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'.0'.S'.m'.'.'.(E,(E,(E,(Q,}(`,{(p,{(,{(,(,(,(,(,(,(,(,)v, )b,)E,),,),),)+(+(+(+(+(,(,,x|@             @       @               @            @         @    @               @           @     @         "1    ? $?4|F̛]h h FXTO""dFHDEO9FO9W 8 e2 C3qC8D6?/;?PEME m6=fK5UcU±sǎ4` ǁRy|wƫZUU؃SLmqhUMJWp DDL..x>\  ɮZ$ G+^\fN8 8+Xf{v4w΄%($MTcx}  [$ ^=.&16nUL̜bdoK;el2Be1ke*;+3 LD"`&04Sǯ  [$ d1:˹ỉl  DMDb39eQ*"`3ل2XB!SL  aq\$ X/<=E,$Ɂ x k1wyoH#˗$$Fכoe1g'$snzPVY&\XxI6Y/ˉ"SI1!"`￯p  ]$ \+li%q1i7VVXVbw]@ : Yj*Uk\*%s{y4-;SQ\.|{  K^$ 9TkQ{$93܂x2XYH9V2%M9η÷  ^$ U&6A3͹ǰ X ]ਨ7!cCHB!`Js  j_$ S   G@  H@@A+BFC0DEhFz1o?A-A-A-A-_A-/A-A-A-@!.@D.@\.@.A.A.7A.QA.Az.A.B- B-B-An-Am-Al-Ak-Ap-Aj-At-A~-A-A-A-A-A-A .A9.Ah.A.A.A/A:/Aq/A/A/AV0A0A0A0A0A0A0A0A0A0A0A0A0A|0AW0A40B02B/B`/B6/C./C,/*C /tB@,sB:,nB4,rB-,wB$,B,B,B,B ,B ,B ,B,B,C,C,C&,!C5,C>,CK,Cd,Bw,B,B,B,B,B,B,~B,{B,zB,{B,B-B-B-B-B-B-C-.C-KC-_C-qC-C-E ,E,E+E,E,E3,EG,E|,E,zE,\E^-;E-EL. E.D.E/E /E5/0E9/QE=/{E%/E/E/E.F.F.F.F.PD-SD-QD-[D-nD-D-D-D-D-E-HE-{E-E-E-E-F-:<@                      @              @         @     "n    ? $>?4|F̛]*hd h FXTO""dFHDEO9FO9W 8 e2 C3qC8$??^Ft{E h4neg'D{ "n.\,fs2x X [UbQP0mF2$1 <岌wƒK9\D JSS$"$Nx  P2r$ Y(K{-!^s[mM˜s˛g  F1\5 ssw7yfmĒL\]~  *s$ =߽V+S7]@w|a.nlyG>xAAθx|8  {s$   C@  @J=A=-BS?C.DEF21o?==>f->Y-s>K-a>C-Q><-A>=-0>=->=-=Z-=u-=-=-t=-R=.8=z.T=.a=.{=.=.=.=.>.7>.^>S.}>%.>.>->->->|->c->T->V->S->V->W->Y->\->^->c->j->n->v->->->->->->.>F.>r.>.>.>.>.>. ?.!?.=?.F?.R?.@                    "Z    ? $*?4|F̛],& h FXTO""dFHDEO9FO9W 8 e2 C3qC8W??^Ft{E Z# 39!;$; +;f -; /; 9; ;; <; >; 9; 6; 1; ;{ ;S :. ::u:X:7::999 9$ 9B 9f 9 9 9 : 3: X: : : : ; ; ; E S{r83r[(YsfmvlͫkjfNzss8k&9o-:l.*j1UF*9͹qԂUuomB"mZ[vUc![6h?  w)$ qHksm6nYa`Dg'32Cπu?nə$s31٭5ePZ LY10TJfe1D ZSTŠTg?  $  j@  @P'AB*C!DEFb1o?{{((((((+(B(z(((0(v(((+ (c ( ( ( ( ( ( ( ( ( ( ( ( (} (b ~(K l(4 Y( A( '( (''' ' s'$ a'> W'] F' X' ' ' '!'!' !( B( g( ( ( &)t!/)x!7)w!A)x!Q)y!b)y!r)x!)w!)z!)w!)u!)s!)q!*V!*K!!*A!&*6!,*'!1*!2*!3* 2* 0* /* -* -* -* .* /* 3* !;* !A*6!G*M!M*d!V*{!]*!d*!g*!i*!l*!h*!e*!b*!Y*!R*!I*!A*!9*!0*!(*!"*!*!*!*!*!*!*!$*!,*!>*!O*!a*!y*!*!*!*!*!*!.0@                  @    @                  "    ? $ǹ?4|F̛]HP h FXTO"dF@HEO9FO9W 8 e2 C3qC8D6?/;?EU+>E M3c͝)lyhܾsvΚ9xg/ɰkUg-lܥb+F*9Q@9{H-,+VXp[w  Г$ 36k4 o= L^k>jg{  g$ iJ;ҩQe,jڶZ"\0f3 ;v#YČq4ΔePKm[cmHc'$[j$6n  V¿$ z k@ <s  `,Non-zero differentials for the trefoil knot.- -H @ 0޽h ? fp=___PPT10i. V+D=' = @B ++0 B:  T(  TH T C AT34& J T <{  nDifferentials for 8 . The bottom row of dots has a-grading 6. The leftmost dot on that row has q-grading -6.To 3-  T <D  419 P T C (Aknot819P H T 0޽h ? fp=___PPT10i. V+D=' = @B +0 p  L(  LN L C &A10_1530  L <`  BDifferentials for 10 . The bottom row of dots has a-grading -2.6C 5  L 0 U  5153 T  L C ,Aknot10153PH L 0޽h ? fp=___PPT10i. V+D=' = @B +4 0 91(  x  c $X    x  c $0     <}g G[S.G., J.Walcher] <  C 4  ,$ 0 Boundaries, corners, & Surface operators Braid group actions on D-branesII @ 6 ` ,$@ 0H  0޽h ? fp=  ___PPT10{ .؂бs+YkD ' = @B D' = @BA?%,( < +O%,( < +D ' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*+8+0+0 +  0   6 (  x  c $      <$ bgauge theory on a 4-manifold X( F p       @A)ABH5CDEF 1o?""H)M);)~K)j)x*++{;,,9--*.U2.id.}.%//=000RT1$1 11A2233v4445G5 @           "f    ? $6? 4|F̛]" h FXTO"!d>DHDEO9FO9W 8 e2 C3qC8A?`Ӳ?EZ y23Ƚ\e3&LI*T٫lLۖ&䈒S7,%L]wzK=Vv bLq1ye*[mkuVۖȓ %Q6E JB,EYe)76T.Ybnf  SpP*     L@4AbB 9CPDEF(1o?..4c4e4c5e5j5o5t05K5g55556!_6C6a6w7H77N8888 988888w8>88 76P6J5 455!50484?4G4J4M4O@               "Y    ? $)? 4|F̛]V> h FXTO"!d>DHDEO9FO9W 8 e2 C3qC8D6?"DHDEO9FO9W 8 e2 C3qC8A?`Ӳ?EZ y23Ƚ\e3&LI*T٫lLۖ&䈒S7,%L]wzK=Vv bLq1ye*[mkuVۖȓ %Q6E JB,EYe)76T.Ybnf  SpP*     L@4AbB 9CPDEF(1o?..4c4e4c5e5j5o5t05K5g55556!_6C6a6w7H77N8888 988888w8>88 76P6J5 455!50484?4G4J4M4O@               "Y    ? 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A!"A!#A!A! @       "     ? $? 4|F̛]$  h FXTO"!dFHDEO9FO9W 8 e2 C3qC8??'ErE BCt˗$Kowװ?=͖*e2~/ 9T6\\ؐ  ~f y   @q&Az$B(C%DEF:1o???