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S 3 /Z n partition function and Dualities

S 3 /Z n partition function and Dualities. Yosuke Imamura Tokyo Institute of Technology. 15 Oct. 2012 @ YKIS2012. Based on arXiv:1208.1404 Y.I and Daisuke Yokoyama. 1. Introduction. L et us consider partition functions of field theories defined on compact manifolds.

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S 3 /Z n partition function and Dualities

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  1. S3/Zn partition function and Dualities Yosuke Imamura Tokyo Institute of Technology 15 Oct. 2012 @ YKIS2012 Based on arXiv:1208.1404 Y.I and Daisuke Yokoyama

  2. 1. Introduction Let us consider partition functions of field theories defined on compact manifolds. Recently, the partition functions of supersymmetricfield 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)

  3. 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.

  4. 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.

  5. 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 ⊂ S3/Zn Wilson line around γ must satisfy

  6. The Wilson line is quantized There are n degenerate vacua labeled by the holonomyh. 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 ).

  7. 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.

  8. Strategy 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.

  9. Example (3d mirror symmetry) Hori et al. hep-th/9702154 A and B are believed to be dual to each other. Theory A Theory B SQED with Nf=1 q q* U(1)gauge +1 -1 U(1)V 0 0 U(1)A +1 +1 Weights Δ Δ Superpotential : W=0 XYZ model No gauge group Q Q* S U(1)V +1 -1 0 U(1)A -1 -1 +2 Weights 1-Δ 1-Δ 2Δ Superpotential: W=Q*SQ dual U(1)V couples to the gauge flux F Has nothing to do with the XYZ spin chain

  10. Let us use this duality to determine the relative phases of Zh(h=0,1,…,n-1) in SQED. Plan of this talk ☑1. Introduction 2. S3 partition function 3. S3/Zn partition function 4. Numerical analysis 5. Summary

  11. 2. S3 partition function The partition function of a 3d N=2 SUSY field theories on S3 Kapustin,Willet,Yaakov (arXiv:1003.5694) Jafferis (arXiv:1012.3210) Hama, Hosomichi, Lee (arXiv:1012.3512) Y.I., D.Yokoyama (arXiv:1208.1404) The data needed: • Gauge group G • Matter representation R • Some parametrs

  12. Integration measure (G=U(N)) Integral over Cartansubalgebraa = diag (a1,a2,…,aN) S0(a) : classical action at saddle points Only CS terms and FI terms contribute to S0(a)

  13. One-loop determinant sb(z) : double-sine function (defined shortly) α : root vector labelling vector multiplets ρ : weight vector labellingchiralmultiplets Δρ: Weyl weight of chiralmultiplet b: squashing parameter (For simplicity we do not turn on real mass parameters)

  14. b: squashing parameter We can deform S3 with preserving (a part of) SUSY. 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 consider squashed S3because we will later consider orbifold by Zn⊂ SU(2).

  15. Double-sine function Φb(z) : ``Faddeev’s quantum dilogarithm’’ q-Pochhammer symbol (Mathematica(ver.7 or later) knows this.)

  16. Infinite product expression 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. p, q : SU(2)R quantum numbers p=j+m, q=j-m (This fact becomes important when we consider orbifold.)

  17. Now we have the general formula 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.

  18. 3d mirror symmetryHori et al. hep-th/9702154 Theory A Theory B SQED with Nf=1 q q* U(1)gauge +1 -1 U(1)V 0 0 U(1)A +1 +1 Weights Δ Δ Superpotential : W=0 XYZ model No gauge group Q Q* S U(1)V +1 -1 0 U(1)A -1 -1 +2 Weights 1-Δ 1-Δ 2Δ Superpotential: W=Q*SQ dual

  19. 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

  20. In fact, this is equivalent to so-called ``Pentagon relation’’ of the quantum dilogarithm. (Faddeev, Kashaev, Volkov, hep-th/0006156) This identity plays an important role in a certain integrable statistical model. (Faddeev-Volkov model)

  21. 3. S3/Zn partition function (Benini, Nishioka, Yamazaki, arXiv:1109.0283) Let us replace M=S3 by its orbifold S3/Zn. The SQED has n degenerate vacua

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

  23. Modifications in the formula 1. Integration measure This is necessary for |ZA|=|ZB| 2. Classical action Chern-Simons term gives extra contribution depends on the holonomy. 3. 1-loop determinant (Orbifolded double sine function)

  24. 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 n h = 0,1,…,n-1 : holonomy Q : charge Survive after the orbifolding.

