S 3 /Z n partition function and Dualities Yosuke Imamura Tokyo Institute of Technology 15 Oct. YKIS2012 Based on arXiv: Y.I and Daisuke Yokoyama

1. Introduction Let 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. S 4 : Pestun (arXiv: ) S 3 : Kapustin,Willet,Yaakov (arXiv: ) S 5 : Kallen,Zabzine (arXiv: )

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 Z i.

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 S 3 /Z n. There is a non-trivial cycle γ ⊂ S 3 /Z n 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 Z tot ).

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 S 3 /Z n Gauge theory Multiple sectors contribute to Z A We need relative phases Theory A on S 3 /Z n Gauge theory Multiple sectors contribute to Z A We need relative phases Theory B on S 3 /Z n Non-Gauge theory Z B can be computed up to overall constant No phase problem Theory B on S 3 /Z n Non-Gauge theory Z B can be computed up to overall constant No phase problem dual We can determine the relative phases in theory A by requiring Z A =Z B. Strategy

Example (3d mirror symmetry) Hori et al. hep-th/ A and B are believed to be dual to each other. SQED with N f =1 q q* U(1) gauge U(1) V 0 0 U(1) A Weights Δ Δ Superpotential : W=0 SQED with N f =1 q q* U(1) gauge U(1) V 0 0 U(1) A Weights Δ Δ Superpotential : W=0 Has nothing to do with the XYZ spin chain XYZ model No gauge group Q Q* S U(1) V U(1) A Weights 1-Δ 1-Δ 2Δ Superpotential: W=Q*SQ XYZ model No gauge group Q Q* S U(1) V U(1) A Weights 1-Δ 1-Δ 2Δ Superpotential: W=Q*SQ Theory ATheory B U(1) V couples to the gauge flux F dual

☑ 1. Introduction 2. S 3 partition function 3. S 3 /Z n partition function 4. Numerical analysis 5. Summary Let us use this duality to determine the relative phases of Z h (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 S 3 The data needed: Kapustin,Willet,Yaakov (arXiv: ) Jafferis (arXiv: ) Hama, Hosomichi, Lee (arXiv: ) Y.I., D.Yokoyama (arXiv: ) 2. S 3 partition function

Integration measure (G=U(N)) Integral over Cartan subalgebra a = diag (a 1,a 2,…,a N ) S 0 (a) : classical action at saddle points Only CS terms and FI terms contribute to S 0 (a)

One-loop determinant s b (z) : double-sine function (defined shortly) α : root vector labelling vector multiplets ρ : weight vector labelling chiral multiplets Δ ρ : Weyl weight of chiral multiplet b : squashing parameter (For simplicity we do not turn on real mass parameters)

b : squashing parameter b=1 : round S 3 b≠1 : deformed S 3 Ellipsoid (b ∈ R) : Hama, Hosomichi, Lee (arXiv: ) U(1)xU(1) symmetric Squashed S 3 (|b|=1) : Y.I and D.Yokoyama (arXiv: ) SU(2)xU(1) symmetric We can deform S 3 with preserving (a part of) SUSY. We consider squashed S 3 because we will later consider orbifold by Z n ⊂ SU(2).

Double-sine function Φ b (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 S 3. Now we have the general formula

3d mirror symmetry Hori et al. hep-th/ SQED with N f =1 q q* U(1) gauge U(1) V 0 0 U(1) A Weights Δ Δ Superpotential : W=0 SQED with N f =1 q q* U(1) gauge U(1) V 0 0 U(1) A Weights Δ Δ Superpotential : W=0 XYZ model No gauge group Q Q* S U(1) V U(1) A Weights 1-Δ 1-Δ 2Δ Superpotential: W=Q*SQ XYZ model No gauge group Q Q* S U(1) V U(1) A Weights 1-Δ 1-Δ 2Δ Superpotential: W=Q*SQ Theory ATheory 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/ )

3. S 3 /Z n partition function Let us replace M=S 3 by its orbifold S 3 /Z n. The SQED has n degenerate vacua (Benini, Nishioka, Yamazaki, arXiv: )

Isometry: SO(4) ~ SU(2) L x SU(2) R Orbifolding by Z n ⊂ SU(2) R The definition of the orbifold

