Preprint typeset in JHEP style - HYPER VERSION

Cheat Sheet for D = 2 N = (2, 2) Abelian Gauge Theories

Daniel S. Park Simons Center for Geometry and Physics Stony Brook University Stony Brook, NY 11794-3636, USA

dpark at scgp.stonybrook.edu

Abstract: This is a cheat sheet for D = 2, N = (2, 2) supersymmetric abelian gauge theories. We explicitly work out the components of chiral, twisted chiral and vector superfields and construct terms of supersymmetric gauge theory Lagrangians using them. We focus on the simple case of abelian gauge theories. Our conventions are based on [1].

Contents 1. Some Conventions

1

2. (2, 2) Superfields 2.1 Chiral Superfields 2.2 Twisted Chiral Superfields 2.3 Vector Fields 2.4 Summary

2 2 3 4 6

3. SUSY Lagrangians From (2, 2) Superfields 3.1 Kinetic Terms 3.2 Superpotential Terms 3.3 Twisted Superpotential Terms 3.4 Summary

6 6 7 7 8

4. Some Theories of Interest 4.1 N = (2, 2) Abelian Higgs Model 4.2 N = (4, 4) Abelian Higgs Model 4.3 N = (4, 4) NS5-brane Model

8 8 9 12

1. Some Conventions We work in Lorentzian signature: ds2 = −(dx0 )2 + (dx1 )2 .

(1.1)

For convenience we define the cooridinates 1 x++ ≡ (x0 + x1 ), 2

1 x−− ≡ (x0 − x1 ) . 2

(1.2)

The derivatives with respect to the bosonic coordinates are defined as following: ∂++ ≡ ∂x++ = ∂0 + ∂1 ,

∂−− ≡ ∂x−− = ∂0 − ∂1 .

(1.3)

The fermionic coordinates are given by the complex grassmann variables θ+ , θ¯+ , θ− , θ¯− .

–1–

(1.4)

We use bars to denote complex conjugation. The derivatives are defined as following:1 ∂+ ≡

∂ ∂ ∂ ∂ , ∂− ≡ − , ∂¯+ ≡ ¯+ , ∂¯− ≡ ¯− . + ∂θ ∂θ ∂θ ∂θ

(1.8)

We set the the integration measure to be d4 θ ≡ dθ− dθ¯− dθ+ dθ¯+ .

(1.9)

Therefore the integral with this integration measure amounts to picking out the coefficient of θ¯+ θ+ θ¯− θ− . (1.10) We note that complex conjugation on grassmann variables reverses order, i.e., (θχ) = χ ¯θ¯ .

(1.11)

In particular, the product χχ ¯ of a complex grassmann variable χ behaves as a real bosonic variable. The SUSY generators Q± and the derivatives D± are defined as Q± ≡ ∂± + iθ¯± ∂±± , D± ≡ ∂± − iθ¯± ∂±± ,

¯ ± ≡ −∂¯± − iθ± ∂±± , Q ¯ ± ≡ −∂¯± + iθ± ∂±± . D

(1.12) (1.13)

It is easy to check that ¯ + } = 2i∂++ , {D+ , D

¯ − } = 2i∂−− , {D− , D

(1.14)

while all other pairs of derivatives anticommute.

2. (2, 2) Superfields 2.1 Chiral Superfields The chiral superfield Φ satisfies the constraints ¯ +Φ = D ¯ −Φ = 0 . D 1

(2.1)

Our conventions are such that the derivative of a grassmann variable θ is given by ∂ (a + θb) = b , ∂θ

(1.5)

where a and b are independent of θ. For example, if θ and χ are two independent grassmann variables, ∂ ∂ (χθ) = − (θχ) = −χ . ∂θ ∂θ

(1.6)

Likewise, we define integration as Z dθ(a + θb) = b .

