KUNS-2538, OU-HET/850 Eternal Higgs inflation andcosmological constant problem

# Kuns-2538, Ou-Het/850 Eternal Higgs inflation and cosmological constant problem

Yuta Hamada,  Hikaru Kawai,  and Kin-ya Oda

Department of Physics, Kyoto University, Kyoto 606-8502, Japan
Department of Physics, Osaka University, Osaka 560-0043, Japan
July 1, 2019
###### Abstract

We investigate the Higgs potential beyond the Planck scale in the superstring theory, under the assumption that the supersymmetry is broken at the string scale. We identify the Higgs field as a massless state of the string, which is indicated by the fact that the bare Higgs mass can be zero around the string scale. We find that, in the large field region, the Higgs potential is connected to a runaway vacuum with vanishing energy, which corresponds to opening up an extra dimension. We verify that such universal behavior indeed follows from the toroidal compactification of the non-supersymmetric heterotic string theory. We show that this behavior fits in the picture that the Higgs field is the source of the eternal inflation. The observed small value of the cosmological constant of our universe may be understood as the degeneracy with this runaway vacuum, which has vanishing energy, as is suggested by the multiple point criticality principle.

## 1 Introduction

The Higgs boson discovered at the LHC [1, 2] beautifully fits into the Standard Model (SM) predictions so far [3]. The determination of its mass [4]

 MH =125.7±0.4GeV (1)

completes the list of the SM parameters, among which the ones in the Higgs potential,

 V =m2|H|2+λ|H|4, (2)

have turned out to be and , depending on the precise values of the top and Higgs masses; see e.g. Ref. [5].

We have not seen any hint of a new physics beyond the SM at the LHC, and it is important to guess at what scale it appears, as we know for sure that it must be somewhere in order to account for the tiny neutrino masses, dark matter, baryogenesis, inflation, etc. In this work, we assume that the Higgs sector is not altered up to a very high scale,111 See e.g. Refs. [6, 7, 8, 9, 10, 11, 12] for a possible minimal extension of the SM with the dark matter and right-handed neutrinos. in accordance with the following indications: The renormalization group (RG) running of the quartic coupling revealed that it takes the minimum value at around the Planck scale and that the minimum value can be zero depending on the precise value of the top quark mass [13, 14, 15, 16, 17, 18, 19, 20, 21, 5, 22, 23, 24, 25, 26, 27]. We have also found that the bare Higgs mass can vanish at the Planck scale as well [18, 19, 20, 28, 21, 29, 30, 31].222 See also Refs. [32, 33, 34] for discussion of quadratic divergences. That is, the Veltman condition [35] can be met at the Planck scale. In fact he speculates, “This mass-relation, implying a certain cancellation between bosonic and fermionic effects, would in this view be due to an underlying supersymmetry.” To summarize, it turned out that there is a triple coincidence: , its running, and the bare Higgs mass can all be accidentally small at around the Planck scale.

This is a direct hint for Planck scale physics in the context of superstring theory. The vanishing bare Higgs mass implies that the supersymmetry is restored at the Planck scale and that the Higgs field resides in a massless string state. The smallness of both and its beta function is consistent with the Higgs potential being very flat around the string scale; see Fig. 1.333 This is indeed suggested by the multiple point criticality principle (MPP) [36, 37, 38], the classical conformality [39, 40, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50], the asymptotic safety [51], the hidden duality and symmetry [52, 53], and the maximum entropy principle [54, 55, 56, 57, 58]. Such a flat potential opens up the possibility that the Higgs field plays the role of inflaton in the early universe [59, 60, 61, 62, 63, 64, 65, 66, 67].444 There are different models of the Higgs inflation involving higher dimensional operators [68, 69, 70, 71, 72].

To understand the whole structure of the potential, it is crucial to investigate its behavior beyond the Planck scale. The calculation based on field theory cannot be trusted in this region. Although it is hard to reproduce the SM completely as a low energy effective theory of superstring, we can explore generic trans-Planckian structure of the Higgs field, under the assumption that the SM is close to a non-supersymmetric perturbative vacuum of superstring theory.

