Varieties of Abelian mirror symmetry on $\mathbb{RP}^2 \times \mathbb{S}^1$

We study 3d mirror symmetry with loop operators, Wilson loop and Vortex loop, and multi-flavor mirror symmetry through utilizing the $\mathbb{RP}^2 \times \mathbb{S}^1$ index formula. The key identity which makes the above description work well is the mod 2 version of the Fourier analysis, and we study such structure, the S-operation in the context of a SL$(2,\mathbb{Z})$ action on 3d SCFTs. We observed that two types of the parity conditions basically associated with gauge symmetries which we call $\mathcal{P}$-type and $\mathcal{CP}$-type are interchanged under mirror symmetry. We will also comment on the T-operation.


Introduction
Three-dimensional N = 2 mirror symmetry [1][2][3][4] is one of the nontrivial classes of dualities among quantum field theories, which states that two distinct theories defined in the UV regime flow into an identical IR fixed point. In order to test such dualities, exact physical quantities protected by supersymmetry are often useful and provide to us understandings of the duality in mathematically well-defined languages.
In this paper, we deepen the study of 3d Abelian mirror symmetry on RP 2 × S 1 performed in [5,6]. We show that the correspondences of loop operators and the multiflavor version of mirror symmetry which were studied on other three-dimensional manifolds [7][8][9] also work even on RP 2 × S 1 in terms of the index [10]. The key mathematical identity to demonstrate them is a mod 2 delta function's formula: This is nonvanishing only if A−B ≡ 0 (mod 2). See (2.23) for details. The mod 2 structure can be regarded as an algebraic representation of the RP 2 unorientable structure.
As considered somewhat systematically in [6], in order to define supersymmetric gauge theories on RP 2 × S 1 , there are two types of parity conditions for the vector multiplet consistent with the unorientable structure which we call P-type and CP-type. The covariant derivative under each parity condition transforms as • P-type vector multiplet ∋ A µ : (∂ µ − iQA µ ) → ±(∂ µ − iQA µ ). (1.2) • CP-type vector multiplet ∋ A µ : (∂ µ − iQA µ ) → ±(∂ µ + iQA µ ). (1. 3) The former simply brings the parity (P) effect, while the latter accompanies charge conjugation (C), +Q ↔ −Q, with parity. In addition, we observe that it is possible to calculate e-charged BPS-Wilson loop W e and m-vorticity BPS-Vortex loop V m . As well known, the Wilson loop is an order parameter of the system. On the other hand, the Vortex loop, which is a codimension-2 defect in 3d theories, is a disorder parameter of the system and is defined by not a standard operator insertion but singular behavior of the fields around the loop [7,8,11]. In fact, there is one simple way [7] to describe the Vortex loop by making use of S-operation of a SL(2, Z) group in 3d CFTs [12]. We systematically find the correspondences of these loop operators based on that fact in the context of 3d N = 2 mirror symmetry with a single matter N f = 1: The Wilson loop on one side is mapped to the Vortex loop on the dual side as seen in the general discussion [7,8] and also in [11] for N = 4. The remarkable point in this duality is to interchange P-type and CP-type for the dynamical (Gauge) or background (Global) gauge field. This can be regarded as the non-trivial effect originated from the unorientable structure.
We also find that the mirror symmetry exchanges P and CP in many ways. One example is multi-flavor mirror symmetry, the duality between the SQED with N f flavors (SQED N f ) and U(1) N f −1 quiver gauge theory, which is realized by the following two ways: CP-SQED N f ⇔ P-U(1) N f −1 gauge theory, (1.6) where leading letters represent the parity type for the vector multiplet in the gauge group. These are new properties in 3d theories equipped with geometrical Z 2 -structure. The rest of this paper is organized as follows. In Section 2, we briefly review the exact results by localization techniques applied to path integrals of Abelian SUSY gauge theories on RP 2 ×S 1 and provide recipe for the computation and the usage of S-operation. In Section 3, we show that the correspondences of the BPS loop operators (1.4) and (1.5) in singleflavor mirror symmetry by utilizing S-operation on RP 2 × S 1 . In Section 4, we check multiflavor mirror symmetry (1.6) by using the localization recipe and the gauging procedure of the global symmetry. We comment on open questions in Section 5. In Appendix, we summarize the bases of the BPS loop operators which are Z 2 -invariant on RP 2 .
2 Features of SUSY gauge theory on RP 2 × S 1