–2–

(1.7)

The bosonic components of the superfields of Φ are then given by √ √ Φ = φ(y) + 2θ+ ψ+ (y) + 2θ− ψ− (y) + 2θ+ θ− F (y)

(2.2)

where the coordinates y are defined to be y ++ ≡ x++ + iθ¯+ θ+ y −− ≡ x−− + iθ¯− θ− One can expand the bosonic fields further and obtain √ √ Φ = φ(x) + 2θ+ ψ+ (x) + 2θ− ψ− (x) + 2θ+ θ− F (x) + iθ¯+ θ+ ∂++ φ(x) + iθ¯− θ− ∂−− φ(x) √ √ + 2iθ+ θ¯− θ− ∂−− ψ+ (x) + 2iθ¯+ θ+ θ− ∂++ ψ− (x) − θ¯+ θ+ θ¯− θ− ∂++ ∂−− φ(x)

(2.3) (2.4)

(2.5)

It is straightforward to see that ¯ = 0. ¯ = D− Φ D+ Φ ¯ are then given by The bosonic components of the superfields of Φ √ √ ¯ y ) − 2θ¯+ ψ¯+ (¯ ¯ = φ(¯ Φ y ) − 2θ¯− ψ¯− (¯ y ) − 2θ¯+ θ¯− F¯ (¯ y) .

(2.6)

(2.7)

The coordinates y¯ are obtained by complex conjugation: y ++ ≡ x++ − iθ¯+ θ+ y −− ≡ x−− − iθ¯− θ− As before, one can expand the bosonic fields further to obtain √ √ ¯ ¯ = φ(x) Φ − 2θ¯+ ψ¯+ (x) − 2θ¯− ψ¯− (x) − 2θ¯+ θ¯− F¯ (x) ¯ ¯ − iθ¯+ θ+ ∂++ φ(x) − iθ¯− θ− ∂−− φ(x) √ + − − √ + + − + 2iθ¯ θ¯ θ ∂−− ψ¯+ (x) + 2iθ¯ θ θ¯ ∂++ ψ¯− (x) ¯ − θ¯+ θ+ θ¯− θ− ∂++ ∂−− φ(x)

(2.8) (2.9)

(2.10)

2.2 Twisted Chiral Superfields The twisted chiral superfield Σ satisfies the constraints ¯ + Σ = D− Σ = 0 . D The bosonic components of the superfields of Σ are then given by √ √ Σ = σ(˜ y ) + 2θ+ χ ¯+ (˜ y ) + 2θ¯− χ− (˜ y ) + 2θ+ θ¯− G(˜ y)

(2.11)

(2.12)

where the coordinates y˜ are defined to be y˜++ ≡ x++ + iθ¯+ θ+ y˜−− ≡ x−− − iθ¯− θ−

–3–

(2.13) (2.14)

One can expand the bosonic fields further and obtain √ √ Σ = σ(x) + 2θ+ χ ¯+ (x) + 2θ¯− χ− (x) + 2θ+ θ¯− G(x) + iθ¯+ θ+ ∂++ σ(x) − iθ¯− θ− ∂−− σ(x) √ √ − 2iθ+ θ¯− θ− ∂−− χ ¯+ (x) + 2iθ¯+ θ+ θ¯− ∂++ χ− (x) + θ¯+ θ+ θ¯− θ− ∂++ ∂−− σ(x)

(2.15)

It is straightforward to see that ¯ = 0. ¯ =D ¯ −Σ D+ Σ

(2.16)

¯ are then given by The bosonic components of the superfields of Σ ¯ =σ Σ ¯ (y¯ ˜) −

√

2θ¯+ χ+ (y¯˜) −

√

¯ y¯˜) . 2θ− χ ¯− (y¯˜) − 2θ¯+ θ− G(

(2.17)

The coordinates y¯ are obtained by complex conjugation: y˜++ ≡ x++ − iθ¯+ θ+ y˜−− ≡ x−− + iθ¯− θ− Expanding the bosonic fields further, we get √ √ ¯ =σ ¯ Σ ¯ (x) − 2θ¯+ χ+ (x) − 2θ− χ ¯− (x) − 2θ¯+ θ− G(x) − iθ¯+ θ+ ∂++ σ ¯ (x) + iθ¯− θ− ∂−− σ ¯ (x) √ + − − √ + + − − 2iθ¯ θ¯ θ ∂−− χ+ (x) + 2iθ¯ θ θ ∂++ χ ¯− (x) + + ¯− − ¯ + θ θ θ θ ∂++ ∂−− σ ¯ (x)

(2.18) (2.19)

(2.20)

2.3 Vector Fields The vector field is defined to be a real superfield V = V¯ .

(2.21)

¯, V →V +Φ+Φ

(2.22)

There is a gauge symmetry

where Φ is a chiral superfield. We can use this gauge symmetry to get rid of lower components of V to arrive at Wess-Zumino gauge. In WZ gauge, the components of V are given by V = −θ¯+ θ+ v++ − θ¯− θ− v−− + θ¯+ θ− σ − θ+ θ¯− σ ¯ √ + + − √ ¯ + − 2iθ¯+ θ+ θ¯− λ+ − 2iθ¯ θ θ λ √ √ ¯ − + 2iθ¯+ θ¯− θ− λ− + 2iθ+ θ¯− θ− λ + 2θ¯+ θ+ θ¯− θ− D .