In four dimensions, string theory has many more tachyon-free non-supersymmetric vacua than the supersymmetric ones. The latest LHC results suggest the possibility of the absence of the low energy supersymmetry, and the research based on the non-supersymmetric vacua is becoming more and more important [73, 74, 75, 76, 77].

In such non-supersymmetric vacua, almost all the moduli are lifted up perturbatively, contrary to the supersymmetric ones which typically possess tens or even hundreds of flat directions that cannot be raised purturbatively. However, there remains a problem of instability in the non-supersymmetric models: The perturbative corrections generate tadpoles for the dilaton and other moduli such as the radii of toroidal compactifications. The dilaton can be stabilized within the perturbation series when  [78], or else by the balance between the one-loop and the non-perturbative potentials when is small [77]. In this paper, we assume that the dilaton is already stabilized. We will discuss other instabilities than the dilaton direction in Sections 2 and 3.

We start from the tachyon-free non-supersymmetric vacua of the heterotic string theory. We assume that the Higgs comes from a closed string and that its emission vertex at the zero momentum can be decomposed into a product of operators whose conformal dimensions are and . This is realized in the following cases for example:

• The Higgs comes from an extra dimensional component of a gauge field [79, 80, 81, 82, 83, 84, 85].

• The Higgs is the only one doublet in generic fermionic constructions [86, 87, 88, 89].

• The Higgs comes from an untwisted sector in the orbifold construction [90, 91]; see e.g. Ref. [73] for a recent model-building example.555 In Ref. [73] the SM-like one Higgs doublet model is constructed, in which Higgs is realized as an extra dimensional gauge field. For example, the model under the orbifold compactification of heterotic string with the shift vector and Wilson lines fits in all the three criteria. We thank the authors of Ref. [73] on this point.

Then we consider multiple insertions of such emission vertices to evaluate the effective potential. It is very important to understand the whole shape of the Higgs potential in order to discuss the initial condition of the Higgs inflation, as well as to examine whether the MPP is realized or not. We will show that, in the large field region, the Higgs potential is connected to a runaway vacuum with vanishing energy, which corresponds to opening up an extra dimension. We find that such potential can realize an eternal inflation.

This paper is organized as follows. In Sec. 2, we show that the potential in the large field limit with fixed radius can be classified into the above three categories. In Sec. 3, we compute the one-loop partition function as a function of a background field in non-supersymmetric heterotic string on , as a concrete toy model [92, 93, 94, 95, 96]. We explicitly check that the limiting behavior of the potential fits into the three categories mentioned above. We argue that physically this corresponds to opening up a multi degrees of freedom space above the Planck scale and that the runaway vacuum is a direction in this space. In Sec. 4, we point out a possibility that the Higgs inflation is preceded by an eternal inflation, which occurs either in a domain wall or in a false vacuum. In Sec. 5, we show a possible explanation for the vanishing cosmological constant in terms of the MPP, and consider a possible mechanism to yield the observed value of the order of . In Sec. 6, we summarize our results. In Appendix A, we summarize our notation for several mathematical functions. In Appendix B, we review the fermionic construction that we use for the heterotic superstring theory. The computation of the partition function is also outlined. In Appendix C, we review the T-duality that we use in this work. In Appendix D, we review the MPP.

## 2 Higgs potential in string theory

In this section, we show how to treat the large constant background of a massless mode in closed string theory. In general, we start from a worldsheet action, say,

 S0=12πα′∫d2zGMN∂XM¯∂XN+⋯, (3)

where is the target space metric, run from to , and is the string tension. In general, a genetic massless string state has the emission vertex

 O(z,¯z)eik⋅X, (4)

where and has conformal dimensions to preserve the conformal symmetry on the worldsheet.

As said in Introduction, we assume in this paper that the emission vertex at the zero momentum of the physical Higgs can be decomposed into a product of the operator and the operator :

 O(z,¯z) =OL(z)OR(¯z). (5)

An operator of this form is exactly marginal: Insertions of the operator can be exponentiated without renormalization, and hence the deformation of the worldsheet action

 S =S0+ϕ∫d2zO(z,¯z) (6)

keeps the theory conformally invariant; see Fig. 2.

