Vacuum Structure of Charge k Two-Dimensional QED and Dynamics of an Anti D-String Near an O1- plane
Adi Armoni Shigeki Sugimoto











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    equationsection
    YITP-18-133
    
     .7cm
    
    =cmr10 scaled4
        =cmmi10 scaled4
        =cmmi7 scaled4
         
          Vacuum Structure of Charge k Two-Dimensional QED 0.3cm
        
          and Dynamics of an Anti D-String Near an O1^--plane 0.3cm
     1cm
    Adi Armoni^⋆ and Shigeki Sugimoto^♮,♯
     0.5cm
           ^⋆Department of Physics, College of Science
     Swansea University, SA2 8PP, UK
    ^♮Center for Gravitational Physics, Yukawa Institute for
    Theoretical Physics,
    
    Kyoto University, Kyoto 606-8502, JAPAN
    ^♯Kavli Institute for the Physics and Mathematics
     of the Universe (WPI),
    
    The University of Tokyo, Kashiwanoha, Kashiwa 277-8583, JAPAN
    
     0.5cm
    ^⋆a.armoni@swansea.ac.uk, ^♮sugimoto@yukawa.kyoto-u.ac.jp
    
     .5cm
     Abstract 20pt
    
     .5cm 
    
    
    We study the vacuum structure of N_f flavour two-dimensional QED
    with an arbitrary integer charge k. We find that the axial symmetry
    is spontaneously broken from _kN_f to _N_f due to
    the non-vanishing condensate of a flavour singlet operator,
    resulting in k degenerate vacua.
    An explicit construction of the k vacua is given by using
    a non-commutative algebra obtained as a central extension
    of the _kN_f discrete axial symmetry and _k 1-form
    (center) symmetry, which represents the mixed 't Hooft anomaly between
    them.
    
    We then give a string theory realization of such a system
    with k=2 and N_f=8 by putting an anti D-string in the vicinity of
    an orientifold O1^--plane and study its dynamics using the
    two-dimensional gauge theory realized on it. We calculate the potential
    between the anti D-string and the O1^--plane and find repulsion in
    both weak and strong coupling regimes of the two-dimensional gauge
    theory, corresponding to long and short distances, respectively. We also
    calculate the potential for the (Q,-1)-string (the bound state of an
    anti D-string and Q fundamental strings) located close to the
    O1^--plane.
    The result is non-perturbative in the string coupling.
     
    empty
    
    plain
    
    
    § INTRODUCTION
    
    
    The study of the dynamics of string theory at strong coupling is
    hard. Both string perturbation theory and non-perturbative string
    phenomenon are difficult to calculate. Field theory calculations are
    often easier especially in the case of two-dimensional gauge theories.
    
    The purpose of this paper is twofold. In its first part we study a
    variant of two-dimensional QED (from now on abbreviated “2 dim QED”)
    with N_f flavours of Dirac fermion fields of
    charge k, where k is an arbitrary integer. (In the following,
    we assume k to be positive without loss of generality.)
    This is a generalization of the well-known multi flavour Schwinger
    model.[The N_f=1 case is recently analyzed in
    <cit.>.]
    Although the k dependence in the action of 2 dim QED can be
    eliminated by rescaling the gauge field and the gauge coupling,
    it enters in the flux quantization condition and
    the charge k is actually physically relevant.
    In fact, the k dependence appears in the symmetry of the system.
    When the fermions are massless,
    it has _kN_f anomaly free discrete axial symmetry and _k
    1-form (center) symmetry, which play crucial roles in our analysis.
    We argue that the _kN_f discrete axial symmetry is spontaneously
    broken to _N_f and, as a result, there are k
    distinct vacua, generalizing the result for N_f=1 given in
    <cit.>.
    Interestingly, when N_f>1, the chiral condensate
    ψ_iψ^j vanishes,
    because the chiral SU(N_f)_L× SU(N_f)_R symmetry
    cannot be spontaneously broken due to the Coleman-Mermin-Wagner
    theorem <cit.>,
    but nonetheless the axial symmetry is spontaneously broken
    because the vacuum expectation value of the determinant
    of the fermion bilinear operator (ψ_Rj^†ψ_L^i) is
    non-vanishing.[
    This scenario was also suggested in <cit.>.
    See <cit.>
    for analogous phenomena in 4 dim gauge theory.]
    The spontaneous break down of the axial symmetry is also
    characterized by the mixed 't Hooft anomaly between
    the _kN_f axial symmetry and the _k 1-form symmetry
    as discussed in <cit.> for N_f=1 case.
    The existence of the mixed 't Hooft anomaly is understood
    as the fact that the axial _kN_f symmetry and
    the _k 1-form symmetry are centrally extended
    in the quantum theory.<cit.>
    The centrally extended algebra is non-commutative
    and gives a stringent constraint on the vacuum structure.
    We give an explicit construction of the k vacua
    by utilizing this non-commutative algebra.
    
    We mainly work in a bosonized description of 2 dim QED.
    We check that all the global symmetry as well as
    the mixed 't Hooft anomaly are realized in the bosonized
    description and the results for N_f=1 given
    in <cit.> are reproduced in a simplified way.
    It turns out to be very efficient in the generalization to N_f>1.
    We also discuss how the vacuum degeneracy is lifted when a small mass
    (with respect to the gauge coupling) is given to the fermion.
    
    In the second part of the paper we use the results obtained in the first
    part to study the dynamics of a non-supersymmetric brane
    configuration. We will focus on a system that consists of an orientifold
    1-plane and a 1-brane. Here, the 1-brane is the anti D-string
    obtained by flipping the orientation of the D1-brane
    in the maximally supersymmetric 1-D1 system.
    The Op- p system is a non-supersymmetric system
    with no tree level tachyon fields.
    It has been studied as a mechanism to break supersymmetry
    in a controlled way
    <cit.>
    and provides interesting playgrounds
    to study non-supersymmetric quantum field theories
    <cit.>.
    We show that the 1-1 system contains a sector that is
    described by 2 dim QED with k=2 and N_f=8.
    
    As an application of the analysis in the first part, we calculate the
    potential between the 1-plane and the 1-brane. The distance
    between them is parametrized by the value of scalar fields that
    correspond to fermion mass in 2 dim QED.
    Using the standard relations between gauge theory and string theory
    parameters, the potential is calculated by evaluating
    the vacuum expectation value of the Hamiltonian of 2 dim QED
    as a function of the fermion mass.
    Since the gauge theory is super-renormalizable, the short distance
    potential is controlled by strong coupling and the long distance
    potential is controlled by weak coupling dynamics. 
    At short distances the potential is calculated by
    using the strong coupling analysis in 2 dim QED. The result contains
    fractional powers of string coupling g_s, which clearly shows
    that the result is non-perturbative.
    In the weak coupling regime, we use the 1-loop Coleman-Weinberg
    potential to calculate the potential. 
    
    We find that the 1-brane is repelled from the
    1-plane at both short and long distance, which suggests that there is
    a run away potential. We also calculate the potential for
    1-brane with Q unit of electric flux on it, which corresponds
    to the bound state of the 1-brane and Q fundamental strings.
    
    The paper is divided into two main parts and it is organized as follows:
    in section <ref>, we discuss the vacuum structure of the
    charge k multi flavour 2 dim QED.
    In section <ref>, we discuss the 1-1 system and its
    dynamics. Section <ref> is devoted to an
    outlook and a discussion about future directions.
    
    
    
    
    § VACUUM STRUCTURE OF CHARGE K MULTI FLAVOUR QED
    
    
    In this section, we wish to find the vacuum structure of 2 dim QED
    with massless N_f flavours of charge k fermions.
    The model is exactly solvable using bosonization. We also consider
    the massive case in section <ref>.
    
    
    
     §.§ Symmetry and anomaly
    
    
    The action of the system we consider is given by
    
        S_ QED=∫ d^2x (
        -1/4 e^2F_μν^2+iψ_iγ^μ(
        _μ+ikA_μ)ψ^i) ,
    
    where A_μ (μ=0,1) is the U(1) gauge field, ψ_i
    (i=1,⋯, N_f) are complex Dirac fermions of charge k∈_>0.
    We take the representation  γ^0=σ^1, γ^1=iσ^2
    and write ψ^i=(ψ^i_R,ψ^i_L)^T, where ψ_L^i and
    ψ_R^i denote the left- and right-handed components of the fermions,
    respectively.
    The U(1) gauge transformation acts on A=A_μ dx^μ and ψ^i as
    
        A A+dλ ,   ψ^i e^-ikλψ^i ,
    
    where λ is a 2π periodic
    ( i.e.λ is identified with λ+2π) real scalar
    field.
    The gauge field is normalized such that it satisfies the usual flux
    quantization condition:
    
        1/2π∫ F∈ ,
    
    where F=dA= F_μνdx^μ dx^ν.
    Because of this quantization condition, we are not allowed
    to rescale the gauge field and the gauge coupling e to eliminate the
    U(1) charge k.
    In fact, as we will shortly see, vacuum structure of the 2 dim QED
    with charge k fermions is completely different from that with charge
    1 fermions.
    
    The classical global symmetry of the theory is[There are other
    symmetries such as Poincaré symmetry, parity, time reversal and charge
    conjugation, which will not be considered in this paper.]
        G_ classical=SU(N_f)_L× SU(N_f)_R × U(1)_A/_2/(ℤ_N_f)_L × (ℤ_N_f)_R ,
    
    where SU(N_f)_L and SU(N_f)_R are the chiral symmetry that act on
    ψ_L^i and ψ_R^i, respectively, as
    
        ψ_L^i (g_L)^i_ jψ_L^j ,   ψ_R^i (g_R)^i_ jψ_R^j
    
    with (g_L,g_R)∈ SU(N_f)_L× SU(N_f)_R,
    and U(1)_A is the classical axial symmetry that acts on the fermions as
    
        ψ_L^i e^-iαψ_L^i ,   ψ_R^i e^+iαψ_R^i ,
    
    with e^iα∈ U(1)_A. Note that 
    U(1)_A is divided by _2 in (<ref>) with the identification
    e^iα∼ -e^iα, because the axial transformation
     (<ref>) with α=π corresponds
    to ψ^i -ψ^i which is an element of the U(1) gauge
    transformation.
    The denominator (ℤ_N_f)_L × (ℤ_N_f)_R
    in (<ref>) acts on (g_L,g_R,e^iα)∈
    SU(N_f)_L× SU(N_f)_R × U(1)_A as
    
        (g_L,g_R,e^iα)
         (ω_Lg_L,ω_Rg_R,e^iαω_L^1/2ω_R^-1/2)
    
    with (ω_L,ω_R)∈ (ℤ_N_f)_L ×
    (ℤ_N_f)_R.
    Though there is a sign ambiguity in the square root
    ω_L^-1/2ω_R^1/2,
    it is well-defined as an element of U(1)_A/_2.
    
