Index theorem on $T^2/\mathbb{Z}_N$ orbifolds

We investigate chiral zero modes and winding numbers at fixed points on $T^2/\mathbb{Z}_N$ orbifolds. It is shown that the Atiyah-Singer index theorem for the chiral zero modes leads to a formula $n_+-n_-=(-V_++V_-)/2N$, where $n_{\pm}$ are the numbers of the $\pm$ chiral zero modes and $V_{\pm}$ are the sums of the winding numbers at the fixed points on $T^2/\mathbb{Z}_N$. This formula is complementary to our zero-mode counting formula on the magnetized orbifolds with non-zero flux background $M \neq 0$, consistently with substituting $M = 0$ for the counting formula $n_+ - n_- = (2M - V_+ + V_-)/2N$.


Introduction
Superstring theory is known as a unique candidate of the unified theory between gauge interactions and quantum gravity. In its formulation, the theory requires the presence of extra dimensions due to conformal anomaly cancellation. A key issue in the long history has been to show that the theory can involve the Standard Model (SM) of particle physics. Indeed in the context of string phenomenology and string cosmology, many frameworks has been used to construct phenomenological models, e.g., heterotic strings [1][2][3][4][5], type I setups [6][7][8], type IIA/B setups [9][10][11][12][13][14] and F-theory [15][16][17].
A smart way to discriminate whether a given setup is chiral or not, is to check the index where n ± are the number of ± chiral zero modes for a Dirac operator / D on extra dimensions. This is the notion of the Atiyah-Singer index theorem [51]. The index is a topological invariant and takes non-zero values if the setup contains lowest-lying states with chirality. The index theorem was applied to a two-dimensional (2d) torus with background magnetic flux [21,52], where M denotes the flux quanta. For M = 0, the index is non-zero and it is easily confirmed that the lowest-lying states are chiral and degenerate (e.g., n + = M and n − = 0 for M > 0) thanks to the presence of magnetic flux. In our previous paper [53], we have discovered a zero-mode counting formula on magnetized orbifolds T 2 /Z N (N = 2, 3, 4, 6) for M > 0, where V + denotes the total winding numbers for positive chirality modes at fixed points. 1 Interestingly, both M/N and V + /N are not integers, in general, but the combination (M − V + )/N becomes an integer in any pattern. Thus, the winding numbers at the fixed points are especially important for the index on the orbifolds. One would suppose that on the magnetized orbifolds, the index should be affected by two sources, i.e. the flux background M = 0 and the orbifold projections. Let us separate Eq. (1.3) into the flux-dependent part M/N and independent one −V + /N + 1. This makes us suppose that the former originates from the total flux on the orbifolds, where the fundamental area is 1/N as much as that on the torus. In this paper, we further pursuit what the fluxindependent term −V + /N + 1 implies. Considering the index theorem on T 2 /Z N for M = 0, we will derive another expression of our zero-mode counting formula (1.4) and find V + + V − = 2N . Here V ± denote the total winding numbers for ± chirality modes at fixed points. These relations keep a consistency with substituting Eq. Then, we find that Eq. (1.5) leads to the index formula (1.4). Our derivation clearly shows that the index n + − n − on T 2 /Z N is determined by the winding numbers at the fixed points. The proof is the main result of this paper. This paper is organized as follows. In Section 2, we start with the Lagrangian of a sixdimensional (6d) Weyl spinor on a 2d torus T 2 . In Section 3, we explicitly construct mode functions on orbifolds T 2 /Z N . The values of n ± for each Z N parity η, the Scherk-Schwarz twist phase (α 1 , α 2 ) and N are computed in Section 3. In Section 4, we evaluate the trace formula (1.5) by using a complete set of the mode functions, and then confirm the relation (1.4) to the index theorem from the viewpoint of winding numbers V ± in Section 5. Section 6 is devoted to conclusion and discussion. In appendix A, we derive a formula used in our discussion.
2 Six-dimensional Weyl fermion on T 2 First, we briefly discuss the mode expansion of a 6d Weyl fermion on a 2d torus T 2 .

