Celebratio Mathematica

Albert S. Schwarz

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My life in science

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N. S. Ramm and A. S. Švarc: “Geo­met­riq blizosti, uni­form­naq geo­met­riq i to­po­lo­giq” [Geo­metry of prox­im­ity, uni­form geo­metry and to­po­logy], Mat. Sb., Nov. Ser. 33(75) : 1 (1953), pp. 157–​180. MR 61368 Zbl 0050.​39101 article

A. S. Schwarz: “A volume in­vari­ant of cov­er­ings,” Dok­lady Akad. Nauk SSSR 105 (1955), pp. 32–​34. In Rus­si­an. Zbl 0066.​15903 article

V. A. Rohlin and A. S. Švarc: “O kom­bin­at­or­noy in­vari­ant­nosti klas­sov Pon­trq­gina” [On the com­bin­at­or­i­al in­vari­ance of the Pontry­agin classes], Dokl. Akad. Nauk SSSR 114 : 3 (1957), pp. 490–​493. MR 102070 Zbl 0078.​36803 article

A. S. Švarc: “Rod rassloen­nogo pros­tran­stva” [The genus of a fiber space], Dokl. Akad. Nauk SSSR 119 : 2 (1958), pp. 219–​222. MR 102812 Zbl 0085.​37703 article

A. S. Schwarz: “On the genus of a fiber space,” Dok­lady Akad. Nauk SSSR 126 (1959), pp. 719–​722. In Rus­si­an. Zbl 0087.​38204 article

A. S. Švarc: “Go­mo­lo­gii pros­tran­stv zamknutjx krivjx” [Ho­mo­logy of spaces of closed curves], Tr. Mosk. Mat. O.-va 9 (1960), pp. 3–​44. MR 133824 Zbl 0115.​17001 article

A. S. Švarc: “Rod rassloen­nogo pros­tran­stva” [The genus of a fibre space], Tr. Mosk. Mat. O.-va 10 (1961), pp. 217–​272. An Eng­lish trans­la­tion was pub­lished in Am. Math. Soc. Transl., Ser. 2 55 (1966),. MR 154284 article

D. B. Fuks and A. S. Švarc: “K go­mo­topicheskoy teorii funk­t­orov v kat­egorii to­po­lo­gicheskix pros­tran­stv” [On the ho­mo­topy the­ory of func­tors in the cat­egory of to­po­lo­gic­al spaces], Dokl. Akad. Nauk SSSR 143 : 3 (1962), pp. 543–​546. An Eng­lish trans­la­tion was pub­lished in Sov. Math., Dokl. 3 (1962). MR 137116 article

A. S. Švarc: “Rod rassloen­nogo pros­tran­stva” [The genus of a fibre space], Tr. Mosk. Mat. O.-va 11 (1962), pp. 99–​126. An Eng­lish trans­la­tion was pub­lished in Am. Math. Soc. Transl., Ser. 2 55 (1966),. MR 151982 article

A. S. Švarc: “Dvoystven­nost’funk­t­orov” [Du­al­ity of func­tors], Dokl. Akad. Nauk SSSR 148 : 2 (1963), pp. 288–​291. An Eng­lish trans­la­tion was pub­lished in Sov. Math., Dokl. 4 (1963). MR 168629 article

B. S. Mityagin and A. S. Shvarts: “Func­tors in cat­egor­ies of Banach spaces” [Funk­t­orj v kat­egoriqx banax­ovjx pros­tran­stv], Us­pekhi Mat. Nauk 19 : 2(116) (1964), pp. 65–​130. An Eng­lish trans­la­tion was pub­lished in Russ. Math. Surv. 19:2 (1964). MR 166593 article

L. M. Kiss­ina and A. S. Švarc: “K vo­prosu ob opis­anii dvoystven­nogo funk­t­ora” [On the ques­tion of de­scrip­tion of the du­al­ity func­tor], Dokl. Akad. Nauk SSSR 167 : 2 (1966), pp. 282–​285. MR 219065 article

R. S. Pokaz­eeva and A. S. Švarc: “Dvoystven­nost’funk­t­orov” [Du­al­ity of func­tors], Mat. Sb., Nov. Ser. 71(113) : 3 (1966), pp. 357–​385. An Eng­lish trans­la­tion was pub­lished Am. Math. Soc., Trans­lat., II. Ser. 73 (1968). MR 220650 article

V. V. Kuznecov and A. S. Švarc: “Dvoystven­nost’funk­t­orov i dvoystven­nost’kat­egor­iy” [Du­al­ity of func­tors and du­al­ity of cat­egor­ies], Us­pekhi Mat. Nauk 22 : 1(133) (1967), pp. 168–​170. MR 204491 article

A. S. Schwarz: “A new for­mu­la­tion of the quantum the­ory,” Dokl. Akad. Nauk SSSR 173 : 4 (1967), pp. 793–​796. In Rus­si­an. article

V. V. Kuznecov and A. S. Švarc: “Dvoystven­nost’funk­t­orov i dvoystven­nost’kat­egor­iy” [Du­al­ity of func­tors and du­al­ity of cat­egor­ies], Sib. Mat. Zh. 9 : 4 (1968), pp. 840–​856. An Eng­lish trans­la­tion was pub­lished in Siberi­an Math. J. 9:4 (1968). MR 236241 Zbl 0186.​03002 article

V. N. Likhachev, Yu. S. Ty­up­kin, and A. S. Shvarts: “Adia­baticheskaq teor­ema v kvanto­voy teorii polq” [Adia­bat­ic the­or­em in quantum field the­ory], Teor. Mat. Fiz. 10 : 1 (1972), pp. 63–​84. An Eng­lish trans­la­tion was pub­lished in The­or­et. and Math. Phys. 10:1 (1972). MR 475468 article

V. A. Fateev and A. S. Shvarts: “On ax­io­mat­ic scat­ter­ing the­ory,” The­or­et. and Math. Phys. 14 : 2 (February 1973), pp. 112–​124. Eng­lish trans­la­tion of Rus­si­an ori­gin­al pub­lished in Teor. Mat. Fiz. 14:2 (1973). Zbl 0274.​47006 article

A. Schwarz, V. Fateev, and Yu. Ty­up­kin: To­po­lo­gic­ally non-trivi­al particles in quantum field the­ory. Tech­nic­al re­port 157, Lebedev In­sti­tute, 1975. In Rus­si­an. techreport