    Quantum mechanically, U(1)_A is broken by the anomaly. In fact
    the path integral measure of the fermions ψψ is
    transformed by
    e^iα∈ U(1)_A as
    
        ψψψψexp(-iα/π k N_f∫ F) ,
    
    and hence, the partition function is invariant only when
    α=πl/k N_f with l=1,2,⋯ 2kN_f.
    Therefore, U(1)_A is
    broken explicitly to (_2k N_f)_A by anomaly and
    the global symmetry G is given by replacing
     U(1)_A/_2 in (<ref>) with
    (_2k N_f)_A/_2 ≡_kN_f^:[
    See <cit.> for similar consideration in 4 dim massless
    QCD.]
        G=SU(N_f)_L× SU(N_f)_R ×ℤ_kN_f^/(ℤ_N_f)_L × (ℤ_N_f)_R .
    
    Below, we will argue that ℤ_kN_f^ is spontaneously broken to
    ℤ_N_f, resulting in k vacua[
    The case with N_f=1 is studied in <cit.>, in which case
    the vacuum structure is similar to that of  4 dim N=1SU(N)
    SYM, where U(1)_A is broken to ℤ_2N by anomaly and
    further broken spontaneously to ℤ_2, resulting
    in N vacua.].
    
    In addition, the system admits a global _k 1-form symmetry,
    denoted by _k^.
    To be explicit, we compactify the spatial direction to S^1
    of radius R. Then,  the elements of _k^ is represented
    by the transformation
    
        A A+1/kdξ ,   ψ^i e^-iξψ^i
    
    with a 2π periodic real scalar field ξ satisfying
    
        ξ(x^0,x^1+2π R)=ξ(x^0,x^1)+2π l ,   (l=1,2,⋯,k)
    
    up to the gauge transformation (<ref>). When we choose ξ=l x^1/R,
    it gives a constant shift of A_1 as
    
        A_1 A_1+l/kR .
    
    Note that this transformation should not be considered as a part of the
    gauge transformation (<ref>) with λ=ξ/k, unless
    l∈ k, because
    (<ref>) is not compatible with the 2π periodicity of
    λ in (<ref>).
    Under this transformation,
    the Wilson loop operator that winds around the spatial circle
    W≡exp(i∫_S^1A) transforms as
    
        W e^i2π l/kW .
    
    
    As discussed in <cit.> for the N_f=1 case,
    this _k^ and the discrete axial symmetry _kN_f^ have
    a mixed 't Hooft anomaly. Indeed, gauging _k^ is equivalent
    to introducing a background gauge configuration with fractional flux
    quantization condition[Following <cit.>,
    one can gauge _k^ as follows.
    Suppose we have a theory with
    a U(1) gauge field A, which has a global U(1) 1-form symmetry
    given by A A+α with α being a closed 1-form.
    One can gauge this 1-form symmetry by promoting α to be
    a 1-form gauge field and introducing a 2-form gauge field B
    that transforms as B B+dα. This U(1) 1-form symmetry can be
    broken to _k by introducing an additional 1-form gauge field C
    that transform as C C+kα and satisfy a constraint
    dC=kB, which is an analogue of
    a would-be Nambu-Goldstone mode of a charge k Higgs field.
    An action that is invariant under the _k 1-form gauge symmetry
    can be obtained by replacing A with A≡ A-1/kC.
    Then, the flux quantization condition for A is given by (<ref>).
    ]
        1/2π∫ F∈1/k .
    
    and this makes (_2kN_f)_A anomalous, because the fermion path integral
    measure (<ref>) is not invariant under generic (_2kN_f)_A
    transformations.
    The anomaly free part of (_2kN_f)_A in such backgrounds
    is a _2N_f subgroup, whose elements are given by
    e^iα∈ U(1)_A with α=πl/N_f with l=1,2,⋯,2N_f.
    Therefore, _kN_f^(=(_2kN_f)_A/_2) is broken to _N_f
    by the mixed anomaly.
    Note that the unbroken subgroup _N_f of _kN_f^
    is equivalent, under the identification (<ref>), to the center
    _N_f of SU(N_f)_L (or SU(N_f)_R).
    
    
    
     §.§ Bosonization and 't Hooft anomaly matching
    
    
      §.§.§ N_f=1
    
    
    Let us start with the one flavour theory.
    We analyze the system by using bosonization.
    For a direct and rigorous argument in terms of the original
    fermionic description, see <cit.>.
    
    It is known that the one flavour 2 dim QED can be mapped to a theory
    with a 2π periodic real scalar field φ with the
    action[See e.g.<cit.> for a review.][
    The action (<ref>) is identical to the dual description of the
    Stückelberg action studied in <cit.> and we can
    borrow some of the arguments given there.
    ]
        S=∫ d^2x  (-1/4e^2 F_μν^2
        +1/8π∂_μφ∂^μφ +k/2πφ F_01 ) .
    
    The correspondence is roughly given by[
    More precisely, the first relation is given as
    ψ_R^†ψ_L=-e^γ/4πμ_μ e^iφ,
    where _μ denotes the normal ordering with scale μ.
    See, e.g., <cit.>.
    ]
        ψ_R^†ψ_L ∼ c  e^iφ ,   ψγ^μψ∼1/2πϵ^μν_νφ ,   ψγ^3γ^μψ∼1/2π^μφ ,
    
    where c is a non-zero constant,
    ϵ_μν is the anti-symmetric tensor (Levi-Civita symbol in
    2 dim) with ϵ^01=-ϵ_01=1, and γ^3=σ^3 is
    the chirality operator in 2 dim.
    
    As discussed in section <ref>, the U(1)_A/_2 symmetry
    is broken by anomaly to ℤ_k.
    This is manifest in the bosonized language, where the classical action
    (<ref>) captures the anomaly.
    The correspondence (<ref>) suggests that 
    e^iα∈ U(1)_A acts on φ as
    
        φφ-2α .
    
    With the generic fluxes with (<ref>), the action (<ref>)
    is invariant (up to 2π shifts) under this transformation only when
    2α=2πl/k with l=1,2,⋯,k, which gives _k^.
    
    The 1-form symmetry _k^ is less obvious.[
    See <cit.> for another explanation of the 1-form symmetry
    in the bosonized theory.
    ]
    At first sight, the action (<ref>) looks invariant under
    any constant shift of the gauge field. In fact, one can construct
    the Noether current associated with this 1-form symmetry
    
        J_μν=1/e^2F_μν
        -k/2πφϵ_μν .
    
    It satisfies the conservation law
    
        ^μ J_μν=0 ,
    
    and formally generate the phase shift of the Wilson loop operator W
    introduced in section <ref> as
    
        e^iα J_01W e^-iα J_01=e^iαW .
    
    (See below for the canonical quantization of the system to show this
    explicitly.)
    However, this current is well-defined only modulo k, because φ is
    2π periodic, and J_01 should be identified with J_01+k. This means
    e^iα J_01 is well-defined only when α=2πl/k
    (l=1,2,⋯,k), reproducing (<ref>).
    Thus, the generator of _k^ is
    
        U≡exp(2π i/kJ_01) .
    
    Note that U^k corresponds to a large gauge transformation
    ((<ref>) with l=k) and the group generated by
    U is _k up to gauge transformations.
    
    Alternatively, one could start from the action
    
        S=∫ d^2x  (-1/4e^2 F_μν^2
        +1/8π∂_μφ∂^μφ-k/2πϵ^μν_μφ A_ν ) ,
    
    which is obtained from (<ref>) by integrating  by parts.
    In this case,  the 1-from _k^ symmetry given by
    A A+1/kdξ with dξ being a closed one-form
    with 2π periods is manifest,[1/2π∫ dφ∧ dξ is an element of 2π.
    ] while
    the axial _k^ symmetry is less obvious.
    The action (<ref>) is invariant under any constant shift of φ
    and one can construct the Noether current associated to this symmetry:
    
        J^A_μ= 1/4π_μφ-k/2πϵ_μν A^ν
    
    The conservation law ^μ J_μ^A=0 follows from the equations
    of motions and reproduces the anomaly equation in the fermionic theory
    via the correspondence (<ref>). However, this current
    is not gauge invariant. The conserved charge
    
        Q≡∫_S^1 dx^1 J_0^A=
        ∫_S^1 dx^1(1/4π_0φ-k/2π A_1
        )
    
    is well-defined only modulo k, because a large gauge transformation
    ((<ref>) with l=k) induces
    
        Q Q-k  .
    
    A well-defined operator can be constructed as
    
        V≡exp(-2π i/kQ) .
    
    This operator gives
    (<ref>) with 2α=2π/k,
    and hence generates the axial _k^ symmetry.[V^k
    induces φφ-2π, which is a trivial transformation
    under the identification φ∼φ+2π.
    In fact, the 2π shift of φ can be understood as a large
    gauge transformation of the 0-form gauge field φ.
    We regard the transformation by V^k as a gauge transformation.]
    
    It is now straightforward to check that the mixed 't Hooft anomaly of
    _k^ and _k^ matches with that in the fermionic
    theory. If one considers a generic background with the fractional flux
    quantization condition (<ref>), the transformation
    (<ref>) leaves the action (<ref>) invariant
     (up to 2π shifts) only when 2α∈ 2π.
    This breaks _k^ to nothing, reproducing
    the mixed 't Hooft anomaly discussed in section <ref>.
    
    Another way of checking the mixed 't Hooft anomaly is to use
    the commutation relation of the generators of 
    _k^ and _k^. In the A_0=0 gauge the
    action (<ref>) becomes
    
        S=∫ d^2x  (1/2e^2 (_0A_1)^2
        +1/8π∂_μφ∂^μφ +k/2πφ_0A_1
         ) .
    
    The canonical momenta conjugate to A_1 and φ are
    
        Π_A≡1/e^2_0A_1+k/2πφ
        =J_01 ,   Π_φ≡1/4π_0φ ,
    
    respectively, and the Hamiltonian is
    
        H=∫ dx^1(
        e^2/2(
        Π_A-k/2πφ)^2+2πΠ_φ^2+1/8π(_1φ)^2
        ) .
    
    Note that the Gauss law equation (equation of motion for A_0) implies
    
        _1 Π_A=0 ,
    
    which will be imposed on the physical states.
    