Lightning review on localization computation on RP 2 × S 1
This subsection is dedicated to the review of [5,6]. What we consider is RP 2 index, where q is fugacity andF ,Ĥ,ĵ 3 are fermion number, Hamiltonian, 3rd angular momentum. H RP 2 is the theory's Z 2 -divided Hilbert space. In order to compute this quantity, we rewrite this definition by using path integral over the supersymmetric multiplets, where φ and φ are complex scalars, σ is real scalar, ψ and λ are spinors, A µ is gauge field 1 . The effect of considering Z 2 -divided Hilbert space should emerge in the boundary conditions for the fields under the reflection, and the (2.1) becomes with appropriate boundary conditions as explained below. In order to represent arbitrary theory's I RP 2 (q), we use quiver diagram.
Vector multiplet Compared with the conventional notation, RP 2 × S 1 quiver diagram is slightly different. One important difference is that we have two types of the vector multiplets V (P) and V (CP) [6], and represent them as nodes with characters P or CP: V (CP) : dynamical = ?>=< 89:; CP , background = s, θ ± where s ∈ Z, θ + = 0 θ − = π .
Matter multiplet We represent matter multiplet degrees of freedom by a rectangular node. As same as the case of vector multiplet, we have basically two possible matter multiplets, Φ s and Φ d , a singlet and a doublet: (2.5) For more detail, see [6]. 1 Vector multiplet V include gauge field Aµ, so the theory with V becomes gauge theory automatically.
Gauge coupling We represent gauge coupling between matter and vector multiplets with arrow → connecting the two nodes. The number of arrowheads means charge Q, and the direction of the arrow means the sign(Q). To apply (2.1), we should make theory Z 2 -invariant at least classically, and the possible couplings are as follows: Note that matter singlet cannot couple to CP-type because it includes charge conjugation C, and singlet is not invariant under the C. Of course, one can couple the background vector multiplets in (2.4) in these way, and in such case, we will use dotted arrow / / . where we use the index "BF" as positive coupling. If the coupling is negative, we use the index "−BF". This coupling will be important in later discussion. One can consider one of them as background factor multiplet, too. then, what we calculate is the following quantity: where S = β 0 dy d 2 x √ g (L mat + L U (1) + L top ) and We write here the auxiliary fields F, F , D (P) , and after integrating out them, we get a nontrivial potential for φ. The ∆ is an arbitrary parameter called R-charge for the matter multiplet. In the final line L top , we use the following configuration parametrized by (s, θ ± ) a(s, θ ± ) = s(cos θ − 1)dϕ + θ ± 2π dy, (ϑ, ϕ): RP 2 coordinates. (2.10) Recipe for the index from localization computation Thanks to supersymmetric localization argument, it turns out that the path integral can be calculated exactly by considering up to one loop. More precisely speaking, the Gaussian calculation is performed around configurations giving S = 0, and all of these contributions should be summed (or integrated) over. The result is summarized in section 2.2 of [6] as simple rule. By using the rule, we can calculate the path integral weighted by the theory characterized by a quiver diagram. The rule is composed by following steps: • If there is node with rigid line, just replace it as sum of possible dotted objects . . .
. . . (2.12) • Then, the resultant quiver diagram is represented by only dotted lines and dotted circler nodes. We regard it as the integrand or summand in (2.11) or (2.12) by using the following rules 3 (2.14) We write here doubly charged matter multiplets in (2.13), (2.14) and (2.15), but if one want to consider singly charged matter, the formula is obtained just by turning off one of charges Q 1 or Q 2 . In addition, we can couple matter rectangular node to circular nodes more than three, in this case, the formula is generalized naturally just by replacing in all expression. 3 We correct the formula (2.14) in the previous version by adding the exponential prefactor.

