Singularity structure and massless dyons of pure N = 2, d = 4 theories with A r and C r

Transcription

1 Singularity structure and massless dyons of pure N = 2, d = 4 theories with A r and C r Disclaimer: all figures are mouse-drawn, not to scale. Based on works w/ Keshav Dasgupta Singularity structure and massless dyons of pure N = 2, d = 4 theories with SU(r + 1) and Sp(2r) 1 / 23

2 Introduction Seiberg-Witten and Argyres-Douglas theories Seiberg-Witten curves, massless dyons (low ranks), and some aspects of singularity in moduli space were studied for SU(2) w/o and w/ matter [Seiberg Witten 94] SU(n) [Klemm Lerche Yankielowicz Theisen 94, Argyres Faraggi 94], SU(n) w/o and w/ matter [Hanany Oz 95] SO(2r) [Brandhuber-Landsteiner 95] SO(2r+1) [Danielsson Sundborg 95], SO with matter [Hanany 95] The hyper-elliptic curve y 2 = f(x) degenerates into a cusp form y 2 = (x a) m when m 3 branch points collide on x-plane. For SU(n) SW curves, this give exotic theories, discovered and studied [Argyres Douglas 95], where we have mutually non-local massless dyons. (Under symplectic transformation, you cannot bring everything purely electronic.) Singularity structure and massless dyons of pure N = 2, d = 4 theories with SU(r + 1) and Sp(2r) 2 / 23

3 Introduction Main idea Want to build Gaiotto s N = 2 theory (see Yuji s talk) in F theory N = 2 Sp(2r) SW theories via r D3-branes compute massless dyon charges for pure SU(r + 1), Sp(2r) study wall crossing provide plethora of Argyres-Douglas (AD) theories physics geometry massless d.o.f. singularity x f = 0 exotic theories (i. e. AD) u x f = 0 Double discriminant captures higher singularity of hyper-elliptic curves. Singularity structure and massless dyons of pure N = 2, d = 4 theories with SU(r + 1) and Sp(2r) 3 / 23

4 Math review: discriminants x, u and singularity of curves Discriminant x : discriminant with respect to x f n (x) = n a i x i = a n i=1 x (f n (x)) = a 2n 2 n n i=1 (x e i ) (e i e j ) 2 Order of vanishing (multiplicity of roots) 2 x = 0 i<j Singularity structure and massless dyons of pure N = 2, d = 4 theories with SU(r + 1) and Sp(2r) 4 / 23

5 Math review: discriminants x, u and singularity of curves Recipe for locating Argyres-Douglas loci Start with hyper-elliptic Seiberg-Witten curve y 2 = f(x; u, v, ) Demanding x f = 0 and u x f = 0 gives two massless BPS dyons. [Argyres Plesser Seiberg Witten 95] Check order of vanishing (o.o.v.) of each solution to u x f = 0 If o.o.v. 3, Argyres-Douglas loci: The hyperelliptic curve degenerates into a cusp-like singularity y 2 = (x a) 3 and two mutually non-local dyons become massless. (checked up to rank 5) Singularity structure and massless dyons of pure N = 2, d = 4 theories with SU(r + 1) and Sp(2r) 5 / 23

6 A 1 = C 1 : original SW curve and massless monopole/dyon, F theory picture of D3/O7 Review of monodromy of SU(2) = Sp(2) SW curve How did they get massless monopole and dyon? y 2 = (x 2 u) 2 Λ 4 Start at a generic point in moduli space, then branch points on x-plane are separated. As you move around a singular locus in u-plane, a pair of branch points approach & change with each other. Their trajectory gives vanishing cycle. Singularity structure and massless dyons of pure N = 2, d = 4 theories with SU(r + 1) and Sp(2r) 6 / 23

7 A 1 = C 1 : original SW curve and massless monopole/dyon, F theory picture of D3/O7 F-theoretic (quantum) 3/7 brane picture of Sp(2) [Sen 96] [Banks-Douglas-Seiberg 96] & [Vafa 96] X 6789 Higgs branch X 0123 Coulomb branch X 45 Two (p, q) 7-branes are monopole and dyon in the original SW theory. D3 brane (1, 1) seven brane (0, 1) seven brane Put r D3-branes as probes to get Sp(2r) gauge theory. [Douglas Lowe Schwarz 96] Singularity structure and massless dyons of pure N = 2, d = 4 theories with SU(r + 1) and Sp(2r) 7 / 23

