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1 TK-NOTE/04-04 since February 8, 004 (September st, 00 last update May 5, 006 Two-dimensional Gauge Field Theory and Mirror Symmetry School of Physics, Korea Institute for Advanced Study, 07-43, Cheongnyangni -dong, Dongdaemun-gu, Seoul 30-7, KOREA Comments Appendix A SUSY algebra (005 6/7 Appendix A N (, supersymmetry (005 7/4 Appendix A N (, supersymmetry (005 8/05

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3 Contents I N = (, Gauged Linear Sigma Model I. Lagrangian I. Renormalization I.3 Classical vacua I.3. Various phases I.3. Classical limit and symmetry restoration I.4 Low energy effective theories I.5 Calabi-Yau/Landau-Ginzburg correspondence I.6 Example I.6. Projective space CP N I.6. O( N bundle on CP N I.6.3 CY hypersurface in CP N I.6.4 Degree l hypersurface in CP N I.7 O( N + l bundle on CP N [l] I.7. Field configuration and supersymmetric vacuum manifold I.7. Calabi-Yau phase I.7.3 Orbifold phase I.7.4 Singularity phase I.8 Squashed GLSM I.A General formulation I.B R-symmetry

4 ii CONTENTS I.B. Charge assignment I.B. R-invariance I.B.3 R-symmetry and U( current in N = (, SCFT I.C Weighted projective space II N = (, Landau-Ginzburg Theory 49 II. Critical point II. Renormalization group flow II.3 Central charges of CFT II.4 Chiral ring II.5 Three significant concepts II.5. Versal deformation II.5. Modality II.5.3 Poincaré polynomial II.6 Modality zero singularities ADE classification II.A Data for ADE II.B Highest charged state II.B. quintic hyperdurface III Two-dimensional N = (, Superconformal Theory 67 III. Superconformal algebras III. NS chiral primary states and R ground states III.3 Spectral flow III.4 Witten index, Poincaré polynomial IV CY/LG Correspondence 77 IV. CY geometry IV. LG minimal model IV.3 Correspondence V Calabi-Yau Data from LG Superpotentials 83

5 CONTENTS iii V. CY data from A n type LG superpotentials V. A few examples V.. CY 3-fold (k i + = (5,5,5,5,5 (quintic hypersurface in CP V.. CY 3-fold (k i + = (8,8,4,4, V..3 K3 surface (k i + = (4,4,4, V.3 CY data from D k type LG superpotentials VI Mirror Symmetry T-duality of GLSM 99 VI. T-duality theories on tori VI. General formulation VI.. Vortex and fermionic zero modes VI.3 Example of mirror pair complex projective space VI.4 Period integral VI.5 Mirror geometry descriptions for various theories VI.6 Mirror dual of O( N bundle on CP N VI.6. mirror description in terms of homogeneous coordinates VI.6. mirror description in terms of local coordinates VI.7 Mirror dual of CP N [N] VI.7. Mirror LG description VI.7. Mirror geometry VI.8 Mirror dual of CP N [l] VI.8. mirror description in terms of homogeneous coordinates VI.8. mirror description in terms of local coordinates VI.9 Mirror dual of O( N + l bundle on CP N [l] VI.9. Field configuration VI.9. Mirror Landau-Ginzburg descriptions VI.9.3 Mirror geometry descriptions VI.9.4 Return to the gauged linear sigma model VI.9.5 Summary and Discussions

6 iv CONTENTS VI.A Linear dilaton CFT and Liouville theory A Conventions 4 A. N = supersymmetry in four dimensions A. N = (, supersymmetry in two dimensions A.. Spinors in this notes A.. Weyl representation in two dimensions A..3 Differential operators A..4 Superfields A..5 Gauging of phase shifts A.3 N = (, supersymmetry A.3. A definition of N = (, supersymmetry A.3. (, superfields reduced from (, superfields A.3.3 Supersymmetry transformations A.3.4 Superspace measure A.4 N = (0, supersymmetry A.4. Differential operators A.4. (0, superfields A.5 N = (0, supersymmetry A.5. A definition of (0, supersymmetry A.6 Propagator and Fourier Transformation A.7 Left/right moving modes B Calabi-Yau Data from Gepner Construction 85 B. Data C Field Theory Realization of SCFT 95 C. Landau-Ginzburg minimal model D Topological Sigma Model 99 D. Supersymmetric nonlinear sigma models

7 CONTENTS v D. Definition of topological twist D.3 Topological twist D.4 Topological gauged linear sigma models D.5 Example Topological GLSM for quintics D.6 Example Topological GLSM for resolved conifold D.7 Problems D.8 A-type branes, B-type branes, boundary CFT

8 Chapter I N = (, Gauged Linear Sigma Model

9 N = (, Gauged Linear Sigma Model Introduction Gauged linear sigma model (GLSM [3] FI parameter phase FI parameter [3] GLSM D = 4, N = supersymmetric abelian gauge theory with chiral matter D =, N = (, dimensional reduction D = spinor irreducible representation Majorana- Weyl chiral field twisted chiral field (twisted chiral field topological field theory T-dualized theory I. Lagrangian superfield level gauge non-abelian (complex U( gauge symmetry gauge coupling e, chiral superfield Φ i U( gauge charge Q i gauge parameter Λ(x,θ,θ (chiral superfield Φ i = e iqiλ Φ i, Φ i = e +iqiλ Φ i, (I..a V = V + i(λ Λ, Σ = Σ. (I..b gauge coupling e field strength normalization gauge coupling Lagrangian component field real U( gauge Wess-Zumino gauge complex U( gauge Wess-Zumino gauge constraint gauge parameter Λ fermion auxiliary field scalar field real part non-zero chiral mutilplet Φ i gauge multiplet V component field gauge δφ i = iq i α φ i, δψ i = iq i α ψ i, δf i = iq i α F i, (I..a δv m = m α, δσ = δλ = δd = 0, (I..b δ(d m φ i = iq i α(d m φ i. (I..c α gauge parameter D m φ i (I..7a gauge covariant derivative GLSM Lagrangian superfield S = d xl GLSM, π { ( L GLSM = d 4 θ e Σ a Σ a + Φ i e P } a Qa i Va Φ i a a i ( ( + d θ W GLSM (Φ i + (h.c. + d θ W(Σ + (h.c. L g + L K + L W + L D,θ. (I..3a (I..3b (I..3c