S-operation
Interpretation for background vector multiplets For later use, let us remind the physical meaning of background vector multiplets, e.g. (2.8) or (2.9). Basically, turning on s ± or s corresponds to changing the Hilbert space because turning on the flux generates Landau levels in the sense of quantum mechanics. On the other hand, turning on θ or θ ± is equivalent to consider twisted boundary condition 4 : . . .

Mirror symmetry via loop operators
Once we regard our index in path integral form like in (2.3), we can insert operator O like In order to apply the supersymmetric localization technique to calculate it, the operator should preserve part of the supersymmetry, called BPS condition.

BPS loop operators
We can find such BPS Wilson loop which wraps S 1 -direction standing at north pole = south pole of RP 2 . To define Wilson loop, gauge field is necessary. In our case, vector multiplet is necessary. As we noted, there are two types of vector multiplet. We find the following operators are BPS in each type On the other hand, one can define disorder operator called Vortex loop operator by considering singular boundary condition along the loop. On S 3 or S 2 × S 1 , these supersymmetric operators are defined in somewhat direct way [8], but one can utilize S-operation to define them as done on S 3 by [7]. We have already observed the S-operation works well in our case, so we use this formula to define vortex loops for two-types: Now, from the point of view that S-operation is "particle-vortex duality", it is understood as P-type vortex = CP-type particle, and CP-type vortex = P-type particle. See Appendix A for more detail.

Loop operator effects on localization computation
First, let us define quiver notation for inserting the above loop operators. We just represent insertion of loop operator O = W e , V m associated symmetry by just adding appropriate subscript to the circular node: then, the recipe for the exact result is as follows: • 1. Follow the previous recipe in (2.11) -(2.16) • 2. For the subscript O = W e or V m , follow the rules: . . .
Generalized index and its loop insertions We can regard the above formulas in (3.7) and (3.8) in the context of generalized indices (2.17) or (2.18). In this case, are associated by the external fields, and we get One can find the Vortex operators act on the indices as finite differential operators. We can regard the gauging procedure as the S-transform itself, i.e.
Therefore, we getŴ Note that this equalities mean the statements in (1.4) and (1.5) because S-transformation exchanges the P and CP.

Mirror symmetry with loop operators
The expectation values of loop operators associated to U (1) A symmetry should be in agreement if the original indices are identical. We can find somewhat non-trivial statement in [8,11] that In addition, we already know that topological U(1) J in the SQED corresponds to flavor U(1) V in the XYZ model under the mirror map. As a consequence, the dualities of the loop operator insertions are claimed between for the gauge symmetry and for the global symmetry as (3.14) and (3.15).