–4–

(2.23)

Here we have defined v++ ≡ v0 + v1 ,

v−− ≡ v0 − v1 .

(2.24)

where vµ is the gauge connection. It is easy to see that ¯ + D− V Σ≡D

(2.25)

¯ 2 = D− 2 = 0 . D +

(2.26)

is a twisted chiral multiplet as

In components,

¯ + D− V = σ(˜ D y) +

√

¯ + (˜ 2iθ+ λ y) −

√

2iθ¯− λ− (˜ y ) + 2θ+ θ¯− (D − iF01 )(˜ y) .

(2.27)

Here we have used the fact that

1 F01 ≡ ∂0 v1 − ∂1 v0 = − (∂++ v−− − ∂−− v++ ) . 2

(2.28)

Comparing with the expressions of section 2.2 we see that

χ± = −iλ± ,

G = D − iF01 .

(2.29)

¯ − D+ V is an anti-twisted chiral multiplet with components Likewise, D

¯ − D+ V = σ(y¯ D ˜) +

√

2iθ¯+ λ+ (y¯˜) −

√

¯ − (y¯˜) + 2θ+ θ¯− (D + iF01 )(y¯˜) . 2iθ− λ

–5–

(2.30)

2.4 Summary Chiral:

Twisted Chiral:

Vector:

√ √ Φ = φ(x) + 2θ+ ψ+ (x) + 2θ− ψ− (x) + 2θ+ θ− F (x) + iθ¯+ θ+ ∂++ φ(x) + iθ¯− θ− ∂−− φ(x) √ √ + 2iθ+ θ¯− θ− ∂−− ψ+ (x) + 2iθ¯+ θ+ θ− ∂++ ψ− (x) − θ¯+ θ+ θ¯− θ− ∂++ ∂−− φ(x) √ √ ¯ ¯ = φ(x) Φ − 2θ¯+ ψ¯+ (x) − 2θ¯− ψ¯− (x) − 2θ¯+ θ¯− F¯ (x) ¯ ¯ − iθ¯+ θ+ ∂++ φ(x) − iθ¯− θ− ∂−− φ(x) √ + − − √ + + − + 2iθ¯ θ¯ θ ∂−− ψ¯+ (x) + 2iθ¯ θ θ¯ ∂++ ψ¯− (x) ¯ − θ¯+ θ+ θ¯− θ− ∂++ ∂−− φ(x) √ √ ¯+ (x) + 2θ¯− χ− (x) + 2θ+ θ¯− G(x) Σ = σ(x) + 2θ+ χ + iθ¯+ θ+ ∂++ σ(x) − iθ¯− θ− ∂−− σ(x) √ √ − 2iθ+ θ¯− θ− ∂−− χ ¯+ (x) + 2iθ¯+ θ+ θ¯− ∂++ χ− (x) + θ¯+ θ+ θ¯− θ− ∂++ ∂−− σ(x) √ √ ¯ ¯ =σ Σ ¯ (x) − 2θ¯+ χ+ (x) − 2θ− χ ¯− (x) − 2θ¯+ θ− G(x) − iθ¯+ θ+ ∂++ σ ¯ (x) + iθ¯− θ− ∂−− σ ¯ (x) √ + − − √ + + − − 2iθ¯ θ¯ θ ∂−− χ+ (x) + 2iθ¯ θ θ ∂++ χ ¯− (x) + + − − + θ¯ θ θ¯ θ ∂++ ∂−− σ ¯ (x) V = −θ¯+ θ+ v++ − θ¯− θ− v−− + θ¯+ θ− σ − θ+ θ¯− σ ¯ √ √ + + − ¯ + − 2iθ¯+ θ+ θ¯− λ+ − 2iθ¯ θ θ λ √ √ ¯ − + 2iθ¯+ θ¯− θ− λ− + 2iθ+ θ¯− θ− λ + 2θ¯+ θ+ θ¯− θ− D .