10 I. Lagrangian 3 index a U( gauge field labeling index i chiral superfield labeling L g gauge coupling overall normalize L g = e d 4 θ ΣΣ e (F 0 = 4e F mnf mn. (I..4 L g = 4 F mnf mn normalization v m ev m twisted superpotential W(Σ twsited chiral superfield FI parameter r Theta angle θ W(Σ = Στ, τ = r iθ, (I..5 W GLSM (Φ chiral superfields Φ holomorphic function superpotential superfield Lagrangian L GLSM component field L g = a L K = i [ ] (D a + (F 0,a σ a ( 0 σ a + iλ +,a ( 0 λ +,a + iλ,a ( 0 + λ,a, (I..6a e a [ F i φ i ( 0 φ i + iψ,i ( 0 + ψ,i + iψ +,i ( 0 ψ +,i + a Q a i { v a,0 ( ψ,i ψ,i ψ +,i ψ +,i + v a, ( ψ,i ψ,i + ψ +,i ψ +,i L W = i σ a ψ,i ψ +,i σ a ψ +,i ψ,i + i(v a,0 φ i 0 φ i + v a, φ i φ i i φ i (λ,a ψ +,i λ +,a ψ,i + i } φ i (ψ +,i λ,a ψ,i λ +,a + φ i { Q a i D a }] Q a i Q b i(v a,m vb m + σ a σ b, (I..6b a,b a { F i W GLSM φ i + W } GLSM ψ,i ψ +,j + (h.c., L D,θ = φ i φ j a { } r a D a + θ a F 0,a. (I..6c gauge covariant derivative D m φ i = ( m + iq i v m φi, D m ψ i = ( m + iq i v m ψi, etc. (I..7a gauge coupling e Lagrangian normalize L g = 4 F mnf mn normalization D m φ i = ( m + iq i ev m φi, (I..7b Lagrangian kinetic term interaction terms L GLSM = L kin + L int interaction terms L int L int = a + i { r a D a + θ a F 0,a + [ e Da a } + i Q a i a Z dθ + dθ θ + θ = F i + i { F i W GLSM φ i v a,0 ( ψ,i ψ,i ψ +,i ψ +,i + v a, ( ψ,i ψ,i + ψ +,i ψ +,i + W } GLSM ψ,i ψ +,j + (h.c. φ i φ j

11 4 N = (, Gauged Linear Sigma Model σ a ψ,i ψ +,i σ a ψ +,i ψ,i + i(v a,0 φ i 0 φ i + v a, φ i φ i i(v a,0 0 φ i φ i + v a, φ i φ i i φ i (λ,a ψ +,i λ +,a ψ,i + i φ i (ψ +,i λ,a ψ,i λ +,a + D a φ i ] i Q a i Q b i φ i (v a,m vb m + σ a σ b. a,b (I..8 auxiliary field D a, F i on-shell condition scalar field potential energy density U(φ,σ e D a = r a Q a i φ i, a i U(φ,σ = { e ( a r a a i F i = W GLSM φ i, (I..9a Q a i φ i } + i W GLSM φ i ( + σ a σ b Q a i Q b i φ i. a,b i (I..9b SUSY vacuum potential energy density fermion gauge field scalar field vacuum fermion gauge field Lorentz symmetry vacuum energy low theory potential density scalar field fermion gauge field Supersymmetric vacua D a = 0, U(φ,σ = 0, du = 0, (I..0 I. Renormalization low energy theory Q a i = 0, (I.. i FI parameter r a one-loop finite (see, ref. [3] section 3; ref. [8] section 3 CY condition low energy theory non-trivial IR fixed point N = (, supersymmetric conformal theory NLSM target space CY manifold FI parameter interaction term component field cut-off theory cut-off scale Lagrangian L GLSM D field potential energy U = R d x U L GLSM ( e D + D Q i φ i,0 r 0 i (I..

12 I. Renormalization 5 U( gauge symmetry gauge coupling e mass dimension + super-renormalizable coupling r 0 cut-off scale bare FI parameter D φ i,0 bare fields cut-off Λ UV scale µ effective theory cut-off φ i,0 = φ i,r + φ, r 0 = r R + r (I..3 effective (µ k Λ UV scale field integrate out index R effective scale µ fields coupling φ one-loop correction 3 φ = φ i (xφ i (x = ΛUV µ d k π (π i k = log appendix A.6 (I..3, (I..4 (I.. L GLSM ( e D + D Q i φ i,0 r 0 i ( e D + D r = i i Q i φ = i Q i φ i,r r R Q i log ( ΛUV µ = e D + D ( i ( ΛUV µ Q i ( φ i,r + φ (r R + r (I..4 (I..5. (I..6 effective theory Lagrangian microscopic (cut-off scale Lagrangian r FI parameter r 0 = r R + i Q i log cut-off theory bare FI parameter Λ UV (I..7 ( ΛUV µ (I..7 log mass dimension dynamical scale parameter Λ r 0 = i Q i log ( ΛUV constant part 4 renormalized FI parameter r(µ scale parameter µ dynamical scale parameter Λ r(µ = i Λ ( µ Q i log Λ (I..8 (classical theory dimensionless parameter r free parameter quantum correction free parameter mass dimension Λ parameter mass dimension dimensional transmutation Gross-Neveu model [43] dynamical scale Λ RG invariant dynamical scale µ Λ µ Λ IR cut-off ( low energy effective theory 3 factor π action (I..3a 4 FI parameter r R constant part corrected part r(µ

13 6 N = (, Gauged Linear Sigma Model scale parameter µ dynamical scale Λ complexified FI parameter τ(µ = r(µ iθ/π τ(µ = { i Q i log(µ/λ iθ r iθ if if i Q i 0 i Q i = 0 (I..9 [54] µ energy scale µ Λ limit low energy limit µ Λ UV high energy limit i Q i > 0 r(µ { if µ ΛUV 0 if µ Λ (I..0 5 i Q i > 0 GLSM asymptotically free FI parameter r(µ IR limit NLSM coupling constant g(µ r(µ /g(µ FI parameter RG flow (I..8 coupling constant g(µ β function µ µ r(µ µ µ g(µ = g(µ µ µ g(µ, µ g(µ β(g = g(µ3 µ 4π Q i. i (I.. FI parameter high energy limit target space target space quantum fields target space (target space free theory asymptotically free theory high energy limit target space low energy limit dynamical compactification (004 3/0 GLSM IR limit NLSM [53] chapter 5.4 CP N model i Q i = 0 FI parameter one-loop i Q i = 0 CY condition CY model R A -symmetry anomalous (path integral measure R A -symmetry supersymmetric vacua FI parameter r a vacua low energy theory low energy theory FI parameter phase FI parameter phase space phase space 6 5 TK-NOTE/0-04 [6] RG flow 6 Morrison-Plesser [8] section 3.

