Aspects of (0,2) theories
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1 Aspects of (0,2) theories Ilarion V. Melnikov Harvard University FRG workshop at Brandeis, March 6, / 22
2 A progress report on d=2 QFT with (0,2) supersymmetry Gross, Harvey, Martinec & Rohm, Heterotic string theory (I) NPB256, received Feb / 22
3 There are many paths to (0,2) theories critical heterotic strings with R 1,3 N=1 superpoincaré heterotic stringy geometry for E X surface defects/strings in d=4 N=1 gauge theory AdS 3 /CFT 2 and (2,0) compactification last refuge of holomorphy in SUSY QFT 3 / 22
4 Plan 1 general properties 2 heterotic geometries 3 linear sigma models 4 accidents 5 outlook 4 / 22
5 (0,2) supercurrents live in an S-multiplet [Dumitrescu&Seiberg, ] R 1,1 : x µ = (x 0, x 1 ) or light-cone x ±± S µ +, S µ + conserved supercurrents Q +, Q + energy-momentum tensor T µν P µ 5 / 22
6 (0,2) supercurrents live in an S-multiplet [Dumitrescu&Seiberg, ] R 1,1 : x µ = (x 0, x 1 ) or light-cone x ±± S µ +, S µ + conserved supercurrents Q +, Q + energy-momentum tensor T µν P µ weak assumptions = T, S, S S-supermultiplet {Q +, S +±± } = T ±±++ ± i ±± j ++ {Q +, S +++ } = 0 {Q +, S + } = ic. 5 / 22
7 (0,2) supercurrents live in an S-multiplet [Dumitrescu&Seiberg, ] R 1,1 : x µ = (x 0, x 1 ) or light-cone x ±± S µ +, S µ + conserved supercurrents Q +, Q + energy-momentum tensor T µν P µ weak assumptions = T, S, S S-supermultiplet {Q +, S +±± } = T ±±++ ± i ±± j ++ {Q +, S +++ } = 0 {Q +, S + } = ic. spin 1 operator j ++ not necessarily conserved C 0 = deformed (0,2) supercurrent algebra, a UV property 5 / 22
8 Many (0,2) QFTs have R-multiplet [Dumitrescu&Seiberg, ] C = 0 conserved R-current j µ j ±± conserved charge R j µ, S µ +, S µ +, T R-multiplet = standard (0,2) SUSY algebra {Q +, Q + } = 4P ++ [R, Q + ] = Q + [R, Q + ] = +Q + {Q +, Q + } = 0 {Q +, Q + } = 0 a key property: Q 2 + = 0 C can be generated by quantum corrections, e.g. (0, 2) P 1 NLSM 6 / 22
9 (0,2) SCFTs are the building blocks for (0,2) QFTs T µ µ = 0 on Euclidean world-sheet Vir c N=2 svir c T (z) ; J(z), G ± (z), T (z) h q h we will discuss unitary and compact SCFTs characterization symmetries, e.g. left-moving Kac-Moody algebra U(1) L current J elliptic genus Z = Tr RR ( 1) F q H q H y J 0 spectrum of marginal operators [depends on moduli!] 2 and 3-point functions [super-primary not sufficient!] 7 / 22
10 Marginal (0,2) deformations are characterized SUSY marginal deformations via conformal perturbation theory [Bertolini et al, ], cf N=1 d=4 [Green et al, ] S = λ d 2 z{q +, U} + h.c. U a chiral primary operator h = 1, h = q/2 = 1/2 unitarity [h q/2] = λ at worst marginally irrelevant 8 / 22
11 Marginal (0,2) deformations are characterized SUSY marginal deformations via conformal perturbation theory [Bertolini et al, ], cf N=1 d=4 [Green et al, ] S = λ d 2 z{q +, U} + h.c. U a chiral primary operator h = 1, h = q/2 = 1/2 unitarity [h q/2] = λ at worst marginally irrelevant obstructions: D-term (J KM (z), U, U) long multiplet = λ breaks KM F-term (U q=1, F q=2 ) long multiplet some consequences: Kähler moduli space no F-term obstructions for c < 6 8 / 22
12 RG flows satisfy some simple constraints relevant S = λ d 2 z{q +, U} + h.c. ; U c.p. with q < 1 (c c) is RG-invariant; ċ < 0 c IR via c-extremization [Benini&Bobev, , ] elliptic genus is RG-invariant holomorphic 1/2-twisted chiral ring of Q + -closed operators [Witten, ],[Tan&Yagi, ],[Borisov&Kaufmann, ] topological heterotic rings generalize A and B model of (2,2) [Adams,Basu&Sethi, ],[Adams,Distler&Ernebjerg, ] but watch out for accidents! 9 / 22