    By using the canonical commutation relations,
    
        [A_1(x^0,x^1),Π_A(x^0,y^1)]=iδ(x^1-y^1)  ,   
        [φ(x^0,x^1),Π_φ(x^0,y^1)]=iδ(x^1-y^1)  ,
    
    we can explicitly check that J_01 and Q introduced in (<ref>)
    and (<ref>) above commute with the Hamiltonian, and 
    _k^ and _k^ generated by V and U defined in
    (<ref>) and (<ref>), respectively, are the symmetry of the system.
    The important point is that U and V do not commute with each other,
    but satisfy the following non-commutative relation:
    
        U V= V U e^2π i/k .
    
    This relation follows from the commutation relation[
    See <cit.> for the derivation without using bosonization.
    See also <cit.> for the realization of this algebra in
    a TQFT describing the IR physics of the system.]
        [J_01, Q]
        =[Π_A,∫ dx^1(Π_φ-k/2πA_1)]
        =ik/2π .
    
    Therefore, if one promotes the transformation by U to
    a gauge symmetry, the operator V is no longer gauge invariant,
    which means that when _k^ is gauged,
    _k^ is not a symmetry of the system any more.
    This is consistent with the mixed 't Hooft anomaly discussed above.
    
    
    
      §.§.§ N_f>1
    
    
    Let us next discuss the generalization to the multi flavour theory.
    The multi flavour theory can be bosonized by using the non-Abelian 
    bosonization rules <cit.> (See e.g. <cit.>
    for a review.)
    
        ψ_Rj^†ψ_L^i∼ c u^i_ j ,   
        J_-j^i∼i/2π(u_- u^-1)^i_ j ,   
        J_+j^i
        ∼i/2π(u^-1_+ u)^i_ j ,
    
    where u is a U(N_f) valued
    scalar field, (J_-,J_+) are the U(N_f)_L× U(N_f)_R currents,
    and _±≡1/√(2)(_0±_1).
    
    We parametrize u as
    
        u=e^iφ g  ,
    
    where φ is a 2π periodic real scalar field
    and g is an SU(N_f) valued scalar field. Since u is invariant
    under the following _N_f transformation
    
        φφ-2π/N_f ,   g e^2π i/N_fg ,
    
    the fields (φ,g)∈ U(1)× SU(N_f) related by this
    transformation are identified. We regard this
    _N_f symmetry as a gauge symmetry.
    
    The bosonization rules (<ref>) imply that
    the U(1)_V current J_μ^V= J_μ is given by
    
        J_μ^V∼N_f/2πϵ_μν^νφ ,
    
    and hence only φ couples with the gauge field.
    The action is given by
    
        S=∫ d^2x  (-1/4e^2 F_μν^2
        +N_f/8π∂_μφ∂^μφ +kN_f/2πφ F_01 )
        +S_ WZW(g) ,
    
    where S_ WZW(g) is the action of the SU(N_f) WZW theory at level 1.
    
    It is not difficult to check that the global symmetry of the system agrees with
    that of 2 dim QED discussed in section <ref>. From the correspondence
    (<ref>), we find that (g_L,g_R)∈ SU(N_f)_L× SU(N_f)_R
    acts on g as
    
        g g_L g g_R^-1 ,
    
    and the generator of _kN_f^ acts on φ as
    
        φφ-2π/kN_f .
    
    One can also check that
    the denominator (_N_f)_L×(_N_f)_R in (<ref>)
    acts trivially on φ and g under the identification
    by (<ref>).
    Repeating the same argument as in section <ref> for the N_f=1
    case with the identification  (<ref>),
    we find that the one form symmetry is _k^ as expected.
    
    The 't Hooft anomaly matching for the mixed anomaly of
    _k N_f^ and _k^ works as well. When we allow
    the fractional flux (<ref>), the unbroken part of the
    _kN_f^ shift symmetry (<ref>) becomes _N_f
    generated by φφ-2π/N_f. This precisely agrees
    with the mixed 't Hooft anomaly discussed in section <ref>.
    
    The discussion below (<ref>) can also be applied to the N_f>1
    cases. Generalizing (<ref>) and (<ref>), the operators that generate
    _k^ and _kN_f^ are obtained as
    
        U   ≡   exp(2π i/kJ_01)
        =
         exp(2π i/kΠ_A) ,
        
         V   ≡   exp(-2π i/kQ)
        =
         exp(-2π i/kN_f∫ dx^1Π_φ
        +i∫ dx^1 A_1
        ) ,
    
    respectively, where
    
        Π_A≡1/2e^2_0A_1
        +kN_f/2πφ ,   Π_φ≡N_f/4π_0φ
    
    are the canonical momenta conjugate to A_1 and φ,
    respectively. They also satisfy (<ref>).
    As discussed in section <ref>,
    when the transformation by U is gauged, V is no longer
    a gauge invariant operator. But, in this case, V^k commutes with U
    and hence it is well-defined. Since V^k generates the _N_f
    subgroup of _kN_f^, we conclude that _kN_f^ is broken
    to _N_f when _k^ is gauged.
    This is again consistent with the mixed 't Hooft anomaly discussed above.
    
    
    
    
     §.§ Spontaneous symmetry breaking and the vacuum structure
    
    
    The existence of the 't Hooft anomaly already predicts that the vacuum
    has to be non-trivial. In fact, because the space of ground states
    has to be a representation of the algebra generated by U and
    V satisfying (<ref>), the vacua have to be at least k-fold
    degenerate.
    In this subsection, we construct the ground states explicitly
    and argue that _kN_f^
    is spontaneously broken down to _N_f via the condensate
    
        (ψ_Rj^†ψ^i_L) 0 ,
    
    which suggests that there are indeed k degenerate vacua
    (apart from the θ vacua)
    in 2 dim massless QED with N_f flavours.
    
    
    
      §.§.§ N_f=1
    
    
    In order to see the vacuum structure more explicitly,
    it is again useful to start with the N_f=1 case.
    For the one flavour 2 dim QED, as it is well-known in the k=1 case
    <cit.>
    and recently shown for general k in <cit.>, the fermion bilinear
    operator ψ_R^†ψ_L has a non-zero vacuum expectation value. As a result,
    _k^ is spontaneously broken to nothing and there are k vacua
    associated with this breaking.
    The axial symmetry _k^ acts as a cyclic rotation of
    these k vacua. From the anomaly relation (<ref>), this fact implies
    that the i-th vacuum is mapped to the (i+1)-th vacuum (mod k) when the
    θ angle is shifted by 2π.
    
    Let us explain how the spontaneous breakdown of _k^ can be
    understood in terms of the bosonized theory (<ref>).[
    See <cit.> for another closely related derivation
    of the k-fold degeneracy of the ground state in the bosonized
    theory (<ref>).]
    To this end, let us parametrize the ground states |θ⟩
    by the eigenvalue θ∈ of the operator
    2πΠ_A.[
    Here we assume that |θ⟩ is the unique vacuum with
    2πΠ_A|θ⟩=θ|θ⟩ and see that this assumption
    is consistent with (<ref>).
    ]
    This is possible because Π_A commutes with the
    Hamiltonian. Note also that θ is a constant
    because of the Gauss law equation (<ref>).
    The operator U
    defined in (<ref>) acts on |θ⟩ as[θ used in <cit.> is θ/k in our notation.
    ]
        U|θ⟩= e^iθ/k|θ⟩ .
    
    The normalization of θ has been chosen such that the eigenvalue
    of the operator U^k, which corresponds to the large gauge
    transformation, is e^iθ.
    [
    Although we have regarded U as the generator of _k^,
    it doesn't satisfy U^k=1 on |θ⟩. It is easy to fix this
    by defining an operator U'≡ e^-iθ/k U
    that satisfies U'^k=1 on the subspace of the Hilbert space we are
    interested in without affecting the non-commutative relation (<ref>).
    See (<ref>).
    ]
    This is compatible with
    the phase factor induced by the θ term
    θ/2π∫ F and this parameter θ is identified
    as the θ angle in 2 dim QED.
    Because V (defined in (<ref>) with N_f=1) satisfies
    the relation VΠ_A V^-1=Π_A-1, we find
    
        V|θ⟩=|θ+2π⟩ .
    
    It is easy to see that (<ref>) and (<ref>) are compatible
    with the relation (<ref>).
    
    On the ground state |θ⟩, the Hamiltonian (<ref>) looks like that of a free massive scalar field with a potential
    
        V(φ)=e^2/8π^2(kφ-θ)^2 .
    
    However, one should be aware that φ is a 2π periodic
    scalar field. In other words, V^k is regarded as a gauge symmetry.
    Therefore, we should mod out the system by the transformation
    
        φφ-2π n ,   |θ⟩|θ+2π kn⟩ ,   (n∈) .
    
    One way to achieve this is to pick |θ⟩
    as a representative of the equivalence class
    {|θ+2π kn⟩| n∈}
    and regard φ as a non-compact scalar field that
    takes values in  without any identifications.
    With this understanding,
    we can regard the system as the theory of
    a free massive real scalar field with the Hamiltonian given by
    setting Π_A=θ/2π in (<ref>):
    
        H=∫ dx^1(
        2πΠ_φ^2+1/8π(_1φ)^2
        +e^2/8π^2(kφ-θ)^2
        ) ,
    
    when we only consider the vacuum |θ⟩ and gauge invariant
    local operators acting on it,  i.e. the superselection sector
    constructed on |θ⟩.
    
    In fact, in various literature on 2 dim QED (with N_f=1 and k=1),
    this Hamiltonian (<ref>) is used as the starting point of the
    bosonized description.
    One new feature that appears in the k>1 case is that there are
    k sectors, {|θ+2π(kn+j)⟩| n∈} labeled by
    j=0,1,⋯,k-1 (k), that are not gauge equivalent.
    These k sectors are related to each other by the action of _k^
    generated by V. This means that the _k^ is spontaneously broken
    down and there are k vacua related by _k^, or, equivalently,
     2π shifts of the parameter θ.
    The theory is invariant under the 2π shift of
    θ, but in order to come back to the original vacuum,
    θ has to be shifted by 2π k.
    These properties cannot be seen if one starts with the system
    defined by (<ref>).
    