A. S. Schwarz, Yu. Ty­up­kin, and V. Fateev: “On to­po­lo­gic­ally non-trivi­al particles in quantum field the­ory,” Pisma Zh. Èksper. Teor­et. Fiz. 22 (1975), pp. 192–​194. In Rus­si­an; trans­lated in­to Eng­lish in JETP Let­ters 22:3 (1975), 88–89. article

A. S. Schwarz, Yu. Ty­up­kin, and V. Fateev: “On the ex­ist­ence of heavy particles in gauge fields the­or­ies,” Pisma Zh. Èksper. Teor­et. Fiz. 21 (1975), pp. 91–​93. In Rus­si­an. article

J. S. Tjup­kin, V. A. Fateev, and A. S. Švarc: “O klas­sicheskom predele mat­ricj rasseq­niq v kvanto­voy teorii polq” [The clas­sic­al lim­it of the scat­ter­ing mat­rix in quantum field the­ory], Dokl. Akad. Nauk SSSR 221 : 1 (1975), pp. 70–​73. MR 378641 article

Yu. S. Ty­up­kin, V. A. Fateev, and A. S. Shvarts: “Con­nec­tion between particle-like solu­tions of clas­sic­al equa­tions and quantum particles,” So­viet J. Nuc­le­ar Phys. 22 : 3 (1975), pp. 321–​325. Eng­lish trans­la­tion of Rus­si­an ori­gin­al pub­lished in Jadernaja Fiz. 22:3 (1975). MR 406170 article

A. A. Be­lav­in, A. M. Polyakov, A. S. Schwartz, and Yu. S. Ty­up­kin: “Pseudo­particle solu­tions of the Yang–Mills equa­tions,” Phys. Lett. B 59 : 1 (1975), pp. 85–​87. MR 434183 article

A. Schwarz, V. Fateev, and Yu. Ty­up­kin: On the particle-like solu­tions in the pres­ence of fer­mi­ons. Tech­nic­al re­port 155, Lebedev In­sti­tute, 1976. In Rus­si­an. techreport

A. S. Schwarz: “Mag­net­ic mono­poles in gauge field the­or­ies,” Us­pekhi Matem­aticheskikh Nauk (Rus­si­an Math­em­at­ic­al Sur­veys) 31 : 3 (1976), pp. 248–​258. In Rus­si­an. article

A. S. Schwarz: “To­po­lo­gic­ally non­trivi­al solu­tions of clas­sic­al equa­tions and their role in quantum field the­ory,” pp. 224–​240 in Non­loc­al, non­lin­ear and non­renor­mal­iz­able field the­or­ies (Proc. Fourth In­ter­nat. Sym­pos. Non­loc­al Field The­or­ies) (Alushta, 1976). Edi­ted by D. I. Blokh­int­sev. 1976. In Rus­si­an. MR 456147 inproceedings

A. S. Schwarz: “Mag­net­ic mono­poles in gauge the­or­ies,” Nuc­le­ar Phys. B 112 : 2 (September 1976), pp. 358–​364. MR 416313 article

J. S. Tjup­kin, V. A. Fateev, and A. S. Švarc: “CHasti­ce­podob­n­je reshe­niq uravn­en­iy kal­ib­ro­voch­njx teor­iy polq” [Particle-like solu­tions of the equa­tions of gauge the­or­ies], Teor. Mat. Fiz. 26 : 3 (1976), pp. 397–​402. An Eng­lish trans­la­tion was pub­lished in The­or­et. and Math. Phys. 26:3 (1976). MR 449236 article

A. S. Schwarz: “On reg­u­lar solu­tions of Eu­c­lidean Yang–Mills equa­tions,” Phys. Lett. B 67 : 2 (1977), pp. 172–​174. MR 443673 article

A. S. Schwarz and I. Fro­lov: “Con­tri­bu­tion of in­stan­tons to the cor­rel­a­tion func­tions of a Heis­en­berg fer­ro­mag­net,” Pisma Zh. Èksper. Teor­et. Fiz. 28 (1978), pp. 274–​276. In Rus­si­an; trans­lated in JETP Let­ters 28, p. 249. article

A. S. Schwarz: “On quantum fluc­tu­ations of in­stan­tons,” Lett. Math. Phys. 2 : 3 (1978), pp. 201–​205. MR 495474 Zbl 0383.​70016 article

V. N. Ro­man­ov, I. V. Fro­lov, and A. S. Švarc: “O sfericheski-sim­met­rich­njx soliton­ax” [Spher­ic­ally sym­met­ric solitons], Teor. Mat. Fiz. 37 : 3 (1978), pp. 305–​318. An Eng­lish trans­la­tion was pub­lished in The­or. Math. Phys. 37:3 (1978). MR 524695 article

A. S. Schwarz: “The par­ti­tion func­tion of de­gen­er­ate quad­rat­ic func­tion­al and Ray–Sing­er in­vari­ants,” Lett. Math. Phys. 2 : 3 (1978), pp. 247–​252. MR 676337 Zbl 0383.​70017 article

A. S. Schwarz: “The par­ti­tion func­tion of a de­gen­er­ate func­tion­al,” Comm. Math. Phys. 67 : 1 (1979), pp. 1–​16. MR 535228 Zbl 0429.​58015 article

A. S. Schwarz, A. Be­lav­in, V. Fateev, and Yu. Ty­up­kin: “Quantum fluc­tu­ations of multi-in­stan­tons solu­tions,” Phys­ics Let­ters B 83 : 3–​4 (1979), pp. 317–​320. article

A. S. Schwarz, V. Fateev, and I. Fro­lov: “Quantum fluc­tu­ations of in­stan­tons in the non-lin­ear \( \sigma \)-mod­el,” Nuc­le­ar Phys­ics B 154 (1979), pp. 1–​20. article

A. S. Schwarz and I. Fro­lov: “On the in­stan­ton con­tri­bu­tion in Eu­c­lidean Green func­tions,” Phys­ics Let­ters B 80 : 4–​5 (1979), pp. 406–​409. article

A. S. Schwarz: “Are the field and space vari­ables on an equal foot­ing?,” Nuc­le­ar Phys. B 171 : 1–​2 (1980), pp. 154–​166. MR 582926 article