13 Plan 1 general properties 2 heterotic geometries 3 linear sigma models 4 accidents 5 outlook 10 / 22
14 (0,2) NLSMs yield a geometric realization geometry [Hull&Witten,PLB 85] : X complex, Hermitian ω, H = i( )ω E X holomorphic bundle 11 / 22
15 (0,2) NLSMs yield a geometric realization geometry [Hull&Witten,PLB 85] : X complex, Hermitian ω, H = i( )ω E X holomorphic bundle anomalies: ch 2 (E) = ch 2 (T X ); c 1 (T X ) = 0 ; c 1 (E) = 0 mod 2; 11 / 22
16 (0,2) NLSMs yield a geometric realization geometry [Hull&Witten,PLB 85] : X complex, Hermitian ω, H = i( )ω E X holomorphic bundle anomalies: ch 2 (E) = ch 2 (T X ); c 1 (T X ) = 0 ; c 1 (E) = 0 mod 2; conformal invariance for dim C X = 3 = [Strominger;Hull;Sen, 86],[Groot Nibbelink&Horstmeyer, ] K X holomorphically trivial and E stable (X, ω, Ω) define SU(3) structure + conformal balance [ dh = α 4 tr R 2 tr F 2] 11 / 22
17 The standard embedding is a heterotic geometry (X, ω, Ω) is CY 3-fold; E = T X ; H = 0 = (2,2) SCFT c = c = 9 q Z E 6 E 8 SUSY compactification of E 8 E 8 string 12 / 22
18 The standard embedding is a heterotic geometry (X, ω, Ω) is CY 3-fold; E = T X ; H = 0 = (2,2) SCFT c = c = 9 q Z E 6 E 8 SUSY compactification of E 8 E 8 string gauge-neutral marginal deformations have canonical split [(2,2)!] H 1 (T X ) H1 (T X ) H 1 (End T X ) 12 / 22
19 The standard embedding is a heterotic geometry (X, ω, Ω) is CY 3-fold; E = T X ; H = 0 = (2,2) SCFT c = c = 9 q Z E 6 E 8 SUSY compactification of E 8 E 8 string gauge-neutral marginal deformations have canonical split [(2,2)!] H 1 (T X ) H1 (T X ) H 1 (End T X ) unobstructed deformations M ck (X ) M c-x (X ) H 1 (T X ) 27 H1 (T X ) 27 W ijk F(t)27 i 27 j 27 k special geometry [Dixon et al,npb 90] 12 / 22
20 The standard embedding is a heterotic geometry (X, ω, Ω) is CY 3-fold; E = T X ; H = 0 = (2,2) SCFT c = c = 9 q Z E 6 E 8 SUSY compactification of E 8 E 8 string gauge-neutral marginal deformations have canonical split [(2,2)!] H 1 (T X ) H1 (T X ) H 1 (End T X ) unobstructed deformations M ck (X ) M c-x (X ) H 1 (T X ) 27 H1 (T X ) 27 W ijk F(t)27 i 27 j 27 k special geometry [Dixon et al,npb 90] W encodes obstructions to H 1 (End T X ) classical via alg-geom quantum via world-sheet instantons 12 / 22
21 Stable bundles over CY generalize standard embedding rank r bundle π : E X [we will take r = 3] 13 / 22
22 Stable bundles over CY generalize standard embedding rank r bundle π : E X [we will take r = 3] in large radius limit (0,2) SCFT c = c = 9 q Z E 6 E 8 SUSY compactification of E 8 E 8 string [Witten,NPB 86] example: X quintic P 4 and E a deformation of T X a zoo e.g. [Anderson et al, ],[anderson&taylor, ] 13 / 22
23 Stable bundles over CY generalize standard embedding rank r bundle π : E X [we will take r = 3] in large radius limit (0,2) SCFT c = c = 9 q Z E 6 E 8 SUSY compactification of E 8 E 8 string [Witten,NPB 86] example: X quintic P 4 and E a deformation of T X a zoo e.g. [Anderson et al, ],[anderson&taylor, ] gauge-neutral marginal deformations do not have canonical split 13 / 22
24 Stable bundles over CY generalize standard embedding rank r bundle π : E X [we will take r = 3] in large radius limit (0,2) SCFT c = c = 9 q Z E 6 E 8 SUSY compactification of E 8 E 8 string [Witten,NPB 86] example: X quintic P 4 and E a deformation of T X a zoo e.g. [Anderson et al, ],[anderson&taylor, ] gauge-neutral marginal deformations do not have canonical split sugra [Anderson et al, ] world-sheet [IVM&Sharpe, ] 13 / 22