&$&$&$&$&$&$&$ '%/'%S'3%r'G%'\%'q%'~%'%'%'%'%'%'z$'{$'|$'~$'$'$'$'$'$|'$v'%q'*%j'O%d'p%\'%S'%N'%I'%C'%>'%?'%?'%s&%u&%&%&%&%&%&%&p%'Y%#'B%C'+%d'%'%'$'$'$'$'$'$($($($@      @       @       "    ? $f? 4|F̛]R"  h FXTO"!dFHDEO9FO9W 8 e2 C3qC8??'ErE DR)Znf̂K.\ dˈ@3ۜب `  `~ A6۽%TeCI&]2Ic;;ng{Kn"!JW7;Ϟ  Z~ D1W*Z\oT"1q|Kz*S[H  }~F  H  0޽h ? fp=___PPT10i.x֩6+D=' = @B +|# 0aN0 {"s"`  "(  R  C *Avortex24@ f `RB  s *Do$L B  # f   8@r A9 B{CDEXFt1o?r u x 2} C U g q z z t j R @5 S q n C  3S 6K 8B :; 5A ,G &O  ; | < Yg)\?R_C{   ^ 5  - < K U ^w j` pT wJ }> ~; }: 9 }@ yH wO of l| h a ^ [ WU T R P O-N_MyON** ***"))(>$S%h# 0C\mw{zO NN M LKKK&L/N@RRWc[zafmsw{cL/6B-IWe s1|=Vt4Pbep78@                          @       @                ""    ? $? 4|F̛]hF h FXTO""dFHDEO9FO9W 8 e2 C3qC8?"*?d*2A*XB+B2+BU+Bi+ @      "c    ? $3? 4|F̛]Z h FXTO"dF@HEO9FO9W 8 e2 C3qC8??HEr&E y"^$#OId`( RX,K(-RĤ\  {T     @1AB2CDE,FB1o?KK17131-1)1'1(1%1)1,101=1K1Y2l92X2s2222222222222222 2&2Dt2be2~V2I2@272/2.2)2&2$2$2"2&2'2"2111111111111212R2t2w2n2d2[2Y2M2N3P2\2a @        @        @        "    ? $o? 4|F̛][  h FXTO"!dFHDEO9FO9W 8 e2 C3qC8??'ErE K&aٸfr۶5wЂceK dc2kV96V,;  P~ C f7fʢᄖY@]$H[ SxD3)33sw>  M~ D&b XP25_ cm`UM\K4Zee{  1~  F f    V    @?A!B@CV"DEpF1o??!?!?!?!?!?!?!?!?!?!?!?!?!@!@!7@!W@ "y@"@*"@6"@A"@L"@O"@R"@U"@Q"@L"@H" @         "    ? $? 4|F̛] h FXTO"!dFHDEO9FO9W 8 e2 C3qC8??'ErE L'z,%ˎ9%fu<==̀fʶ͹K8\$A{z  );~v N f   fS   @?Ap!B@C"DEpF1o?@p!@r!@o!@q!@r!@r!@s!@v!@~!@!@!@!@!g@!M@!5@"@#"@B"?_"?p"?"?"?"?"?"?"?"?" @         "    ? $? 4|F̛] h FXTO"!dFHDEO9FO9W 8 e2 C3qC8??'ErE I%Wfږ3$w<](&G#ݶ{ซXLW7||%  ̕~r 0   @?A!B#AC$"DEXF1o??!"?!"?""?!"? 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A!"A!#A!A! @       "     ? $? 4|F̛]$  h FXTO"!dFHDEO9FO9W 8 e2 C3qC8??'ErE BCt˗$Kowװ?=͖*e2~/ 9T6\\ؐ  ~f y   @q&Az$B(C%DEF:1o???&$&$&$&$&$&$&$ '%/'%S'3%r'G%'\%'q%'~%'%'%'%'%'%'z$'{$'|$'~$'$'$'$'$'$|'$v'%q'*%j'O%d'p%\'%S'%N'%I'%C'%>'%?'%?'%s&%u&%&%&%&%&%&%&p%'Y%#'B%C'+%d'%'%'$'$'$'$'$'$($($($@      @       @       "    ? $f? 4|F̛]R"  h FXTO"!dFHDEO9FO9W 8 e2 C3qC8??'ErE DR)Znf̂K.\ dˈ@3ۜب `  `~ A6۽%TeCI&]2Ic;;ng{Kn"!JW7;Ϟ  Z~ D1W*Z\oT"1q|Kz*S[H  }~a^   <b@F  *a-brane corresponds to the static configuration of surface operators below ( time direction not shown)8l j  B`iwO K [a-brane8 H  0޽h ? fp=___PPT10i.x֩6+D=' = @B +# 0aN0 ""p !"(  X  S 0;Avortex26 `L B  # f   8@r A9 B{CDEXFt1o?r u x 2} C U g q z z t j R @5 S q n C  3S 6K 8B :; 5A ,G &O  ; | < Yg)\?R_C{   ^ 5  - < K U ^w j` pT wJ }> ~; }: 9 }@ yH wO of l| h a ^ [ WU T R P O-N_MyON** ***"))(>$S%h# 0C\mw{zO NN M LKKK&L/N@RRWc[zafmsw{cL/6B-IWe s1|=Vt4Pbep78@                          @       @                ""    ? $? 4|F̛]hF h FXTO""dFHDEO9FO9W 8 e2 C3qC8?"*?d*2A*XB+B2+BU+Bi+ @      "c    ? $3? 4|F̛]Z h FXTO"dF@HEO9FO9W 8 e2 C3qC8??HEr&E y"^$#OId`( RX,K(-RĤ\  {T     @1AB2CDE,FB1o?KK17131-1)1'1(1%1)1,101=1K1Y2l92X2s2222222222222222 2&2Dt2be2~V2I2@272/2.2)2&2$2$2"2&2'2"2111111111111212R2t2w2n2d2[2Y2M2N3P2\2a @        @        @        "    ? $o? 4|F̛][  h FXTO"!dFHDEO9FO9W 8 e2 C3qC8??'ErE K&aٸfr۶5wЂceK dc2kV96V,;  P~ C f7fʢᄖY@]$H[ SxD3)33sw>  M~ D&b XP25_ cm`UM\K4Zee{  1~  F f    V    @?A!B@CV"DEpF1o??!?!?!?!?!?!?!?!?!?!?!?!?!@!@!7@!W@ "y@"@*"@6"@A"@L"@O"@R"@U"@Q"@L"@H" @         "    ? $? 4|F̛] h FXTO"!dFHDEO9FO9W 8 e2 C3qC8??'ErE L'z,%ˎ9%fu<==̀fʶ͹K8\$A{z  );~v N f   fS    @?Ap!B@C"DEpF1o?@p!@r!@o!@q!@r!@r!@s!@v!@~!@!@!@!@!g@!M@!5@"@#"@B"?_"?p"?"?"?"?"?"?"?"?" @         "    ? $? 4|F̛] h FXTO"!dFHDEO9FO9W 8 e2 C3qC8??'ErE I%Wfږ3$w<](&G#ݶ{ซXLW7||%  ̕~r 0   @?A!B#AC$"DEXF1o??!"?!"?""?!"? "? "?"?"?"@"3@ "d@"@!@!@!@! A!A! A!"A!#A!A! @       "     ? $? 4|F̛]$  h FXTO"!dFHDEO9FO9W 8 e2 C3qC8??'ErE BCt˗$Kowװ?=͖*e2~/ 9T6\\ؐ  ~f y   @q&Az$B(C%DEF:1o???&$&$&$&$&$&$&$ '%/'%S'3%r'G%'\%'q%'~%'%'%'%'%'%'z$'{$'|$'~$'$'$'$'$'$|'$v'%q'*%j'O%d'p%\'%S'%N'%I'%C'%>'%?'%?'%s&%u&%&%&%&%&%&%&p%'Y%#'B%C'+%d'%'%'$'$'$'$'$'$($($($@      @       @       "    ? $f? 4|F̛]R"  h FXTO"!dFHDEO9FO9W 8 e2 C3qC8??'ErE DR)Znf̂K.\ dˈ@3ۜب `  `~ A6۽%TeCI&]2Ic;;ng{Kn"!JW7;Ϟ  Z~ D1W*Z\oT"1q|Kz*S[H  }~a^   # B`CxDEFo H0`x@  S" .  <r@%F  as(a)-brane corresponds to the static configuration of surface operators with a half-twist*b ^  B^x t P s(a)-brane* H  0޽h ? fp=___PPT10i.x֩6+D=' = @B +l% 0aN0 k$c$ #(  X  S 00Avortex27 p  # BCPDE(Fo P|(  @   S" ^.$L B  # f   8@r A9 B{CDEXFt1o?r u x 2} C U g q z z t j R @5 S q n C  3S 6K 8B :; 5A ,G &O  ; | < Yg)\?R_C{   ^ 5  - < K U ^w j` pT wJ }> ~; }: 9 }@ yH wO of l| h a ^ [ WU T R P O-N_MyON** ***"))(>$S%h# 0C\mw{zO NN M LKKK&L/N@RRWc[zafmsw{cL/6B-IWe s1|=Vt4Pbep78@                          @       @                ""    ? $? 4|F̛]hF h FXTO""dFHDEO9FO9W 8 e2 C3qC8?"*?d*2A*XB+B2+BU+Bi+ @      "c    ? $3? 4|F̛]Z h FXTO"dF@HEO9FO9W 8 e2 C3qC8??HEr&E y"^$#OId`( RX,K(-RĤ\  {T     @1AB2CDE,FB1o?KK17131-1)1'1(1%1)1,101=1K1Y2l92X2s2222222222222222 2&2Dt2be2~V2I2@272/2.2)2&2$2$2"2&2'2"2111111111111212R2t2w2n2d2[2Y2M2N3P2\2a @        @        @        "    ? $o? 4|F̛][  h FXTO"!dFHDEO9FO9W 8 e2 C3qC8??'ErE K&aٸfr۶5wЂceK dc2kV96V,;  P~ C f7fʢᄖY@]$H[ SxD3)33sw>  M~ D&b XP25_ cm`UM\K4Zee{  1~  F f    V    @?A!B@CV"DEpF1o??!?!?!?!?!?!?!?!?!?!?!?!?!@!@!7@!W@ "y@"@*"@6"@A"@L"@O"@R"@U"@Q"@L"@H" @         "    ? $? 4|F̛] h FXTO"!dFHDEO9FO9W 8 e2 C3qC8??'ErE L'z,%ˎ9%fu<==̀fʶ͹K8\$A{z  );~v N f   fS   @?Ap!B@C"DEpF1o?@p!@r!@o!@q!@r!@r!@s!@v!@~!@!@!@!@!g@!M@!5@"@#"@B"?_"?p"?"?"?"?"?"?"?"?" @         "    ? $? 4|F̛] h FXTO"!dFHDEO9FO9W 8 e2 C3qC8??'ErE I%Wfږ3$w<](&G#ݶ{ซXLW7||%  ̕~r 0   @?A!B#AC$"DEXF1o??!"?!"?""?!"? "? "?"?"?"@"3@ "d@"@!@!@!@! A!A! A!"A!#A!A! @       "     ? $? 4|F̛]$  h FXTO"!dFHDEO9FO9W 8 e2 C3qC8??'ErE BCt˗$Kowװ?=͖*e2~/ 9T6\\ؐ  ~f y   @q&Az$B(C%DEF:1o???&$&$&$&$&$&$&$ '%/'%S'3%r'G%'\%'q%'~%'%'%'%'%'%'z$'{$'|$'~$'$'$'$'$'$|'$v'%q'*%j'O%d'p%\'%S'%N'%I'%C'%>'%?'%?'%s&%u&%&%&%&%&%&%&p%'Y%#'B%C'+%d'%'%'$'$'$'$'$'$($($($@      @       @       "    ? $f? 4|F̛]R"  h FXTO"!dFHDEO9FO9W 8 e2 C3qC8??'ErE DR)Znf̂K.\ dˈ@3ۜب `  `~ A6۽%TeCI&]2Ic;;ng{Kn"!JW7;Ϟ  Z~ D1W*Z\oT"1q|Kz*S[H  }~a^   <8 ;3   <L@UF  gs (a)-brane corresponds to the static configuration of surface operators with three half-twists*h cL @ 2p  # H r  <@ 2p  P s(a)-brane*   <|p   ;3  H  0޽h ? fp=___PPT10i.x֩6+D=' = @B +? 0aN0 >> U>(  RB  s *Do  # BCPDE(Fo P|(  @   S" ^.$L B  # f   8@r A9 B{CDEXFt1o?r u x 2} C U g q z z t j R @5 S q n C  3S 6K 8B :; 5A ,G &O  ; | < Yg)\?R_C{   ^ 5  - < K U ^w j` pT wJ }> ~; }: 9 }@ yH wO of l| h a ^ [ WU T R P O-N_MyON** ***"))(>$S%h# 0C\mw{zO NN M LKKK&L/N@RRWc[zafmsw{cL/6B-IWe s1|=Vt4Pbep78@                          @       @                ""    ? $? 4|F̛]hF h FXTO""dFHDEO9FO9W 8 e2 C3qC8?"*?d*2A*XB+B2+BU+Bi+ @      "c    ? $3? 4|F̛]Z h FXTO"dF@HEO9FO9W 8 e2 C3qC8??HEr&E y"^$#OId`( RX,K(-RĤ\  {T     @1AB2CDE,FB1o?KK17131-1)1'1(1%1)1,101=1K1Y2l92X2s2222222222222222 2&2Dt2be2~V2I2@272/2.2)2&2$2$2"2&2'2"2111111111111212R2t2w2n2d2[2Y2M2N3P2\2a @        @        @        "    ? $o? 4|F̛][  h FXTO"!dFHDEO9FO9W 8 e2 C3qC8??'ErE K&aٸfr۶5wЂceK dc2kV96V,;  P~ C f7fʢᄖY@]$H[ SxD3)33sw>  M~ D&b XP25_ cm`UM\K4Zee{  1~  F f    V    @?A!B@CV"DEpF1o??!?!?!?!?!?!?!?!?!?!?!?!?!@!@!7@!W@ "y@"@*"@6"@A"@L"@O"@R"@U"@Q"@L"@H" @         "    ? $? 4|F̛] h FXTO"!dFHDEO9FO9W 8 e2 C3qC8??'ErE L'z,%ˎ9%fu<==̀fʶ͹K8\$A{z  );~v N f   fS   @?Ap!B@C"DEpF1o?@p!@r!@o!@q!@r!@r!@s!@v!@~!@!@!@!@!g@!M@!5@"@#"@B"?_"?p"?"?"?"?"?"?"?"?" @         "    ? $? 4|F̛] h FXTO"!dFHDEO9FO9W 8 e2 C3qC8??'ErE I%Wfږ3$w<](&G#ݶ{ซXLW7||%  ̕~r 0   @?A!B#AC$"DEXF1o??!"?!"?""?!"? "? "?"?"?"@"3@ "d@"@!@!@!@! A!A! 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( _   >v*a4" }Y y u r n+!o^!p!s!!-"?">"(>"E3"c)"{!""",0@                   @       @            "    ? $? 4|F̛]\^ h FXTO""dFHDEO9FO9W 8 e2 C3qC8)R??8FcEE _ \$s&dEvݝl[ Fule(jULI#&ffLm׷[vwkYcC"RbIE` A@L\Dԓ%>}  PjvĂ D:FQՌ/C(nU0Qg||_`  -Ă `;,, Mɹ*$2d."{$eɜs31d䒥]{{w^eKZf0"$MF  Ă  4  <0 BClosing the braid gives = space of a - s (a) stringsbC 1  <t@  ;3 e   @c#A B-&C^DEFV1o?ssc#Zg#Ze#^j#]o#\q#Zu#X}#S}#S#L#F#?#7### # # # # # # # # # # # # # # # # # # $ $ $ $ !