    Note also that the Hamiltonian (<ref>)
    cannot be used if one wants to consider operators that do not
    commute with U, because such operators
    change the eigenvalue of U and map a state to
    a different sector.  The Wilson loop operator W
    that winds once around the spatial circle is such an example.
    It changes the eigenvalue of U by a factor of e^2π i/k.
    In general, there is no gauge invariant operator that
    changes the eigenvalue of U by a factor
    other than e^2π i/kn with n∈, because
    the gauge invariant operator should commute with the generator of the
    large gauge transformation U^k.
    Therefore, we may restrict
    the whole Hilbert space to the subspace that is constructed
    by acting gauge invariant operators on the k vacua represented by
    |θ+2π j⟩ (j=0,1,⋯, k-1) with fixed θ.
    The operators U and V are represented on the
    k dimensional vacuum space by the clock and shift matrices as
    
        U'≡ e^-iθ/k U=
        (
        [     1;           ω;           ⋱;       ω^k-1 ])
         ,    V=
        (
        [   1; 1;   ⋱;   1 ]) ,
    
    where ω≡ e^2π i/k.
    
    We can also argue the spontaneous breakdown of _k^
    by using order parameters characterizing it.
    (<ref>) implies that the operator U transforms as
    
        U V U  V^-1=e^-2π i/k U
    
    by the generator of _k^. Therefore, the fact that
    ⟨θ| U|θ⟩=e^iθ/k is non-zero
    implies the breaking of _k^.
    Another natural order parameter is the vacuum expectation value of
    e^iφ, which is related to ψ_R^†ψ_L by the
    bosonization rule (<ref>).
    Since the minimum of the potential in (<ref>)
    is φ=θ/k, the classical value is
    
        ⟨θ|e^iφ|θ⟩= e^iθ/k .
    
    Quantum mechanically, the operator e^iφ should be defined
    by taking the normal ordering. Because φ in (<ref>) is a massive
    free scalar field, this can be easily done and the result (<ref>)
    is unchanged, which again shows the breaking of _k^.
    
    
    
    
      §.§.§ N_f>1
    
    
    Since the non-Abelian part involving the SU(N_f) valued field
     g in (<ref>) decouples from φ and the
    gauge field, most of the arguments in section <ref> go through with a
    little modification.
    The main difference is that (φ,g) is identified by
    the transformation (<ref>) rather than a simple
    2π shift of φ. The transformation (<ref>) is induced by
    the operator V^k ω_L, where ω_L is the operator
    that induces the transformation  (<ref>) with
    g_L=e^2π i/N_f.
    Therefore, this operator V^k ω_L is the generator
    of the _N_f gauge symmetry (<ref>)
    and plays the same role as V^k
    in the N_f=1 case in section <ref>.
    
    A crucial point is that, because of the Coleman-Mermin-Wagner
    theorem,<cit.> the continuous
    chiral symmetry SU(N_f)_L× SU(N_f)_R cannot be spontaneously broken.
    This means that the ground states are the singlet state as a representation
    of SU(N_f)_L× SU(N_f)_R and vacuum expectation values of any
    operators that are not invariant under SU(N_f)_L× SU(N_f)_R have
    to vanish.
    Therefore, we can basically forget about the field g for the
    consideration of the vacuum structure. Then, as in the N_f=1 case
    discussed in section <ref>, the vacuum is parametrized by
    θ satisfying (<ref>) and (<ref>). This also implies
    that the action of the operator V^kω_L
    on the ground states is equivalent to the action of V^k,
    which is the generator of the _N_f subgroup of the discrete axial
    symmetry _kN_f^. Then, we should identify the system by the
    transformation
    
        φφ-2π n/N_f ,   |θ⟩|θ+2π kn⟩ ,   (n∈) .
    
    that generalizes (<ref>) to the cases with N_f>1.
    Therefore, under this identification, _kN_f^ is spontaneously
    broken to _N_f, because the vacuum |θ⟩ is not invariant
    under the action of V (the generator of _kN_f^),
    but it is invariant under V^k  (the generator of _N_f⊂_kN_f^).
    Just as in the N_f=1 case, there are k vacua represented by
    |θ+2π j⟩ with j=0,1,2,⋯,k-1 (k),
    which are related by the 2π shift of the parameter θ.
    Then, generalizing (<ref>), Hamiltonian for the superselection
    sector constructed on |θ⟩ with φ regarded as a
    non-compact scalar field is obtained as
    
        H=∫ dx^1(
        2π/N_fΠ_φ^2+N_f/8π(_1φ)^2
        +e^2/8π^2(kN_fφ-θ)^2
        )+H_ WZW ,
    
    where H_ WZW is the Hamiltonian for the SU(N_f)
    valued field g
    induced from S_ WZW(g) in (<ref>).
    
    As mentioned above, the Coleman-Mermin-Wagner theorem
    implies that SU(N_f)_L× SU(N_f)_R non-singlet
    operators have vanishing vacuum expectation values and hence an order
    parameter that characterizes the vacuum structure
    has to be the vacuum expectation value of
    an SU(N_f)_L× SU(N_f)_R singlet operator.
    In addition, such an operator has to be invariant under (<ref>).
    As discussed in section <ref>, the vacuum expectation value
    of the operator U is one of the order parameters
    that shows the breaking of _kN_f^ to _N_f.
    Another natural operator is e^iN_fφ.
    The argument around (<ref>) implies
    
        ⟨θ|e^iN_fφ|θ⟩= e^iθ/k ,
    
    which again shows the breaking of _kN_f^ to _N_f.
    Since e^iN_fφ= u, this operator is related by
    the bosonization rules (<ref>) to (ψ_Rj^†ψ_L^i).
    Therefore, (<ref>) suggests (<ref>), even though
    the vacuum expectation value of
    ψ_Rj^†ψ_L^i vanishes for N_f>1.
    
    
    
    
    
     §.§ Mass deformation
    
    
    In this subsection, we consider adding a fermion mass term 
    M_0ψ_jψ^j to the action (<ref>) and
    study the chiral condensate, vacuum energy and string tension for the
    cases with general N_f and k. 
     As we will see, the effect of non-zero M_0
    drastically alters the qualitative features of the vacuum structure
    for N_f>1 considered in section <ref>.
    In the bosonized description, an interaction term proportional to
    M_0(u+u^†), which corresponds to the fermion mass term via
    the bosonization rule (<ref>), is added and the system
    is no longer exactly solvable. However, we are able to get quite
    non-trivial results thanks to various powerful techniques developed
    in 2 dim QFT. The results in this subsection will be used in section
    <ref> to give non-perturbative predictions in string theory.
    In this subsection, we focus on the small mass (strong coupling) regime
    M_0≪ e and the large volume limit M_0R∞.
    We also assume M_0>0 throughout this paper.
    
    
    
    
      §.§.§ Chiral condensate and vacuum energy
    
    
    As mentioned in section <ref>, the chiral condensate
    ψ_iψ^j vanishes for N_f>1
    due to the Coleman-Mermin-Wagner theorem in the massless limit.
    When the mass term M_0ψ_jψ^j is added,
    however, the continuous chiral symmetry
    SU(N_f)_L× SU(N_f)_R is explicitly broken to the
    diagonal SU(N_f) subgroup and there is no reason for the chiral
    condensate to vanish. In fact, it was shown in
    <cit.> that
    the trace part of the chiral condensate is non-zero and behaves as
    
        ψ_jψ^j∝ e^2/N_f+1M_0^N_f-1/N_f+1 .
    
    This is clearly a non-perturbative effect,
    since it has fractional powers of e and M_0.
    
    Let us first outline the derivation of (<ref>) using the bosonized
    Hamiltonian (<ref>).
    It is convenient to redefine the scalar field φ
    as
    
        h≡√(N_f/π)(φ-θ/kN_f)  ,
    
     so that  the kinetic term is canonically normalized and
    the mass term for h is simplified.
    The Hamiltonian is given by
    
        H=∫ dx^1
        _μ[
        Π_h^2+(_1 h)^2
        +m_h^2/2h^2
        ]+H_ WZW+H_ mass ,
    
    where
    
        m_h^2≡e^2k^2N_f/π ,
    
    and
    
        H_ mass≡∫ dx^1 μ_μ[
        c̃ M_0 e^i(2√(π/N_f)h+θ/kN_f)(g)+ h.c.] .
    
    Here, we have regularized the system by taking the normal ordering
    at scale μ[See <cit.> for the normal-ordering
    prescription.], denoted by the symbol _μ.
    The last term H_ mass in (<ref>) comes from
    the fermion mass term via the bosonization relation[
    See, e.g., <cit.> for a review.]
        ψ_Rj^†ψ_L^i=c̃ μ_μ u^i_ j
        =c̃ μ_μ e^iφg^i_ j ,
    
    where c̃ is a numerical constant.
    In this section, we are interested in the small mass (strong coupling) regime
    M_0≪ e and treat H_ mass as a small deformation of the Hamiltonian
    for the M_0=0 case used in the previous subsections.
    In this case, as we can see from (<ref>) and (<ref>),
    the scalar field h is much heavier than g, and we first try to integrate
    it out.
    Using the formula <cit.>:
    
        _m[e^iβϕ]=(μ^2/m^2)^β^2/8π_μ[e^iβϕ] ,
    
    where ϕ denotes a canonically normalized free scalar field,
    and the fact that if the mass of ϕ is m,
    vacuum expectation value of the left hand side is 1, we obtain
    
        ⟨θ|_μ[e^i 2√(π/N_f)h]|θ⟩
        =(m_h/μ)^1/N_f .
    
    Replacing the operator _μ[e^i 2√(π/N_f)h]
    with its vacuum expectation value (<ref>), we obtain the low energy
    effective Hamiltonian for the light field g:
    
        H^ low_ mass=∫ dx^1 μ^1-1/N_fm_h^1/N_f_μ[ c̃M_0
         e^iθ/kN_f(g)+ h.c.] .
    
    We find that the mass scale of the light field g is given by
    
        m_l^2=μ^1-1/N_fm_h^1/N_f M_0 .
    
    Choosing the normal ordering scale as μ=m_l, we get
    
        m_l^2=m_h^2/N_f+1M_0^2N_f/N_f+1 ,
    
    and
    
        H^ low_ mass=∫ dx^1 m_l^2
         _m_l[ c̃ e^iθ/kN_f(g)+ h.c.] .
    From this expression, we find that m_l is the only mass scale of the low
    energy effective Hamiltonian in the R∞ limit.
    Then, a simple dimensional analysis implies
    
        M_0⟨θ|ψ_jψ^j|θ⟩
        =f(θ)  m_l^2
        =f(θ) m_h^2/N_f+1M_0^2N_f/N_f+1
    
    with some function of θ denoted by f(θ), reproducing
    (<ref>).
    
