A novel method for locating Argyres-Douglas loci in pure N = 2 theories

Transcription

1 A novel method for locating Argyres-Douglas loci in pure N = 2 theories May 1, 2011, Great Lakes Strings 2011, University of Chicago Based on work w/ Keshav Dasgupta (see his talk after coffee) A novel method for locating Argyres-Douglas loci in pure N = 2 theories 1 / 31

2 Outline A novel method for locating Argyres-Douglas loci in pure N = 2 theories Motivation Hyperelliptic SW curves y 2 = f (x; u, v, ) for pure A r and C r Root structure on x-plane Identify massless dyons of A r and C r curves at x f = 0 Higher singularity structure at x f = 0 & u x f = 0 Example: Sp(4) = C 2 Order of vanishing of u x f is high (3) at Argyres-Douglas loci. Conclusion and open questions A novel method for locating Argyres-Douglas loci in pure N = 2 theories 2 / 31

3 Outline A novel method for locating Argyres-Douglas loci in pure N = 2 theories Motivation Hyperelliptic SW curves y 2 = f (x; u, v, ) for pure A r and C r Root structure on x-plane Identify massless dyons of A r and C r curves at x f = 0 Higher singularity structure at x f = 0 & u x f = 0 Example: Sp(4) = C 2 Order of vanishing of u x f is high (3) at Argyres-Douglas loci. Conclusion and open questions A novel method for locating Argyres-Douglas loci in pure N = 2 theories 2 / 31

4 Outline A novel method for locating Argyres-Douglas loci in pure N = 2 theories Motivation Hyperelliptic SW curves y 2 = f (x; u, v, ) for pure A r and C r Root structure on x-plane Identify massless dyons of A r and C r curves at x f = 0 Higher singularity structure at x f = 0 & u x f = 0 Example: Sp(4) = C 2 Order of vanishing of u x f is high (3) at Argyres-Douglas loci. Conclusion and open questions A novel method for locating Argyres-Douglas loci in pure N = 2 theories 2 / 31

5 Outline A novel method for locating Argyres-Douglas loci in pure N = 2 theories Motivation Hyperelliptic SW curves y 2 = f (x; u, v, ) for pure A r and C r Root structure on x-plane Identify massless dyons of A r and C r curves at x f = 0 Higher singularity structure at x f = 0 & u x f = 0 Example: Sp(4) = C 2 Order of vanishing of u x f is high (3) at Argyres-Douglas loci. Conclusion and open questions A novel method for locating Argyres-Douglas loci in pure N = 2 theories 2 / 31

6 Literature Review Seiberg-Witten curves, massless dyons (rank 2), and some aspects of singularity 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] For SU(n), exotic theories were discovered and studied [Argyres Douglas 95], where we have mutually non-local massless dyons. (Vanishing cycles have non-zero intersection number. You cannot bring everything purely electronic, under symplectic transformation.) The SW curve degenerates into y 2 = (x a) m namely, m 3 branch points collide together on x-plane. - SU(m) A novel method for locating Argyres-Douglas loci in pure N = 2 theories 3 / 31

7 Main idea and motivation geometry physics singularity x f = 0 massless d.o.f. u x f = 0 exotic theories (i.e. AD) compute dyon charges for all massless fields for SW curves for SU(r + 1), Sp(2r) provide plethora of Argyres-Douglas (AD) theories systematic method to study hyper-elliptic curve brane construction of N = 2 gauge theory in F theory with D3/D7/O7 (WIP with K. Dasgupta and A. Wissanji. See Dasgupta talk) Double discriminant captures higher singularity of hyper-elliptic curves y 2 = f (x). (This technique can be useful in any theory involving Riemann surfaces.) A novel method for locating Argyres-Douglas loci in pure N = 2 theories 4 / 31

8 Main idea Recipe for locating Argyres-Douglas loci guaranteed to work (checked up to rank 5) for pure N = 2 A r, C r theories Start with hyper-elliptic Seiberg-Witten curve y 2 = f (x; u, v, ) Demand x f = 0 and u ( x f ) = 0 (codim 2 C ) Check order of vanishing (o.o.v.) or degeneracy of each root to u x f = 0 If o.o.v. 3, you are at Argyres-Douglas loci. The hyperelliptic curve degenerates into y 2 = (x a) 3 (cusp-like singularity) here. two (or more, at some non-generic choice of other moduli) massless dyons coexist and are mutually non-local. A novel method for locating Argyres-Douglas loci in pure N = 2 theories 5 / 31

9 Outline A novel method for locating Argyres-Douglas loci in pure N = 2 theories Motivation Hyperelliptic SW curves y 2 = f (x; u, v, ) for pure A r and C r Root structure on x-plane Identify massless dyons of A r and C r curves at x f = 0 Higher singularity structure at x f = 0 & u x f = 0 Example: Sp(4) = C 2 Order of vanishing of u x f is high (3) at Argyres-Douglas loci. Conclusion and open questions A novel method for locating Argyres-Douglas loci in pure N = 2 theories 6 / 31