Mode functions on T 2
The mode functions f ±,n (z) on T 2 are taken as eigenfunctions of the differential operator −4∂ z ∂z , i.e.
3 Mode functions on T 2 /Z N orbifolds

Z N eigen mode functions
Let us now proceed to T 2 /Z N orbifolds. The T 2 /Z N orbifold is defined by the torus identification (z ∼ z + 1 ∼ z + τ ) and an additional Z N one It has already been known that there exist only four kinds of the orbifolds: T 2 /Z N (N = 2, 3, 4, 6). For N = 2, there is no limitation on τ except for Imτ > 0. For N = 3, 4 and 6, τ must be equivalent to ω because of crystallography [56]. For convenience, we will use both τ and ω. It should be noticed that in order to be consistent with the orbifold identification (3.1), the SS twist phase (α 1 , α 2 ) has to be quantized [25] such that Mode functions on the T 2 /Z N orbifold are classified by Z N eigenvalues η = ω k (k = 0, 1, · · · , N − 1) under the Z N rotation z → ωz such as We emphasize that if the Z N eigenvalue of f +,n+α (z) is η, then that of f −,n+α (z) has to be ωη. This additional factor ω comes from a rotation matrix acting on 2d spinors [25], and is necessary to be compatible with the supersymmetry relations (2.14) and (2.15).
The set {ξ η n+α (z) | n + α ∈ Λ/Z N } of the Z N eigen modes satisfies the complete orthonormal condition: with the normalization constant We point out that the normalization constant (3.18) is important to derive Eq. (4.5) in Section 4.

Number of zero modes on T 2 /Z N
The mode functions f ±,n+α (z) on T 2 /Z N with the Z N transformation properties (3.3) and (3.4) are written in terms of ξ η n+α (z) as The eigenvalue m 2 n+α of f ±,n+α (z) is still given by Eq. (2.21). Thus, the chiral zero modes such that m n+α = 0 can appear only when n + α = 0 . The lists of the zero modes are summarized in Tables 1 -4.
From Tables 1 -4, we find that the index n + −n − can be non-zero and the lowest-lying state in the KK spectrum is chiral. This property has been used to construct phenomenologically semi-realistic models [18,19,[57][58][59]. We will prove a nontrivial formula: as an index theorem. Its nontriviality is that even if n + and/or n − take zero, the sum of the winding numbers V ± can take non-zero values. Our derivation clearly shows that the index n + − n − on T 2 /Z N can only be determined by the winding numbers at the fixed points, as we will see. Table 3: The number of the zero modes f ±,0 on T 2 /Z 4 such that m n+α = 0.