$ +$ 4$ >$ G$ T$]$ f$q$u$~$$$$$ $$ $ $ $ $ $ $ $ $ $ $ $y $k $^ $? $ $ $ $ $ $ $ $ $ $ $ % %( %7 %K !%a 1%q 9%y @% J% \% l% ~% % %v %l %_ %S %E %8 %* % % % % % % % % % & & & "& %& (& ,& (,@                                      "    ? $f? 4|F̛]R8" h FXTO"dF@HEO9FO9W 8 e2 C3qC8:?"(UYcB&"Mgrٳc, BDdˑͅgbjն%ʂDIr16Qh%bK(%,ՖTdYibIJ,K(I`(DYio  pw,B c   @4A B5C%DE$F:1o?II4 444 4)414: 4E4N4U4])4_54_S4^v4M4F4B4>4=4=4;4@4E4F4N4W4`4i4{444444444 44$4,41464:4;4;4;484541 5,5'65T5s555555!515G5c5uz5j5\5O5:5;5 A5B5D5$@                        "X    ? $(? 4|F̛]$, h FXTO"dF@HEO9FO9W 8 e2 C3qC8:?"<1B1AZ>AO>AB>A,>A>A>B=UB=B=B=C=rC=C=8D=D=E=LE=E=E=E=E=E>B>B>B?B8?Bc?B?B?B?B?wB@dB7@OBY@=BX@>BS@@BS@ABQ@GBP@LBP@OBO@SBP@ZBO@`BD@B@@B=@$C:@dC8@C7@C6@#D.@`D%@D@D@D?E?/E?CE?AE??E?$El?EZ?DZ?DS?DO?CS?CI?C[?Cq?C?C?C?C?C#@C[@C@C@C@C'ACUACkACpAC}ACA6BA,BA)BABA$BABA BAt#*RД~N  ĕ!s E   <X0p  = D (q) Alexander polynomialf!   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"2;]62 6=3@ &!!LO*2 20T [*]^"  ek9mnoNp/qZcuuBw x{|} !=*2 p   0NO!=2d 0d !=*2  b$e ^5(Bb$؄ Լ'jn ovteAb$@O@G4Bb$4kHRKs%?vR$)6Nwyqngb$Y&u{/L _Ob$;{Pwa_Z5 Xb$'E?zVzab$ܻ|8OӝGͬm'ub$I|7kP}W ʸb$b&9^gsz "$&ƌ8nwLvn 2b$@OO ۧ'Pjb2${_Gr&(Zb$ֿA:ɢ'ѯvřAR$a[:LMy%{E2$tLYR& b$qg0 `yPz_b$֖w!!ifb$7' 2f9m_@b$h'u-&P%fH)Tb$y &I>Jqlb$Cb^*l!l:~8b$sϭf]erbb$,`B1,O`d R$dQ@sE ؉ [j=mb$ EG] ib$CvH.sD)b$v7 :*3_,/K,b$კS(jC5b$u֦jo=wfN:b$p<>`vPLوUP<ɇ 0AA 333MMM|@8Qʚ;aݫ8ʚ;g4BdBd`T0Fppp@ <4dddd!0L/ <4!d!dO$0L(PREAMBLE\documentclass{article} \pagestyle{empty} \usepackage{xspace,amssymb,amsfonts,amsmath} \usepackage{color} \usepackage{TeX4PPT}   MAGPC200$FONTSIZE180___PPT10 ?  %`}'Topological Strings and Knot Homologies Sergei Gukov|OutlineTopological Strings:4Applications of Top. StringsSUSY sector: F-terms SUSY Gauge Theory [Dijkgraaf-Vafa] Black Hole Physics [OSV] Math: & ( (Perturbative Top. 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"2;]62 6=3@ &!!LO*2 20T [*]^"  ek9mnoNp/qZcuuBw x{|} !=*2 p   0NO!=2d 0d !=*2  b$e ^5(Bb$؄ Լ'jn ovteAb$@O@G4Bb$4kHRKs%?vR$)6Nwyqngb$Y&u{/L _Ob$;{Pwa_Z5 Xb$'E?zVzab$ܻ|8OӝGͬm'ub$I|7kP}W ʸb$b&9^gsz "$&ƌ8nwLvn 2b$@OO ۧ'Pjb2${_Gr&(Zb$ֿA:ɢ'ѯvřAR$a[:LMy%{E2$tLYR& b$qg0 `yPz_b$֖w!!ifb$7' 2f9m_@b$h'u-&P%fH)Tb$y &I>Jqlb$Cb^*l!l:~8b$sϭf]erbb$,`B1,O`d R$dQ@sE ؉ [j=mb$ EG] ib$CvH.sD)b$v7 :*3_,/K,b$კS(jC5b$u֦jo=wfN:b$p<>`vPLوUP<ɇ 0AA 333MMM|@8Qʚ;aݫ8ʚ;g4BdBd`T0Fppp@ <4dddd!0L/ <4!d!dO$0L(PREAMBLE\documentclass{article} \pagestyle{empty} \usepackage{xspace,amssymb,amsfonts,amsmath} \usepackage{color} \usepackage{TeX4PPT}   MAGPC200$FONTSIZE180___PPT10 ?  %}'Topological Strings and Knot Homologies Sergei Gukov|OutlineTopological Strings:4Applications of Top. StringsSUSY sector: F-terms SUSY Gauge Theory [Dijkgraaf-Vafa] Black Hole Physics [OSV] Math: & ( (Perturbative Top. String>A-model/B-model Z_top Holomorphic Anomaly eqn. Mirror Symmetry>  Computng Z_top$4Non-compact CY : localization Dualities Gauge TheoryD-branes1Categories Matrix Factorizations Open Top. StringLarge N Dualities1What is Topological String Theory? What is Z_top?+Open/Closed Table1What is Topological String Theory? 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Mirror Symmetry,Perturbative Topological String ( Perturbative Topological String ( Perturbative Topological String (Holomorphic Anomaly Wave Function Interpretation Computing Z_top$ 4Non-compact CY : localization Dualities Gauge TheoryD-branes1Categories Matrix Factorizations Open Top. StringLarge N Dualities1What is Topological String Theory? What is Z_top?+Open/Closed Table1What is Topological String Theory? 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What is Z_top?