From single-flavor to multi-flavor mirror symmetry
Abelian 3d N = 2 mirror symmetry [1][2][3][4] with a single matter is the duality between the SQED and the XYZ model, namely, they flow to the same fixed point in IR. The moduli parameters in the SQED, the combination of the scalar in the vector multiplet and the dual photon e ±(σ+iρ) for Coulomb branch, and the product of mattersQQ for Higgs branch, correctly match (X, Y ) and Z which are matter multiplets in the XYZ model, respectively. Furthermore, there are two global symmetries U(1) J (as a topological one) and U(1) A in addition to an R-symmetry in the SQED, and the XYZ model has exactly corresponding flavor symmetries U(1) V and U(1) A acting on the matter fields as for usual dualities. For the neat review of this simplest version of mirror symmetry and charge assignments on the fields, see [5,9]. This symmetry can be directly checked as the equality of the superconformal indices computed by the localization [9,13], for the SQED I N =2 SQED and the XYZ model I N =2 XYZ (from now on, we omit the superscript RP 2 on the index).
We can enlarge supersymmetry to N = 4 [1][2][3] by adding an adjoint chiral multipletS to the N = 2 SQED in such a way that it couples with the matters through a superpotential QSQ. Correspondingly, in the XYZ model, we need to couple a chiral multipletZ to Z, which is equivalent to introducing the mass termZZ for them. Thus, they are integrated out, and the resultant matter theory becomes the free theory only with X and Y . In the language of the index which is known to be a protected quantity along the RG flow, those processes can be done simply by multiplying the inverse of the one-loop determinant Z Z 1-loop for Z, or equivalently, the one-loop determinant ZZ 1-loop forZ, by both sides of the equality (4.1). This is just the observation that integrating out Z in the XYZ model is to simply remove its one-loop determinant from the index, and, on the other hand, the new adjoint fieldS in the SQED has charges under the global symmetries corresponding to these ofZ which can be read off from the superpotential. Thus, the contribution ZS 1-loop forS to the index can be viewed as the one ZZ 1-loop forZ under mirror symmetry, and then schematically, N = 4 mirror symmetry should be written as where I N =4 SQED and I N =4 free are the indices for the N = 4 SQED and for the free matter theory, respectively, and under the suitable identification of parameters. The index formula (4.2) of N = 4 mirror symmetry with a single flavor is a key to demonstrate N = 2 mirror symmetry with many flavors. We will show these explicit forms depending on the parity conditions in the next subsection. Now, we aim to briefly review the statement of N = 2 multi-flavor mirror symmetry. The one side of the duality is the SQED with N f a general number of matters. The dual side is the quiver theory with an U(1) N f −1 gauge symmetry, and there are an adjoint matter associated with each gauge node, bifundamental matters connecting two neighboring gauge nodes, and fundamental matters at the ends of the quiver diagram. The former can be obtained from the theory with N f free matters by gauging the sum of N f U(1) V 's equipped with each single-flavor free theory. To construct the correct dual theory, we start with N f sets of the N = 4 SQED comprised of the N = 2 SQED and an adjoint matter coupled to the fundamental matter via the superpotential, then gauging the sum of N f U(1) J 's. Note that this gauging process is equivalent to ungauging the diagonal U(1) part of the U(1) N f gauge symmetry, which results in the U(1) N f −1 quiver gauge theory with appropriately redefining the gauge charges of the bifundamental fields. Actually, the above theories which we use as starting points to get multi-flavor mirror symmetry are exactly connected to each other by N = 4 mirror symmetry: as explained, the N f = 1 free matter theory is dual to the N = 4 SQED with a single flavor. Therefore, we take the following strategy [9] based on that fact to build multi-flavor mirror symmetry in terms of the index: 1. We set the index I N f as the collection of N f free single-matter theories, that is, free (a i ) where a i is some parameter of the i-th free theory. This clearly enjoys N = 4 mirror symmetry as In the following, we say a P-type or a CP-type U(1) symmetry for short if the background gauge field for that U(1) respects the P-type or the CP-type parity condition, respectively.

Single-flavor N = 4 mirror symmetry
In this subsection, we will give the formulas for N = 4 mirror symmetry which play a central role to derive the multi-flavor version of N = 2 mirror symmetry. As explained, they can be readily derived from N = 2 mirror symmetry whose detailed discussions including the charge assignment under the symmetries and rigorous proof can be seen in [5,6].
There are two types of Abelian mirror symmetry on RP 2 × S 1 depending on the case where the dynamical vector multiplet in the SQED is set to be P-type or CP-type. We name the former P-SQED (±) and the latter CP-SQED (±) . The superscript (±) represents a fixed value of the background field coupled to the dynamical one though a BF term. N = 4 P-SQED (±) vs The free matter theory. (4.7) The index formula for N = 4 P-SQED (±) is given by ( (4.8) The other version of N = 4 mirror symmetry is described by the indices (4.8) for N = 4 CP-SQED (±) and (4.9) for the dual free theory. In the former, the parameter s is a holonomy and w is a fugacity for P-type U(1) J , on the other hand, (s ± ,w) are a holonomy and a fugacity for P-type U(1) V in the dual theory. N = 4 mirror symmetry with one flavor is realized as the equality of (4.8) and (4.9) under identifications a =ã and (s ± , w) = (s ± ,w).
The mathematical proof to show equivalence of the indices in both cases is done in [5,6].
We should note that the background gauge holonomy for U(1) A is fixed to be even in all above index formulae. Actually, we can also consider its odd-holonomy by using the revised index formula (2.14), namely, one more parameter for U(1) A is brought into the indices of the SQED and the XYZ model. We now skip to show detailed expressions, but the statement of Abelian mirror symmetry on RP 2 × S 1 still holds at the level of the index.