Index theorem on T 2 /Z N orbifolds
In Sections 4 and 5, we derive the index formula (3.21) by use of the trace formula This is our main subject of this paper. In terms of the complete orthonormal sets of the mode functions {f ±,n+α (z)}, the trace lim ρ→∞ tr[σ 3 e / D 2 /ρ 2 ] can be represented as where N ±,n+α denote the numbers of the mode functions f ±,n+α and n ± ≡ N ±,0 . The right-hand-side in the second line of (4.2) can reduce to n + − n − because of the relation N +,n+α = N −,n+α for n + α = 0.
By using the relation and a fact that the integration measure d 2 z and (ξ η(ωη) Eq. (4.5) is proved in the appendix.
In the limit of ρ → ∞ and z → z, the l = 0 term could diverge like δ 2 (0), but it actually vanishes thanks to the coefficient (1 − ω l ). Therefore, we can take the limit of ρ → ∞ and z → z without any divergence or singularity. Then, taking the limit leads to Here, we have replaced the integral T 2 d 2 z by Imτ T 2 dy 1 dy 2 , where Imτ corresponds to the area of the 2d torus T 2 .
One may take the integral T 2 dy 1 dy 2 to be 1 0 dy 1 1 0 dy 2 , as usual. However, it is more convenient to choose the fundamental domain of T 2 , as depicted in Figure 2, in order to avoid troublesome treatment of delta functions appearing on y 1 = 0, 1 or y 2 = 0, 1. 3 To sum up n 1 and n 2 in Eq. (4.6), it is useful to introduce y (l) = (y (4.7) For l = 1, (y 2 ) is explicitly given by for T 2 /Z 6 . Then, after summing up n 1 and n 2 in Eq. (4.6), the index n + − n − is expressed as where we have used the formula n∈Z e i2πny = m∈Z δ(y − m). (4.10) To evaluate Eq. (4.9) further, we will examine T 2 /Z N orbifolds with N = 2, 3, 4 and 6, separately.
A. Index for T 2 /Z 2 Let us first discuss the T 2 /Z 2 orbifold. In this case, (y 2 ) is given by (−y 1 , −y 2 ). Inserting it into Eq. (4.9) with N = 2 and ω = −1, we have Since the fundamental domain of T 2 has been taken to be −ε ≤ y 1 , y 2 < 1 − ε, the values of (m 1 , m 2 ), which remain in the summation of Eq. (4.11) after the y-integration, are given by (m 1 , m 2 ) = (0, 0), (1, 0), (0, 1) and (1, 1). Then, we find where y f j (j = 1, 2, 3, 4) are defined by An important observation is that y f j given in Eq. (4.13) is just the position of the fixed points on T 2 /Z 2 , as explained below. Fixed points on T 2 /Z N in the complex plane z = y 1 +τ y 2 are defined by where ω = e i2π/N for the T 2 /Z N orbifold (N = 2, 3, 4, 6). The orbifold fixed points, which are invariant under the Z N rotation up to torus lattice shifts, are found as Thus, y f j (j = 1, 2, 3, 4) in Eq. (4.13) corresponds to the position of the fixed points on T 2 /Z 2 in the complex coordinate. This fact implies that the index n + − n − can only be determined by information on the fixed points.
The explicit values of W j (j = 1, 2, 3, 4) are summarized in Table 5. We can then confirm that the formula (4.12) correctly gives the index n + −n − in Table 1, as it should be. However, in the derivation of Eq. (4.12), the physical meaning of W j is less clear. In the next section, we reveal a geometrical meaning of W j .

C. Index for T 2 /Z 4
Let us discuss the index for the T 2 /Z 4 orbifold. In this case, (y 2 ) = (−y 2 , y 1 ), (y 1 , y 2 ) = (−y 1 , −y 2 ), (y 1 , y 2 ) = (y 2 , −y 1 ). (4.23) After the integration of T 2 dy 1 dy 2 , the delta functions δ(y 1 + y 2 − m 1 )δ(y 2 − y 1 − m 2 ), δ(2y 1 − m 1 )δ(2y 2 − m 2 ) and δ(y 1 − y 2 − m 1 )δ(y 2 + y 1 − m 2 ) in Eq. where Here, y f j (j = 1, 2, 3, 4) in Eq.  The y f 1 and y f 2 correspond to the fixed points on T 2 /Z 4 given in Eq. (4.15). Interestingly, we found additional contributions from the points y f 3 and y f 4 . Since the Z 4 group includes Z 2 as its subgroup, there are additional two "Z 2 fixed points" that are not invariant under the Z 4 rotation but invariant under such a subgroup Z 2 (z → ω 2 z = −z) up to torus lattice shifts. Indeed, y f 3 and y f 4 are the "Z 2 fixed points". The explicit values of W j (j = 1, 2, 3, 4) are summarized in Table 7. We can then confirm that the formula (4.25) correctly gives the index n + − n − in Table 3, as it should be. In the next section, we show that W j is related to the winding numbers at the fixed points (4.30).