  25. Spherical harmonics on S3 q p

  26. Spherical harmonics on S3/Z3 (h=0) q p-q = 0 mod 3 p

  27. Spherical harmonics on S3/Z3 (hQ=1) q p-q = 1 mod 3 p

  28. Spherical harmonics on S3/Z3 (hQ=2) q p-q = 2 mod 3 p

  29. We define new function (orbifolded double sine function) by restricting the product with the condition. p, q ⊂ Z+ p – q = h

  30. This function can also be represented by the original function [m]n is the remainder of m/n 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).

  31. Local and global We can turn on non-trivial holonomies not only for the gauge symmetry but also for global symmetries, too. h=(hlocal,hglobal) The general formula becomes Let us repeat the analysis of the duality between A and B.

  32. 4. Numerical analysis A: N=2SQED + Nf=1 This factor comes from the coupling of U(1)V to the gauge flux. Holonomies U(1)gauge h = 0,…,n-1 U(1)VhV= 0,…,n-1 U(1)AhA =0,…,n-1 the U(1)gaugeholonomy sum

  33. B: XYZ model Holonomies U(1)VhV= 0,…,n-1 U(1)AhA =0,…,n-1 No holonomy sum

  34. S3/Z3 (n=3) Holonomies: (hV,hA) = (0,0) Parameters: (b,Δ) = (e0.2i,0.3) ZSQED h=0 1.16561 h=1 0.32343 h=2 0.32343 ------------------- Sum 1.81247 ZXYZ = 1.81246 Agree !!

  35. Turn on the non-trivial holonomy S3/Z3 (n=3) Holonomies: (hV,hA) = (0,1) Parameters: (b,Δ) = (e0.2i,0.3) ZSQED h=0 0.32172 + 0.02704 i h=1 0.40491 - 0.02862 i h=2 0.40491 - 0.02862 i ----------------------------------- Sum 1.13156 - 0.03020 i ZXYZ = 0.48809 – 0.08429 i Not agree !!

  36. S3/Z3 (n=3) Holonomies: (hV,hA) = (0,1) Parameters: (b,Δ) = (e0.2i,0.3) 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 ZXYZ = 0.48809 - 0.08429 i Phase factors Agree !!

  37. More check S3/Z10 (n=10) Holonomies: (hV,hA) = (4,3) Parameters: (b,Δ) = (e0.2i,0.3) 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 + 0.01522 - 0.03570 i Not agree !! ZXYZ = 0.68440 + 0.28454 i

  38. S3/Z10 (n=10) Holonomies: (hV,hA) = (4,3) Parameters: (b,Δ) = (e0.2i,0.3) 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!! ZXYZ = 0.68440 + 0.28454 i

  39. We can guess the general formula for the sign factor

  40. We have obtained formula for the phases ina 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.

  41. 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

  42. We can rewrite ZXYZ=ZSQED as We can absorb the sign into the definition of sb,h(z)

  43. We can rewrite ZXYZ=ZSQED as We can absorb the sign into the definition of sb,h(z)

  44. If we define ``modified orbifoldeddouble sine function’’ by ZXYZ=ZSQED becomes No extra sign factor

  45. Suggestion 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. Jafferis-Yin duality (arXiv:1103.5700) A chiralmult⇔ SU(2) gauge theory + adjointchiralmult.

  46. 5. Summary We can determine relative phases of Z in the holonomy sum by matching Z of dual pairs. Odd n case: 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.

  47. Open questions: 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)?

  48. Thank you

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