Multi-flavor N = 2 mirror symmetry
We will demonstrate systematically the multi-flavor version of Abelian mirror symmetry on RP 2 ×S 1 following the prescription explained in Section 4.1. The starting point is the index I N f of N f free matter multiplets coupled to the background gauge fields associated with N f U(1) V flavor symmetries which rotate a phase to each matter multiplet. We should notice that even though a priori each background field can take freely a Por a CP-type parity condition, it seems to be only allowed to set all of parity conditions being either Por CP-type because we may gauge the diagonal part of U(1) N f V if all of them are the same. It is apparently unable to construct multi-flavor mirror symmetry with the combination of an arbitrary number of Pand CP-type U(1) V 's. This is why, in this paper, we have two kinds of the SQEDs in the context of multi-flavor mirror symmetry on RP 2 × S 1 , that is, P-SQED N f which is the theory obtained by gauging the sum of P-type U(1) N f V in the N f -free matter theory and CP-SQED N f by gauging the sum of CP-type U(1) We first consider P-SQED N f and its dual theory. We begin with the index I (P) N f for the free theory with N f P-type U(1) V 's given by where s ± i corresponds to a holonomy on RP 2 and w i is a fugacity associated to each U(1) V . A fugacity a i contains a parameter associated to each P-type U(1) A symmetry as defined in single-flavor N = 4 mirror symmetry. We rewrite this index (4.10) as follows: The parameters (m ± , z) are ones for the diagonal part of U(1) N f V which we will denote by U(1) diag . Gauging U(1) diag in this free theory make it P-SQED N f whose index is where we introduce parameters m J ∈ Z and θ J± for a CP-type U(1) J topological symmetry from gauging U(1) diag because a background field for U(1) J should be coupled to a P-type dynamical one via the BF term. This is the index for P-SQED N f .
• CP-U(1) N f −1 gauge theory: We refer to the theory dual to P-SQED N f as CP-U(1) N f −1 which is comprised of (N f − 1) CP-type U(1) gauge groups with global symmetries corresponding to these in P-SQED N f . Returning back to I (P) N f (4.10), we apply the identity between (4.6) and (4.7) representing N = 4 mirror symmetry to re-express I (P) N f as an equivalent but quite different form I CP-U(1) N f ,

(4.13)
This is simply the index for N f copies of CP-SQED (±) 's having N f adjoint matters and N f P-type U(1) J global symmetries whose labels are (s ± i , w i ). Here, we set new variables (m ± i , u i ) for these global symmetries such as where (m ± , z) correspond to a holonomy and a Wilson line phase for the sum of N f P-type U(1) J 's which will be denoted by U(1) diag . Then, by gauging this P-type U(1) diag , we acquire the index for CP-U(1) N f −1 as where m J ∈ Z and θ J± are a monopole flux and a Wilson line phase, respectively, for a topological U(1) J which emerges from gauging U(1) diag . Now, we can evaluate the summations over m ± and z. The sum over m ± gives the delta function (2.22) defined by mod 2 m ± e im ± (θ J ± + i θ i± ) = 2δ mod 2 (θ J± + i θ i± ), (4.16) and the integral over z reduces to the ordinary Kronecker delta dz 2πiz z m J + i s i = δ m J + i s i ,0 . (4.17) Table 1. The symmetries and their fugacities in P-SQED N f and CP-U(1) N f −1 .

