S 3 /Z n partition function and Dualities

S3/Zn partition function and DualitiesYosuke ImamuraTokyo Institute of Technology

15 Oct. 2012 @ YKIS2012

Based onarXiv:1208.1404 Y.I and Daisuke Yokoyama

1. IntroductionLet us consider partition functions of field theories defined on compact manifolds.

Recently, the partition functions of supersymmetric field theories on various backgrounds have been computed exactly.

S4: Pestun (arXiv:0712.2824)S3: Kapustin,Willet,Yaakov (arXiv:1003.5694)S5: Kallen,Zabzine (arXiv:1202.1956)

For the definition of the partition function, we usually consider Eucludean space.In such a background, we do not have ``time’’ direction, and the ``unitarity’’ of the theory is lost. (At least not manifest.)The partition function is not guaranteed to be real.

In a Euclidean space, ψ and ψ* are treated as independent fields.There is no canonical way to fix the phase of the path integral measure.

is complex

In general the partition function Z is complex.

In the literature, the phase of Z is often neglected, and only the absolute value is focused on.

However, there is a situation we need to take account of the phase.

If the theory has many sectors and the total partition function is given by

We need to fix the relative phases of Zi.

This is the case if we consider a gauge theory on a manifold with non-trivial fundamental group.

As an example, let us consider U(1) gauge theory defined on the orbifold S3/Zn.

There is a non-trivial cycle γ S⊂ 3/Zn

Wilson line around γ must satisfy

The Wilson line is quantized

There are n degenerate vacua labeled by the holonomy h.

In this case, we need to carefully determine the phases of contribution of each sector to obtain the total partition function (even if we want only the absolute value of Ztot ).

Question:How should we determine the phase of the partition function?

Unfortunately, I do not have the answer to this question.

In this talk, I focus on a specific theory, and show that it is possible with the help of a duality to determine the phases of the contributions of multiple sectors.

I hope this provides useful information to look for a general rule to determine the phases.

We consider two N=2 3d SUSY theories A and B dual to each other.

Theory A on S3/Zn

Gauge theory

Multiple sectors contribute to ZA

We need relative phases

Theory B on S3/Zn

Non-Gauge theory

ZB can be computed up to overall constant

No phase problem

dual

We can determine the relative phases in theory A by requiring ZA=ZB.

Strategy

Example (3d mirror symmetry) Hori et al. hep-th/9702154

A and B are believed to be dual to each other.

SQED with Nf=1 q q*U(1)gauge +1 -1U(1)V 0 0U(1)A +1 +1Weights Δ Δ

Superpotential : W=0

Has nothing to do with the XYZ spin chain

XYZ modelNo gauge group Q Q* SU(1)V +1 -1 0U(1)A -1 -1 +2Weights 1-Δ 1-Δ 2Δ

Superpotential: W=Q*SQ

Theory A Theory B

U(1)V couples to the gauge flux F

dual

☑1. Introduction 2. S3 partition function 3. S3/Zn partition function 4. Numerical analysis 5. Summary

Let us use this duality to determine the relative phases of Zh (h=0,1,…,n-1) in SQED.

Plan of this talk

• Gauge group G• Matter representation R• Some parametrs

The partition function of a 3d N=2 SUSY field theories on S3

The data needed:

Kapustin,Willet,Yaakov (arXiv:1003.5694)Jafferis (arXiv:1012.3210)Hama, Hosomichi, Lee (arXiv:1012.3512)Y.I., D.Yokoyama (arXiv:1208.1404)

2. S3 partition function

Integration measure (G=U(N))

Integral over Cartan subalgebra a = diag (a1,a2,…,aN)

S0(a) : classical action at saddle points

Only CS terms and FI terms contribute to S0(a)

One-loop determinant

sb(z) : double-sine function (defined shortly) α : root vector labelling vector multiplets ρ : weight vector labelling chiral multiplets Δρ : Weyl weight of chiral multiplet b : squashing parameter

(For simplicity we do not turn on real mass parameters)

b : squashing parameter

b=1 : round S3

b≠1 : deformed S3

Ellipsoid (b R∈ ) : Hama, Hosomichi, Lee (arXiv:1102.4716) U(1)xU(1) symmetric

Squashed S3 (|b|=1) : Y.I and D.Yokoyama (arXiv:1109.4734) SU(2)xU(1) symmetric

We can deform S3 with preserving (a part of) SUSY.