V. N. Ro­man­ov, V. A. Fateev, and A. S. Schwarz: “Mag­nit­nye mono­pol­ii v ed­inykh teor­iyakh ehlektro­mag­nit­nogo, slabogo i sil’nogo vzaimod­ejstviya” [Mag­net­ic mono­poles in the uni­fied the­or­ies of elec­tro­mag­net­ic, weak, and strong in­ter­ac­tions], Yadernaya Fiz. 32 : 4 (1980), pp. 1138–​1141. An Eng­lish trans­la­tion was pub­lished in Sov. J. Nucl. Phys. 32:4 (1980). MR 624047 article

A. S. Schwarz: “The­or­ies with non­loc­al elec­tric charge con­ser­va­tion,” Pisma Zh. Èksper. Teor­et. Fiz. 34 (1981), pp. 555–​558. In Rus­si­an; trans­lated in JETP Let­ters 34: 10 (1981), 532–534. article

A. V. Gay­duk, V. N. Ro­man­ov, and A. S. Schwarz: “Su­per­grav­ity and field space demo­cracy,” Comm. Math. Phys. 79 : 4 (1981), pp. 507–​528. MR 623965 article

V. A. Fateev, I. V. Fro­lov, A. S. Schwarz, and Yu. S. Ty­up­kin: “Quantum fluc­tu­ations of in­stan­tons,” Sov. Sci. Rev., Sect. C 2 (1981), pp. 1–​51. MR 659266 Zbl 0535.​58026 article

A. S. Schwarz: “Su­per­grav­ity and com­plex geo­metry,” Yadernaya Fiz. 34 : 4 (1981), pp. 1144–​1149. An Eng­lish trans­la­tion was pub­lished in So­viet J. Nuc­le­ar Phys. 34:4 (1981). MR 660226 article

A. S. Schwarz: “Su­per­grav­ity, com­plex geo­metry and \( G \)-struc­tures,” Comm. Math. Phys. 87 : 1 (1982), pp. 37–​63. MR 680647 Zbl 0503.​53048 article

A. S. Schwarz: “Field the­or­ies with no loc­al con­ser­va­tion of the elec­tric charge,” Nuc­le­ar Phys. B 208 : 1 (1982), pp. 141–​158. MR 682897 article

A. A. Roslyĭ and A. S. Schwarz: “Geo­met­riya nemin­im­al’noj i al’ternat­ivnoj min­im­al’noj su­per­ggrav­it­at­sii” [Geo­metry of non­min­im­al and al­tern­at­ive min­im­al su­per­grav­ity], Yadernaya Fiz. 37 : 3 (1983), pp. 786–​794. An Eng­lish trans­la­tion was pub­lished in So­viet J. Nuc­le­ar Phys. 37:3 (1983). MR 719875 article

O. M. Khu­daverdy­an and A. S. Shvarts: “Nor­mal’naq kal­ib­rovka v su­per­grav­itacii” [Nor­mal gauge in su­per­grav­ity], Teor. Mat. Fiz. 57 : 3 (1983), pp. 354–​362. An Eng­lish trans­la­tion was pub­lished in The­or. Math. Phys. 57:3 (1983). MR 735394 article

A. A. Rosly and A. S. Schwarz: “Geo­metry of \( N=1 \) su­per­grav­ity,” Comm. Math. Phys. 95 : 2 (1984), pp. 161–​184. MR 760330 Zbl 0644.​53077 article

A. A. Rosly and A. S. Schwarz: “Geo­metry of \( N=1 \) su­per­grav­ity, II,” Comm. Math. Phys. 96 : 3 (1984), pp. 285–​309. MR 769349 Zbl 0644.​53078 article

A. S. Shvarts: “On the defin­i­tion of su­per­space” [K opredel­eniü su­per­pros­tran­stva], Teor. Mat. Fiz. 60 : 1 (1984), pp. 37–​42. An Eng­lish trans­la­tion was pub­lished in The­or. Math. Phys. 60:1 (1984). MR 760438 article

A. A. Rosly and A. S. Schwarz: “Geo­met­ric­al ori­gin of new un­con­strained su­per­fields,” pp. 308–​324 in Pro­ceed­ings of the third sem­in­ar on quantum grav­ity (Mo­scow, 23–25 Oc­to­ber 1984). Edi­ted by M. A. Markov, V. A. Berez­in, and V. P. Fro­lov. World Sci­entif­ic (Singa­pore), 1985. MR 829729 incollection

M. A. Baran­ov and A. S. Shvarts: “O mno­go­petle­vom vk­lade v teor­iyu struny” [Mul­tiloop con­tri­bu­tion to string the­ory], Pis’ma Zh. Ek­sper. Teor­et. Fiz. 42 : 8 (1985), pp. 340–​342. An Eng­lish trans­la­tion was pub­lished in JETP Lett. 42:8 (1985). MR 875755 article

M. A. Baran­ov, A. A. Roslyĭ, and A. S. Shvarts: “Geo­met­ric as­pects of su­per­grav­ity,” Prob­lemy Yadern. Fiz. i Kosm. Lucheĭ 24 (1985), pp. 25–​33. MR 887675 article

A. S. Schwarz and Yu. S. Ty­up­kin: “Grand uni­fic­a­tion, strings, and mir­ror particles,” pp. 279–​289 in Group the­or­et­ic­al meth­ods in phys­ics (Zvenig­orod, USSR, 24–26 Novem­ber 1982), vol. 2. Edi­ted by M. A. Markov, V. I. Man’ko, and A. E. Shabad. Har­wood Aca­dem­ic (Chur, Switzer­land), 1985. MR 989351 incollection

M. A. Baran­ov, A. A. Roslyĭ, and A. S. Shvarts: “Su­per­light­like geodesics in su­per­grav­ity,” Yadernaya Fiz. 41 : 1 (1985), pp. 285–​287. An Eng­lish trans­la­tion was pub­lished in So­viet J. Nuc­le­ar Phys. 41:1 (1985). MR 804665 article

M. A. Baran­ov, A. A. Roslyĭ, and A. S. Shvarts: “O svqzi grav­itacii i su­per­grav­itacii” [On the con­nec­tion between grav­ity and su­per­grav­ity], Teor­et. Mat. Fiz. 64 : 1 (1985), pp. 7–​16. An Eng­lish trans­la­tion was pub­lished in The­or. Math. Phys. 64:1 (1985). MR 815093 article