We want to know the effective potential for the background: . At the tree-level, the potential vanishes

 Vtree(ϕ)=0. (7)

This is because the one-point function of any emission vertex, especially that of the graviton, vanishes on the sphere as it has non-zero conformal dimension. At the one-loop level and higher, we have non-zero effective potential.666 On the whole plane that is mapped from the sphere, an operator with the scale dimension satisfies and the translational invariance reads . Hence we get for . On the other hand, for torus and surfaces with higher genera, we cannot define the scale transformation, unlike the plane.

The -dimensional energy density is given by

 Vg-loop =−ZgVD, (8)

where is the volume of -dimensional spacetime and is the partition function on the worldsheet with genus after moduli integration. We note that the potential (8) is given in the Jordan frame that does not yet make the gravitational action canonical; we will come back to this point in Secs. 2.1 and 3.2.

We emphasize that in string theory, the partition function can be obtained even for the field value larger than the Planck scale, unlike the ordinary quantum field theory where infinite number of Planck-suppressed operators become relevant and uncontrollable.

Before generalizing to arbitrary compactification, we first analyze two simple examples to build intuition: In Sec. 2.1, we study the large field limit of the radion, namely an extra dimensional component of the graviton under the toroidal compactification. This limit corresponds to the large radius limit of the compactified dimension. In Sec. 2.2, we further turn on the Wilson line and the anti-symmetric tensor field. We can analyze this setup by considering the corresponding boost in the momentum space [97, 98]. From the analysis of the spectrum of these modes, we argue that the effective potential in the large field limit can be classified into three categories, namely, runaway, periodic, and chaotic. (In Sec. 3, we will confirm it by a concrete computation for the toroidal compactification of the heterotic string theory.)

In Sec. 2.3, we discuss more general compactifications, and show that the same classification holds.

As said above, we start from the toroidal compactification of the th direction: . The emission vertex of the radion, , is

 ∂XD−1¯∂XD−1eik⋅X. (9)

Its constant background is given by setting the momentum .

We want the partition function with the radion background :

 Sworldsheet=12πα′∫d2z(1+ϕ)∂XD−1¯∂XD−1+⋯. (10)

In this case, we can transform the action into the original form with by the field redefinition

 X′D−1=√1+ϕXD−1, (11)

which however changes the periodicity as

 X′D−1∼X′D−1+2π√1+ϕR. (12)

 R′ :=√1+ϕR. (13)

Therefore if the compactification radius is large, the effective action is proportional to it, and the -dimensional effective action for large becomes

 Seff ∼∫dD−1x√−gR′(R−C−2R′2(∂R′)2) =∫dD−1x√−g√1+ϕR(R−C−12(1+ϕ)2(∂ϕ)2) (14)

up to an overall numerical coefficient, where we have taken the units, is the Ricci scalar in -dimensions, and is a -independent constant that is generated from loop corrections in the non-supersymmetric string theory. can be viewed as the -dimensional cosmological constant.

This can be confirmed at the one-loop level as follows. The radius dependent part of the one-loop partition function before the moduli integration is

 ∞∑n,w=−∞exp[2πiτ1nw−πτ2α′((nR′)2+(R′wα′)2)], (15)

where and are the Kaluza-Klein (KK) and winding numbers, respectively, and is the moduli of the worldsheet torus. In the large radius limit , we can rewrite Eq. (15) by the Poisson resummation formula:

 R′√πτ2α′∑m,wexp[−πR′2α′τ2|m−wτ|2]. (16)

We see that the partition function becomes indeed proportional to in the large limit. Note that in the large limit, only the modes contribute, and hence that the winding modes are not important here.

