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In the bosonic theories the short-distance repulsion between loops

become the 𝜙 4 coupling, which in the massless limit flow in the IR into the

WF fixed point

Example: Fermion particle-vortex duality

• Case for opposite mass signs (QH phase) follows from the same logic

• Current mapping also natural upon integrating out b:

Use loop models to derive the duality between free Dirac fermion and QED 3 with (quantized) Chern-Simons terms

Z

d 3 x J µ A µ + ⇡ [J] ⇡ [J] +

Z

d 3 x

J µ a µ 1

2⇡ adb + 2

4⇡ bdb 1

2⇡ bdA

Integrate out a, b

¯ µ ! 1

µ⌫ @ a

i ¯ / D[A] 8⇡ 1 AdA $ i ¯ / D[a] + 8⇡ 1 ada 2⇡ 1 adb + 4⇡ 2 bdb 2⇡ 1 bdA

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M 0 , ¯ M 0 > 0

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M ¯ , M < 0

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Z F [A; M < 0] = Z QED 3 [A; M 0 > 0], Z F [A; M > 0] = Z QED 3 [A; M 0 < 0]

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