    The function f(θ) in (<ref>)
    was calculated in <cit.> for the k=1
    case. (See Appendix <ref>.)
    Since k always appears in the combination θ/k or m_h in the
    Hamiltonian (<ref>), the k dependence of the function
    f(θ) can be included simply by rescaling θθ/k in
    the expression for k=1 and we obtain
    
        f(θ)=-N_f/4π(2exp(γ)cos(
        1/N_f(θ/k)))^2N_f/N_f+1 ,
    
    where γ≃ 0.577 is the Euler's constant and
    
        x≡ x-2π[x+π/2π]
    
    with [x] being the floor function
    that gives the greatest integer less than or equal to x.
    Note that (<ref>) implies x=x for -π≤ x<π
    and x+2π=x. Therefore, the expression
    (<ref>) is manifestly invariant under the 2π k shift of
    θ, though it has cusp singularities at θ/k=π
    (2π) for N_f>1.[
    These cusp singularities exist even for k=1. The existence of
    the cusps can be understood from the mixed anomaly between
    the vector-like flavour symmetry SU(N_f)_V/(_N_f)_V
    and the charge conjugation symmetry discussed in
    <cit.>. We thank the anonymous
    referee for pointing this out to us.] The 2π k periodicity 
    of θ follows from the fact that 2π k shift of θ can
    be absorbed by the redefinition of g as
    g e^-2π i/N_fg in (<ref>).
    
    So far, we have implicitly assumed that |θ⟩ is the vacuum state
    of the system. However, this is not always true.
    As we have seen in section <ref>, there are k degenerate ground states
    |θ+2π j⟩ with j=0,1,2,⋯,k-1 (k) for the M_0=0 case.
    When the fermion mass M_0 is turned on, the degeneracy is lifted, because
    the discrete axial symmetry _kN_f^ is explicitly broken by
    H_ mass.
    In fact, it can be shown that the energy density (expectation value of the
    Hamiltonian density ) is proportional to the chiral condensate
    (<ref>) as
    
        (θ)≡⟨θ||θ⟩=
        N_f+1/2N_ff(θ)  m_h^2/N_f+1M_0^2N_f/N_f+1 ,
    
    up to some irrelevant terms that are independent of
    θ and M_0.[
    The overall factor in (<ref>) is different from the
    expression for the energy density given in <cit.>.
    In <cit.>, only the
    contribution from the fermion mass term is taken into account. We found
    that the kinetic term also has a contribution of the same order and
    included in (<ref>). See Appendix <ref> for details.
    ]
    (See Appendix <ref>.)
    Note that comparing (<ref>) and (<ref>),
    we get a relation
    
        (θ)/ M_0=⟨θ|ψ_jψ^j|θ⟩ .
    
    The expression for the energy density (<ref>) implies that
    |θ⟩ (up to the identification
     |θ⟩∼|θ+2π k j⟩ (j∈))
    is the lowest energy state among the k states |θ+2π j⟩
    with j=0,1,⋯,k-1 for generic θ.
    For θ=π (2π), the lowest energy states
    are two-fold degenerate and given by |±π⟩.
    (See Figure <ref>.)
    
    
    Then, the vacuum energy density
    _ vac(θ) is given by
    
        _ vac(θ)=min_j∈(θ+2π j)=(θ) ,
    
    which is a 2π periodic function with cusps at θ=π
    (2π).[These cusps can be understood from the mixed
    anomaly between the 1-form symmetry and the charge conjugation
    symmetry.]
    
      §.§.§ String tension
    
    
    The formula (<ref>) can be used to obtain the string
    tension as it was done in <cit.> for k=1.
    Let us consider an electric flux created by a pair of external point
    particles of charge ± Q placed at x^1=∓∞.
    We call it a Q-string, though it fills up the 2 dim space-time.
    This amounts to adding
    
        S_ int
        = Q ∫ dx^0 A_0|_x^1=-∞-  Q∫ dx^0 A_0|_x^1=+∞
        = Q ∫ F
    
    in the action, which is equivalent to shifting θ to θ+2π Q.
    Therefore, the tension σ(Q) of the Q-string is estimated as
    
        σ(Q)= (θ+2π Q)-(θ) ,
    
    where we have assumed -π<θ<π so that |θ⟩ is
    the vacuum state.
    Though (<ref>) can be formally used for Q∉,
    since the last expression in (<ref>) is manifestly gauge invariant
    for any Q∈,
    we impose Q∈ so that the external point particles
    are also consistent with the U(1) gauge symmetry.
    For k=1, this restriction is too strong and one always gets
    σ(Q)=0, because of the 2π periodicity of the function
    (θ). For k>1, however, the periodicity of (θ)
    becomes 2π k and we get non-trivial results for Q=1,2,⋯,k-1
    (k). (See Figure <ref>.)
    The Q-strings with Q=nk (n∈) are tensionless,  i.e.σ(Q=nk)=0, which is a consequence of the screening by the
    dynamical charge k fermions.
    
    There are some interesting special cases worth mentioning.
    First, consider the case with θ=±π. Then, because
    |θ⟩ and |θ∓ 2π⟩ are degenerate, we have
    σ(Q=∓ 1)=0, which means that strings with the unit flux
    can become tensionless. This is probably not too surprising.
    From (<ref>), we see that θ=±π can be interpreted as a flux
    with Q=± 1/2. By adding Q=∓ 1, we end up with a flux with Q=∓ 1/2,
    which is obtained by the charge conjugation from the original configuration.
    
    A possibly more surprising case is the M_0 0 limit. The formula
    (<ref>) implies that the tension σ(Q) vanishes for
    any Q in the M_0 0 limit.<cit.>
    This is a consequence of the fact that the θ-dependence of
    the energy density disappears in the massless limit because of the
    anomaly relation (<ref>). However, if one tries to
    understand the vanishing of the Q-string tension intuitively as a
    screening phenomenon, this looks very strange, because the charge of any
    combination of the charge k fermions belongs to k and it doesn't
    look possible to completely screen a charge that doesn't belong to k.
    In our case, the Q-string state |θ+2π Q⟩ becomes one of the
    ground states in the M_0 0 limit. As discussed in section <ref>,
    this state is an eigenstate of the operator Π_A, which contains the
    electric flux.
    The energy contribution from the eigenvalue of Π_A is diminished
    (in the M_0=0 case) by a constant shift of the scalar field, as the operator
    Π_A appears in the Hamiltonian in the combination
    (2πΠ_A-kN_fφ)^2 (see (<ref>) for the N_f=1 case).
    However, unlike the usual screening phenomenon, the Q-string state does not
    loose the information of the flux Q in the process of making M_0 0,
    although it becomes completely tensionless. There is a conserved _k charge
    associated with the operator U' defined in (<ref>)
    that characterizes the Q-string state.
    
    
    
    
    § APPLICATION TO STRING THEORY
    
    
    In this section, we propose a way to realize 2 dim QED
    with k=2 in string theory.[
    A string theory realization of 2 dim QED with k=1 using
    a D1-D9-9 system was studied in <cit.>. ]
    Many of the properties of 2 dim QED studied in section <ref>
    have natural interpretations in string theory
    and the results in section <ref> are applied
    to give a new non-perturbative calculation on the brane dynamics.
    
    
    
    
     §.§ 1-1 system
    
    
    Consider a system of n coincident 1-branes (anti-D1-branes) on top
    of an 1-plane (a negative tension orientifold 1-plane).
    The low energy effective theory realized on the 1-brane world-sheet
    is a 2 dim non-supersymmetric SO(2n) gauge theory with
    the following massless fields:
    
        [          SO(2n) SO(1,1)   SO(8);     A_μ     adj       2       1;     Φ_I     adj       1     8_v;   λ_+^i     sym     1_+     8_+;   λ_-^i     sym     1_-     8_-;         ]
    
    Here, A_μ (μ=0,1), Φ_I (I=1,⋯,8)
    and λ_±^i  (i=1,⋯,8) are the gauge field,
    scalar fields and fermions, respectively.
    SO(2n) is the gauge group, SO(1,1) is the 2 dim Lorentz group
    and SO(8) is the global symmetry associated to the rotation
    in the transverse 8 dim space. The labels “adj” and
     “sym” refer to the adjoint and rank 2 symmetric tensor
    representations of the gauge group, respectively.[Here,
    “sym” representation is a n(2n+1) dimensional reducible representation
    and it can be decomposed to a singlet and rank 2 traceless symmetric tensor
    representations.]1_± denotes the positive/negative chirality Majorana Weyl spinor
    of SO(1,1). 8_v and 8_± are the vector and positive/negative chirality
    spinor representations of SO(8), respectively. 
    The field content (<ref>) is obtained by replacing the adjoint
    fermions with that of the rank 2 symmetric tensor representation
    in the 2 dim supersymmetric gauge theory realized in
    the 1-D1 system,<cit.>
    which is given by the dimensional reduction of
    the 10 dim =1 supersymmetric SO(2n) Yang-Mills theory.
    
    The gauge coupling e of this 2 dim gauge theory is related to the
    string coupling g_s and the string length l_s=√(α') by
    
        e^2=g_s/2πα' .
    
    We take the field theory limit  α' 0 and g_s 0
    with e kept fixed, so that stringy massive excitations become
    infinitely heavy and interactions with closed string fields including
    the gravitational interaction decouple.[This limit may cause some
    divergences in the effective field theory. For example, the one-loop
    analysis suggests that a tachyonic mass term for the scalar fields
    Φ_I will be generated and the mass scale will diverge
    in the field theory limit.
    In renormalizable quantum field theory, such divergences can be
    canceled by introducing counter terms and setting the renormalization
    conditions to make physical quantities finite. However, it is not
    clear whether such counter terms can always be introduced in string theory.
    To avoid this problem, we actually keep l_s finite, though we assume
     e^2≪ 1/α',
    and consider the system as a theory with the cut-off scale 1/l_s
    regularized by string theory.  (See sections <ref>
    and <ref>.)
    ]
    
    Let us focus on n=1. In this case, since SO(2) is equivalent
    to U(1), the theory (<ref>) becomes a U(1) gauge theory.
    The massless fields can be written as
    
        A_μ=,-a_μ,a_μ, ,  Φ_I=,-ϕ_I,ϕ_I, ,  λ^i_±=λ_±^(0)i+λ_±^(1)i,λ_±^(2)i,
        λ_±^(2)i,λ_±^(0)i-λ_±^(1)i .
    
