JHEP08(2017)101
2.3 Handle-gluing operator and flux operator
The A-twisted theory is a topological field theory [29], whose local observables are fully
determined by the topological action:
S
TFT
=
Z
Σ
g
d
2
x
√
g
−2f
1
¯
1
a
∂W
∂σ
a
+
e
Λ
a
¯
1
Λ
b
1
∂
2
W
∂σ
a
∂σ
b
− 2F
1
¯
1
(F )
∂W
∂m
F
+
i
2
ΩR
, (2.24)
which is given in terms of W and Ω. Here f
a
= da
a
and F
(F )
= dA
(F )
. We refer to [34]
for a more thorough discussion.
As in any topological field theory, there exists a local operator H, the handle-gluing
operator, whose insertion corresponds to “adding a handle” to the Riemann surface:
hOHi
g
= hOi
g+1
. (2.25)
For A-twisted N = (2, 2) gauge theories, H was first computed explicitly in [31] — see
also [17, 32, 33]. It is given by:
H(σ) = exp (2πiΩ(σ)) H(σ) , (2.26)
where Ω is the effective dilaton (2.22) and H is the Hessian determinant (2.13). This latter
contribution comes from the gaugino zero-modes in the twisted theory, which couple to W
as indicated in the second term in (2.24). It is clear from (2.24) that H corresponds to a
local operator insertion one obtains by concentrating the curvature of a single handle at a
point, with a δ-function singularity.
Similarly, there exists local operators whose insertion changes the background fluxes
for the flavor symmetries. These so-called “flux operators” [34] are simply given by:
Π
F
= exp
2πi
∂W
∂m
F
, (2.27)
in term of the effective twisted superpotential W = W(σ, m
F
).
We should also note that the coupling of the GLSM to curved space introduces a
“gravitational” anomaly for the axial R-symmetry U(1)
ax
[14, 15], with coefficient:
ˆc
grav
= −dim(g) −
X
i
(r
i
− 1)dim(R
i
) . (2.28)
This corresponds to the U(1)
ax
−U(1)
R
‘t Hooft anomaly:
b
R
0
= −ˆc
grav
. (2.29)
This anomaly is reproduced by the handle-gluing operator, since
H → e
−2iαb
R
0
H (2.30)
under a U(1)
ax
rotation, corresponding to an anomalous shift of θ
R
. When b
I
0
= 0 and if
the theory flows to a conformal fixed point, c = 3 ˆc
grav
is the central charge of the infrared
SCFT [1].
– 8 –
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