8 Seiberg-Witten curves for pure A r and C r SW curve for pure SU(r + 1) = A r From [Klemm Lerche Yankielowicz Theisen 94] y 2 = f SU(r+1) ( x r+1 + u 1 x r 1 + u 2 x r u r ) 2 Λ 2r+2 = f + f f ± x r+1 + u 1 x r 1 + u 2 x r u r ± Λ r+1 f + r (x P i ) f i=0 r (x N i ) Note that f ± do not share roots (f + f = 2Λ r+1 0). On the x-plane, only P i s (or N i s) can collide among themselves. Discriminant factorizes x f SU(r+1) = # ( x f + ) ( x f ) i=0 Singularity structure and massless dyons of pure N = 2, d = 4 theories with SU(r + 1) and Sp(2r) 8 / 23

9 Seiberg-Witten curves for pure A r and C r SW curve for pure Sp(2r) = C r By taking no-flavor limit of [Argyres Shapere 95], obtain ( r y 2 ( ) ) ( ) r ( ) = f Sp(2r) x φ 2 a x x φ 2 a + Λ 2r+2 = f C f Q a=1 a=1 f C r ( ) r x φ 2 a = (x C i ) a=1 i=1 = x r + u 1 x r 1 + u 2 x r u r r f Q f C x + Λ 2r+2 = (x Q i ) Similarly as in SU(r+1) curve, x f Sp(2r) = # ( x f C ) ( x f Q ) Singularity structure and massless dyons of pure N = 2, d = 4 theories with SU(r + 1) and Sp(2r) 9 / 23 i=0

10 Identify massless dyons of A r and C r curves at xf = 0 Vanishing cycles of SW curve for pure SU(r+1) SU(3) [Klemm Lerche Yankielowicz Theisen 94] Singularity structure and massless dyons of pure N = 2, d = 4 theories with SU(r + 1) and Sp(2r) 10 / 23

11 Identify massless dyons of A r and C r curves at xf = 0 For SU(r + 1), in a region of moduli space given by u 1 = = u r 2 = 0 u r /Λ r+1 I, monodromy satisfies ν P i ν P i+1 = ν N i ν N i+1 = 1 ν P i ν N i = 2 ν P i ν N i+1 = 2 all other intersection numbers vanish. Singularity structure and massless dyons of pure N = 2, d = 4 theories with SU(r + 1) and Sp(2r) 11 / 23

12 Identify massless dyons of A r and C r curves at xf = 0 ν P i = β i+1 β i α i ν N i = β i+1 β i + α i+1 2α i i = 1,, r 1 ν0 P = β 1 r 1 νr P = β r + ν N 0 = β 1 + i=1 α i r α i + α 1 i=1 ν N r = β r 2α r Singularity structure and massless dyons of pure N = 2, d = 4 theories with SU(r + 1) and Sp(2r) 12 / 23

13 Identify massless dyons of A r and C r curves at xf = 0 Sp(2r) monodromy In a moduli region by u 2 = = u r 1 = 0 and u r = const: choose small enough u r to keep ν Q ν C = 0 (checked up to rank 5). Singularity structure and massless dyons of pure N = 2, d = 4 theories with SU(r + 1) and Sp(2r) 13 / 23

14 Identify massless dyons of A r and C r curves at xf = 0 With the same choice of symplectic basis as SU(r + 1) before, the vanishing cycles are written as ν Q 0 = β 1 ν Q r = β r r α i α r i=1 ν C r = β 1 β r r i=2 α i ν Q i = β i+1 β i α i νi C = β i+1 β i + α i+1 i = 1,, r 1 whose non-vanishing intersection numbers come from only ν Q i ν Q i+1 = 1, νq r ν Q 0 = νc i ν C i+1 = 1 The spectra change as we move in moduli space. (C 2 example) Singularity structure and massless dyons of pure N = 2, d = 4 theories with SU(r + 1) and Sp(2r) 14 / 23