P-SQED
WIth these constraints, we can split (θ i± , s i ) into other (hatted) variables as follows: Note that m J should be a multiple of N f due to gauge invariance so thatŝ i are still integers. Then, the index for the dual theory CP-U(1) N f −1 can be rewritten as
The equality of the indices (4.12) for P-SQED N f and (4.19) for CP-U(1) N f −1 gauge theory is just Abelian N = 2 mirror symmetry with general N f flavors. We summarize the symmetries and its fugacities contained in both theories in Table 1. Although all of the global symmetries are mapped in the one-to-one manner between two theories, we find that types of the parity conditions for the gauge groups are exchanged with each other. Surely, the gauge symmetries in several theories which share the common IR fixed point do not necessarily match at UV, but this observation is an interesting, specific feature reflecting the fact that the theories are put on an unorientable manifold. Note that we can appropriately set the holonomy for U(1) N f A by applying the modified formula (2.14) as mentioned above, even then the equality of the indices realizing mirror symmetry is valid. 5 There is an additional shift symmetry acting asŝi →ŝi + (const.) for all i at the same time, and we can fix one ofŝi's by this symmetry.
CP-SQED N f vs P-U(1) N f −1 gauge theory. Let us turn to multi-flavor mirror symmetry between the SQED N f of the CP-type gauge symmetry and the dual quiver theory. The index I (CP) N f of N f free matters coupled to N f CP-type U(1) V 's is written by where s i is a magnetic flux and θ i± is a Wilson line phase associated to each CP-type U(1) V .
• CP-SQED N f : As done in the previous example, in order to describe CP-SQED N f starting from where (m, σ ± ) are a monopole flux and a discretized Wilson line for CP-type U(1) diag . Thus, we obtain the index I CP-SQED N f for CP-SQED N f with summing over both m and σ ± where m ± J and w are a holonomy and a fugacity, respectively, associated to a topological U(1) J coupling to the gauged U(1) diag via the BF term.
• P-U(1) N f −1 gauge theory: Let us call P-U(1) N f −1 the quiver theory of (N f − 1) P-type U(1) gauge groups with appropriate global symmetries. We can apply equivalence between (4.8) and (4.9) to rewriting the free matter index (4.20), then the resultant index I P-U(1) N f is This shows the contributions of N f copies of N = 4 P-SQED (±) . To repeat the argument mentioned in Section 4.1, we change the variables (s i , θ i± ) such as where (m, σ ± ) correspond to saddle values for the sum of U(1) N f J which we will call U(1) diag . Then, gauging U(1) diag produces the index I P-U(1) N f −1 of this dual theory where m ± J and w also are a holonomy and a fugacity associated to U(1) J originated from gauging U(1) diag . Now, we focus on the consequences of gauging the parameters m and σ ± . The summation over m results in an usual delta function (4.26) and the summation over σ ± gives a mod 2 delta function (2.23) As demonstrated in the previous case, to obtain the standard description of the dual theory to CP-SQED N f , we redefine the variables by using above constraints as (4.28) Accordingly, the index for P-U(1) N f −1 is written by (4.29) In this expression, one can see the correct contributions of bifundamental and adjoint fields attached to suitable nodes in the quiver diagram. : As a result, the other example of N = 2 multi-flavor mirror symmetry is described as the equality between the indices (4.22) for CP-SQED N f and (4.29) for P-U(1) N f −1 . Table 2 briefs the symmetries with variables in these thoeies. Again, though the global symmetries on one side correspond one-to-one to these on the other side, mirror symmetry accompanies the interchange of the parity type of the gauge symmetries in two theories. We note agian that the holonomy for U(1) N f A can be taken into account by using the modified formula (2.14). Since the arguments above to derive equivalence of the indices do not change in any way, we claim to find strong evidence for Abelian mirror symmetry on RP 2 × S 1 with general numbers of flavors.