A. A. Rosly and A. S. Schwarz: “Su­per­sym­metry in a space with aux­il­i­ary di­men­sions,” Comm. Math. Phys. 105 : 4 (1986), pp. 645–​668. MR 852094 article

M. A. Baran­ov, Yu. I. Man­in, I. V. Fro­lov, and A. S. Shvarts: “The mul­tiloop con­tri­bu­tion in the fer­mi­on string,” So­viet J. Nuc­le­ar Phys. 43 : 4 (1986), pp. 670–​671. Eng­lish trans­la­tion of Rus­si­an ori­gin­al pub­lished in Yadernaya Fiz. 43:4 (1986). MR 857425 article

A. A. Roslyĭ, O. M. Khu­daverdy­an, and A. S. Shvarts: “Su­per­sim­met­riq i kom­pleksnaq geo­met­riq” [Su­per­sym­metry and com­plex geo­metry], pp. 247–​284 in Com­plex ana­lys­is: Sev­er­al vari­ables, III. Edi­ted by G. M. Khen­kin and R. V. Gamkrelidze. Sovre­mennye Prob­lemy Matem­atiki. Fun­da­ment­al’nye Napravlen­iya 9. VIN­ITI (Mo­scow), 1986. An Eng­lish trans­la­tion was pub­lished in 1989. MR 860614 incollection

A. Schwarz and M. Baran­ov: Su­per­con­form­al geo­metry and mul­tiloop con­tri­bu­tion to the string the­ory. Pre­print, MI­FI (Mo­scow), 1987. techreport

M. A. Baran­ov, I. V. Fro­lov, and A. S. Shvarts: “Geo­met­riya dvumernykh su­per­kon­formnykh teor­ij polya” [Geo­metry of two-di­men­sion­al su­per­con­form­al field the­or­ies], Teor. Mat. Fiz. 70 : 1 (1987), pp. 92–​103. An Eng­lish trans­la­tion was pub­lished in The­or. Math. Phys. 70:1 (1987). MR 883786 Zbl 0624.​58002 article

A. M. Baran­ov, Yu. I. Man­in, I. V. Fro­lov, and A. S. Schwarz: “A su­per­ana­log of the Sel­berg trace for­mula and mul­tiloop con­tri­bu­tions for fer­mi­on­ic strings,” Comm. Math. Phys. 111 : 3 (1987), pp. 373–​392. MR 900500 Zbl 0624.​58033 article

M. A. Baran­ov and A. S. Schwarz: “On the mul­tiloop con­tri­bu­tion to the string the­ory,” In­ter­nat. J. Mod­ern Phys. A 2 : 6 (1987), pp. 1773–​1796. MR 913613 article

A. S. Shvarts: “The fer­mi­on string and a uni­ver­sal mod­u­lus space,” JETP Lett. 46 : 9 (1987), pp. 428–​431. Eng­lish trans­la­tion of Rus­si­an ori­gin­al pub­lished in Pis’ma Zh. Èksper. Teor­et. Fiz. 46:9 (1987). MR 940601 article

A. S. Schwarz and Yu. S. Ty­up­kin: “Meas­ure­ment the­ory and the Schrödinger equa­tion,” pp. 667–​675 in Quantum field the­ory and quantum stat­ist­ics: Es­says in hon­our of the six­tieth birth­day of E. S. Fradkin, vol. 1. Edi­ted by I. A. Batal­in, C. J. Isham, and G. A. Vilko­visky. Hil­ger (Bris­tol, UK), 1987. MR 943165 incollection

A. A. Rosly, A. S. Schwarz, and A. A. Voro­nov: “Geo­metry of su­per­con­form­al man­i­folds,” Comm. Math. Phys. 119 : 1 (1988), pp. 129–​152. MR 968484 Zbl 0675.​58010 article

A. S. Schwarz: “Fer­mi­on­ic string and uni­ver­sal mod­uli space,” Nuc­le­ar Phys. B 317 : 2 (1989), pp. 323–​343. MR 1001895 article

M. A. Baran­ov, I. V. Fro­lov, and A. S. Schwarz: “Geo­met­riq su­per­kon­form­nogo pros­tran­stva mod­u­ley” [Geo­metry of the su­per­con­form­al mod­uli space], Teor. Mat. Fiz. 79 : 2 (1989), pp. 241–​252. An Eng­lish trans­la­tion was pub­lished in The­or­et. and Math. Phys. 79:2 (1989). MR 1007798 Zbl 0679.​53058 article

A. S. Schwarz: “New to­po­lo­gic­al in­vari­ants arising in the the­ory of quant­ized fields” in Baku In­ter­na­tion­al To­po­lo­gic­al Con­fer­ence, ab­stracts (Baku, Oc­to­ber 3–9, 1987). Edi­ted by B. R. Nur­iev. Elm (Baku, Azerbaijan), 1989. MR 1347199 inproceedings

A. A. Rosly, A. S. Schwarz, and A. A. Voro­nov: “Su­per­con­form­al geo­metry and string the­ory,” Comm. Math. Phys. 120 : 3 (1989), pp. 437–​450. MR 981212 Zbl 0667.​58008 article

V. Kac and A. Schwarz: “Geo­met­ric in­ter­pret­a­tion of the par­ti­tion func­tion of 2D grav­ity,” Phys. Lett. B 257 : 3–​4 (1991), pp. 329–​334. MR 1100639 article

A. S. Schwarz and A. Sen: “Glu­ing the­or­em, star product and in­teg­ra­tion in open string field the­ory in ar­bit­rary back­ground fields,” Int. J. Mod. Phys. A 6 : 30 (1991), pp. 5387–​5407. MR 1137572 Zbl 0802.​53032 article

A. Schwarz: “Geo­metry of fer­mi­on­ic string,” pp. 1377–​1386 in Pro­ceed­ings of the In­ter­na­tion­al Con­gress of Math­em­aticians (Kyoto, Ja­pan, 21–29 Au­gust 1990), vol. 2. Edi­ted by I. Satake. Spring­er (Tokyo), 1991. MR 1159322 Zbl 0751.​53024 incollection