    Then, a_μ is the U(1) gauge field, ϕ_I are neutral real
    scalar fields, λ_±^(0)i are the neutral Majorana-Weyl
    fermions and the complex Weyl fermions defined by
    
        ψ^i_L≡λ^(1)i_-+iλ^(2)i_- ,
           ψ^i_R≡λ^(1)i_++iλ^(2)i_+
    
    are the charge 2 fermions.
    The neutral fermions λ_±^(0)i do not interact with
    other fields in the low energy effective action
    (in the α' 0 limit) and will be neglected
    in what follows.
    Then, the table (<ref>) for n=1 becomes
    
        [              U(1)  charge      SO(1,1)        SO(8);          a_μ            0            2            1;          ϕ_I            0            1          8_v;        ψ^i_R            2          1_+          8_+;        ψ^i_L            2          1_-          8_-;              ]
    
    This is almost like a 2 dim QED with N_f=8 and k=2,
    [
    It is possible to introduce an external (infinitely heavy) charge
    by putting one end point of a fundamental string on the 1-brane,
    which has the unit charge. Since ψ_±^i have twice of
    this charge, we obtain k=2.]
    but couples with 8 massless scalar fields ϕ_I. Though
    ϕ_I do not have gauge interaction, they couple with the fermions
    through the Yukawa interaction
    
        S_ Yukawa=∫ d^2 x (
        y Γ^I_ij ϕ_Iψ_R^i†ψ_L^j+ h.c.)
         ,
    
    where y is a constant and (Γ^I_ij) is the invariant tensor of
    the 8_v⊗ 8_+⊗ 8_- representation of SO(8).
    (See  Appendix <ref> for an explicit form.)
    Because of this Yukawa interaction, the SU(N_f)_L× SU(N_f)_R
    chiral symmetry of 2 dim massless QED is explicitly broken and only
    the SO(8) symmetry is manifest.
    In the string theory construction, the Yukawa coupling y is not an
    independent parameter. When the scalar fields ϕ_I are canonically
    normalized, y is given as the gauge coupling e times some numerical
    constant.
    
    The partition function of the full system is given as
    
        Z_ Full   =   ∫ϕ  e^i∫ d^2x (_μϕ_I)^2Z_ QED[ϕ_I] ,
        
        
        Z_ QED[ϕ_I]   ≡   ∫ aψψ  e^iS_ QED[a_μ,ψ^i]+iS_ Yukawa[ϕ_I,ψ^i] .
    
    In the following, we will focus only on Z_ QED[ϕ_I] and
    treat ϕ_I as parameters of the system. Furthermore, we assume that
    ϕ_I are all constant, in which case
    
        V_ eff(ϕ_I)≡ -i/V_2log Z_ QED[ϕ_I] ,
    
    where V_2≡∫ d^2 x is the volume of 2 dim space-time, is
    interpreted as the effective potential for the scalar fields ϕ_I
    that correspond to the position of the 1-brane in the transverse
    8 dim space.
    Using the SO(8) symmetry, we can set ϕ_I=0 (I=1,⋯,7) and assume
    that only ϕ_8≡ϕ can take a non-zero value without loss of generality.
    According to (<ref>), Γ^8 is the 8× 8 unit matrix
    and (<ref>) is simplified as
    
        S_ Yukawa=∫ d^2 x (
        yϕ ψ_Ri^†ψ_L^i+ h.c.) .
    
    This is nothing but the mass term for the fermions considered in section
    <ref> with the identification M_0=yϕ.
    When ϕ is non-zero, the 1-brane is separated from
    the 1-plane. Since the fermions ψ^i_L and ψ^i_R
    are created by the open strings that hung on 1-plane
    with end points attached on the 1-brane, this mass M_0
    is interpreted as twice the string tension 1/(2πα')
    times the distance Y between the 1-brane and the 1-plane:
    
        M_0=yϕ=Y/πα' .
    
    
    We can also introduce a non-trivial θ angle in our system.
    The parameter θ is identified as the value of RR 0-form field
    C_0, which is normalized to be a 2π periodic scalar field.
    However, because the _2 orientifold action maps C_0 to -C_0
    (2π), the allowed value on top of the
    1-plane is either C_0=0 or C_0=π (2π).
    
    
    
     §.§ (Q,-1)-strings and the short distance potential
    
    
    Since the electric flux on the 1-brane is interpreted as
    the fundamental string, Q-strings considered in section
    <ref> should be interpreted as a bound state
    of Q fundamental strings and the 1-brane, which is called
    a (Q,-1)-string.[In general,
    a bound state of p fundamental strings and q D1-branes
    is called a (p,q)-string.] Note that the _2 orientifold action
    flips the orientation of the fundamental strings,  i.e. it maps
    a fundamental string (F-string) to an anti-fundamental string (F-string)
    in the covering space (the space before modding out the space
    by the _2 orientifold action).
    Therefore, in the covering space, we have a pair of (Q,-1)-string and
     (-Q,-1)-string that are mapped to each other by the orientifold
    action, and hence there is no net F-string charge.
    
    As we have seen in section <ref>, Q-string and (Q+k)-string
    are identified, because k unit of electric flux can be screened
    by the charge k fermions. In our k=2 case, it implies that
    the only non-trivial one is the (1,-1)-string.
    In terms of the pair of (Q,-1)-string and (-Q,-1)-string
    in the covering space, this can be understood from the fact
    that an even number of F-string - F-string
    pairs can be annihilated by reconnection. (See Figure <ref>.)
    
    
    On the other hand,
    a single  F-string - F-string pair cannot be annihilated.
    In order for the single F-string - F-string pair
    to be reconnected, the F-string has to go through the orientifold
    plane to be consistent with the orientifold action.  However,
    the  p-plane does not allow such configurations.[
    It is possible to have such configurations for the  p-plane,
    on which F-strings can end.
    ]
    
    The Q-string tension (<ref>) corresponds to the
    difference between the (Q,-1)-string tension, denoted as
    T_(Q,-1), and the 1-brane tension T_(0,-1).
    Using
    (<ref>), (<ref>), (<ref>) and
    (<ref>), the (Q,-1)-string tension is obtained as
    
        T_(Q,-1)
        =T_0-C_Qg_s/α'(Y^2/g_sα')^8/9 ,
    
    where T_0 is a constant that does not depend on θ, Q and Y,
    and C_Q is given by
    
        C_Q=
        18/π^3(
        exp(γ)/2cos(1/8(θ/2+π Q))
        )^16/9 .
    
    This expression is valid when M_0≪ e, which is equivalent to
    Y^2≪ g_sα'.
    Note that the second term in (<ref>) contains the string coupling
    with a fractional power as g_s^1/9.
    This is a non-perturbative prediction at
    short distances.
    
    
    When θ=±π, (<ref>) implies
    
        C_Q=
        18/π^3(
        exp(γ)/2cos(π/16)
        )^16/9 ,
    
    which is independent of Q. Note that
    T_(∓ 1,-1)=T_(0,-1) for θ=±π  follows from the fact
    that σ(Q=∓ 1) vanishes when θ=±π as
    discussed in section <ref>.
    This is true even when the 1-brane is far away from the 1-plane,
    in which case the well-known formula for the (p,q)-string tension
    
        T_(p,q)=1/2πα'√((p-qθ/2π)^2+q^2/g_s^2)
    
    with θ=±π implies the same conclusion.
    
    For θ=0, (<ref>) implies
    
        C_Q=
        18/π^3(exp(γ)/2)^16/9×{[             1     (Q= even); cos^16/9(π/8)      (Q= odd) ].
         .
    
    In particular, we obtain
    
        T_(1,-1)-T_(0,-1)
        =C g_s/α'(Y^2/g_s α')^8/9 ,
    
    where
    
        C≡18/π^3(exp(γ)/2)^16/9(
        1-cos^16/9(π/8)
        )
    
    is a positive constant.
    This result is in contrast to the behavior at long distances
    (α'≪ Y^2) obtained from (<ref>):
    
        T_(1,-1)-T_(0,-1)=1/2πα' g_s(√(1+g_s^2)-1)
        ≃g_s/4πα' ,
    
    where we have used g_s≪ 1.
    
    As noted in the last paragraph of section <ref>,
    the right hand side of (<ref>) vanishes in the Y 0 limit.
    Therefore, when the (1,-1)-string is placed on top of the 1-plane,
    the energy contribution from the fundamental string completely
    disappears and the tension become the same as the 1-brane
    without the electric flux. This phenomenon should not be interpreted
    as the annihilation of the F-string - F-string pair,
    as emphasized in section <ref>.
    
    The short distance potential between the (Q,-1)-string
    and the 1-plane is given by (<ref>).
    This formula is obtained by evaluating the energy density
     in (<ref>) as a function of Y related
    to the fermion mass M_0 by (<ref>). Note that
    this agrees with the effective potential V_ eff(Y)
    defined in (<ref>) up to Y independent constant terms, as
    
        Y V_ eff(Y)/ Y
        =-i/V_2M_0_M_0Z_ QED/Z_ QED
        =M_0ψ_iψ^i=Y(Y)/ Y ,
    
    where we have used (<ref>) in the last step.
    
    The expression (<ref>) shows that there is a repulsive force between
    the (Q,-1)-string and the 1-plane. It is interesting to note that
    this force is proportional to the chiral condensate in 2 dim QED
    as (<ref>) and (<ref>) implies
    
        ψ_jψ^j=πα'(Y)/ Y .
    
    
     §.§ Coleman-Weinberg potential
    
    
    Within the field theory limit, the long distance potential between the
    1-plane and the 1-brane can be calculated using a Coleman-Weinberg
    potential <cit.>. Indeed, in our system, perturbation
    theory (with respect to the gauge coupling e) can be trusted when the
    vacuum expectation value
    of the scalar field ϕ is large and satisfies e≪ M_0=yϕ,
    which corresponds to g_sα'≪ Y^2.
    In order for the field theory description to be valid,
    e^2≪ 1/α' and M_0^2≪ 1/α' have to be satisfied,
    which is equivalent to g_s≪ 1 and Y^2≪α'.
    
    The one-loop Coleman-Weinberg potential in 2 dim is written in general as
    
        V_ eff(ϕ)=V_ tree(ϕ)
        +1/2∫d^2 k/(2π)^2[
        ∑_b: bosonlog(k^2+m_b^2(ϕ)/k^2)
        -∑_f: fermionlog(k^2+m_f^2(ϕ)/k^2)
        ] ,
    
    where V_ tree(ϕ) is the tree level potential, 
    k is the momentum in the Euclidean space, and m_b and m_f
    are the mass for the bosonic field b and the fermionic field f, respectively.
    