We consider squashed S3 because we will later consider orbifold by Zn SU(2).⊂

Double-sine function

Φb(z) : ``Faddeev’s quantum dilogarithm’’

q-Pochhammer symbol (Mathematica (ver.7 or later) knows this.)

Infinite product expression

p, q : SU(2)R quantum numbers

p=j+m, q=j-m

This product is obtained by the path integral of the partition function.Each factor corresponds to the spherical harmonics specified by the quantum number p, q.

(This fact becomes important when we consider orbifold.)

With this formula, we can compute the partition function of any N=2 field theory (if we know its Lagrangian.)

As an exercize, let us use this to check the duality on S3.

Now we have the general formula

3d mirror symmetry Hori et al. hep-th/9702154

SQED with Nf=1 q q*U(1)gauge +1 -1U(1)V 0 0U(1)A +1 +1Weights Δ Δ

Superpotential : W=0

XYZ modelNo gauge group Q Q* SU(1)V +1 -1 0U(1)A -1 -1 +2Weights 1-Δ 1-Δ 2Δ

Superpotential: W=Q*SQ

Theory A Theory B

dual

According to the general formula, the partition functions are

If two theories are really dual, the partition function should agree.

This is highly non-trivial

In fact, this is equivalent to so-called ``Pentagon relation’’ of the quantum dilogarithm.

This identity plays an important role in a certain integrable statistical model.

(Faddeev-Volkov model)

(Faddeev, Kashaev, Volkov, hep-th/0006156)

3. S3/Zn partition function

Let us replace M=S3 by its orbifold S3/Zn.

The SQED has n degenerate vacua

(Benini, Nishioka, Yamazaki, arXiv:1109.0283)

Isometry: SO(4) ~ SU(2)L x SU(2)R

Orbifolding by Zn SU(2)⊂ R

The definition of the orbifold

Modifications in the formula1. Integration measure

2. Classical action

Chern-Simons term gives extra contribution depends on the holonomy.

3. 1-loop determinant

(Orbifolded double sine function)

This is necessary for|ZA|=|ZB|

The double sine function in the 1-loop determinant in S3:

Contribution of spherical harmonics with SU(2)R quantum numbers (p,q).

p= j+m, q=j-m.

Only modes with

p - q = hQ mod nSurvive after the orbifolding.

h = 0,1,…,n-1 : holonomyQ : charge

p

q

Spherical harmonics on S3

p

q

Spherical harmonics on S3/Z3 (h=0)

p-q = 0 mod 3

p

q

Spherical harmonics on S3/Z3 (hQ=1)

p-q = 1 mod 3

p

q

Spherical harmonics on S3/Z3 (hQ=2)

p-q = 2 mod 3

We define new function (orbifolded double sine function)

by restricting the product with the condition.

p, q Z⊂ +

p – q = h

The one-loop determinant Z1-loop for the orbifold is obtained by replacing all sb(z) in Z1-loop on S3 by sb,h(z).

[m]n is the remainder of m/n

This function can also be represented by the original function

We can turn on non-trivial holonomies not only for the gauge symmetry but also for global symmetries, too.

The general formula becomes

h=(hlocal,hglobal)

Let us repeat the analysis of the duality between A and B.

Local and global

A: N=2SQED + Nf=1HolonomiesU(1)gauge h = 0,…,n-1U(1)V hV= 0,…,n-1U(1)A hA =0,…,n-1

This factor comes from the coupling of U(1)V to the gauge flux.

the U(1)gauge holonomy sum

4. Numerical analysis

HolonomiesU(1)V hV= 0,…,n-1U(1)A hA =0,…,n-1

B: XYZ model

No holonomy sum

S3/Z3 (n=3) Holonomies: (hV ,hA) = (0,0) Parameters: (b,Δ) = (e0.2i,0.3)

ZSQED h=0 1.16561h=1 0.32343h=2 0.32343-------------------Sum 1.81247 ZXYZ = 1.81246

Agree !!