K. N. Anagnosto­poulos, M. J. Bowick, and A. Schwarz: “The solu­tion space of the unit­ary mat­rix mod­el string equa­tion and the Sato Grass­man­ni­an,” Com­mun. Math. Phys. 148 : 3 (1992), pp. 469–​485. MR 1181066 Zbl 0753.​35073 ArXiv hep-​th/​9112066 article

A. Schwarz: “Geo­metry of Batal­in–Vilko­visky quant­iz­a­tion,” Com­mun. Math. Phys. 155 : 2 (1993), pp. 249–​260. MR 1230027 Zbl 0786.​58017 ArXiv hep-​th/​9205088 article

A. Schwarz: “Semi­clas­sic­al ap­prox­im­a­tion in Batal­in–Vilko­visky form­al­ism,” Com­mun. Math. Phys. 158 : 2 (1993), pp. 373–​396. MR 1249600 Zbl 0855.​58005 ArXiv hep-​th/​9210115 article

M. Atiyah, A. Borel, G. J. Chaitin, D. Friedan, J. Glimm, J. J. Gray, M. W. Hirsch, S. MacLane, B. B. Man­del­brot, D. Ruelle, A. Schwarz, K. Uh­len­beck, R. Thom, E. Wit­ten, and C. Zee­man: “Re­sponses to ‘The­or­et­ic­al math­em­at­ics: To­ward a cul­tur­al syn­thes­is of math­em­at­ics and the­or­et­ic­al phys­ics’, by A. Jaffe and F. Quinn,” Bull. Am. Math. Soc., New Ser. 30 : 2 (April 1994), pp. 178–​207. Zbl 0803.​01014 ArXiv math/​9404229 article

A. Schwarz: “Sym­metry trans­form­a­tions in Batal­in–Vilko­visky form­al­ism,” Lett. Math. Phys. 31 : 4 (1994), pp. 299–​301. MR 1293493 Zbl 0802.​58027 ArXiv hep-​th/​9310124 article

M. Pen­kava and A. Schwarz: “On some al­geb­ra­ic struc­tures arising in string the­ory,” pp. 219–​227 in Per­spect­ives in math­em­at­ic­al phys­ics (Taiwan, sum­mer 1992 and Los Angeles, winter 1992). Edi­ted by R. C. Pen­ner and S.-T. Yau. Con­fer­ence Pro­ceed­ings and Lec­ture Notes in Geo­metry and To­po­logy 3. In­ter­na­tion­al Press (Bo­ston), 1994. MR 1314668 Zbl 0871.​17021 ArXiv hep-​th/​9212072 incollection

D. Fuchs and A. Schwarz: “Mat­rix Vi­eta the­or­em,” pp. 15–​22 in Lie groups and Lie al­geb­ras: E. B. Dynkin’s sem­in­ar. Edi­ted by S. G. Gindikin and E. B. Vin­berg. Amer­ic­an Math­em­at­ic­al So­ci­ety Trans­la­tions. Series 2 169. Amer­ic­an Math­em­at­ic­al So­ci­ety (Provid­ence, RI), 1995. This book is also no. 26 in the Ad­vances in Math­em­at­ics series. MR 1364450 Zbl 0837.​15011 ArXiv math/​9410207 incollection

M. Pen­kava and A. Schwarz: “\( A_{\infty} \) al­geb­ras and the co­homo­logy of mod­uli spaces,” pp. 91–​107 in Lie groups and Lie al­geb­ras: E. B. Dynkin’s sem­in­ar. Edi­ted by S. G. Gindikin and E. B. Vin­berg. Amer­ic­an Math­em­at­ic­al So­ci­ety Trans­la­tions. Series 2 169. Amer­ic­an Math­em­at­ic­al So­ci­ety (Provid­ence, RI), 1995. This book is also no. 26 in the Ad­vances in Math­em­at­ics series. MR 1364455 Zbl 0863.​17017 ArXiv hep-​th/​9408064 incollection

A. Schwarz: “Sigma-mod­els hav­ing su­per­man­i­folds as tar­get spaces,” Lett. Math. Phys. 38 : 1 (1996), pp. 91–​96. MR 1401058 Zbl 0859.​58004 ArXiv hep-​th/​9506070 article

M. Al­ex­an­drov, A. Schwarz, O. Za­boron­sky, and M. Kont­sevich: “The geo­metry of the mas­ter equa­tion and to­po­lo­gic­al quantum field the­ory,” Int. J. Mod. Phys. A 12 : 7 (1997), pp. 1405–​1429. MR 1432574 Zbl 1073.​81655 ArXiv hep-​th/​9502010 article

A. Connes and A. Schwarz: “Mat­rix Vi­eta the­or­em re­vis­ited,” Lett. Math. Phys. 39 : 4 (1997), pp. 349–​353. MR 1449580 Zbl 0874.​15010 article

A. Schwarz and O. Za­boron­sky: “Su­per­sym­metry and loc­al­iz­a­tion,” Com­mun. Math. Phys. 183 : 2 (1997), pp. 463–​476. MR 1461967 Zbl 0873.​58003 ArXiv hep-​th/​9511112 article

A. Connes, M. R. Douglas, and A. Schwarz: “Non­com­mut­at­ive geo­metry and mat­rix the­ory: Com­pac­ti­fic­a­tion on tori,” J. High En­ergy Phys. 1998 : 2 (1998). art­icle no. 3, 35 pages. MR 1613978 Zbl 1018.​81052 ArXiv hep-​th/​9711162 article

A. Schwarz: “Grass­mann and string the­ory,” Com­mun. Math. Phys. 199 : 1 (1998), pp. 1–​24. MR 1660223 Zbl 0921.​58078 ArXiv hep-​th/​9610122 article

A. Schwarz: “Mor­ita equi­val­ence and du­al­ity,” Nucl. Phys., B 534 : 3 (1998), pp. 720–​738. MR 1663471 Zbl 1079.​81066 ArXiv hep-​th/​9805034 article

N. Nekra­sov and A. Schwarz: “In­stan­tons on non­com­mut­at­ive \( \mathbb{R}^4 \), and \( (2,0) \) su­per­con­form­al six di­men­sion­al the­ory,” Com­mun. Math. Phys. 198 : 3 (1998), pp. 689–​703. MR 1670037 Zbl 0923.​58062 ArXiv hep-​th/​9802068 article