D. Index for T 2 /Z 6
Let us finally discuss the index for the T 2 /Z 6 orbifold. In this case, (y 2 ) = (−y 2 , y 1 + y 2 ), (y 2 ) = (−y 1 − y 2 , y 1 ), (y 2 ) = (y 2 , −y 1 − y 2 ), (y 2 ) = (y 1 + y 2 , −y 1 ). (4.31) Inserting Eq. (4.31) into Eq. (4.9) with N = 6 and computing in the same way, we arrive at Here, y f j (j = 1, 2, · · · , 6) in Eq. The y f 1 corresponds to a single fixed point on T 2 /Z 6 given in Eq. (4.15). Since the Z 6 group includes its subgroups Z 3 and Z 2 , there are additional two "Z 3 fixed points" and three "Z 2 fixed points" that are not invariant under the Z 6 rotation but invariant under such Z 3 and Z 2 rotations up to torus lattice shifts, respectively. The two Z 3 and three Z 2 fixed points are just given by y f 2 , y f 3 and y f 4 , y f 5 , y f 6 in Eq. (4.36), respectively. The explicit values of W j (j = 1, 2, · · · , 6) are summarized in Table 8. We can then confirm that the formula (4.32) correctly gives the index n + − n − in Table 4, as it should be. In the next section, we show that W j is related to the winding numbers at the fixed points (4.36).

Winding numbers at fixed points on T 2 /Z N
In this section, we compute the winding numbers at fixed points on T 2 /Z N and clarify the geometrical meaning of the coefficients W j in front of the delta functions in Eqs. (4.12), (4.18), (4.25) and (4.32).
Let us define the winding number for the Z N eigen modes ξ η n+α (z) as where C j denotes a sufficiently small circle encircled anti-clockwise around a fixed point z = p j . The line integral along the contour C j gives a winding number (or occasionally called vortex number), i.e. how many times ξ η n+α (z) wraps around the origin, as illustrated in Figure 3. Note that if ξ η n+α (z) does not vanish at z = p j , the winding number χ j (η, α) We are now ready to define the winding numbers for the mode functions f +,n+α (z) and f −,n+α (z). Then, we define the winding numbers χ ±j for the mode functions f ±,n+α (z) around the fixed point z = p j , as 4 One might define the winding number χ −j for f −,n+α by χ j (ωη, α), instead of χ j (ωη, −α), since f −,n+α = ξ ωη n+α . This is not, however, the case. As we will see later, the definition (5.3) for χ −j leads to the expected result otherwise we will not obtain any meaningful relation. Another reason to adopt the definition (5.3) may be explained as follows. To this end, let us consider the 6d charge conjugation C to the 6d fermion Ψ(x, z): The 6d charge conjugation matrix C is represented as where C (4) is the 4d charge conjugation matrix. Under this charge conjugation, the mode functions f ±,n+α transform as 5 Then f The above transformation properties bring another reason to adopt Eq. (5.3) as the winding number for f −,n+α . In the following, we will define the winding numbers χ ±j on the fundamental domain of T 2 even for the orbifold T 2 /Z N . If one defines the winding numbers on the fundamental domain of the orbifold T 2 /Z N , instead of T 2 , the sum of the winding numbers χ ±j at fixed points should be divided by N , i.e. j χ ±j /N due to 1/N reduced area and the deficit angles around the fixed points in comparison with that of the torus.

