf kl it is conformal − → solve it in flat space range of radius is ¯ ρ ∈ [2−1/3ρ0, ∞) Effectively Non zero density is the same as zero density, but with ball excised

Jo˜ ao N. Laia Moduli spaces of cold holographic matter

Introduction Technique Conclusions D3/D7 D3/D5

Solution is Aα(z) =

∞

• l=1
• bl ¯

ρl+1Yl,+

α

+ cl ¯ ρl+1 Yl,−

α

• ,

A¯

ρ(z) = 0 ,

Normalizable solutions: bl = 0 (no sources, other than chemical potential) Recall, zero density: ¯ ρ = ρ ∈ [0, ∞) non zero density: ¯ ρ(ρ) ∈ [2−1/3ρ0, ∞)

Aα(z) = c 22/3 ρ2

• 1 + ρ6

0/ρ6 − 1

• 1 + ρ6

0/ρ6 + 1

1/3 Y1,−

α

1 2 3 4 5 0.25 0.5 0.75 1 1.25

f(ρ/ρ0) ρ/ρ0

Infinite series changes position of singularity → multipole expansion

Jo˜ ao N. Laia Moduli spaces of cold holographic matter

Introduction Technique Conclusions D3/D7 D3/D5

A pictorial way to see what is going on is

ρ0 21/n Rn\Bn

Non zero density de-singularizes the solution Summing up no moduli space at zero density non-singular solutions at nonzero density: moduli space

Jo˜ ao N. Laia Moduli spaces of cold holographic matter

Introduction Technique Conclusions D3/D7 D3/D5

Meaning of the cls

In the field theory, cls are related with expectation values of

• perators

O−

l ∼ cl,

O−

l

= Q†σIχQ

(χ: product of l − 1 symmetrized adjoint scalars)

The instanton number is

• F ∧ F ∼ sum of c2

l

Also, it is the number of D3s dissolved into D7s SWZ = T7

• P[C4] ∧ F ∧ F

Should be quantized?

Jo˜ ao N. Laia Moduli spaces of cold holographic matter

Introduction Technique Conclusions D3/D7 D3/D5

Outline

1 Introduction

Motivation Moduli spaces Results

2 Technique

D3/D7 D3/D5

3 Conclusions

Summary Future directions Speculation

Jo˜ ao N. Laia Moduli spaces of cold holographic matter

Introduction Technique Conclusions D3/D7 D3/D5

The D3 background - d´ ej` a vu

Consider a stack of Nc D3s x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 D3 X X X X Low energy physics is N = 4 SYM with SU(Nc) gauge group. Describe this brane using holography Nc → ∞, gYM → 0, λ = g2

YMNc fixed

λ → ∞ The background is (R4 = 4πgsNcα′2) ds2 = Z −1/2ηµνdxµdxν + Z 1/2 dr2 + r2ds2

S5

• ,

Z(r) ≡ R4/r4, F5 = 4 R (volAdS5 + volS5) , F5 = dC4

Jo˜ ao N. Laia Moduli spaces of cold holographic matter

Introduction Technique Conclusions D3/D7 D3/D5

Probe D5

x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 D3 X X X X D5 X X X X X X

Low energy description has 4d N = 4 SYM with gauge group SU(Nc) 3d N = 4 fundamental hypermultiplets living in the intersection Dual is D5 in the D3 background S7 = −T5

• d6ξ
• det(−P[G]ab + Fab) + T5
• P[C4] ∧ F.

Ansatz

relabel (z1, z2, z3) = (x4, x5, x6)

x3(ξ) = x3(z), A(ξ) = A0(z)dx0 + Ai(z)dzi

Jo˜ ao N. Laia Moduli spaces of cold holographic matter

Introduction Technique Conclusions D3/D7 D3/D5

Probe D5

No dependence in field theory coordinates integrate them We are lead to

s5 = −T5

• d3z
• det(gij + Z −1/2fij + Z −1∂ix3∂jx3) − 1

2Z −1˜ ǫijk∂ix3fjk

• ,

gij is a 3d effective metric, and fij a field strength in R3 gij ≡ δij − ∂iA0∂jA0, fij ≡ ∂iAj − ∂jAi. Recall We have done something very similar before. We wrote the action as in 4d, and used an inequality.

Jo˜ ao N. Laia Moduli spaces of cold holographic matter

Introduction Technique Conclusions D3/D7 D3/D5

Cosmetic uplift

Recall: we will use at some point the inequality For general symmetric G and antisymmetric F

• det (Gij + Fij) ≥
• det Gij + 1

8

• ˜

ǫijklFijFkl

• .