Modifications in the formula 1. Integration measure 2. Classical action Chern-Simons term gives extra contribution depends on the holonomy loop determinant (Orbifolded double sine function) This is necessary for |Z A |=|Z B |

The double sine function in the 1-loop determinant in S 3 : 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 n Survive after the orbifolding. h = 0,1,…,n-1 : holonomy Q : charge

p q Spherical harmonics on S 3

p q Spherical harmonics on S 3 /Z 3 (h=0) p-q = 0 mod 3

p q Spherical harmonics on S 3 /Z 3 (hQ=1) p-q = 1 mod 3

p q Spherical harmonics on S 3 /Z 3 (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 Z 1-loop for the orbifold is obtained by replacing all s b (z) in Z 1-loop on S 3 by s b,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=(h local,h global ) Let us repeat the analysis of the duality between A and B. Local and global

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

Holonomies U(1) V h V = 0,…,n-1 U(1) A h A =0,…,n-1 B: XYZ model No holonomy sum

S 3 /Z 3 (n=3) Holonomies: (h V,h A ) = (0,0) Parameters: (b,Δ) = (e 0.2i,0.3) S 3 /Z 3 (n=3) Holonomies: (h V,h A ) = (0,0) Parameters: (b,Δ) = (e 0.2i,0.3) Z SQED h= h= h= Sum Z SQED h= h= h= Sum Z XYZ = Agree !!

Z SQED h= i h= i h= i Sum i Z SQED h= i h= i h= i Sum i Not agree !! S 3 /Z 3 (n=3) Holonomies: (h V,h A ) = (0,1) Parameters: (b,Δ) = (e 0.2i,0.3) S 3 /Z 3 (n=3) Holonomies: (h V,h A ) = (0,1) Parameters: (b,Δ) = (e 0.2i,0.3) Z XYZ = – i Turn on the non-trivial holonomy

Z SQED h=0 - ( i ) h=1 + ( i ) h=2 + ( i ) Sum i Z SQED h=0 - ( i ) h=1 + ( i ) h=2 + ( i ) Sum i Agree !! S 3 /Z 3 (n=3) Holonomies: (h V,h A ) = (0,1) Parameters: (b,Δ) = (e 0.2i,0.3) S 3 /Z 3 (n=3) Holonomies: (h V,h A ) = (0,1) Parameters: (b,Δ) = (e 0.2i,0.3) Z XYZ = i Phase factors

More check Z SQED h= i h= i h= i h= i h= i h= i h= i h= i h= i h= i Sum i Z SQED h= i h= i h= i h= i h= i h= i h= i h= i h= i h= i Sum i Not agree !! S 3 /Z 10 (n=10) Holonomies: (h V,h A ) = (4,3) Parameters: (b,Δ) = (e 0.2i,0.3) S 3 /Z 10 (n=10) Holonomies: (h V,h A ) = (4,3) Parameters: (b,Δ) = (e 0.2i,0.3) Z XYZ = i

Z SQED h=0 + ( i ) h=1 - ( i ) h=2 + ( i ) h=3 - ( i ) h=4 - ( i ) h=5 - ( i ) h=6 - ( i ) h=7 - ( i ) h=8 + ( i ) h=9 - ( i ) Sum i Z SQED h=0 + ( i ) h=1 - ( i ) h=2 + ( i ) h=3 - ( i ) h=4 - ( i ) h=5 - ( i ) h=6 - ( i ) h=7 - ( i ) h=8 + ( i ) h=9 - ( i ) Sum i Agree!! S 3 /Z 10 (n=10) Holonomies: (h V,h A ) = (4,3) Parameters: (b,Δ) = (e 0.2i,0.3) S 3 /Z 10 (n=10) Holonomies: (h V,h A ) = (4,3) Parameters: (b,Δ) = (e 0.2i,0.3) Z XYZ = 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/N f =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 Z n 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 Z XYZ =Z SQED as We can absorb the sign into the definition of s b,h (z)

We can rewrite Z XYZ =Z SQED as

If we define ``modified orbifolded double sine function’’ by No extra sign factor Z XYZ =Z SQED becomes

When n is odd, we obtain partition function with ``correct’’ phase by replacing s b,h (z) in Z 1-loop by ^s b,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: )

5. Summary We 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 s b,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. s b (z) plays an important role in integrable models. How about s b,h (z)? Open questions:

Thank you