We then rewrite the action (14) in the Einstein frame. In -dimensions, the field redefinition by the Weyl transformation, , gives us the volume element and the Ricci scalar in the Einstein frame as

 √−gE =e(D−1)ω√−g, (17) RE =e−2ω[R−2(D−2)∇2ω−(D−3)(D−2)gμν∂μω∂νω], (18)

respectively. By choosing , we get the Einstein frame action:

 Seff =∫dD−1x√−gE(R% E+(D−3)(D−2)gEμν∂μω∂νω−e−2ωC−2R′2(∂R′)2) =∫dD−1x√−gE(R% E−D−4D−3gEμνR′2∂μR′∂νR′−CR′2/(D−3)) =∫dD−1x√−gE(R% E−gEμν2∂μχ∂νχ−Cexp(−√2χ√(D−3)(D−4))), (19)

where the second term in Eq. (18) has become a total derivative and we have defined . When and , we see that the last term, the potential, becomes runaway for large or .777 The small radius limit is the same as the large radius limit due to the T-duality:  [99, 100, 101].

To summarize, the large field limit of the radion , the extra dimensional component of the graviton, leads to the decompactification of the corresponding dimension. This decompactified vacuum corresponds to the runaway potential if the cosmological constant is positive [102, 103]. Since the large radius limit is equivalent to the weak coupling limit, the runaway vacuum corresponds to a free theory. Therefore this runaway nature is not altered by the higher order corrections. We will see in Section 6 that this argument also applies to the dilaton background.

### 2.2 Boost on momentum lattice

As the second example, we turn on the backgrounds for graviton, gauge, and anti-symmetric tensor fields. Let and be the numbers of the compactified dimensions in the left and right moving sectors of the closed string, other than our four dimensions. We take without loss of generality. The spectrum of -dimensional momenta of the non-oscillatory mode is restricted to form an (even self-dual) momentum lattice, due to the modular invariance [97, 98]; see Appendix B.4. Different lattices that are related by the rotation of correspond to different compactifications, up to the rotation that leaves and invariant. Therefore the compactifications are classified by the transformation

 SO(p,q)SO(p)×SO(q). (20)

This is the moduli space of the theory at the tree level, which is lifted up by the loop corrections in non-supersymmetric string theory.

The boost in the momentum space corresponds to putting constant backgrounds for the degrees of freedom that are massless at the tree-level [97, 98]:

 Cij∂XiL¯∂X¯jR, (21)

where and run for and , respectively. In terms of -dimensional fields, they can be interpreted as the symmetric tensor (metric), antisymmetric tensor, and gauge fields (Wilson lines), whose total number is

 q(q+1)2+q(q−1)2+q(p−q)=pq. (22)

Indeed, this agrees with the number of degrees of freedom of the coset space (20):

 (p+q)(p+q−1)2−p(p−1)2−q(q−1)2=pq. (23)

We are interested in switching on the background of a single field. If the emission vertex of the field is given by , this corresponds to adding

 λcij∂XiL¯∂X¯jR (24)

to the worldsheet action, where represents the strength of the background. In general, the rotation can make into the diagonal form

 ci¯j →⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣∗∗⋱∗⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦, (25)

where the blank slots stand for zero. This background corresponds to the combination of boosts in the -, …, - planes. That is, the -dimensional vector

 k=(k1L,…,kpL;k¯1R,…,k¯qR) (26)

is transformed by

 ⎡⎣k′iLk′¯iR⎤⎦ =[coshηisinhηisinhηicoshηi]⎡⎣kiLk¯iR⎤⎦, k′jL =kjL, (27)

for and .

Let us first consider the effect of a boost in a single plane:

 [k′Lk′R] =[coshηsinhηsinhηcoshη][kLkR]. (28)

Then one of is contracted and the other expanded:

 k′L+k′R =eη(kL+kR), k′L−k′R =e−η(kL−kR). (29)

The effective potential in the large limit depends on whether or not there exists a lattice point on the light cone in this plane, as is illustrated schematically in Fig. 3. There are two possibilities in the infinite boost limit:

• If a point in the initial momentum lattice sits on the light cone as in the left panel in Fig. 3, infinite amount of its integer multiplications on the light cone are contracted to form a continuous spectrum. This behavior is the same as that of the KK momenta in the large radius limit discussed in Sec. 2.1. The resultant partition function becomes proportional to the radius . The same argument as Sec. 2.1 gives us the runaway potential.

