intoning algebras and minimal models One laa-algebrt ( from algebraic topology, theaimofthistalkisto describe the minimal model construction Dee An Aa

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1 algebra miffmy Aa algebras and minimal models intoning own theaimofthistalkisto describe the minimal model construction and give an example seethe end for references Outline a algebras 2 the minimal model theorem 3 Example from singularity theory laaalgebrt from algebraic topology Dee An Aa isak graded vector space homotopy algebras see [ K ] for background A0r@A0soA0tcf16ms61suchthatfornZ1AoraAaA6t0ILdnt5tstmrnttfIramso1otjofmrHttADefIAmoThnfiABisfniAohBst A One >< An with operation mn A A n 1 K linear degree 2 n m A A deg +1 Mz A@A A deg O Mz A 3 A deg 1 A

2 isotoendoce intoning 2 my Examp says A m mz satisfies \ mile ab no mi ab Mila b+ tjlalam b mzis associative rex PYM d^ AmimDisaDGa1gebraMmanifo1djIngeh_lm3isahomotopymzl10MDemzCmz@11Examp1e_Forcb31eK1AQkH1e2kokemniOfrnCts3cBmzmnHiplicationlfA4isaZzgradedAaoa1gebramdleoiQejtljdt1WhIaEm1wnviangulatedcategoyofinteresteg ftp Dblwhx hmflw Fukl Y P Da enhancement of J egby lkinjecliveres I< Generator allobjects bniltfom E gahingthmedmepnimal Endple DG algebra usually a dim/ algebra quasi Knows everything about J H*EhdpF {mn}n>z Aa fdimk

3 To study moduli of Jitsetf or objects AT Lazawiu Lerche relations closed intwaihfzf Why find minimal models? 3 Topological string theory boundary sector Aa minimal cyclic strictly unital An categories Herbst Costello cyclicity unit constraints form s axiomsofopen TCFT monoidalfuncton Rie Gmp Suspendedforwardwmposihons technical point Let A { minaz bean An algebra define S A AN SC a a Ha A A kn a oan ftpj/9ill9jlmn1an@oa Cn A[D A[ Knarr n[k} D Cn so Kuo s cr+i+t nosignslfkesearethe suspended forward compositions 501 [ 1*0 0 degree +1 Lemont The data A { cn }naz satisfy for all n rotten * suspended forward An

4 d e st iopela C A fd ihtoain@ 2 the minimal model theorem A B qis 4 Let A J m bea DG algebra suspended forward products i A { Qm } sat * Astnithomotopyretacti of Aisa 2 graded vector space B and linear maps 2 such that it GAF B p I are degveezuomorphismsofcpxs where Bisgiuenzew differential HJTZH C ii IB poi C iii la iop ie B±H* A with a particular choice ofhowtopwject i2pas Zipa elements in A onto cocydes Th o oriented and connected planar trees with ntl leaves } 2 3 ^ { KK l % } linanot inflation Internal edge valency 3

5 intoning 0 Defy Given TEJN we define Pt lge B B by example 0Oy 0 BaB B / tloioi 1 of \ T Botten B ioh BH B[ k 13 # intedges C 1 pt porno Cia H 130 m 13 aioi Pn PT D theorem Minimal model B { A ]n3z is an Aa suspended forward products and there is an An algebra with quasi isomorphism A m 2 > B { A }na e% the minimalmodt recall B±H*A

6 ok n txnw 2 nw ][4i* ooh example tahr JNKOAINFC 3 Example 6 Let WEME Qki ] have an isolated singularity at 0 c dime #±Y Jaw write W x w t < N W Ny y3tz2 then #g n D 4 Dee Aw is the Be graded Dai algebra with underlying module Aw Ende Gica 164 a Clifford algebra generated by Ti satisfying [ xiig Yin 45*30 [ Yi 4*3 Ti*Y*s tj tgmdedwmmutubr with the usual algebra structure and differential 2cL [ Eixilittsiwik operator on Nato oc[x] %xi[yi* a ] + With ;D Examp W xd e EH d 32 W x xd W xd Yi 9 Aw Ende Eoe4o [ 2C ± x ] Mz [x] a x[4* a ]+xd [ Y y & ]! re x[ to! dtx t d D

7 aeon [ modules oak intoainfe O theorem Dyckerhoff There is a generator E for the category of matrix factorisation hmflw with DG Hence there is an equivalence endomouphism algebra Aw perttw ± hmffw perfect Aw ~Dk±2M PEAKH Theminimalmodd of Aw Tm Cio a Daalg with to S0 tw # SQEAWIS E It C End c Neha Theater CM There is a E linear strict homotopy retraction with EE 1 E Io 1 2 it ] precise form of TIE H from the perturbation lemma the minimal model of Ende 14 s with higher multiplications computed by eg sums over trees like End ml Endlnh Endkt \% } set Tie * Yat m % A sot A End End 114

8 ae4n* oak intoning 8 oh Nafta c End c Naha / embedded as contractions [ tf] acbaswntaction on uh MY Thm All trees map M EEHDHYSF to MEENDHD and the induced An algebra M { minimal model of Aw for WEM? Skekhofteynmanmle see [ }n>z is the If only Went there are some small changes Higher products compute scattering of particles through trees Almon precisely E and It can be written as sums of products of bosonic fermionic creation and annihilation operators x 2 D set More creation and annihilation operators QOYEE ] acting on NEO s ozne4* s qe[ operatdfeynmatgmianymonomialtc7naoydwtd2akyoaegy9inexff@wntainssot0aftpak # ^ ns W Ejxjwj weft Tawil 1 Ot 2 gte of

9 attn exautyoneateachint vertices attn intwainfg O at#exfoi ynmanmles computing pn Moh ul To compute pn 40 # vertices at inputs and internet edges 40 gw@ BO at int each Twmpule scattering of Itt ;n 0 7/0 edge sum overall trees T and for with interaction vertices ie sum overall possible kinds of interaction Example W X z M 2 ^ with vacuum bdvywnd W X foobosonsx and 1e4*jc fermions O as %jwilo fry ;o I to %ci

10 To compute A 14*04*04*1 e M of oilseed %eefme#qeamenm tiger are an escaping O violating vacuum bay Hence the only contributing collection of interaction vertices is \ ±lg feee oo BO p 4*04%4*1 1 One can check only Pub are no now an% determines A the minimal model of AW for Wx3 is A 3 from earlier So

11 intoning 0 Conclusion hmflw triangulated category 8 Aw mflw ^ I DG category generator Endpl fp E Da algebra fminimalmodd >z for W xdwefindthe An algebra A Heyer M {ffr astheminimalmodel

12 Lazaro Keller Mnrfet in Yan References K Costello Topological conformal field theories and Calabi categories Advances 2007 [ K ] B A brief introduction to An algebras H C in Generating the super potential on a D brine category [ D ] T Dyckevhoff Compact generators in categories of matrix factorisation Duke 2011 [ M ] D Mnrfet Cut systems and matrix factorisation I au Xix [ MY D work progress githubcom/dmurfet/ainfmf