¯ + D− V is a twisted chiral superfield with the identification D χ± = −iλ± ,

G = D − iF01 .

(2.31)

3. SUSY Lagrangians From (2, 2) Superfields 3.1 Kinetic Terms Kinetic terms for fields are constructed by integrating a superspace action over the fermonic R coordinates d4 θ. This amounts to picking out the bosonic coefficients of the term θ¯+ θ+ θ¯− θ− of the superspace action.

–6–

(3.1)

The canonical kinetic term for a chiral superfield is Z 1 ¯ = −∂µ φ∂ µ φ¯ + iψ¯+ ∂−− ψ+ + iψ¯− ∂++ ψ− + F F¯ d4 θΦΦ 4

(3.2)

up to integration by parts. We have used the fact that ∂++ φ∂−− φ¯ = (∂0 + ∂1 )(φ1 + iφ2 )(∂0 − ∂1 )(φ1 − iφ2 ) = −(∂µ φ1 )2 − (∂µ φ2 )2 = −∂µ φ∂ µ φ¯ ,

(3.3)

as we are using the signature ds2 = −(dx0 )2 + (dx1 )2 . The chiral superfield may couple to gauge fields. The relevant superspace integral is then Z 1 ¯ ¯ 2qV Φ = − Dµ φDµ φ¯ + iψ¯+ D−− ψ+ + iψ¯− D++ ψ− + F F¯ + qφφD d4 θΦe 4 (3.4) ¯ − − ψ¯− λ ¯ + ) + iq φ(ψ ¯ + λ− − ψ− λ+ ) + iqφ(ψ¯+ λ − qσ ψ¯− ψ+ − q¯ σ ψ¯+ ψ− − q 2 σ¯ σ φφ¯ where Dµ = ∂µ + iqvµ is the covariant derivative. We note, in particular, that D++ = ∂++ + iqv++ ,

D−− = ∂−− + iqv−− .

The canonical kinetic term for a twisted chiral superfield is given by Z 1 ¯ = −∂µ σ∂ µ σ ¯. − d4 θΣΣ ¯ + iχ ¯+ ∂−− χ+ + iχ ¯− ∂++ χ− + GG 4

(3.5)

(3.6)

up to integration by parts. It is worth noting the minus sign relative to the kinetic term ¯ + D− V , this term yields the kinetic for the chiral superfields. In particular, when Σ = D term for the gauge multiplet: Z 1 ¯ + ∂−− λ+ + iλ ¯ − ∂++ λ− + D2 + F 2 . ¯ = −∂µ σ∂ µ σ − d4 θΣΣ ¯ + iλ (3.7) 01 4 3.2 Superpotential Terms The (2, 2) theory can have a superpotential term: Z 1 dθ− dθ+ W (Φc ) + (h.c.) 2

(3.8)

where Φc are chiral superfields. Expanding in terms of components, the superpotential term becomes Z 1 dθ− dθ+ W (Φc ) = Fa ∂a W (φc ) + ∂a ∂b W (φc )ψa,− ψb,+ . (3.9) 2 3.3 Twisted Superpotential Terms The (2, 2) theory can also have twisted superpotential terms: Z 1 ˜ (Σt ) + (h.c.) dθ¯− dθ+ W 2

(3.10)

where Σt are twisted chiral superfields. Expanding in terms of components, the twisted chiral superpotential term becomes Z 1 ˜ (Σt ) = Gr ∂r W (σt ) + ∂r ∂s W ˜ (σt )χr,− χs,+ . dθ¯− dθ+ W (3.11) 2

–7–

3.4 Summary 1 4

D-Terms:

Z

¯ 2qV Φ = − Dµ φDµ φ¯ + iψ¯+ D−− ψ+ + iψ¯− D++ ψ− + F F¯ d4 θΦe ¯ − − ψ¯− λ ¯ + ) + iq φ(ψ ¯ + λ− − ψ− λ+ ) + iqφ(ψ¯+ λ ¯ − qσ ψ¯− ψ+ − q¯ σ ψ¯+ ψ− − q 2 σ¯ σ φφ¯ + qφφD