    The integral can be evaluated as follows:
    
        ∫d^2 k/(2π)^2log(k^2+m^2/k^2) = 1/4π∫_0^Λ d k k
        log(k^2+m^2/k^2)
        
           =   1/8π[
        m^2log(Λ^2/m^2)
        +(m^2+Λ^2)log(1+m^2/Λ^2)
        ] ≃m^2/8π[
        log(Λ^2/m^2)+1
        ] ,
    
    where Λ is the cut-off scale.
    In the last step, we neglected the terms that vanishes in the
    Λ∞ limit.
    
    In our system (<ref>),
    only the fermions ψ^i have ϕ dependent mass via
    the Yukawa term (<ref>) and hence we obtain
    
        V_ eff(Y) ≃ -M_0^2/8π[
        log(Λ^2/M_0^2)+1]
        =-Y^2/8π^3α'^2[
        log(π^2α'^2Λ^2/Y^2)+1
        ]
          ,
    
    which again shows a repulsive force.
    
    
    
     §.§ The potential from the Möbius strip amplitude
    
    
    For completeness, let us review the calculation of the potential
    between an orientifold plane and an anti D-brane using string theory.
    This calculation takes into account the stringy tower and it is
    valid at long distances, where Y^2 is of order α' or larger
    and the effective field theory description breaks down.
    
    Consider a system with an  p-plane and a  p-brane
    placed in parallel with distance Y. The potential between
    the  p-plane and  p-brane is given by
    an open string vacuum amplitude corresponding to
    the world-sheet with topology of the Möbius strip at one-loop level.
    The open string one-loop amplitude can be interpreted
    as a closed string tree level amplitude and the potential is
    generated by the exchange of graviton, RR fields and so on.
    The explicit calculation of the Möbius strip amplitude
    for a system with parallel  p-plane and  p-brane
    was carried out in <cit.>, and the result is
    
        V^_ eff(Y)   =   ∓∫^∞_0dt/2t(8π^2α't)^-p+1/2
        e^-t(2Y)^2/2πα'F_O(t) ,
    
    where
    
        F_O(t)≡Z^0_1(2it)^4Z^1_0(2it)^4/η(2it)^8Z^0_0(2it)^4 .
    
    The functions η(it) and Z^α_β(it) are defined as
    
        η(it)   =   
        q^1/24∏_m=1^∞(1-q^m) ,
        
         Z^0_0(it)   =   
        q^-1/24∏_m=1^∞(1+q^m-1/2)^2 ,
        
         Z^0_1(it)   =   
        q^-1/24∏_m=1^∞(1-q^m-1/2)^2 ,
        
         Z^1_0(it)   =   
        2q^1/12∏_m=1^∞(1+q^m)^2 ,
    
    where q=e^-2π t.
    They satisfy
    
        η(it)=t^-1/2η(i/t) ,   
         Z^α_β(it)^4=Z^β_α(i/t)^4 ,
    
    and
    
        Z^0_0(it)^4-Z^0_1(it)^4-Z^1_0(it)^4=0 .
    
    One can check that
    F_O(t) is a positive monotonically increasing function that interpolates
    F_O(t=0)=0 and F_O(t=∞)=16. For small t, it behaves as
    F_O(t)≃ 2^8 t^4 and hence the integral over t in (<ref>)
    is convergent when p<7.
    The potential is positive (negative) monotonically decreasing (increasing)
    function of Y for Op^--p system (Op^+-p system, respectively).
    In particular, for the 1-1 system we find a repulsion.
    See Figure <ref> for its shape for the 1-1 system.
    
    
    
    A few comments are in order.
    The function F_O can be written as
    
        F_O(t)= ∑_b: bosone^-t(2πα')m_b^2
        - ∑_f: fermione^-t(2πα')m_f^2 ,
    
    and the expression (<ref>) is interpreted as a sum of
    the contributions from all the fields in the spectrum with
    
        m_b^2(ϕ)=m_b^2+(2Y/2πα')^2 ,   
        m_f^2(ϕ)=m_f^2+(2Y/2πα')^2
    
    in the Coleman-Weinberg potential ((<ref>) for the p=1 case).
    
    In particular, F_O(t=∞)=16 corresponds to the contribution
    from the 16 complex massless Weyl fermions (ψ_R^i and
    ψ_L^i in our case).
    The contributions from these massless modes are dominant at short
    distances and the potential smoothly approaches the Coleman-Weinberg
    potential ((<ref>) for p=1) obtained in field theory.
    
    One can easily show that the potential (<ref>) behaves as
    
        V^_ eff(Y)≃∓ 2^p(8π^2α')^-p+1/2(2πα'/Y^2)^7-p/2Γ(7-p/2)
    
    for α'≪ Y^2. This is interpreted as the potential
    generated by the exchange of supergravity fields.
    It is known that there is no force between an  p-plane
    and a Dp-brane due to supersymmetry. When we replace
    the Dp-brane with the  p-brane, the sign of the RR charge
    is flipped from positive to negative and the force balance
    is broken. Because  p-plane ( p-plane) has negative
    (positive) RR-charge, the  p-brane is repelled from 
    (attracted to) the  p-plane ( p-plane, respectively).
    
    
    
    § SUMMARY AND DISCUSSION
    
    
    In this paper, we first considered an interesting variant of 2 dim
    QED with N_f flavours of fermions with charge k. We found a rich
    vacuum structure of k vacua, parametrized by
    the vacuum expectation value of (ψ_Rj^†ψ_L^i)
    as an order parameter. We also studied the Q-string tension
    including the quark mass.
    
    In section <ref>, we used the results of the first part to
    learn about non-perturbative string dynamics. The vacuum energy
    as a function of the quark mass can be interpreted as
    the potential between the 1-plane and the 1-brane.
    In addition, we could also use the energy density for the Q-string
    to learn about the dynamics of the (Q,-1)-string placed
    near the 1-plane.
    We found that the (Q,-1)-string tension (<ref>)
    admits a non-perturbative dependence on the
    string coupling of the form g_s^1/9.
    
    One might think that the existence of k vacua would imply domain walls
    that interpolate between these vacua.
    If such an object exists, it would make the Q-string states unstable
    by the creation of domain wall - anti domain wall pairs.
    However, it is not allowed in our case. As we have seen
    in section <ref>, the vacuum is parametrized by θ,
    which is an eigenvalue of the operator 2πΠ_A.
    Because of the Gauss law equation (<ref>), we cannot have
    an object that interpolates different values of θ
    in x^1-∞ and x^1+∞.[
    It is possible to set a boundary condition that the scalar field
    φ in the bosonized description (<ref>)
    approaches different values at the boundaries x^1-∞
    and x^1+∞. However, in that case, the electric flux F_01
    will be inevitably induced (at least) at one of the boundaries, due to
    the Gauss law equation (<ref>). Therefore, it does not describe a
    domain wall that connects two vacua related by the discrete axial
    symmetry.]
    In terms of string theory, if one wants to consider a configuration
    with a (Q,-1)-string at x^1-∞ and 
     (Q+1,-1)-string at x^1+∞, there must be a fundamental
    string attached on it. Since the other end point of the fundamental
    string should escape to spatial infinity, it is infinitely heavy.
    
    Part of the discussion of the previous sections could be used in
    higher dimensional cases. 
    Consider a system of  p-plane and a  p-brane. Similar to
    our 2 dim case, the low energy theory of this system is described by a
    (p+1) dim QED with charge 2 fermions coupled with neutral scalar
    fields via Yukawa coupling.
    The expressions analogous to (<ref>) would also hold in higher
    dimensional cases. In general, we expect that the vacuum expectation
    value of the fermion bilinear operator is non-zero when
    the fermion mass is turned on and hence the relation
    (<ref>) implies a force between them.
    This is a typical relation that connects a vacuum expectation value
    of an operator in quantum field theory and a quantity that characterizes
    the brane dynamics. It would be interesting to explore the
    generalization of such relations and find more fruitful interplay
    between quantum field theory and string theory.
    
    
    
     §.§ Acknowledgements
    
    
    We thank Sinya Aoki, Aleksey Cherman, Satoshi Iso, Zohar Komargodski,
    Erich Poppitz, Satoshi Yamaguchi, Piljin Yi and Yang Zhou for
    discussions and helpful comments.
    We also appreciate useful discussions during the workshops
    “Simons Summer Workshop 2017” held at the Simons Center for Geometry and
    Physics at Stony Brook and
    “New Frontiers in String Theory 2018” Ref. YITP-T-18-04
    held at the Yukawa Institute for Theoretical Physics, Kyoto University.
    SS is also grateful to the participants of the workshops
    “East Asia Joint Workshop on Fields and Strings 2018” held
    at KIAS and “KEK Theory Workshop 2018” held at KEK, in which
    he made presentations on this work and got
    a lot of valuable feedback.
    The work of AA has been supported by STFC grant ST/P00055X/1.
    The work of SS has been supported by JSPS KAKENHI (Grant-in-Aid for Scientific
    Research (C)) Grant Number JP16K05324.
    
    
     
    
    
    § THE EXPLICIT FORM OF Γ^I_IJ
    
    
    In this section,
    we list an explicit form of Γ^I_ij used in (<ref>).
    The 8× 8 matrices Γ^I=(Γ^I_ij)
    are related to the SO(8) gamma matrices Γ^I as
    
        Γ^I=,Γ^I,^tΓ^I, ,
    
    and an explicit form is given as
    
        [         Γ^1=ϵ⊗ϵ⊗ϵ ,     Γ^2=1_2⊗τ_1⊗ϵ ,     Γ^3=1_2⊗τ_3⊗ϵ ,    Γ^4=τ_1⊗ϵ⊗ 1_2 ,;    Γ^5=τ_3⊗ϵ⊗ 1_2 ,    Γ^6=ϵ⊗ 1_2⊗τ_1 ,    Γ^7=ϵ⊗ 1_2⊗τ_3 , Γ^8=1_2⊗ 1_2⊗ 1_2 , ]
    
    where ϵ≡ iτ_2 
    and τ_j (j=1,2,3) are the Pauli matrices.[
    See p.288 of <cit.>.]
    
    § ABELIAN BOSONIZATION, CHIRAL CONDENSATE AND ENERGY DENSITY
    
    
    In this section, we outline the calculation of the chiral condensate
    and the vacuum energy density in the strong coupling regime
    for the k=1 case, using the Abelian bosonization.
    