A. Konechny and A. Schwarz: “On \( (k\oplus l|g) \)-di­men­sion­al su­per­man­i­folds,” pp. 201–​206 in Su­per­sym­metry and quantum field the­ory (Kharkov, Ukraine, 5–7 Janu­ary 1997). Edi­ted by J. Wess and V. P. Ak­u­lov. Lec­ture Notes in Phys­ics 509. 1998. Pro­ceed­ings of the D. Volkov Me­mori­al Sem­in­ar. MR 1677320 Zbl 0937.​58001 incollection

L. Fried­lander and A. Schwarz: “Grass­man­ni­an and el­lipt­ic op­er­at­ors,” pp. 79–​88 in High­er ho­mo­topy struc­tures in to­po­logy and math­em­at­ic­al phys­ics (Pough­keep­sie, NY, 13–15 June 1996). Edi­ted by J. Mc­Cle­ary. Con­tem­por­ary Math­em­at­ics 227. 1999. Con­fer­ence to hon­or the 60th birth­day of Jim Stasheff. MR 1665462 Zbl 0923.​58053 ArXiv funct-​an/​9704003 incollection

M. A. Rief­fel and A. Schwarz: “Mor­ita equi­val­ence of mul­ti­di­men­sion­al non­com­mut­at­ive tori,” Int. J. Math. 10 : 2 (1999), pp. 289–​299. MR 1687145 Zbl 0968.​46060 ArXiv math/​9803057 article

A. Konechny and A. Schwarz: “BPS states on non-com­mut­at­ive tori and du­al­ity,” Nucl. Phys., B 550 : 3 (1999), pp. 561–​584. MR 1693262 Zbl 0949.​58004 ArXiv hep-​th/​9811159 article

B. Pi­oline and A. Schwarz: “Mor­ita equi­val­ence and \( T \)-du­al­ity (or \( B \) versus \( \Theta \)),” J. High En­ergy Phys. 1999 : 8 (1999). art­icle no. 21, 16 pages. MR 1715542 Zbl 1060.​81612 ArXiv hep-​th/​9908019 article

A. Konechny and A. Schwarz: “\( 1/4 \)-BPS states on non­com­mut­at­ive tori,” J. High En­ergy Phys. 1999 : 9 (1999). art­icle no. 30, 15 pages. MR 1720696 Zbl 0957.​81086 ArXiv hep-​th/​9907008 article

A. Schwarz: “Quantum ob­serv­ables, Lie al­gebra ho­mo­logy and TQFT,” Lett. Math. Phys. 49 : 2 (1999), pp. 115–​122. MR 1728307 Zbl 1029.​81064 ArXiv hep-​th/​9904168 article

A. Schwarz: “Non­com­mut­at­ive su­per­geo­metry and du­al­ity,” Lett. Math. Phys. 50 : 4 (1999), pp. 309–​321. MR 1768707 Zbl 0967.​58004 ArXiv hep-​th/​9912212 article

A. Schwarz: “Non­com­mut­at­ive al­geb­ra­ic equa­tions and the non­com­mut­at­ive ei­gen­value prob­lem,” Lett. Math. Phys. 52 : 2 (2000), pp. 177–​184. MR 1786861 Zbl 0972.​15001 ArXiv hep-​th/​0004088 article

A. Konechny and A. Schwarz: “Mod­uli spaces of max­im­ally su­per­sym­met­ric solu­tions on non-com­mut­at­ive tori and non-com­mut­at­ive or­bi­folds,” J. High En­ergy Phys. 2000 : 9 (2000), pp. article no. 005, 24 pages. MR 1789106 Zbl 0989.​81623 ArXiv hep-​th/​0005174 article

A. Konechny and A. Schwarz: “Com­pac­ti­fic­a­tion of M(at­rix) the­ory on non­com­mut­at­ive tor­oid­al or­bi­folds,” Nuc­le­ar Phys. B 591 : 3 (2000), pp. 667–​684. MR 1797572 Zbl 1042.​81580 ArXiv hep-​th/​9912185 article

A. Konechny and A. Schwarz: “The­ory of \( (k\oplus l|q) \)-di­men­sion­al su­per­man­i­folds,” Se­lecta Math. (N.S.) 6 : 4 (2000), pp. 471–​486. MR 1847384 Zbl 1001.​58001 article

A. Schwarz: “Non­com­mut­at­ive in­stan­tons: A new ap­proach,” Comm. Math. Phys. 221 : 2 (2001), pp. 433–​450. MR 1845331 Zbl 0989.​46040 ArXiv hep-​th/​0102182 article

A. Schwarz: “Theta func­tions on non­com­mut­at­ive tori,” Lett. Math. Phys. 58 : 1 (2001), pp. 81–​90. MR 1865115 Zbl 1032.​53082 ArXiv math/​0107186 article

A. Schwarz: “To­po­lo­gic­al quantum field the­or­ies,” pp. 123–​142 in XIIIth In­ter­na­tion­al Con­gress on Math­em­at­ic­al Phys­ics (Lon­don, 17–22 Ju­ly 2000). Edi­ted by A. Grigory­an, A. Fo­kas, T. Kibble, and B. Zegar­l­in­ski. In­ter­na­tion­al Press (Bo­ston), 2001. MR 1883299 Zbl 1043.​81066 ArXiv hep-​th/​0011260 incollection

A. As­tashkevich and A. Schwarz: “Pro­ject­ive mod­ules over non-com­mut­at­ive tori: Clas­si­fic­a­tion of mod­ules with con­stant curvature con­nec­tion,” J. Op­er. The­ory 46 : 3 (supplement) (2001), pp. 619–​634. Ded­ic­ated to Pro­fess­or D. B. Fuchs on his 60th birth­day. MR 1897158 Zbl 0996.​46031 ArXiv math/​9904139 article

A. Konechny and A. Schwarz: “In­tro­duc­tion to M(at­rix) the­ory and non­com­mut­at­ive geo­metry,” Phys. Rep. 360 : 5–​6 (2002), pp. 353–​465. MR 1892926 Zbl 0985.​81126 ArXiv hep-​th/​0012145 article

M. Di­eng and A. Schwarz: “Dif­fer­en­tial and com­plex geo­metry of two-di­men­sion­al non­com­mut­at­ive tori,” Lett. Math. Phys. 61 : 3 (2002), pp. 263–​270. MR 1942364 Zbl 1019.​58004 ArXiv math/​0203160 article