25 Stable bundles over CY generalize standard embedding rank r bundle π : E X [we will take r = 3] in large radius limit (0,2) SCFT c = c = 9 q Z E 6 E 8 SUSY compactification of E 8 E 8 string [Witten,NPB 86] example: X quintic P 4 and E a deformation of T X a zoo e.g. [Anderson et al, ],[anderson&taylor, ] gauge-neutral marginal deformations do not have canonical split sugra [Anderson et al, ] world-sheet [IVM&Sharpe, ] H 1 (E ) 27 H 1 (E) 27 W C ijk (t)27 i 27 j 27 k 13 / 22
26 Stable bundles over CY generalize standard embedding rank r bundle π : E X [we will take r = 3] in large radius limit (0,2) SCFT c = c = 9 q Z E 6 E 8 SUSY compactification of E 8 E 8 string [Witten,NPB 86] example: X quintic P 4 and E a deformation of T X a zoo e.g. [Anderson et al, ],[anderson&taylor, ] gauge-neutral marginal deformations do not have canonical split sugra [Anderson et al, ] world-sheet [IVM&Sharpe, ] H 1 (E ) 27 H 1 (E) 27 W C ijk (t)27 i 27 j 27 k world-sheet instantons can ruin conformal invariance zoo or bestiary? 13 / 22
27 Torsional geometries are more mysterious X topologically non-kähler e.g. odd b 1 string-scale cycles = α expansion formal known examples: X = T 2 principal bundle over K3 [Dasgupta et al, ],[fu&yau, ],[becker et al, ] 14 / 22
28 Torsional geometries are more mysterious X topologically non-kähler e.g. odd b 1 string-scale cycles = α expansion formal known examples: X = T 2 principal bundle over K3 [Dasgupta et al, ],[fu&yau, ],[becker et al, ] torsional geometries with N=2 spacetime supersymmetry [IVM,Minasian&Theisen, ] classified exhibit (0,2+4) worldsheet susy cf [Banks&Dixon,NPB 88] conjecture: IIA duals are K3-fibered but not elliptically fibered 14 / 22
29 Plan 1 general properties 2 heterotic geometries 3 linear sigma models 4 accidents 5 outlook 15 / 22
30 Linear sigma models offer key insights two-dimensional gauge theories with (0,2) susy [Witten, ] simple UV physics chiral bosonic multiplets Φ and fermi multiplets Γ A = γ A + gauge group G and holomorphic potentials E A (Φ), J A (Φ), J E = 0 phases determined by Fayet-Iliopoulos parameters geometric phases include X V toric & E X monad bundle 16 / 22
31 Linear sigma models offer key insights two-dimensional gauge theories with (0,2) susy [Witten, ] simple UV physics chiral bosonic multiplets Φ and fermi multiplets Γ A = γ A + gauge group G and holomorphic potentials E A (Φ), J A (Φ), J E = 0 phases determined by Fayet-Iliopoulos parameters geometric phases include X V toric & E X monad bundle rich IR physics vast classes of (2,2) and (0,2) SCFTs interpolate between disparate descriptions CY/LG correspondence the art of the linear sigma model : lift IR queries to UV fields 16 / 22
32 Developments in (0,2) linear sigma model technology elliptic genus [Gadde&Gukov, ],[Benini et al, ] = probe novel (0,2) SCFTs, e.g. [Gadde et al, ],[haghighat et al, ] 17 / 22
33 Developments in (0,2) linear sigma model technology elliptic genus [Gadde&Gukov, ],[Benini et al, ] = probe novel (0,2) SCFTs, e.g. [Gadde et al, ],[haghighat et al, ] vanishing theorem for spacetime superpotential [Bertolini&Plesser, ] = (X, E) geometries safe from world-sheet instantons 17 / 22