• If no point sits on the light cone in the initial momentum lattice, as in the right panel in Fig. 3, then the continuum is not formed by the infinite boost. For a given amount of boost, the closest point to the origin contributes the most to the partition function. Then the potential becomes either periodic or chaotic for larger and larger boost.

The fate of the large field limit depends on whether or not a lattice point sits on the light cone of the boost plane in the momentum space.

In the case of the multiple boosts (25), the boost in each plane is independent from the others. However, if there are several degenerate massless states as in Eq. (21), we should better consider all of them simultaneously. As we will see in Sec. 3 in a concrete model, the asymptotic behavior of the potential remains essentially the same.

### 2.3 General compactifications

We discuss the large field limit in more general setup including compactification on a curved space, possibly involving orbifolding etc., or even the case without having a geometrical interpretation. We will show that the classification still holds: runaway, periodic, and chaotic.

As said above, the emission vertex of a massless field must be written in terms of a operator, and we assume that this operator separates into the holomorphic and anti-holomorphic parts,

 O(1,1)=O(1,0)×O(0,1), (30)

on the worldsheet. Then we can write at least locally,

 O(1,0)=∂Y, O(0,1)=¯∂Z, (31)

where and are free worldsheet scalars. If we further assume that the Higgs field is uniquely identified, i.e., that it does not mix with other massless states at the tree level, then it suffices to consider a single background as in Eq. (6). In this case we may not need to consider the multi-field potential discussed above.

We can show that and are periodic at least in one sector: In fact, if we insert the graviton emission vertex near the Higgs emission vertex, the latter is single valued in the neighborhood of the former. This is because and are independent of the spacetime coordinates . Therefore, and are periodic in the graviton sector; see Fig. 4.

In such a sector, we can mode-expand and . Let us consider the simultaneous eigenvalues of the constant modes of and . The set of the pairs of eigenvalues form a momentum lattice : If there exist states and with momenta and , respectively, there is a state with the momentum ; such a state appears when and merge. If contains a non-zero vector, it forms the momentum lattice. Then the same argument applies as in Sec. 2.2. Putting a constant background for corresponds to the momentum boost. If there is a point on the light cone with being a rational number, then a runaway direction emerges in the infinite boost limit. If not, namely if there is no such point, then the potential becomes chaotic.

## 3 SO(16)×SO(16) heterotic string

We verify the argument in the previous section in the concrete model: the heterotic string theory [92, 93]. This model breaks supersymmetry at the string scale but, unlike the bosonic string theory in 26 dimensions, the tachyonic modes are projected out as in the ordinary heterotic superstring theories. In the fermionic construction, the modular invariance of the partition function restricts the allowed set of the fermion numbers in Neveu-Schwarz (NS) and Ramond (R) sectors. The classification of the ten dimensional string theories is completed in Ref. [104]. The model [92, 93] is the only one that has neither a tachyon nor a supersymmetry in ten dimensions.

We write the uncompactified dimensions (), and the compactified ones () for the left movers. We then compactify this model on  [94]:

 X9 ∼X9+2πR. (32)

We further turn on a Wilson line for the gauge field , and compute the one-loop partition function.

In Appendix B, we spell out the construction of the model and the computation of the partition function; the notations for the theta functions are put in Appendix A. In Sec. 3.1, we review the partition function in the heterotic string theory in 10 dimensions. In Sec. 3.2 we compute the one-loop partition function of this model for the case described above.

### 3.1 Partition function of SO(16)×SO(16) string

We first review the computation of the partition function in the non-supersymmetric heterotic string [92, 93]. Here we have chosen a non-supersymmetric string as a toy model because, as discussed in Introduction, the low energy data at the electroweak scale suggests via the Veltman condition that the supersymmetry is broken at the Planck scale. In such a non-supersymmetric theory, the flat direction of the effective potential is raised perturbatively. Detailed procedure of the fermionic construction of the model is explained in Appendix B.1 and B.2.