We need to write the action in a more 4d fashion. introduce extra direction z4 uplift the fields to R4 ˆ gij = gij + δ4

i δ4 j ,

ˆ a = Ai(z)dzi + x3(z)dz4 , ˆ f = dˆ a, demand they don’t depend on z4 We get to s5 = −T5

• d4z
• det(ˆ

gij + Z −1/2ˆ fij) − 1 8Z −1˜ ǫijklˆ fijˆ fkl

• ,

Looks very much like previous D3/D7 action (reason: T-duality)

Jo˜ ao N. Laia Moduli spaces of cold holographic matter

Introduction Technique Conclusions D3/D7 D3/D5

Using the inequality, we get s5 ≤ −T5

• d4z
• det ˆ

gij + 1 8Z −1 |˜ ǫijklˆ fijˆ fkl| − ˜ ǫijklˆ fijˆ fkl

• .

Bound is saturated for self dual ˆ f wrt ˆ g s5 = −T5

• d3z
• det gij,

independent of Ai(z) and x3(z) Self dual equation in terms of original variables is vector/scalar duality ∂ix3 = 1 2ǫijkf jk, All very similar with D3/D7 We’ll have a moduli space IF we find normalizable solutions

Jo˜ ao N. Laia Moduli spaces of cold holographic matter

Introduction Technique Conclusions D3/D7 D3/D5

We go throw the same procedure solve electrostatic problem: find A0(z) find solution for vector scalar duality equation Possible A0(z) for a compressible state is A′

0(ρ) =

1

• 1 + ρ4/ρ4

, ρ4

0 related to density

The effective metric is now gijdzidzj = ρ4 ρ4 + ρ4 dρ2 + ρ2ds2

S2.

metric is again conformal, but... cannot solve equation in flat space: equation is not conformally invariant

Jo˜ ao N. Laia Moduli spaces of cold holographic matter

Introduction Technique Conclusions D3/D7 D3/D5

Solving the vector/scalar dual equation

Form notation We want to solve dx3 = ⋆F In particular, it is true d ⋆ dx3 = dF If there are no sources dF = 0 is the Bianchi identity both Ai(z) and x3(z) are harmonic (d ⋆ d = 0) If there are sources monopole source gives dF = 0 and

• F = 0

WZ term tells us it is D3 brane charge within the D5 T5

• P[C4] ∧ F

Jo˜ ao N. Laia Moduli spaces of cold holographic matter

Introduction Technique Conclusions D3/D7 D3/D5

Zero density

Solutions (l = 0) are of the form x3(z) =

• m

Cm |z − z′

m|,

and corresponding F they are singular they have a physical interpretation

x3

r

x3

r

x3

r

Punchline At zero density, D3/D5 has a moduli space! (already known...)

Jo˜ ao N. Laia Moduli spaces of cold holographic matter

Introduction Technique Conclusions D3/D7 D3/D5

The field theory side

Field theory contains two scalar spin-l operators O±

l , dual to

ϕ+

l (ρ) ≡ l Al(ρ) + ρ x3 l (ρ)

ϕ−

l (ρ) ≡ (l + 1)Al(ρ) − ρ x3 l (ρ)

Operators O±

l

are a sandwich of squarks, with symmetrized adjoints in the middle For the zero density: ϕ+

l (ρ) = 0 and O+ l = 0

O−

l = 0 in general

Jo˜ ao N. Laia Moduli spaces of cold holographic matter

Introduction Technique Conclusions D3/D7 D3/D5

Non zero density

Effective metric is different, hence solutions will be different still exist these spiky solutions also new solutions with kink in the origin Properties of these solutions Depend on a parameter cl ϕ±

l (ρ) = 0 in general

O+

l ∝ cl and O− l ∝ cl

So... Moduli space survives the introduction of density

Jo˜ ao N. Laia Moduli spaces of cold holographic matter

Introduction Technique Conclusions D3/D7 D3/D5

Outline

1 Introduction

Motivation Moduli spaces Results

2 Technique

D3/D7 D3/D5

3 Conclusions

Summary Future directions Speculation

Jo˜ ao N. Laia Moduli spaces of cold holographic matter

Introduction Technique Conclusions Summary Future directions Speculation

Summary

We studied D3/Dp systems (p = 3, 5, 7), with 4ND directions at non zero density We constructed normalizable solutions with the same free energy

D3/D7: excising a ball desingularizes solution D3/D5: new kinky, non spiky, solutions

Dual scalar operators get VEVs that parametrize the Higgs branch It is very surprising that there is a Higgs branch at non zero

• density. Large-N artifact?

Jo˜ ao N. Laia Moduli spaces of cold holographic matter

Introduction Technique Conclusions Summary Future directions Speculation

Future directions

Is there any relation between effective metric and moduli space metric? Turn on T, turn on B. Do they lift the moduli space? Are there states with lower energy? (eg. Multi BIon)

Jo˜ ao N. Laia Moduli spaces of cold holographic matter

Introduction Technique Conclusions Summary Future directions Speculation

Speculation

Can we classify holographic matter? From observations: 4ND 6ND 8ND Higgs branch X Striped instability X SUSY X: (0 + 1)d fermion Chiral anomaly X: D4’s in ABJM X: (1 + 1)d fermion Correlation between number of ND’s and character of compressible state related with form of WZ terms are these just special cases? Can anomalies classify holographic matter? K-theory?

Jo˜ ao N. Laia Moduli spaces of cold holographic matter