1 − 4

Z

¯ = − ∂µ σ∂ µ σ ¯ d4 θΣΣ ¯ + iχ ¯+ ∂−− χ+ + iχ ¯− ∂++ χ− + GG ¯ + ∂−− λ+ + iλ ¯ − ∂++ λ− + D2 + F 2 = − ∂µ σ∂ µ σ ¯ + iλ 01

1 2

Z

Sup. Pot.:

dθ− dθ+ W (Φ) =Fa ∂a W (φ) + ∂a ∂b W (φ)ψa,− ψb,+

Z

T-Sup. Pot.:

1 2

˜ (Σ) =Gr ∂r W (σ) + ∂r ∂s W ˜ (σ)χr,− χs,+ dθ¯− dθ+ W

The covariant derivatives are defined as Dµ = ∂µ + iqvµ ,

D++ = ∂++ + iqv++ ,

D−− = ∂−− + iqv−− .

(3.12)

4. Some Theories of Interest 4.1 N = (2, 2) Abelian Higgs Model Let us consider a theory with one chiral multiplet Φ coupled to a U (1) vector multiplet V with unit charge. We assume there is an F.I. term, i.e., a linear twisted superpotential ˜ (Σ) = 1 (−2πr + iθ)Σ , W 2

(4.1)

−2πrD + θF01

(4.2)

which contributes the term to the Lagrangian. Hence the Lagrangian for the theory is given by Z
1 ¯ 2V Φ − ΣΣ ¯ L(2,2)AH = d4 θ Φe 4 Z Z 1 1 − + ¯ ¯ + dθ dθ (−2πr + iθ)Σ + dθ¯+ dθ− (−2πr − iθ)Σ 4 4 2 ¯ = − Dµ φDµ φ¯ − ∂µ σ∂ µ σ ¯ + D2 + F01 + F F¯ − σ¯ σ φφ¯ + φφD ¯ + ∂−− λ+ + iλ ¯ − ∂++ λ− + iψ¯+ D−− ψ+ + iψ¯− D++ ψ− + iλ

(4.3)

¯ − − ψ¯− λ ¯ + ) + iφ(ψ ¯ + λ− − ψ− λ+ ) − σ ψ¯− ψ+ − σ + iφ(ψ¯+ λ ¯ ψ¯+ ψ− − 2πrD + θF01 We can promote the F.I. parameter into a superfield Θ with components √ √ Θ = (−2πr + iθ) + 2θ+ χ ¯+ + 2θ¯− χ− + 2θ+ θ¯− G

–8–

(4.4)

in which case the last line of the above Lagrangian is modified to

1 ¯ − 2πrD + θF01 + (σG + σ ¯ G) 2 i ¯+ + λ ¯−χ − (χ− λ+ + λ− χ+ + χ ¯− λ ¯+ ) . 2

(4.5)

4.2 N = (4, 4) Abelian Higgs Model

Let us consider a theory with one (4, 4) hypermultiplet — consisting of two chiral fields ˜ of positive and negative unit charge — coupled to a (4, 4) vector multiplet that Q and Q consists of a vector multiplet V and a chiral multiplet Φ. We rename components of the superfields to be compatible with [2] as the following:

√ ˜ + + 2iθ− λ ˜ − + 2θ+ θ− (D1 + iD2 ) 2iθ+ λ √ + √ − Q = q + 2θ ψ+ + 2θ ψ− + 2θ+ θ− F √ √ ˜ = q˜ + 2θ+ ψ˜+ + 2θ− ψ˜− + 2θ+ θ− F˜ Q √ √ ¯ + − 2iθ¯− λ− + 2θ+ θ¯− (D3 − iF01 ) Σ = σ + 2iθ+ λ Φ=φ+

√

(4.6)

We also add the superpotential and twisted superpotential terms

1 ˜ − 2π(r1 − ir2 )Φ), W = (2QQΦ 2

˜ = 1 (−2πr3 + iθ)Σ W 2

(4.7)

to the Lagrangian. These contribute the following terms:

2 Re(q q˜)D1 − 2 Im(q q˜)D2 − 2π(r1 D1 + D2 r2 + r3 D3 ) + θF01 + φ(F˜ q + F q˜) + φ(F˜ q + F q˜) ¯ ψ¯˜− ψ¯+ + ψ¯− ψ¯˜+ ) + φ(ψ˜− ψ+ + ψ− ψ˜+ ) − φ( ¯ ¯˜ ¯˜ ¯˜ ˜+ + λ ˜ − ψ˜+ ) + i¯ + iq(ψ˜− λ q (ψ˜− λ + + λ− ψ+ ) ¯˜ ¯˜ ¯ ˜+ + λ ˜ − ψ+ ) + iq¯˜(ψ¯− λ + i˜ q (ψ− λ + + λ− ψ+ )

–9–

(4.8)

Hence the Lagrangian for the theory is given by   Z 1 1 ¯ ¯ 4 2V −2V ˜ ¯ ¯ ˜ L(4,4)AH = d θ Qe Q + Qe Q + 2 (ΦΦ − ΣΣ) + (Potential Terms) 4 g µ = − Dµ qD q¯ + iψ¯+ D−− ψ+ + iψ¯− D++ ψ− ¯ ¯ − Dµ q˜Dµ q¯ ˜ + iψ˜+ D−− ψ˜+ + iψ˜− D++ ψ˜−  1  ¯ ¯˜ ˜ + ∂−− λ ˜ + + iλ ˜ ∂ λ + 2 −∂µ φ∂ µ φ¯ + iλ − ++ − g
1 ¯ + ∂−− λ+ + iλ ¯ − ∂++ λ− + F 2 + θF01 + 2 −∂µ σ∂ µ σ ¯ + iλ 01 g ¯ 2 + |F¯ ¯ ˜ + qφ|2 − (˜ ¯φ| + |F + q˜ q q˜¯ + q q¯)(σ¯ σ + φφ) (4.9)

1 2 D − 2π(ri Di ) + 2 Re(q q˜)D1 − 2 Im(q q˜)D2 + (|q|2 − |˜ q |2 )D3 g2 i ¯ ¯ + σ(ψ˜− ψ˜+ − ψ¯− ψ+ ) − σ ¯ (ψ˜− ψ˜+ − ψ− ψ¯+ )

+

¯ ψ¯˜− ψ¯+ + ψ¯− ψ¯˜+ ) + φ(ψ˜− ψ+ + ψ− ψ˜+ ) − φ( ˜+ + λ ˜ − ψ˜+ − λ ¯ − ψ¯+ − ψ¯− λ ¯+) + iq(ψ˜− λ ¯ ¯ ¯ ˜+ + λ ˜ − ψ¯ ˜+ − λ− ψ+ − ψ− λ+ ) + i¯ q (ψ˜− λ ˜+ + λ ˜ − ψ+ + λ ¯ − ψ˜¯+ + ψ˜¯− λ ¯+) + i˜ q (ψ− λ ¯ ¯ ¯ ˜ ˜ ˜ ˜ + iq¯ ˜(ψ¯− λ + + λ− ψ+ + λ− ψ+ + ψ− λ+ ) This action can be simplified using R-symmetry SU (2)R × SO(4)R . The scalars of the hypermultiplet transforms as a doublet under the first SU (2)R — let us define the doublet qa as ! q=

q q¯˜

.

(4.10)

The scalars of the vector multiplet transform as a vector of the SO(4)R group — let us define the vector φm as φ1 = Re σ,

φ2 = Im σ,

φ3 = Re φ,

φ4 = Im φ .

(4.11)

\ It is useful to define SO(4)R = SU˙(2)R × SU (2)R spinors

Ψ+,a˙ =

¯ ψ˜+ ψ+

! ,

Ψ−aˆ =

¯ ψ˜− ψ−

! ,

Ψ+ Ψ−

Ψ=

!

¯  ψ˜+ ψ   + = ¯  ψ˜−  ψ−

(4.12)

and a rather pecular choice of gamma matrices of SO(4): 1

Γ =

! 0 −I , Γ2 = −I 0

0 iσ 3 −iσ 3 0

! 3

, Γ =

– 10 –

0 iσ 2 −iσ 2 0

! 4

, Γ =

0 iσ 1 −iσ 1 0

! .

(4.13)

Let us define the γ m aˆ ˙ a matrices m

Γ

≡

0 γˇ m γm 0

! .

(4.14)

It is clear that γˇ mˆaa˙ = γ m aˆ ˙a .