10 A 1 = C 1 :Rank 1 case - original SW curve and massless monopole/dyon Review of monodromy of SU(2) SW curve How did they get massless monopole and dyon? y 2 = (x 2 u) 2 Λ 4 = (x 2 u + Λ 2 )(x 2 u Λ 2 ) As you move around u-plane, branch points on x-plane move around. If you move around a singular locus in u-plane (on a non-contractible loop), your branch points collide with each other and then change with each other. (vanishing cycle) A novel method for locating Argyres-Douglas loci in pure N = 2 theories 7 / 31

11 A 1 = C 1 :Rank 1 case - original SW curve and massless monopole/dyon We use this same idea for more complicated curves for other gauge groups. We move around in a moduli space, surrounding a singular locus, so that you are on a non-contractible loop, and then you observe how branch points move around. This can give us all the vanishing cycles. Before we apply this technique, let us first review SW curves themselves. A novel method for locating Argyres-Douglas loci in pure N = 2 theories 8 / 31

12 Seiberg-Witten curves for pure A r and C r SW curve for pure SU(r+1) 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 4r+4 = f + f f ± x r+1 + u 1 x r 1 + u 2 x r u r ± Λ 2r+2 f + r (x P i ) f i=0 r (x N i ) Note that f ± do not share roots. On the x-plane, only P i s (or N i s) can collide among themselves. x f SU(r+1) = #( x f + ) ( x f ) i=0 A novel method for locating Argyres-Douglas loci in pure N = 2 theories 9 / 31

13 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 ) A novel method for locating Argyres-Douglas loci in pure N = 2 theories 10 / 31 i=0

14 Root structure on x-plane Root structure on x-plane x f SU(r+1) = ( x f + ) ( x f ) = u 2(r+1) + = ( u r+1 + ) ( u r+1 + ) x f Sp(2r) = ( x f C ) ( x f Q ) = u 2r+1 + = (u r + ) ( u r+1 + ) (r + 1) + (r + 1) vanishing cycles for SU(r+1), for bringing two P (or N) points together on x-plane. r + (r + 1) vanishing cycles for Sp(2r), for bringing two C (or Q) points together on x-plane. Argyres-Douglas loci: Bring n 3 branch points together on x-plane, such that y 2 = (x a) n. Maximal AD points (n = r + 1) occur SU(r + 1) set all P i = P j or N i = N j at 2 pts in mod. space (f + f ) Sp(2r) set all Q i = Q j at r + 1 pts in moduli space (x x exp 2πi r+1 ) A novel method for locating Argyres-Douglas loci in pure N = 2 theories 11 / 31

15 SU(3) Vanishing cycles of SW curve for pure SU(3) 6 solutions to the x f = 0, 6 vanishing cycles [Klemm Lerche Yankielowicz Theisen 94] A possible arrangement for some choice of moduli v (WCF?). A novel method for locating Argyres-Douglas loci in pure N = 2 theories 12 / 31

16 SU(r + 1) SU(r + 1) curve: possible arrangement somewhere in moduli space νi P = β i+1 β i α i νi N = β i+1 β i + α i+1 2α i (for 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 A novel method for locating Argyres-Douglas loci in pure N = 2 theories 13 / 31

17 Sp(8) = C 4 Vanishing cycles of SW curve for pure Sp(8) = C 4 9 solutions to the x f = 0, 9 vanishing cycles A possible arrangement for some part of moduli space A novel method for locating Argyres-Douglas loci in pure N = 2 theories 14 / 31

18 Sp(2r) Sp(2r) curve: possible arrangement somewhere in moduli space νr C ν Q 0 = β 1 ν Q i = β i+1 β i α i r νr Q = β r α i α r r νi C = β i+1 β i + α i+1 = β r β 1 α i α 1 i=1 i=1 A novel method for locating Argyres-Douglas loci in pure N = 2 theories 15 / 31

19 Outline A novel method for locating Argyres-Douglas loci in pure N = 2 theories Motivation Hyperelliptic SW curves y 2 = f (x; u, v, ) for pure A r and C r Root structure on x-plane Identify massless dyons of A r and C r curves at x f = 0 Higher singularity structure at x f = 0 & u x f = 0 Example: Sp(4) = C 2 Order of vanishing of u x f is high (3) at Argyres-Douglas loci. Conclusion and open questions A novel method for locating Argyres-Douglas loci in pure N = 2 theories 16 / 31

20 Sp(4)=C 2 Example: Sp(4) = C 2 identify singularity locus in 3 R 2 C moduli space A novel method for locating Argyres-Douglas loci in pure N = 2 theories 17 / 31

21 Sp(4)=C 2 Two cycles ν Q 0 and νq 2 vanish at two different moduli loci uq 0 and u Q 2 respectively. A novel method for locating Argyres-Douglas loci in pure N = 2 theories 18 / 31