+E?HBPolynomial Knot InvariantsIDPolynomial Knot InvariantsKEPolynomial Knot InvariantsLFPolynomial Knot InvariantsMGPolynomial Knot Invariants>8sjCategorification /-Categorification YPCategorification Knot homologyOHKnot HomologiesRKKnot HomologiesUMKnot Homologies $A general picture of knot homologies%% TL Physical Interpretation   qSurprisingly, this physical interpretation leads to a rich theory, which unifies all the existing knot homologiesrr?9WN XOFamilies of Differentials@:6Matrix Factorizations, Deformations, and Differentials77(neripg[RWhat s Next?Generalization to other groups and representations The role of matrix factorizations Finite N (stringy exclusion principle) Realization in topological gauge theory~k!Gauge Theory and Categorification""$l!Gauge Theory and Categorification""$m!Gauge Theory and Categorification""$n oGauge Theory with Boundaries(pGauge Theory with Boundaries(qGauge Theory with Corners(rGauge Theory with Corners(sFrom Lines to SurfacesA line operator lifts to an operator in 4D gauge theory localized on the surface where the gauge field A has a prescribed singularity,wt Sometimes a surface operator (a.k.a.  impurity or  singular vortex ) can be also described using:cbuBraid Group Actions on D-branes (Any four-dimensional topological gauge theory which admits supersymmetric surface operators provides (new) examples of braid group actions on D-branes,f.v  w  x  y  z  {'Topological Twists of SUSY Gauge Theory((  ~  qhw 0  (  r  S     X  0A#?  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"2;]62 6=3@ &!!LO*2 20T [*]^"  ek9mnoNp/qZcuuBw x{|} !=*2 p   0NO!=2d 0d !=*2  Rob$e ^5(Bb$؄ Լ'jn ovteAb$@O@G4Bb$4kHRKs%?vR$)6Nwyqngb$Y&u{/L _Ob$;{Pwa_Z5 Xb$'E?zVzab$ܻ|8OӝGͬm'ub$I|7kP}W ʸb$b&9^gsz "$&ƌ8nwLvn 2b$@OO ۧ'Pjb2${_Gr&(Zb$ֿA:ɢ'ѯvřAR$a[:LMy%{E2$tLYR& b$qg0 `yPz_b$֖w!!ifb$7' 2f9m_@b$h'u-&P%fH)Tb$y &I>Jqlb$Cb^*l!l:~8b$sϭf]erbb$,`B1,O`d R$dQ@sE ؉ [j=mb$ EG] ib$CvH.sD)b$v7 :*3_,/K,b$კS(jC5b$u֦jo=wfN:b$p<>`vPLوUP<ɇb$u9;Y2$b*A^DHW b$s|դuhb&0R$NaRA G dcb$->k ; VR$FE-Qڛb>"&n 0AA 333MMM|@8ߋʚ;>9ʚ;g4BdBdTaT0Lppp@ <4dddd!0L0 <4!d!dO$0L(PREAMBLE\documentclass{article} \pagestyle{empty} \usepackage{xspace,amssymb,amsfonts,amsmath} \usepackage{color} \usepackage{TeX4PPT}   MAGPC200$FONTSIZE180___PPT10 ?  %}'Topological Strings and Knot Homologies Sergei Gukov|Outline)Introduction to Topological String Theory:4Applications of Top. StringsSUSY sector: F-terms SUSY Gauge Theory [Dijkgraaf-Vafa] Black Hole Physics [OSV] Math: & ( (Perturbative Top. String>A-model/B-model Z_top Holomorphic Anomaly eqn. Mirror Symmetry,Perturbative Topological String ( Perturbative Topological String ( Perturbative Topological String (Holomorphic Anomaly Wave Function Interpretation Computing Z_top 4Non-compact CY : localization Dualities Gauge Theory ComputingD-branes1Categories Matrix Factorizations Open Top. StringLarge N Dualities1What is Topological String Theory? What is Z_top?+Enumerative InvariantsOpen/Closed Table1What is Topological String Theory? What is Z_top?+E?HBPolynomial Knot InvariantsIDPolynomial Knot InvariantsKEPolynomial Knot InvariantsLFPolynomial Knot InvariantsMGPolynomial Knot Invariants>8sjCategorification /-Categorification YPCategorification Knot homologyOHKnot HomologiesRKKnot HomologiesUMKnot Homologies $A general picture of knot homologies%% TL Physical Interpretation   qSurprisingly, this physical interpretation leads to a rich theory, which unifies all the existing knot homologiesrr?9WN XOFamilies of Differentials@:6Matrix Factorizations, Deformations, and Differentials77(neripg[RWhat s Next?Generalization to other groups and representations The role of matrix factorizations Finite N (stringy exclusion principle) Realization in topological gauge theory~k!Gauge Theory and Categorification""$l!Gauge Theory and Categorification""$m!Gauge Theory and Categorification""$n oGauge Theory with Boundaries(pGauge Theory with Boundaries(qGauge Theory with Corners(rGauge Theory with Corners(sFrom Lines to SurfacesA line operator lifts to an operator in 4D gauge theory localized on the surface where the gauge field A has a prescribed singularity,wt Sometimes a surface operator (a.k.a.  impurity or  singular vortex ) can be also described using:cbuBraid Group Actions on D-branes (Any four-dimensional topological gauge theory which admits supersymmetric surface operators provides (new) examples of braid group actions