14 I.3 Classical vacua 7 Kähler cone K V phase space FI parameter region low energy theory Kähler manifold V NLSM Kähler form FI parameter phase space region 7 SUSY cone K c [8] cone SUSY condition (I..0 region K V K c CY condition K c phase space multi abelian gauge symmetry Lagrangian low energy theory example single U( gauge theory multi gauge theory I.3 Classical vacua low energy effective theory vacua supersymmetric theory SUSY condition (I..0 FI parameter r a CY condition (I.. 8 I.3. Various phases Calabi-Yau sigma model FI parameter r a massless classical field Higgs mechanism mass term chiral superfield chiral superfield vector superfield massive U( gauge symmetry massive fields integrate out low energy effective theory supersymmetric nonlinear sigma model 7 Morrison-Plesser [8] compact toric geometry V GLSM low energy theory NLSM on V region in phase space K V additional fields non-compact toric geometry V + vector bundle on V NLSM phase space Kähler cone V Kähler cone 8 P CY condition (I.. one-loop low energy theory CFT i Q i > 0 axial R-symmetry anomalous [3, 54]

15 8 N = (, Gauged Linear Sigma Model CY condition (I.. CY sigma model 9 superconformal sigma model ( Greene [38] CY sigma model r a Kähler cone CY phase Orbifold model FI parameter supersymmetric vacua multi gauge theory single gauge theory r < 0 CY sigma model orbifold model sigma model chiral superfield Higgs mechanism chiral superfield vector superfield massive U( gauge symmetry discrete symmetry massive fields integrate out effective theory orbifold model Landau-Ginzburg (LG theory LG theory nontrivial infrared fixed point minimal model LG theory sigma superpotential dynamics minimal model superfield chiral primary field central charge LG model [78, 04, 7, 56, 86] phase orbifold phase others CY phase orbifold phase Higgs branch FI parameter r = 0 Coulomb branch chiral superfield vector superfield phase CY phase LG phase CY/LG correspondence phase multi gauge theory FI parameter FI parameter supersymmetric vacua CY sigma model LG orbifold model hybrid model phase hybrid phase I.3. Classical limit and symmetry restoration CY phase orbifold phase FI parameter r a vacuum manifold ( VEVs FI parameter vacuum manifold parameter (CY condition (I.. parameter (I..8 dynamical scale parameter Λ vacuum manifold SUSY vacuum fluctuation field theory vacuum manifold Coleman symmetry restoration [7] ( + -dimensional spacetime theory continuous symmetry breaking no-go theorem ( + -dimensional 9 N = sigma model target space Kähler manifold [6] Ricci-flat sigma model one-loop finite [3] Ricci-flat Kähler manifold Calabi-Yau manifold target space sigma model Calabi-Yau sigma model

16 I.4 Low energy effective theories 9 spacetime massless field quantum theory well-defined (propagater IR divergence global symmetry breaking massless Goldstone modes 0 ( gauge symmetry breaking Higgs mechanism massless Goldstone mode field Coleman s no-go theorem GLSM SUSY vacua fluctuation theory NLSM classical limit classical limit vacuum manifold fluctuation mode vacuum manifold FI parameter r r CY phase NLSM finite r NLSM I.4 Low energy effective theories GLSM one-loop FI parameter parameter dynamical scale Λ parameter FI prameter RG flow (I..8 r 0 classical theory asymptotic free low energy RG flow FI parameter scale µ dynamical scale Λ r(µ = 0 energy scale FI parameter FI parameter NLSM target space Kähler form RG flow FI parameter I.5 Calabi-Yau/Landau-Ginzburg correspondence GLSM FI parameter low energy theory GLSM phase transition CY/LG correspondence CY sigma model LG orbifold model CFT level central charge partition function massless field LG orbifold model ring structure R cc, R ac CY complex structure, (complexified Kähler class chapter IV 0 Coleman [7] 973

17 0 N = (, Gauged Linear Sigma Model I.6 Example I.6. Projective space CP N toric geometry GLSM for non-comact CY manifold K V K c toric variety projective space V = CP N low energy effective theory field space GLSM GLSM toric variety field theory [3] GLSM for CP N configuration Φ i = ( S,S,,S N, Qi = (,,,, (I.6.a { L = d 4 θ N e ΣΣ + } ( S i S i e V + d θ W(Σ + (h.c., W(Σ = τ Σ. (I.6.b auxiliary D field potential energy density (contained only by scalar fields N e D = r s i, (I.6.a U(s,σ = e D + σ Kähler cone K V SUSY cone K c N s i, (I.6.b K V = K c = { r > 0 }. (I.6.3 classical vacua SUSY vacua (I..0 K V D = 0 r = U(s i,σ = 0 N s i, (I.6.4a N 0 = σ ( s i. (I.6.4b σ = 0 vacuum manifold M = {(s,s,,s N C N r = N }/ s i > 0 U( = CP N (I.6.5 (U( gauge symmetry vacuum manifold field non-zero classical field φ φ quantum fluctuation φ, φ φ, vacuum manifold M (tangent modes (non-tangent modes

18 I.6 Example vacuum manifold M M = { } (s i,,s N C N Fa (s,,s N = 0 for a =,,k. classical fields s i vev s i, tangent modes s i, non-tangent modes ŝ i vev, tangent modes F a ( s,, s N = 0, N s i F a(s j s i M = 0. F a (s i = 0 tangent modes non-zero s F a M 0 N fields s i N k non-tangent modes ŝ i GLSM vacuum manifold M (I.6.5 classical fields s i s i = s i + s i + ŝ i, F(s i = r N s i, σ = σ + σ + σ, σ = 0, (I.6.6a F( s i = 0, N s i F s i M = 0, (I.6.6b N s i s i = 0, σ + σ σ. (I.6.6c σ + σ σ σ σ massive field potential energy density { U = e r { = e N } si + s i + ŝ i ( N + σ si + s i + ŝ i N ( si + ŝ i + Re[ s i ŝ i + ŝ i s i ]} + σ N ( s i + s i + ŝ i + Re[ s i ŝ i + ŝ i s i ]. (I.6.7 VEV r = i s i > 0 ŝ i, σ i s i mass m e r s i mass term N s i N mass dimension gauge coupling constant e (IR limit massive fields effective theory classical theory r 0 M target space supersymmetric NLSM