    
    
     §.§ Abelian bosonization
    
    
    Here, we briefly review the Abelian bosonization.[
    See, e.g., <cit.> for a review]
    The idea is to bosonize the N_f flavours of fermions one by one
    and map the system to a theory with
    N_f real scalar fields. This description has a disadvantage
    that the SU(N_f)_L× SU(N_f)_R symmetry is not manifest.
    But, it is often used because it is simpler and useful for calculations.
    
    Let us consider 2 dim QED with N_f massive Dirac fermions of charge k=1.
    The bosonization rules are <cit.>
        ψ_iγ^μψ^i
           =   1/√(π)ϵ^μν_νφ_i ,
        
        ψ_iψ^i
           =   -c μ_μcos(2√(π)φ_i) ,
    
    where the index i=1,2,⋯,N_f in the left hand side is not summed over,
    φ_i are canonically normalized real scalar fields and  c is
    a constant
    
        c≡exp(γ)/2π .
    _μ denotes the normal ordering with respect to an arbitrary
    scale μ. Useful identities (see <cit.>) are
    
        _μ[(Π^2+(_1φ)^2)]
           =   _μ'[(Π^2+(_1φ)^2)]
        +1/8π(μ'^2-μ^2) ,
        
        _μ[ m^2φ^2 ]
           =   _μ'[ m^2φ^2 ]
        -m^2/8πlog(μ'^2/μ^2) ,
    
    where Π is the canonical momentum operator conjugate to φ,
    and
    
        _μ e^iβφ=(μ'/μ)^β^2/4π_μ'e^iβφ .
    
    (<ref>) implies
    
        μ_μcos(2√(π)φ)=μ'_μ'cos(2√(π)φ) .
    
    
    Then, after integrating out the gauge field, the bosonized Hamiltonian (density)
    for the N_f flavour massive Schwinger model is
    
        =   _μ[
        ∑_i=1^N_f(Π_i^2+(_1φ_i)^2)
        +e^2/2π(∑_i=1^N_fφ_i)^2
        -c μ M_0∑_i=1^N_fcos(2√(π)φ_i
        +θ/N_f)
        ]
        
              +N_f/8π(μ^2-e^2/πlogμ^2)
         ,
    
    where M_0 is the fermion mass.
    Note that the Hamiltonian (<ref>) does not depend on the
    renormalization scale μ because of
    the relations (<ref>), (<ref>) and (<ref>).
    This is the reason that we put the constant term in (<ref>).
    
    In the Hamiltonian (<ref>), the 2π periodicity of θ
    is not manifest. But, it is easy to see that the Hamiltonian is
    invariant under 2π shift of θ together with the transformation
    
        2√(π)φ_1      2√(π)φ_1-2π/N_f+2π ,
        
        
        2√(π)φ_i      2√(π)φ_i-2π/N_f ,   (i=2,3,⋯,N_f) .
    
    Therefore, physics is invariant under the 2π shift of θ.
    
    
    
     §.§ Chiral condensate and energy density
    
    
    We are interested in the cases with e^2≫μ M_0 and try to integrate
    out the heavy combination ∑_i=1^N_fφ_i first.
    To do this, it will be convenient to introduce a matrix notation:
    
        Φ≡(φ_1,φ_2,⋯,φ_N_f) ,
    
    and
    
        h   ≡   (T^0Φ) ,
        
        
        l_i   ≡   (T^i Φ)
         ,   (i=1,2,⋯,N_f-1) ,
    
    where T^0≡1/√(N_f)1_N_f,
    and
     T^i(i=1,2,⋯,N_f-1) are the generators
    of the Cartan subalgebra of su(N_f) (diagonal
    traceless Hermitian matrices) normalized as
    
        (T^iT^j)=δ^ij .
    
    Then, Φ can be expanded as
    
        Φ=hT^0+∑_i=1^N_f-1l_iT^i ,
    
    and the Hamiltonian becomes
    
        =   _μ[
        (Π_Φ^2+(_1Φ)^2)
        +e^2/2π(Φ)^2
        -c μ M_0(cos(2√(π)Φ
        +θ/N_f))]
        
              +N_f/8π(μ^2-e^2/πlogμ^2)
        
           =   _μ[(Π_h^2+(_1h)^2)
        +∑_i=1^N_f-1(Π_l_i^2+(_1l_i)^2)
        +m_h^2/2h^2
        +N_f/8π(μ^2-e^2/πlogμ^2)
        
              
        -c μ M_0/2e^i√(4π/N_f)h+iθ/N_f(e^i 2√(π)∑_i=1^N_f-1l_iT^i)+ h.c.] ,
    
    where we have defined
    the mass scale of the heavy component h as
    
        m_h^2≡e^2N_f/π .
    
    
    For e^2≫μ M_0, the heavy field h can be treated as a free
    massive scalar field of mass m_h and the operator
    _μ e^i√(4π/N_f)h in the Hamiltonian (<ref>)
    can be replaced with its vacuum expectation value in the low energy
    effective theory for the light fields l_i.
    To evaluate the vacuum expectation value, it is convenient to choose
    the scale of the normal ordering to be m_h using the formula
    
        _μ e^i√(4π/N_f)h
        =(m_h/μ)^1/N_f_m_he^i√(4π/N_f)h ,
    
    which follows from (<ref>). Using (<ref>), we obtain
    
        _μ e^i√(4π/N_f) h
        =(m_h/μ)^1/N_f .
    
    Similarly, using (<ref>), (<ref>) and
    _m_h[(Π_h^2+(_1h)^2+m_h^2h^2)]=0,
     we get
    
        _μ[(Π_h^2+(_1h)^2+m_h^2h^2
        )]=
        1/8π(m_h^2-μ^2-m_h^2log(m_h^2/μ^2)
        ) .
    
    
    Then, the effective Hamiltonian for the light fields l_i is
    
        _l
           =   _μ[∑_i=1^N_f-1(Π_l_i^2+(_1l_i)^2)
        -c μ M_0/2(m_h/μ)^1/N_f
        e^iθ/N_f(
        e^i 2√(π)∑_i=1^N_f-1l_iT^i)+ h.c.]
        
              
        +N_f/8π(μ^2-e^2/πlogμ^2)
        +1/8π(m_h^2-μ^2-m_h^2log(m_h^2/μ^2)
        )
        
           =   _μ[∑_i=1^N_f-1(Π_l_i^2+(_1l_i)^2)
        -c μ M_0(m_h/μ)^1/N_fcos(θ/N_f)
        (
        N_f-2π∑_i=1^N_f-1l_i^2
        )+(l_i^3)
        ]
        
              
        +N_f-1/8πμ^2
        +m_h^2/8π(1-log m_h^2)
         .
    
    In the last expression in (<ref>), we expanded _l
    with respect to l_i to extract the mass term for l_i.
    Here, we have assumed that l_i=0 is the vacuum
    configuration. This is true for -π≤θ≤π,
    but not for general θ. To see the vacuum configuration
    for general θ, let us assume
    that the vacuum configuration of l_i satisfies
    
        exp(i 2√(π)∑_i=1^N_f-1l_iT^i)=e^iα1_N_f ,
    
    where α∈ and 1_N_f is the unit matrix of size N_f,
    so that the flavour symmetry is not broken. Because the left hand side
    of (<ref>) is a diagonal element of SU(N_f), α has to be
    of the form α=2πn/N_f (n=1,⋯,N_f). Then, the potential
    term for l_i in the first line of (<ref>) is given by a positive constant
    times -cos((θ+2π n)/N_f), which is minimized when n is chosen
    so that it satisfies -π≤θ+2π n≤π. Then, expanding
    the fields l_i around this configuration is equivalent to (<ref>)
    with θ replaced by
    
        θ≡θ-2π[θ+π/2π] ,
    
    where [x] is the floor function
    that gives the greatest integer less than or equal to x.
    By definition, the value of θ is restricted to be in the
    interval -π≤θ≤π and ensures the 2π periodicity
    of the θ parameter.
    Then, the mass scale of the light fields l_i is given by
    
        m_l^2≡ 4π c μ
        M_0(m_h/μ)^1/N_fcos(θ/N_f) .
    
    
    In order to find the ground state, we employ the variational method
    used in <cit.>. We first assume that the ground state is
    given as |0;μ⟩, which is the state annihilated by
    the annihilation operators defined by the fields l_i with
    the mass scale μ in the Schrödinger picture
    (see <cit.>) and then find the value of μ
    that minimizes the expectation value of the Hamiltonian.
    Although we will not try to prove that there is no state
    with lower energy, it gives a candidate for the ground state.
    In fact, we will show that the expectation value of the fermion
    bilinear operator with respect to the state |0,μ⟩,
    denoted as ψ_iψ^i_μ≡⟨0,μ|ψ_iψ^i|0,μ⟩, reproduces
    the results in <cit.>.
    
    The expectation value of the Hamiltonian (<ref>)
    with respect to the state |0,μ⟩ is evaluated as
    
        _μ[∑_i=1^N_f-1(Π_l_i^2+(_1l_i)^2)
        ]+N_f-1/8πμ^2_μ
        =N_f-1/8πμ^2 ,
    
        _μ[
        -c μ M_0/2(m_h/μ)^1/N_f
        e^iθ/N_f(
        e^i 2√(π)∑_i=1^N_f-1l_iT^i)+ h.c.]_μ
        =-N_f/4π m_l^2 ,
    
    and
    
        _l_μ=
        N_f-1/8πμ^2-N_f/4πm_l^2
        +m_h^2/8π(1-log m_h^2) .
    
    Then, it is easy to see that the value of μ
    that minimizes this expression satisfies
    
        μ^2= m_l^2 ,
    
    which implies
    
        m_l= (4π c cos(θ/N_f)
         M_0)^N_f/N_f+1m_h^1/N_f+1 ,
    
    and
    
        _l_μ=m_l=
        -N_f+1/8π(4π
         c cos(θ/N_f)M_0)^2N_f/N_f+1
        m_h^2/N_f+1
        +m_h^2/8π(1-log m_h^2)
         .
    
    The last term of (<ref>) can be omitted because it doesn't depend
    on θ and M_0. Therefore, the energy density 
    is obtained as
    
        =-N_f+1/8π(4π
         c cos(θ/N_f)M_0)^2N_f/N_f+1
        m_h^2/N_f+1 .
    
    
    (<ref>) with the condition (<ref>)
    corresponds to the vacuum expectation value
    of the fermion mass term and therefore we conclude
    
        M_0ψ_iψ^i
        =-N_f/4π(4π
         c cos(θ/N_f)M_0)^2N_f/N_f+1
        m_h^2/N_f+1 ,
    
    which agrees with <cit.>.
    
    
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