22 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) A novel method for locating Argyres-Douglas loci in pure N = 2 theories 19 / 31

23 Sp(4)=C 2 Now u Q 0 and uq 2 are separated, but u Q 0 runs toward another singularity locus u2 C. (ν Q 2 = ν Q 0 + ν Q 2 ; Not related by sympl. transf.; Wall crossing?) A novel method for locating Argyres-Douglas loci in pure N = 2 theories 20 / 31

24 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) A novel method for locating Argyres-Douglas loci in pure N = 2 theories 21 / 31

25 Sp(4)=C 2 Now u Q 0 and uc 2 are separated, but the vanishing cycles did not change, unlike (b) (c) A novel method for locating Argyres-Douglas loci in pure N = 2 theories 22 / 31

26 u x f Massless dyons coexist at a codimension 2 C loci x f = 0 & u x f = 0, 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 ) A novel method for locating Argyres-Douglas loci in pure N = 2 theories 23 / 31

27 u x f Factorization of double discriminant u x f Order of vanishing of each root of u xf tells types of singularity ( ) u x f Sp(2r) = # v (2r+1)2 + ( ) 2 = # v r(r+1) + (v r + ) 3 ( v r+1 + )3 ( ) 2 ( ) 2 v r(r 3)/2 + v (r+1)(r 2)/2 + ( ) u x f SU(r+1) = # v (2r+2)2 + ( ) 2 (v r+1 + )3 ( v r+1 + )3 = # v r(r+1) + ( v (r+1)(r 2)/2 + ) 2 ( ) 2 v (r+1)(r 2)/2 + A novel method for locating Argyres-Douglas loci in pure N = 2 theories 24 / 31

28 u x f Roots of u x f Sp(2r)=Cr Picture: an example with Sp(8) = C 4 = 0 and curve degeneration Two pairs of branch points on x-plane collide each other pairwise. y 2 = (x e C ) 2 (x e Q ) 2 #( of choices: ) ( r r u x f = ) = r(r + 1) ( v r(r+1) + ) 2 A novel method for locating Argyres-Douglas loci in pure N = 2 theories 25 / 31

29 u x f Roots of u x f Sp(2r)=Cr Picture: an example with Sp(8) = C 4 = 0 and curve degeneration Two pairs of branch points on x-plane collide each other pairwise. y 2 = (x e C1 ) 2 (x e C2 ) 2 # of choices: ( ) ( r r u x f = ) /2 = r(r 3)/2 ( v r(r 3)/2 + ) 2 A novel method for locating Argyres-Douglas loci in pure N = 2 theories 26 / 31

30 u x f Roots of u x f Sp(2r)=Cr Picture: an example with Sp(8) = C 4 = 0 and curve degeneration Two pairs of branch points on x-plane collide each other pairwise. y 2 = (x e Q1 ) 2 (x e Q2 ) 2 # of choices: ( ) ( r+1 r u x f = ) (r + 1)(r 2) /2 = 2 ( v (r+1)(r 2)/2 + ) 2 A novel method for locating Argyres-Douglas loci in pure N = 2 theories 27 / 31

31 u x f Roots of u x f Sp(2r)=Cr Picture: an example with Sp(8) = C 4 = 0 and curve degeneration Three branch points on x-plane collide all together. y 2 = (x e C ) 3 # of choices: ( r 1 ) = r u x f = (v r + ) 3 A novel method for locating Argyres-Douglas loci in pure N = 2 theories 28 / 31

32 u x f Roots of u x f Sp(2r)=Cr Picture: an example with Sp(8) = C 4 = 0 and curve degeneration Three branch points on x-plane collide all together. # of choices: ( r +1 1 y 2 = (x e Q ) 3 ) = r + 1 u x f = ( v r+1 + )3 A novel method for locating Argyres-Douglas loci in pure N = 2 theories 29 / 31

33 Outline A novel method for locating Argyres-Douglas loci in pure N = 2 theories Motivation Hyperelliptic SW curves y 2 = f (x; u, v, ) for pure A r and C r Root structure on x-plane Identify massless dyons of A r and C r curves at x f = 0 Higher singularity structure at x f = 0 & u x f = 0 Example: Sp(4) = C 2 Order of vanishing of u x f is high (3) at Argyres-Douglas loci. Conclusion and open questions A novel method for locating Argyres-Douglas loci in pure N = 2 theories 30 / 31

34 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 o.o.v. (degeneracy of root) 3, then Argyres-Douglas. Open questions Behaviour in the Argyres-Douglas neighborhood non-argyres-douglas & non-local loci ( conformal limit?) Wall crossing? (codim 2 C 1 R, AD CMS?) Compare against solutions to Picard-Fuch equations. Test resemblance A r C r (h V = r + 1) [Kuchiev 09] N = 2 1 vacua A novel method for locating Argyres-Douglas loci in pure N = 2 theories 31 / 31