on D-branes,f.v  w  x  y  z  {'Topological Twists of SUSY Gauge Theory((  ~  qh! 0   4 (  4r 4 S t/   / n 0  74 #"   0  / 4 Bd/?pp  WKnot Homologies @` 4 B\/? p p VBPS invariants @` 4 BH /?p  fopen GW (relative stable maps) @` 4 BQ/?p  LOpen @`  4 BH/?pP p  S Equivariant   @`  4 B@/? Ppp  b DT (ideal sheaves) @`  4 B`8/?P p  _ GW (stable maps) @`  4 B /?Pp  NClosed @`  4 h/A&?Parchment" p0 P X Refinement   @`( 4 h0/A&?Parchment"  0pP z"Integer (gauge theory, embeddings)&# @` 4 h)/A&?Parchment" 0 P gRational (maps)&  @` 4 h.A&?Parchment" 0P @ @`fB 4 6o ?0 0`B 4 01 ?P P`B 4 01 ?p p fB 4 6o ? fB 4 6o ?0`B 4 01 ?0`B 4 01 ? 0 `B 4 01 ?p0pfB 4 6o ? 0  >4  @AX(BC.DEF 1o?""I)A))S P)\ )y (~ o( ?(k()I)*|+|,GA--X.$ .4 . /h/.Z.F.~-,,,A+=) (((O);) @           "    ? $ǀ?4|F̛]l h FXTO"!dFHDEO9FO9W 8 e2 C3qC8B?_8?BXE ^}3UKRpT(&213333''IrXV]w{ۮ˦۫ee;Zg]lٱ+"L39",]Mmiu֮3WZFYU.2rd32fgr13I^m*B!*J,(,IbĹRK(Ye,%"K,go(  'A&   B4  N@A+BC,DEF,1o?//,,,,,,,,,,2,a,,, ,I,,,i,v<,,+A+su+H+7+$++++++l+K+,+ ++++\+-++,2,L,d,,@         @      "c    ? $3?4|F̛]T( h FXTO"!dFHDEO9FO9W 8 e2 C3qC86?_8?BPE T&DTbXÄb*RsǛgVeیg7b:H[J"*"RV@  P{$.A D  #:WSt H*0M&f.&ao@  .A {.  C4  @Ax*B C,DEF$1o?## x* * * ***+Y+ +O + $, T, s,J , , , ,C , s, ++ 0+ 7+ D+ T+< g+W +k +z +r +j +M ,8 1,% =, @, K,@      @     "Q    ? $!?4|F̛] 4. h FXTO"!dFHDEO9FO9W 8 e2 C3qC86?_8?BPE HlJcUX` iD&7-oy@لf L&wd  0/A >߯js|ȃ F8 eАI4QC=,  H0A   V4  8@A:B' C?DEF&1o?++;; };;;;*?e?J???z?h?,?>C [> = V= <# {<- &<" ; ; (; ; :s; n;;"?2?@S@t@@@k@P@Z@ ? ?h ? u? Z? D? H?m K?T??l?W?@?,?%?? >>$5><I>U^>rl>v>>>8>^>>>}>r> d>&X>AH>Z7>>====s=Y=<= ===<<<>),@                @           @        "r    ? $B?4|F̛].| h FXTO"!dFHDEO9FO9W 8 e2 C3qC86?_8?B@%F 3 <,f s.%˕B-R͖eTY͔j֭vlUTEMR-ʶͥJB$A!,TMJϠ`{˂YbڊeYE6[*%IbbI"DȂU˼-f(j+e)yVnl%Ŗ,@TTY.X\o%%5pAl,XBŅ,eT`ŋTè`T`,`,,\e`e-  A V2I%d,gyܶjq,k[kVUٵ@LO;ۀJf2(.d  @A D!V[>w6Ur\Ђn #*Ѳɐy 5  iy!! "2;]62 6=3@ &!!LO*2 20T [*]^"  ek9mnoNp/qZcuuBw x{|} !=*2 p   0NO!=2d 0d !=*2  Rob$e ^5(Bb$؄ Լ'jn ovteAb$@O@G4Bb$4kHRKs%?vR$)6Nwyqngb$Y&u{/L _Ob$;{Pwa_Z5 Xb$'E?zVzab$ܻ|8OӝGͬm'ub$I|7kP}W ʸb$b&9^gsz "$&ƌ8nwLvn 2b$@OO ۧ'Pjb2${_Gr&(Zb$ֿA:ɢ'ѯvřAR$a[:LMy%{E2$tLYR& b$qg0 `yPz_b$֖w!!ifb$7' 2f9m_@b$h'u-&P%fH)Tb$y &I>Jqlb$Cb^*l!l:~8b$sϭf]erbb$,`B1,O`d R$dQ@sE ؉ [j=mb$ EG] ib$CvH.sD)b$v7 :*3_,/K,b$კS(jC5b$u֦jo=wfN:b$p<>`vPLوUP<ɇb$u9;Y2$b*A^DHW b$s|դuhb&0R$NaRA G dcb$->k ; VR$FE-Qڛb>"&n 0AA 333MMM|@8ߋʚ;>9ʚ;g4BdBdTaT0Lppp@ <4dddd!0L0 <4!d!dO$0L(PREAMBLE\documentclass{article} \pagestyle{empty} \usepackage{xspace,amssymb,amsfonts,amsmath} \usepackage{color} \usepackage{TeX4PPT}   MAGPC200$FONTSIZE180___PPT10 ?  %k}'Topological Strings and Knot Homologies Sergei Gukov|Outline)Introduction to Topological String Theory:4Applications of Top. StringsSUSY sector: F-terms SUSY Gauge Theory [Dijkgraaf-Vafa] Black Hole Physics [OSV] Math: & ( (Perturbative Top. String>A-model/B-model Z_top Holomorphic Anomaly eqn. Mirror Symmetry,Perturbative Topological String ( Perturbative Topological String ( Perturbative Topological String (Holomorphic Anomaly Wave Function Interpretation Computing Z_top 4Non-compact CY : localization Dualities Gauge Theory ComputingD-branes1Categories Matrix Factorizations Open Top. StringLarge N Dualities1What is Topological String Theory? What is Z_top?+Enumerative Invariants1What is Topological String Theory? What is Z_top?+E?HBPolynomial Knot InvariantsIDPolynomial Knot InvariantsKEPolynomial Knot InvariantsLFPolynomial Knot InvariantsMGPolynomial Knot Invariants>8sjCategorification /-Categorification YPCategorification Knot homologyOHKnot HomologiesRKKnot HomologiesUMKnot Homologies $A general picture of knot homologies%% TL Physical Interpretation   qSurprisingly, this physical interpretation leads to a rich theory, which unifies all the existing knot homologiesrr?9WN XOFamilies of Differentials@:6Matrix Factorizations, Deformations, and Differentials77(neripg[RWhat s Next?Generalization to other groups and representations The role of matrix factorizations Finite N (stringy exclusion