19 N = (, Gauged Linear Sigma Model quantum vacua one-loop RG flow effective theory i Q i > 0 (I.. asymptotic free theory (I..8 r(µ 0 Λ µ Λ UV energy scale RG flow low energy µ Λ r(µ 0 low energy r < 0 r < r = 0 r < 0 r(µ = 0 potential energy density U s i = 0, σ = arbitrary (I.6.8 σ gauge multiplet mass dimension low energy σ VEV σ potential energy density U chiral superfields S i σ effective theory integrate out gauge multiplet Σ massless effective theory Σ S i integrate out effective Lagrangian ( ( L eff = d 4 θ K eff (Σ,Σ + d θ Weff (Σ + h.c., (I.6.9a W eff = Στ(µ N { ( Qi Σ } Q i Σ log µ { ( Σ } = Στ(µ NΣ log, (I.6.9b µ effective Kähler potential K eff (Σ,Σ critical point SUSY vacua supersymmetric U e eff σ W eff vacua W eff = 0 critical point 0 = σ Weff (σ = N log ( σ/µ τ(µ (I.6.0 ( σ/µ N = e τ(µ (I.6. σ = σ cr N N r(µ = 0 µ = Λ low energy µ < Λ r(µ < 0 quantum theory r(µ < 0 r(µ sigma model on CP N low energy limit quantum vacua vacuum manifold M = CP N N Figure I. r(µ < 0 N CP N cohomology ring N = dimh (CP N. (I.6. I.6. O( N bundle on CP N phase NLSM on CY manifold low energy theory GLSM superpotential chiral superfield U( charge + CY condition

20 I.6 Example 3 µ < Λ µ = Λ µ > Λ r(µ < 0 r(µ > 0 Figure I. Quantum Sigma Model on CP N. (I.. field content U( charge assignment Φ i ( S,S,,S N,P, Q i ( +,+,,+, N. (I.6.3 bosonic field auxiliary D potential energy density U(φ,σ N e D = r s i + N p, (I.6.4a SUSY condition D = U(φ = 0 U(s,p,σ = N e D + σ ( s i + N p, (I.6.4b D = 0 r = U(s i,p,σ = 0 N s i N p, (I.6.5a N 0 = σ ( s i + N p. (I.6.5b FI parameter r D = 0 r 0 case CY phase r 0 phase low energy effective theory Higgs branch chiral superfield (gauge theory hypermultiplet low energy limit phase i s i 0 σ = 0 p D = 0 = r i s i +N p vacuum manifold M CY p s i U( gauge du = 0

21 4 N = (, Gauged Linear Sigma Model symmetry M CY = {(s,s,,s N ;p C N+ r = N }/ s i N p > 0 U( (I.6.6 = O( N bundle over CP N s i CP N homogeneous coordinates p fiber coordinate total space noncompact CY manifold massless effective theory massless field vacuum manifold φ quantum fluctuation part φ + φ φ = φ + φ + φ N s i = s i + s i + ŝ i, p = p + p + p, σ = σ + σ + σ, (I.6.7a F(s i,p,σ = r N s i + N p, (I.6.7b σ = 0, F( s i, p, σ = 0, (I.6.7c F s i + p F = 0 = s i MCY p MCY potential energy density { U = e r { = e N s i s i N p p, σ + σ σ. (I.6.7d N si + s i + ŝ i } + N p + p + p ( N + σ si + s i + ŝ i + N p + p + p N ( si + ŝ i + Re[ s i ŝ i + ŝ i s i ] + N ( p + p + Re[ p p + p p ] } N + σ { ( si + s i + ŝ i + Re[ s i ŝ i + ŝ i s i ] + N ( p + p + p + Re[ p p + p p ] }. (I.6.8 vector multiplet scalar field σ m = e i s i = e r Higgs mechanism gauge field v i Higgs mechanism superpartner λ (gaugino field ŝ i, p m = e r massive field integrate out (e limit gauge field kinetic term freeze massless effective theory U( gauge symmetry s i 0 N = (, supersymmetric NLSM CY manifold target space conformal sigma model phase CY phase moduli space vacuum manifold [53]

22 I.6 Example 5 r 0 case orbifold phase r 0 phase Higgs branch low energy limit hypermultiplet phase moduli p 0 σ = s i r < 0 free vacuum manifold M orbifold M orbifold = {(s,s,,s N C N r = N }/ s i N p < 0 U( (I.6.9 effective theory vacuum manifold p 0, s i = 0 U( gauge symmetry Z N s i 0 Z N gauge symmetry s i = 0 vacuum manifold quantum fluctuated field effective potential U(φ,σ ( U = e N s i + N ( p p + p p + p N + N p σ + σ ( s i + N ( p p + p p, (I.6.0 σ, p m = e N p = N e r superpartner Higgs mechanism r fixed moduli p r = i s i N p S i flat space C N orbifold phase low energy effective theory C N /Z N orbifold theory (sigma model r = 0 case Coulomb branch and singularity Theta angle θ (zero/non-zero low energy theory Coulomb branch phase supersymmetric condition (I.6.5 r = 0 Higgs branch

23 6 N = (, Gauged Linear Sigma Model In the Higgs branch, M r>0 M r<0 r ±0 s i 0, p 0 (I.6.a s i p = O( (I.6.b CY phase CY vacuum manifold M r>0 CY geometry well-defined r = 0 Kähler [46] FI parameter parameter ([46] b base space well-defined geometry FI parameter r zero base space CY geometry {s i,p} fiber bundle C N+ space constraint i s i = N p (mod U( gauge symmetry Coulomb branch CY phase/lg phase vector multiplet scalar field σ nonzero s i = p = 0 auxilary field D = 0 potential energy density U = 0 condition σ σ 0 configration U( gauge symmetry CY phase FI parameter r FI parameter r CY manifold Kähler class moduli CP N size CP N r = 0 CP N Coulomb branch CY phase gauge symmetry enhance Coulomb branch Coulmob s i = p = 0 σ σ chiral multiplets S i, P σ massive fields integrate out low energy effective theory chiral multiplets vector Σ effective theory scalar σ, gaugino λ free field quantum effect gauge field ( F 0 electric field gauge field effective Lagrangian L eff = e (F 0 + θf 0, (I.6. θ = θ mod πz two dimensional gauge theory Theta term Theta angle θ constant electric field F 0 vacuum energy (I.6. constant electric field F 0 = θe, (I.6.3

24 I.6 Example 7 vacuum energy density E vacuum = L constant field = e θ, (I.6.4 e Coulomb branch r 0 vacuum energy density r 3 { E vacuum = e r + θ }, (I.6.5 r 0, θ 0 (mod πz Coulmob branch Higgs branch ( σ = 0 FI parameter r = 0 phase transition smooth [3] Theta angle θ nonzero string theory Theta angle NS-NS -form field B worldsheet θ 0 r = 0 CP N NS-NS -form flux smooth θ = 0 singular Aspinwall [6] Higgs branch singularity (I.6.a effective theory ϕ a = σ = 0 Coulomb branch σ 0 r = θ = FI parameter effective theory r 0 Coulomb branch Higgs branch {ϕ a } = 0 Coulomb branch CY phase r = 0 singular phase Higgs branch (I.6.b orbifold phase phase CY r > 0 singular r = 0 orbifold r < 0 connection (order r Higgs Coulomb Higgs connection (arbitrary order Higgs Higgs Higgs Coulomb branch at r = 0 CY phase r > 0 σ is free singularity φ = O(r r +0 r 0 φ = O( r φ is free orbifold phase r < 0 Higgs branch at r = 0 3 effective potential U s i = p = 0