(4.15)

Since the fundamental representation 2 of SU (2) is complex, we must be mindful of the placement of SU (2) indices on the fields. Our conventions are such that we first fix a fundamental representation for each SU (2)R and take its indices to be lower indices. We use upper indices for the conjugate representation. Therefore, in our conventions Ψ+ \ transforms as a 2 in SU˙(2) while Ψ− transforms as a ¯ 2 in SU (2). We note that we have paired up the fields so that the charged multiplets q and Ψ have positive unit charge. The fermionic fields of the (4, 4) vector multiplet — or rather, the (4, 4) gauginos — \ transform as a (¯ 2, 2, 1)+(¯ 2, 1, ¯ 2)+(c.c.) under SU (2)R × SU˙(2)R × SU (2)R . It is convenient to define ! ! ¯˜ ˜ − λ− λ a ˙ λ −λ † Λ− a˙a = , Λ− a = ¯ − ˜ − , ¯˜ ¯− λ −λ λ− λ− − ! ! (4.16) ¯˜ ˜ + λ+ λ λ −λ † + + Λ+ aˆa = ¯+ λ ˜¯ + , Λ+ aˆa = λ ¯+ λ ˜+ −λ to make the action of the symmetry group manifest. Integrating out the auxiliary fields by setting Di + g 2 (q† σi q − πri ) = F¯˜ + qφ = F¯ + q˜φ = 0 ,

(4.17)

we obtain L(4,4)AH = Lkin + Lpot + Lf erm 1 2 1 ~ µ~ · ∂ φ + 2 F01 Lkin = − (Dµ q)† Dµ q − 2 ∂µ φ g g   i + iΨ†+ D−− Ψ+ + iΨ†− D++ Ψ− + 2 Tr Λ†+ ∂−− Λ+ + Λ†− ∂++ Λ− 2g † 2 † 2 ~ · φ) ~ − g (q σi q − πri ) + θF01 Lpot = − (q q)(φ

(4.18)

Lf erm =(Ψ†+ γ m Ψ− + Ψ†− γ¯ m Ψ− )φm − i(Ψ†+ Λ− − Ψ†− Λ+ )q + iq† (Λ†− Ψ+ − Λ†+ Ψ− ) . We note that the fermionic interaction terms can be rewritten in various different forms — a˙

† a ˆ Ψ†+ γ m Ψ− + Ψ†− γ¯ m Ψ− = Ψ†+ γ m aˆ ˇ mˆaa˙ Ψ+,a˙ = Ψ† Γm Ψ ˙ a Ψ− + Ψ−,ˆ aγ

Ψ†+ Λ− q = Ψ†− Λ+ q =

a˙ Ψ†+ Λ− a˙a qa Ψ†− aˆ Λ+ aˆa qa

(4.19) (4.20) (4.21)

Recall that our conventions are such that q†a = qa ,

a˙

Ψ† + = Ψ+,a˙ ,

– 11 –

Ψ† −,ˆa = Ψ−aˆ .

(4.22)

While the (2, 2) auxiliary fields for the hypermultiplet do not transform covariantly in the R-symmetry group of the (4, 4) theory, the Lagrangian is invariant under the symmetry once these auxiliary fields are integrated out. In contrast, the (2, 2) auxiliary fields for the vector multiplet come in representations of the R-symmetry group. In fact, we could have retained the D-fields, i.e., −g 2 (q† σi q − πri )2

→

1 2 D + 2Di (q† σi q − πri ) , g2 i

(4.23)

and still have an R-symmetry covariant Lagrangian.

4.3 N = (4, 4) NS5-brane Model Let us consider a theory where we couple a twisted hypermultiplet to the (4, 4) abelian Higgs model described in the previous section through a linear coupling. To be more precise we couple the twisted hypermuliplet that consists of a chiral superfield P and a twisted chiral superfield Θ — whose components are given by √ √ P = (−r1 + ir2 ) + 2θ+ χ+ + 2θ− χ− + 2θ+ θ− (−G3 + iG4 ) √ √ ¯˜+ + 2θ¯− χ Θ = (−r3 + iθ) + 2θ+ χ ˜− + 2θ+ θ¯− (−G1 + iG2 )

(4.24)

through the chiral and twisted chiral superpotentials 1 ˜ + P Φ), W = (2QQΦ 2

˜ = 1 ΘΣ . W 2

(4.25)