ZSQED

h=0 0.32172 + 0.02704 ih=1 0.40491 - 0.02862 ih=2 0.40491 - 0.02862 i----------------------------------- Sum 1.13156 - 0.03020 i

Not agree !!

S3/Z3 (n=3) Holonomies: (hV ,hA) = (0,1) Parameters: (b,Δ) = (e0.2i,0.3)

ZXYZ = 0.48809 – 0.08429 i

Turn on the non-trivial holonomy

ZSQED

h=0 - ( 0.32172 + 0.02704 i )h=1 + ( 0.40491 - 0.02862 i )h=2 + ( 0.40491 - 0.02862 i )-------------------------------------- Sum 0.48810 - 0.08428 i

Agree !!

S3/Z3 (n=3) Holonomies: (hV ,hA) = (0,1) Parameters: (b,Δ) = (e0.2i,0.3)

ZXYZ = 0.48809 - 0.08429 i

Phase factors

More check

ZSQED

h=0 + 0.34103 + 0.11345 ih=1 - 0.24392 + 0.07820 ih=2 + 0.02112 - 0.00808 ih=3 - 0.06320 - 0.17364 ih=4 - 0.17344 - 0.01453 ih=5 + 0.41407 - 0.18050 ih=6 - 0.06741 + 0.16045 ih=7 - 0.05094 + 0.17763 ih=8 - 0.01234 + 0.01895 ih=9 - 0.14975 - 0.20782 i--------------------------------------Sum + 0.01522 - 0.03570 i

Not agree !!

S3/Z10 (n=10) Holonomies: (hV ,hA) = (4,3) Parameters: (b,Δ) = (e0.2i,0.3)

ZXYZ = 0.68440 + 0.28454 i

ZSQED

h=0 + ( + 0.34103 + 0.11345 i )h=1 - ( - 0.24392 + 0.07820 i )h=2 + ( + 0.02112 - 0.00808 i )h=3 - ( - 0.06320 - 0.17364 i )h=4 - ( - 0.17344 - 0.01453 i )h=5 - ( + 0.41407 - 0.18050 i )h=6 - ( - 0.06741 + 0.16045 i )h=7 - ( - 0.05094 + 0.17763 i )h=8 + ( - 0.01234 + 0.01895 i )h=9 - ( - 0.14975 - 0.20782 i )-------------------------------------Sum + 0.68440 + 0.28452 i

Agree!!

S3/Z10 (n=10) Holonomies: (hV ,hA) = (4,3) Parameters: (b,Δ) = (e0.2i,0.3)

ZXYZ = 0.68440 + 0.28454 i

We can guess the general formula for the sign factor

We have obtained formula for the phases in a specific example. (N=2 SQED w/Nf=1)

From this result, we want to guess a universal rule which can be applied to arbitrary theories.

When the order n of the orbifold group Zn is odd, it is possible to give a simple rule to determine the phase factor.

Let us consider odd n case, and define.

([h]n is the remainder of h/n)

The functions f and g are related to this by

We can rewrite ZXYZ=ZSQED as

We can absorb the sign into the definition of sb,h(z)

We can absorb the sign into the definition of sb,h(z)

We can rewrite ZXYZ=ZSQED as

If we define ``modified orbifolded double sine function’’ by

No extra sign factor

ZXYZ=ZSQED becomes

When n is odd, we obtain partition function with ``correct’’ phase by replacing sb,h(z) in Z1-loop by ^sb,h(z).

We confirmed that this prescription works for another example of duality.

Suggestion

A chiral mult SU(2) gauge theory + adjoint chiral mult.⇔Jafferis-Yin duality (arXiv:1103.5700)

5. SummaryWe can determine relative phases of Z in the holonomy sum by matching Z of dual pairs.

In two examples (N=2 mirror, Jafferis-Yin duality) of dual pairs, we found that we can get ``correct’’ phases by modifying the function sb,h(z).

This may be universal. We should check this in other examples of dual pairs.

Odd n case:

For even n, we have no general rule to fix the phases.

Our results are a kind of ``experimental results’’.No derivation from the first principle.

It is important to find nice criteria for the ``correct’’ phase which do not rely on dualities.

sb(z) plays an important role in integrable models. How about sb,h(z)?

Open questions:

Thank you