Many faces of Mirror Symmetries

Transcription

1 Many faces of Mirror Symmetries Huajia Wang University of llinois at Urbana Champaign S. Kachru, M. Mulligan, G.Torroba, H. Wang, arxiv: S. Kachru, M. Mulligan, G.Torroba, H. Wang, arxiv:

2 ntroduction/motivation Duality: equivalence between descriptions non-perturbatively related Examples: 2+1 dimensions: Peskin-Dasgupta-Halperin (PDH) duality (1978, 1981) S XY = Z d 3 x (@ µ ia µ ) 2 V ( ) S A.H. = Z U(1) J : J µ / µ f d 3 x (@ µ ia µ ) 2 Ṽ ( )+ 1 2 µ A a dynamical gauge field Particle-vortex duality: Monopole operator f µ =2

3 ntroduction/motivation Recently: particle-hole symmetry + composite fermion + 1/2 Landau level conjecture (Son, 14): S free = Z dx 3 i µ (@ µ ia µ ) S QED = Z U(1) J : J µ / µ f dx 3 i µ (@ µ ia µ ) µ A a (fermionic version) dynamical gauge field Particle-vortex duality: Monopole operator (2-fluxes) f µ =4 Can be understood as arising at the boundary of 3+1 bulk topological order under S duality. (Wang, Senthil; Metlitski and Vishwanath, 15)

4 ntroduction/motivation Bosonization/Fermionization dualities: (Aharon, 16) N f fermions + U(k) N N+ f 2, N+ N f 2 N f scalars + SU(N) k N f fermions + SU(k) N+ N f 2 N f scalars + U(N) k,k N f fermions + U(k) N N+ f 2, N k+ N f 2 N f scalars + SU(N) N+ N f 2 Main Evidences: large N, k limit: dual to the same higher-spin (Vasiliev) theories of gravity in AdS4 massive deformations: level-rank dualities

5 ntroduction/motivation Bosonization/Fermionization dualities: (Aharon, 16) N f fermions + U(k) N N+ f 2, N+ N f 2 N f scalars + SU(N) k N f fermions + SU(k) N+ N f 2 N f scalars + U(N) k,k N f fermions + U(k) N N+ f 2, N k+ N f 2 N f scalars + SU(N) N+ N f 2 special case of interest: N f = N = k =1 i /D a i /D A 1 8 ada 1 2 Ada 1 8 AdA D A 2 V ( )+ 1 4 AdA D a ' 2 V (')+ 1 4 ada 1 2 Ada

6 ntroduction/motivation Web of dualities : (E. Witten, et al; A. Karch, at al 16) Z scalar+flux (A) =Z fermion (A) e i 8 R AdA Z scalar-qed (A) =Z scalar (A) PDH s duality Z QED (A) =Z fermion (A) Son s duality Z fermion+flux (A) =Z scalar (A) e i 4 R AdA by formal manipulation of path-integrals proving one automatically proves other three!

7 ntroduction/motivation Web of dualities : (E. Witten, et al; A. Karch, at al 16) Proving any one of the dualities is difficult, due to lack of control parameters in the strongly coupled theories. numerical evidence (S. Geraedts, et al, 15); wire construction (D. Mross, et al, 17); lattice construction (J. Chen, et al, 17) Goa of the talk: can they be connected with better understood dualities (mirror symmetry)? Peskin duality? Mirror Symmetry Son s duality scalar+flux $ fermion fermion+flux $ scalar

8 Mirror symmetry: Exact dualities Key ingredient: SUPERSYMMETRY (SUSY) make fun of SUSY? An organizing principle for quantum corrections: Symmetry: boson fermion + = 0 boson fermion SUSY in condensed matter systems: S.S. Lee 06; T. Grover, at al 13;

9 Mirror symmetry: Chiral mirror symmetry: (D. Tong, 00; S. Kachru et al, 16) Theory A Theory B L A µ 2 + i /@ Free theory! L B / 1 g 2 K kin(a, ada +,,D)+ D a ' 2 + i /D a SUSY completions SUSY-multiplets hypermultiplet: {, } {', }, SUSY extension of charged field vectormultiplet: {a µ,,,d} SUSY completions of interactions SUSY extension of gauge field e.g. 1/2 CS gauge coupling! 1 8 ada +2D +

10 Mirror symmetry: dentify global symmetries: Theory A Theory B U(1) J U(1) R U(1) J U(1) R U(1) g ' e 2 i /g Background vector multiplet:  µ! {µ, ˆ, ˆD}

11 Deforming away from Mirror symmetry: Break SUSY: Mirror Symmetry? Peskin duality Son s duality scalar+flux $ fermion fermion+flux $ scalar

12 Deforming away from Mirror symmetry: Break SUSY: Mirror Symmetry? Peskin duality Son s duality scalar+flux $ fermion fermion+flux $ scalar Turn on backgrounds: {ÂJ µ, ˆJ, } + {ÂR µ, ˆR, ˆD R }