Conclusion and outlook
In this paper, we systematically construct Abelian mirror symmetry on RP 2 × S 1 with the insertions of loop operators by consistently defining the S-operation and general N f flavors. The unorientable structure of RP 2 is translated into mod 2 delta functions which are analogies to the Fourier transformations of the usual delta functions on an orientable manifold. The characteristic aspect of our mirror symmetry implemented here is to interchange the two types of the parity conditions under it just as one can see in Table 1, 2 for gauge symmetries of the multi-flavor theories and (3.14), (3.15) for the map of the loop operators. Moreover, we should remark that the index formula (2.14) is updated to be multiplied by the exponential prefactor from the previous version. This refinement leads to the introduction of the holonomy for U(1) A to the indices of the SQED and the XYZ model, and the statement of our mirror symmetry supported from the index viewpoint can get more strong than in the previous version.
Although we could construct consistent physical dualities on the unorientable manifold, yet there are several interesting questions which we would like to approach in the near future as listed below.
T-operation (Chern-Simons term). Let us comment on defining the T-operation of a SL(2, Z) group [12]. The T-operation T generates the action of adding the Chern-Simons (CS) term of a background gauge field A at the level k = 1 to the original Lagrangian L(Φ) coupled with it: A∧dA .
(5.1) Therefore, we need to have the CS term compatible under the parity action in order to set the T-operation on RP 2 × S 1 as well as the BF term for the S-operation. However, it seems impossible to obtain such T-operation because the CS term made of either a P-type or a CP-type vector multiplet becomes parity-odd. At the present moment, there is no appropriate complement to make it parity-even.
Parity anomaly. It might be related to the above comment on the T-operation, if we consider one matter multiplet coupled to a vector multiplet, then, the half-integer level Chern-Simons term for the vector multiplet emerges usually. We do have such situation for the P-type vector multiplet: which is obtained by taking Q 1 = 0 and Q 2 = 1 in (2.13). The factors e ± iθ 4 appearing in the so-called Casimir energy sector are indeed very similar to the Chern-Simons term contribution in the usual index [10]. We have no physical explanation for this "anomalous factor" which is cancelled in all our examples with the gauge symmetry. For a global symmetry, however, our check is verified under the existence of this factor as a fugacity, hence it should have a certain physical meaning, but we now do not have any plausible interpretation. Recently, "parity" anomaly on an unorientable manifold is discussed in [14], in which a similar anomalous structure to our Casimir energy e ± iθ 4 is found. It may be interesting to study this factor.
3d-3d correspondence. The geometrical aspects of the SL(2, Z) action are naturally encoded in the 3d-3d correspondence [15,16] which is one interesting application of the index on RP 2 × S 1 . The fundamental observation to establish this correspondence is a duality where T 1 is the theory containing a free chiral multiplet (of charge +1) and the CS term of a background field at the level k = − 1 2 . This is derived from N = 2 mirror symmetry with N f = 1, and of course, we can directly check this statement by utilizing the superconformal indices. Clearly, to discuss the 3d-3d correspondence for the theories on RP 2 × S 1 based on this viewpoint, the CS term is needed to define in the parity-even way under the parity condition on our geometry. Also, we must find the correct action of the T-operation on the index consistent with parity on RP 2 . Now, we do not have any resolution and would like to keep approaching this problem.
Factorization. Another recent development in supersymmetric gauge theories on 3d curved manifolds is so-called factorization [17][18][19][20], which is to factorize exact partition functions on such manifolds into basic building blocks called holomorphic blocks. In fact, this property is related to some basic arguments to realize the 3d-3d correspondence. Thus, to find the factorised form of our indices on RP 2 × S 1 might give us somewhat important points to establish the 3d-3d correspondence.
More general quiver gauge theories. It is known that 3d Abelian mirror symmetry can be realized for more general quiver theories containing a number of matters charged under gauge symmetries. For instance, an U(1) k gauge theory with N matter multiplets Φ i (i = 1, · · · , N ) is mirror to an U(1) N −k gauge theory with N matter multipletsΦ j (i = 1, · · · , N ) [4,21]. In the former, the matters Φ i have charges Q a i (a = 1, · · · , k) under the a-th U(1) gauge symmetry, and the Chern-Simons couplings are expresed by In the latter, the mattersΦ j have chargesQ p j (p = 1, · · · , N − k) under the p-th U(1) gauge symmetry, and the Chern-Simons couplings are given bŷ We are wondering whether we can construct such kinds of quiver theories showing mirror symmetry by appropriately connecting the Pand CP-type gauge symmetries.
Non-Abelian gauge theories. In addition, an extension which we should consider is to derive the index formulas of non-Abelian gauge symmetries. The difficulty to do this is lying on determining the Jacobian which arises from projecting the integration measure of the vector multiplet onto the Cartan subalgebra. The solution to this question requires us to find the precise form, especially, of the holonomy on RP 2 . We hope that these will become clear and provide some new insights into the theories on unorientable manifolds.
Brane construction. Besides the above viewpoints on the quantum field theories, it is a natural question if the brane construction in type IIB string theory [22] can be constructed for mirror symmetry even on RP 2 × S 1 . Two theories mirror to each other realized as the worldvolume theories on D3-branes are connected by S-duality. Since the key observation of mirror symmetry on RP 2 × S 1 is the interchange of P-type on one side with CP-type on the other side, we expect that this phenomena in the field theory could be accomplished by S-duality in string theory.
A BPS loop operators on RP 2 × S 1 Before proceeding to the discussion of loop operators, we note the geometrical data. The vielbein for the round two-sphere is taken to be e 1 = dϑ, e 2 = sin ϑdϕ, e 3 = dy. (A.1) The gamma matrices are denoted by γ a = σ a (a = 1, 2, 3) where σ a s are Pauli matrices, and the Killing spinors are given by ǫ = e where v µ = (v ϑ , v ϕ , v y ).

