and Localization Kazutoshi Ohta (Meiji Gakuin University)

Transcription

1 Volume Calculation of Moduli Spaces and Localization Kazutoshi Ohta (Meiji Gakuin University) A. Miyake, KO and N. Sakai, Prog. Theor. Phys. 126 (2012) 637 [arxiv: [hep-th]] KO, N. Sakai and Y. Yoshida, arxiv: JSPS/RFBR Collaboration Synthesis Of Integrabilities Arising From Gauge-string Duality, 3/23/2013

2 Introduction Volume of moduli space of BPS solitons (instantons, monopoles, vortices, domain-walls,...) is important to Statistical mechanics of the BPS solitons Non-perturbative corrections in supersymmetric gauge theory Topological string amplitudes (topological invariants) We calculate the volume of the moduli space of the BPS solitons by the localization method (equivariant cohomology)

3 Introduction In particular, I consider today the volume of moduli space of the domain-walls Why? Extension of the localization method to the case with boundaries (cf. CS on S 3, N=(2,2) SYM on S 2,... without boundary) Boundary conditions are important! Important to solve BFSS like matrix quantum mechanics ((1+0)-dim)

4 Moduli space Moduli space: parameter space of solutions of the BPS solitons M = \ iµ 1 i (0) {gauge sym.} µ i : moment maps e.g. U(N) instantons on C 2 (z,w) M k = µ 1 r (0) \ µ 1 c (0) \ µ 1 c (0) U(N) µ r = F z z + F w w µ c = F zw self-dual equation Z k = F ^ F

5 Volume of the moduli space In general, Vol(M) = Z M d n x p g g : metric of M, n =dimm Moduli space metric : difficult Roughly speaking, we can formulate the volume as a path integral of the supersymmetric gauge theory Vol(M) Z D(fields) Y i J i (µ i ) Z SYM Jacobians

6 BPS domain-walls We consider the BPS domain-wall system: µ r D y g2 2 c1 N HH =0 c µ c D y H + H HM =0 µ c D y H + H MH =0 D-term F-term G=U(N c ) global U(N f )! Adj. 1 g: gauge coupling, c: FI parameter M = diag.(m 1,m 2,...,m Nf ) H 1 N f H 1 N f

7 Boundary condition domain-walls Vac. ~ A Vac. ~ B = diag.(m A1,m A2,...,m AN c ) = diag.(m B 1,m B2,...,m BN c ) -L/2 L/2 y Moduli space: M N c,n f ~A! B ~ = µ r 1 (0) \ µ c 1 (0) \ µ c U(N c ) 1 (0)

8 BRST symmetry We introduce a fermionic symmetry (a part of 4 SUSY) QA y = y, Q y = D y, Q =, Q = i[, ] Q =0 QH =, Q = i H QY c = i c, Q c = Y c Q 2 = gauge ( ) Equivariant cohomology

9 Localization Vol M N c,n f ~A! B ~ = Z D(fields)e S 0 tq = e S 0 SYM S SYM where QS 0 =0 S 0 = i and Z L/2 L/2 dy Tr [ µ r ] + (fermions) ) µ r =0 impose Independent of the coupling t 1-loop exact Localized at saddle (fixed) point

10 Contour integral Fixed point (with a gauge fixing): Q(fields)=0 & "c =" c =0 G=U(Nc ) U(1) Nc #: const. on y #=diag.($1, $ 2,..., $ Nc ) 1-loop det. Vol M N c,n f ~A! ~ B = X 2S Nc YN c a=1 Z 1 1 d a 2 ( 1) n (i a ) ind P ei a ˆL a where ˆL g2 c 2 L P a H D y H + H HM (mb (a) " c =" c =0 m A a ) o S 0 ind P a = # of zero modes of H = (# of domain-walls) + 1

11 Example Let us consider the case that N c =2 and N f =4 (m 1 <m 2 <m 3 <m 4 ) The boundary condition: ( L/2) = diag.(m 1,m 2 ) ~A =(1, 2) (L/2) = diag.(m 3,m 4 ) ~B =(3, 4) Total # of domain-walls = 4

12 Example Using a residue formula: Z 1 i 1 i d 2 i 1 n+1 ei B = ( 1 n! (ib)n if B 0 and n 0 0 otherwise we find Vol M 2,4 (1,2)!(3,4) = Z d 1 2 = 4 ( d 2 2 Z d 1 2 e i 1( ˆL (m 3 m 1 )) e i 2( ˆL (m4 m2)) (i 1 ) 3 (i 2 ) 3 d 2 2 e i 1( ˆL (m 4 m 1 )) e i 2( ˆL (m3 m2)) (i 1 ) 4 (i 2 ) (ˆL (m 3 m 1 )) 2 (ˆL (m 4 m 2 )) (ˆL (m 4 m 1 )) 3 (ˆL (m 3 m 2 )) )

13 Example Vol M 2,4 (1,2)!(3,4) = - =det m 1 m 2 m 3 m 4 1 2! (ˆL (m 3 m 1 )) 2 1 3! (ˆL (m 4 m 1 )) 3 ˆL (m 3 m 2 ) 1 2! (ˆL (m 4 m 2 )) 2 Transition matrix

14 Seiberg like duality Vol M N c,n f ~A! B ~ = Vol M N f N c,n f ~B! ~ A ˆL!1 (g!1) Q Nc j=1 (j 1)! Q Ñ c Q Nf (i 1)! i=1 k=1 (k 1)! ˆL Nc (N f N c ) Volume of the Grassmaniann: G Nc,N f U(N f ) U(N c ) U(Ñc)

15 Seiberg like duality

16 T-duality We consider domain-walls on a cylinder S 1 I S 1 vortices on a cylinder S 1 I (k=3 with SS mass and holonomy) I

17 T-duality In particular, the N c =N f =2 case (local vortex) Vol M 2,2 0 (S1 I) =1 Vol M 2,2 1 (S1 I) = Â 1+ˆ" ˆ"2 Vol M 2,2 2 (S1 I) = 1 5 2Â2 ˆ" +ˆ" 2 Â Vol M 2,2 3 (S1 I) = 1 6Â3 1 7 ˆ" +ˆ" 2 Â ˆ" ˆ"2 3 ˆ" ˆ"4 Â Vol M 2,2 4 (S1 I) 5 3 ˆ" 2 +2ˆ"2 3 ˆ" ˆ" ˆ" ˆ" ˆ"3 12 ˆ" ˆ"5 45 ˆ"6 24Â ˆ" +ˆ"2 Â 3 = ˆ" ˆ"2 2 3 ˆ" ˆ"4 Â 2 Â : area of S 1 I Vol M N,N k Âk k! ˆ" ˆ"2 18 ˆ" ˆ"4 15 ˆ" ˆ"6 Â ˆ" ˆ"2 5 ˆ" ˆ"4 45 ˆ" ˆ"6 315 ˆ" ˆ"8 In the ˆ"! 0 limit, the volume agrees with that of the local vortices on S 2

18 Conclusion and Discussion Results: We exactly evaluate the volume of the moduli space of the domainwalls via the localization method The volume is given by the simple contour integral We find the dualities between the moduli spaces Problems: Relation to integrable systems (spin chain, etc.) Relation to string/m theory (topological invariants)