We have almost written out this theory in the previous section — we merely need to promote the (4, 4) F.I. terms to superfields. These contribute the following terms 2 Re(q q˜)D1 − 2 Im(q q˜)D2 − (r1 D1 + D2 r2 + r3 D3 ) + θF01 + φ(F˜ q + F q˜) + φ(F˜ q + F q˜) ¯ ψ¯˜− ψ¯+ + ψ¯− ψ¯˜+ ) + φ(ψ˜− ψ+ + ψ− ψ˜+ ) − φ( ¯˜ ¯˜ ¯ ¯˜ ˜+ + λ ˜ − ψ˜+ ) + i¯ + iq(ψ˜− λ q (ψ˜− λ + + λ− ψ+ ) ¯˜ ¯˜ ¯ ˜+ + λ ˜ − ψ+ ) + iq˜¯(ψ¯− λ + i˜ q (ψ− λ + + λ− ψ+ )
i ¯+ + λ ¯−χ ¯˜− λ ¯˜+ − χ ˜− λ+ + λ− χ ˜+ + χ 2  i ˜ ¯˜ ¯˜ ˜ − χ+ + χ + χ− λ + + λ ¯− λ ¯ + + φm G m + + λ− χ 2

(4.26)

which is identical to (4.8) except for the last two lines coming from promoting the F.I.

– 12 –

parameters to superfields. The Lagrangian for the full theory is given by   Z 1 1 ¯ 1 ¯ ¯ 4 2V −2V ˜ ¯ ¯ ¯ ˜ L(4,4)N S5 = d θ Qe Q + Qe Q + 2 (ΦΦ − ΣΣ) + 0 2 (P P − ΘΘ) 4 g g + (Potential Terms) 1 1 = − 0 2 (∂µ ri )(∂ µ ri ) − 0 2 (∂µ θ)(∂ µ θ) g g i ¯˜+ ∂−− χ ¯˜− ∂++ χ ¯+ ∂−− χ+ + iχ ¯− ∂++ χ− + iχ ˜+ + iχ ˜− ) + 0 2 (χ g ~ · φ) ~ + 1 D2 + 2Di (q† σi q − 1 ri ) + θF01 + 1 G ~·G ~ ·G ~ −φ ~ − (q† q)(φ g2 i 2 g 02
i ¯+ + λ ¯−χ ¯˜− λ ¯˜+ − χ ˜− λ+ + λ− χ ˜+ + χ 2  i ˜ ¯˜ ¯˜ ˜ − χ+ + χ + ¯+ χ− λ+ + λ ¯− λ + + λ− χ 2 + LAH,kin + LAH,f erm

(4.27)

where LAH,kin and LAH,f erm are given by equation (4.18). \ Notice that under the SU (2)R × SO(4)R = SU (2)R × SU˙(2)R × SU (2)R symmetry of the theory the scalars of the twisted hypermultiplet transform as (3 + 1, 1, 1) while the auxiliary fields Gm transforms as a vector of the SO(4)R . The fermions in the twisted \ hypermultiplet transform as (2, ¯ 2, 1) + (2, 1, 2) + (c.c.) under SU (2)R × SU˙(2)R × SU (2)R . This can be made manifest by labeling the fields ! ! ¯ ¯ χ χ ˜ χ χ ˜ + + − − X+ aa˙ = , X− aˆa = , (4.28) −χ ˜+ χ ¯+ −χ ˜− χ ¯− in which case the Lagrangian can be succinctly written as   1 i 1 † † µ µ (∂ r )(∂ r ) − (∂ θ)(∂ θ) + Tr X ∂ X + X ∂ X µ i i µ + −− + − ++ − g02 g02 2g 0 2 ~ · φ) ~ + 1 D2 + 2Di (q† σi q − 1 ri ) + θF01 + 1 G ~·G ~ ·G ~ −φ ~ − (q† q)(φ g2 i 2 g 02 i + Tr (Λ− X+ + X− Λ+ ) 2 + LAH,kin + LAH,f erm

L=−

References [1] K. Hori and C. Vafa, “Mirror symmetry,” hep-th/0002222. [2] D. Tong, “NS5-branes, T duality and world sheet instantons,” JHEP 0207, 013 (2002) [hep-th/0204186].

– 13 –

(4.29)