A.1 Wilson loop
BPS condition. Let us consider the Wilson loop along a curve C charged under a U(1) gauge symmetry, where e is an electric charge. The BPS condition, δ ǫ W (±) e (C) = 0, which is necessary to apply localization technique (see for example [23]), turns to be (ǫγ µ ǫ)ẋ µ ± (ǫǫ)|ẋ| = 0, (ǫγ µ ǫ)ẋ µ = 0. (A.5) It is found by simple calculation with the Killing spinors (A.2) and the relations (A.3) that the BPS Wilson loops wrap the S 1 direction with sitting on the poles (ϑ = 0, π) of the two-sphere. We express these configurations in terms of the curve C as To argue concretely the loop operators respecting the Z 2 -invariance on RP 2 , we here recall the relevant parts of the Pand CP-type parity condition for the vector multiplet (see [6] for the complete list): y , σ (P) → −σ (P) . We can interpret it from the index point of view (2.17) just by ungauging it, and we denote this as somewhat "trivial" insertion of the Wilson loop associated with a P-type global symmetry as if it is an operation on the index with background:

A.2 Vortex loop
Definition via S-operation. As studied in [7], a vortex loop operator associated with a global symmetry can be defined by using the S-operation which we explain in Section 2.2, where m is a magnetic charge of the gauge symmetry. This means that the BPS vortex loop also extends to S 1 and is located at the poles of the two-sphere, which has been directly derived from the boundary condition preserving supersymmetry in [7,8]. Now, it is quite convenient to accept this definition in constructing the Z 2 -invarient vortex loops on RP 2 × S 1 because we already saw that the S-operation does work well. For the vortex loop associated with the gauge symmetry is defined bŷ (q 2 ; q 2 ) ∞ (q; q 2 ) ∞ I RP 2 (q; s ± 1 , θ 1 ) . . .

(A.24)
Note that, if m ∈ Z, it is identical to the original index because θ is defined modulo 2π. To make it easy to use this insertion in our applications, we write the insertion of the P-type vortex loop with a U(1) global symmetry as the second quiver diagram in (3.7). (q; q 2 ) ∞ (q 2 ; q 2 ) ∞ I RP 2 (q; s 1 , θ ± 1 )