34 Developments in (0,2) linear sigma model technology elliptic genus [Gadde&Gukov, ],[Benini et al, ] = probe novel (0,2) SCFTs, e.g. [Gadde et al, ],[haghighat et al, ] vanishing theorem for spacetime superpotential [Bertolini&Plesser, ] = (X, E) geometries safe from world-sheet instantons {holomorphic parameters}/{redefinitions} for deformations of (2,2) [Kreuzer et al, ] = (0,2) mirror map for pairs (X, E), (X, E ) [IVM&Plesser, ] 17 / 22
35 Developments in (0,2) linear sigma model technology elliptic genus [Gadde&Gukov, ],[Benini et al, ] = probe novel (0,2) SCFTs, e.g. [Gadde et al, ],[haghighat et al, ] vanishing theorem for spacetime superpotential [Bertolini&Plesser, ] = (X, E) geometries safe from world-sheet instantons {holomorphic parameters}/{redefinitions} for deformations of (2,2) [Kreuzer et al, ] = (0,2) mirror map for pairs (X, E), (X, E ) [IVM&Plesser, ] A/2 heterotic ring for deformations of (2,2)[McOrist&IVM, ] = quantum sheaf cohomology for toric varieties[guffin et al, ] = 27 3 couplings for deformations of (2,2) SCFT 17 / 22
36 Developments in (0,2) linear sigma model technology elliptic genus [Gadde&Gukov, ],[Benini et al, ] = probe novel (0,2) SCFTs, e.g. [Gadde et al, ],[haghighat et al, ] vanishing theorem for spacetime superpotential [Bertolini&Plesser, ] = (X, E) geometries safe from world-sheet instantons {holomorphic parameters}/{redefinitions} for deformations of (2,2) [Kreuzer et al, ] = (0,2) mirror map for pairs (X, E), (X, E ) [IVM&Plesser, ] A/2 heterotic ring for deformations of (2,2)[McOrist&IVM, ] = quantum sheaf cohomology for toric varieties[guffin et al, ] = 27 3 couplings for deformations of (2,2) SCFT torsional linear sigma models[adams et al, , ],[quigley et al, , ] 17 / 22
37 (0,2) marginal deformations via (2,2) linear sigma model marginal deformations depend on the phase [Kachru&Witten, ],[Distler&Kachru, ],[Aspinwall et al, ],[bertolini et al, ] hybrid: LG over base B FI 2 smooth CY Landau-Ginzburg orb orbifold FI 1 a rich laboratory for explorations, e.g. (0,2) McKay [Aspinwall&Gaines, ] 18 / 22
38 Plan 1 general properties 2 heterotic geometries 3 linear sigma models 4 accidents 5 outlook 19 / 22
39 Life without accidents would be nice (0,2) Landau-Ginzburg theory with U(1) L U(1) R symmetry L = free kinetic term + dθ A ΓA J A (Φ; α) + h.c. quasi-homogeneity: J A (t q Φ; α) = t Q AJ A (Φ; α) charges (q, Q) determine family of LG theories generic J A = U(1) L U(1) R symmetry IR SCFT with (c, c) & elliptic genus determined by (q, Q) 20 / 22
40 There are accidents in (0,2) RG flows [Bertolini et al, ] LG kinetic term irrelevant and slaved to superpotential J A (Φ; α) and J A (Φ; α ) are IR-equivalent if linked by field redefinition symmetries of J A (Φ; α) depend on α = c-extremization can lead to c(α) c(α )! = UV parameter space stratified according to basin of attraction = naive c may not have realization for any α how can we ensure that accidents do not occur? 21 / 22
41 Outlook impressive results for just two supercharges many challenges some key directions when does a (0,2) linear sigma model yield a heterotic CFT? what is (0,2) mirror symmetry beyond deformations of (2,2)? when is a (0,2) RG-flow accident-free? which torsional geometries yield heterotic CFTs? what sorts of geometric transitions occur in (0,2)? are there nice local models? is there a class of (0,2) minimal models? hope: new sources of (0,2) questions may also yield new (0,2) answers a little (0,2) book is in the works! 22 / 22
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