25 8 N = (, Gauged Linear Sigma Model I.6.3 CY hypersurface in CP N O( N bundle on CP N configuration (I.6.3 superpotential W GLSM W GLSM = P G N (S i. (I.6.6 G N (S i homogeneous polynomial of degree N G N (S i = G N S = = G N S N = 0 implies S = S = = S N = 0. (I.6.7 Fermat type G N (S i = setup potential energy density CY phase N Si N. (I.6.8 U(s,p,σ = N e D + σ ( s i + N p N + G N (s i + p i G N, (I.6.9a N e D = r s i + N p. (I.6.9b r 0 phase supersymmetric vacuum U(φ = 0 (I.6.9 N s i 0, σ = 0, G N = 0, p i G N (s k = 0 (I.6.30 p 0 U(φ = 0 (I.6.9 G N = G N = G N = = N G N = 0 (I.6.7 s i = 0 (I.6.30 i s i 0 p = 0 (I.6.3 vacuum manifold M CY M CY = {(s,s,,s N C N r = N = hypersurface CP N [N] = CY (N -fold }/ s i > 0, G N (s i = 0 U( (I.6.3 moduli space effective theory (r, e s i CP N homogeneous coordinates vacuum manifold s i = s i + s i + ŝ i, p = p. (I.6.33

26 I.6 Example 9 CP N 4 massive field ŝ i, σ superpartner effective potential p p ŝ i, σ integrate out field p massive integrate out massless field low energy theory vacuum manifold CP N hypersurface G N = 0 compact manifold ( CY manifold N = 5 quintic hypersurface in CP 4 CY target space NLSM orbifold phase r 0 phase SUSY cone Kähler cone low energy effective theory sigma model superpotential SUSY condition moduli p 0 σ = 0, (I.6.34 (I.6.7, (I.6.9 s i = 0 (I.6.35 vacuum manifold M orbifold M orbifold = { }/ (s,s,,s N = (0,0,,0 U( (I.6.36 effective theory vacuum manifold p 0 U( gauge symmetry Z N p, σ superpartner Higgs mechanism massive field homogeneous function G N degree potential energy ŝ i homogeneous function G N degree N field massive massless field low energy theory 5 N 3 ŝ i low energy theory ŝ i flat kinetic term superpotential W = rg N (ŝ i ( p = r Z N gauge symmetry LG orbifold theory superpotential W = p G N (S i = r/n G N (S i (I.6.37 r/n superfield Si LG model IR limit CFT minimal model central charge central charge orbifolding Lagrangian (chiral ring orbifolding superpotential (I.6.37 chiral superfield S i chiral primary field LG model scale invariant S i scale worldsheet coordinate 4 O( N + l bundle on CP N [l] 5 massive field low energy theory

27 0 N = (, Gauged Linear Sigma Model z z z = λ z (λ C superspace coordinates θ α measure d zd θ superfield S i z = λ z, z = λ z, θ α λ / θ α ( y m = x m + iθσ m θ, dz λ dz, dθ α λ / dθ α, d z d θ λ d z d θ. S i(z,z = ( z z h i ( z z h isi (z,z = λ hi S i (z,z, (I.6.38 h i, h i left/right conformal weight 6 F-term scale invariant h i superpotential W (S i W (S i W (λ hi S i λw (S i (I.6.39 = h i N, h i = N, (I.6.40 superpotential W left/right U( charge (, chiral superfield S i (q i,q i = (/N,/N S i chiral superfield homogeneous superpotential minimal model conformal weight h i U( charge h i = q i model highest charged state h ρ = c/6 total central charge c total (appendix III; ref. [05] c total = 6β, β = ( i q. (I.6.4 i c total = 3(N, (I.6.4 N = 5 c total = 9 r > 0 CP 4 [5] CY 3-fold conformal sigma model central charge c = 9 CY/LG correspondence LG model ring structure CY cohomology class chiral ring chiral chiral primary field OPE Φ ( (zφ ( (w C (3 (z wφ (3 (w C (3 ring structure constant chiral ring R cc = C(S i [dw (S i ], (I.6.43 C(S i S i chiral superfield (c,c-ring appendix V LG model CY geometry 6 S i chiral superfield left/right conformal superfield lowest component field scalar spin s = h i h i = 0

28 I.6 Example I.6.4 Degree l hypersurface in CP N configuration S S N P U( l W = P G l (S i. (I.6.44 G l (S i homogeneous function of degree l (3 l N G l (S i = i G l = 0 implies S = S = = S N = 0. (I.6.45 setup potential energy density { U(s,p,σ = e r r 0 phase N s i + l p } ( N + σ s i + l p N + G l (s i + p i G l. (I.6.46 r 0 phase supersymmetric vacuum U(φ = 0 (I.6.46 N s i 0, σ = 0, G l = 0, p i G l (s j = 0 (I.6.47 p 0 U(φ = 0 (I.6.46 G l = G l = G l = = N G l = 0 (I.6.45 s i = 0 (I.6.47 i s i 0 p = 0 (I.6.48 vacuum manifold M CY M CY = {(s,s,,s N C N r = N = hypersurface CP N [l] }/ s i > 0, G l (s i = 0 U( (I.6.49 moduli space effective theory (r, e s i CP N homogeneous coordinates vacuum manifold s i = s i + s i + ŝ i, p = p. (I.6.50 CP N massive field ŝ i, σ superpartner effective potential p p ŝ i, σ integrate out field p massive integrate out massless field effective theory vacuum manifold CP N hypersurface G l = 0 compact manifold target space NLSM

29 N = (, Gauged Linear Sigma Model r 0 phase r 0 phase SUSY cone Kähler cone low energy effective theory sigma model superpotential SUSY condition moduli (I.6.45, (I.6.46 p 0 σ = 0, (I.6.5 s i = 0 vacuum manifold M orbifold M orbifold = { }/ (s,s,,s N = (0,0,,0 U( (I.6.5 (I.6.53 effective theory vacuum manifold p 0 U( gauge symmetry Z l p, σ superpartner Higgs mechanism massive field effective theory ŝ i flat kinetic term superpotential W = rg l (ŝ i ( p = r Z l gauge symmetry LG orbifold theory superpotential W = p G l (S i = r/l G l (S i (I.6.54 r/l superfield Si effective theory classical Q a = N l > 0 a asymptotic free FI parameter r (I..8 Λ µ Λ UV r(µ > 0 sigma model on CP N [l] RG flow r(µ 0 µ Λ r(λ = 0 sigma model description sigma model on CP N low energy theory r(µ = 0 r = 0 potential energy density U (I.6.46 σ flat direction σ mass dimension low energy theory mass dimension σ S i, P massless effective theory integrate out massless limit Σ effective Lagrangian L eff = W eff (Σ = Στ(µ a d 4 θ { K eff (Σ,Σ } ( + d θ Weff (Σ + h.c., (I.6.55a { ( Qa Σ } Q a Σ log. (I.6.55b µ sigma model on CP N r = 0 massless effective theory effective Lagrangian CP N CP N [l] GLSM classical Lagrangian chiral superpotential W GLSM = P G l (S i

