The A-type gauge theory is constructed by twisting theU(1)E rotational symmetry by theU(1)V
vector R-symmetry of the N = (2,2) SYM theory (see Table 4.5 for the charges of field in the N = (2,2) SYM theory),
E→E+V.
Field U(1)E U(1)A
Az 2 0
Az¯ −2 0
ψz 2 1
η 0 −1
χz¯z 0 −1 ψz¯ −2 1
φ 0 2
φ 0 −2
,
Table 4.6: Fields in A-type gauge theory
This theory consists of an adjoint complex scalar fieldφ, an adjoint fermionic scalar field η, an adjoint fermionic 1-form ψ, an adjoint fermionic 2-formχ, and a gauge fieldA. The BRST charge is
QA=Q−+Q+,
which is a scalar after twisting (see Table 4.2 for the charges of theN = (2,2) supercharges). The
BRST variations follow from the corresponding supersymmetry transformations (4.13), δA=ξψ,
δη=ξ φ, φ
, δψ=iξDφ, δχ=ξF, δφ= 0, δφ=ξη.
Notice that the variations are nilpotent (up to gauge transformations) except forχ, δ2A=iξ1ξ2Dφ,
δ2η=ξ1ξ2
φ, η , δ2ψ=ξ1ξ2
φ, ψ], δ2χ=ξ1ξ2Dψ, δ2φ= 0, δ2φ=ξ1ξ2
φ, φ .
It will be convenient to introduce an auxiliary bosonic 2-formP in the adjoint representation. We require thatP satisfy the following on-shell constraint,
P =F, (4.14)
so that we can write the BRST variations as δA=ξψ,
δη=ξ φ, φ
, δψ=iξDφ, δχ=ξP, δφ= 0, δφ=ξη, δP =ξ
φ, χ .
(4.15)
It is not difficult to construct an action equivalent to original A-model action that enforces the on-shell constraints on the auxiliary fields (4.14) and respects the BRST symmetry (4.19),
SA= 1 g2
Z
Σ
n
QA, iψ∧?Dφ+ 4χ∧? P−2F−?1 2
φ,φ¯o
. (4.16)
We now construct the category of branes for the A-type gauge theory with gauge groupU(1) [21].
Neumann boundary conditions require that?F and?dφvanish along the boundary, while Aandφ are free along the boundary. BRST-invariance requires?χ and the restriction of?ψ vanish on the boundary, while η and the restriction ofψ remain unconstrained. The algebra of BRST-invariant observables on the Neumann boundary is spanned by powers of φ, the algebra O of holomorphic functions onC. One can construct a more general boundary condition by placing additional degrees of freedom on the boundary which live in a vector spaceV graded by theU(1)Acharge. The BRST operator gives rise to a degree-1 differentialT :V →V which may depend polynomially onφ. Thus we may attach a brane to any free DG-moduleM = (V ⊗ O, T) over the graded algebra O. The space of morphisms between any two such branes M1 = (V1⊗ O, T1) and M2 = (V2⊗ O, T2) is the cohomology of the complex HomO(M1, M2), which agrees with the space of morphisms in the categoryDb(C).
In the A-type 2d gauge theory one may also consider the Dirichlet boundary condition which setsφ= 0 on the boundary and requires the restriction of the gauge field to be trivial. One might guess that it corresponds to the skyscraper sheaf at the origin ofC, and indeed one can verify that the space of morphisms from any of the branes considered above to the Dirichlet brane agrees with the space of morphisms from the corresponding complex of vector bundles on Cto the skyscraper sheaf.
Another way to approach the problem is construct an isomorphism between the A-type gauge theory and the B-model with targetC. From the physical viewpoint, an isomorphism between two- dimensional TQFTsXandYis an invertible topological defect lineAbetween them. In the present case, there is a unique candidate for such a defect line. Recall that a B-model with target Chas a bosonic scalar σ, a fermionic 1-formρ, and fermionic 0-formsθ andξ. The BRST transformations read
δφ= 0, δφ¯=η, δη= 0, δθ= 0, δρ=dφ.
The fieldσhas ghost number 2, the fields ρhas ghost number 1, and the fields η andθhave ghost
number−1. The action of the B-model is
S=−1 2δ
Z
M2
ρ∧?dφ¯+ Z
M2
θ∧dρ.
Obviously, the ghost-number 2 bosonsσandφmust be identified on the defect line, up to a numerical factor which can be read of the action. Similarly, the fermionic 1-formsψandρmust be identified, as well as the fermionic 0-forms β and η. Finally, one must identify ?χand θ. BRST invariance then requires?F to vanish on the boundary, which means that the gauge field obeys the Neumann boundary condition.
Note that this defect line is essentially the trivial defect line for the fermionic fields andσ. Since the zero-energy sector of the bosonicU(1) gauge theory is trivial, the invertibility of the defect line is almost obvious. Let us show this more formally. First, consider two parallel defect lines with a sliver of the A-type 2d gauge theory between them. The sliver has the shape R×I, where R parameterizes the direction along the defect lines. The statement that the product of two defect lines is the trivial defect line in the B-model is equivalent to the statement that the U(1) gauge theory on an interval with Neumann boundary conditions on both ends has a unique ground state.
This is obviously true, because the space-like component of Ain the sliver can be gauged away by a time-independent gauge transformation, and therefore the physical phase space of theU(1) gauge theory on an interval is a point.
Second, consider the opposite situation where a sliver of the B-model is sandwiched between two defect lines. We would like to show that this is equivalent to the trivial defect line in the A-type 2d gauge theory. The sliver has the shapeS1×I. For simplicity we will assume that the worldsheet with a sliver removed consists of two connected components. Each component is an oriented manifold with a boundary isomorphic to S1, and the path-integral of the A-type 2d gauge theory defines a vector in the Hilbert spaceV corresponding to S1. Any topological defect line in the A-type gauge theory defines an element in V∗⊗V ' End(V). We would like to show that the B-model sliver corresponds to the identity element in V∗⊗V. First we note that V can be identified with the tensor product of the Hilbert space of the zero-energy gauge degrees of freedom and the Hilbert space of the zero-energy degrees of freedom ofσand the fermions. As mentioned above, the defect line separating the A-type 2d gauge theory and the B-model acts as the trivial defect line onσand the fermions, so in this sector the statement is obvious. As for the gauge sector, the corresponding Hilbert space of zero-energy states is one-dimensional, so the B-model sliver is proportional to the identity operator. The argument of the preceding paragraph shows that its trace is one, so the sliver must be the identity operator.