Let us write down the contribution from the momentum lattice after the bosonization in each sector:

 ^ZT2,α→w (33)

In our case, they are

 ^ZT2,→0 =−18((¯ϑ00)4−(¯ϑ01)4)((ϑ00)8+(ϑ01)8)2, ^ZT2,→w0 =−18(¯ϑ10)4(ϑ10)16, ^ZT2,→w1 =−18((¯ϑ00)4−(¯ϑ01)4)(ϑ10)16, ^ZT2,→w2 =−18((¯ϑ00)4+(¯ϑ01)4)(ϑ10)8((ϑ00)8−(ϑ01)8), ^ZT2,→w0+→w1 =−18(¯ϑ10)4((ϑ00)8−(ϑ01)8)2, ^ZT2,→w0+→w2 =−18(¯ϑ10)4((ϑ00)8+(ϑ01)8)(ϑ10)8, ^ZT2,→w1+→w2 =−18((¯ϑ00)4+(¯ϑ01)4)(ϑ10)8((ϑ00)8−(ϑ01)8), ^ZT2,→w0+→w1+→w2 =−18(¯ϑ10)4((ϑ00)8+(ϑ01)8)(ϑ10)8, (34)

where and are basis vectors for the boundary conditions on the fermions; see Appendix B.3 for details.

Let us sum up all the above contributions, multiplied by those from the oscillator modes in the bosonization. Including also the spacetime momentum and oscillator modes from the bosonic (), we get the one-loop vacuum amplitude [92, 93]:

 ZT2 =V10α′512(2π)10∫Fdτ1dτ2τ621|η(τ)|16η(τ)16¯η(¯τ)4∑% sector α→w^ZT2,α→w =V10α′514(2π)10∫Fdτ1dτ2τ621|η(τ)|16η(τ)16¯η(¯τ)4 (35)

where represents the fundamental region,

 F :=\Set(τ1,τ2)−1/2≤τ1≤1/2, |τ|=|τ1+iτ2|≥1, (36)

and we have used the Jacobi’s identity:

 (¯ϑ00)4−(¯ϑ01)4−(¯ϑ10)4=0. (37)

We can see from this identity that the contributions between and cancel. By the numerical calculation, we obtain [92, 93]

 ρ10 =−ZT2V10≃(3.9×10−6)1α′5. (38)

### 3.2 S1 compactification with Wilson line

Now we compactify the direction on with radius :  [94]. Here we consider a large field limit of an extra dimensional component of the gauge field, . We will find three possible large field limits discussed in the previous section.

The emission vertex for the gauge field with the polarization and momentum and , respectively, is

 ϵm(i¯∂Xm+α′2(k⋅ψR)ψmR)∂XILeik⋅X, (39)

where indices run such that and . We see by putting in Eq. (39) that a constant background corresponds to adding

 AIm∫d2z¯∂Xm∂XIL, (40)

to the worldsheet action.888 In obtaining the constant background by putting , it is again important that the is massless at the tree-level. In particular, we switch on the component of and , and write :

 A∫d2z¯∂X9∂X1L. (41)

Turning on the Wilson line background does not affect the oscillator modes since Eq. (40) is a total derivative in the worldsheet action; only the momentum lattice of the center-of-mass mode is changed by .

Let be the momentum of . After fermionization, we have

 lL =√2α′m, (42)

where and for the NS (anti-periodic) and R (periodic) boundary conditions, respectively. Let and be the spacetime momenta of the -compactified direction for the left and right movers, respectively:

 pL =nR+Rwα′, pR =nR−Rwα′, (43)

where and are the KK and winding numbers, respectively.