15 Example: Singularity structure of Sp(4)=C 2 Example: Sp(4) = C 2 x f = 0 at Σ which intersect at u x f = 0 Singularity structure and massless dyons of pure N = 2, d = 4 theories with SU(r + 1) and Sp(2r) 15 / 23

16 Example: Singularity structure of Sp(4)=C 2 Two cycles ν Q 0 and νq 2 vanish at two different moduli loci uq 0 and respectively. u Q 2 Singularity structure and massless dyons of pure N = 2, d = 4 theories with SU(r + 1) and Sp(2r) 16 / 23

17 Example: Singularity structure of Sp(4)=C 2 As we vary in a moduli space (where u Q 0 and uq 2 intersect tangentially), we hit a locus where they vanish simultaneously. The curve degenerates into y 2 (x a) 3. (Argyres-Douglas) The red curve on the right does not give a well defined cycle. Singularity structure and massless dyons of pure N = 2, d = 4 theories with SU(r + 1) and Sp(2r) 17 / 23

18 Example: Singularity structure of Sp(4)=C 2 Now u Q 0 and uq 2 are separated, but u Q 0 runs toward another singularity locus u C 2. (ν Q 2 = ν Q 0 + νq 2 ; Not related by symplectic transformation) Singularity structure and massless dyons of pure N = 2, d = 4 theories with SU(r + 1) and Sp(2r) 18 / 23

19 Example: Singularity structure of Sp(4)=C 2 This time, singularity loci u Q 0 and uc 2 intersect (node-like crossing). Vanishing cycles ν Q 0 and νc 2 seem mutually non-local, but the curve looks like y 2 = (x a) 2 (x b) 2 and it is not Argyres-Douglas form. ( generalized AD) Singularity structure and massless dyons of pure N = 2, d = 4 theories with SU(r + 1) and Sp(2r) 19 / 23

20 Example: Singularity structure of Sp(4)=C 2 Now u Q 0 and uc 2 are separated, but the vanishing cycles did not change, unlike (b c) Singularity structure and massless dyons of pure N = 2, d = 4 theories with SU(r + 1) and Sp(2r) 20 / 23

21 u xf captures higher singularity Massless dyons coexist at a codim-2 C loci { x f = u x f = 0} [Argyres Plesser Seiberg Witten 95], where the curve looks like either of following two: y 2 = (x a) 3 curve has cusp-like singularity (Argyres-Douglas) x f = 0 locus also intersects at u x f = 0 (o.o.v 3) with cusp-like singularity (tangential intersection) two massless dyons are mutually non-local. y 2 = (x a) 2 (x b) 2 curve has node-like singularity x f = 0 locus also intersects at u x f = 0 (o.o.v 2) with node-like singularity two massless dyons are mutually local, with some exception (generalized AD: non-argyres-douglas & non-local loci) occurring at SU(5 ), Sp(4 ) Singularity structure and massless dyons of pure N = 2, d = 4 theories with SU(r + 1) and Sp(2r) 21 / 23

22 u xf captures higher singularity Generalized Argyres-Douglas loci curve degen.n y 2 = (x a) 3 y 2 = (x a) 2 (x b) 2 o.o.v of u x 3 2 shape of curve cusp node shape of x = 0 cusp node intersection mutually non local local non-local name Argyres-Douglas ML gen AD generalized Argyres-Douglas: non-argyres-douglas & non-local loci ( conformal limit?) Singularity structure and massless dyons of pure N = 2, d = 4 theories with SU(r + 1) and Sp(2r) 22 / 23

23 Conclusion and Future Directions Conclusion Discriminant x f = 0 of the SW curve y 2 = f: identified all 2r + 1 and 2(r + 1) massless dyons for Sp(2r) and SU(r + 1) At double discriminant u x f = 0: massless dyons coexist. If order of vanishing 3, then Argyres-Douglas. (checked up to rank 5) Questions still remain... Behaviour in the Argyres-Douglas neighborhood? Rank r curve classification as rank 2 of [Argyres Crescimanno Shapere Wittig 05][Argyres Wittig 05] Global behaviour in moduli space: wall crossing [ShapereVafa99, GaiottoMooreNeitzke09] Singularity structure and massless dyons of pure N = 2, d = 4 theories with SU(r + 1) and Sp(2r) 23 / 23