principle) Realization in topological gauge theory~k!Gauge Theory and Categorification""$l!Gauge Theory and Categorification""$m!Gauge Theory and Categorification""$n oGauge Theory with Boundaries(pGauge Theory with Boundaries(qGauge Theory with Corners(rGauge Theory with Corners(sFrom Lines to SurfacesA line operator lifts to an operator in 4D gauge theory localized on the surface where the gauge field A has a prescribed singularity,wt Sometimes a surface operator (a.k.a.  impurity or  singular vortex ) can be also described using:cbuBraid Group Actions on D-branes (Any four-dimensional topological gauge theory which admits supersymmetric surface operators provides (new) examples of braid group actions on D-branes,f.v  w  x  y  z  {'Topological Twists of SUSY Gauge Theory((  ~  qh! 0   4 (  4r 4 S t/   / n 0  74 #"   0  / 4 Bd/?pp  WKnot Homologies @` 4 B\/? p p VBPS invariants @` 4 BH /?p  fopen GW (relative stable maps) @` 4 BQ/?p  LOpen @`  4 BH/?pP p  S Equivariant   @`  4 B@/? Ppp  b DT/GV (ideal sheaves) @`  4 B`8/?P p  _ GW (stable maps) @`  4 B /?Pp  NClosed @`  4 h/A&?Parchment" p0 P X Refinement   @`( 4 h0/A&?Parchment"  0pP z"Integer (gauge theory, embeddings)&# @` 4 h)/A&?Parchment" 0 P gRational (maps)&  @` 4 h.A&?Parchment" 0P @ @`fB 4 6o ?0 0`B 4 01 ?P P`B 4 01 ?p p fB 4 6o ? fB 4 6o ?0`B 4 01 ?0`B 4 01 ? 0 `B 4 01 ?p0pfB 4 6o ? 0  >4  @AX(BC.DEF 1o?""I)A))S P)\ )y (~ o( ?(k()I)*|+|,GA--X.$ .4 . /h/.Z.F.~-,,,A+=) (((O);) @           "    ? $ǀ?4|F̛]l h FXTO"!dFHDEO9FO9W 8 e2 C3qC8B?_8?BXE ^}3UKRpT(&213333''IrXV]w{ۮ˦۫ee;Zg]lٱ+"L39",]Mmiu֮3WZFYU.2rd32fgr13I^m*B!*J,(,IbĹRK(Ye,%"K,go(  'A&   B4  N@A+BC,DEF,1o?//,,,,,,,,,,2,a,,, ,I,,,i,v<,,+A+su+H+7+$++++++l+K+,+ ++++\+-++,2,L,d,,@         @      "c    ? $3?4|F̛]T( h FXTO"!dFHDEO9FO9W 8 e2 C3qC86?_8?BPE T&DTbXÄb*RsǛgVeیg7b:H[J"*"RV@  P{$.A D  #:WSt H*0M&f.&ao@  .A {.  C4  @Ax*B C,DEF$1o?## x* * * ***+Y+ +O + $, T, s,J , , , ,C , s, ++ 0+ 7+ D+ T+< g+W +k +z +r +j +M ,8 1,% =, @, K,@      @     "Q    ? $!?4|F̛] 4. h FXTO"!dFHDEO9FO9W 8 e2 C3qC86?_8?BPE HlJcUX` iD&7-oy@لf L&wd  0/A >߯js|ȃ F8 eАI4QC=,  H0A   V4  8@A:B' C?DEF&1o?++;; };;;;*?e?J???z?h?,?>C [> = V= <# {<- &<" ; ; (; ; :s; n;;"?2?@S@t@@@k@P@Z@ ? ?h ? u? Z? D? H?m K?T??l?W?@?,?%?? >>$5><I>U^>rl>v>>>8>^>>>}>r> d>&X>AH>Z7>>====s=Y=<= ===<<<>),@                @           @        "r    ? $B?4|F̛].| h FXTO"!dFHDEO9FO9W 8 e2 C3qC86?_8?B@%F 3 <,f s.%˕B-R͖eTY͔j֭vlUTEMR-ʶͥJB$A!,TMJϠ`{˂YbڊeYE6[*%IbbI"DȂU˼-f(j+e)yVnl%Ŗ,@TTY.X\o%%5pAl,XBŅ,eT`ŋTè`T`,`,,\e`e-  A V2I%d,gyܶjq,k[kVUٵ@LO;ۀJf2(.d  @A D!V[>w6Ur\Ђn #*Ѳɐy?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|0Dcmti10arrow0LL(0ԖT0Ԗ"@DSymbolarrow0LL(0ԖT0Ԗ C .  @n?" dd@  @@``   8 > 5  iy!! "2;]62 6=3@ &!!LO*2 20T [*]^"  ek9mnoNp/qZcuuBw x{|} !=*2 p   0NO!=2d 0d !=*2  Rob$e ^5(Bb$؄ Լ'jn ovteAb$@O@G4Bb$4kHRKs%?vR$)6Nwyqngb$Y&u{/L _Ob$;{Pwa_Z5 Xb$'E?zVzab$ܻ|8OӝGͬm'ub$I|7kP}W ʸb$b&9^gsz "$&ƌ8nwLvn 2b$@OO ۧ'Pjb2${_Gr&(Zb$ֿA:ɢ'ѯvřAR$a[:LMy%{E2$tLYR& b$qg0 `yPz_b$֖w!!ifb$7' 2f9m_@b$h'u-&P%fH)Tb$y &I>Jqlb$Cb^*l!l:~8b$sϭf]erbb$,`B1,O`d R$dQ@sE ؉ [j=mb$ EG] ib$CvH.sD)b$v7 :*3_,/K,b$კS(jC5b$u֦jo=wfN:b$p<>`vPLوUP<ɇb$u9;Y2$b*A^DHW b$s|դuhb&0R$NaRA G dcb$->k ; VR$FE-Qڛb>"&n 0AA 333MMM|@8ߋʚ;>9ʚ;g4BdBdTaT0Lppp@ <4dddd!0L0 <4!d!dO$0L(PREAMBLE\documentclass{article} \pagestyle{empty} \usepackage{xspace,amssymb,amsfonts,amsmath} \usepackage{color} \usepackage{TeX4PPT}   MAGPC200$FONTSIZE180___PPT10 ?  %k}'Topological Strings and Knot Homologies Sergei Gukov|Outline)Introduction to Topological String Theory:4Applications of Top. StringsSUSY sector: F-terms SUSY Gauge Theory [Dijkgraaf-Vafa] Black Hole Physics [OSV] Math: & ( (Perturbative Top. String>A-model/B-model Z_top Holomorphic Anomaly eqn. Mirror Symmetry,Perturbative Topological String ( Perturbative Topological String ( Perturbative Topological String (Holomorphic Anomaly Wave Function Interpretation Computing Z_top 4Non-compact CY : localization Dualities Gauge Theory ComputingD-branes1Categories Matrix Factorizations Open Top. StringLarge N Dualities1What is Topological String Theory? What is Z_top?