Turning on the background corresponds to the boost on the momentum lattice [98]:

 [l′Lp′R]=[coshηsinhηsinhηcoshη][lLpR], (44)

since there appears only and in Eq. (41). This boost necessarily changes the radius of the compactification too.

we will see that the identification

 A =sinhη, (45)

gives the correct answer below. Let us define by

 r :=Rcoshη, (46)

which will turn out to be the compactification radius in the presence of . Note that in the language of Sec. 2.2, we have 17 left-moving and 1 right-moving internal dimensions ( and ). The non-trivial transformations on the compactified space are

 SO(17,1)SO(17). (47)

Among them, we have chosen the boost between the left and right dimensions with the momenta and , respectively. The left momentum of the dimension, , is untouched. We will soon use the rotation between and that belongs to .

We now show the validity of the identification (45). In terms of and , we have

 p′R =pRcoshη+lLsinhη =nr−rwα′(1+A2)+√2α′mA, (48) l′L =lLcoshη+pRsinhη =√2α′m√1+A2+nrA√1+A2−rwα′A√1+A2, (49) p′L =pL =nr1√1+A2+rwα′√1+A2. (50)

We further rotate by a part of in Eq. (47),

 [l′′Lp′′L] =[cosθsinθ−sinθcosθ][l′Lp′L], (51) p′′R =p′R,

with

 cosθ =1√1+A2, sinθ =−A√1+A2, (52)

to get

 l′′L =√2α′m−2rwα′A, (53) p′′L =nr+rwα′(1−A2)+√2α′mA. (54)

The spectrum becomes

 ∑all modes(l′′2L+p′′2L+p′2R) =∑m,n,w[(√2α′m−2rwα′A)2+(nr+rwα′(1−A2)+√2α′mA)2 =∑m,n,l[+(nr−rwα′(1+A2)+√2α′mA)2]. (55)

As promised, this result (55) correctly reproduces that in Ref. [98, 94], which is obtained from the quantization of the scalar field under constraints. Furthermore, from Eq. (55), we see

 (l′′2L+p′′2L)∣∣m=w=0=p′′2R∣∣m=w=0=n2r2, (56)

which indicates that is the physical radius of .

Now let us discuss the T-dual transformations that can be read off from the above result.

• We can see that the shift

 A →A+√2α′r (57)

leaves the spectrum (55) unchanged.999 After the shift of , redefine the mode numbers by , , and .

• From Eq. (43), we see that the spectrum is invariant under the T-dual transformation [99, 100, 101]

 R →α′R, (58)

or in terms of and , .

By defining

 ~τ =~τ1+i~τ2:=rA√α′+ir√α′, (59)

we can write down the enlarged T-dual transformation:101010 The -transformation is the transformation (58) composed with , while the is Eq. (57).

 S:~τ→−1~τ T:~τ→~τ+√2. (60)

The general form of the T-dual transformation is

 ~τ′=a~τ+bc~τ+d, (61)

where and , , , and are either

 a ∈Z, b ∈√2Z, c ∈√2Z, d ∈Z, (62)

or

 a ∈√2Z, b ∈Z, c ∈Z, d ∈√2Z. (63)

The fundamental region is . More details can be found in Appendix C.

### 3.3 Effective potential under Wilson line

Let us write down the contribution from the momentum lattice after the bosonization in each sector ; this time we include the momentum (43) of the -compactified which is modified by the Wilson line as in Eqs. (48) and (54):

 ~ZT2,α→w =Trα→we2πiτ1(L0−¯L0)−2πτ2(L0+¯L0)∣∣∣% momentum lattice. (64)

Concretely,

 ~ZT2,→0 ~ZT2,→w0 =−18(¯ϑ10)4∑m∈Z+1/2gm(η,R)(ϑ10)15, ~ZT2,→w1 =−18((¯ϑ00)4−(¯ϑ01)4)∑m∈Z+1/2gm(η,R)(ϑ10)15, ~ZT2,→w2 =−18((¯ϑ00)4+(¯ϑ01)4)∑m∈Z+1/2gm(η,R)(ϑ10)7((ϑ00)8−(ϑ01)8), ~ZT2,→w0+→w1 =−18(¯ϑ10)4∑m∈Zgm(η,R)((ϑ00)7−(−1)m(ϑ01)7)((ϑ00)8−(ϑ01)8), ~ZT2,→w0+→w2 =−18(¯ϑ10)4∑m∈Zgm(