30 I.6 Example 3 U(σ continuous spectrum discrete spectrum U eff = e eff τ(µ llog( l 0 σ Figure I. Effective potential around r(µ = 0. effective Kähler potential K eff (Σ,Σ twisted chiral superpotential W eff (Σ (c,c-ring (a,c-ring ( W eff (Σ = Σ τ(µ llog( l (I.6.56 potential energy density U eff (σ = e eff σ Weff (σ = e eff τ(µ llog( l (I.6.57 effective Kähler potential K eff e eff ( 7 Σ supersymmetric potential energy density potential energy Figure I. log( l = πi + log l + πin n Z (I.6.58 θ U eff Figure I.3 potential ( θ θ θ πln (I.6.59 CY sigma model on CP N [N] Coulomb phase θ τ(µ = llog( l scale parameter µ worldsheet cylinder S R field theory discrete spectrum t(µ = llog( l singular τ = 0 singular 7 [53] chapter 5

31 4 N = (, Gauged Linear Sigma Model U 0 θ Figure I.3 Relation between θ and U eff. τ(µ llog( l (mod πz dicrete spectrum RG flow Figure I.4 θ µ Λ sigma model on CP N [l] singular point τ(µ = l log( l µ Λ LG theory with W LG Figure I.4 Parameter space τ(µ. I.7 O( N + l bundle on CP N [l] I.7. Field configuration and supersymmetric vacuum manifold Now we are ready to analyze massless low energy effective theories in the GLSM for O( N +l bundle on CP N [l]. We consider a U( gauge theory with N + chiral superfields Φ a of charges Q a. We set the field configuration to chiral superfield Φ a S... S N P P U( charge Q a... l N + l (I.7. In addition we introduce a superpotential W GLSM (Φ = P G l (S, where G l (S is a function of chiral superfields S i. This is a holomorphic homogeneous polynomial of degree l. Owing to the homogeneity, this polynomial has a following property if G l (s = G l (s =... = N G l (s = 0 then s i = 0. (I.7.

32 I.7 O( N + l bundle on CP N [l] 5 By definition, the numbers N and l are positive integers l,n Z >0. We assume that these two integers satisfy l N and N. The sum of all charges Q a vanishes (I.. in order to obtain non-trivial SCFTs on the CY manifold. Now we consider the potential energy density and look for supersymmetric vacua. Imposing the Wess-Zumino gauge, we write down the bosonic part of the potential energy density U(ϕ U(ϕ = e D + G l (s + D = r N p i G l (s + U σ (ϕ, (I.7.3a N s i + l p + (N l p, (I.7.3b N U σ (ϕ = σ { s i + l p + (N l p }. (I.7.3c Imposing zero on them, we obtain the supersymmetric vacuum manifold M. Since the Lagrangian has N = (, supersymmetry and the single U( gauge symmetry, the vacuum manifold becomes a Kähler quotient space { }/ M = (ϕ a C N+3 D = Gl = p i G l = U σ = 0 U(, (I.7.4 In attempt to study effective theories, we choose a point on M as a vacuum and give VEVs of scalar component fields ϕ a ϕ a. Then we expand the fluctuation modes around the vacuum. In general, the structure of M is different for r > 0, r = 0 and r < 0 and there appear various phases in the GLSM. The phase living in the r > 0 region is referred to the CY phase, and the phase in r < 0 is called to the orbifold phase. A singularity of the model emerges in the phase at r = 0. Thus we sometimes call this the singularity phase. We will treat these three cases separately. We comment that in each phase the vacuum manifold is reduced from the original M. We often refer the reduced vacuum manifold to M r M. The VEVs of the respective phases can be set only in M r. I.7. Calabi-Yau phase In this subsection we analyze the CY phase r > 0. In this phase, D = 0 requires some s i cannot be zero and therefore σ must vanish. If we assume p 0, the equations G l (s = i G l (s = 0 with the condition (I.7. imply that all s i must vanish. However this is inconsistent with D = 0. Thus p must be zero. The variable p is free as long as the condition D = 0 is satisfied. Owing to these, the vacuum manifold M is reduced to M CY defined by { }/ M CY = (s i ;p C N+ D = Gl (s = 0, r > 0 U(. (I.7.5 Here we explain this manifold in detail. This is an (N -dimensional noncompact Kähler manifold. The components s i denote the homogeneous coordinates of the complex projective space CP N. The constraint G l (s = 0 reduces CP N to a degree l hypersurface expressed to CP N [l]. we find that p is a fiber coordinate of the O( N + l bundle on CP N [l]. Furthermore the vanishing sum of U( charges indicates that the FI parameter r is not renormalized. This is equivalent to c (M CY = 0. Thus we conclude that the reduced vacuum manifold M CY is nothing but a noncompact CY manifold on which a non-trivial superconformal field theory is realized.

33 6 N = (, Gauged Linear Sigma Model Let us consider a low energy effective theory. We choose a vacuum and take a set of VEVs of the scalar component fields. Because s i 0, the U( gauge symmetry is spontaneously broken down completely. Next, we expand all fields in terms of fluctuation modes such as ϕ a = ϕ a + ϕ a + ϕ a. We set p, σ and σ to be zero. Substituting them into the potential energy density (I.7.3, we obtain { [ U = e Re N ] ŝ i s i + (N l p p N si + ŝ i + l p + (N l } p + p N l + ŝ i i G l ( s + ( s + ŝ i ( s + ŝ ik i ik G l ( s k! k= i,,i k N + p l i G l ( s + ( s + ŝ j ( s + ŝ jk i j jk G l ( s k! j,,j k k= N + σ { si + s i + ŝ i + l p + (N l } p + p + p. Fluctuation modes s i and p remain massless and move only tangent to M CY because they are subject to δd VEV = δg l VEV = 0. The variation δ(p i G l VEV = 0 indicates p = 0. The modes σ, p, ŝ i and p have mass m = O(e r. The gauge field v m also acquires mass of order O(e r by the Higgs mechanism. The fermionic superpartners behave in the same way as the scalar component fields because of preserving supersymmetry. In the IR limit e and the large volume limit r, the massive modes decouple from the system. Thus we obtain N = (, supersymmetric NLSM on M CY (I.7.6 as a massless effective theory. Notice that this description is only applicable in the large volume limit because the NLSM is well-defined in the weak coupling limit from the viewpoint of perturbation theory. This effective theory becomes singular if we take the limit r +0 because the decoupled massive modes becomes massless. This phenomenon also appears in the Seiberg-Witten theory [95, 96], the black hole condensation [00, 39], and so on. Let us make a comment on the target space M CY. By definition, the number l means the degree of the vanishing polynomial G l (s = 0, which gives a hypersurface in the projective space CP N. We can see that if l =, G l= (s = 0 gives a linear constraint with respect to the homogeneous coordinates s i and the hypersurface CP N [l = ] is reduced to (N -dimensional projective space CP N. This reduction does not occur if l N. Here we summarize the shape of the target space M CY in Table I. degree l vacuum manifold M CY l = O( N + bundle on CP N l N O( N + l bundle on CP N [l] Table I. Classification of O( N + l bundle on CP N [l]. Although the l = case has been already analyzed in the original paper [3], the other cases l N are the new ones which have not been analyzed.