A. Schwarz: “Gauge the­or­ies on non­com­mut­at­ive Eu­c­lidean spaces,” pp. 794–​803 in Mul­tiple fa­cets of quant­iz­a­tion and su­per­sym­metry: Mi­chael Marinov me­mori­al volume. Edi­ted by M. Olshanet­sky and A. Vain­shtein. World Sci­entif­ic (River Edge, NJ), 2002. MR 1964925 Zbl 1026.​81062 ArXiv hep-​th/​0111174 incollection

A. Schwarz: “Non­com­mut­at­ive su­per­geo­metry, du­al­ity and de­form­a­tions,” Nuc­le­ar Phys. B 650 : 3 (2003), pp. 475–​496. To Alain Connes on his 55th birth­day. MR 1952770 Zbl 1006.​81083 ArXiv hep-​th/​0210271 article

A. Pol­ishchuk and A. Schwarz: “Cat­egor­ies of holo­morph­ic vec­tor bundles on non­com­mut­at­ive two-tori,” Comm. Math. Phys. 236 : 1 (2003), pp. 135–​159. MR 1977884 Zbl 1033.​58009 ArXiv math/​0211262 article

A. Schwarz: “A-mod­el and gen­er­al­ized Chern–Si­mons the­ory,” Phys. Lett. B 620 : 3–​4 (2005), pp. 180–​186. MR 2149784 Zbl 1247.​81441 ArXiv hep-​th/​0501119 article

Y. Chen, M. Kont­sevich, and A. Schwarz: “Sym­met­ries of WDVV equa­tions,” Nuc­le­ar Phys. B 730 : 3 (2005), pp. 352–​363. MR 2180009 Zbl 1276.​81096 ArXiv hep-​th/​0508221 article

S. Gukov, A. Schwarz, and C. Vafa: “Khovan­ov–Roz­ansky ho­mo­logy and to­po­lo­gic­al strings,” Lett. Math. Phys. 74 : 1 (2005), pp. 53–​74. MR 2193547 Zbl 1105.​57011 ArXiv hep-​th/​0412243 article

M. Kont­sevich, A. Schwarz, and V. Volo­god­sky: “In­teg­ral­ity of in­stan­ton num­bers and \( p \)-ad­ic B-mod­el,” Phys. Lett. B 637 : 1–​2 (2006), pp. 97–​101. MR 2230876 Zbl 1247.​14058 ArXiv hep-​th/​0603106 article

A. Schwarz and X. Tang: “Quant­iz­a­tion and holo­morph­ic an­om­aly,” J. High En­ergy Phys. 2007 : 3 (2007), pp. article no. 062, 18 pages. MR 2313939 ArXiv hep-​th/​0611281 article

A. Schwarz and V. Volo­god­sky: “Frobeni­us trans­form­a­tion, mir­ror map and in­stan­ton num­bers,” Phys. Lett. B 660 : 4 (2008), pp. 422–​427. MR 2396265 Zbl 1246.​14053 ArXiv hep-​th/​0606151 article

A. Schwarz and I. Sha­piro: “\( p \)-ad­ic su­per­spaces and Frobeni­us,” Comm. Math. Phys. 282 : 1 (2008), pp. 87–​113. MR 2415474 Zbl 1202.​14017 ArXiv math/​0605310 article

A. Schwarz and I. Sha­piro: “Twis­ted de Rham co­homo­logy, ho­mo­lo­gic­al defin­i­tion of the in­teg­ral and ‘phys­ics over a ring’,” Nuc­le­ar Phys. B 809 : 3 (2009), pp. 547–​560. MR 2478121 Zbl 1192.​81311 ArXiv 0809.​0086 article

A. Schwarz and V. Volo­god­sky: “In­teg­ral­ity the­or­ems in the the­ory of to­po­lo­gic­al strings,” Nuc­le­ar Phys. B 821 : 3 (2009), pp. 506–​534. MR 2547214 Zbl 1203.​81155 ArXiv 0807.​1714 article

M. Movshev, A. Schwarz, and R. Xu: Ho­mo­logy of Lie al­gebra of su­per­sym­met­ries. Pre­print, 2010. ArXiv 1011.​4731 techreport

A. Schwarz: “Space and time from trans­la­tion sym­metry,” J. Math. Phys. 51 : 1 (2010), pp. article no. 015201, 7 pages. MR 2605834 Zbl 1245.​81070 ArXiv hep-​th/​0601035 article

M. V. Movshev, A. Schwarz, and R. Xu: “Ho­mo­logy of Lie al­gebra of su­per­sym­met­ries and of su­per Poin­caré Lie al­gebra,” Nuc­le­ar Phys. B 854 : 2 (2012), pp. 483–​503. MR 2844329 Zbl 1229.​81117 ArXiv 1106.​0335 article

J.-M. Li­ou and A. Schwarz: “Equivari­ant co­homo­logy of in­fin­ite-di­men­sion­al Grass­man­ni­an and shif­ted Schur func­tions,” Math. Res. Lett. 19 : 4 (2012), pp. 775–​784. MR 3008414 Zbl 1300.​14025 ArXiv 1201.​2554 article

M. V. Movshev and A. Schwarz: “Su­per­sym­met­ric de­form­a­tions of max­im­ally su­per­sym­met­ric gauge the­or­ies,” J. High En­ergy Phys. 2012 : 9 (2012), pp. article no. 136, 77 pages. MR 3044913 Zbl 1397.​81390 ArXiv 0910.​0620 article

J.-M. Li­ou and A. Schwarz: “Mod­uli spaces and Grass­man­ni­an,” Lett. Math. Phys. 103 : 6 (2013), pp. 585–​603. MR 3054647 Zbl 1276.​14016 ArXiv 1111.​1649 article

A. Mikhail­ov, A. Schwarz, and R. Xu: “Co­homo­logy ring of the BRST op­er­at­or as­so­ci­ated to the sum of two pure spinors,” Mod­ern Phys. Lett. A 28 : 23 (2013), pp. article no. 1350107, 6 pages. MR 3085958 Zbl 1279.​81052 ArXiv 1305.​0071 article