+Enumerative Invariants1What is Topological String Theory? What is Z_top?+E?HBPolynomial Knot InvariantsIDPolynomial Knot InvariantsKEPolynomial Knot InvariantsLFPolynomial Knot InvariantsMGPolynomial Knot Invariants>8sjCategorification /-Categorification YPCategorification Knot homologyOHKnot HomologiesRKKnot HomologiesUMKnot Homologies $A general picture of knot homologies%% TL Physical Interpretation   qSurprisingly, this physical interpretation leads to a rich theory, which unifies all the existing knot homologiesrr?9WN XOFamilies of Differentials@:6Matrix Factorizations, Deformations, and Differentials77(neripg[RWhat s Next?Generalization to other groups and representations The role of matrix factorizations Finite N (stringy exclusion principle) Realization in topological gauge theory~k!Gauge Theory and Categorification""$l!Gauge Theory and Categorification""$m!Gauge Theory and Categorification""$n oGauge Theory with Boundaries(pGauge Theory with Boundaries(qGauge Theory with Corners(rGauge Theory with Corners(sFrom Lines to SurfacesA line operator lifts to an operator in 4D gauge theory localized on the surface where the gauge field A has a prescribed singularity,wt Sometimes a surface operator (a.k.a.  impurity or  singular vortex ) can be also described using:cbuBraid Group Actions on D-branes (Any four-dimensional topological gauge theory which admits supersymmetric surface operators provides (new) examples of braid group actions on D-branes,f.v  w  x  y  z  {'Topological Twists of SUSY Gauge Theory((  ~  qh! 0 ! 4 (  4r 4 S t/   / z 0  4 #"   0  / 4 Bd/?pp  WKnot Homologies @` 4 B\/? p p VBPS invariants @` 4 BH /?p  fopen GW (relative stable maps) @` 4 BQ/?p  LOpen @`  4 BH/?pP p  S Equivariant   @`  4 B@/? 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"2;]62 6=3@ &!!LO*2 20T [*]^"  ek9mnoNp/qZcuuBw x{|} !=*2 p   0NO!=2d 0d !=*2  ,RB b$e ^5(Bb$؄ Լ'jn ovteAb$@O@G4Bb$4kHRKs%?vR$)6Nwyqngb$Y&u{/L _Ob$;{Pwa_Z5 Xb$'E?zVzab$ܻ|8OӝGͬm'ub$I|7kP}W ʸb$b&9^gsz "$&ƌ8nwLvn 2b$@OO ۧ'Pjb2${_Gr&(Zb$ֿA:ɢ'ѯvřAR$a[:LMy%{E2$tLYR& b$qg0 `yPz_b$֖w!!ifb$7' 2f9m_@b$h'u-&P%fH)Tb$y &I>Jqlb$Cb^*l!l:~8b$sϭf]erbb$,`B1,O`d R$dQ@sE ؉ [j=mb$ EG] ib$CvH.sD)b$v7 :*3_,/K,b$კS(jC5b$u֦jo=wfN:b$p<>`vPLوUP<ɇb$u9;Y2$b*A^DHW b$s|դuhb&0R$NaRA G dcb$->k ; VR$FE-Qڛb>"&n$$R$O\ӜyYt 0AA 333MMM|@8ߋʚ;>9ʚ;g4BdBdTaT0Lppp@ <4dddd!0L0 <4!d!dO$0L(PREAMBLE\documentclass{article} \pagestyle{empty} \usepackage{xspace,amssymb,amsfonts,amsmath} \usepackage{color} \usepackage{TeX4PPT}   MAGPC200$FONTSIZE180___PPT10 ?  %}'Topological Strings and Knot Homologies Sergei Gukov|Outline)Introduction to Topological String Theory:4Applications of Top. StringsSUSY sector: F-terms SUSY Gauge Theory [Dijkgraaf-Vafa] Black Hole Physics [OSV] Math: & ( (Perturbative Top. 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String>A-model/B-model Z_top Holomorphic Anomaly eqn. Mirror Symmetry,Perturbative Topological String ( Perturbative Topological String ( Perturbative Topological String (Holomorphic Anomaly Wave Function Interpretation Mirror SymmetryComputing Z_top 4Non-compact CY : localization Dualities Gauge Theory ComputingD-branes1Categories Matrix Factorizations Open Top. StringLarge N Dualities1What is Topological String Theory? What is Z_top?+Enumerative Invariants1What is Topological String Theory? 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String>A-model/B-model Z_top Holomorphic Anomaly eqn. Mirror Symmetry,Perturbative Topological String ( Perturbative Topological String ( Perturbative Topological String (Holomorphic Anomaly Wave Function Interpretation Mirror SymmetryComputing Z_top 4Non-compact CY : localization Dualities Gauge Theory ComputingD-branes1Categories Matrix Factorizations Open Top. StringLarge N Dualities1What is Topological String Theory? What is Z_top?+1What is Topological String Theory? 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