34 I.7 O( N + l bundle on CP N [l] 7 I.7.3 Orbifold phase Here we consider the negative FI parameter region r < 0. In this region the total vacuum manifold (I.7.4 is restricted to a subspace defined by M orbifold = { }/ (p,p ;s i C N+ D = Gl = p i G l = 0, r < 0 U(. (I.7.7 Since D = 0 does not permit p and p to vanish simultaneously, σ must be zero. This subspace is quite different from M CY in the CY phase. In addition, the shape of M orbifold is sensitive to the change of the degree l because of the existence of the constraints G l = p i G l = 0 and the property (I.7.. Thus let us analyze M orbifold and study massless effective theories on it in the case of 3 l N, l = and l =, separately. Effective theories of 3 l N Here we analyze the vacuum manifold M orbifold and massless effective theories of 3 l N. Owing to the constraints G l = p i G l = 0 and their property (I.7., the manifold M orbifold is decomposed into the following two subspaces M 3 l N orbifold = M r<0 M r<0, }/ M r<0 = {(p,p C D = 0, r < 0 U(, { }/ M r<0 = (p ;s i C N+ D = Gl = 0, r < 0 U(. (I.7.8a (I.7.8b (I.7.8c In the former subspace the condition (I.7. is trivially satisfied whereas in the latter subspace it is satisfied nontrivially. Both of the two subspace include a specific region p = s i = 0. The subspace M r<0 is defined as a one-dimensional weighted projective space WCP l,n l represented by two complex fields p and p of U( charges l and (N l, respectively. The precise definition of the weighted projective space is in appendix I.C. Let us choose a supersymmetric vacuum and set VEVs of all scalar fields. Then we expand all the fields around the VEVs. Expanding the potential energy density (I.7.3 in terms of VEVs and fluctuation modes, we obtain the following form { [ ] U = e Re l p p + (N l p p i + Gl ( s + ŝ + p + p + p i s i + ŝ i + l p + p + (N l p + p } i G l ( s + ŝ + σ { si + ŝ i + l p + p + p + (N l } p + p + p, i where p and p are VEVs of scalar components of P and P, respectively. They live in the weighted projective space (I.7.8b. Because the VEVs of s i are all zero, the U( gauge symmetry is spontaneously broken to Z α, where α is the great common number between l and N l α = GCM{l,N l}. This potential energy density provides that all fluctuation modes s i and ŝ i appear as linearly combined forms such as s i +ŝ i, which do not acquire any mass terms. The modes p and p remain massless and move tangent to the subspace (I.7.8b. The other

35 8 N = (, Gauged Linear Sigma Model fluctuation modes acquire mass of order m = O(e r. Thus, in the IR limit e, all the massive modes are decoupled from the system. Thus we obtain the following massless effective theory N = (, supersymmetric NLSM on WCP l,n l { }/ coupled to LG theory with W LG = ( p + P G l (S Z α, (I.7.9 where P and S i are massless chiral superfields. Note that the sigma model sector also contains the Z α orbifold symmetry coming from the property of WCP l,n l. As is well known that the term p G l (S forms an ordinary LG superpotential. Thus in the IR limit we can interpret that this term is marginal and flows to the N = (, minimal model. The second term P G l (S is somewhat mysterious. Since this term has not any isolated singularities we might not obtain well-defined unitary CFT. This difficulty causes the noncompactness of the manifold M CY which appears in the CY phase. There are two specific points in the subspace WCP l,n l. One is the point p = 0 and the other is p = 0. In the former point the gauge symmetry is enhanced to Z l. Furthermore the mode p disappears and p becomes massless, which combines with a massless fluctuation modes p linearly. This combined mode is free from any constraints. The other massless modes s i + ŝ i in (I.7.9 remain massless and are also free from constraints. Thus in the IR limit and the large volume limit, the massless effective theory becomes an N = (, supersymmetric theory as { }/ CFT on C LG theory with W LG = p G l (S Z l. (I.7.0 This effective theory consists of N + massless chiral superfields such as P and S i, which live in the free and the LG sectors, respectively. Since we take the IR limit, this effective theory becomes an SCFT. The LG sector flows to a well-known LG minimal model [3]. Thus the sigma model sector is also a superconformal field theory. Here we notice that we did not integrate out but just decomposed all massive modes in the above discussion because it is generally impossible to calculate the integration of them. Thus the above effective theory is merely an approximate one. If we will be able to integrate out all massive modes exactly, the obtaining effective theory will be different from the above one. In later section we will discuss the exact form of the effective theory. Next let us consider the latter point p = 0 in the space WCP l,n l. On this point the broken gauge symmetry is partially restored to Z N l. The massless fluctuation mode p becomes zero whereas the massive mode p becomes massless, which combines with p being free from any constraints. Thus P appears as a massless chiral superfield. In the IR limit we obtain the supersymmetric massless effective theory such as { LG theory with W LG = P G l (S on C N+}/ Z N l, (I.7. which consists of N + massless chiral superfields such as P and S i. This theory is not a well-defined LG theory because the superpotential W LG has no isolated singularities. We interpret the defect of isolated singularities as a noncompactness of the manifold M CY in the CY phase via CY/LG correspondence (if this correspondence is satisfied in the case of sigma models on noncompact CY manifolds. This property prevents from calculating a chiral ring of this model in the same way as unitary LG minimal models describing compact CY manifolds [7]. Here we study massless effective theories on the subspace M r<0 defined in (I.7.8c. As mentioned before, there are non-trivial constraints in M r<0. Thus, as we shall see, the effective theories are also under these constraints.