13 Deforming away from Mirror symmetry: Break SUSY: Mirror Symmetry? Peskin duality Son s duality scalar+flux $ fermion fermion+flux $ scalar Lorentz symmetry Turn on backgrounds: {ÂJ µ, ˆJ, } + {ÂR µ, ˆR, ˆD R }

14 Deforming away from Mirror symmetry: Break SUSY: Mirror Symmetry? Peskin duality Son s duality scalar+flux $ fermion fermion+flux $ scalar Lorentz symmetry Turn on backgrounds: {ÂJ µ, ˆJ, } + {ÂR µ, ˆR, ˆD R } One simple choice: {ˆJ, } ˆ2J Theory A 2 +ˆJ minimal coupling / ˆ/A J Theory B ˆDJ +ˆJD B.F. coupling / daâj ˆJ : : SUSY mass mass-split tad-poles for gauge multiplet

15 Deforming away from Mirror symmetry: Strategy: Step1: identify massive phases (,,, V) Easy Difficult V ˆJ V ˆJ Theory A Theory B characterized by topological responses: L A,B / K ij  i dâj,i,j2 {J, R}

16 Deforming away from Mirror symmetry: Strategy: Step2: identify critical phases (/, /, /V, V/) V ˆJ V ˆJ Theory A Theory B { L A (/) L B (/) L A (/) L B (/), {? web of dualities

17 Massive phases Theory A is free, easy to identify responses L A / DÂJ +ÂR m 2 2 m 2 + i /DÂJ 1 8 ÂJd ˆ A J m 2 < 0 m < 0 m 2 < 0 m > 0 m 2 =ˆ2J m =ˆJ m 2 > 0 m < 0 sgn(ˆj)ˆ2j m 2 > 0 m > 0 integrating out massive fermion m > 0! L A / 0 m < 0! L A / 1 4 ÂJdÂJ integrating out massive scalar m 2 > 0! nothing m 2 < 0! ÂJ + ÂR =0

18 Massive phases Theory A is free, easy to identify responses L A / DÂJ +ÂR 2 + i /DÂJ V m 2 2 m 1 8 ÂJd ˆ A J m 2 =ˆ2J m =ˆJ sgn(ˆj)ˆ2j Responses: : : L A =0 L A = 1 4 ÂJdÂJ V : L A = b : 2 b 2 ÂJ ÂR + L A = 1 4 ÂJdÂJ ÂJ + ÂR

19 Massive phases Theory B follows from duality: Z A ˆJ, = Z B ˆJ, L B / 1 g 2 K kin (a,,,d)+ D a ' 2 +i /D a ˆDJ +ˆJD + ' + h.c. ada +2D + 2 ' a 2dÂJ + dâr difficult to analyze in its own! V sgn(ˆj)ˆ2j Responses: : L B =0 : : V : L B = 1 4 ÂJdÂJ L B = 1 4 ÂJdÂJ L B = b 2 b 2 ÂJ + ÂR ÂJ + ÂR

20 Critical phases Theory A m 2 > 0 m 2 < 0 m < 0 m 2 < 0 m > 0 V sgn(ˆj)ˆ2j m 2 > 0 m < 0 m > 0 / : / : /V : V/ : m =0 m 2 =0 m =0 m 2 =0 free fermion Wilson-Fisher free fermion Wilson-Fisher V / : Theory B sgn(ˆj)ˆ2j / : /V : V/ : }? Clue: consistent with transitions in topological responses

21 Critical phases Theory B : parametrize the R by effective mass parameters: L B / 1 g 2 K kin (a,,,d)+ D a ' 2 +i /D a + 1 ada +2D ˆDJ +ˆJD 2 ' ' + h.c. 4 a 2dÂJ + dâr R L B /... + m 2 ' ' 2 + f M f f eigenvalues f =(, ) m f+,m f Responses (in terms of R mass parameters): m 2 ' > 0 : m 2 ' < 0:a! 0 L B / sgn(m f + ) 8 + sgn(m f ) 8 Â R dâr (a + ÂR)d(a + ÂR)+ 1 8 ada 1 4 a 2dÂJ + dâr L B / 1 8 sgn(mf+ ) + sgn(m f ) Â R dâr

22 Critical phases Theory B : consistency with responses in massive phases uniquely fix the signs: m 2 ' > 0 m < 0 m < 0 m 2 ' > 0 m < 0 m > 0 V / : m 2 ' =0 sqed / : m =0 fqed m 2 ' > 0 m > 0 m < 0 sgn(ˆj)ˆ2j m 2 ' < 0 m f+ > 0 m f < 0 /V : V/ : m =0 m 2 ' =0 free fermion sqed

23 Critical phases Matching: Theory A Theory B / : free fermion sqed scalar+flux $ fermion / : Wilson-Fisher fqed fermion+flux $ scalar /V : free fermion free fermion trivial duality V/ : Wilson-Fisher sqed Peskin duality Conclusion: V web of dualities" can be embedded as deformations of a multi-critical parent duality: mirror symmetry sgn(ˆj)ˆ2j

24 Possible extensions: More general SUSY-breaking deformations Other realizations of mirror symmetries (e.g. deconfined criticality?) Non-relativistic limits Thank you!