C. Winding numbers for T 2 /Z 4
In the following, we examine the winding numbers χ ±j for the mode functions f + (z) = ξ η (z) and f − (z) = ξ ωη (z) on T 2 /Z 4 at the fixed points.
As noted in the previous section, there are two Z 4 fixed points and additionally two "Z 2 fixed points" Z 2 fixed points : which are not invariant under the Z 4 rotation but invariant under the Z 2 one (z → ω 2 z = −z) up to torus lattice shifts. The winding numbers not only at the Z 4 fixed points (5.32) but also at the "Z 2 fixed points" (5.33) contribute to the formula (4.25). Under the Z 4 rotation z → ωz around the fixed points p 1 and p 2 , and under the Z 2 rotation z → ω 2 z around the fixed points p 3 and p 4 , the Z 4 eigen mode function ξ η (z) is found to satisfy the relations ξ η (ωz) = ω k ξ η (z), (5.34) ξ η (ωz + 1/2 + τ /2) = ω k+4α 1 ξ η (z + 1/2 + τ /2), where η = ω k (k = 0, 1, 2, 3). From Eqs. (5.34) -(5.37) the winding number χ j (η, α) around the fixed point p j is found as Table 11: The winding numbers χ ±j at the fixed points p j on T 2 /Z 4 and their sums V ± = j χ ±j . All the values of (−V + + V − )/8 exactly agree with the index n + − n − for the chiral zero modes.

Conclusion and discussion
In this paper, we have derived the index formula on the T 2 /Z N orbifold from the trace formula (4.1). In Section 3, we have explicitly constructed the mode functions on T 2 /Z N and counted the numbers n ± of the chiral zero modes.  The winding numbers χ ±j at the fixed points p j on T 2 /Z 6 and their sums V ± = j χ ±j . Here, we omit the column of (α 1 , α 2 ) due to the fact that (α 1 , α 2 ) = (0, 0) for the T 2 /Z 6 orbifold. All the values of (−V + + V − )/12 exactly agree with the index n + − n − for the chiral zero modes.
In Sections 4 and 5, we have succeeded in evaluating the trace formula (4.1) and clearly shown that the index n + − n − is determined by the winding numbers at the fixed points on T 2 /Z N .
We have emphasized that the dependence of n ± on N , η and (α 1 , α 2 ) is rather simple, as shown in Tables 1 -4, but the equality in Eq. (6.1) is nontrivial. This is because the values of V ± (or χ ±j ) can be non-vanishing even if n + and/or n − are zero, as seen in Tables 9 -12. Furthermore, V ± /2N are not integer-valued in general, but the difference (−V + + V − )/2N becomes an integer in any case.
It is interesting that from Tables 9 -12 the sums V ± of the winding numbers at the fixed points satisfy the relation which may be regarded as an expression of the index theorem. Then, from Eqs. (6.1) and (6.2) we have This can be understood as a special case of M = 0 in the zero-mode counting formula [53] n There the formula (6.4) with the quantized magnetic flux M has been confirmed only for M > 0. Since we can show that the relation (6.2) holds also for M = 0, the formula (6.4) can be rewritten as which leads to a generalization of the formula (6.1) to M = 0.
In this paper, we found that the orbifold projections bring the chirality of the massless level in the KK tower from the vewpoint of the index theorem (even if no flux is turned on). In addition, the orbifold fixed points and zero points there play important roles in the trace theorem, as we expected from the previous paper [53]. Thus, these evidences let us conclude that the term −V + /N + 1 or (−V + + V − )/2N reflects the contribution from the orbifold geometry, i.e. the singularity of the fixed points.
There are two possibilities to interpret the term −V + /N + 1 or (−V + + V − )/2N . One is that the term originates from some singular spin connection or curvature at the orbifold fixed points, which should be regarded as "geometric flux". Another possibility is that there exist localized Wilson-line sources at the fixed points, as discussed in [60,61]. In this case such sources should be regarded as "gauge flux". Our result suggests that the two-dimensional orbifolds T 2 /Z N are equivalently described by setups with localized fluxes at fixed points.
A remaining task is to derive the formula (6.5) for M = 0 from the trace formula. In order to evaluate the trace formula, we may need a complete orthonormal set of Z N eigen mode functions on the magnetized T 2 /Z N orbifold, as we have done in this paper. That, however, seems to be hard since mode functions on the magnetized T 2 /Z N are given by Jacobi theta functions [12] and their Z N transformation property is quite complicated [25,62].