36 I.7 O( N + l bundle on CP N [l] 9 In the same way as discussed before, we choose one point in the subspace M r<0 and make all the scalar fields fluctuate around it. Then we write down the expanded potential energy density (I.7.3 in terms of VEVs and fluctuation modes ϕ a, ϕ a and ϕ a { [ U = e Re ] ŝ i s i + (N l p p si + ŝ i + l p } + (N l p + p i i l + ŝ i i G l ( s + ( s + ŝ i ( s + ŝ ik i ik G l ( s k! i k= i,...,i k + p l i G l ( s + ( s + ŝ j ( s + ŝ jk i j jk G l ( s k! i k= j,...,j k + σ { s i + s i + ŝ i + l p + (N l p } + p + p. i This potential energy density indicates the following The fluctuation modes ŝ i, p and p are massive; s i and p move tangent to M r<0. Thus, taking e and r, we obtain N = (, supersymmetric NLSM on M r<0. (I.7. In this theory there exist P and S i as massless chiral superfields, which move tangent to M r<0. Notice that in general points in M r<0 the U( gauge symmetry is completely broken because of the existence of non-zero VEVs s i. However, taking s i = p = 0 and p 0 in the subspace M r<0, we find that the gauge symmetry is partially restored to Z N l. Even though the vacuum manifolds M r<0 and M r<0 are connected on p = s i = 0, the effective theories given by (I.7. and (I.7. are quite different from each other. The reason is that while the subspace M r<0 is free from constraints G l = p i G l = 0, in the subspace M r<0 these constraints are still valid on the region p = s i = 0. On account of the existence of these constraints, a phase transition occurs when the theory moves from one to the other. Thus we conclude that a new phase appears on the subspace M r<0, which has not been discovered in well-known GLSMs such as the models for O( N bundle on CP N, for CP N [N], for resolved conifold, and so on. We refer this phase to the 3rd phase. Here we refer the phase on M r<0 to the orbifold phase, as usual. Effective theories of l = Let us consider the orbifold phase of l =. In the same way as the previous analysis, the constraints G l = p i G l = 0 and the property (I.7. decompose the manifold M orbifold into two subspaces M l= orbifold = M r<0 M r<0, }/ M r<0 = {(p,p C D = 0, r < 0 U( WCP,N, { }/ M r<0 = (p ;s i C N+ D = G = 0, r < 0 U(. (I.7.3a (I.7.3b (I.7.3c These two subspaces are glued in the region given by p = s i = 0. Although this situation is same as to the case of 3 l N, the appearing massless effective theories are quite different.

37 30 N = (, Gauged Linear Sigma Model Here let us analyze the effective theories on the subspace M r<0 = WCP,N. We choose a point in this subspace as a supersymmetric vacuum and take VEVs of all scalar fields. Then we make all scalar fields fluctuate around the VEVs. Fluctuation modes p and p are subject to the constraints such that they move only tangent to WCP,N. The fluctuation modes s i have no degrees of freedom because of the variation of the constraint p i G = 0. (In the case of (I.7.8b, the equations G l = 0 and p i G l = 0 are trivially satisfied in WCP l,n l. These variations are also trivial. However the case of l = is quite different. By definition, some i j G must have non-zero values. Thus even though the above equations are trivially satisfied in the subspace WCP,N, their variations give non-trivial constraints on the fluctuation modes. Under these conditions we write down the potential energy density (I.7.3 in terms of VEVs ϕ a and fluctuation modes ϕ a and ϕ a { [ ] U = e Re p p + (N p p i + G (ŝ + p + p + p i i G (ŝ ŝi + p + p + (N p + p } + σ { ŝ i + 4 p + p + p + (N p } + p + p. i This function denotes the following The fluctuation modes ŝ i, p and p acquire masses m = O(e r ; the modes p and p remain massless and move tangent to WCP,N. Thus taking e and r, we obtain the massless effective theory described by N = (, supersymmetric NLSM on WCP,N. (I.7.4 This sigma model has Z α orbifold symmetry coming from the property of WCP,N, where α = GCM{,N }. This effective theory does not include massless LG theory. The reason is that the degree two polynomial G generates mass terms such as p i ig. (See, for example, [05]. Now we consider the effective theory on two specific points in WCP,N like (I.7.0 and (I.7.. Expanding the theory on the one point (p,p = (p,0, the gauge symmetry is partially restored to Z. Thus we obtain the effective theory on this specific point as N = (, SCFT on C /Z. (I.7.5 Note that this theory can possess the LG theory with a quadratic superpotential W LG = p G (S, which gives massive modes of S i. The effective theory drastically changes if we expand the theory on another point (p,p = (0,p in WCP,N. On this point, the broken gauge symmetry is enhanced to Z N and the fluctuation modes ŝ i become massless with being free from any constraints. Both p and p are massless and linearly combined in the potential energy density. The remaining field p becomes zero because there exists a non-trivial variation of the constraint D = 0. Summarizing these results, we find that the following massless effective theory appears in the limit e, r { N = (, LG theory with W LG = P G (S on C N+}/ Z N. (I.7.6 Although this superpotential also has no isolated singularities, this theory should describe a non-trivial SCFT. We shall return here in later discussions.

38 I.7 O( N + l bundle on CP N [l] 3 We next study the massless effective theories on the subspace M r<0 defined in (I.7.3c. The potential energy density is obtained as { [ U = e Re ] ŝ i s i + (N p p si + ŝ i + p } + (N p + p i i + ŝ i i G ( s + ( s + ŝ i ( s + ŝ j i j G ( s! i i,j + p i G ( s + ( s + ŝ j i j G ( s i j + σ { si + s i + ŝ i + 4 p + (N } p + p + p i under the following constraints on fluctuation modes The fluctuations s i and p move tangent to M r<0; the other tangent mode p is zero; the fluctuations ϕ a are all massive of m = O(e r. Thus the effective theory expanded around generic points in M r<0 becomes N = (, supersymmetric NLSM on M r<0 (I.7.7 in the IR and large volume limit e, r. The U( gauge symmetry is completely broken if some s i 0 exist. On the other hand, if we expand the theory on a specific point p = s i = 0, the gauge symmetry is partially restored to Z N. So far we have studied the effective theories on all regions of the vacuum manifold M orbifold of l =. From the same reason discussed in the case of 3 l N, there exists a phase transition between the theories (I.7.6 and (I.7.7 because of the non-trivial constraint coming from the variation of the equation G = 0. Thus we find that the GLSM for the O( N + bundle on CP N [] also includes two phases in the negative FI parameter region. The phase on (I.7.4 is called the orbifold phase, and we refer the phase on (I.7.7 to the 3rd phase. Here we illustrate the relation among the phases in the GLSM schematically in Figure I.5 s i p p 3rd phase CY phase p s i orbifold phase p Figure I.5 Various phases in GLSM for O( N + l bundle on CP N [l] with l N. The axes with thin/thick lines represent the vacuum space coordinates in the positive/negative FI parameter regions, respectively.