D. Kre­fl and A. Schwarz: “Re­fined Chern–Si­mons versus Vo­gel uni­ver­sal­ity,” J. Geom. Phys. 74 (2013), pp. 119–​129. MR 3118578 Zbl 1283.​58018 ArXiv 1304.​7873 article

X. Liu and A. Schwarz: Quant­iz­a­tion of clas­sic­al curves. Pre­print, March 2014. ArXiv 1403.​1000 techreport

R. Xu, M. Movshev, and A. Schwarz: “In­teg­ral in­vari­ants in flat su­per­space,” Nuc­le­ar Phys. B 884 (2014), pp. 28–​43. MR 3214872 Zbl 1323.​81093 ArXiv 1403.​1997 article

A. Schwarz: “Quantum curves,” Comm. Math. Phys. 338 : 1 (2015), pp. 483–​500. MR 3345383 Zbl 1318.​81042 ArXiv 1401.​1574 article

J.-M. Li­ou, A. Schwarz, and R. Xu: “Wei­er­strass cycles and tau­to­lo­gic­al rings in vari­ous mod­uli spaces of al­geb­ra­ic curves,” J. Fixed Point The­ory Ap­pl. 17 : 1 (2015), pp. 209–​219. To Pro­fess­or An­drzej Granas. MR 3392990 Zbl 1345.​14042 ArXiv 1308.​6374 article

A. Schwarz, V. Volo­god­sky, and J. Wal­cher: “Fram­ing the di-log­ar­ithm (over \( \mathbb{Z} \)),” pp. 113–​128 in String-Math 2012 (Bonn, Ger­many, 16–21 Ju­ly 2012). Edi­ted by R. Don­agi, S. Katz, A. Klemm, and D. R. Mor­ris­on. Pro­ceed­ings of Sym­po­sia in Pure Math­em­at­ics 90. Amer­ic­an Math­em­at­ic­al So­ci­ety (Provid­ence, RI), 2015. MR 3409790 Zbl 1356.​81196 ArXiv 1306.​4298 incollection

M. V. Movshev and A. Schwarz: “Gen­er­al­ized Chern–Si­mons ac­tion and max­im­ally su­per­sym­met­ric gauge the­or­ies,” pp. 327–​340 in String-Math 2012 (Bonn, Ger­many, 16–21 Ju­ly 2012). Edi­ted by R. Don­agi, S. Katz, A. Klemm, and D. R. Mor­ris­on. Pro­ceed­ings of Sym­po­sia in Pure Math­em­at­ics 90. Amer­ic­an Math­em­at­ic­al So­ci­ety (Provid­ence, RI), 2015. MR 3409803 Zbl 1356.​81200 ArXiv 1304.​7500 incollection

A. Schwarz: “Ax­io­mat­ic con­form­al the­ory in di­men­sions \( > 2 \) and AdS/CT cor­res­pond­ence,” Lett. Math. Phys. 106 : 9 (2016), pp. 1181–​1197. MR 3533564 Zbl 1347.​81058 ArXiv 1509.​08064 article

M. T. Luu and A. Schwarz: “Four­i­er du­al­ity of quantum curves,” Math. Res. Lett. 23 : 4 (2016), pp. 1111–​1137. MR 3554503 Zbl 1354.​81040 ArXiv 1504.​01582 article

A. Schwarz, V. Volo­god­sky, and J. Wal­cher: In­teg­ral­ity of fram­ing and geo­met­ric ori­gin of 2-func­tions. Pre­print, February 2017. ArXiv 1702.​07135 techreport

A. Mikhail­ov and A. Schwarz: “Fam­il­ies of gauge con­di­tions in BV form­al­ism,” J. High En­ergy Phys. 2017 : 7 (2017), pp. article no. 063, 25 pages. MR 3686735 Zbl 1380.​81371 ArXiv 1610.​02996 article

J.-M. Li­ou and A. Schwarz: “Wei­er­strass cycles in mod­uli spaces and the Krichever map,” Math. Res. Lett. 24 : 6 (2017), pp. 1739–​1758. MR 3762693 Zbl 1393.​14022 ArXiv 1207.​0530 article

M. Movshev and A. Schwarz: “Quantum de­form­a­tion of planar amp­litudes,” J. High En­ergy Phys. 2018 : 4 (2018). art­icle no. 121, 20 pages. MR 3801153 Zbl 1390.​81613 ArXiv 1711.​10053 article

A. Schwarz: Scat­ter­ing mat­rix and in­clus­ive scat­ter­ing mat­rix in al­geb­ra­ic quantum field the­ory. Pre­print, August 2019. To Max­im Kont­sevich on the oc­ca­sion of his 55th birth­day with love and ad­mir­a­tion. ArXiv 1908.​09388 techreport

M. Luu and A. Schwarz: It­er­ated in­teg­rals on af­fine curves. Pre­print, December 2019. ArXiv 1912.​09506 techreport

A. Schwarz: “In­clus­ive scat­ter­ing mat­rix and scat­ter­ing of qua­si­particles,” Nuc­le­ar Phys. B 950 (2020). art­icle no. 114869, 14 pages. MR 4039472 ArXiv 1904.​04050 article

A. Schwarz: “Geo­met­ric ap­proach to quantum the­ory” in Spe­cial is­sue on al­gebra, to­po­logy, and dy­nam­ics in in­ter­ac­tion in hon­or of Dmitry Fuchs, published as SIGMA, Sym­metry In­teg­rabil­ity Geom. Meth­ods Ap­pl. 16. Issue edi­ted by B. Khes­in, F. Ma­likov, V. Ovsi­en­ko, and S. Tabach­nikov. Na­tion­al Academy of Sci­ences of Ukraine (Kiev), 2020. Pa­per no. 020, 3 pages. MR 4080799 Zbl 1436.​81018 ArXiv 1906.​04939 incollection

A. Schwarz: Geo­met­ric and al­geb­ra­ic ap­proaches to quantum the­ory. Pre­print, February 2021. ArXiv 2102.​09176 techreport

A. Schwarz: Scat­ter­ing the­ory in al­geb­ra­ic ap­proach: Jordan al­geb­ras, 2021. In pre­par­a­tion. misc

A. S. Schwarz: Scat­ter­ing in geo­met­ric ap­proach to quantum the­ory. Pre­print, 2021. ArXiv 2107.​08557v1 techreport